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Gromov-Witten theory in dimensions two and three Gholampour, Amin 2007

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Gromov-Witten Theory in Dimensions Two and Three lrnin Cholampour  M.Sc., SL.iriIUniveisity of Technology. 2002  If SIS SUIJMJTTED IN PAUTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Mathematics)  The University Of British Goiumbf November 2007 ©  Amin Gholampour, 2007  Abstract in this the^  so me important classes^ is and^ udy their Aationships to other branches of matlien. The first object is the class of P 2 -buililles over a smooth curve C of genus g. Our bundles are of the form P(1, 0 ® L i tCL2) for arbitrary line bundles L o , L 1 and L2 over C. We compute the partition functions of these invariants for all classes of the form 8 -I- nf, where a is a section, f is a fiber and n is an integer. In the case where the class is Calabi-Yau, i.e., K • (a+ = 0, the partition function is given by u)2g -.2  3 9 (2 sin  2  As an application, one can obtain a series of hill predictions for the equivariant Donaldson Thomas invariants for this family of non-tone Secondly, we compute the C-equivariant quantum co o ology Y, the minimal resolution of the DuVal singularity C 2 /C where C is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADD root system canonically associated to C. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an a parameter Runny of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifbld Grornov-Witten potential of [C 2 /CI. Thirdly, for a polyhedral group C, that is a finite subgroup of 50(3), we completely determine the Gromov-Witten theory of Nakamura's 67 Hilbert scheme, which is a preferred Calabi-Yau resolution of the polyhedral singularity C 3 /G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of C. We express the Cromov-Witten potential in terms of an ADD root system associated to C. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Grornov-Witten invariants of [C 3 /671. C is the group A. 1^compute the -  A bstract integral of A 0 on the Hurwitz locus Liu C X14 of curves admitting a degree 4 cover of P i having monodromy group C. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E 6 and D n root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds iC 3 /.1 41 and ['C33/(2S2 x Zadi.  fable of Contents Abstract^  ii  Table of Contents ^ List of Tables^  .......  iv vii  List of Figures ^  viii  Acknowledgements^.^.....^.........^. .^ix Dedication ^ Statement of Co-Authorship .^.  iv  xi  1 Preliminaries 1.1 Introduction  ^1  1.2 GromomWitten invariants and quantum colic }logy ^2 1.2.1 Crepaut Resolution Conjecture ^ 1.3 McKay correspondence ^  5  Bibliography ^ 2 On the OW theory of plane bundles over curves 2.1^Introduction^..............^.^.^ 8 2.1.1 Donaldson-Thomas invariant prediction .^12 2.1.2 Organization of this chapter ^ 12 2.2 Relative invariants and the Gluing theorem ^ 13 2.3^Calculations^.^.^... 2.3.1^Calculations via localization ^ 18 2.3.2 Calculations via gluing techniques . . m .... 2.1^Proof of Theorem 2.1.6^... .^........... . 2.5^Proof of Theorem 2.1.7^......................4 2.6 Proof of the Gluing theorem^........... , .^35  Table of Contents  2.7 Some special case Bibliography . .  .^..............^45  3 Quantum Cohomology of ADE resolutions .^47 3.1 introduction ^47 3.1.1 Over view ^47 3.1.2^Results,^ 48 3.1.3^Relationship to other work . .^........^.^49 3.2 Gromov-Witten theory of Y ^ 50 3.2.1^Invariants of non-zero degree.^.^.^. ^ 50 3.2.2^Degree 0 invariants. .^.^.^.^......^.^53 3.3 Proof of the main theorem ^.^......... . .^. 3.4 The algebra for arbitrary root systems ^ 57 3.4.1 Root system notation ^ 57 3.4.2^The algebra QH.11^...........^.. . . ^ 3.4.3 The proof of Theorem 3.1.3 . .^.^.^.. 60 3.4.4 Proofs of Lemma 3.4.5 and Lemma 3.4.6.^63 3.4.5 The root theoretic formula for triple intersections. ^. 64 3.5 Predictions for the orbifold invariants via the Crepant Resolution Conjecture^ 65 3.51 The statement of the conjecture ^ 66 3.5.2 The prediction.^. . .^ 67 Bibliography ........^.^.^....... . .^72 4 The Quantum Mckay Correspondence . ^. . ...... 74 4.1^Introduction^.^.^.^. .^................^74 4.1.1^Overview ^ 4.1.2^Notation and Results^.^.........^75 4.1.3 Appli cations^ 80 4.1.1 Example^ . . ^ 83 4.2 Proof of the main result^ 84 4.2.1 Geometry of Y =^..^.^84 4.2.2^Proof of Theorem 4.1.6 ....... .^. 86 4.3 Crepant Resolution Conjecture ^ 93 Bibliography ^  101  ^  Table of Contents  5 Hurwitz-Hodge integrals and the CRC ^ 1111 5.1 Introduction ^ 101 5.2 Notation and Resulis^ 107 5.2.1 Outline of the proof ^ 110 5.3 The WDVV equations ^ 110 5.1 Computing / 1 L 2:K 7c,^ 1 12 5.5 Computing I l id 5.5.1 The WDVV relations for /1 4 -Hurwitz llodge integrals116 5.6 Computing 34 -Hurwitz-Hodge integrals ^ 120 5.7 'Hie Relationship with orbifold Gromov-Witten theory^126 Bibliography ^  129  •^•^...........^ 6 Conclusion . ......^131 6.1 Introduction ^ 131 6.2 Importance of the results and the future researel ^13 1 Bibliography^. .^............^.^135  List of Tables Reducing non-simply laced case to simply laced case^ 61  6.1 Cuyrant lies  ^  Conjecture versus McKay correspondence 134  List of Figures 2.1 2.2 2.3 2.5 2.6  (0, — )-tube = (0, — 1 )-cap + (0, 0)-pants (0, 0)-tube = (0, —1)-tube^(0. 1)-tube ^ (0, 1 )-cap = (0, 0)-cap + (0, 1)- tub e ^ (0, 1)-tube = (0, 1)-cap + (0, 0)-pants^.^. Frobenius relation: all pants are level (0,0) ^ Attaching (0, 0)-pants^.........^.  4.1 4.2 4.3 4.4  Surface fi brat ion of 3. Classical McKay correspondence Intersection graphs:^(0),,,1 (left)^I (0) r ( r i g ht) smooth K3 Iihral ion ^  •^.^......  25 26 27 29 29 73t7',  1  78 83 87  Acknowledgements I would like to thank Jim Bryan, my supervisor, for ^ ous help, advising and enthusiasm on the whole thesis project and for ^ valuable comments On the draft. I am grateful to Tom Crawl , Rabid Pandharipande for ^ ful discussions on Chapter 2, and to Davesh Maulik for his crucial help with the proof of the main theorem in Chapter 4. .  British Columbia Vancouver September 2007  ^  Amin Cholamtx  To my parents  Statement of Co-Authorship Chapters 1, b arc e.po story.Chapters 3, 1, 5 are m joint works Bryan. We shared ideas and work in defining ^ e fo ing the research and preparing the manuscript. British Columbia, Vancouver September 2007  Amin Cholampour  ^  Chapter 1  Preliminaries LI Introduction  ^H  Gromov-Witten theory and in particular quantum cohomology arose from a. very curious interaction between mathematics and the physics of string theory. Gromov-Witten theory studies maps of Riernann surfaces (or algebraic nodal curves) into higher dimensional complex spaces with possibly orbifold rularities. This has been inspired by the study of bow world-sheets sit inIce-time by string theorists. In particular, Gromov-Witten invariants e X are defined by integrating cohomology classes of X over the al class of the moduli space of maps from the Riemann surThese invariants carry some nice properties; most importantly they are invariant under deformations of the complex structure on X. A certain subset of the Gromov-Witten invariants are elegantly packaged as a deformation of the cohomology ring of X. called quantum cohomology of X. That is the Gromov-Witten invariants give rise to a product (the quantum cup product) in (X,Q)^Q111 0^1.„11] which when the (the quantum parameters) are all set to zero one get, the ordinary cup product. Associativity of the quantum cup product plays an important role in enumerative geometry. In the case that X is a point, the Gromov-Witten theory is already nieresting since it turns into the study of the moduli space of Rie surfaces. Gromov-Witten theory has provided a strong tool towards the study of intersection theory of moduli space of Rieman surfaces. In the case^ .bifold singularities, there is a beautiful conjeceraction betty en the Gromov-Witten theory of X and the Gromovtheory of some^ resolutions, called crepant resolution We start by briefly renewing some of the materials ^ In the later chapters^ tu^ important classes of spaces, study ^ )^resolution^  'llater. ()lies of some re for  ^  S  (  with polyhedral section ttheo  S, and prov ide  an example of the interactions of o uli space of curves and the crepant. resolution  1.2 Gromov-Witten invariants and quantum cohoinology Let X be a smooth complex pro ,lc ct rve vari tv. By a ()onus (C,pi,...,p„) we mean a complex curve C of arithmetic genus g with at most nodal singularities with a distinct, ordered, smooth pon A special point of C is either a node or one of p t ,...,p n . A g enus g, ted stable map f:C  from a genus g, n-pointed curve as bove suclm that if f contracts a component^C then C, cannot be  (i) a smooth rational curve with less than 3 special points , (ii) of arithmetic genus 1 with no special points. Given a Homology class /3 C TI9(X,Z) and non-negative integers n and g (with a + 2g 3 if /3 0), we denote by Al g .„(XM3) the moduli space of stable maps as above such that f',.[C1= /1. See [7] and [2] for the construction  and properties of Af g ,, (X, /3). What is essential for its is that the moduli space admits a universal curve C 4 Al y ,„ (X, /3) and a universal map .72 : C > X. Using this, one can define the evaluation morphisnis -  evi : Tig ,„(3Y, /3) by evaluating^at time i-th marked point. The other key fact is that Ill y „ ^/3) admits a (virtual) fundamental class ying in the^Chow group of Al ga , (X,$), where d^(3)( 3  (  di  is the expected (or virtual) dime? by [Al ga , (^(3)] 3 T. Giving Gramm, -  3)( —^a [2]). We denote  Chapter 1. Preliminaries  Definition 1.2.1. Given classes 7 1 ,^d c 11 (X. Q), the co respond Gromov-Witten invariant s defined as , (x,133DP .  ev't (21 ) U • •^el4e7 a ).  Gromov-Witten invai  njoy several remarkable properties Ale and Mania. We mention a few that we will use leader can find more about theaxis and their proofs in [3, 91. ioms of Gromov-Witten inv  s linea with respect  rity axiom. molo gv variable. o PfJ'ecii ity axiom . (7 • Point axio  (71,  ^7,a)xs^ 0 if /3 is not aneffect, effecve class.  ,  ' 700 .0  i x-  c3  =3 0^otherw ise  2y > 4, and^  • Divisor aziorrt.  en  , •^  -yr&ovc3  U 72 U 73 n  C ./12 ( Q) then  ^ -y-  • Deformation axiom. Let 1,1 : X ) 13 be a smooth proper map of varieties, with fibers^For locally constant co)homology class sections Y t C 11 2 (X 1 ,21) and 71,1, .^C 11*(X t , .)^.  ^(  •  is constant Next, we describe the quantnmr cohomology ring associated to X via Gromov-Witten invariants. We let 7homogenous basis for the cohomology groups of X, such that To = 1 and T„ is its Poincare dual. Let ,t,, be supercommuting variables parameterizing these basis elements. Define =i TUT] and It is a ft  (13 ) aer series  e Gro ov Witten potential fu intiot (21[ 1 ()^given by  r I where we set the terms corresponding to zero. Let PU k be the partial deriva Definition 1.2.2. We define the  ^n„, < 3 equal and tk.  i/noe l  7; * t i =, >7  1.1  k,!  ® Q UA)^t.„11. It is easy^ identity eie.ment , and X has no odd^ ology. However, associativity is a deeper fact resulted from the relations ong the isors in the module space of genus zero curves with 4 marked points. The associativity . is equivalent to the relations Fukg Ft  = E Fipkg kJ.  called WDVV equations. Thus, (IP (X, Q ) 0 Q[ko ,^ t„,1],*) forms a ring called the quantum cohomology ring of X, The Cromov-Witten theory for an orbifbld is ^ loped along the same lines, but it involves more technicalities. We refer the reader to 1 . 1,^for the details. )  1.2.1 Crepant Resolution Conjecture This section is needed for Chapters 3-5. The Crepant Resolution Conjecture, having its origins in physics, was first conjectured by Cumrun Vafa (see [12]) in 1989 and Eric Zaslow (see [13]) in 1993, and later on in mathematics by Wiemin Chen and Yongbin Ruan in [6]. The physical version of Crepant Resolution Conjecture. claims that strin g theory of a Gorenstein orbifbld X is equivalent to the s t ring kny of any of its crepant resolution Y. By the crepaut resolution we mean a map f X which is the resolution of singularities of X and furthermore the pullback of the canonical sheaf of A' under f is isomorphic to the canonical bundle of Y. The mathematical consequences of this prediction are the equivalence of the quantum cohomology of X and Y for the A-model string theory, and the equivalence of the derived categories of X and Y for the 13-model string theory. In the case that A' satisfies^ conditi Craber have given an elegant formulation of the Crepant Resolution Conifokis  Chapter I. Preliminaries  Let 1717  C  QUM),^  C  0{1 1 0, ...  Y7r)^•  ,x Trt ,91 r ,  '  CCU,  ...  be the (equivai^ genus 0 Gromov-Witten potential fun ctions X. where yi and^  )ararneterize cohomology bases for Y and X, res  tively, and gi's are quantum parameters. It is known that  I r s > 0. The Gromov-Witten invariants of Y (respectively, the orbifold Gromov-Witten invariants of X) are multilinear functions of yi's (respectively, rtis). Then the new formulation of CRC claims that there exists a graded linear isomorphi sm between the cohomology rings of X and V, and there exist roots of unity 0.2 1 , ,w e which induce the equality  FX = under q z =  1.3 McKay correspondence This section is not , eded in Chapters 2, 5. The classical McKay Correspondence first observed by McKay (see [10]) in 1980, gives a natural correspondence among the nontrivial irreducible representations of a finite subgroup (7 C (2, C), and irreducible components of the exceptional set ,  of the minimal resolution Y of the singularity of C 2 /6 (in other words, a basis of /1 2 (Y, Z)), and also the simple roots of the corresponding simple Lie algebra of type ADE. G. Gonzalez-Sprinberg and 5. L. Verdier (see [8]) in 198:3 interpreted the McKay Correspondence as a natural isomorphism between the G-equivariant K-theory of C 2 , and the ordinary K-theory of the minimal resolution of C 2 /6. Based on their result, and other physical calculations by C. \Taft,/  n  .  ..for symmetry, Miles Reid in [11] conjectured an existence of similar  correspondence. when Y is the preferred crepant resolution of C1 16 for finite subgroup G C SL(n, C). The preferred crepant resolution in this context is Cliilb(C") which is the moduli space of 6-clusters (i.e. "scheme theoretic 6-orbits") of C". This is a highly nontrivial fact that G-Hilb(C3 ) is always a crepant resolution for C3 16. Note that 6-i1ilb(r' ) may not give a resolution when 71 > 3. Reid's conjecture has been verified in dimension 3 in a v ery concise form in the language of derived -.)ries by T. Bridgeland, A. K ing and M. Reid (see C4D.  Bibliography ^ Dan Ahranrov:cl toll. Gromov-Witten theory of Deligne-Mtnnford stacks. 41^ath.AG/0603151. [2J N. Behrend. Gromov-Witten invariants in algebraic geometry. Invent. Math, 127(3):601 617, 1997. end and Yu. Mania. Stacks of stable maps and Gromov-Witten 85(1):1 60, 1096.  :ia nts. Duke Math. J.,  [4) Tom Bridgeland, Alastair King, and Miles Reid. The McKay conespondence as an equivalence of derived categories. J. Am er. Math. Soc., 14(3):535 554 (electronic), 2001. )5] Jim Bryan and Tom Graben. The erepant resolution conjecture. To appear in Algebraic Geometry Seattle 2005 Pmecedings, arXiv: mat1LAG 10610129. [6) Wehnin Chen and Yougbin ninth Orbifold Gromov-Witten theory. In Orbit olds in mathematics and physics ( Madison, WI, 2001), volume 310 of Coniemp. Math., pages 25-85. Amer. Math. Soc., Providence, 111, 2002. [7] William Fulton and RahulPandharipande. Notes on stable n a quantum cohomology. In Algebraic geometry—Santa Cruz time 62 of Proc. Sympos. Pure Math., pages 45-96, Providence, RI. 1997. American Mathematical Society. [8) G. Gonzalez-Sprinberg and J.-L. Verdier. Construction geometrique de ht correspondance de McKay. Ann. SAL E colc Nor rn. Sup. (4), 16(3):409-449 (1984), 1983. 19) M. Kontsevieh and Yu. Mania. C, uantr Invent. Math., 124(1-3):313 -339, 1996. With an appendix by ann.  Bibliography  [10] Jelin McKay. Graphs, singularities. and finite groups. In The Santo Cruz Conference on Finite Groups (Univ. California , Santa Cruz, Calif., 1979), volume 37 of Proc. Sympos.^Math., p ages 183-186. Amer. Math. Soc., Providence, RI., 1980. Reid. McKay correspondence. Proc of Kinosaki conference 1996), and Warwick preprint. 1997^ geom/9702016). [12] Cunirtin Vafa. String vacua and orbifoldized LC models, Mode rn Lett. A, 4(12):1169 1 85 19 89. ^ [13] Eric Zaslow. Topological orbilbld lels and quantum coho logy ifs. Comm. Alath. Phys., 15^1993_  Chapter 2  On the equivariant Gromov-Witten theory of plane bundles over curves  2.1 Introduction Let X be a I9 2 -bundle over a smooth complex projective curve C of genus be its canonical class. We denote the cohomology class of the Definition 2.1.1. A class /3 e IL' ( X , 73) is called section class if F • /3 -= We say ;3 is a Calabi Yitu class if K x • 13 = 0 ^is called Calabi 'Van section class if both conditions hold. Remark 2.1.2. A section class is not necess  section of the bundle X. It could be, for example, of fiber curves attached to it. Now let X be a. pc bindleof the form Lim (Ido ED Li ei ,b2)^C, where C is a^ m C is a line bundle of degrees ki. Without loss of generality,^ assume that 120 is the trivial bundle, As re use the word level to refer to (k i , k,). It can be shown that the lass of X is given by -= —311 3- (2g — 2 k i -- kdJ. whe  ^  2 A version of geometry Volume  Cher n. class of 0(1)^X. Cl IS In  ai  Chapte 2. On the G1.4.7 theory of plane bundles over curves  Definition 2,1.3. There is a di nition the locus of (1 : 0 : 0) in „. in /1 1 (X, Z), which is represente  s by deliology cla ss  f = If . P c 11 1 (X.).  Note that,^ is a set of generators for II I (X, a ) and^ relations a . „30 = 0 and I%. o = I hold in the cohomology ring. The latter implies that. 00 is a sectic Remark 2.1.4. One  ^  p2 bundlesof this form  C^V, Z)  c^ (see Dfinition 2.1.1) if and only^  ')  = fin +74  for^  integer 7, (see also Remark 2.1.2). The complex torus T (04 3 acts on X by (zo, :LA , 7,4 (xo  Let „ , C 11(X, Z) be a section class. The partition function of the degr /3, Gromov-Witten invariants is given by  Ea 00  f3.,(9I k ,k2) = ,  h= 0  where Mh(X,„3„) is the Ann. space of degree 13 3 , genus It st able maps 3 to X. and [Mh(X,f1±,)1" 2 AT)(Alh(X,13:) is in the Di,h equivariant Chow  aroun for  /7 = -^/38^vird^Os). Since we a orking equivariantly (the invariants are defined by equig forward to a point), our definition makes sense even for Y1 neerative m3 of D (c.f. Section 2.1 of [N)). Remark 2.1.5. The equivariant Gromov-Witten partition ^  tions are er equivariant deformations. The space X that we work with is determined up to equivariant deformationthe ^ is of C, and the level (k,. k 2 ) and so in this^ ?^to X by specifying only^ , pararneters. \ VI? aSSUIlle  ^  flee Remark 2.2 4)  Chapter 2. On the OW theory of plane bundles over curves goner^e equ ivariant Chow group of a point: t,2  1-  Z3 (21^k2) is a homogeneous polynomial m to, $ , t 2 of degree 0 w ith coefficients in Q((u)). In particular. it is zero if D is positive, an d Laurent series in a, independent of to, / 1 ,1, 2 , when D = 0 (this happens when xis is a Calahi-Yau section class). In the later case, Z3, (g k 1 , k 2 ) is equal to the usual Gromov-Witten partition fu ncti o n, ( c. f . Section 2.1 o f [ND . Thu partition function of the section class Cromov-Witten invariants is given by: .5  y  I kJ. k2).  Z4.0 ki,k2) = 01  We will define the relative vt sr rnof Z kl , k2) and prove a gluing theorem for it. The gluing formula allows us to compute the partition function in the general case in terms of the basic partition functions for the case of g 0, relative to one two or three fibers. We will compute these basic partition functions via localization techniques combined with relations Mg formula. These give rise to explicit 3 x 3 matrices C, U1 and U9 with entries in Q((u))(to, t7 , t 2 ). We will then prove the main result of this paper, which gives a formula for Z(g kl,k2) for any given genus g and level (A: 1 , k2). Theorem 21.6. Let X be a 7 2 -brendie over a 01717C C of genus g of the form Pie e L t G.) L9). where L i and L1 are two line bundles of degrees ky and lts, respectively. Then (g k4 , k2) = tr (0 9 = - Lt whew) G, U 1 and Lt2 are given by  IV  Chapt^  the OW theory of plane^  r cc  ) (to — ^to)(11  —  In)^0 0^(02 —1 0112  2(2/ 0 —^—^i^_ _^tO + t2 - 2I —1 0)(tu — (i0 -^)( 1 u + 1 2) ( 1 0 + /O( 1 0 + 1 2) - 21,2 2(21i^—^—1 2) ti 12 2/1 - IOWA - 12)  111^- tOilli - t2)  (11  10)(11 -1 2) tO +12 - 21 1^t7 -E 12— 2/0^2(2/2 — to — 11) (12 —1 ))(12 — ti)^(12 — to)(12 — t].)^(02 — 00(12  ( 1 0 + 1 0( 0 22 1 2) 420 2 + (21 1^t2)(/) 10)2( 7 ^1  ^1  U)  0 )2(1  =  2  - 1o)(111 - 12)  ( t i — 1: 00 1 o) ( 1 2^I t ) 112^LIN  0  (to - 1 0( 1 0 -1 2) U2 =  1/  1t2 (11 -- to)(ti^t2)  ^10) 1 (1 2 — t1) 2 (10-2^(212  ^(1 t2^11  ^  lo)(f2 —  where bb^2sit As an application of Theorem 2.1.6 we prove the following result: Theorem 2.1.7. Let X be any IP; let^E II 1 (X„Z) be a Calabi- You^C  ( g)  C of g enus  C apter 2. On ttre C'N theory of p lane where Z3,,(g) is the usual partition function for the degree fi e , G lOMOVWitten invariants of X, given by 9/  =  /1 11 h(X,X5,9)1"'  2.1.1 Donaldson-Thomas invariant prediction Another application of Theorem 2.1.6 is the prediction for eq uivariant  Donaldson Thomas partition fun( Section 9 of [14] and a l so see [22, 231). As an example, by the assumptions of '19reorem 2 . 1 . 7 , and ta king e' ----q, we can write Zg7:(g) = 3 113(q q -^• 2 ) ( -q) where  "( q )  = 11 (1 7>J  Moreover , by the same notations as in Theo^2 6 if we define the t itioit ftnrction for the class 13, equivariant Donals si -Thomas inyaria as  z,c/ (gl k l,k2)=  (  -  o (/  - 1 2) '  3  E  Lin  (X,34in  then Theorem 2.1.6 gives t h e full prediction for 7 D1 (q I ki, ku) for any section class fi s . The CW/DT correspondence in the case^f- f has recently been worked out to the first order in [16]. See also section 2.7 for more explicit o n the Cromov-Witten theory s ide.  2.1.2 Organization of this chapter In Section 2.2, we define the partition function of the relative. Groner Witten invariants of the space LP (0 p t, e L 2 ). Then^ x^gluing theorem for these partition functions. In Section 2,3, we compute some of the basic pi titiou f actions we defined in Section 2.2 in the case g^0. There are some basic partition 12  Chapter 2. On the Girt' theory of plane bund les  functio ns in this case that we can compute via localization, w e them in 2.3.1. We use the gluing theorem of Section 2.2 to compute those that we cannot compute via localization. This will be done in Section 3. In Section 2.4, using the results of Section 2.3, we construct the matrices U t and U9, which appeared in Theorem 2.1.6 and then we prove Theorem  2. 1.6. In Section 2.5, we first prove (Lemma 2.5.1) that any d 2 bundleover a curve C is deformation equivalent to a P 2 -bundle over C of the form P(0 ED 0 (1., L). Having this, we use Theorem 2.1.6 to prove Theorem 2.1.7. In 2.6, we first prove that it is enough in this paper to only consider the moduli space of maps with connected domains (Lemma 2.6.1). After that we give a proof for the gluing theorem expressed hi Section 2.2. In 2.7, we provide some more formulas for the partition function of Nun - sit Cromov-Witten invariants in some special cases.  2.2 Relative invariants and the Gluing theorem Let (C,Th^pP) be a non-singular curve of genus g with^  ,s  Following the notationsof Section 2.1, we take X = P(0 ED Lt C L2) ^) (C, ,... p r .). We will review the definition of the section class equivariant Gromov-Witten invariants relative ' ors F t , , F,,, where F, is the fiber over the point. pi. For a treatment of the foundations of equi variant relative Gromov-Witten theory, see 1181. The complex torus^(1C* ) 3 acts on X as in Section 2.1. We need to fix a basis, B p , for the equivariant. Chow group of each fiber, Ft„ which is a copy of P 2 :  ZjH , fo t t2j / ,  Let /E s E F3' (X, F) he a section class (defined in Section 2.1) We take  4,(g I k j . to be degree g3,,, g enus to the divisors F Fr , with restr te take  it of X relati ve by a i,  C  13 0 , one for each  =^(t7t_;1jr, 13  Chapter 2. On the OW theory of plane bundles over curve ollowm Section 2 of [23]. let X LI he the i,, step degeneration of X along ealet Mh1X1F,^be the moduli space of relative stable maps q :^X[L] from nodal genus It curvest C", to XIA, for some which are representing the class /3,. Then. Mh(X// .I.1,) is a DM-stack of 1 dimension —Ky /3, (see also j2011. r each p =^ h ave^ evaluation map which ive pointy and is 1-equuv ^ t (see [21]): )  h( I  Then zj,(g e kIlkAt1^(1 7  . = • [Li h(/ ,3 6 )11  where h(z I  el, (a l )^U^o,.),  os )^Mh(x1P,,(30\  s in the Dth egnivariant Chow group for D = —Kx^= virdim^(X/F, Note that the invariants can be non-zero even for negative values of D (c.f. Section 2.1 of [1 4], and also see Remark 2.1.5). Then the partition function of the degree /3„ relative, Gromov-Witten invariants is given by ,  =EZ  8  0 f k1 k2)0•1•-•O r^(2.2.1)  1000  We can also write th e partition function of the section class, relative Gro nov-Witten invariants as Z (g  ZO)ki,k2 sci . on  It is e ‘. de t that when r = 0, we get the partition unction for the ord invariants defined in Section 2.1. -  Remark 2.2.1. Z3, (g kl^a homogeneous polynon of degree  N =2.  E^ le^—D =^2g — 2^— 0  p=1  u/ViIl)c that  2  P=1  ^  ,  s are cormeeLed (see Re^Is 2.2.4).  — 377,  .  Chapter 2. On the GIV theory of plane bundles over curves with coefficients^ Q((u))^ a Laurent sen s in u. independent of t o , t m ,^ Section 2.1).  zero if IV < 0, and it it when N = 0 (c.f. 111j.  Remark 2.2.2. We can reexpress the definition of the partition function lot the section class invariants as^Remark 2.1.4): :7 (s i kf . k2)c e -• or^-,30 i ni(q 1 kl, k2)0, - ti, . to because by Remark 2.2.1 ^clear that the sum is terminated from ^o 'e and it is also terminated from below because for the ^ large negative values 0 there is no curve representing the class )3 0 + It f which means that  h(X/1"„(3(^nit ) _0 for n < 0. To see the last claim, let If^0 Q 11, 1 W L9, and notice that for n «0 there is a one to one correspondence between geometric sections representing /3 0 4- n f (note that for It < 0, a curve class 0 0 + 71.1 must be a geometric section, see Remark 2.1.2) and degree sub-line bundles of E. E has no sub-line bundle of degree greater than k 1 k2. Therefore, for < 0, E has no sub-line bundle of degree n,, which proves oar claim. Before expressing the gluing theorem, we fix a basis, B, for the equivadant Chow group of y 2 . We take  (a— 1 )(b" — t 2 ), = (H t o )(H = (if — to)(// —1 2) , are in fact eluivariant classes represented by three fixed points of the torus action on P 2 . We define  B {X0, x It is easy to see that B is a basis for At'  ^ro  Convention. From now on, we assume that each ^ definition of relative partition functions belongs to this is set, B.  1(,  We take  — fiq ta -(0)(( 7^t2).  -  15  Chapter 2. On the G117 theory of trlane bone We have these relations: T(xi)xi, = 0 for i  ri j.^  (2.2.2)  Then we raise the indices for the relative partition functions by the following rule:  z 9 k i, k 2 );y.: ; , (  Then we have  ^  1(2) .)]^2  T h .p)  7  2. • 2,  o lowing gluing rules similar to Theorem 3.2 in  Theorem 2.2.3. For any choices of elements 02,^, a s and ^t ] ,^ from  the set B and integers satisfying g^g",^k/i k7 and k2^+4, we have  z(grk,^7t  ^  E 72,0 1M, 0 ) 1  2 a ,^A  z( g"  Act3  and k],  E^ki 1( ) 2  A  a A  AGS  The proof of this theorem will he given in Section 2.6.  Remark 2.2.4. In most of the contexts in which the relative Cromov-Witten invariants are being used, maps with disconnected domain curves are considered as well as ones with connected domains. In Lemma 2.6.1, we prove that in our case, where we only deal with section classes, we do not need to consider disconnected domain curves.  Remark 2.2.5. In exactly the same way as in HAL one can prove by using Theorem 2.2.3 that the partition functions Z (g 1 0, 0),,.., ‘,„ give rise to a 1 -t- 1-dimensional TQFT taking values in the ring R = Q((n)) (to, i i t 2 ). The Frobenius algebra corresponding to this TQFT (see [15b Theorem 2.1) ,  Chap( 2. On the 014' theory of piano bundles We will prove that this Frobenius algebra and hence the TQFT is semi-simple (Proposition 2.3.11). In Section 2.5, we proving Theorem 2.1.6 and 2.1.7 for the case g = 0.  urrespondirig e this fact for  We will use the following corollary of Theorem 2.2.3 in or Corollary 2.2.6. With the same notation as in Theorem 2.,  Z.3 0 n f(9  I kb k2),,,, z3E, nif  2.3 Calculations We wil l work with the space  X = P(0 . L, ...................r)^) throughout 1 section. In accordance with the notations in 111] w e will use the words cap. tube arul pants to refer to the case where the base curve ; C. is a genus zero curve with one, two and three marked points, respectively (see Remark 2.1.5). We sometimes refer to the partition :functions by referring to the space to which they correspond. We will use the notation = 2 sin -2 in the later calculations. Similar to Section 4.3 in [14], one can see that the following partition functions determine the theory completely: Zia j 0, 0),„^240^0, 0),,,„ 2 ,,,^Z(01^1, 0) 0 , Z(010,-1) 0 ,^Z(011,0),,^Z(010, 1)„. We refer to the partition functions ove as the basic lrartitii 13y the discussion given in Rem 2.2.2, one can prove le Lemma 2.3.1. The basic partition functions are given by Z^0, 0)e, = Z80 (0 1 0, 0), Z(0;0.0) Z(0 r 0)& Z 3 0 (01 —1. 0) ,„ • Z( 0 ! 0, ,  Z(01 1, 0),-, Z(0  Z^f (0 ; 1. 0) „ ,  Z(010, h) &  = Z 3 5 (0 0, 0)„, (003  =^(0^0 )«l a„ = Z,3 0 (010, --1),„ /g rip^J.(010, 1 )a, , f (0 I 0.  Chapter 2. On the^theory plane bundles^  curves  Proof. We prove the last equality^  ows. In the right hand side, we do not have any partition function of degree S o n f for it < 0, because O e 7,, 1^L2 does not have any sub-line bundle of a positive degree, as Li and 1Z are of degree zero. We also do not^functic degree /10 a-- ?I f for ra > 1 because 3  E deg (a ) — D y  p,= i  — (2 1- 2 2)^(311 4- 2F) (So + 71f = --- 3n, proved si  tive for rt > 1 (see Remark 2.2.1). The other equalities re 0  The rest of this sc^ nn puting the terms that. appeared es of^ 2.3.1 •  2.3.1 Calculations via localization  The complex torus acts on X as before. We define So: the locus of (1 : fl : 0) in X^1)(0 C U 1 Lo i C L21,6 S,: die locus of (0 : 1 : 0) in X^IP(Loti I ED 0 ID 1,21,,7 I ), 82: I he locus of (0 • U • 1) in K C P(L 0 1;2 I G /A L; I CD 0). It is clear that So, S i and S9 are fixed under the torus . One can also see that So, S i and 89 represent the classes S o , /30 — k d /So — k2 f , respectively. As before, let fl be a section class. The torus action o an action on M1 1 (X/III ,,51). We denote the fixed locus of I:. by Hi, (X/ /I I , ) By notations of Section 2.2, we let^C XT; be the ystep de'ation of Si along intersection points ri p^n Fr, for p = 0,1,2, such that^is still fixed under the induced actions on XIIL Then, Al1,(X/12100 2 ' parameterizes maps q : C'^X11:4 for some L, whose^ ( re either of Si[1,11Uni i f„ for i = 0,1 or 2, where by the last expression mean with m i .t-invariant fiber curves, fn represents the Class bf for some hcZ 1 ), are attached to it at some points. Note that the choice of i E I, 2}, and also the number of fibers which are attached to Sh^eve constrained by the class ;),,.  Chapter 2. On the GW theory of plane bundles over curves In general. the moduli space Ath(X/1 2 ,13,7' can be quite complicated because of the existence of the fibers attached to each Si [i]. However, in the special case where 711. = 0 for some i there is no fiber attached to S,110, it is evident that the corresponding component of Ai n (X/È, ,C30 7 (parameterizing maps with images equal to Si LLD is the moduli space of degree one relative stable maps to curves, which we denote by 21711,(.9iii=i 9 1), where 7 j = (71 . -4  Assumption 1. For the rest of Section 2.3.1. w assume that ^ PC/P4 4,34 11 illh(82/±i. 1).^(2.3.1) where 1 c {0  21, depending on the class tit s .  Then one c n see that the 7-fixed part of the I of M b (X/1' „13 9 is exactly the usual obstruction theo and therefore  M 1 ,(x/to s )"  ) truction theory AIL (Siffi , 1),  MaSitfi, 1)]  cl  the special case where nth ^ 0 lob all possible^ ASS UM [olds. One can see easily that this is the case for all the partition functions in the right hand sides of equations in Lemma 2.3.1. except for Zo o; (0 0, 0),,,, 2 „ 3 . In this section, we will use localization for calculating the former partition functions. 43 07 1(0 0, 0)„, a t„, 3 will be calculated in Section 2.3.2 by combining the results of this section with the gluing techniques. Applying the relative virtual localization formula (see Section 3 of P8) 5 ), we can write  Zd s (g I A;1, k2)ot• evi )  st  ) n • • • n  e(Nor  eit;.(ar) ^(2.3.2)  where Nonni" . is the equivariant virtual normal ^ of the component of the 7-fixed loci in illh(X/1 7 , (6,), and c(Norn)' tr l is its equivariant Euler class. Let 71 : U -^h(X / P SO and q U — the universal map, respective in X. Then we have the following is 01, the authors assume for CO oven locus but it is suaight forward to adapt their methp(  :amp ter 2. On the CW theory of plane bundles over curves Lemma 2.3.2. For each i E I, the equity^class of the virtual normal bundle of the component of the fixed tom ^1 th is isomorphic to At 1 490'1,1), is given by .  c(Norrrir)= c($rxr,q N51X)= e (11C7r,91(ti^)) ! 1  when tj and k are two distinct elements of {1, 2, 3} f. Following the notations of [18j, for a normal crossings divisor D contained e smooth locus of a variety V, by Tv (— log D), we mean the dual of the sheaf of Kidder differentials with logarith Cale poles along D. Let Si[L]^X[L] be a relative map^  sponding to an element of Alk(Si. fit ) and : X[I] ---> X  be the natural contraction map. For the divisor D C^as above, and a given integer I > 0, let^be the infinity section of the ith component of  the Tstep degeneration of V along D (see Section 2.1 of 118 1, for more details). Note that when t = 0, D oc. is D. Then by the description of Tv log I)) given in Section 2.8 of [18], we have the following equality in Iff-theory —  Y: too'  Ts., H^E log rim ,  p= 1  J cE N G,.,,,,v ,  Pni  inclusion cif S [U ^to X[iti, and hence  C'  ^ ,  -  E  log Fp ,  p=1 =^, 4T9,1^  E lo g Ting  et' ,^c 11t / ). 1  ^ The first equality in the lemma follows from to CO pa rts of the obstruction/deformation sequences (X/P, (3 9 ) ant^Si/f.i, 1) under the induced torus action (See (2) a in (18)), and noting that we consider degree 1 maps to Si[]. second equality in the lemma follows from the ^o Thism :  NiLiN CD  211  Chapter 2. On the CSV theory of plane bundles e Remark 2.3.3. The last expression in Lemma 2.3.2 is consistent with fire notations of Section 2.2 of [14], where the authors considered the u space of maps from curves to the total space of a direct, sum of two line bundles over a curve.  For manipulating the evaluation ^ in (2.3.2), we use the followi ng Cartesian diagram for each p^I^r and i C I: ,  ii  ,, (X/ P  4  )^f-c) FT„  where two vertical maps are inclusions, and  Ti p 's the intersection point of Si with Pp , which is the fixed point of the torus action on _Pp representing the class xi C B. Tram this diagram, it is clear that ev;(a p ), restricted to Al' I), is a class of pure weight for each p and can be taken out of the integrals. We summarize all the discussion above in the folio  Z,3(g^1, k2)g,^Eii2h--2  /-  hi 0  e(—P771 * (C(NSi/v )) .^(2.3.3) -  f3y chasing the diagram above, one can see easily that T(rt i ) if i^k  ("pc'^ (Xk) =  0^otherwise  (2.3.4)  for k C {0, I 2} (see Section 2.2 for the definition of T( )). ,  Computing degree ,3 0 , level (0,0) cap, tube and pants Lemma 2.3.4. Partition functions for the degree ala. level (0,0 . and pants are given by 43 0 (01 0,^= 1 if a b otherwise  = T( { a;„  )  2  0 21  On the CVII theory of plane bundles over curves  Chapter^  E {0 1,2). Proof. Since kj^k 2^0. all So. S i and S., represent the class Po, and (2.3.3) we have I = {0,1,2}  so  in  .  We use the results of Sections 6.2 and 6.1.2 in 11 it to evaluate the integrals in (2.3.3), for the cap, tube and pants. We prove the fliffinula for the tube, and the other cases are similar. By Lemma 6.1 in [147 (for d = 1), and Lemma 2.3.2, and also by (2.3.3) and (2.3.4), we have ^ -  u2h-2--h_^  0  30) * (xa)(ev2 0 30 * (3 0  h=0  1  1,17 h(So/froa  ^(-1e73,„q*(L i L t^  L0 t ))  (evt oil) * big)(ev^vbrd frirq* (L o^L  ^  /AI  +  k  T'n  (ev1^'V2 0n2)*(4) ^  gl4 h(. 5 2/(721,r22),1)r  (Ir7r,q* (1..0", " 1 OS L 1 1,  1 ))  2 = EOri/ T ( X inOti T (Xi  Note that the weights of the torus action on the first and the second factors of LiLT' e L k LT' are ti — ti and ti — C , respectively (which they play the roles of t i and tg in Lemma 6.1 in [1117). El Computing degree 00, level (0, —1) and (-1,0) and degree ,3 0 — f, level (0, 1) and (1,0) caps Lemma 2.3.5. Partition functions for the deg re) ri g level (0^)^, caps are given by ,  )  Z3 0 (010,  7 (0 I —1, O ra  .2.  22  ^  Chapter 2. On the OW theory of plane bundles over curves Proof. We prove the first formula, and the second one is proved in a similar We have k 1 = 0 and k 2^-1, So, 5' r represent the class fi b , but 82 represents the class 00 f Therefore in (2.3.3) we have I = {0,1}.  By Lemma 6.3 in HI (for d = 1). and also Lemma 2.3.2 and (2.3.4), we can rewrite (2.3.3) as X'30 10 1 U  ^ora  ^E  , L 2),^  0=0  (---Ir7r,q 1 (1;  ^  -f- (evi o j i y(r„)  (eV,  ' C L2L0 1 )  MY/  • e (-1r7r,t1II 0 L 1 1 0 =^  )  L9  r))  5,1.17'(:00)^5,1,7t(I1.: I)) , 1.0 = 11^1,1 =  to  = (1:0^12)01  Lemma 2.3.6, Partition ,nctions ,for h degree (1,0) (nits are given by  0  -^level (0,1) and  130^Oro = Ita -^Ito --- 1 00 “2 ^4  :2„^I 1,0)^=(ta^10)(f. — t2)(1)  for a = 0,1 2. Proof. For the first relation , note that only^ its the class po and so we have I = {2} Note also that^ mndle of 82 in X is level (-1, -^in this case. The rest of the proof is quite similar to the proof of Lemma 2.3.5, except that this time the relevant integral is obtained by applying Theorem 3.2 in [14] to the level (0, -1) and (-1,0) caps (given by Lemma 6.3 in (11]) and the level (0, 0) pants (given in Section 6.4.2 in [14]) Eo get the level (-I, -1) cap. ^  2.3.2 Calculations via gluing techniques  In this section, we use Corollary 2.2.6 (which is referred^ as the gluing t tt of Section 2.3.1 to find Z3, 110 r 0. 0j a ,„„m8 . For tt th e gl mng t heorem , see 2.6.  formula) ,^ t reatment of^  23  Chapter 2. On the C111 theory of plane families over curves We first need to find the following partition functions of tubes: Z(0 0, —0,,  2  = Z3„(01 0, —1)(tia2  Z;30( f^— 0a00 .  Z(0 0, I)(0a2 ^Z30- f^0, 00012^Z,30 ( 0 F 0, 1)a0120  ^  (2.3.5)  relat ions hold after swapping the degrees. These equaht s can be  proven^  the proof of Lemma 2.3.1.  Computing degree O n , level (0, —1) and (-1, 0) and degree ,3 0 — f, level (0, 1) and (1, 0) tubes Lemma 2.3.7. Partition functions for the degree Pia, level (0,-1) a m^0)  tubes are given by (to — ti) (to — t2) 2 0°'  ZA (0  10,  ( t1 — (o) (11  —  12 2 0' )  0  = b = 0,  if a=b= 1, otherwise.  ( -*)( im -- ) 0 2  Zt o^j -- I,0  f  -1  — 1 0)(02 — 1:1 ) 2 0  0  if a=b= 0,  f  b re- 2,  otherwise  ,for a, b E (0,1, 2}. Proof. The first relation is siinply proved ^  tt^the level (( to the level (0, 0) pants and applying the gluing formula.  ^  cap  Z3,,,(0 1 0 ,^Za„ (0 P 0, — , 4 7 3 o( 0 „  This is schematically indicated iii the first row of Figure 2.1.  The result is now obvious by applying Lemm as 2.3.5 and 2.3.6. The proof of the second relation is similar. Similar to the proof of Lemma 2.3.7, we can prove  ^jj  Chapter 2. On the OW theory plane int  x acj^Ljx„ = j  —11  ( 0. 0  Po  (0,0)  (0,9/ Th^ ( 0, —1) \ 4^‘b^ s  It  004  Figure 2.1: ( -1)-tube^  ,  11/^-1-  ^  11 ,01  (0, -1)-cap -f (0, 0)-pants  Lemma 2.3.8. Partition functions for the degree Ego^eve! (0, 1) and (1,0) tubes are given by Zi^( 0j 0,1)  (^■0) 2 12 -  a=b  (  9  otherwise. 19) 2 (1 ] -10 2 0  2  if a= 0=1, otherwise  for a b E (0, 1, 2).  Computing degree 0 0 , level (0,1) and (1,0), and also degree 0 0 f, level (0, -1) and (-1,0) tubes We do the calculations for degree 130, level (0, 1) and degree 0 f level (0, -1) tubes, the (1, 0) and (-1,0) cases are similar. attach two tubes of levels (0,-1) and (0,1) to get a tube of level (0.0) (see the picture). Now applying gluing formula and using Lemmas 2.3.7 and 2.3.8, we get (see the first row in Figure 2.2): 2) 0 (0 I 0.0 i t ^0, j  ^0,  1Lj^f ( 1)  ),(0^, 1)%  - 1),T„),„ Z^f ( 0 I 0 ,  bi a =. 0, 1 2 By using Lemmas 2.3 2.3.7 and 2.3.8, we can soly< the equations above  ^  Chapter 2. On the CI4 7 theory ci plane b e  for the other unknowns: ^Z3„  (0 0, 1)^— 3 ,, (0  ^Z  L sa I(010.^= zsz By changing^ row in Figure 2.2):  conditions, we can get more relations (s ee  0 , Z 0 (0 1 0, 0),„ 02 .,^Z3 0 (01 0, —1), 05.^o(0 i 0, 1) , which implies that ' 41 (0  We can also write ^  F 0.  0.  1) :r oz;  ng Lemmas 2.3.7 and 2.3.8 (see the third row in  Figure 2.2): 0 = Z3 0 (010,0) F2x „ = 3,(0 ,^  Z3 0 ( 0 1 0  ,  1)  + Z3„„"(010,1) r , x2 Z3 0 f(HO,  for a = 0,1. (0,0)  ^  0, 1) 'L (0,1)^(0.-1) -  ■^f"`,^?"--\ 0 rx Po^  Po  ,  on 19.  (0,0)  3,  -  1^  ;  ,,  '  tio^fio  (0,(L PO  0,04  ^1  ^A  0, I),^=^- WO,  Pox° X2^1 30 0 1) ^x„j0,-14,0^(0,1) X, (0,71) 7 .  )^  0 0 +r  '^  T a o)  zure 2.2:  ti  Po hi' -  XTY^  Chapter 2. On the O W theory of plane bundles over curves -  This with Lemmas 2.3.7 and 2.3.8 implies that  Z 5 0 (0 10, 1) h o .h 2  = 410 El  ( 0 0, = 1 ) . 0)3. '2( 1 2 —  Z03„ (0 I 0, 1)1.2^= Z3 0 f^I^— nr,„22 0,2 — 1,00  1  ,^(2.3.6)  Attaching the level (0,0) cap to the level (0, 1) tube, we get three relations (see Figure 2.3): 0 = Z (0 0,^Z30(0 10, ])11 1  -  h Z3 0 (0 I 0, 0'1;2,  for a = 0,1 , 2.  (0.1)^(0.(,),4., (0,1)^(0.01,,N (0 1)_^(00)^(0,1) , 2  (.  —}^Dx, ^1.^4>a Po^i3o^1:30 x ^1 1 0 4  Figure 2.3: (0, 1)-cap = (0, 0)-cap ± (0, 1)-tube  We already knOW that  0 I 0,1)% = Z 30 (0 I 0.^= 0,  ^(  so we get  zo o (OF 0, 1),0772 = ( 1 2 = Z,,30^0, 1-)71.(2 = ( t2 — 400, Z)) 0 (0 0, 02 2 x2 = ( 2(2 o — MO' ,  —  Combining with (2.3,6), we find  Z5„ if (0 I 0, — ) ) . 0 , (2  =^f  (0 F 0,^=  By writing more relations in a similar way (see lie forth and the fifth rows in Figure 2.2. a = 0. 1). we can prove  1) 2 0 , 0^43 0^0 1! 0^(0 0,^I^= (1) 2 . -  -  We now summarize all we have proven in this section into the following lemma:  27  Chapter 2. On the Gil/ theory of plac e bundles ova Lemma 2.3.9. Partition functions for the degree /3 0 , level (0. 1) an d  tubes and also for the degree /30 -I- f, level (0,-1) and (— ^tubes given by — E.1  Zg o (0 0. 1  - 1 0^12 — to —  X212-10-11 11  0  17, 3 0 (0 1 , 0),„„„  2/. 1 — to — / 9^ti — t o (/). ti — lo^12 — N  i  f (0 ( 0, — 1)„.„„. 6 ]  (0 —1,0  =  for a b c {0,1,2}, where partition lanai as with the ^ (a -I- 1,b,- 1) entry of the matrices above. ,  are the  Computing degree I3 ^f , level (0, 0) pants Lemma 2.3.10. Partition functions for the degree S o f level (0, 0) pc  are given by 01^= 0,^Zo o f (0 I 0, 0).,„„ 2 xs = (t52 — 11)(i5 3 , Z^f ( 0 I 0, 0 ),E^= (t2 = t0)0 3^z«,, ( 0 IQ,^— (10^11W, 2;3, bf (0 F 0,^1.2^(t1 — 10)0 3 ,^z30^!0).r 2 .r. — 1 00 ^ = (212 — to— ZJO f^= ("I — /2)0 3 Z3, f (O 0,1-)ooro.c, rr-- (to ^ 4 30 f (DI 0, O)io:taro = ( 21 i) — t t — (°! 0 , Cixix, 43 0  , ,[  = (21,1 -^t2)03.  ^  Proof. We attach th to the level (0, 0) pants to obtain the degree ibo, level (0, ^ ogether Lemma s 2.3.6 and 2.' ^e get the following relation (see Figure 2.4): 0. 0),„,^  —  43() ( 0 1 0 . Or a 28  Chapter 2. On the C11 ,17 theory of plane bundles over curves 0111  ±h I) x- 9^lx -1  ( 0,1)  C ' 1=211-1\11x,  30  [  Figure 2.1: (0,1)-tube = (0 1)-cap 1 (0  From this, we can get all Z3,„ f .101 0 , O ^with at least one of a, h. c equal to 2. If we attach T. le level (0, —1) cap to the level (0, 0) pants to obtain the degree /3 + f, level (0, —1) tube, we will get (see the second row i 2.1) .  Z;3 0 + f( 0^o).„ T o ,  — z3„ 1 T^o.  0 1r0 1 3 13 = ( t 0^t )0 3 ) Z3 0 F^0, 0 ),tozo,to — Z30-1/(0 0, 0 )ro.no,r; _ (t o ^t I )0 3 , 73 0 I"^0 0 )xtst,^7X30 11(0 0, 0 )2'1,1 - 1,,m =^—^) 03'^(2, _,  1^  as follows (see Figure 2.5):  U =Z3of f(U FU9 0 Yro/9 Z30( 0 0 , O)r; 0 ,2 Z 3 0 0 Q 21^Z 3 0 1( 0 (), (0<;;; •  where the left ^s zero by Lemma 2.3.4.  e 2.5: Frobenius relation: all pants are level (0,1.0  0  0 0)  ^  (11^/2) 2( 3o  29  )  Chapter 2. On the CW theory of plane bundles over curves  Combining^ with (2.3.7), we eve will find the rest of the in the lemma.^ We now know prove the semi-sin  ^  its. and s o  QV' (se  'k 2.2.5) :  Proposition 2.3.11.^ level^0) TQFT resulted from our selling and Theorem 2.2.3 is SCTIV(SZ Proof. By Lemmas 2.3.1 ^we have  Z(0 0, 0) 2  1, ()^2g0(0 0, 0  = IT(:t c,) if a^c, otherwise.  This means that for a = 0, the basis ic,„„fr(xo),ex l /T(xi), a 2, 2 /7 0'21_1 of the corresponding Frobenins algebra (see Remark 2.2.5) is idempotent: 1  ex, -,r ^ ® 7 1 (Xi)^T(r j )  Tbr i )  This proves the semi-simplicity when a = 0 (see [151. Section 2). Now tile proposition follows from Proposition 2.2 in [15). ^^  2,4 Proof of Theorem 21.6 We now know everything we need in order to prove 'theorem 2.1.6. We first find the first and second level creation operators and also germs adding operator, which are by definition f%t  =[z(0i1,x){j..^Le) = {Z (0^C^TZ(110,0) .  n,  respectively. here partition^ ;tions with the lower index x a and the upper ndex d b are the (b + 1, a + 1) entry of the matrices above. We can find U 1 and /72 by simply raising the indices in Lennnas 2.3.8 and 2.3.9:  CI  ^  = fz30  On the CR' theory of plane bundles over curves  U U  ^ r^[Zsoto  0, 1)2: =^0^0 ,  () U (2  — 1 o)(2  — I) (  t2 (  /I/ —^  l —^)( 1 0 — t2) —  12  I] —1 2^— to)( 1 1 —1. >) 212 — to — t i — ,)^(t2^1 0)( 1 2 — t3) -  (to -  ^  0 ( 1 0 - / 2)  (t2  tE -12^ (12  - t o ) (1) -10 (ti - (2)  -^( 1 2 - - 1 1) 2 0 °2 ( 1 21 2 — 1 0 —^111  12 — /2)^  (12^1 0)( 1 1  01 is obtained in a similar way, and it is as in Theorem 2.1.6.  13y Lemma 2.3.4, 1Z, 3„ 010, 0):2 l is the identity matrix, and by the gluing formula, we can write 201 0,0)1'1  .  1 = [Z(0 0,1)2:  [Z(0^— 112L1  Therefore, by Lemmas 2.3.7 and 2.3.9, we have  iz3o^0,^1zs o^i —  ( 1 0 —^12/^(to —^(10 °^)(to —  ^1 ^—  ^1^1  /OH^— 1 2)^  )(11^/9)  _( 1 2^—^— to)(T2 —^(12-- 10(12  (2.4.2)  ^ ^  (anipter 2. On the GIV theory of plan e bundles over curves ( 10  —  1 1)( 1 0 — 1 2 2 0 )  ^I tte2-  to - - 1t )(to - 12)^  ^(  )(10 -  02^  (I• t10 — 1 41 0^122)  1/1 - /01 -1-1 2) 2 0 I 95 2 -^-^(t1 —10)(h,^12)^(ti - 11 0(ti 12) ^02^ 02^ (//1 2 -( 0)( 1 2^tl)^(12^4)( 1 2 -1 1)^(12 -1 0)(t2^/I) -  1:2 I I is obtain ed in a similar way.^(1.22 I are the^ It I second level annihilat.ionoperators, respec Now we are going to find the matrix G. fly the same arguvient given for Lemma 2.3.1, one can prove that  Z(1 I 0,^= Z 3,^0, 0),„^Z30± f 0 I 0, D I.1 1- 0 for a, b C {0,1, 2}. Thus, we have [43 0 (110, 0)2} + {Z /3,, *f (1^,  For calculating the terms in the right hand side of this, we attach two pants at two points (see Figure 2.6) and apply the gluing formula: Zit o ( 1 I 0 ,^=z,30 ( 0 I 0 , 0 )/:,x,:ra Z.30 ( 0 I 0, 0 ): 11::" • Z,3 0 , / (1 0,^= Z3 o (0 I 0.^(010,0)20'1x' f (0(0, 0) 3 „ .0 bib 2 /3„ (0 0, 0)",:y r b  whi ch implies tZi„( 1 I 0, 0)I2 (10  —  ]  11WD — 1 2)^0^0 0^(11^0)(11^2)^0 ( 11 - t0)( 1 2 -1 l  and  l z,3o^o,^j  ^"  ^2  2(2t o —^— t 2^2t2^to^12 — 2t (to --- 1 1)(E0 — 1 2)^(to^/.1)(tu — (2)^(to —^/ to^— 21.2^2(2t1^— (9)^ t2^2t o — to)(1i — 12)^(11 — tu)(1,1^12)^-I /WW , t2^2 11^tr + 12 - 210^2(21:2 — to — I -  —  1 0)( 1 2 — 11)^(12^(01(12—^("2 -- (ol(t>— ti  32  Chapter 2. on the GU' theory of ^ (0.0)  bundles  (0.0)^(OM _  x^ kd^+^I  Rt  0,0+—k^ (0)(9,----,t+),p,o) ___+-^) )^)).(0.0)^ _  ..._  , )), ^+^  1)-- x txli(li^fi::1-----:-L--zr3, 1  r  "Th‘b  a  2.6: Attachi^)ant  Now Theorem 2.1.6 for g > 1 is a d irect application of the gluing rules (Theorem 2.2.3). The same formula holds for g = 0, which follows from the semi simplic ity of the level (0, 0) TQFT (see Remark 2.2.5 and Proposition 2.3.11, and sc Section 2 in [15], and Section 5 in [III).  Chapter 2. ()(1 the CW theo{v of^  e bundles over^  S  2 5 Proof of Theorem 21.7 .  We first prove Lemma 2.5.1.^ .P2-bundle over a curer C is do formation equivalent to LP(0 ED 0 eL)^— C,  where 0^C is the tt rural bundle and L ^C is a brae bundle. Proof. First, we show that every P 2 -bundle over a curve C is of the form P(E) C, where Cis a rank 3 bundle. A rank 3 vector bundle (resp. 15°-bundle) over C is classified by an element in 11 (C,(71(3)) (resp. H (C, PC1(3))), where C1(3) (resp. lill(3)) is the (non-Abelian) sheaf of G1(3) (resp. PG1(3)) valued holornorphic functions on C. From the exact sequence of sheaves  0 — 0" —401(3) --) PCI(3) --i we get a nap  11 1 (C,PCI(3))^cry  By examining e cocycle s, one can see that a F 2 -bundle over C is of the form 12(E) C if and only if the corresponding element in li t (C. PC1(3)) goes to zero under the above map. This element is represented by the (inch cocycle obtained from the transition functions of the bundle. But I/ 2 (C, 0') = 0 for a curve C: this completes the first part of the proof of the lemina. Next. we show that IP(E)^Cis deformation equivalent to  standard^  for a rank 3 bundle E over a curve, we  following exact sequence of bundles over C (see [19], Example 5.0.  0—> 0 0 0 --r E(tn)^---> 0, for some line bundle L and sonic in > 0 such that E(7n) is globally generated by its sections. Thus E(m) corresponds to an element v E Ext t (L., 0 0 0). n E(m) by deforming the extension iass v to 0 inside this 0 C Ext i (L, 0 (1) 0) corresponds to 0 (4)0 L C.  Chapter 2. On the (sTT theory of plane bundles  over curve  So we have proven that I3(nt) is deformation equiiadent to 0 EL 0 ci L. Now we use the isornor'phisrn '(13)  = ! (L(1 it.))  to complete the Proof of the lemma  Ti  By this lemma, we assume that the space X = 2(0 0 C + L), and we let cm (L) = k. For simplicity, we use the following notations in this section:  B = [L  1 11Jo,Of.hh].  C_ o leo— r(ulu.11 ^]  13^[ 40(0 F 0 1)i;], N= [4 o (0]0,-1)2]^Al^[23,,.. 1 (0 0^1)rp1.  A and 13 were given at! the end of Section 2.4, and C and E (respectively, N and Al) were the first and the second matrices in the right hand side of (2.41) (respectively, (2.4.2)). By using the notations of the previous sections, we can write C = A i- B,.^U2 =CIE,^U; = N + Af,  and hence the formula in Theorem 2.1.6 (for k r = 0 and k2 = k) reads as follows: Z(g ]0.k) = ts((A + B)°( + E)). Now we are looking for those terns in this forrnnla that correspond to CalahiYan section class. If we denote this class by lts = uf, then n must satisfy 1iy^=0 --: 29--2—k-3n=0.  If for given y and A: there is an integral solution for n in tins equation, t the Calabi-Yau class exists. We write the ahove equation ire terms c instead of k: Z(9 ] 0. k)  = tr((A + B)9 ' (C + —  Now by the gluing formula, C = A-t- B coimnutes with U2 = C+ 14, and so we have Z(g 0,k) =tr((( A r-B)(C -L 13) 2 ? (C -F-B) Notation, her two matrices U and  V, by (U`°, V t ) for the sun of the all the products that we can write coup and b copies of V. For example  in  (2.5.1)  !re. rne'itn of U  U 2 1^F- VU2  35  ^ ^  Chapter 2. On the GliV theory of plane bundles over curves  We first assume that g > 0.distinguish two cases: (i) vi < 0. One can see that E j 0 and 13E 2 = 0, and so we have Z(g 10,0 = tr(((il B)(c2 E2^on9--1 (E•  • ((E 2 ,C)+ (E, C 2 ) + C 3 ) - ")  tr(44E 2 1 B(E, C)^• .) 11 ((E 2 ,C)  )•  f and One can see that only those terms that have been written in th e above contribute to make the class tics = fie + n f. Thus, A, B, C and If correspond to the classes 00, ,(30±f , i3 o -  (g 10, k)^tr((4E 2 B(E, C)) 9 ((E`, C)) t,r((AE 2 BEC i BC Er I (E 2 C t CE ECE) -n )  tr( (21E 2 ) 9- (CE 2 ) ") 1 tr((BEC)Y -I (E 2 C) - ) + tr((BCE) 9-1 (EC E) -7I ).^  (2.5.2)  For the last equality, we only used the fact that E 3 BE 2 a gain , an d also tr(il.")^tr(VU),^(2.5.3) for Vow one can see^ by induction that For any non nega t ive t 11 AE , (c ,„,2 ) ,, _ ,, „f 2 = I 1  Therefore ,^first term in (2.5.2) is 1^1^" I  ^((A E 2 ) 9— (CE 2 )  r  ) =tr  fJ  1 1 1^020-  I  Again induction on non-negati Lions imply that  1^1 .^ Lg2 (2.5.4)  = 32 ^ ^  0 (.B.EC) (.E 2  =3  2  b together with simple calculaI  lo 10  0  --  e 36  Chapter 2. On the OW theory of plane bundles Therefore, the the second term in (2.5.2) is tr ( ( BEC)g (E 2 C)^= 3ft 912 9 - 2 ^(2.5.5) .  Powers of BCE are more difficult, to compute and so for coin ing the third term in (2.5.2) we first notice that ,  0 0 0-  C EB = l0 and also ( EC E) = EEL for any can write  0  0 to^> 1, we  JO,  (BCE) 6 (ECE)" = (B(CEB) b-1 0E)(ECE) = 3, t)- ( BcE) ( Li c Fi)0 2b - 2  .  An easy calculation shows that  tr ((BCE)(ECE)) = tr(BCE) = 30 2 . Putting all together, we can find the third term in (2.5.2): tr((fiCE)g - i (ECE)  -  Ii )  39  -  1 .13 2 9 -2 .^(2.5.6)  By (2.5.2)-(2.5.6), we find 7 (9 10, k) = 3 9 (1) 29 , which proves the theorem in this case. (ii) n > 0. We have U., I = Al N, and so we can rewrite (2.5.1) as  Z(g10,k)^trifiA +.13)(Cl^E) 2 ) (1- I (M^Nr 1 "). One can check that (Al 2 , N) = 111 3 = 0, and so  Z(g 0, k) = tr((AE 2 B(E, C)^• • )g -I ((t N 2) N 3 )n) , By the same reason as in the last case, we have Z (910, k)^r((A E 2 ) 9  1  (N 2 /1/) , 9 s- tr((BEC) 9- I (NA  tx((3CE) 11- '(AlN 2 )"). The rest of the proof is similar to the last case and is omitted. For g = 0, the result is deduced from the semi-sarnplicity of the TQPT (Proposition 2.3.11, see also Remark 2.2.5).  Remark 2.5.2. The partition function for a Calabi-Yau section class is a priori independent of the equivariant parameters. Hence, the calculation for Theorem 2.1.7 can be done with any choice of equivariant parameters. For example, the choice of io = 0,^  t^0,  C]linpter  2  On the GIV theory^  undies over^  ves  2,6 Proof of the Gluing theorem We first prove the assertion of Remark 2.2.4, which deals with the fact that we do not need to consider maps with disconnected domains: Lemma 2.6.1. Tice con tribution of snaps with disconnected (10711Will curves in the section class e(^iant Cminov.- Witten invariants of the space 1F(0 3 ,  E, L 2 ) is zero. Proof. A disconnected^ -e whose image represents the class $o 1 of f is a onion of a distinguished connected component, whose image repre sents the class 00 in n' f , and a number of other components, whose images represent, the class 71 f for some positive integer ii . We have  -  "  "  virdimM(X/F, 77 f) = —(-311 + (2g — 2 — k.1 — k2 )F) , n" f = 3n" > 0. "  So by a discussion similar to Remark 2.2.1, one can see that  1= 0. Disconnected^ can be expressed in terms of the products of the connected invariants,^ so the lemma, follows from the vanishing above. Now we return to the proof of Theorem 2.2.3. We prove the first fornmla, and the proof of the second one is similar. For simplicity, we prove the case s = 0 and t = O. Extending the argument to the general case is raightfbrward. Let Co be a connected curve of genus g with two irreducible components, C' and C" of genera g' and g" respecticrely, which are attached together at one point p. In other words  C p=p' =I/  °  ^  For non-rela is space of disconne quotients of di of disconnected Ma connected maps; howeve , ants to the connected invariants still w here the authors prove that the relat the non relative invariants 1261.  o ^e lementary tip oily be expressed in terms of product as. For the relative^ , the roednli si ple description in terms of e moduli spi in telating the disconneered ir is This can be move r us,rw the nsults of tt,rariants can always It  Chapter 2. Ou the G theory of plane bundles over curve  where p i C C and p" C C". Now we consider two i c bun dles X` mill(0 @LC /}2)^C',^X" = P(0 e e^  C",  where^t" and^are line bundles of degrees e, k(21 1g and kl; respectively. We attach these two spaces by identifying the fibers. P' and F" over TY and p", respectively, such that the resulting space is t0 = INC9M L 1 ED L,) Co, where L 1 and L2 are line bundles of degrees k 1 I and k9 =^+^respectively. In other words Wo = X'  L^„  where F is the fiber over p. Let W a Lt be a generic, 1-parameter deformation of Wo for which fibers Wt for t ^. 0 C A l are 740 Q3 L1 0- L9)^C, where C is a smooth curve of genus g, and 1. 1 and L2 are line bundles of degrees k l , k9. We follow Jun Li's proof of the degeneration formula in [21]. Let 211 be the stack of expanded degenerations of W. with central fiber 2/1 0 , and let 114(211, /3) be the stack of non-degenerate, pre-deformable, genus h, degree /3 maps to 911, where is a section class [20]. We have the evaluation maps ev :^F' = F  ,^  ev" :^(X'/F", /3") —  con structs a map  4- (X 1 tr, Oil ) xF^(X"/F”, 0")^.711(2/30, /3),  when rf in cludes a pair of classes (13', /3"), such that fi^)3' +Q" and a pair a^h"), such that h =^+ h". Then he gives a virtual cycle which in our case is pif (2)30 ),^4'7 A l M(X71'",^x rAjl(X"/F", 0") (2.6.1) where A :^>12xFis the diagonal map. Remark 2.6.2. The torus action on the family W m A l gives an action on the stack of expanded degeneration, TB. One can check that pre-deMrmability condition is a ariant under this action, and so it induces (canonically) an action on each VW, In, Al m ( X"/F",,{3") and i ,(211,13). Therefore. Li's formula holds^ groups.  ^  Chapter 2. On the Cif theory of plane bundles over curves  If we work with the basis elements xo, XI z 2 (introduced in Section 2.2). for the equivar lain'. Chow group of the fiber F. by using (2.2.2). one can see easily that ,  X6 - x^T 1  =  XIX X2 ^2V^ =I ^  T 0) .^'.1(.ti)'^TO:2)  is its dual basis, and so the cohomology class of the diagonal of F x F is given by [25. Theorem 11.111 2  im(A) = ^X0  Exx i  Xi^Xi X0 ^ + E X t IX) X ^  X  T(X0)^rT (3: )^(17(x2)  Using this, we can rewrite (2.6.1) as [M(211 0 . /3)]  7:4 ^=  Th  (ev')' ern n fyr (X'/];".  s')  "Zr  71  [ev")*(2:,)\ x -^nHI (x"/F", /3")  trGro  We now have Z4(g k 1 , A:9)=^1 ipi(213^if 02110,Th% ■' 7 ,  =  .  (ev")*Cla)  (ev') (xi) i=0 ' P I CA-F M q31 )i''”  DTA- f0"  ^ = E i=0 (g' WI , 02 ) x 4, (g"iq, 1  Then we can write  ,3"t i2'  Chapter 2. On the CI theory ofhlane bundles over curve  Z(9 I^k2) =  tt 2h2  2-Ay 3  73  (q k  t.  k2)  E E^ E E ,1 ,0' IIke kf2 h^ E u - -,3z3j; 0/^, k 2)„, t  (9" I k7. h iP`'  1  )—^  ;3^  // i=0  2h _2 ,  /Cy r  ^2  1  3, h,  G(9 I ki^s Z (.9"^k'zy i. m in  2.7 Some special cases In this section, we provide a few more applications of Theorem 2,1.6. The proofs of the first three theorems are straightfOrward. One first expands the formula in Theorem 2.1.6 in each case to get a polynomial in to, t t and 12. The partition function that appeared in the statement of each theorem then corresponds to the terms in this polynomial of a specific degree, determined by the given curve class in that theorem (see Remark 2.2.1, and also the proofs of Theorem 2.1.7 and Theorem 2.7.1). Theorem 2.7.1. Assume th.at kr > 0 and A:2^There Cite degree )3o — ti l l, level^— k 2 equivariant Gromov-Witten partition function of X is given by 3„^(9 O^k2)^(ti - tor W-1 (II^t2)9  k1 h2 -1  (2 sin 1112  k > O. Then the deg ree^v (k, k) equivariant Grotnov^n partition function of X is giv en  Theorem 2.7.2. k f(9  Assume that  k, k) =^- 1 0 )9  k-  (I ^t2y1 )  f - (12 — kO) 9 k^(k2^1)11  Theorem 2.7.3. Ass rune that k i > 0 and A: 9 > 0 -  .)  ^ ^  ^2  Chapter 2. On the OW theory of plane bundles over curves ariailt Gromov-Witten partition function of X is gi Z;olg — k 1 , - -k2) !  sin  )(  ,^> 0, k2 > 0, 2 2  ^k1 > 0, k2=  )k2 -r) ( 1, 1 — 10 9 1 (ti — in ) 11 ' K2- 1 ) (2 sin -uk2 2 (t0 — /I)g I (to — t2) 9-1 + 1 1 1 - ( M g  k1 = 0, k2 > 0,  (ti - t2)g -1 ± (t2 - 10) 9°1 (t2 ti)g -1^k 1 = 0, k2 = 0.  2.7.4. Let a be the greatest integer that satisfies 3n < 2g - 2. Then the degree P a + a, f,, level (0, 0) eau; variant Groin on- Witden partition function of X is give n b y Theorem  { 0^ Z ik f (g 10,0)  g 3k,  3 11 (2 sin 1 ) 2 3112( g -^+ JOI  = 3k d I,  u 2f —^— 12) 2 sin —  21^  g = 3k 2.  Proof. The case g = 0 again follows from the semi-simplicity of the level  (0,0) TUT (Corollary 3.2.5, see also Remark 2.2.5), once one knows the result for g > 0. So we will assume that g > 0. Applying Theorem 2.1.6 to this case, we can write Z (g ! 0, 0)^tr ((.-1 1- 13) 9- I ),  where 4 and B were defined in Section 2.5. We prove tch case in theorem separately: (i) g = 3k. In this case, n = 2k — I, and one can see that (by using the notation introduced in Section 2.5): Z3„ 4 (2k_..])f (g 10,0) =^((A k B 2k-1 )). We Ilave :113  :19  Chapter 2. On the OW theory of plane bundles ove r  so for any positive integer^  we have  (AB 2 )"^311-3^132.^(2.7.1) One can prove easily that /3 3 = 0.^(ABA/32)2 = 0.^(2.7.2) Applying (2.5.3) a few times to each term of tr(($k. B2k^and using the^ ualities above, we can prove that each of these terms is or is equal to 10 AB (21/3 2 ) k^33k- 6  ^  2 tr(ABAB 2 ).  However, an easy calculation shows that tr(ABAB 2 ) = 0. This prov that each term of tr ((il k , B 2ik )) is zero. So, 23 0 (21c-^(g 10,0)^0.  (ii)  In this case /3p of is the Calabi-Vau sectio n :,l the theorem fox iris a e in more generality in Sectio n  (iii) g = 3k + 2. In this case a= 2k and this time we have .  Zg,42,1.•f (k1 1 0 , 0 ) = 13-3(ii k I , -82k ))•  The cases k = 0, 1 can be proved by easy calculations, and so we assume that k > 1. Applying (2.5 .3) a few times to each term of this, and using (2.7.2), we can prove that each term of tr(oao, /32kk )1) ) is either zero or is equal to either of tr( A (A /13 2 ) k ),^tal(A8 ) 2 (A13 2 ) t^),^0.(A2132( 5 2 ) k ) 1 ), where the number of the terms of the first. the second and t are 2k -h 1, 3k -- I and k. respectively. So we can u ite  ^  rd kinds  ^1- 1) 1.421(2113 2 ) 1(' ) F (3k -t-1)^(()4B) 2^2)k-1) (g 1 0, 0) = 8 2 (AB 2 ) A-1 )^ ^2 (2.7.3) 713 0 -2k;  On the OW theory of plane bundles over curves Using (2^vt (AB 2 ) 2 ) = 332- 3062 43 tr(21(AB 2 ))  = 3 32-^+ lZ + tZ - tott - tel2^ 12) = 3 32- '0" k- 12 tr(OBY2 (AB 2 )) -23 • 3^ 3  .6= -,--^tot,^tote --ti )061(,,  tr (A .^B2)k-1 = 33k -6  /4  2B2 (A BM  = 33k --  \\e de we get  told - tot2^(2)06k•  - lit 2 . Putting all these into (2.7.3), Z3(g^)^3 3k-1 (2k + 1 2c 2(3k -4 1) d k)Q55 6k  = 3 32 (3k ± 1 ) (2 02 3 g 2(2 -  and this proves ,he theorem in this case.  1)(202g-4 ,  Bibliography -f curves. To  andh aripande The local Gromov^ theor y in J. Amer. Math. Soc. arXimmath G /0111037.  ;Jim Bryan and Rahul Pandharipande. Curves in Calabi- \Tau 3-folds and Topological Quantum Field Theor, Duke Mathematical Journal, 126(2):369 396, 2005. Preprint version: 17 LAG/0306316. 'L I 6; Amin Cholampour and Yinan Song. Evidence for the GromovWitten/Donaldson-Thomas correspondence. Math. Res. Lett., 13(4):623-630, 2006. [17] Tom Graber. Personal co^.ion [18] Tom. Craber and Ravi Vakil. Relative virtual localization ^ vanishing of tautological classes on moduli spaces of curves. Duke Math. J., 130(1):1 37, 2005. [191 Daniel Huybrechts and Manfred Lam. The geometry of moduli of sheaves. Friedr. Vieweg & Sohn, Braunschweig, 1997. [20] Jun Li.^Stable morphisms to singular schemes acid relative stable morphisms.^J. Differential Geom., 57(3):.509-578, 2001. arXiv:math.AG/0009097. [21] :Jun Li. A degeneration formula of OW-invariants. J. Differential Geom., 60(2):199 293, 2002. [22] D Manlik, N. Nekrasov, A. Okounkov, and R. Pandharipande. Gromov Witten theory and Donaldson-Thomas theory. ^ CoMpos. 142(5):1263-1285, 2006. [23] D. Manlik, N. Nekrasov, A. Okounko , and R. Gromov-Witten theory and Donaldson-Thomas theory. IL Compos. Math., 112(5):12861304, 2006.  I3iLliography  [241 D. Malink and R. Pandharipau le. A topological view 1 0 ^lovW tten theory.^ology, 45(5):887 918, 2006. .  [25] John^ ys D. Stasheff. Clutmei n su e classes . Princeton University Press , Princeton. N. J., 1974. Annals of 1 Studies. No. 76. [26] Rahn' Pandliaripande. Personal ^ 11  Chapter 3  Root Systems and the Quantum Cohomology of ADE resolutions  3.1 Introduction' 3.1.1  Overview.  Let C be a finite subgroup of SUL2), and let Y C2 /G be the mini^solution of the corresponding DuVal singularity. The classical '.McKay correspondence describes the geometry of V in terms of the representation theory of G [36, 39, 4[J]. The geometry- of Y gives rise to a Dynkin diagram of ADE type. The nodes of the diagram correspond to the irreducible components of the exceptional divisor of Y. Two nodes have a connecting edge if and only if the corresponding curves intersect.  Remark 3.1.1. This Dynkin diagram also arises from the re vs( ory of G. The nodes of the diagram correspond to non-trivial irreducib representations of G. Two nodes corresponding to representations p and p' have a connecting edge if and only if p is a summand of V ® p'. Associated to every Dynkin diagram of ADE type is a simply laced root system. In this paper, we describe the C-equivariant quantum cohomology of V in terms of the associated root system. This provides a quantum version of the classical McKay correspondence. A version of this chapter 11 as been submit  Chapter 3. Quantum Cohonlalogy of ADE resolutions 3.1.2 Results.  The set, {E1, ... ,Z;,,}^ ,( u b^)^:s y forms a basis of H 9 (Y, Z). The intersect^iij defin e 0 on If9(1 ,Z). Let R be the simply laced root system associa ie Dynkin diagram of Y. We can identify I Y, Z) with the root of R in a way so that E l , ... , En correspond to simple roots a l , ... , a^( the intersection matrix is minus the Cattail matrix Ei • Ej = —(n i (r . i. ) lasing the above pairing, we identify if 2 (YE) with ff)(Y,Z) (and hence with the root lattice). Since the scalar action of Cl on Cr commutes with the action of G, C* acts on C 2 /G and this action lifts to an action on Y. The cycles E 1 ,...,E„ are Cl invariant, and so the classes on ,a„ have natural lifts to equi variant (co)homology. Additively, the equivariant quantum cohomology ring is thus a free module generated by the classes {1. a l , The ground ring is if^, g„]] where I is the equivariant parameter and ,q„ are the quantum parameters associated to the curves E l , So additively we have QIC,(1) )^(Y, Z)^, q„l]. We extend the^ ( , ) to a Q[I, r 1 11,1q 1 ^ jj valued pairing on (211,l.,, ( Y) g I orth ogonal to ni and setting --1 t 2 IGj  (1. 1)  .  The product st ructure^determined by Theorem 3^C^(Y,17„) 'which we ider  y with the root lattice  of I? as above. Then  g iven by the formula:  ^  v*w=  2  K11  duct of v and w  ^ ci  where the sum i8 over the pom defi ned  by  of  0) Of  1 + (f3 3  1 --  3 /'  ^  CI I^3. Quanta I cohomology of sill)K cosolufions The quantum product satisfies the fq'obenias condition v xw , u)^Kg , * v.)  making QII,* 01 a Frobenins algebra over Qft t 11[[(11-, ,  ^Note that by a standard^root theory [29, VI.1.1 Prfrposit o^and V.6.2 Corollary to Theorem ^form ula in Theorem 3.1.2 can anemiatively be written as vow =  E (v, iT)(w> 13 )  where tc = A is the Conger number of R. We remark that we can regard 11 0 0) ce H 2 Y ) as the root latti ce affine root system and consequently, we can regard QH„;`, (Y) as defining a family of algebra structures on the affine root lattice depending on variables ,q„. We also remark that even though the product in Theorem 3.1.2 is expressed purely in terms of the root system, we know of no root theoretic proof of associativity, even in the "classical" limit q; a 0. In section 3.4, which can be read independently from the rest of this paper, we will generalize our family of algebras to root systems which are not simply-laced (Theorem 3.1.3). We will prove associativity of the product hi the non-simply laced case by reducing it to the simply laced case. Our fornmla also allows us to prove that the action of the Wry' group induces automorphisms of the Frobenius algebra (Corollary 3.4.4). Our theorem is formulated as computing small quantum cohomology, but since the cohomology of Y is concentrated in degree 0 and degree 2, the large and small quantum cohomology rings contain equivalent information. The proof of Theorem 3.1.2 requires the computations of genus 0 equivariant Gromov-Witten invariants of Y. This is done in section 3.2. In section 3.5, we use the Crepant Resolution Conjecture [31j and our computation of the Gromov-Witten invariants of Y, to obtain a prediction for the orbifold Gromov-Witten potential of [6 2 /61 (Conjecture 3.5.1). (  3.1.3 Relationship to other work  A certain dization of the Ifrobenius algebra QM,. (11 appears as the quantum cohomology of the C-IIilbert scheme resolution of (2 3 /G for GI C 80(4 (see [301). The equi arrant. Gromov-Witten theory of 1 7 in higher genus has been determined by recent work of Maulik [381.  Chapter 3. Quantum Cohomolot3y of 9DE resolutions  3.2 Gromov-Witten theory of Y  .  section We compute the equivariant genus ^ Gromov-Witten in^^u its of non-zero degree are computed by relating them to the invariants ^certain threefold IV constructed as the total space of a family of deformations of Y. The invariants of W are computed by the method of Bryan, Katz, and Leung [341. The degree zero invari are computed by localization.  3.2.1 Invariants of non-zero degree. Def(Y), the versa! space of Cnequivariant deformations of Y is naturally identified with the complexified root space of the root system 1? [37]. A generic deformation of Y is an aline variety and consequently has no compact curves. The hyperplane D,3 C Def(Y) perpendicular to a positive root  reter izes (hose deformationsY for which the cur b hi ----- IDEri [  ]  also deforms. Moreover, for a generic point t E Dg , the corresponding curve is a smooth P I which generates the Picard group of the corresponding surface ([3 ,1,, Prop. 2.21 and [37, Thm. 1]). Let. C Def(Y) be a generic linear subspace. We obtain a threefold W by pulling back the universal family over Def(Y) by z. The embedding ? can be ma de C"by defining^ tion on C to have weight 2. This follows from U after^ action in l37I is the square of the iced by the action on C2 /C? Clearly Y C IV and the degree of the normal bundle is c1 (ATI^= 2t The threefold W is Calabi^ and its Gromov-Witten invariants are well defined in the non-equivar^This assertion follows from the ace of stable maps o W is compact. This in turn from the fact that lt" adn.its a, birati.^contracting 50  ^  AVE r rsoluti if ADP • UP„ such that^is is an aline variety (see [31, 37]). Consequently all non-constant stable maps to IF must have image contained ceptional set of IF^Hia tt and thus, in particular, all non-constant stable maps to IV have their image contained in Y. There is a standard technique in Gromov-Witten theory ^ comparing the virtual class for stable maps to a submanifold to the virtual class for the stable maps to the ambient manifold when all the maps have image contained in the submanifold [27]. This allows us to compare the GromovWitten invariants of IV and F. For any non-zero class A e 8 2 (Y) cH>(1V) let )  denote the genus zero. legree A, zero insertion Gromov-Wittenin and W respectively. We have  (^= 110  .1 0  ^;(—leir,..f*N y/o ,)  where 0 , 0^A) is the mod uli^ ,e^C^A10,0(Y, A) is the universal curve, f :C^Y is the universal^the Euler class. Since the line bundle Ny711z is trivial up to the C action, and famil y of genus zero curves, we get 17. ° 71- * PNy/ T. F R (1 7r,f'Nyftv^0 0 02/ where C.) t is the^representation of weight 2 so that we have el ( 0 0 C 2t) = -  2t  .  Consequently, we have ./VY/w) and so  2t  Chapter 3. Quantum Cohomology of ADE resolutions  To compute O we use the deformation^ Gt01110V-Witten invariants. Although W is non-compact, the moduli space of stable maps is compact. and the deformation of W is done so that the stable map moduli spaces are compact throughout the deformation. The technique to the deformation argument used in [341 where it is presented in greater We  deform W to^follows. Let : C Def(11  be a generic affine linear embedding and let I, be the pullback by i of the universal family over Def(Y). The threefold W' is a deformation of W since is a deformation of Lemma 3.2.1. The compact curves of W' consist of isolated P 1 s, each having normal bundle 0(-1) ci 0(-1), one in each homology class /f e 111(11") 119( 7 ) corresponding to a positive root.  PROOF: The map z' intersects each hyperplane Do transversely in a point t. The surface S t over the point t contains a single curve C,f t of normal ban dle Not/s 0(-2) and this curve is in the class 1.1. There is a short exact sequence --> Ned si /Vo w 0 -4  ()  and since i intersects Dg transversely, C does not have any deformations (even infinitesimally) inside tr. Consequently, we must have AT)))3 /^)2-= 0( ^ 1) (D 0( ^ 1).^ ^ Since all the curves in W' are isolated (-1, —I) curves, we can compute the Cromov-Witten invariants of Hy using the Aspinwall-Morrison multiple cover formula. Combined with the deformation invariance of GrornovWitten invariants, we obtain: Lemma 3.2.2. For nt ^ 0 we have  O  2t^2t O,14  2^if A^dfi where (3 is a positivie positive root 0^otherwise.  Since^ co b onrology of Y is in H ° (Y) and 11 2^n-point Cromov - Wit ter inva tarts of nor. zero de gree are determi 'ntal class an  Chapter 3. Quantum Cohothology of ADE resolutions  3.2.2 Degree 0 invariants. The only non-trivial degree zero rnvarianls have 3 insertions and are determined by classical integrals on Y. They are given in the following lemma. Lemma 3.2,3. Let I he the generator of 'E. (Y) and let fa h . a n } be the basis for 11 n. 'i',,(Y) which is also identified with the simple roots of R as in sec tion 3.1. Then. the degree 0, 3-point GI071101)- Witten invariants of Y are given as follows:  (1, 1  1  (3.2.1)  ()^t 2 ICI  (ai, 1, 1) 0 = 0, aj , 1) 0 = (oh a j )  (3.2.2) (3.2.3) (3.2.4)  PROOF: The degree^point Gram( Witten invariants are given by integrals o (f:,(11 ,^=  ^X U ,t) U Z.  Because Y is non-compax t, the integral must be defirted b via^ i ttion and takes values in Oft,^the localized equivariant cohornolog . ring of a point:  Uff t '1 -  •  fie  e(Np/y)'  Here F C Y is the (compact) fixed point locus of the action of C on Y. By correspondence of residues [28], integrals over Y can be computed by first pushing forward to €12 1G followed by (orbifold) localization on VIC. Equation (3.2.1) follows immediately: ^e  1= /^1 ^ .  wic^titIG [  We remark that this en pact space / tie tt: will holds and See [31, se  Gronaw-54 ies of (i 11 exists with the iovelty  Chapter 3. Quantum Cohothology of :)E resat  The factor 2 is the^er class of the normal bundle of [0/G C [C2 /C] and t he ITyr ac c ounts for the automorphisms of the point [0/GT Let L1 -4 Y be t eC equivariaut line bundle with n ( 10  = Oi  k  Since^was defined to be dual to Ei via the interne tionTiring we have  Computing the left hand side using localizatior ^ he weight^of the C; action on Li at a fixed point p C Ez must be the sail he - ght of the C action on the normal bundle Ar E ay at p, and the weight of the action on L i is 0 over fixed points not on Equation (3.2.3) and Equation (3.2.2) then easily follow from localization. To prove Equation (3.2.4), we compute le left hand side by localization to get U E Ek = 0. „h: ) 0 = 8E,^if i^= k, if 1$ j = k and Ei U^0 where ei(Np /Y [pe:) .  is the weight of the C* action on the normal bundle of Ej at the point =E U Ej . The norma weiglu s w z t1 satisfy^ following^ mditions: 1. Since By is the trivial bundle with a^action of weight 21 th e sum  of the normal weights at p E, n E j is 21, and so m 21 when Ei Ej  ^  0 and^j.  2. Since Ei is C invariant, the smn of the tang distinct fixed points on E, is zero. Combined that the sum of the normal weights t any two 41 so 4/ when E. n Ek . 0. Ei ^  54  Chapter 3. Quantum Cohomology of ADP; resolutions  3. Since automorphisms of the Dynkin diagrams induce equivariant antomorphisms of Y, the normal weights are int under such automorphisms. The normal weights are completely determined by the above three conditions.^ it is clear ghat once one normal weight is known, then Pt(^(2) determine he rest. Moreover, in the case of Dynkin of type^ or E. the curve corresponding to the trivalent vertex thin graph mthst be fixed and so its tangent weights are zero. In the^condition (3) provides the needed extra equation. To summarize the above, the three point degree zero invariants (a„ ak) satisfY the following conditions and are completely determined by them. (i) (ou,a3,00 0 is symmetric in {i, j, k}, (ii) (iii) (iv)  at, ak.) 0 is invariant under any permutation^ a Dynkin diagram automorphism,  'dices induced by  ai,ok)0 = 0 if (,t,), (..uk) = 0, (liVi ,^, aky,^—81, if i^k.  (v) (ai,ai,o5) 0 +^21 if (ai, ay) = —I,  (vi) (nu, ak, ak) c ,^(a 3 ,ak,ak) 0^4t, if i^j and (ai, ak)^(a3, EEO = So to finish the proof of Lenuna 3.2.3, it suffices to show that the right hand side of equation (3.2.4) also satisfies all the above properties. This is precisely the content of Proposition 3.4.7, a root theoretic result which we prove in section 3.4. ^  3.3 Proof of the main theorem Having computed all the Gromov-Witten invarian to compute the quantum product and prove our main theorem. The quantum product * is defined in terms of the genus 0, 3-point invariants of Y by the formula: — (x * y, = where the st range looking minas sign is winch coincides with the Caftan pairin col tomological pai ring  icing (, t of the  0  Chapter 3. Quantum Cohornal  ^  [  ADE solutions  To prove our ^it suffices to check that the 11 ^ula holds both sides^ h 1 and with any r C 1 2 ( By definition and Lemma 3.2.3 we have  = (o, w,1) () = ( >OP) which is in z^with the right hand side of the fornmla in Theo^ rem 3.1.2 who ired with 1 since 1 is orthogonal to 11 2 (Y) and (1,  ^(2K-,t For a C 112( 7 ) we apply the divisor^  o get  — (v to, 32)^E^(1 3.13, Iran (  v, w,a) 0  ^ —  (v, A) (r, A) (n, A) (^.  Applying Lemma 3. 2. 3 and Lemma 3.2.2 w (33, 13) (w, 13) (a„):3) —  = —t  31, di]) (w. 18) 01,3() 21  E („,(3)0 0 /3 ) E^ 3) (u, 3) )  )  (V  (al  33  ,CERA  Pairing^the right hand side of the formula in Theorem 3.1 9 agreement with the above and the formula for * is proved. To prove that the Frobenius condition holds. we only need ^ e pairing on (21123 (Y) is induced by the three pun f 1: —^  w e find  ob,  , it) = 1,33. 3. I \3 0  This indeed follows from equation. (3.2.1). (3.2.2). and (3.2.3). 56  Chapter 3. Quantum Cohomology of ADE resolutions  3.4 The algebra for arbitrary root systems In this^ xmstruct a hrobenius algebra QTIR associated ,o any irreducible, reduced root , ^It (Theorem 3.4.3). This section can read independently from the rest of the paper. 3.4.1 Root system notation  In this section we let I? be an irreducible, reduced, rank  7  R= {R,  consists of a finite subset 1? of a real inner product space V of dimension satisfying L I? spans V , 2. if a C R then kit e I? implies = +1, 3. for all a C R,, the reflection s o, about a' , the hyperplane perpendicula to a leaves I? invariant, -  C if, the num 5. V is irreducible as a representation of IV, the Weyl group (i.e. the group generated by the reflections 8 c„ a C R). We will also assume that the inner product ( ) takes values in Z on R. Let^ ,a„} be a system of simple roots, namely a subset of R spanning V and such that for every /1^itt I? the coefficients bi are either all non-negative or all non-positive. As is customary, we define 2a (a, a) We will also require a certain constant c i? which epends on the root system and scales linearly with the inner product. Definition 3.4.1. Let n i be the Ali coefficient of the^  =  a) .  gf,root  Ghat^urn Cohomok^)13 resolutions Note that in the case where R is as in ection 3.1, namely of ADE type and the roots have norm square 2, then ck = 1+ 1 and we have that  ETL n2  R^I Cl  ^(  G^ subgroup f SU(2). This is a consequence of the Mc s correspondence , part^•^, ^ are the dimensions of the irreducible representationsof G (see [36. page Lemma 3.4.2. For any irreducible root^)) the following identity holds:  E RER, where^  1!!)  w,  `)  ^ the Cotecter number.  Pao0F: The linear map v^  5 (s,v)s"  ;tot,  t e s with the action of TV (this is easily seen using [29, VI.1.6 Corollary I]). Since V is an irreducib l e 7T 7 iepresenta.tion , S(1 'tint .)t such that  (13^1.1V  (3.4.1)  CI&  Taking traces, we o btai u,k^((hi) eh where c(),3 (v)^(0,^Choosing^ orthonormal basis e t ^ e,, of V such that. f = [Me,, we find that the only non-zero diagonal entry of ct),3 is the first one which is 2. Therefore IR) A = = TI,  and the L ens f bilows fromequation (3.4.1) with  Chapter 3. Quantum Cohomology of ADE resolutions  3.4.2 The algebra QM  ?  Let H R = 22:; ip Zn1 ED • J ZIT?  be the affine root latti ce and let QI/A be the free module ore generated by 1, am • . a n  [MI •  ,  =H  gal a„)] valued pairing on CI  extend the p  aking 1 o  setting  ( 1 , 1 ) =(-^  t' en  •  For =^biai, we use the notat li  =  Theorem 3.4.3. Define a product operation on QHR by letting be the  identity a.rtd defirChty  1+ q/3 1—  (cr i , f3) Kai,,(3)/ ^  a) t !  3c it+  Then the product is associative, and moreover, it satisfies the Frobenius condition y ,^= (x, y * z) ^ making QI^into a Frobenius algebra over the ri (ul t ,^' 11 1/2 1 • • • • , (ha Corollary 3.4.4. The liVeyt group acts on Q^(and thus  ^))  by automorphisms. Namely, if we define  fl(Ii (1 ) then for v,/tv C QHR we have = (ye)  59  Chapter 3. Quantum Cohomology of ADE resoluti ons  PROOF: Let Sk be he reflection about the hyperplane orthogonal to By [29. V1.1.6 Corollary j, sk permutes the positive roots other than ak. And since the terms 4 (/ 3 ,)  Kai, /3) (^0 .1 )  1 — 1/  remain unchanged raider F-- 13, the effect ofapplying^ sum: for ai * a,/ is to permute the order of the  ^(PS Ka fin t ^ 1 — tip  ^1  = — OCR^a1) ^/3)  —PeR(skai,s3a;) 8 k( 0  to^for h  )  (a  0* Sk(Oi  and the Corollary follows.^  ^  3.4.3 The proof of Theorem 3.4.3  When I? is of ADE type and the pairing is normalized so that the roots have a norm square of 2, then QHR coincides with Q/1„?.., (Y) and so Theorem 3.4.3 for this case then follows from Theorem 3.1.2. For any R, the Frobenius condition follows immediately from the forrnm las fOr * and ( So what needs to be established in general is the associativity of the * product. This is equivalent to the expression 1  — t2 ((t * y)* u, v) being ful^n ,r, y, u,^Writ,ten o ut, we have —  CR (2:,  (3, 3 E  11) (3,39) (u, 7)  I -I- q 3^( ^  e  \  1—q"  J  )airing, we see that  60  Chapter 3. Quantum Cohothology of ADE resolutions  prove the associativity of QIIR for root systems not of ADE type, we reduce the non-simply laced case to the simply laced case. Let {R, V, , )} be an ADE root system and let (II) be a group of automorphisms of the Dynkin diagram. We construct a new root, system {/?4, ( ) 4,} as follows. A somewhat similar construction can be found in fill, Section 10.3.1]. Let V`u CV ^ be the 1 invariant subspace equipped with (, ) 4„ the restriction o ) to 1[1) , and let the roots of R0 be the 4> averages of the roots of R:  1  114) ---z-- Zi = g€  Then t is easily checked that^4', , specifically of type given in Table 3  (1)  }  is an irreducible root system,  A.2. 7] -1^L 2 D„+1 Z2 B„ E6^Z9 1. D4^Z3 G2  Table 3.1: Reducing r on-simp y laced case to simply laced case  Thus all the irreducible, reduced root systems arise in this way. will^ fact that if y C y e , then (x,Y)^(f]Y)^ (3.4.2)  which easily follows from (x, y)^(gx,gy)^(gx,y) for g C <D. We will also have need of the following two lemmas which we will p rove at the end of the section. Lemma 3.4.5. The constants defined in Defin ition  Tool systems R and lit:  Chapter 3. Quantum Cohomology of ADE resol uti Lemma 3.4.6. Let 13 6 R f and let 4)/i lie the Cl) orbit of /1. Then  = E  The simple roots of R4, are given by rei, the averages of the simple oots of R. Thus if —  is the index set for the simple roots of R, then 4 acts on I and  1I4) is the natural dex set for the simple roots of R4), For^  E J we let  Tip} E R,j, denote the simple root given by at.  We specialize the variables {T}, e r to variables {ij iii }[ i i ci by setting (li =  ^  (3.4.3)  and it is straightforward to see that under the above specialization, fi =^.  Now let 1? be an ADE root sys era whose roots have norm square 2. Then 2^is fully symmetric in x, y, v, n }. We specialize the q variables to the 4 variables as in equation (3.4.3) and we assume that :1,3,1),11 c  Then A.^- r^(x,y) (thy) =  E (x,i(3)^/3 ) ( 11 ,7)^)  = E (x),Th (mg) = E (r,g)  (y , 14 ) (1, 7)^7)  3 15'0  E 0, 7 c4  (x 1 71)„, (Y. 7-3 )  is it, l [ ± eR 62  ^  Chapter 3. Quantum Cohomology of ADD resolutions and thins It 8 8 nut  ,  is fully symmetric in {x, y, a, v} and the theorem is proved once we establis Lem ma 3.4.5 and Lemma 3.4.6.  3.4.4 Proofs of Lemma 3.4.5 and Lemma 3.4.6. We prove Lemma 3.4.6 first. If is fixed by F, the lemma is ima mate. We claim that if /3 is not fixed then ((3, g/3) = 0 for nontrivial g E For simple roots, this follows from inspection of the Dynkin diagrams and automorphisms which occur in the table: a node is never adjacent to a node in its orbit. For other roots this can also be seen from a. direct inspection of the positive roots (listed, for example, in [29, Plates LIV-VIP). For /3 not fixed by 'F we then have:  E^/to)  (,(3,o)  s,  _^ t(1) 12  V  (  (JO g  0  2  1(1, 1 and Lemma 3.4.6 follows. Note that the above formula generalizes to all roots 3 by  ( -73. , 73)^2 stab(i3) 1 4) 1 where stab(0) is the order of the stabilizer of the action of (I) on /3. TO prove Lemma 3.4.5 we must find the coefficients of the longest roo t R. Since the longest root of R is unique, it is fixed by (D . and so it coincides with the lo ngest root of R4:  _=E  IVY —  E E  Cti  id/^ME,/^i'ECi  ^=  27,,„  70H z E^ =^ trY[i],e Cli i  CY  63  Chap ter 3. Quantum Cohomology of ADE resolutions IS  tab(ad 474  HD I^(CI" di)  et  (d, &) +  2ni2i  20n and Lemma 3.4.5 is proved.^  ^  3.4.5 The root theoretic formula for triple intersections. Here we prove the root theoretic result required to finish the proof of equation (3.2.4). Recall that I? is a root system of ADE type normalized so that the roots have norm square 2. We write  Proposition 3.4.7. Gel  G ijk = —  ^  (a  -  [3 ) J. 8)^8)  SE  then Gijk satisfies the following properties. (i) Gijk  is symmetric in^:Lk},  Gijk is invariant under any permutation of ind icestitdneed by a Dynk•n  diagram automorphism,  (is)  Gijk = 0  if gjk = 0,  Gijk =  8 if i =j  (v) Giij (%)) Gikk  —  =  E- 2 if = — kk =  4 :Ur  6. 1  Chapter 3. Quantum Cohomology of ADE resolutions  PROOF: From the definition of 135, properties (i) and (ii) are clear satisfied. Let sk be reflection about the hyperplane p er pendicularto ak so that skai  Since s k permutes the positive roots other than ni c [29, VI. I.6 Corollary 1], we get the following xpression for Gijk: -2gikg^  E  (ai,  so)  (ai ,  ;300 = —4 9ikgjk^  so)^  ski3  )  E^- gik^;(3)(cri — g^k, (3 ) ( - ak, 13 )  :3ERG =  —  4thkflik Giik gikG jkk g35G155 ilikgjkG kkk  and so Glik = —2 gikgik + — 2 (giker jkk gikGikk gikgjkGkkk) ,  (3.4.4)  Settingi=j=k= rt we obtain property (iv): Grunt nt —8  which we can substitute back into equation (3.4.4) = a to get Cans = 2 9 24 + gukGakk  d then specialize  (3.4.5)  Property (iii) then follows from equation (3.4.4) and equation (3.1.5) and property (v) follows from equation (3.4.5). For property (vi), observe that if gik = gjk^---1 then 9q = 0 and so Gus = 0 and equation (3.4.4) then simplifies to prove property (vi). ^^  3.5 Predictions for the orbifold invariants via the Crepant Resolution Conjecture her C c SU(2) be a finite subgroup and let X^ff(Y/G1 be the orbifold quotient of C 2 by G. Recall that :^X  is the minimal re solution X.  Chapter 3. Quantum Colwyn°logy of ADE resoiutioi The Crepant Resolution Conjecture [31] asserts t.^Fy, the genus z ero Gromov-Witten potential of Y, coincides with Fix, the genus zero orbifold Gromov-Witten potential of X after specializing the quantum parameters of Y to certain roots of unity and making a linear change of variables in the cohomological parameters. Using the Gromov-Witten computations of section 3.2, we obtain a formula for Fy. By making an educated guess for the change of variables and Dots of unity, and then applying the conjecture, we obtain a prediction for the orbifold Gromov-Witten potential of X (Conjecture 3.5.1). This prediction has been verified in the cases where G is Z9, Z3, Z4 in [31-33] respectively, and recently it has been verified for all Z by Coates, Corti, Iritani, and Tseng [35].  15.1 The statement of the conjecture The variables of the potential function Fy are the quantum parameters {^q.„} and cohomological parameters {ye, ^, y n } correspondin g the generators {1,ca y ,^ a n } for^(Y). The potential function is the natural generating function for the genus 0 Gromov- Witten invariants of Y. It is defined by  Y  =  The potential function for the orbifold X = [C 2 /G] depends on variables {3:0 • which correspond to a basis {1, , } of H0* }b (X), the orbifold cohomology of X. The orbifold cohomology of [C 2 /0] has a natural basis which is indexed by conjugacy classes of G. If g E G is an element of the group, we will write x,[ 9 ] for the variable corresponding to the conjugacy class of g. There are no curve classes in X and hence no quantum parameters so the potential function is given by  "n = ^ )  The conjecture states that there exists roots of lytic continuation of Fy to the points 110  =  Chapter 3. Quantum Cohomology of'ADEresolutions  such that the equality FY (W I^ton, no ^ thi)= Fa/(ch... ,x„)  holds after making a (grading preserving) linear change of variables  Thus to obtain a prediction for the potential Fx, we in s roots of unity wi and the^ e of variables rnatrix 9 L. 3.5.2 The prediction.  The only m-trivial invariants involving I are degree zero three point invariants. We split up the potentials FX and Fy into terms involving t ) and yo respectively and terms without J.. 0 and yo respectively. Let fit, be the part of Fy with non-zero y o terms. It follows from Lemma 3.2.3 that 11. is given by 3  ho 01613! 1  Let FXo be the part of F,with non-zero xo terms. An easy localization computation shows that Ft. is given by 1 :r03^xo 1 ^ '^t,21(71 3! + 2 IG1 Since the change of^ "Thles respects the grading, the terms^ v ic are linear and cubic in y o must match up with^ is in FA, which are linear and cubic in xo. Consequently we must have = (110  and moreover, the change of variables must take the quadratic form (3.5.1)  67  Chapter 3. Quantum Cohomology of ADE resolution. to the quadratic fors  ^E  ^a1) Yi Yj •^(3.5.2)  j=1  We can rewrite^ bove quadratic form in terms of the representation theory of G using the classical McKay correspondence [39] as follows. The simple roots a l , , a, which correspond to nodes of the Dynkin diagram, also correspond to non-trivial irreducible representations of G, and hence to their characters Xi , x„. Under this correspondence, the Caftan paring can be expressed in terms of (1.), the natural pairing on the characters of  C:  -( aj) = ((x - 2) xi I 26) ^=  —^(g) - 2)x i (g)36 (g)  ICI 9EG  where V is the two dimensional :n; ^ induced by le embedding  C SL 42). This discussion leads to an obvious candid ate for the change of variable s. Namely, if we substitute .  [r =  1Xv — 2^Xi OA Yi  (3.5.3)  into equation (3.5.1) we obtain equation (3.5.2). Since xv(g) is always real and less than or equal to 2, we can fix the sign of the square root by making it a positive multiple of i. Thus we've seen that  un der the change of variables given by equation (3.5.3) and fro m here on out, we set = yo 0 and deal with just the part of the potentials Ex and Fyn not involving xo and y u . We apply the divisor axiom and the computations of section 3.2:  Chapter 3. Quantum Cohomology of ADE resolutions  E 72  6  )0 11011,1:14  (a/a  :  id,k=11  —t  , 13 ) (a j, 13 ) (air /3 ) ye  1,3,k=1 3e )  'faking triple deriva Ives we get 03Fy  (ai,^(a1, 1 ) ((kk, 13 )^1 OMOYi^ (n .j, )(ak )  2gfier-i  111  ^ 1 qfizza 309 -  + (13 e>:, i giscYli  We specialize the quantum parameters to ro  qj exp  27rini  (3.5.4)  where n.1 is the :Ali coefficient of the largest root as in Definition 3.4.1. Note that nj is also the dimension of the corresponding representation. After specializing the quantum parameters, the triple derivatives of the potential Fy can be expressed in terms of the function 1 7 1^e i( u - ' )\^1 -u) - tan — 2i^1^ €.4"^ 2 2  H () u as fo llows 013 Fy  0 11.11 0 1.1jOYk where for =  Er!=  - 2i  Uja, j  (t-i,,d)^d)^k, /3 ) JEW )  we deft  f)3= a  GJ  Chapter 3. Quantum Cohomolo y of ADP resolutions  It then follows that y ) 2t „  2  fie  h(Q5)  where h(n) is a series satisfying  111 (u) = — tan ^ 2 \. 2 ./ We can now make the change of variables given by equation (3.5.3).  E  It  (bk ( 0/klati}tit  9=t  as-^  .3_  Pi  ts.  (X49^2 )(k(ti)X;(9)Y/  bk  I EkEEG  ^2 — X49)  Yk(9):1:fg1'  Substituting this back into (2 /3 we arrive at our ;onject^formula for / Conjecture 3.5.1. Let Fa, (x , x„) denote the C equivariant  9672118 zero orbifold Gromov-Witten potential of the orbifold X = C 2 /G] where we have set the unit parameter xo equal to zero. Let R be the root system associated to C as in section 3.1. Then  ( ^ =2t^h(Q /3) ;3€  where h(u) is a series 1 hm^ (u) = tan  (—u 2  and 2arrik^V2— xi( ElkG  -where bk are the^ of t3 C root, anti V is the two dimensional representation ^ G C SU(2).  by the er  70  Chapter 3. Quantum Cohomology of FADE resolu tion Note that, the index set t 1 , ... , n) in OIL above formulacorre sponds to 1. simple roots of 2. non-trivial irreducibl e representati ns of G, and 3. non-trivial conjugacy classes^  G.  The index of a conjugacy class cgmtaining a group element g is denoted by [g]. Finally note that the terms of degree less than three are i11-defined both the potential Fr aud our conjectural formula for it. The above conjecture has been proved in the cases where G is Z2. Z3, Z4 in [31-33] respectively, and recently it has been verified for all Z.„ by Coates, Corti, Iritani, and Tseng [35]. We have also performed a number of checks of the conjecture for nonAbelian G. Many of the orbifold invariants must vanish by monodromy considerations, and our conjecture is consistent with this vanishing. One can geometrically derive a relationship between some of the orbifold invariants of [C2 /(7] and certain combinations of the orbifold invariants of [C 2 /1/ when II is a normal subgroup of G. This leads to a simple relationship between the corresponding potential functions which we have checked is consistent with our conjecture.  Bibliography [27] Kai Behrend and Barbara Pantechi. In preparation. [28] Aaron Bertram. Another way to enumerate rational curves with orlls actions. Invent. Math., 142(3):487'512, 2000. [29] N. Bourbaki. Element s de mathematique. Fare. XXXIV. Grouper et &gams de Lie. Chapitre IV: Grouper de Corder et systêmer de Tits. Chapitre V: Grouper engendre5 par des reflezions. Chapitre VI: systêmes de racines. Actualitds Scientifiques et Industrielles, No. 1337.  Hermann, Paris. 1968. [30] Jim Bryan and Amin Gtholin mur. The Quantum McKay corresp o ndence for polyhedral singularities. In preparation. [31] Jim Bryan and Tom Graben The crepant resolution conjecture. To appear in Algebraic Geometry^Seattle 2005 Proceedings, arXiv: math.AG/0610129. [32] Jim Bryan, Torn Graber, and Rahul l andharipa e. The orbifold quantum cohomology of C 2 /Z3 and Hurwitz Hodge integrals. arXivmmth.AG/0510335, to appear in Journal of Alg. Geom. [33] Jim Bryan and Yunfeng Jiang. The Crepant Resolution Conjecture for the orbifold C 2 / Z4. In preparation. [34] Jim Bryan, Sheldon Katz, and Naichung Conan . in g . Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds. ,L Algebraic Cre0M., 10(4519-568, 2001. Preprint version: math.AG/9911056. Tom Coates, Alessi() Corti, Ihroshi^ b and 1 m-Hua Tseng. The Crepant Resolution Conjecture for Type A Surface Singularities. arXiv:0704.2034v1 [math,AG].  72  Bibliography  1 361 G. Gonzalez-Sprinberg and J.-L. Verifier. Construction gOon ^que de la correspondance de McKay. Ann. Sci. Ecole Norm. Sup. (4), 16(3):409-449 (1984), 1983. .  [37) Sheldon Katz and David R.^Gorenstein threefold singularities with small ^via invariant theory for Wey1 groups. S. Algebraic Georg., 1(3):449--530, 1992. [38) Davesh Maulik. Gromov-Witt^o f^ esolutions. In^pc :a tion. .  1391 John McKay. Graphs, singularities, and finite groups. In The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif, 1979), volume 37 of Proc. Sympos. Pure Math., pages 183 - 186.  Amer. Math. Soc., Providence, RI., 1980. [40) Miles Reid. La correspondance de McKay. Astelrisque, (276):53-72, 2002. SOminaire Bourbaki, Vol. 1999/2000. T. A. Springer. Linear algebraic groups, volume 9 of Progress^ lath ernatics. Birkhauser Boston Inc., Boston. MA, second edition, 1998.  -  Chapter 4  The Quantum Mckay Correspondence for polyhedral singularities  4.1 Introduction 4.1,1 Overview The relationship between varieties with orbifold singularities and their resolutions has been studied by a lot of mathematicians and physicists. One of the outcomes in string theory is a principal which has different names and interpretations in various types of string theory, but it basically asserts the equivalence of string theories on an orbifold and its crepant resolutions. Accordingly, this principal has different interpretations in the mathematics side, which are all known by Crepant Resolution Conjecture among mathematicians. A crepant resolution of a Gorenstein singular variety X is a smooth variety Y and a hirational projective morphism  : Y^X  such that wy =  fr *  (wfv),  where wX and coy are dualizing sheaves of X and Y respectively. The case of the Calabi-Yau threefolds is particularly important in both mathematics and physics. The first step towards studying the Crepant Resolution Conjecture for the Calabi-Yau threefolds is to understand the local Calabi-Yan case X C3 /G where G is a finite subgroup of SU(3) acting on C 3 via its natural 3-dimensional representation. In this case, X has a preferred crepant olution, which is Nakamura's C.-Hilbert scheme, denoted by G-Ifilb(C) (see Section 4.2.1 for more details). of this chapter will be prepared for puhiicati01  Chapter 4. The Quantum Mckay Correspondence  An outcome thematics which is closely related to the Crepant Resolution Conjecture is McKay correspondence (see [651 for a survey). A version of iMcKay correspondence, relat es the orbifold cohomology of C 3 /0 to the cohomology of its crepant resolutions via the representation theory of G (see for example [584). Such a relation was first noticed by some string theorists (see j531). Another version of McKay correspondence, related to the B-model string theory, identifies the derived categories of coherent sheaves on C3 /0 and on G-Hilb(0) (see [44]). Recently, Bryan and Graber, using Gromov-Witten theory, have given an explicit formulation of the Crepant Resolution Conjecture which is related to the A-model string theory (see [46]). They conjecture an isomorphism between the orbifold quantum cohomology of a Gorenstein orbifold satisfying hard Lefschetz condition, and the quantum cohomology of its crepant resolutions (see Sections 4.1.3 and 4.3 for the details). This conjecture has been verified in some cases where the orbifold is a global quotient orbifold. The orbifold [C3 /G] satisfies the hard Lefschetz condition if and only if C 3 /0 has a semi-small crepant resolution, i.e. any fiber of 71 is at most one dimensional. It turns out that in this case, the hard Lefschetz condition is equivalent to C; being a finite subgroup of SU(2) or 80(3). We studied the former case in [45]. The finite subgroups of 80(3) arise as the symmetries of the Platonic solids. The singularities of C 3 /G are sometimes referred to as polyhedral singularities. In this paper we study the Gromov-Witten theory of Y = G-Hilb(C3 ), where G is a finite subgroup of 80(3). Besides motivations from the McKay correspondence and Crepant Resolution Conjecture, studying the GromovWitten theory of Y, which is a Calabi-Yau threefold, is interesting on its own: the virtual dimension of the moduli space of genus g stable maps to Y representing any curve class fi E BAY) is zero, and therefore the corresponding Gromov-Witten invariant, which we denote by A/(1 1 ), gives the virtual count of genus g curves in the class /3. In analogy with the classical McKay correspondence, we will associate a natural simply laced root system R to Y. The main result of this paper is the following: Quantum Mckay Correspondence. The Grumov-Witten theory of Y is  completely determimgl by the  7'0W SyStM711, R.  We will give a precise formulation of this statement the ne sect io n (Theorem 4.1.6) after providing preliminaries.  i5  Chapter 1. The Quantum Mckay Correspondence 4.1.2 Notation and Results Let, Al be a smooth quasi-projective Calabi-Yau threefold, c assume bl has a 0-action with compact fixed loci. Assume that 112 (M, Z) is torsion free of rank it with a basis {p h^For any nonnegative integer g and any curve class $^ci (where di > 0), the genus g degree equivariant Gromov-Witten invariant of Al, denoted by M(A1), is defined via localization with respect to the C" action. (See Section 2.1 in [471). Definition 4.1.1. With the notation above, we define the reduced GromovWitten partition function of Al to be given by  (in,^exl)  N3(M)qii •  d 2g-2  Throughout the paper, G will be a finite subgroup of SO(3), acting on C 3 via its natural 3-dimensional representation. We will refer to this representation by V. The main object of the study in this paper is G-Hilb(C3 ) that we will always denote by Y. Our main geometric tool is to realize Y as a family of non-compact K3 surfaces as follows. Consider the family of surfaces E : C 3 --r C given by  ^get,  y,^= X 2 2- p 2 + 2 2 .  Let  Qt =^+ y 2 + z 2 = c be the fiber over^  C. Note that the central fiber is a quadric cone  ^Qu^C2 /ii 1 1. 2 Since G c SO(3), it preserves the form x 2 + z 2 , and hence it acts fiberwise on this family. We will see in Section 4.2.1 that Y inherits a fibration  whose general fiber, S t , is fiber, So, is a partial resolution  Q0/G C2  ition^t/G, and whose cents  Chapter 4. The Quantum kkay Corres pon dence where G is the binary version of G, namely the preimage of G un der the double covering j:  Figure 4.1: Surface fibration of Y  .  Let r :^c2 KC be the minimal resolution. Boissiere and Sal^(43) construct a morphism fo^4 Y that makes the diagram in Figure 4.1 commute. It is classical that there are ADE classifications for both G and CT, and that the geometry of S gives rise to a Dynkin digram of ADE type. In other words, the irreducible components of the exceptional set of '77 is an ADE configuration of smooth rational curves. Moreover, associated to G is a root system of ADE type that we denote by R(G). Our strategy is to relate the equivariant Gromox ^ theory of Y to that of S. We studied the equivariant Gromov-Wit ^of S (45j, where we expressed the quantum cohomology of S in term^R(d). ;  Chapter 4. The Quantum Mckay Correspondence As we will see, our answer for the Gromov-Witten theory of Y is also in terms of R(G). In F igure4.2 we summarize a set of known relations and identifications that we will use in this paper. The first bijection in Figure 4.2 is the classical McKay correspondence in dimension three (see [55], [56], [58] and [64]). The second inclusion is obtained by pulling back irreducible representations of G by j. The last two bijections are the classical McKay correspondence in dimension two, which it also gives the intersection pairing of S as minus the Caftan matrix of the root system 11(G) (see for example  Irre ucil  components of R -1 (0) r „d  I Injection 5 I Nor  vial irred  ^  representations of G inclusion 2  {Nontrivial irreducible represe itations of or I bijection t: 3  educible components of FIT` (0)„,d bijection p 4 plc roots of the ADE root system R(0)}  Figure 4.2: Classic( IcKav correspondence  Definition 4.1.2. We denote the second and the^fig 4.2 by r'(G) and Ire (GT respectively. We drop the star whe we want to include the trivial irreducible representation. For any p e Irr(G), by x p (—) we mean the irreducible character corresponding to the representation p. Using the first correspondence in Figure 4.2, we denote by (C) the set of irreducible components of TT - (0) re d. It is also proven in [561 {^  Chapter 4. The Quantum Icay Correspondence  (Theorem 3.1) that each Cp is a smooth rational curve, and for any p , pi E Ire^C 1, intersects C1, transversally at a node if and only if p appears as an irreducible summand of V Op'. One can observe that tr -1 (0)„, is a tree configuration of rational curves, and at most three curves meet at each node. Note that 71 - (0) re d is a deformation retract of an open subset of Y 711 , and hence tr -I (0) re d is a retract of its arbitrarily small neighborhoods which be chosen to be homeornorphic to Y by scaling. This implies that (1', 11',) 125.^or N. (0),.„,,,,, ,71). In pa ocular, ICA represents a basis for 11 2 (Y,7). Let {ap}pe.,(0) also be the set of irreducible components of if -1 (0),d, which gives rise to a basis for 112(15-1 ,7). Our next task is to relate the root system R(0) to the geometry of We first need the following definition: Definition 4.1.3. By the virtue of the third and the forth bijections in Figure 4.2, we denote by lo p } pem .,. ( a ) the set of simple roots of R(G). Let 4(61 ) and /1 1 (0) be respectively the root lattice and the set of positive roots in this root system. We will use the notation 1?, a, and 1? 1 when the connection to C. is clear. We identify H2 (S T) with the root lattice A. define a surjective homomorphism c : A , N T72 (Y,7G), by  c ( c p ) re  ,  Cp p C Irr s (G), 0^p C Irr 1 (G)\,Irr*(G).  The homomorphism c is implicit in our expression of the Gr ov-Witten theory of Y in terms of the root system R. The diagonal C*-action on U3 commutes with the action of C. and can be lifted naturally to an action on Y. Thus we can define the reduced Gromov-Witten partition function of Y as in Definition 4.1.1. ;  Definition 4.1.4. We need the following notation in the future: la dle pair (1 7,7r -1 (0),„h) can be triangulated, and hence ir -1 (0)„„h is the deformation retract of its second star neighborhood in Y (see [61]).  Chap ter  nt  e^  Adckay Co rresponden t  (i) For any a E a. we define the integers cep to be the coordinates of a with respect to the basis { e p }: a  (ii) Let and be sets of variables indexed by the irreducible representations of and G respectively. In this paper, we use p and if to parameterize homology bases of the threefolds which are indexed by In' (Ci) and Irr * (C) respectively. For any a E A we define -  and ir^4„aP pE  Remark 4.1.5. Note that  p`t  =^the specialization p E Irr'(G)  (ip  The  ^  p E Irr f (a) \ Irr*(61 ).  heorem of this paper can then be written as follows.  Theorem 4.1.6. Let C be a finite subgroup of 50(3). The  reduced Gramm ,  Witten partition function of Y = C-Ifilb(0) is given by  z Y (q, A) =^—elAy1/2, cte^  , q "¢ 1  where AIV =  ^— xy d1  )  ---  This theorem will be proven in Section 4.2.2.  4,1.3 Applications Theorem 4.1.6 gives direct predictions for ie BPS states, Donaldson-Thomas  inva riants of the Calabi-Yau threefold Y as well as for the genus zero orbifold Gromov-Witten invariants of c3 IG. In this section we give a brief discus sion of these applications.  Chapter 4. The Quantum Mckay Correspondence  Donaldson-Thomas theory Donaldson-Thomas invariants of a threefold is defined in terms of the  mo  spaces of its ideal sheaves. It was conjectured in [62] (Conjecture 3) and (Conjecture 3R) that the reduced partition function of Gromov-Witten and Donaldson-Thomas theory are related via the change of variables —e iA = Q. Using this, Theorem 4.1.6 immediately gives a prediction for the reduced Donaldson-Thomas partition function of Y:  43 7- (q, (2 ) = 7  nE  h  BPS states The result of Theorem 4.1.6 in terms of the BPS states is remarkably simple. The most common mathematical definition of Copakumar-Vafa BPS invariants of a Calabi-Yau threefold is via Gromov-Witten invariants (See Definition 1.1 in [48]). The genus g, degree (3 BPS invariants of Y, denoted by Tay is defined by the formula ,  EEnywA29-2=EE s-o e=o^ Theorem 4.1.6 implies  7r  F  ) 2g 2 -  11 (^  2  9=0 3-;20^fix)  (J O  (See also Proposition 4.3.8) that BPS states  of C-Hilb(0) are given by )2u I CI  (0) n R+1  = 0,  0^ > The map c :^112 Y^defined in the last section. Remark 4.1.7. It has been observed that for any f3 e 112 (Y, Z) as above, c -1 (3) n R can only be either of 1, 2, 4. or 8. In particular, for some of the curve classes the BPS state is 1/2. This does not contradict the conjectural integrality of the BPS states of the Calabi-Yau threefolds. This is because of the non compactness of Y that causes the ordinary Gromov-Witten  nvariant corresponding to some particular curve classes not to be a priori well defined. We use the C action on Y and define the equivariant version of the invariants for such curve classes (See Definition 4.1.1). The equiv-  ariant Gromov-Witten invariants agree with the ordinary Gromov-Witten invariants whenever that the latter are well-defined.  Chapter 4. The Quantum Mckay Correspondence Orbifold Gromov-Witten theory of C3 /G Let X be the orbifold associated to the singular space 3 /C. Theorem 4.1.6, together with classical invariants of Y (see Section 4.3) and the Crepant Resolution Conjecture (See 46)), give a prediction for the genus zero GromovWitten invariants of X. To express this we need to introduce some more notation. The inertia orbifold, /X is a union of con tractible^col c omponents, PrX(9)}(9)G Conj(G) indexed by Conj(G), the set of the conjugacy classes of G. Recall that the (equivariant) orbifold cohomology of X is by definition (see [51D  11,;,b(71)) =  ^ (g)E  of  X91)),  (0)  where z( g ) is the degree shifting ^ bet (age) of the component indexed by (g) E Conj(G). One can see that t( g ) = 1 for (g)^(e), and therefore there exists a canonical basis {(5( g )}( 9 ) E coni(G ). for 11:;,, b (X) where 5 (e) 6 H2eb( -1) )  ^  and^6 (9) E 11,2„ b (cr for (9)^(c).  Let a] fx(oloy Conj(G) be a set of variables parameterizing this cohomology basis. For any given vector in = (n(0) (OE co„i(G) of nonnegative integers, we use the following notation  6" =  ^II^dr tgJe eonj(G  and  Xr  n!^  11  g)e Con . (G)  )  X,  n(g) (9)  71 \ 91.  Suppose =^  L^ it(g)•  (g) E. Con] (C)  We denote the genus zero, inbpoint, equivariant ^ fold Gromov-Witten invariants of X corresponding to the vector 91^ri• (see [50]). These invariants take values in Q(t) (see [46D. Definition 4.1.8. Using the notation above, we write the genus ero orbTold Gromov-Witten potential function of X as Tn.  F x br) =^(5') Lb mi  where r = IConj(G) I. For simphrrty,^ Remark 4.3.10).  the^to ze  89  ^  Chapter 4. The  Quantum Mckay Correspondence  In Section 4.3, ve establish the following prediction for Fii (x). The reader can find a similar prediction for the orbifold Gromov-Witten invariants of [C2 /G] given by Conjecture 11 in [45]. Conjecture 4.1.9. Let G be finite subgroup of SO(3) The genus zero orbifold Gromov-Witten potential function of X = [C3 /0] is given by (P)  + 1 GI  E^— A v(9) AR(g) ,c G  where h 1 (s) = 2  tan  4)1.4 Example  Let S3 C SO(3) be the group of permutations of three letters. We have Ire(S3) =^,172} where 17; and 172 are respectively the alternating and the standard representations of S3. S3 acts on C3 via its natural 3-dimensional representation ^V  ^e 112 .  his case Y^lb(0) and 7r :Y^C 3 /S3  is a crepant resolution. Correspond to S3 is the generalized quater io of order 12 that we denote by S3. We have irr * (S3) =^U2, U3} where U1^(§3) Irr*(S3) (See Figure 4.2). S3 acts on C2 via 112 Let : S-4 C2 /S3 be the minimal resolution. S corresponds to Ds in the ADE ela ssificatio The intersection graphs of the exceptional sets If(0) r e d and 7r -I drawn in Figure 4.3. The vertices in the graphs are correspondin g irreducible components of 1( -1 (0) r „,1 and 71 - I (0),„d. In each graph two vertices are connected if and only if the corresponding components meet at one point. All the components of if --1 (0)„,, d are smooth rational curves with self-intersection -2. The components of 71-1 (0)„,„; corresponding to 1. 71 and 172 are smooth rational^ mal bundles respectively isomorphic to Oft (-1) (1) a4-1) and O i pi (I) a 0,,))(a-3).  Chapter 4. The Quantum tl4ckay Correspondence  0  v,^'a,  Figure 4,3: Intersection gi P s: 77--- t ( 0 )red (rig ht  The morphism ,t o : S -a Y in this case maps the left graph to the right one by contracting the components corresponding to the solid vertices. Let C14 C 192(Y, 74 be the curve class corresponding to v; for i = 1, 2. We denote the homology class di.Cv, ± d2Cv2 by (d ,d2). The left graph above, also corresponds to the D5 root system R(83) in the ADE classifications with the vertices stand for the simple roots. There are 20 positive roots in this root system: R" ( 5 3) = 0 0 (1000, 0100, 0 1 0111, 0110,  0 0010, 1 0011,  0 0001 0 1111  1 0000, 1 1110,  0 1100, 1, 0111,  0 0110, 1 0121,  0 0011, 1 1111,  1 0010, 1 1121,  0 1110, 1 1221)  where for each element, the numbers demonstrate the multiplicity of the corresponding simple root in the graph above. By Theorem 4.1.6, the partition function for the Gromov-Witten invari ants of Y is given by ' q1,12, A) = A1 (qvi^MBiB^)-2119 (4  ^_ eza ) .1 1,r ( q?,2, _ elA) 1,2 1,r 9vi^, _ e iA) (  Then one can see easily that the nonzero BPS states are it o ) = 1.^n c(110 -8= 2,^7q0,1 .) = 4,^n i(,0,,, ) ) = 1/2,  4.2 Proof of the main result This section is divided into two parts. In the first part, we provide mo details on the K3 fibration Y C 3 /G mentioned e tilier (see Figure 4.1). The proof of the Theorem 4.1.6 will be given iu the second part. 84  Chapter 4. The Quantum Mckay Correspondence  4.2.1 Geometry of Y G-Hilb(C)  We start by recalling the definition of Y, our main object of the study: Definition 4.2.1. Y = G-Hilb(C3) is defined as the subscheme of Hilbl G i (C13 ) parameterizing closed suhschemes Z c C 3 with the property that 11 0 (0z) s is omorphic to the regular representation of C. Here Hilbi G (C13 ) denotes the Hilbert scheme of length subschemes of C3 . Consider the Hilbert-Chow morphism from 11111) 16 ^to the IG-th symmetric product of C 3 (C3) —> Sy 9C3 ) that sends a closed subscheme Z a C3 to the zero cycle [Z] associated to it. It is a projective morphism. By restricting to the G-fixed loci of the source and the target, h factors through a projective morphism 71  Y  Theorem 4.2.2. (Theorem 1.2 in I-441)Y is smooth, reduced and irreducible variety, and a gives a crepant resolution of C 3 IC. Remark 4.2.3. In general, one can define the II-Hilbert scheme H-Hilb(M), where M is any smooth n-dimensional quasi-projective variety, and H is a finite group of the automorphisms of Al, such that the canonical bundle of Al is a locally trivial H-sheaf.  (i) /1-Hilb(M) represents the moduli functor of H-clusters (see Section 3 in [43]), so it has nice functorial properties. (ii) Theorem 1.2 in [44] proves a more general fact: whenever^< 3. Theorem 4.2.2 remains valid if one replaces Y by H-Ifilb(M), and C3 /G by M/11. Remark 4.2.4. Crepant resolutions of C 3 IG are not unique. However, in  the context of the McKay correspondence, Y has been proven to be the preferred one (see [44]). We now return to the diagram in Figure 4.1. By Theorem 1.1 in [43], : Y^Ii 3 /G restricts^  artial resohnto o : Se  — 85  Liter 4. The Quantum M ckay Correspondence whose reduced exceptional locus ^ i^with n " 1 (0) re d and the uniqueness of the minimal solution we have  Borssiere and Sarti^ e the geometry of S and Y^ Theorem 8.1 in [43]) by proving that the morphism to:^Y is the minimal resolution of So C Y, it maps O p isomorphically to CI, if p e hi* (C), and t contracts Cip to an ordinary node if p C Ire (0) \ (C). 13y the functoriality of the G-Hilbert scheme (see Remark 4.2.3 OH, for each t 0  Sr C±'-' Ggliffi(QI), and by Remark 4.2.3 (ii), x restricts to the minimal resolution  Q t /C. t le libration of Y C by non-compact K3 surfaces as mentioned in the introduction.  Remark 1.2.5. By the naturality of the construction, the C*-action on C 3 lifts to an action on the family c. Note that the weight of the (7-action is 2 on the base of the family, because it acts with weight 1 on the coordinate functions x y and z of C3 . ,  4.2.2 Proof of Theorem 4.1.6 ^ Let C be the family of surfaces constructed in the last section, and let C be its double cover branched over the central fiber 50 -4. 0. fjs- i ng has conifold singularities over the singular points of So, and it is smooth away from these points. Let Y -King be a small resolution of conifold singularities. We obtain a C-equivariant family  "(TT :Y -- } C  Chapter 4. The Quantum Ntekay Correspondence  ?sing  2:1  2:1  C  Figure 4.4: 1: smooth K3 fibration  For any 0 t C C, —1 (t) "Sr.< e (t2)^Ste  .  Moreover, by the uniqueness of the minimal resolution in dimension two, the central fiber 1 -1 (0) is isomorphic to (§, the minimal resolution_ of c 2 /0. Hence,1is a K3 fibration with all the fibers being smooth. Thus, {C IA ocirrych forms a basis for RAC', 2,)^119(51 , 21 21 ) The Gromov-Witten theory of Y is determined in the following proposition. Remark 4.2.6. Note that because the base of the family /double covers the  base of e, the C*-action has weight 1 on the base of /(see Remark 4.2.5). Proposition 4.2.7. The reduced G7'0714011- Witten partition function of - C7 is (  rilveri, by Z ( A) ae  P30 DP: The pro^sed on the deformation in varian ce of the Gr omovWitten invariants. See 019] and [45] for a similar argument. Since Y is smooth, /above defines a flat 'family of C-equivariant deformations of S. Thus it induces a classifying morphism :  C^Def(88),  87  Chapter 4. The Quantum Mckay Correspondence  where Def(S) is the versal space of C*-equivariant deformations of 5'. Def(S) is naturally identified with the complexified root; space associated to the root system R (see [59]). There is bijection between the set of positive roots Ili and the irreducible components of the discriminant locus in Def(S). Each irreducible component is a hyperplane iu Def(S) perpendicular to the corresponding positive root. A generic point of Def(S) corresponds to an affine surface (with no compact curve), whereas a generic point in the component of the discriminant locus perpendicular to fi C IV, corresponds to a smooth surface with only one smooth rational curve with self-intersection-2 representing the homology class 44 Since p, is a 0-equivariant map, and since the 'eights of the 0-action are equal to 1 on its both source and target (see Remark 4.2.6 and Theorem I in [59]), p has to be a linear map, and hence its image is a line passing through the origin. We first equivariantly deform 42 to fti, by egnivariantly deforming p to the linear map : C Def(S), whose image is a generic hue passing through the origin. Since the uivari-ant Gromov-Witten invariants are invariant under equivariant deformation we have 721(4, A) = (441,A). Now the image of pi meets the discriminant locus only at the origin, hence all the curve classes of P: 1 are represented only by the curves supported on the central fiber of the family 1,1 C. Thus, for any /3 ^ 0, N3(17.0 (and hence ZYI (FLA)) has a nonequivariant limit. Next, we make a nonequivariant deformation of 141 to V2 by defor ming p i to the linear map p9 : C Def(S), whose image is a generic affine line meeting the discriminant locus transversely. Since the o ov-Witten invariants are invariant under deformations we have 4ZC2.(74 A) = Z i '(14, A). By the description of the discriminant locus given above it is evident that the image of P i meets transversally each component of the discriminant locus once at a generic point. This implies that the threefold V2 contains a unique isolated smooth rational curve with normal dle isomorphic to C941(-1) 42) 0-4 ,(-1) $8  Chapter 4. The Quantum Mckay Correspondence n the curve class correspondingo each element of ^12. 'flirts I  OCR  See the proof of Theorem 3.1 in [421 fora similar calculation to derive the last equality. From this the proposition immediate.  Remark 4.2.8. We have observed that a canonical candidate of the map p in the proof of Proposition 4.2.7 is given by vector Gr^pep pERC' (a)  \ Irr* (C)  where pP is the multiplicity of e p in the expression of the largest positive Pot terms of the simple roots. Any line obtained from this one by the action subgroup Weyl group (generated by reflection with_respect to the ots corresponding to p C Irr*(G)\ lr?(G)) on Def(S) is the image p C Def(S) defining a threefold which is a small resolution of fferent lines give rise to the different possible small resolutions p),.. This fact is not needed in the proof of Proposition 4.2.7. Next, we relate the Gromov-Witten theory of Y to the Gromov-Witten theory of Y. This is given in the following proposition: Proposition 4.2.9. 'Die reduced GMTTIOV- Witten partition function of  and Y satisfy the relation ((7, A) = Z  /7  (q, A) 2  after the specialization (s^rk 4.1.5) p^Irr*(C) 1^p E IrC(a) \ 117*(6').  (1.2.1)  PROOF: In the course of proof, we will use some auxiliary threefolds which are not necessarily Calabi-Yau. However, we will see that we only need the Gromoy-Witten theory of the Calabi-Y-au classes of these auxiliary threefolds, i.e. the curve classes for which the virtual dimension of the corresponding moduli space of stable maps is equal to 0. The Calabi-Yau class The claim about the normal bundle follows from the transversality of the intersection of the image of fi e with the components of the discriminant locus. This implies that the corresponding rational curves do not have any infinitesimal defornmtions inside 172. "  Chapter 4. The Quanimn Illeicay Correspondence Grornov-Witten partition function of any smooth quasican be defined as in Definition 4.1.1. We first compactify the base of r and construct a new  ariety nily of surfaces  :  as follows. IT W=  c3  L €2  c3  \4a  where the two copies of C 3 are glued along C 3 \20 via the rna0 fir,^144 The two copies of fibration  y2 ± z2 x 2 „t„ y 2 +^y2 + z2  e^  e :^C patch^  On  the  :  The general fibers of -^2 are smooth (they are^-e s the fibers over 0 and 44 ° (0) 4,2;4 41 (00) =44- C2 /1+11 are isomorphic to the quadric cone inside C. 3 . The actions of^and G extends naturally to^Define (see Remark 4.2.3) Y^64 1101)(14 4 ). Then V is the crepant resolution of W/G. Clearly, this construction Wes rise to the K3 fibration Y .  with the compact base. Over the^  C-patches of^Y is isomorphic to Y but with opposite 0-weights. Define the subspace of fiber classes of Y , denoted by H 2 (14- 4 74)f, to be the kernel of  112 ( z) -> H1 ( P Mayer-Vietoris argument 1 3 for the res 'C-patches of 1P 1 shows that  lions of E t o the two  H2 (Y, :T) 2-44 112(Y,Z): 13  Note that 1/1 (Y\e -1 (0), Z )^7, and I/2 WV)) (a), Z Vtorsion  77, 0", 2).  Z  Chapter 4. The Quantum Mckay Correspondence All the fiber classes of Y are Calabi-Yau classes. We denote by  Zf (q, A) the  reduced partition function for fiber class Cromov-Witten invariants of Y. The C -fixed part of the moduli space of stable maps to Y representing fiber classes has two isomorphic components, corresponding to maps with -r images in either of E _ t (0) or E (eo). Each component is isomorphic to the C -fixed part of the moduli space of stable maps to Y. Since the dimension of the moduli space of stable maps to Y representing (3 E 112(1', Z)1 is zero, the corresponding Cromov-Witten invariant N. (Y) is independent of the k  weight of the C action. Thus, by localization  N4 (Y) = 2N' (Y) and hence  4(q, A) = Z Y (q, A) 2 . Since Y is a family over a compact base, and each fiber of E is a non-compact surface but with a finite number of compact curves, the moduli space of stable maps to Y representing a fiber class is proper. Thus. Z1 (q, A) has a non-eginvariant limit. Let ,1 Y5i„g -r , be the double cover of the family E branched over the E Let Y -> Ym 2  -t  (0) and E -' (eo)  be a small resolution of the conifold singularities. One obtains the family Y --> IF of non-compact 1(3 surfaces that make the following diagram commute. Y  -  -  Chapter 4. The Quantum Mckay Correspondence  Let 119(Y s h ig ,Z)f and^A)1 be respectively the subspace of the fiber classes of Y s i ng and Y. They can be defined analogous to /79(Y, Z)1, and one can show that 11 2( 1'si  Z)^ll20Ing,^1:1-1 11 20Z:4  112(C.^f c.14 H 2(f7^CZ' A.  All the elements of 112(Y, Alf are Calabi-Yau classes, and by the same argument the corresponding Gromov-Witten partition function, Zi (7, A), has a non equivariant limit, and again a localization calculation shows that  47(11, A) = Z I/ (4,A) 2 . Thus, in order , o finish the proof of Proposition it suffices to prove ZN, A) = I,f (g, A) 2  after the specialization (4.2.1). Let : Y g. ,, be the double cover of the family E branched over two generic fibers. Yg is related to 1' by conifold transitions (see [60]): Moving the branch points in -_-. FP 1- to 0 and (se defines a deformation of Y g to Y s i ng , which it then limits the small resolution Y. Let 1/2(Y g , A) f be the subspace of the fiber classes of Yg. We have 112(Y Z) f Z=./ 112(Y si • Z) f c  .1?-1/  HAY, ) •  where the first isomorphism is proven in [60] page 167. The Gr iov-Witten theory of Y g and Y are related by a theorem of Li and Ruan (Theorem B in [60]): There is a surjective homomorphism 11 2( 1 Z)c - 112 (Yg , A) 1 ,  so tha to Groyne ^of Y g and Y have the followin flow NO  (  'I  1'^'^!  4 2.2)  92  Chapter 4. The Quantum Mckay Correspondence  00_ for any 11 E .112(Y g-,Z)f. Note that by the identifications of the source and target of 0 with respectively D and Y, Z) given above, 0 is identified with the homomorphism --/ 112(Y, Z) defined in Section 4.1.2. Hence on t:he level of reduced partition functions, (4.2.2) means that the fiber class partition function of Y s is obtained from that of Y by setting the homology parameters corresponding to classes in ker(c) = ker(0) equal to I. In other words 4(4, 1 A)^(q. A) after the specialization (4.2.1). The rest of the proof is very similar to the Mayer-Vietoris argument used by Maulik and Pandharipande for the computations of the fiber class Gromov-Witten theory of Enriques Calabi-Yau threefold (See [63], Sections 1.1 and 1.4). By degenerating the K3 fibration Eg ,we form a "good degeneration" of Y g to Y110 V. the union of Y with itself along a smooth fiber. For any fiber class /3 c /12(Y g ,Z)f , and any non negative integer g the degeneration formula for Gromov-Witten invariants implies that 1V:(Y 5 ) 2A1pY/F), where the right hand side is the corresponding Gromov-Witten invaria Y relative to F. However, since F is a K3 surface (see the proof of he 2 in [63]) ATY/F) = Npy). The last two equalities immediately imply that ZJ (q. A) =^(g, A) 2 . The proof of Proposition is now complete.^ Theorem 4.1.6 follows immediately from Proposition 4.2.7 and 9  ^ (  93  Chapter 4. The Quantum Mckay Correspondence  4.3 Crepant Resolution Conjecture In order to study the Crepant Resolution Conjecture in the next section, we need to extend the Definition 4.1.1 to also contain k-point Gromov-Witten invariants. Define a basis {D p } pel „s( G ) for the cohomology group H 2 (Y, Q) that satisfies Epp, for any p,^Ire (C). For any p E Irt*(C) let L i he a line bundle on Y defined by ci (L p ) = D p . We choose a lift of the ^action of Y to each 14 1, such that  c1(4) = 0 where the tegral is evaluated via localization. Denote by -yp the equivariant them class of L p with the chosen lift. We take 70, corresponding to the trivial 1-dimensional representation of G, to be the lift of the class of the identity to the equivariant cohomology. We have that {-y p } p on .(c) is a set of generators for the equivariant cohomology ring of Y. Let y = {y i,} pciri ( m be a set of variables corresponding to this cohomology basis. For any given vector In^(tri p ) pe: i ri .(G) of nonnegative integers, we use the following notation  (re),  1  ,,.??1^  Y  and —I  nip  yp  i rri ,^ mn! ,E. liT(0)^'  Let /3 c I/2(Y, Z), g E Z 1 _, and suppose that  ,r,^Tap. pEirr(G)  The genus g, degree 13,equivariant Gromov-Witten invariant of Y corresponding to the vector in is denoted by (7 g . These invariants take values in Q(t) (see [471). Note that in Section 4.1.2, we showed 0-point invariants. (by- AT9(Y) ' ,  Definition 4.3.1. Using the notation above and Definition 4.1.4, we w rite the potential function for Gromov-Witten invariants of Y . as („y rn ).1"  Chapter 4. The Quantum Mckay Correspondence where r^li (01. We call the^0 part of F Y the classical part, the [3 ^ 0 part of F Y the quantum part.. We denote the former by ^and the latter by 1 1',/pC. F)-/ involves only the classical triple equivariant in sections on Y. In other words, FY is the cubic polynomial p given by  If /3 ^ 0 then using the point and divisor axioms, one can write all the Cromov-Witten invariants in terms of 0-point invariants O Y 9, 0'  Remark 4.3.2. One can see that for any p C Ire (G), the variables p i, and q 1, always appear in Fq/,1 as the product g o Thus if we set p 0 in F Y no information will be lost from its quantum part. Recall that in Definition 4.1.1 we defined the reduced partition function, Z r (q, A), as exp(F Y^= 0, A) ). Definition 4.3.3. We define a new set of variables q' = N iCIpcirr• k co where  qp = g o epe. By the remark above the quantum part of F Y is really a function of g' and A. Note that q is obtained from (I by specializing to y = 0. We determine the clan ^id the^n parts of propositions. Proposition 4.3.4. The classical triple eguivaria  separate  section numbers  on Y are given by (4.3.1) (4.3.2) (4.3.3)  Coteter number of the root Si  Chapter 4. The Quantum Mckay Correspondence PROOF: We use part of the construction given in Section 1.2.2. Let f he the composition  Y in Figure 4.4. f is an equivariant map with respect to the C*-actions o n and  2:1  C  For any p E Irr*(0), the 0-equivariant line bundle L p on Y defined above, pulls back via f to a C-equivariant line bundle on Y denoted by L. We have  1 .  CI(L  a  = pp'  I  ,;:,(31(1,,,) = 0  for any p, p' c lre(C7). Extend {E p } pcir . c(G) to the set of C-e line bundles {L p } oein ,„ (a) satisfying the two relations above for any p, p` c Tre(G). Define ii, = c 1 1134, and let /0 be the class of identity in the equivariant cohomology of ,§, denoted by ./Trt, (§,Q). fly the construction { 34 purr( o ) gives a basis for I.U.C:5,Q). -  Now let i 6  HY,(2)  be the pull back of some ry E PP( Y,Q) via the  i  v = 12. I,'^r‘i  All the integrals are defined via localization. ^ factor 1/2 in the first equality appears because f is generically ^ ON^ 1 /l factor in the second equality , of 8 ?— Y is 1. Proposition 4.3.4 follows imme Lemma 4.3.5. 0  96  Chapter 4. The Quantum Mckay Correspondence Lemma 4.3.5. Let {-10 1 pchrt mbe the basis for 117,(§,Q) chosen above. Then we have 4  1;j 7:27tH 0,  E  h  ^yP ^  cC R+  I  7p^p" =  2  2_,  aP a P  (I C It +  ^P110 oF: This is basically^ ences  in 1451 with only the follow-big differ-  1. The basis chosen for HE' ) , Q) is obtained from the basis chosen in 145] by the inverse of the Cartan matrix of R( ). The reason for this is that the cohomology basis chosen in 45] is the dual of -{Ci p l pefri .„ ( a ) with respect to the intersection pairing (which is the Cartan matrix by McKay correspondence), while in this paper it is the dual of {C p } pciril a ) with respect to the equivariant Poincare pairing. )  2. The weights of the induced C-action on the canonical bundle of S and the cohomology basis elements -7„, are half of the corresponding weights in 144 0  Remark 4.3.6. Note that (4.3.2) in Proposition 4.3.4 is the direct result of the definition of {-yd. (4.3.1) in Proposition 4.3.4 can also be deduced by the same argument as in the proof of Lemma 4 in [45], by pushing forward the class of the identity to H. (X). Remark 4.3.7. Define the matrix G = (4.4 0p ,),, ,t4Eh.,4 K0 by gPP I  =I  7P 711  Then one can stow that (7 = it/^with =( V— (  G -1 is the analogous of the Cartan matrix 97  Chapter 4. The Quantum Mckay Correspondence Proposition 4.3.8. The Titanium part of the Gro70,00- Witten potential fuucLion of^is ,given (it A) 88-  E^E  ize  IA  C  11+ d=1 g=0  2  \ 2  /l  Ida  (.1  18 .R.00F: This is a direct corollary of Theorem 4.1.6 and Remark 4.3.2 (See the proof of Theorem 3.1 in [42] for a similar calculation). Let PT (q', y) be the genus zero Gromov-Witten potential function. We will need the following immediate corollary in the next subsection. Corollary 4.3.9. The genus zero Cron ton- Witten potential function of Y given by  -P(n(1,y)  =  As it was mentioned in Section 4.1.1, the Crepant Resolution Conjecture has been explicitly formulated in [44 with the assumption that the orbifold X satisfies the hard Lefschetz condition. The Crepant Resolution Conjecture has been studied in [52] without this assumption, however the relation between the quantum cohomologies of the orbifold and its crepant resolution is more elegant in hard Lefschetz cases. The hard Lefschetz condition means that the canonical involution of the inertia orbifold IX preserve the degree shifting numbers (age) of its connected components (see Section 4 in {54]). The canonical involution of IX when X is the orbifold [C83 /0], simply swaps the connected components TX( 9 ) and IX0-1 ) for any (g) C Conj(G) (See [51]). Our next task in this section, is to establish the prediction for Fo x given in Conjecture 4,1.9 by using the Crepant Resolution Conjecture and Corollary 1.3.9 Define a new set of variables \'  f  (11))(0)E Goof (6') and p pe^KO^  (c. P) (le Irr(G),  toi tin v  98  Chapter 4. The Quantum Mckay Correspondence where Conr(C) is the set of nontrivial conlugacy classes of C, C(g) is the centralizer of the group element 9, and V is the natural representation of C. It was conjectured in [46) (Conjecture 3.1) that F61( (1) is obtained from .17,1T (w, y) after replacing  We start by inserti Lary 4.3.9, we get  Fci (x) aE It'  ^ ^ e  isformation above into F(ny,q). Using Corol-  aP  2,  (b)(a)14,()9)/(9  p  I(4.3.5)  where in the summations above p and (g) run over b (CI) and Cony (C), respectively. If we take the third partial derivatives of (4.3.5) with respect to the variables 3)( k ), x (1,4) and x( k e) corresponding to three nontrivial conjugacy classes, and use (4) in Proposition 4.3.4 we get (4.3.6)  L .)  e2i Q(a)  E,,,,o)L(k") ^ e21 Q(01)  where  a p (a) c p p  e summations above p, p', p" and (g) run over b(C) and C) r esp ectiv ely. Using the identity  (  2( 9^2^tan 6+ 9  99  Chapter 4. The Quantum Ildckay Correspondence we c^simplify ( 3.6) to  a p Ml k)^ap(P)  (  i f/  0 )i.:,  '")  \  (4.3.7) Integrating (4 .3.7) with re. replacing for (2(a), we find tl  to the^  les a,; (),), ft (k ,) and in Conjecture 4.1.9 fox only nontrivial terms h  Remark 4.3.10. By the point containing 3;( 0 are .  ( 6 (ic ) 3  ^  and^( 8( e)^[r( e x (9) x (g -i ) for g^e  corresponding to the classical invariants, which are easily evaluated to / s3 (e)^t3  and (8  B y the similar argument given in [14 up with the part of F r with^  (  6( 9 -1)  — ^  flc(01  one can check that these terms match  zero yo terms (see Remark 4.3.7).  1 00  Bibliography [42] Kai Behrend and Jim Bryan. Super-rigid Donaldson-Thomas invariants. arXiv:matILAG/0601203. [43] Samuel Boissiere and Alessandra Sarti. Contraction of excess fibres between the McKay correspondences in dimensions two and three. arXiv:math.AG/0504360. [441 Bridgeland, T. and King, A. and Reid, M. Mukai in flies McKay: the McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc., 14:535-554, 2001. [45] Jim Bryan and Amin Gholampour. Root systems and the quantum cohomology of ADE resolutions. arXiv:0707.1337v1 knath.AGI. [46] Jim Bryan and Tom Graben The Crepant Resolution Conjecture. arXiv: m ath. AG /0610129. [47] Jim Bryan and Rahul Pandharipande. The local Gro ov itten theory of curves. arXiv:math.AG/0411037. ] Jim Bryan and Rahul Pandharipande. BPS states of curves in CalabiYau 3-folds. Geom. Topol., 5:287-318 (electronic), 2001. J. and Katz, S. and Leung, N. C. Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds. of Algebraic Geom., 10(3):549-568, 2001. [50] Weimin Chen and Yongbin Ruan. Orbifold Gromo -Witten theory. In Orbifolds in mathematics and physics (Madison, 1471, 2001), volume 310 of Contemp. Math., pages 25- 85. Amer. Math. Soc., Providence, R.I. 2002. [51] Weimin Chen and Yongbin Ruan. A new^ Comm. Math. Phys., 248(1):1 31, 2004.  log theory of j  10l  Bibliography  [52] Tom Coates,^ sio Corti, Hiroshi Wall-Crossings in `fork Gromov-Witten Theory Crepant Ex arXiv:math.AG/0611550. [53] L. Dixon, J. A. Harvey, C. Vafa, and E. A\itten. Strings ^ Nuclear Phys. B, 261(4):678 686, 1985.  orbifolds.  [54] Javier Fernandez.^Hodge structures for orbifold cohomology. arX i v: math/ 0311026v2 [math.AG]. [55] Yasushi Gomi, Iku Nakamura., and Ken-ichi Shimoda. Hilbert schemes of 64 -orbits in dimension three, Asian J. *lath., 4(1)'51-70, 2000. Kodaira's issue. [56] Yasushi Gomi, Iku Nakamura, and Ken-ichi Shimoda. Coinvariant algebras of finite subgroups of SL(3,C). Cant!. J. Math., 56(3):495-528, 2004. [57 Y. Ito and I. Nakamura. Hilbert schemes and simple singularities. In New trends in algebraic geometry (Warwick, 1996), volume 264 of London Math. Soc. Lecture Note Ser., pages 151 233. Cambridge Univ. Press, Cambridge. 1999. [581 Yukari Ito and Miles Reid. The McKay correspondence forfinite subgroups of SL(3, C). In Higher-dimensional complex (Trento, 199¢), pages 222-240. he Gruyter, Berlin, 1996. [59] Sheldon Katz and David R. Morrison. Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups. J. Algebraic Geom., I(3):449 530. 1992. [60] An-Min Li and Yongbin Ruan. Symple tic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds. In vent. Math., 145(1):151-218, 2001. [61] S. Lojasiewicz. Triangulation of ;3( Manalytic sets. Ann. Scuola Norm. Sup. Pisa (3), 18:449-474, 1964. [62] D. Maulik, N. Nekrasov, A. Okounkov,^ Gromov-Witten theory and Donaldson-Thomas^ 142(5):1263-1285, 2006.  h ripande. Campos.  [63] Davesh Maulik and Rahn' Pandharipande New calculationsGromovWitten theory.^ 0/0601395. 1 i)2  Bibliography [6 11 Iku Nakamura. Hilbert schemes of abelian group orbits.^Ilyc roic Geom., 10(4):757-779, 2001. ,  Miles Reid. La correspondanee de, McKay. As tGrisque. (270):53--72, 2002. Sdminaire Bourbaki, Vol. 1999/2000.  L03  Chapter 5  Hurwitz-Hodge integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture  5.1 Introduction In his seminal 1983 paper [78], Mumford developed an enumerative geometry for the moduli space of curves analogous to Schubert calculus. On the Grassmannian, one can integrate the Chern classes of the tautological bundle over Schubert cycles, namely cycles given by the loci of linear spaces satisfying various incidence conditions. On the moduli space of stable curves, one can integrate Chern classes of the Hodge bundle over Hurwitz cycles, namely cycles defined by the loci of curves satisfying some Hurwitz conditions. Such integrals (and their variants) are called Hurwitz-Hodge integrals and they arise in various contexts, notably in orbifold Gromov-Witten theory, e.g. [70-73]. In this paper we consider Hurwitz-Hodge integrals^ s of g dimensional cycles on Mg which are defined as follows. Let C be either A4, the alternating group on 4 letters, or 7G2 x Klein four group. Let HG C Mg ,  be the locus of genus p curves C admitting a d : C^P I whose snonodromy group is contained in C. A  ITI'S;011  of this chapter has been submitted in arXiv:07C ^244.  ^  Chapter 5. Hurwitz-.Hodge integrals and the CRC The branch po^C P r of f are of exactly 2 poi nts. 13,  =  consists  — 3  and consequently HG has dimension g. In modern terms, IIG can be described as 1.1:0,„(BG), the moduli stack of twisted maps to the the classifying stack BC. As such, there is a natural compactification H0 C M y given by twisted stable maps Mo,„(BC). The C-Hurwitz space  f3G) has components indexed by the monodromy around the ri points. These are given by n-tuples of non-trivial conjugacy classes in G. Since each component has dimension g, we can evaluate the Hodge class (-1) 9 A 9 on each component to obtain a rational number. There are three non-trivial conjugacy classes in G, so the natural generating functions for these G-Hurwitz-Hodge integrals are formal power series FG(Xi,X9, X3) in three variables (defined in detail in §5.2). Our main result is an explicit formula for F0 written in terms of the E6 and /9, 1 root systems for G equal to A4 and 7L2 x Z2 respectively. To write our expression for _PG, we will need to introduce some concepts which will relate conjugacy classes of G to the F6 and D, root systems. Both A4 and Z9 x 7L2 are naturally subgroups of 80(3) (they are the y groups of the tetrahedron and the prism over the 2-gon). Let G Binary version of G, that is the preimage of G in SU(2) (namely the ary tetrahedral group .4 1 and the quaternion 8 group J). c ^ SU(2)  ^G  r^ 80(3)  By t he class}cal lcl{ay correspondence [76, 77 79], finite^ SU(2) admit an ADE classification where the non-trivia]^ sentations of a group naturally correspond to the nodes of the  oups of repre-  Dynkin diagram. In this classification, the binary tetrahedral group 1 4 a the quaternion 8 group Q correspond to the E6 and D4 Dynkin diagrams respectively. The non-trivial irreducible representations of A4 and^that pullback from representations of A4 and Z2 x Z2 correspond to the white nodes in the Dynkin diagrams below: 105  Chapter 5. Hurwitz-Hodge integrals ^  0^0  a  the CRC  0  ADE Dynkin diagrams also correspond to simply^ root systems where the nodes of the diagram correspond to simple oots of ^ root system. Let R be the E6 or D4 root system, let p be a non-trivial irreducible representation of G, and let e y be the simple root which corresponds to the same node in the Dynkin diagram as p. For any positive root a C RT , let no denote the coefficient of e p in a. We index non-trivial conjugacy classes of G by i E {1,2,3} and we let be the value of the character of a representation p on the ith conjugacy class. Let z i be the order of the centralizer of the ith conjugacy class, and let V be the 3 dimensional representation of G arising from the embedding G C SO(3). We define the following matrix which is a modification of the character table of G: 1 Li = — \I 3 —  Out main^  is the following  Theorem 5.1.1. Let G be A4 or 7L2 x Z and let I? be the F.% or 124 root system respectively. The generating function for the G-Ilarwity-Hodge integrals is given by  &(l:), 12,13  ) I )^  2  where RI" is the set of positive roots of I? ; irreducible representations of G, and h(u  over p is over a  ivial  s the series defined by  lim(u) = 2— tan (— 9 The above formula s expanded out explicitly for Z 9 X 7G2 Lion 5.2.1 and for A4 in oposition 5.2.2. Note that since the linear, and quadratic • of the series h(u) are undefined true for 1' G (Xi , 11, 13) . s corresponds to the defined for n < 3. We prove in Propo 0 5.7.1 that .PG(11 r2 . equal to the (monclassical part of the) genus zero Oromov-Witten r the orbifold 1  106  Gin^z-Liodge  ^  C  [C3 /G]. In [691 it is conjectured that fora crepant resolution Y X of an orbifold X satisfying the Hard Lefschetz condition, the Gromov-Witten potentials of Y and X are related by a linear change of variables and a specialization of quantum parameters of Y to roots of unity. The singular space X = C 3 /G underlying the orbifold X [C 3 /01 admits a preferred Calabi-Yam resolution IT :  G-Hilb(C 3 ) -- X  given by Nakamura's Hilbert scheme of G clusters [67]. In [68], we completely compute the Gromov-Witten theory of G-Hilb(C 3 ) for all finite subgroups G C 80(3) and we use the Crepant Resolution Conjecture to obtain a predicition for the germs zero Gromov-Witten potential of the orbifold X = [C 3 /G]. The change of variables matrix for the conjecture in this example is given by j-1/,;, and the roots of unity are given by / qp = exp  2iri dim p) ICI  Note that the matrix Po , the roots of unity qi„ and the formula in Theorem 5.1.1 make sense for any finite subgroup G C 80(3) (although the number of variables differs from three in general). Indeed, the conjectural fornnila for Fx, the (non-classical part of the) Gromov-Witten potential of X = iC 3 /G] is exactly the same as the formula for l' in Theorem 5.1.1 (see the conjecture in [68]). An immediate consequence of Proposition 5.7.1 and Theorem 5.1.1 is the proof of the conjecture in [68]. In particular, we have proven the following. Theorem 5.1.2. The genus zero Crepant Resolution Conjecture is true for the orbifold [C.3 /G] with its crepant resolution given by G-Hilb(C 3 ) when G is Z9 72x74, or Al. The Gromov-Witten invariants for X = [C 3 /G] in general can also be described as integrals over 61 -Hurwitz loci in Mg but for other G C 80(3), the integral of Ag is replaced with a slightly more exotic Hodge classes obtained from Chern classes of eigen-subbundles of the Hodge bundle.  5.2 Notation and Results Let G be a finite group, and Mo,„(BG) be the moduli space of genus 0, n-marked twisted stable maps to BG. The evaluation maps, denoted by evi 107  Chapter 5. thirwitz-Hodge integrals and the CRC fOr i E^n}, take values in the inertia tack IBG. The coarse space of fi nite collection of points, one fbr each conjugacy class in G. Let 8 (c 1 ,^ an) be an n-tuple of conjugacy classes in G. We define the following open and closed substack of M0.„ (BC):  MARCO = Prril Concretely, Al ABC) parametrizes G covers of an n marked genus zero curve with monodromy ei around the ith marked point. In this paper we will deal with the cases that the group C is either Z9 X 242, .44 or Sp We fix a notation for the conjugacy classes in these groups that we will use throughout the paper: • Let 1, r, a, p, (, denote the conjugacy classes in 84 corresponding to the elements (1), (12), (1 2 3), (1 2 3 4), (1 2)(3 4), respectively. ® Let 1, al, a 2 , (, denote the conjugacy classes in 11 4 corresponding to the elements (1), (1 2 3), (1 3 2), (1 2)(3 4), respectively. e Let 1, (I, (•, , denote the conjugacy classes in L„ x 249 corresponding to its four elements. All the above groups have a natural ti on the set of four elements. Thus to any element LI : C BC] E s(BG) we can associate a degree 4 cover C C with monodromy type c over the^  point. t  C Ms(LIG) be the universal family of the four fold covers and let R I T * (00 be the dual Hodge bundle of this family. We now define the G-Hurwitz-Hodge integrals as folio  Chapter ,5. Hurwitz-Hodge integrals d the CRC  When G is A4 or 22 x 12 and C is connected, it is genus^ the intecirand n the above definition is (-1) 9 4 The conjugacy classes Gi n (c1 • • • e n ) are called insertions and the total number of insertions will be called the length. We will drop the superscript C, when it is understood from context. We define PIG, the generating functions for these integrals, by 0-32(n3y14  F3.2.2 33  E«  2hti  x 1 ^W.7;3 11, 2 1 n2! li,3!  \23,2x73 2^3 I  ro-Fo2i-os>3  n3(14).533  2,^1) =  ' 2^3  n2! n3 1 x,752.3 zr23 2 23:73 nq £3 n 1 ! n9l n3! n4!  Our use of Fs, is auxiliary to our computationsFz, we do not determine it completely. For concreteness, we write out the formula in Theorem 5.1.1 explicitly for the two cases of A. and 12 x 12. As before, we define the series h(u) by hm(u) 9— tan (— 9 By a theorem of a^and Pandharipande [75], h(u) is the generating series 15 for 19-Hurwitz-Hodge integrals, namely the integral of —A g A g _i over the hyperelliptic locus Hz, c M g . Proposition 5.2.1. The generating function for 12 x 12-Hurwitz Hodge  integrals is given by the formula  1 J —h(ti^ + —111:c2)^— h(it3 3).  2  '  Note that our series^  FabewPandharipande [75).  is equal  ^  w het^  left  ^  Chapter 5. Hu "t ^integrals and the CRC -  -  Proposition 5.2.2.^e2ri/3. The , generating function,for 1 4 -Hurwitz  Hodge integrals is given by the formula FA4  at^) +  ^  5R  -  2^6  1  h  a)  4 –h (:v3) 2  5.2.1 Outline of the proof We prove Theorem 5.1.1 by first computing F, ,x2, 2 to prove Proposition 5.2.1 and then by computing Fit, to prove Proposition 5.2.2. For each of Fz 2x 27, 2 and Ft, we first prove that F is uniquely determined by the WDVV equations along with certain specializations. We then prove that our formulas for F satisfy the WDVV equations and specialize correctly. The required specialization for F  2 X ; 2 uses the Faber-Pandharipande computation of F7 2 The required specializations for F,4 4 uses the previously proven formula for Fz. x z, as well as certain generating series for 84-Hurwitz-Hodge integrals. These are in turn determined by a WDVV argument in §5.6 use the Z3Hurwitz-Hodge integrals computed hi [701. .  5.3 The WDVV equations (1;  satisfy he following version of the WDVV equations. These equations are the primary tools of is paper.  The C-Hurwitz-Hodge integrals (e, • e n )  Theorem 5.3.1. For^any n-tuple^,c„) of conjugacy classes of G and  any subset I C^,^of eardinality III, let ci denote the corresponding III-tuple of conjugacy classes and let I' he the complement of T. For g let (g) denote the corresponding conjugacy class and let z(g) be the order of the centralizer of g. Then (el  .  • ca^a21a3a^  v2a4 110  Chapter 5. Horwitz Ifodgeintegrals and the CRC where (e l^e, ( „ajlako„)) G is given by: z(g)(cjaia j (g)) a (0  1) k  \0  torG From this Theorem one equations:  PDE version of the WDVV  ,  Corollary 5.3.2. Let Fc be the !lodge integrals. Let  function for the  - Hurwitz-  0:3 FG  and let gij  1  — 6[4,  where ai is the order of the centralizer of the ith conjugacy class and if (g) is the Ph conjugacy class then (g -1 ) is the ith conjugacy class. Then the following expression is symmetric in {i,j, it, rn}:  grifintalGi k,1  The constant term in the above expression corresponds to tenns contain-  ing insertion of the trivial conjugacy class. These terms occur separately from the derivative terms since our variables only correspond to non-trivial conjugacy classes. The proof Theorem 5.3.1 is substantially no different than the proof of the WDVV equations in orbifold Gromov-Witten theory given in [66, §6.2). The only difference is that we are integrating the total Chem class of the dual Hodge bundle and so we need to check that the Hodge bundle behaves well on the boundary. Indeed, the Hodge bundle restricted to the boundary component where two domain curves are glued along a marked point is equal n K-theory (up to a trivial :factor) to the sum of the Hodge bundles of each factor.  Remark 5.3.3. We may assume that the series PIG given in Theorem 5.1.1 satisfies the WDVV equations, before we actually prove that it is the generating function for the C-Hurwitz-Hodge integrals. The formula in Theorem 5.1.1 for Fc was obtained from Fy , the Gromov-Witten potential for the Calabi-Yau threefold Y = G-Hilb(C 3 ) by a linear change of variables and a specialization of the quantum parameters (see [681). Since the change  Chapter 5. Hurwitz-Hodge integrals and the CRC of variables transforms the Poincaró pairing on Y to the pairing g, J defined above, it transforms the WDVV equations for Ey into the WDVV equaons Fe. Thus the predicted formula for PG automatically satisfies the WDVV equations. This is a feature common to all predictions obtained via the Crepant Resolution Conjecture.  5.4 Computing  FL 2 ,<E,  In this section, we fix ^= Z2 x Z2. For the integral (e l • e n ) to be nonzero, we must have the monodromy condition satisfied: ^ product of the insertions must be trivial. This is equivalent to  (c,"^= o  unless a l^n3 mod 2.^(5.4.1)  Consequently, the only non-trivial integrals of length three are ((i (2(3) = (10 = (2(2) = (30) = We also note the integrals ((V' (.2"(") are symmetric under pernu ations of (2, (2)  (c.< 2-2((`r) are uniquely deter^by the length three integrals, the integrals (GT and the ^0718.  Lemma 5.4.1. The integrals  PROOF: We proceed by induction on length (total number of insertions). The length three integrals start the induction and so we fix n > 4 and we assume that all integrals of length less than a are known. We introduce the notation Length (< a) to stand for any combination of integrals of length less than a. Fix (h i , k2, k3) with kt + k2 + k3 = n — 3  and consider the WDVV relation  Kc ikl (2k2  C^(3'i 16(2  ^  C1 ()2 ( 3 (.(1.(21(1 ( 2) -  Expanding out eachinto a sum of products of integrals and applying the monodromy condition (5.11), we find that there is only one non-zero term of length a. This elds: Length(< n).  112  Chapter 5. Hurwitz^ integrals and the CRC Since the integrals (C) are known by hypothesis, the only unknowns of length n are of the form  K91  r sr s7"r!) where i^/ and a -4- b= rt.  Now consider the following^  elation with k 1 + k 2^— 3  ((i"^(C- 1(1 IC2(3))^  Kt  0.2 (C1(2KIC3)  Expanding out we obtain ( 511 -1 3 (- 2k2 )^(chi +l Solving for  ^  '2  I 2) + Kei  + Length(< n).  using the previous equation to write ((tit 1(-k2a)  in terms of Length(< n) and setting k 1 = a — 1 and k2 = b — 2 we get 06) ((1+2(2-2  ^  Length(< 71)  for any a > 1^2 with a b = n. By the monodromy condition (5.1.1), we have that a and b must both be even, so we can use the above equation to inductively solve for all ((ND in terms of (C 1 ) and integrals of length less than n and the lemma is proved. To prove Proposition 5.2,1, we now show that the series in Proposition 5.2.1 is the unique solution to the WDVV relations which has the correct cubic terms and the correct specialization F4, (x, 0, 0). Up to symmetry, there are two distinct WDVV relations for the Z2 x Z2Hurwitz-Hodge integrals. In generating function form (Corollary 5.3.2), the relations are 1,22  1^[22^[123^11  -11  + F1 , . 2 F9 22 + F113 E322 +  16  and Fl 21 /4'133 F122F233 F123 F333 = 1'131F123 + Fl 32 F223 + F133 F323.  It is a straight forward but tedious exercise in trigonometry to prove that the series given in Proposition 5.2.1 satisfies the above WDVV equations (see Remark 5.3.3 for a conceptual proof). To finish the proof of Proposition 5.2.1, it remains to check that the formula for F given in Proposition 5.2.1 has the correct specializations, namely that (0, 0 , 0 )^(C-i(j(k)  Chapter 5. Hurwitz Hodgeint grals and the CRC  0,0) =  The first is easy to check and the second is equ^t o the following: Lemma 5.4.2. Let P(t h x2, 2 3 ) be the series given in Proposition 5.2.1. Then  u r• t 3  CC  F 1 1 1 Fu  —  0, 0) =  g  • n - 3)!  PROOF: We compute Fl u (u, 0, 0) to get 1^ ^^1 u^ ) - (tan^ - - tan (--7r,-tanu - ) = - tan(--2 8 4 4 4 2 Since for S (C1 ) all the monodromies are equal, the universal cover 2143(13G) is disconnected and is the union of two copies of the universal double cover over the hyperelliptic locus H g where 2g+2 n. Consequently, the dual Hodge bundle of C is two copies of the dual Hodge bundle on the hyperelliptic locus. Denoting both by F. V we can then write: 00^ ,-3^x 'ti n ' -3  _,_^ E m5 ^ en - 3)!^- /! iir.x. ( 3  0=3^ n=3  -  9  0^2g---1  U  -s-1 (2g - I tr  c,(Ev)  :(E" )c(lE y )  29-1  (2g - 1)! ri-19,  Y=1^  =  1  —  2  tan^. 2  The last equality follows from the computation of Faber-Pandharipande [75] and the lemma is proved.^ ^ This completes the proof of Proposition 5.2.1.  5.5 Computing PA, In this section, we compute the l t -llurwitz-Hodge Lite  114  Chapter 5. Hurwitz Ilodge integrals and the CRC using the WD\TV equations along with certain 8 4 -Hurwitz-Hodge integrals that will be computed in §5.6. The main technical result of this section is the following.  Proposition 5.5.1. The generating function F1 4 (:1:1, x2, x2) for the Atflurwilz-Hodge integrals is uniquely determined by the ITT ITV equations, the cubic coefficients of FA, , and the specializations FA, (x, x 0) and FA :1 (0, 0, tr). ,  We will prove this proposition in 0.5,1. The specializations that appear n th e above proposition can be expressed in terms of 8 4 -Hurwitz-Hodge integrals and Z2  x 7,2-Hurwitz-Hodge integrals by the following lemma.  Lemma 5.5.2. The following equalities hold:  31*'A,, (0, 0, x) = l2x7/52( 1' x, 21Ps, (0, x, 0, 0) = F) t 4 (x, x, 0). PROOF: The integrals that appear as coefficients of the specialization FA., (0, 0, x) are (C) Al . These correspond to A4 covers whose monodromy around every branched point is in the conjugacy class of disjoint pairs of two cycles. The structure group of such a cover reduces to Z2 x Z2 C A4 and so the integral is given as a sum of 7Z2 x Z2 integrals as follows:  3 (C „ ) a ' = rt:^r  1  CT I (.17 2  CZ 3 2 X Y.; 2  The muitinomial coefficient takes into account all the possible 11)k:es of the distribution of the monodromy among the n marked points. The factor of 3 occurs because the degree of the map  Air o ,(sz 2 x Z 2 )^M o ,,,(B is 3. 13y a similar argument, we derive  2(  n) Tin) .4 al  '  The lemma follows easily.  Corollary 5.5.3. The validity of Props mulct for FA,, follows feona Proposition 5. sition 5.6.1.  2, ea h -it  115  Chapter 5. Hurwitz Ilodge irmtcgrals and the CRC  Pilaw?: By Proposition 5.5.1 we only ed to show that the explicit formula for FA, (xi ,19,13) given in aif^5.2.2 ,  . satisfies the WDVV equations, 2. has the correct cubic terms, and 3. has the correct specializations 1. 17,(7, x, 0) and FA ,(0 0,4 )  The fact that the predicted formula satisfies the WDVV equations is once again a tedious but straightforward exercise in trigonometry, or for a more conceptual proof see Remark 5.3.3. The cubic terms correspond to the three point .f ^wHodge integrals. These are simply counts of As covers which can be evaluated using group theory and TQFT methods [74, section 4). The non-zero values are given by, 4^1 (5.5.1) (f/T12() = 1 ,^( }^K c'D^7 3 ,^( C 3 ) = 2. It is easy to check that cubic terms of the predicted formula for FT, agrees with the above values. Finally, in light of Lemma 5.5.2, we must check that when we specialize the predicted formula for EA,(x 1 ,19,13) to FA, (0, (), X) and Ft, (x, T, 0) we get 1 wff.Tx, it) and Fs I (0, x,0, 0) which are determined by Proposition 5.2.1 and Proposition 5.6.1 respectively. With the use of the following trigonometric identities: 27r )^97r 1^ -h(3u)^h(tt) + h^+ ^h (7t -^,  3  -11(2n) = h (it +  Tr^7 + h (li - 2  2  )  this is a straightforward check.^  (5.5.2)  , ^  5.5.1 The WDVV relations for A i -Hurwitz-Hodge integrals In this subsection we give the proof of Proposition 5.5.1. As before, we use the notation Length(< ft)  y combination of integrals of length^ n. rill use induction on the length to prove that the integrals are deter, WDVV relations from the^  als (which start  116  ^  Chapter .5. Hurwitz-Hodge integrals and the CRC the induction) and the integrals (or combinations of integrals) which occur as the )efficients of the specializations Fi„ 0, 0, x) and FA., (5, x, 0). The WDVV relations we need are given in the following: Lemma 5.5.4. Let it^a l + a.> b + 3. We have the following vlatitons among the 114-1Thrwitz-Rodge integrals: z, 4(4 1+2 ot C b-}-1 ) = 41(4. ' aP -h ( 11^Length(< a), ^2 , c-b-21)^4( a illi 1 a ,202 ( 64 2)^ 4(ar Length(< a), ^4(01“+'  a 2a24-1 (b+^tl(aV rt2 2 .7, 11  iv) 4K u p  3 42  (I) + 4(41 or  ) + Length(<  3 (I) = 8( 01, ±1  a a2 1. 1 (1+1)  Length(< a).  PROOF: We prove the above relations using the following WDVV rela-  tions which are expressed using using/the the notation of Theorem 5.3.1. i) (4 1 01 2 ( b (al ( ai^=  4 2 ( I (al at I (: 0) ,  ii) ( 01 1^(b (a2 (1 0- 2^= (4' 0t ( I) (a2 0. 21(0), (aig (4 2 co (10,2 ^c o ( al a2^0) , iii) c)} (al  iv) ( 0 11 0. 2a2 (I (at al a2 a2)) = ( ft  (7 22  )  (I) (a1 0. 2 al a m .  After expanding i) for a ) r a2^> 0, the resulting equation is  E  7 a1\  'al  \a2 2  7  L\  (,a(ii ^a2  ( 111+2 ( (ra( ilf^022(6 ( 171+2 ) 4)  bn p 3  3( a ai  Cha.^itz- Hodge integrals^  he CRC  Ter a`T. +^((!2 + 4 = a2, f. I b' //' = b. the integrals of length a in the above expression are multiplied1), ( a integrals of length 3 which are given ^ conta n only terms of length less than . Substituting^ es of the we obtain the relation il. The proof of^ orbs ii), iii), and iv) is similar. Remark 5.5.5. One can m easily by monodromy considerations that where^  (a)'  0  12 e,i  is nonzero^ al E., a2 (mod 3). Vote also that the above integral is symmetric in a l and a2 due to the fact that A 4 has a nontrivial outer automorphism which exchanges (-x i and a2. We will now use the relations i)-iv) in Lemma 5.5.4 to show that all the A t integrals can be inductively recovered from the length three integrals, the integrals {4) A4 (which are the coefficients of Ft, (0, 0, x)), and the integrals (a") S1 (which by virtue of Lemma 5.5.2 are the coefficients of FA , (a; x, 0)). The following relations are direct consequences of relations i) and iii) in Lemma 5.5.4: ^ 0 177 ( b)^(4-2 0. 2 (.7b ■ ) + Length(< k + b > 0,^> 2, (  (a)'  .  ^( y = Ka),  a  '2(t1•2) + Length (<^a2^a2, b > 0.  The following lemma is proven readily by a direct repeated application of two relations above: Lemma 5.5.6. Assume that b > 0. We have the following relation:  (a)' af; 2 (" C ) = ((a 11- " 2-h ) + Length (<^a2 - h b). LI  In the next lemma we deal with the Lemma 5.5.7. Let a = al - a a2. All the integrals of the form (a)' 42 )  can be written in terms of (<35 51 , length a integrals having at least one ( insertion, and Length(< n). PROOF: Let 0 < k < 2 be so that a E-s7 k (mod^One can see easily  that (see Remark 5.5.5): al  -4 -  a2^77}  k-^n k  a2)  ,  118  ^  Chapter 5. iferwitz1iodge integrals and the CRC elements  This is a set of^  /^— 2,1(7)/3, For simplicity we write the elements of this set^ le same order form left to Note that by symmetry of the integrals (see Rentarlc 5.5 ^ x' (5.5.3) ^ Applying^ t na 5.5.11 times, we get the^ 'ing set of relations : k 2  ar^Length(  xi 1 x2 = 2(cr(` —k15 (4 "0 Length(<  =2 Now we consider^  ^  -2  0 +  Length(<^(5.5.4)  cases:  i) a is odd: We see that 1 (the number of relations in (5.5.4)) is odd as well, and then (5.5.3) and (5.5.4) boil down into the following set of independent equations among la's:  ^ze^= 2(0) 1'2-2 :"^ 4 Length(< + ix2^k^k -1^ f Length(< a)  2(pr li i-k- 2^4+1+3a0  Length(<  where p = (1 1 )72. From these equations we get  =  ^k- 2-3p a2k^3p 0 0 + Length(<  and the Le mr^n this case.  e th at one of e relations in .5.3) is  Chapter 5. Hurwitz-Hodge integrals and e CRC  edundant. However, we the coefficients of ^ are known by hypothesis, we get the following ^  -k  xo  k—3  7rt\  )(T:  0) 2Fs 4 (0, x, 0 1 relation:  rt -+^(5.5.5)  Adding (5.5.5) to (5.5.1), we get a system of 1+7 Ninth) unknowns (we call xi's unknowns). One can see that the of the matrix of coefficients is given by (note that n is eve n ,n —  1.\\ \ 71 -  k—3  5)  )  .  which is nonzero by [70, Lemma A.6]. This means that we can express xi's in terms of the right hand side of the system of l + 1 equations, and the Lemma is proven in this case. We may now pro Propos  1011 5.5.1  hi the following equivalent fo  Proposition 5.5.8. The A4^  y deterintegrals  m iraed by the WDVTI equations , he length^  ((+n)A 1 and (a rk ) s t.  PROOF: We use induction on the length T1 of the integrals. The length three integrals are known by hypothesis. Let (aV a 2a2 ( 1 ) 114 be an arbitrary integral of length it > 3. By Lemmas 5.5.6 and 5.5.7, we can write this integral in terms of (C") A4 , (a") 54 , and Length(< n). Both ((") A1 and (e) .94 are known by the assumption, and Length(< n) is also known by the induction hypothesis. Therefore (ar 52'" ( 6 ) -41 is determined, and the Proposition is proven. ^  5.6 Computing 5 1 -Hurwitz-Hodge integrals In this ection we prove the following proposition, which is needed to complete the proof of Proposition 5.2.2. Proposition 5.6.1. Let F3 4 (1; 1 ,1790; 3 ,1 4 ) he the getowatiug function for the Sp fhtrwitz-Hody e integrals. Then  F ^—  ^8  1  h  2x^27r)  \ V3^3  2h  x^rr  120  ^  Chapter 5. Thinv Hedge integrals and^  CRC  This follows^ely from the identity in equations (5.5.2) and the following: Theorem 5.6.2.  F5 4 (0, )1, 0, v)= -K(2u^K(—q ,  2  where K(u,t,)^h (  ^v^\ 2^6 /  / u^v h ^ 2  +2h I- 2h  u^)  v2  z  2 iv 370 3^ 2^2 To prove Theorem 5.6.2, the basic strategy is once again to use WDVV to inductively determine the integrals. We will not determine all the S 1° Hurwitz-Hodge integrals but we will have to determine a certain set of integrals that have a small number of 7 and p insertions. We use the following generating functions. For convenience, we define unstable integrals (those have fewer than three insertions) to be zero. • TB ),^= Bic (0, U, 0, v) =  (  b  U" )  a! b!  • X ev) = (L  00  •  17b(U)^(,(In ( b )  rt=0  unfb  ^ b^3)!  '  • B(v )  121  Chapter 5. Hurwitz^integrals and the CRC  The length three integrals can be valuated using group theory and FT methods [74, section ]. The ) zero values are given by :  (a 2 () = (r 2 a) =  (Tpa)  = (p 2 a) = 1,  ( lc3 ) = ( 0 20 = (7.2() = ( r 21) = ( p 21) ^(  4^1  73 )  =^(7P(}^(<721) = 3 .(4, 2 U =  We now determine the series X0, X 1 , and B. Lemma 5.6.3. The series X00.0, Xi(a), and 13(a) are  given  by  1^u^) 1^u ^ = 6—h -3,7-- —^+ -,-,h( + h (u), ) 4 \^L Xj(u) =0,  X°(u)  —a B(a)=-- tr^ /73V12 PROOF: The proof ( si 'on 5.2.1 and the forum].  Ir  lv from  P ror  xo ( u)^—^, (11 , u, The proof of the above formula is almost identical to the proof of the first formula in Lemma 5.5.2; the only difference being that the 3 (the index of Z2 X Z2 in A4) is replaced with 6 (the index of Z2 X Z2 in 84). The Lemma's second formula is a consequence of (an =  which follows from monodromy considerations (see Remark 5.5.5). To prove the Lemma's third formula, we need to show that the generating function B(a) has the same coefficients, up to an alternating sign, as the generating function with the same name computed in [70, Proposition A.31. The coefficients of the generating function in [70] are the A 0 integrals over the Hurwitz locus of curves admitting a degree three cover of 1P 1 with 2 ordinary ramification points and g f 1 double ramifications. The identification of this integral with the integral (-1) 9 (7-2 a 9 + 1 ) 5 is because the only chance for a degree four cover to contribute to (7 2 0-3 I 1 ) 54 is if the cover is disconnected. Indeed, if it is not disconnected, then the genus is g 1 and hence the gth - -  192  Chapter 5. Hurwitz Ilodge integrals and the CRC Chem class of V is zero. The sign is because we work with the dual Hodge bundle instead of the Hodge bundle. To determine T. v), it clearly suffices to determine X„(/) for all a. We will do this using an induction on a. The following lemma provides the basic relations that are needed in the induction. Lemma 5.6.4. The series T(1,v), X„va), Nu),^Y9(u), and .5e11 W11 the [01101fling relations. .  (i) 311722ut , + 811;122 , v  (ii) 3 (X2) 2 — 8 E2X8 1 1 88- 0, (iii) (6Y2B + 4Y-3 + 1)(3Y0B + 2Y; — 4B 2 + 2) = 2(3Y1 B  2 Y2  — B) 2 .  (iv) (6.2C2 — 8 iX1r) rat2 — 3aA721 X„. 4„^G (X0,^. ,^4 .1), where in item (iv) a > 0, and G(X0 , X, ,^) is a function of the ser Xe(u) for0<k <a+ 1. PROOF: These relations are all consequences of the WDVV equa o For fixed a and b, consider the WDVAT relation  (e( b (a(10- C"  ))  —  (a"( b (0- 0- 10()  t is given by 0 = So „50,/, +  2(-b, 1) K a a28-2(1/2  8 K a o r i 1(/,  f 3(by ) K o,a2^2^8  - -  ) ( n a2  K a u' 12(bi  ^  bz  b.-, 13)  Multiplying the above equation by 74 and then sumn  yield s the PDE in (i). Fixing a = 0, multiplying by '2 r . b yields (ji). To prove v), we fix a > 0, we multiply the above e u 1 we move to the right a side of the equation all the terms con Tin grals with e^a + 2 insertions of a, and then we sum over b.  Chapter 5. Hurwitz-Hodge a tegr sand the CRC  The relation (iii) follows easily from the 61f2 B^411-3 -i- 1 = 8C2,  3Y1 B 2Y2 = 4CD ± B, 3Yo - I - 2Y1 I- 2 = 4B 2 4D 2 winch  ^  derived in a fashionsimilar to the above from th e  ^  7 relations:  (a"(TTECO) = (a" f,TCHT(:)), (a"(Tria())^Ka"(Tair()), (caerrlau)) = aa(raIra)).  In the derivation , we use the following crucial fact: ( 7.2 07 a)  0 if a > 0,  = 0. This follows from the fact that a four oh, cover of ]P with 3 branched points of monodromies T, 7, and a branch points of monodromy a is connected and of genus a 1. Consequently c„(132/) = 0, namely the ath Chern class of the dual Ilodge bundle vanishes, and so the corresponding Hurwitz-Hodge integral is zero. 0  Proposition 5.6.5.  The functions X„(n) are uniquely determined for all a by the series B(u), X 0 (u) „ X1(1), the length three integrals, and the relations (ii), (iii), and (iv) in Lemma 5.64  Pnoor: The relation (i i) of Lemma 5,6.4 is a quadratic equation for (u) whose solution is fixed by the the condition  Xf2 (0) = (a 2 () = 1. Since the constant term of X2 (n) is an unstable integral (an convention), X10.0 is then uniquely determined. We now proceed to determine X" (u) by induction on a. We assume that Xk (u) is known for all k < a +2 and we need to show that we can determine X„ 2 (u), Since X0, X i , and X2 are known, we may assume that a > 0. Then relation (iv) in Lemma 5.6.4 is a first order ODE for Xi a+ 2(u). Since the coefficient of X a1 2 in the ODE is an invertible series, the ODE has a solution which is uniquely determined by specifying X„4 2(0). NOW X a _ i_2(0) is equal to the coefficient of ue -1 in the se so we need to determine this coefficient. Since the series X 0 ,.  124  Chapter 5. Hurwitz-Hbdge ^-!orals and the CRC 0 known by the induction hypothesis. we know the coefficients of Ys(u) up to the uldb -2 term. By examining udj-I term in the relation (iii) from Lemma 5.6.4, we find that the only unknown is the coefficient of tej m of Yo which appears exactly once with non-zero coefficient. Hence we can uniquely solve for this coefficient which^ i s the initial condition which uniquely determines X„..1..9(u).^ Thus to complete the proof of Theorem 5.6.2, we must, show that the formula for T(u, v) in the theorem, yields series X.,(u) ^  1. which predict the correct length three integrals, 2. which agree with the formulas for X0 and X 1 given in Lemma 5.6.3, 3. and which are solutions to the relations (ii),^and (iv) of Lemma 5.6.4. The first two are straightforward checks. The compatibility of the solution with relation (iii) is also a straightforward check. The compatibility with relations (ii) and (iv) is equivalent to the formula for 7' satisfying the PDE (i). This compatibility can be checked directly (with Maple fbr example), but there is a more conceptual proof, along the lines of Remark 5.3.3 which we Outline below. The formula for T(u, v) can be derived from the formula for Fm(d: via the relation 7 (n, v)^Fs, (0, ?I, 0, v) 1  rn,^14  (a,  which is proved by the same method as the proof of Lemma 5.5.2. The fact that T(u, v) satisfies (i) can be seen to be a consequence of the fact that FA, satisfies the A4 WDVV equations, a fact which we proved in §5.5. Recall that our formula for FA, was derived via the Crepant Resolution Conjecture from the Gromov-Witten potential of the crepant resolution Y C3 /A4 given by the A4 Hilbert scheme. The derived formula for FA, thus automatically satisfies the WDVV equations since the Gromov-Witten potential of Y satisfies the WDVV equations and the Crepant Resolution Conjecture is compatible with the WDVV equations. Thus the fact, that T satisfies relations (i) in Lemma 5.6.4 is ultimately a consequence of the fact that the Gromov-Witten potential of Y (which was computed in [681) satisfies the WDVV equations. 1.71 This completes the proof of Theorem 5.6.2 and consequently it completes the proof of Proposition 5.2.2 and hence it completes the proof of our main result, Theorem 5.1.1.  125  Chapter 5. IThrwitii-11^ the CRC  5.7 The Relationship with orbifold Gromov-Witten theory Let G (which is Al or 71.2 x Z2) act on C3 by the representation^ ai from the embedding G C 80(3) C SU(3). Let X be the orbifold given the quotient: X = [C 3 /G]. The singular space X = IG underlying the orbifold^ d Calabi-Yau resolution Y—X  preferred  given by Nakamura's G-Hilbert scheme [67]. The Crepant Resolution Conjecture [69] states that the Gromov-Witten potentials FY(y0,1/1^1/3,^; (121 9 3) and^(to, I :1221 ;1:3 .  are equal after a linear change of variables and a specialization of the quantum parameters qi to certain roots of unity. In [68], we compute all the Gromov-Witten invariants for the G-Hilbert scheme resolution of C. 3 /G for all G C 80(3). There we use the Crepant Resolution Conjecture to find a prediction for the orbifold Gromov-Witten potential of X. Indeed, we reduce the (genus zero) Crepant Resolution Conjecture for the pair (Y, X) to the following prediction for the of the Gromov-Witten potential of X. The orbifold cohomology . 11f,rf (X) has a canonical basis labelled by conjugacy classes of G. Consequently, the insertions for the orbifold GromovWitten invariants of X are conjugacy classes, and the genus zero GromovWitten invariants of X take the form (c 1 c rt ) d . In this section we prove: Proposition 5.7.1. The genus zero  orbifold G l07110(27- Witten invariants t o of X are given by the G-Hurwitz-Hodge integrals, namely  tie = (el " ' cli) G for any^ uple of non-trivial conjugacy classes. Consequently, the (nonclassical part of the) genus zero Grornov-Witten potential of X is equal to the generating function of the G-Hurillitzillodge in tegrals: Fx(ai l , To £3)^FG (XI , X2, £3). :lv speaking, some of the orbifold Gromov-IVitten invariants of X are not well the^ twisted stable rnaps is non-compact. invariants^.t1^ the C action on  126  Chapter 5. Hurwitz-Hodge integrals  ^  CRC  Puoot By definition, the, orbifold invariants ) U U^(CO  Using virtual local the above integral ii which is ilgt o , k (BG):  'pect to the C action on X we c an integral over the C fixed locus of Mo ,k (A)  and N' i r is the virtual normal bundle of Ms(BG) in 110,h,(X) regarded as an element in K-theory. So to prove the Proposition, we need to show that • ck)  e(—Arvir)=  C9 )^(3.7.1)  Let V BG  be the bundle given by the three dimensional representation of G induced from the embedding G C 50(3). By the standard argument in GromovWitten theory, the virtual normal bundle is given by Nvir^R•7r,  ry  where a C --> M 0 k (BC) is the universal curve and^C^BC` is the universal map. Let H C G be the subgroup Z3 C A4 or {O} C Z2 x Z2 respectively. Then the action of G on the cosec space G/H is the usual permutation action of G on the set of four elements. Consequently, we can construct the universal degree 4 cover p:e C by pulling back the map ,  : BH BC,  127  :grain and the CRC  Chapter 5 That is , we have the^  ving  1311  C  ^  M (B G We now^  BYG  )  K-theory:  Irtr,.(9,7 = /?.*^En * f 0 B  * (.1 ti^nil) r 71,1 ' ( C Cl) =^+ Tr.0c. The equality on the top line uses the fact that p is finite. The equality on the second line uses the fact that the commutative square in the diagram is Cartesian. The equality on the third line uses the fact that the G representation induced by the trivial representation is V plus the trivial representation. The equality on the fourth line uses the fact that 71 : C om (BG) is genus 0. Finally, we apply the total Chern class to both sides of the above equaly and integrate over Ms(/3G). The C.' equivariant Euler class is the same as the total Chern class with the appropriate power of the equivariant parameter appearing in each degree. Since the virtual complex dimension of .11.10,k(X) is 0, and all the insertions et have (non-shifted) degree zero, the integral is degree 0 in 1 the equivariant parameter and is hence independent of 1. Equation (5/.1) is proved. ^  Bibliography [66] Dan Abramovich, Torn Graber, and Angelo Vistoli. Gromov-Witten theory of Deligne-Mumford stacks. arXiv:math.AG /0603151. [67] Tom. Bridgeland, Alastair King, and Miles Reid. The McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc., 14(4535-554 (electronic), 2001. [68] Jim Bryan and Amin Gholampour. The Quantum McKay correspondence for polyhedral singularities. In preparation. [69] ,Jim Bryan and Tom Graber. The crepant resolution conjecture. To appear in Algebraic Geometry ^ Seattle 2005 Proceedings, arXiv: mat h .AG /0610129. [70] Jim Bryan, Tom Craber, and Rahul Pandharipande. The orbifold quantum cohomology of C 2 /Z3 and Hurwitz Hodge integrals. arXiv:math.AG/0510335, to appear in Journal of Alg. Geom. [71] Renzo Cavalieri. Generating Functions for Hurwitz-Hodge Integrals. matILAG/math/0608590. [721 Tom Coates, Alessi() Corti, Hiroshi Iritani, and ^sian-Hua Tseng. Computing Genus-Zero Twisted Gromov-Witter Invariants. arXiv:thath.AG/0702234. [731 Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng. The Crepant Resolution Conjecture for Type A Surface Singularities. arXiv:0704.2034v1 [math.AG]. [741 Robbert Dijkgraaf. Mirror symmetry and elliptic curves. In The moduli space of curves (Tenet Island, 1994), volume 129 of Prvgt 119 163. Birkhiiuser Boston, Boston, MA, 1995. C. Faber and R. Pandharipande. Log ^'c series and Hodge mtegrals ^ gical ring. Al z. T. 48:215 252, 2000. Wit 12 9  Bibliography appendix by Don Zagier, Dedicated to William Fulton on the occasion of his 60th birthday. [76] G. Gonzalez-Sprinberg an I J.-L. Verdier. Construction gOometrique de la correspondance de McKay. Ann. Sci. Ecole Norm. Sup. lip, 16(3 ):409 449 (1984), 1983. .  ,  [77 John McKay. Graphs, singularities, and finite groups. n The Santa  Cruz Conference on Finite Groups (Un iv. Cruz, Calif., 1979), volume 37 of Pine. Sympos. Pure Ma^ pa ges 183 186. -  Amer. Math. Soc., Providence, R.I., 1980. [78] David Mumford. Towards an enumerative geometry of the moduli space of curves. In Arithmetic and geometry, Vol. II, pages 271-328. Birkhiinser Boston, Boston, Mass., 1983. [79] Miles Reid. La conespondance de McKay. Asterisque, (276):53-72, 2002. Seminaire Bourbaki, Vol. 1999/2000.  Chapter 6  Conclusion 6.1 Introduction The first chapter of this thesis ^introduction^ GromovWitten theory and McKay correspondence. It reviews the lite rature covers some of the basic notions assumed known in the later chapters. By reading the first chapter, a reader with the background of algebraic geometry obtains sufficient information for following the next chapters with no difficulties, and is guided to the literature for finding about more details. Chapters 2-5 are written in the publishing format, and hence can be read independently. On the other hand, they all follow the thematic of this thesis which is the computation of the Gromov - Witten invariants. Chapter 2, is devoted to solve for a part of Gromov-Witten theory of 7 2 -bundles over curves which the latter define a class of non-toric compact threefolds. Chapters 3 and 4 have a more evident coherence by having applications in the Crepant Resolution Conjecture and providing a quantum version of the McKay correspondence in dimensions two and three. Chapter 5 provides solution for a certain Hurwitz-Hodge integrals on the moduli space of curves and as application proves some cases of the prediction given in Chapter 4 based on the Crepant Resolution Conjecture.  6.2 Importance of the results and the future research Chapter 2 gives a complete answer for the section class Cromov-Witten invariants of 7 2 -bundles over curves. It provides an example of passing from local to global Gromov-Witten theory as it uses the results of local Gromov-Witten theory of curves of [831 as one of the main ingredients. The partition functions of the Gromov-Witten invariants of these spaces, give rise to a semi-simple 1+1-dimensional topological quantum field theory in the same way as the partition functions of local Gromov-Witten invariants studied in [83, 841. It also uses localization and degeneration techniques 1 31  Chapter 6. Conclusi most powerful computational techniques in the subject. The results of Chapter 2 are also interesting as they can be used for verifying the Gopakumar-Vafa-Pandbaripande conjecture, and studying the correspondence of Gromov-Witten theory and Donaldson-Thomas theory. Gopakumar-Vafa invariants were introduced by two string theorist R. Gopakumar and C. Vafa (see [871) in 1998 as BPS state counts in a study of Type IIA string theory on a Calabi-Yau threefold via M-theory. Gopakumar and Vafa also proposed a mathematical construction of the invariants using moduli space of sheaves. These invariants are conjecturally closely related to Gromov-Witten invariants. The Gopakumar-Vafa conjecture, predicts a strong integrality and vanishing condition on the Gromov-Witten invariants. This was generalized for an arbitrary threefold by R. Pandharipande (see [91]). This conjecture is one of the most famous conjectures in GromovWitten theory. Donaldson-Thomas invariants were defined for Calabi-Yau threefolds by S. Donaldson and R. Thomas in 1998 via the moduli space of deal sheaves. In 2004, D. Maulik, N. Nekrasov, A. Okounkov, R. Pandharipande generalized Donaldson-Thomas invariants for an arbitrary threefold, and they presented a precise mathematical conjecture relating the GromovWitten and Donaldson-Thomas theories of a threefold (see [89, 90D. They conjectured that these two theories are related via the change of variables = where u is the parameter in Gromov-Witten theory and q is the holomorphic Euler characteristic parameter in Donaldson-Thomas theory. Gopakumar-Vafa conjecture can be regarded as a special case of the GromovWitten/Donaldson-Thomas correspondence. [86] provides a nontrivial evidence for such a correspondence. It mainly uses the result of Chapter 2 for the Gromov-Witten theory side of the correspondence. Chapters 3-5 are mainly motivated by the Crepant Resolution Conjecture and McKay correspondence, which were reviewed briefly in Chapter 1. Table 6.1 contains a summary of relationships of materials covered in Chapters 3-5 and demonstrates the future plans. In Table 6.1, CI is a finite subgroup of SO(3) (respectively SU(2)), and V is the corresponding complex 3-dimensional (respectively 2-dimensional) representation of Cr. In each classical case (rows 2 and 3), the geometric problem has been solved by giving a purely algebraic description, and so the connection between is a solid arrow. The generalized McKay correspondence (row 2) as an equivalence of derived categories has been proven in [80]. The A-model Crepant Resolution Conjecture (given row 4) has been verified for the case X = [C 2^82, 85j). C 132  Chapter 6. Conelusw conjecture 2 , >)], and X = [C3 /A 4 ]. The proof of the [0 AZ2 CD 2 conjecture wolves solving for new Hurewitz-Hodge integrals by exploiting the interplay be groups Z2, Z3, 71,2 e Z9, A4 and S4, and the result of Chapter 4 for the Gromov-Witten theory of the (trepan) resolutions of the orbifolds. Chapters 3, 4 gives a quantum version of McKay Correspondence. They provide a complete description of the quantum cohornology of C-Hill)(17) in terms of the ADE root system associated to C, which sits in the bottom right; corner of the table above. Chapters 3, 4 also give prediction for the orbifbld Gromov-Witten invariants of X = [V/G] (for all V and (.7 as in table above). This is the first step in the proof of the Crepant Resolution Conjecture in these cases. In all the cases, the matrix of the linear isomorphism between the cohomology groups (described in 1) is obtained by a simple modification of the character table of the group G. One goal in the future is to prove these predictions for all the groups in Table 6.1, along the same lines that was followed in Chapter 5 for the cases Z2 2 and A4. Carrying this out would make the A-model QMC a solid arrow. The other goal is to make the B-model QMC a solid arrow. One way to achieve this may be by looking for an interpretation of the results men' d in the last paragraph in terms of derived categories. This will allow transition from the A-model to B-model. This transition will be in of Gopakumar-Vafa invariants which have been recently rigourously defined in genus 0 by Sheldon Katz (see [88]) in terms of the moduli space of sheaves along the lines of Donaldson-Thomas theory. One plan is to study the Gopakumar-Vafa invariants of C-Hilb(V) by directly understanding the sheaves involved in each case. Results of Chapters 3, 4 give a full prediction for the invariants based on the Gopakumar-Vafa conjecture. One of the outcome of this would be verifying the Gopakumar-Vafa conjecture for C-Hilb(V). The other outcome would be proposing Gopakumar-Vafa invariants and conjecture for [V/G] through the QMC. Another plan is to find a purely algebraic description for the B-model QMC as we did for its A-model counterpart.  ez  ^  PHYSICS  BY GEOMETRY ()repai, Orbilold/Siadarity^ to^ Resolution VA/^ Conjecture  G-Hilbert Scheme it G-Hilb( V)  ALGEBRA  Moduli Space Moduli space of^QMC 5.?? -›-^of Sheaves on (I-Sheaves on V^G V /DT G-Hilb( V) Repre Derived Category 13-Model Derived Category of CMC a sentation of Sheaves ott Classical G-Sheaves Ott V G-Hilb(V) Tlteory O Orbifold Groin} 1-Model Cohontology Collontology^'MC Classical Theory of G-Hilb(V) of V Quantum Orbifold Quantum Quant ADIS A-Model QMC Cohontology^<^ >^Colunnology Root, Quantum GVV of^1 1 /G1 of G-Hilb( V) Theory GVV• Gromov-Witten theory^CV: Copakuntar-Vaftt theory ^DT: Donaldson-Thu tuts theory C: Classical McKay Cotavitpondettce^QMC: Quantum McKay Corres p ondence B-Model Quantum  Table 6.1: Crepant Resolution Conjecture versus McKay correspondence  2  Bibliography [80] Tom Bridgeland, Alastair King, and Miles Reid. The McKay correspondence as an equival enc e of derived categories. J Ame r. Math. Soc., 14(4535-554 (electronic), 2001. [81] Jim Bryan, Tom Graber, and Rahul Pandharipande. The orbifold quantum cohomology of C 2 /Z3 and Hurwitz Hodge integrals. AG/0510335, to appear in Journal of Alg. Geom [82] Jim Bryan and Yunfeng Jiang. The Crepant Resolution Conjecture fbr the orbifold C 2 /Z 4 . In preparation. [83] Jim Bryan and Rahn] Pandharipande. The local Grongw-Witten theory of curves. To appear in J. A mer. Math. Soc. arXiv:math.AG/04:11037. [84] Jim Bryan and Rahul Pandharipande. Curves in Calabi-Yau 3-folds and Topological Quantum Field Theory. Duke Mathematical Journal, 126(2):369- 396, 2005. Preprint version: matkAG/0306316. [85] Torn Coates, Alessi() Corti, Hiroshi Iritani, and llsian-Hua Tseng.  The Crepant Resolution Conjecture for Type A Surface Singularities. arXiv:0704.2034v1 imath.AG1. [86] Amin Gholampour and Yinan Song. the GromovWitten/Donaldson-Thomas correspondence.;Ma th. Res. Lett., 13M:623-630, 2006. [87] Rajesh Gopakumar and Curium' Vafa. M-theory and topological s--II, 1998. Preprint, hep-th/9812127. [88] Sheldon Katz. Gromov-Witten, Gopakumar-Vida, and DonaldsonThomas invariants of Calabi-Yau threefolds. In Snowbird lectures on string geometry, volume 101 of Contemp. Math., pages 43 52. Amer. Math. Soc., Providence, RI, 2006.  135  Bibliography  [89 D. ivIaul1k, N. Nekrasov, A. Okounkov, and B. Pandharipande. Gromov-Witten theory and Donaldson-Thomas theory. I. Compos. Math 112(5):1263--1285, 2006. [901 D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande. Gromov-Witten theory and Donaldson-Thomas theory. II. Compos. tie., 142(5):1286-1304, 2006. Pandharipande. Three questions in Gromov-Witten theory. In PMgs of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 503--512, Beijing, 2002. Higher Ed. Press.  36  

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