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Fusion algebras and cohomology of toroidal orbifolds Duman, Ali Nabi 2010

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Fusion Algebras and Cohomology ofToroidal OrbifoldsbyAli Nabi DumanB.Sc., Bilkent University, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinThe Faculty of Graduate Studies(Mathematics)The University of British Columbia(Vancouver)April, 2010c© Ali Nabi Duman 2010AbstractIn this thesis we exhibit an explicit non-trivial example of the twisted fusionalgebra for a particular finite group. The product is defined for the groupG = (Z/2)3 via the pairing θg(φ)R(G) ⊗θh(φ) R(G) →θgh(φ) R(G) whereθ : H4(G,Z) → H3(G,Z)istheinversetransgressionmapandφisacarefullychosen cocycle class. We find the rank of the fusion algebra X(G) =summationtextg∈Gθg(φ)R(G) as well as the relation between its basis elements. We also givesome applications to topological gauge theories.We next show that the twisted fusion algebra of the group (Z/p)3 isisomorphic to the non-twisted fusion algebra of the extraspecial p-group oforder p3 and exponent p.The final point of my thesis is to explicitly compute the cohomologygroups H∗(X/G;Z) where X/G is a toroidal orbifold and G = Z/p for aprime number p. We compute the particular case where X is induced bythe ZG-module (IG)n, where IG is the augmentation ideal.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . v1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 A Twisted Fusion Algebra . . . . . . . . . . . . . . . . . . . . 62.1 Projective representations . . . . . . . . . . . . . . . . . . . . 82.2 The twisted fusion product for finite groups . . . . . . . . . . 102.3 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 2-cocycles in G with values in U(1) . . . . . . . . . . 142.3.2 The projective representations . . . . . . . . . . . . . 182.3.3 The relations. . . . . . . . . . . . . . . . . . . . . . . 262.4 Topological gauge theories . . . . . . . . . . . . . . . . . . . 293 The Fusion Algebra of an Extraspecial p-group . . . . . . . 323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Finite group modular data . . . . . . . . . . . . . . . . . . . 343.3 Twisted example G = (Z/p)3 . . . . . . . . . . . . . . . . . . 363.4 Modular data for extraspecial p-group . . . . . . . . . . . . . 39iiiTable of Contents4 Cohomology of Toroidal Orbifolds . . . . . . . . . . . . . . . 484.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 The case L = IGm . . . . . . . . . . . . . . . . . . . . . . . . 52Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59ivAcknowledgmentsI am thankful to the most gracious, most merciful God who has given to menumerous bounties. If I want to count these bounties I cannot manage tocomplete.I sincerely acknowledge my supervisor Alejandro Adem for his support,patience and guidence during my research. I am also grateful to my col-leagues Jose Cantarero, German Combariza and Jose Manuel Gomez.FinalIy I would like to thank my family for their tremendous support.vChapter 1IntroductionIn the past twenty years there has been a great influence of physical ideason mathematics. One such example is the subject of string theory whichincreased the interest on some topological objects such as orbifolds, a verynatural generalization of manifolds, and also on the invariants related tothese objects, for example, the Euler characteristic, homotopy groups, ho-mology groups and cohomology rings. The study of orbifolds in physicsbegan with the introduction of the conformal field theory model on singularspaces by Dixon-Harvey-Vafa-Witten in 1985.Orbifolds are spaces that locally look like a quotient of an open set of avector space by the action of a group such that the isotropy group at eachpoint is finite. Chen and Ruan [19] discovered a cohomology theory for theseinspired by their ideas in quantum cohomology and orbifold string theory.The interest in K-theory is also introduced from the consideration of a D-brane charge on a smooth manifold and the notion of discrete torsion onan orbifold by Vafa. K-theory relates to equivariant theories if the orbifoldsare the quotient of a smooth manifold M by a compact Lie group actingon M. Adem and Ruan [7] defined the twisted orbifold K-theory to studythe resulting Chern isomorphism. In [8], Adem, Ruan and Zhang give anassociative stringy product for the twisted orbifold K-theory of a compact,almost complex orbifold. This product is defined on the twisted K-theory1Chapter 1. Introductionof the inertia orbifold ∧X where the twisting gerbe τ is assumed to bein the image of the inverse transgression map H4(BX) −→ H3(B ∧ X).If the orbifold is X = ∧[∗/G] and G is a group then the resulting ringτKorb(∧[∗/G]) is called fusion algebra.One way to approach the fusion algebra is the finite group modular datawhich is mainly explored in [20]. In our context, when we mention the termmodular data one should understand two symmetric matrices associated to afinite group G. This modular data was originally introduced, in Lusztig’s de-termination of the irreducible characters of the finite groups of Lie type [31],[32]. To describe the unipotent characters, he considered the modular datafor some particular finite groups. The primary fields of the fusion algebraparametrize the unipotent characters associated to a given 2-sided cell in theWeyl group. Lusztig interprets this fusion algebra as the Grothendieck ringfor G-equivariant vector bundles; in other words, the equivariant K-theory.Themostphysicalapplicationofthismodulardataisin(2+1)-dimensionalquantum field theories where a continuous gauge group has been sponta-neously broken into a finite group [13]. Non-abelian anyons (i.e. particleswhose statistics are governed by the braid group rather than the symmetricgroup) arise as topological objects. The effective field theory describing thelong distance physics is governed by the quantum group of [23].A set of modular data (i.e. matrices S and T) may be obtained for anychoice of finite group G. Much information about a group can be recov-ered easily from its character table including whether it is abelian, simple,solvable, nilpotent, etc. For instance, G is simple if and only if for all ir-reducible χ negationslash≡ 1, χ(a) = χ(e) only for a = e. Thus it may be expectedthat finite group modular data, which probably includes the character ta-ble, should provide more information about the group, i.e. be sensitive to a2Chapter 1. Introductionlot of the group-theoretic properties of G.One way to generalize this data is to twist with a cocycle from the co-homology group. One can obtain topological (e.g. oriented knot) invariantsfrom this twisted data, as explained in [9]. These invariants are functionsof the knot group (i.e., the fundamental group of the complement of theknot). Although non-isomorphic knots can have the same invariant, theseinvariants can distinguish a knot from its inverse (i.e., the knot with oppo-site orientation), unlike the more familiar topological invariants arising fromaffine algebras.Another important example of orbifolds is obtained by group actions ontori induced by the integral representations of finite groups. These orbifoldsappear as examples of interest in physics. For this reason we are interestedin computing the cohomology groups H∗(X/G;Z) where X/G is a toroidalorbifold.One can give an integral presentation ϕ : G → GLn(Z) for finite groupG. In this way G acts linearly on Rn preserving the integral lattice Zn,thus inducing a G-action on the torus Xϕ = X := Rn/Zn. The quotientX → X/G naturally has the structure of an orbifold as a global quotient,and these kind of orbifolds are usually referred to as toroidal orbifolds. Thegoal of this chapter is to compute the cohomology groups H∗(X/G;Z) forthe particular case where G = Z/p for a prime number p.The quotients of the form X/G appear naturally in different contexts.For example, given a topological space Y, the m-th cyclic product of Y isdefined to be the quotientCPm(Y) := Y m/Z/m,3Chapter 1. Introductionwhere Z/m acts by cyclically permuting the product Y m. In the particularcase where the representation ϕ : G → GLn(Z) induces the ZG-module(ZG)n, the associated torus X is (S1)p)n, where G acts cyclically on each(S1)p and diagonally on the product ((S1)p)n. In this caseX/G = (((S1)p)n)/Z/p ∼= ((S1)n)p/Z/pwhere now Z/p acts cyclically on the ((S1)n)p. ThereforeX/G ∼= CPp((S1)n).The homology groups of quotient spaces of the form Xm/K, where K ⊂ Σm,have long been studied. In particular, in [38] Swan formulated a method forthe computation of cyclic products of topological spaces.In Chapter 2, the stringy product on twisted orbifold K-theory in [8]is reviewed, and an example for non-trivial twistings is calculated. For afinite group G this reduces to the twisted fusion algebra. Here we exhibitan explicit non-trivial example of the twisted fusion algebra for a particularfinite group. The product is defined for the group G = (Z/2)3 via the pairingθg(φ)R(G)⊗θh(φ) R(G) →θgh(φ) R(G) where θ : H4(G,Z) → H3(G,Z) is theinverse transgression map and φ is a carefully chosen cocycle class. Wefind the rank of the fusion algebra X(G) = summationtextg∈G θg(φ)R(G) as well as therelation between its basis elements. We also provide some applications totopological gauge theories.In Chapter 3, we study the twisted fusion algebra for the finite group(Z/p)3. One way to calculate the coefficients in the fusion algebra is to useconformal field theory. The modular data in conformal field theory provides4Chapter 1. Introductionthe Verlinde formula, which gives the fusion coefficients for fusion algebras.The modular data associated to finite groups is introduced in the work ofCoste, Gannon and Ruelle [20]. They also consider the data arising from acohomological twist, and write down the modular data for a general twistin terms of quantities associated directly with the finite group. We utilisethe results from the conformal field theory, namely finite group modulardata, to prove this algebra is isomorphic to the non-twisted fusion algebraof extraspecial p-group with exponent p and of order p3.Chapter 4 is devoted to computing the cohomology of orbit space X/Gwhere G = Z/p and X = ((S1)p−1)s and the action is induced by the actionon the ZG-module H1(X) = L = IGs, where IG is the augmentation idealof ZG.5Chapter 2A Twisted Fusion AlgebraInspired by the Chen-Ruan cohomology for orbifolds, it has been shown byAdem, Ruan, Zhang [8] that there is an internal product in twisted orbifoldK-theory αKorb(X). The information determining this stringy product lies inH4(BX,Z) instead of H3(BX,Z): Given a class φ ∈ H4(BX,Z), it inducesa class θ(φ) ∈ H3(B ∧X,Z) where ∧X is the inertia stack. As a result, onecan define a twisted K-theory on θ(φ)K(∧X). The map θ can be thought ofthe inverse of the classical transgression map.The construction of this internal product is motivated by the so-calledPontryagin product on KG(G) for a finite group G. Indeed, if the orbifoldis X = ∧[∗/G], one obtains the same product for the orbifold K-theoryin the untwisted case. There is also an explicit calculation of the inversetransgression map θ for the cohomology of finite groups (see [8]). Using theseresults, we exhibit a non-trivial product structure in the case of G = (Z/2)3.We use an integral cohomology class φ ∈ H4(G,Z) such that under theinverse transgression it maps non-trivially for every twisted sector, yieldinga product structure on the algebra θ(φ)KG(G) = X(G) = summationtextg∈G θg(φ)R(G)defined via the pairing θg(φ)R(G)⊗θh(φ)R(G) →θgh(φ) R(G). In this chapter,we derive the relations between the basis elements of the algebra X(G), andwe prove the uniqueness of this product in this particular case. G = (Z/2)3is indeed the abelian group of smallest rank such that it has non-trivial6Chapter 2. A Twisted Fusion Algebratransgressions (see [8], section 5).These twisted rings have also been worked out in the conformal fieldtheory literature. In [35], the modular invariant (i.e. S and T matrices)of this group G = (Z/2)3 is calculated. As a result, one can calculatethe relations between basis elements of X(G) using the Verlinde formula.Moreover, the same example is considered in [15], where a decompositionformula for twisted K-theory is given and the product is calculated aftertensoring by rational numbers.There is also a physical counterpart of this theory. In our case thismap is the inverse transgression map, which is actually the map comingfrom the correspondence between the Chern-Simons action and the Wess-Zumino terms that arise in connecting a specific three-dimensional quantumfield theory to its related two-dimensional quantum field theory. One cansee that the Chern-Simons theory associates to each group element gi ∈G a 2-cocycle βi of the stabilizer group Ngi, which is G = (Z/2)3 in ourabelian case. We use the formulations in [25] to calculate the partitionfunction Z(S1×S1×S1). It is also worth mentioning that the algebra X(G)corresponds to a fusion algebra in this physical context.In this chapter, we first introduce projective representation and its basicproperties. We next give some preliminaries and the definition of our fusionalgebra. In the third section, we calculate the rank and the uniqueness ofthis algebra as well as the relation between the basis elements which are theprojective representations. Finally, we present the application to topologicalgauge theories by using the formulation in [25].7Chapter 2. A Twisted Fusion Algebra2.1 Projective representationsFor the entirety of this section, G is a finite group. More information onprojective representations can be found in [30].Definition 2.1.1. Let V be a complex vector space. A projective represen-tation of G is a map ρ : G → GL(V) such that there exists α : G×G → C∗with• ρ(x)ρ(y) = α(x,y)ρ(xy) for all x, y ∈ G, and• ρ(1) = I.One can easily see that α is a 2-cocycle of G. The associativity ofmultiplication in GL(V) combined with the first condition gives the cocyclecondition, and α(g,1) = α(1,g) = 1 for all g ∈ G by the second condition.Any projective representation associated to α is called an α-twisted rep-resentation of G. As in the case of complex representations of G, we have anotion of isomorphism between two representations.Definition 2.1.2. Two α-twisted representations ρi : G → GL(Vi), i = 1,2,are called linearly equivalent if there is a vector space isomorphism f : V1 →V2 withρ2(g) = fρ1(g)f−1 for all g ∈ G.Clearly, the direct sum of two α-twisted representations is an α-twistedrepresentation. Thus we can form the monoid of isomorphism classes of α-twisted representations of G. Rα(G) is the associated Grothendieck group.Note that if α is the trivial cocycle, then Rα(G) is R(G), the complexrepresentation ring of G.8Chapter 2. A Twisted Fusion AlgebraIt is not hard to see that the tensor product of two α-twisted represen-tations is no longer an α-twisted representation. Note however that if α andβ are two cocycles, then the tensor product of an α-twisted representationand a β-twisted representation is an (α + β)-twisted representation. Thiscan be extended to a pairingRα(G)⊗Rβ(G) → Rα+β(G).In order to study Rα(G), we introduce the α-twisted group algebra CαG.We denote by CαG the vector space over C with basis {¯g}g∈G and withproduct ¯g¯h = α(g,h)gh extended by linearity. This makes CαG into a C-algebra with ¯1 as the identity.Definition 2.1.3. If α and β are cocycles of G, then we say that CαG andCβG are equivalent if there exists a C-algebra isomorphism φ : CαG → CβGand a mapping t : G → C∗ such that φ(¯g) = t(g)˜g for all g ∈ G, where ¯gand ˜g are the bases for the two twisted group algebras.This defines an equivalence relation on such twisted algebras, and wehave the following result.Lemma 2.1.4. There is an equivalence of twisted group algebras, CαG similarequalCβG, if and only if α is cohomologous to β. In fact, α mapsto→ CαG induces abijective correspondence between H2(G,C∗) and the set of equivalence classesof twisted group algebras of G over C.In [30], it is proved that these twisted group algebras play the same rolein determing Rα(G) that CG plays in determining R(G):Theorem 2.1.5. There is a bijective correspondence between α-twisted rep-resentations of G and CαG-modules. This correspondence preserves sums9Chapter 2. A Twisted Fusion Algebraand bijectively maps linearly equivalent representations into isomorphic mod-ules.2.2 The twisted fusion product for finite groupsIn this section, we review a special case for the product in twisted orbifold K-theory which is formalised by Adem, Ruan and Zhang in [8]. We considerthe inertia orbifold ∧[∗/G] where G is a finite group. In this case, theuntwisted orbifold K-theory of ∧[∗/G] is simply KG(G), which is additivelyisomorphic to summationtext(g) R(ZG(g)), where ZG(g) denotes the centraliser of g inG, and the sum is taken over conjugacy classes. The product in KG(G) isdefined as follows. An equivariant vector bundle over G can be thought of asa collection of finite dimensional vector spaces Vg with a G-module structureon summationtextg∈G Vg such that gVh = Vghg−1. The product is defined as(V starW)g =circleplusdisplayg1g2=gVg1 ⊗Wg2.One can give an alternative definition. We first define the maps e1 :G × G → G, e2 : G × G → G and e12 : G × G → G as e1(g,h) = g ,e2(g,h) = h and e12(g,h) = gh respectively, which are G-equivariant up tothe conjugation action. If α, β are elements in KG(G) the product is definedasαstarβ = e12∗(e∗1(α)e∗2(β)).We now need to review the inverse transgression map for finite groupsto extend the latter definition to twisted K-theory. In order to define theproduct in twisted K-theory, Adem, Ruan and Zhang [8] define a map tomatch up the levels which appear in the twistings. This cochain map θ is10Chapter 2. A Twisted Fusion Algebracalled inverse transgression map, and it induces the homomorphismθ∗ : Hk+1(BG,Z) → Hk(B ∧G,Z).If the orbifold G is [∗/G], where G is a finite group, the inverse trans-gression has a classical interpretation in terms of the shuffle product. Re-call that ∧[∗/G] is equivalent to unionsqtext(g)[∗/ZG(g)] (see [8]). Hence we wouldlike to focus on the map θg : Ck(G,U(1)) → Ck−1(ZG(g),U(1)). If Gis a finite group then the cochain complex C∗(G,U(1)) is in fact equal toHomG(B∗(G),U(1)), where B∗(G) is the bar resolution for G (see [18], page18). If t is the generator of Z, the shuffle product is Bk(ZG(g))⊗B1(Z) →Bk+1(G) given by[g1|g2|...|gk]star[ti] =summationdisplayσσ[g1|g2|...|gk|gk+1],where gk+1 = gi, σ ranges over all (k,1)–shuffles andσ[g1|g2|...|gk+1] = (−1)sgn(σ)[gσ(1)|gσ(2)|...|gσ(k+1)]. (2.1)A (k,1)–shuffle is an element σin the symmetric group Sk+1 such that σ(i) <σ(j) for 1 ≤ i < j ≤ k.We can dualize this using integral coefficients. Given a cocycle φ ∈Ck+1(G,Z), one can see that the inverse transgression θg(φ) ∈ Ck(ZG(G),Z)can be defined asθg(φ)([g1|g2|...|gk]) = φ([g1|g2|...|gk]star[g]) (2.2)where g1,g2,...,gk ∈ G. Hence it induces a map in integral cohomology.11Chapter 2. A Twisted Fusion AlgebraWe can now induce the inverse transgression map for H∗(G,F2) forG = (Z/2)3 using Bockstein homomorphism. We want to find a non-trivialcocycle in the image of the inverse transgression map. Notice that H∗(G,F2)is a polynomial algebra on three degree one generators x, y and z. Ingeneral, for an elementary abelian 2-group, the modulo 2 reduction mapfor k > 0 is a monomorphism Hk(G,Z) → Hk(G,F2) which is the kernelof the Bockstein homomorphism Sq1 : Hk(G,F2) → Hk+1(G,F2). In orderto get nontrivial cocycle in the image of the inverse transgression map, wechoose α = Sq1(xyz) = x2yz +xy2z +xyz2, which represents a non-squareelement in H4(G,Z). The following lemma is proved in [8] by analyzing themultiplication map in the cohomology.Lemma 2.2.1. Let g = xaybzc be an element in G = (Z/2)3, where weare writing in terms of the standard basis (identified with its dual). Let usconsider α = Sq1(xyz) = x2yz+xy2z+xyz2 which represents a non-squareelement in H4(G,Z). Thenθ∗g(α) = a(y2z +z2y)+b(x2z +xz2)+c(x2y +xy2),and so θ∗g(α) is non-zero on every component except the one correspondingto the trivial element in G.Proof. See [8], Lemma 5.2.This implies that for all g, h ∈ G, θ∗g +θ∗h = θ∗gh in the cohomology up tocoboundaries. This also implies that the correspondence g mapsto→ θg(α) definesa homomorphism. In the case of G = (Z/2)3, we have the isomorphismθ?(α) : G → H3(G,Z) = G.We now define the product as follows:12Chapter 2. A Twisted Fusion AlgebraDefinition 2.2.2. Let τ be a 2-cocycle for the orbifold defined by the con-jugation action of a finite group G on itself which is in the image of theinverse transgression. The product on τKG(G) is defined by the followingformula: if α, β ∈τ KG(G), thenαstarβ = e12∗(e∗1(α)e∗2(β)).If τ = θ(φ) then we have the following formula proved in [8]:e∗1τ +e∗2τ = e∗12τup to coboundary. Hence the product e1(α)e2(β) lies ine∗1τ+e∗2τKG(G) =e∗12τ KG(G).Applying e12∗, this is mapped to τKG(G), which gives the product inthe twisted K-theory.Using the identification θ∗g + θ∗h = θ∗gh, the following product is definedon the algebraθ(φ)KG(G) = X(G) = summationdisplayg∈Gθg(φ)R(G)via the pairingθg(φ)R(G)⊗ θh(φ)R(G) → θgh(φ)R(G).In the next section, we investigate the properties of this algebra whilecalculating its rank and the relations between the irreducible projective rep-resentation.13Chapter 2. A Twisted Fusion Algebra2.3 Calculations2.3.1 2-cocycles in G with values in U(1)We will assume that G = (Z/2)3 for the remainder of this chapter. Werecall that, for a finite dimensional complex vector space, a mapping ρ :G → GL(V) is called a projective representation of G if there exists a U(1)-valued 2-cocycle α ∈ Z2(G;U(1)) such that ρ(x)ρ(y) = α(x,y)ρ(x,y) forall x,y ∈ G and ρ(1) = IdV . Hence in order to compute θg(φ)R(G) wefirst need to find the 2-cocycles in H2(G,U(1)) corresponding to θg(φ) inH3(G,Z) where both cohomology groups are isomorphic to G. For thispurpose, we consider the isomorphismH2(G,U(1)) → H3(G,Z)induced by the natural coefficient sequence 0 → Z → R → U(1) → 1. AsH3(G,Z) ∼= G, we need to find the eight non-cohomologous 2-cocycles inH2(G,U(1)) corresponding to each θg(φ) for all g ∈ G.We now determine the relations in order to obtain the 2-cocycles inC2(G,U(1)). Any 2-cocycle β in C2(G,U(1)) should satisfy:δβ = 1.By the boundary formula of the bar resolution of G, we derive:β(g2,g3)β(g1g2,g3)−1β(g1,g2g3)β(g1,g2)−1 = 1for all gi ∈ G, i = 1,2,3. Some interesting relations result when we plug in14Chapter 2. A Twisted Fusion Algebrag1 = g3 = g and g2 = 1 into this formula and we obtainβ(g,1) = β(1,g). (2.3)Moreover, for g1 = g2 = g, we haveβ(1,g3)β(g,g) = β(g,g3)β(g,gg3). (2.4)Asβ isdefiningaprojectiverepresentation, sayρ, itshouldsatisfyρ(1)ρ(g) =β(1,g)ρ(g). This implies β(1,g) = 1 for all g ∈ G. Now we consider the fol-lowing tables for 2-cocycles βi : G×G → U(1) which satisfies the identities(3.3) and (2.4). We call these cocycles fundamental cocycles.We choose β1as the trivial co-cycle. Here, xi, yi and zi’s are in U(1), and they will bedetermined later.β2 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 1 x1 x2 x1 x2 x3 x3g3 1 x1 −1 x4 −x1 x6 −x4 −x6g4 1 x2 −x4 1 x5 x2 −x4 x5g5 1 x1 −x1 −x5 −1 x8 −x8 x5g6 1 x2 −x6 x2 −x8 1 −x8 −x6g7 1 x3 x4 x4 x8 x8 1 x3g8 1 x3 x6 −x5 −x5 x6 x3 115Chapter 2. A Twisted Fusion Algebraβ3 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 −1 y1 y2 −y1 −y2 y3 −y3g3 1 y1 1 y4 y1 y6 y4 y6g4 1 −y2 y4 1 y5 −y2 y4 y5g5 1 −y1 y1 −y5 −1 y8 −y8 y5g6 1 y2 y6 y2 −y8 1 −y8 y6g7 1 −y3 y4 y4 y8 y8 1 −y3g8 1 y3 y6 −y5 −y5 y6 y3 1β4 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 −1 z1 z2 −z1 −z2 z3 −z3g3 1 −z1 1 z4 z1 z6 z4 z6g4 1 z2 z4 −1 z5 −z2 −z4 −z5g5 1 z1 z1 z5 1 z8 z8 z5g6 1 −z2 z6 −z2 −z8 1 −z8 z6g7 1 −z3 z4 −z4 −z8 z8 −1 z3g8 1 z3 z6 −z5 z5 −z6 −z3 −1As the multiplication of two cocycles gives us another cocycle, one can con-struct five more cocycles, one of which is the trivial cocycle. We recall thatfor β ∈ Z2(G;U(1)), an element g ∈ G is called β-regular if β(g,x) = β(x,g)for all x ∈ CG(g) (see [30], page 107). Thus all of the 2-cocycles have two16Chapter 2. A Twisted Fusion Algebraβ-regular elements, one of which is 1, the other one is different for eachcocycle. For example, one can immediately see that the β-regular elementsfor β2, β3 and β4 are g2, g3 and g4, respectively.Ontheotherhand, fromtheboundaryformulawenotethata2-coboundaryshould satisfy β(g1,g2) = σ(g1)σ(g2)σ(g1g2)−1 where σ is in C1(G,Z). AsG is abelian, we haveβ(gi,gj) = β(gj,gi)for all gi and gj in G. This implies that all of these eight cocycles representdifferent cohomology classes, as multiplication with a coboundary does notchange the β-regular elements. Thus, we have following proposition.Proposition 2.3.1. The rank of θg(φ)R(G) is 2 if θg(φ) is nontrivial.Proof. A basic result of projective representations states that αR(G)is a free abelian group of rank equal to the number of distinct α-regularconjugacy classes of G (see [30], theorem 6.7). So, the ranks of βiR(G) is 2for any non-trivial βi.On the other hand, we obtained eight non-cohomologous cocycles whichshould correspond to θg(φ)’s because H2(G,U(1)) and H3(G,Z) are isomor-phic to G. The result follows.We can therefore conclude:Corollary 2.3.2. The rank of X(G) is equal to 22.Proof. The ranks of βiR(G) are 2 for any non-trivial βi accounting for 14and beta1R(G) has rank 8.17Chapter 2. A Twisted Fusion Algebra2.3.2 The projective representationsIn order to compute the irreducible projective representations of G, it ishelpful to determine the xi, yi and zi’s. From the boundary formula, wehave the following relations in β2:−1 = x1x3x4x5−1 = x2x3x5x81 = x6x8x3x1.By a routine calculation, one can check that the other relations dependon these three relations. We can choose x1 = x2 = x3 = x4 = −x5 = x6 =x7 = x8 = 1 that obviously satisfy these relations. Similarly, we find yi’sand zi’s. The other cocycles are computed by multiplying β2, β3 and β4.We will later show that the choice of xi, yj and zk from the set {±1} doesnot change our representations. Here are our eight cocycles.β2 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 1 1 1 1 1 1 1g3 1 1 −1 1 −1 1 −1 −1g4 1 1 −1 1 −1 1 −1 −1g5 1 1 −1 1 −1 1 −1 −1g6 1 1 −1 1 −1 1 −1 −1g7 1 1 1 1 1 1 1 1g8 1 1 1 1 1 1 1 118Chapter 2. A Twisted Fusion Algebraβ3 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 −1 1 1 −1 −1 1 −1g3 1 1 1 1 1 1 1 1g4 1 −1 1 1 −1 −1 1 −1g5 1 −1 1 1 −1 −1 1 −1g6 1 1 1 1 1 1 1 1g7 1 −1 1 1 −1 −1 1 −1g8 1 1 1 1 1 1 1 1β4 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 −1 1 1 −1 −1 1 −1g3 1 −1 1 1 −1 −1 1 −1g4 1 1 1 −1 1 −1 −1 −1g5 1 1 1 1 1 1 1 1g6 1 1 1 1 1 1 1 1g7 1 −1 1 −1 −1 1 −1 1g8 1 1 1 −1 1 −1 −1 −119Chapter 2. A Twisted Fusion Algebraβ5 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 −1 1 1 −1 −1 1 −1g3 1 1 −1 1 −1 1 −1 −1g4 1 −1 −1 1 1 −1 −1 1g5 1 −1 −1 1 1 −1 −1 1g6 1 1 −1 1 −1 1 −1 −1g7 1 −1 1 1 −1 −1 1 −1g8 1 1 1 1 1 1 1 1β6 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 −1 1 1 −1 −1 1 −1g3 1 −1 −1 1 1 −1 −1 1g4 1 1 −1 −1 −1 −1 1 1g5 1 1 −1 1 −1 1 −1 −1g6 1 −1 −1 −1 1 1 1 −1g7 1 −1 1 −1 −1 1 −1 1g8 1 1 1 −1 1 −1 −1 −120Chapter 2. A Twisted Fusion Algebraβ7 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 1 1 1 1 1 1 1g3 1 −1 1 1 −1 −1 1 −1g4 1 −1 1 −1 −1 1 −1 1g5 1 −1 1 1 −1 −1 1 −1g6 1 −1 1 −1 −1 1 −1 1g7 1 1 1 −1 1 −1 −1 −1g8 1 1 1 −1 1 −1 −1 −1β8 1 g2 g3 g4 g5 g6 g7 g81 1 1 1 1 1 1 1 1g2 1 1 1 1 1 1 1 1g3 1 −1 −1 1 1 −1 −1 1g4 1 −1 −1 −1 1 1 1 −1g5 1 −1 −1 1 1 −1 −1 1g6 1 −1 −1 −1 1 1 1 −1g7 1 1 1 −1 1 −1 −1 −1g8 1 1 1 −1 1 −1 −1 −1By considering the 2-cocycles that we obtained, it is obvious that thereis no 1-dimensional projective representation whenever the 2-cocycle is nottrivial. For the trivial cocycle, we have eight 1-dimensional representationswhich are just the irreducible linear representations of G. For the othercases, we find two 2-dimensional irreducible representations for each of the2-cocycles (see [30], Theorem 6.7). Let ρi1 and ρi2 be the irreducible rep-resentations corresponding to the cocycle βi. It is also enough to give the21Chapter 2. A Twisted Fusion Algebramatrices corresponding to the generators of G. Without loss of general-ity, we assume g2 = (1,0,0), g3 = (0,1,0), g4 = (0,0,1), g5 = (1,1,0),g6 = (1,0,1), g7 = (0,1,1), g8 = (1,1,1). Here, βiR(G) is the projectiverepresentation for 2-cocycle βi. We give only the matrices corresponding tothree elements the rest can be obtained from these.β2R(G) :ρ21 : g2 mapsto→ 1 00 1g3 mapsto→ 0 1−1 0g4 mapsto→ 0 11 0ρ22 : g2 mapsto→ −1 00 −1g3 mapsto→ 0 −11 0g4 mapsto→ 0 −1−1 0β3R(G) :ρ31 : g2 mapsto→ 0 1−1 0g3 mapsto→ −1 00 −1g4 mapsto→ 0 11 0ρ32 : g2 mapsto→ 0 −11 0g3 mapsto→ 1 00 1g4 mapsto→ 0 −1−1 0β4R(G) :ρ41 : g3 mapsto→ 0 11 0g5 mapsto→ 0 i−i 0g6 mapsto→ 1 00 −122Chapter 2. A Twisted Fusion Algebraρ42 : g3 mapsto→ 0 −1−1 0g5 mapsto→ 0 −ii 0g6 mapsto→ −1 00 1β5R(G) :ρ51 : g2 mapsto→ 0 1−1 0g4 mapsto→ 0 11 0g5 mapsto→ 1 00 1ρ52 : g2 mapsto→ 0 −11 0g4 mapsto→ 0 −1−1 0g4 mapsto→ −1 00 −1β6R(G) :ρ61 : g2 mapsto→ 0 1−1 0g3 mapsto→ 0 ii 0g6 mapsto→ 1 00 1ρ62 : g2 mapsto→ 0 −11 0g3 mapsto→ 0 −i−i 0g6 mapsto→ −1 00 −1β7R(G) :ρ71 : g2 mapsto→ 0 11 0g3 mapsto→ 0 i−i 0g6 mapsto→ 1 00 −123Chapter 2. A Twisted Fusion Algebraρ72 : g2 mapsto→ 0 −1−1 0g3 mapsto→ 0 −ii 0g6 mapsto→ −1 00 1β8R(G) :ρ81 : g2 mapsto→ 0 11 0g5 mapsto→ 0 i−i 0g6 mapsto→ 1 00 −1ρ82 : g2 mapsto→ 0 −1−1 0g5 mapsto→ 0 −ii 0g6 mapsto→ −1 00 1The projective representations ρ1 and ρ2 are not linearly isomorphic toeach other. Indeed, for the 2-cocycles β2, β3, β5 and β6, there exist gi suchthatρ1(gi) = 1 00 1whileρ2(gi) = −1 00 −1. Thus there is no M ∈ GL2(Z) such that Mρ1(gi)M−1 = ρ2(gi). The otherrepresentations of β4, β7 and β8 are obtained from the book of Karpilovski[30] (see page 120 and 124 Theorem 7.1 and 7.2). Thus none of them arelinearly isomorphic to each other.One question that needs to be answered is whether these representationsdepend on the choice of xi, yi and zi. One can check by calculation that24Chapter 2. A Twisted Fusion Algebrathe representations only depend on the values of βi(g,g). More precisely,ρ(g)ρ(g) should be equal to β(g,g) for all g ∈ G. For example,ρ(g3)ρ(g8) = β(g3,g8)ρ(g7)⇔ ρ(g3)β(g2,g7)ρ(g2)ρ(g7) = β(g3,g8)β(g2,g4)ρ(g2)ρ(g4)⇔ ρ(g3)β(g2,g7)ρ(g2)β(g3,g4)ρ(g3)ρ(g4) = β(g3,g8)β(g2,g4)ρ(g2)ρ(g4)⇔ ρ(g3)ρ(g3)β(g2,g7)β(g3,g4) = β(g3,g8)β(g2,g4)Iwhich is true if and only if ρ(g3)ρ(g3) = β(g3,g3)I by the relations we getfrom the boundary formulas. The other elements can be checked in a similarmanner.Thus we found the basis of our algebra. We will show that this productis unique up to coboundary. First we prove the following result.Proposition 2.3.3. If φ and φprime are cohomologous cocycles in H4(G,Z) thenthe fusion algebras corresponding to these cocycles are isomorphic.Proof. If φ and φprime are cohomologous cocycles then θg(φ) and θg(φprime) rep-resent the same cohomology class in H3(G,Z). In order to compute the2-cocycle corresponding to θg(φ) ∈ H3(G,Z) ∼= G, we will use the isomor-phism induced from the short exact sequence0 → Z → R → U(1) → 1.Thusθg(φ) is mapped to a certain class of 2-cocycles inH2(G,U(1)) ∼= G.As we found eight non-cohomologous 2-cocycles in G, it is enough to checkhow the representations change if we multiply our fundamental 2-cocyclesby a 2-coboundary. This is a basic result of projective representation theory25Chapter 2. A Twisted Fusion Algebra(see page 72 in [30]). After multiplying our fundamental 2-cocycles by a2-coboundary, the new projective representation of this cocycle becomeslinearly isomorphic to the former one. The result follows from the aboveargument.2.3.3 The relations.Now we are able to calculate the relation of this basis using the pairingθ(φ)gR(G) ⊗ θ(φ)hR(G) → θ(φ)ghR(G). The calculations are nothing butsolving linear equations. Namely one should prove that ρki ⊗ρli’s are linearlyisomorphic to a sum of some basis elements. Here ρ1i denotes the irreducibleregular representations of G for i = 1,2,...,8.Let us start with ρ1i ⊗ρji which is linearly isomorphic to ρjk for some k ∈{1,2}where j negationslash= 1 because ρ1i⊗ρji should be 2 dimensional βj representation.Here is the results of these multiplications:26Chapter 2. A Twisted Fusion Algebra⊗ρ1 1ρ1 2ρ1 3ρ1 4ρ1 5ρ1 6ρ1 7ρ1 8ρ2 1ρ2 2ρ3 1ρ3 2ρ4 1ρ4 2ρ5 1ρ5 2ρ6 1ρ6 2ρ7 1ρ7 2ρ8 1ρ8 2ρ1 1ρ1 1ρ1 2ρ1 3ρ1 4ρ1 5ρ1 6ρ1 7ρ1 8ρ2 1ρ2 2ρ3 1ρ3 2ρ4 1ρ4 2ρ5 1ρ5 2ρ6 1ρ6 2ρ7 1ρ7 2ρ8 1ρ8 2ρ1 2ρ1 2ρ1 1ρ1 5ρ1 6ρ1 3ρ1 4ρ1 8ρ1 7ρ2 2ρ2 1ρ3 1ρ3 2ρ4 1ρ4 2ρ5 2ρ5 1ρ6 2ρ6 1ρ7 1ρ7 2ρ8 2ρ8 1ρ1 3ρ1 3ρ1 5ρ1 1ρ1 7ρ1 2ρ1 8ρ1 4ρ1 6ρ2 1ρ2 2ρ3 2ρ3 1ρ4 1ρ4 2ρ5 2ρ5 1ρ6 1ρ6 2ρ7 2ρ7 1ρ8 2ρ8 1ρ1 4ρ1 4ρ1 6ρ1 7ρ1 1ρ1 8ρ1 2ρ1 3ρ1 5ρ2 1ρ2 2ρ3 1ρ3 2ρ4 2ρ4 1ρ5 1ρ5 2ρ6 2ρ6 1ρ7 2ρ7 1ρ8 2ρ8 1ρ1 5ρ1 5ρ1 3ρ1 2ρ1 8ρ1 1ρ1 7ρ1 6ρ1 4ρ2 2ρ2 1ρ3 2ρ3 1ρ4 1ρ4 2ρ5 1ρ5 2ρ6 2ρ6 1ρ7 2ρ7 1ρ8 1ρ8 2ρ1 6ρ1 6ρ1 4ρ1 8ρ1 2ρ1 7ρ1 1ρ1 5ρ1 3ρ2 2ρ2 1ρ3 1ρ3 2ρ4 2ρ4 1ρ5 2ρ5 1ρ6 1ρ6 2ρ7 2ρ7 1ρ8 1ρ8 2ρ1 7ρ1 7ρ1 8ρ1 4ρ1 3ρ1 6ρ1 5ρ1 1ρ1 2ρ2 1ρ2 2ρ3 2ρ3 1ρ4 2ρ4 1ρ5 2ρ5 1ρ6 2ρ6 1ρ7 1ρ7 2ρ8 1ρ8 2ρ1 8ρ1 8ρ1 7ρ1 6ρ1 5ρ1 4ρ1 3ρ1 2ρ1 1ρ2 2ρ2 1ρ3 2ρ3 1ρ4 2ρ4 1ρ5 1ρ5 2ρ6 1ρ6 2ρ7 1ρ7 2ρ8 2ρ8 127Chapter 2. A Twisted Fusion AlgebraAnother type of multiplication is ρji ⊗ ρji, which is linearly isomorphicto a sum of four irreducible regular representations ρ1k as βj(g,h)2 = 1for all j. We calculate all of these using the associativity of our algebraX(G) and investigating the eigenvalues of the matrices. Of course, onecan also calculate them by defining the linear isomorphism explicitly. Asρi1(gj) = −ρi2(gj) for three gj’s in the definitions of representations, we haveρi1 ⊗ρi1 = ρi2 ⊗ρi2 as well as ρi2 ⊗ρi1 = ρi1 ⊗ρi2.⊗ ρ21 ρ22ρ21 ρ11 +ρ13 +ρ14 +ρ17 ρ12 +ρ15 +ρ16 +ρ18ρ22 ρ12 +ρ15 +ρ16 +ρ18 ρ11 +ρ13 +ρ14 +ρ17⊗ ρ31 ρ32ρ31 ρ11 +ρ12 +ρ14 +ρ16 ρ13 +ρ15 +ρ17 +ρ18ρ32 ρ13 +ρ15 +ρ17 +ρ18 ρ11 +ρ12 +ρ14 +ρ16⊗ ρ41 ρ42ρ41 ρ14 +ρ16 +ρ17 +ρ18 ρ11 +ρ12 +ρ13 +ρ15ρ42 ρ11 +ρ12 +ρ13 +ρ15 ρ14 +ρ16 +ρ17 +ρ18⊗ ρ51 ρ52ρ51 ρ11 +ρ15 +ρ16 +ρ18 ρ12 +ρ13 +ρ14 +ρ17ρ52 ρ12 +ρ13 +ρ14 +ρ17 ρ11 +ρ15 +ρ16 +ρ18⊗ ρ61 ρ62ρ61 ρ11 +ρ13 +ρ16 +ρ18 ρ12 +ρ14 +ρ15 +ρ17ρ62 ρ12 +ρ14 +ρ15 +ρ17 ρ11 +ρ13 +ρ16 +ρ1828Chapter 2. A Twisted Fusion Algebra⊗ ρ71 ρ72ρ71 ρ13 +ρ14 +ρ15 +ρ16 ρ11 +ρ12 +ρ17 +ρ18ρ72 ρ11 +ρ12 +ρ17 +ρ18 ρ13 +ρ14 +ρ15 +ρ16⊗ ρ81 ρ82ρ81 ρ12 +ρ13 +ρ14 +ρ18 ρ11 +ρ15 +ρ16 +ρ17ρ82 ρ11 +ρ15 +ρ16 +ρ17 ρ12 +ρ13 +ρ14 +ρ18The last type of multiplication that we have to consider is ρji ⊗ρnm, wheredistinct i, m are in {1,2} and j and n are in {2,3,...,8}. As ρji ⊗ρnm is four-dimensional, it should be linearly isomorphic to 2ρl1, 2ρl2 or ρl1 + ρl2. Noneof −2ρl1, −2ρl2 nor −ρl1 −ρl2 are possible as they are not βl representationsas indicated by the list of our representations in the previous section. Therepresentations 2ρl1 and 2ρl2 are also impossible by the following associativityargument.Suppose ρji ⊗ρnm = 2ρl1 by the table above we can always find ρ1k suchthat ρ1k⊗ρji = ρji and ρ1k⊗ρl1 = ρl2. This gives a contradiction if we multiplyeach side of ρji ⊗ρnm = 2ρl1 by ρ1k.We can conclude ρji ⊗ρnm = ρl1 +ρl2. We have finished calculating all therelations.2.4 Topological gauge theoriesIn [25], Dijkgraaf and Witten show that three dimensional Chern-Simonsgauge theories with a compact gauge group can be classified by the integercohomology group H4(BG,Z). Wess-Zumino interactions of such groups Gare classified by H3(G,Z). The relation between three dimensional sigmamodels involves a certain natural map of H4(BG,Z) to H3(G,Z) which is29Chapter 2. A Twisted Fusion Algebrathe inverse transgression map defined in the second section. Our calcu-lations provide an example of three-dimensional topological theories withfinite gauge group. In this context, our algebra X(G) is a fusion algebra. InQFT (Quantum Field Theory) one can associate to a (d + 1)-dimensionalmanifold M a certain number Z(M), the partition function. A detaileddiscussion is provided in [25].We consider the partition function of the 3-torus S1 ×S1 ×S1. If g, hand k are three commuting gauge fields, the partition function is evaluatedto giveZ(S1 ×S1 ×S1) = 1|G|summationdisplayg,h,k∈GW(g,h,k),where [g,h] = [h,k] = [k,g] = 1. We define W asW(g,h,k) = α(g,h,k)α(h,k,g)α(k,g,h)α(g,k,h)α(h,g,k)α(k,h,g).for α ∈ H3(BG,U(1)).The Chern-Simons theory associates to each group element gi ∈ G a2-cocycle βi, which we calculated above. Again, by the result of Witten andDijkgraaf [25], we may express βi in terms of a 3-cocycle α ∈ H3(G,U(1)):βi(h1,h2) = α(gi,h1,h2)α(h1,h2,gi)α(h1,gi,h2).This may also be obtained by the formula for inverse transgression map (2.2):βi(h1,h2) = θgi(α)([h1|h2]) = α([h1|h2]star[gi]) = α([gi|h1|h2])α([h1|h2|gi])α([h1|gi|h2]).Note that the shuffle product (2.1) is defined via additive notation.30Chapter 2. A Twisted Fusion AlgebraThus the action W can be written in terms of 2-cocycles:W(gi,h,k) = βi(h,k)βi(k,h)−1.We define epsilon1g(h) = β(g,h)β(h,g)−1 for fixed g which is a one dimensionalrepresentation of G. Thus an element g is β-regular if and only if epsilon1g = 1.This impliesr(G,β) = 1|G|summationdisplayg,h∈Gβ(g,h)β(h,g)−1where r(G;β) denotes the number of irreducible projective representations ofβR(G). Our 2-cocycles defined in the second section satisfies this condition.Comparing all these results we obtain the following result for the parti-tion function of the 3-torus where G = (Z/2)3:Proposition 2.4.1.Z(S1 ×S1 ×S1) =summationdisplayir(G;βi) = 22.Using our representations, we find that the basis elements να of Hilbertspace corresponds to 3-torus in QFT. These basis elements are given in [25]asνα(gi,h) = Trρi(h).In this context, our algebra X(G) can be regarded as the smallest twistednon-trivial fusion algebra for abelian groups.31Chapter 3The Fusion Algebra of anExtraspecial p-group3.1 IntroductionIn this chapter, we give an application of the finite group modular data whichis mainly explored in [20]. This modular data was originally introduced, inLusztig’s determination of the irreducible characters of the finite groups ofLie type [31], [32]. To describe the unipotent characters, he considered themodular data for some particular finite groups. The primary fields of thefusion algebra parametrize the unipotent characters associated to a given2-sided cell in the Weyl group. Lusztig interprets this fusion algebra asthe Grothendieck ring for G-equivariant vector bundles; in other words, theequivariant K-theory.Themostphysicalapplicationofthismodulardataisin(2+1)-dimensionalquantum field theories where a continuous gauge group has been sponta-neously broken into a finite group [13]. Non-abelian anyons (i.e. particleswhose statistics are governed by the braid group rather than the symmetricgroup) arise as topological objects. The effective field theory describing thelong distance physics is governed by the quantum group of [23].A set of modular data (i.e. matrices S and T) may be obtained for any32Chapter 3. The Fusion Algebra of an Extraspecial p-groupchoice of finite group G. Much information about a group can be recov-ered easily from its character table including whether it is abelian, simple,solvable, nilpotent, etc. For instance, G is simple if and only if for all ir-reducible χ negationslash≡ 1, χ(a) = χ(e) only for a = e. Thus it may be expectedthat finite group modular data, which probably includes the character ta-ble, should provide more information about the group, i.e. be sensitive to alot of the group-theoretic properties of G.One way to generalize this data is to twist with a cocycle from the co-homology group. One can obtain topological (e.g. oriented knot) invariantsfrom this twisted data, as explained in [9]. These invariants are functionsof the knot group (i.e., the fundamental group of the complement of theknot). Although non-isomorphic knots can have the same invariant, theseinvariants can distinguish a knot from its inverse (i.e., the knot with oppo-site orientation), unlike the more familiar topological invariants arising fromaffine algebras.In this chapter, we consider the internal product in twisted orbifold K-theory αKorb(X) formalised by Adem, Ruan and Zhang [8] employing resultsfrom the associated finite group modular data. Our aim is to exhibit thatthe non-trivial product structure for αKG(G) for G = (Z/p)3 is the sameas the product structure for KH(H) where H is the extraspecial p-groupof order p3 and exponent p. The product structure in these theories is thesame as in the fusion algebras associated to the same finite group.We first give some preliminaries and the definition of finite group mod-ular data, then mention the results of Coste, Gannon and Ruelle [20] whocalculated the S matrix for (Z/p)3 with the particular twist. After this, wecalculate the S matrix of the extraspecial p-group and show that it givesthe same fusion algebra as the twisted case of (Z/p)3.33Chapter 3. The Fusion Algebra of an Extraspecial p-group3.2 Finite group modular dataIn this section, we concentrate on the modular data associated with anyfinite group G. We first fix a set R of representatives of each conjugacyclass of G. The identity e of G is in R, and more generally the center Z(G)of G is a subset of R. For any a ∈ G, let Ka denote the conjugacy class ofa in G and CG(a) be the centralizer of a in G. We have |G| = |Ka||CG(a)|.The primary fields of the G modular data are labeled by pairs (a,χ),where a ∈ R, and where χ is an irreducible character of CG(a). We willwrite Φ for the set of these pairs. We define the S matrix and T matrix asS(a,χ),(b,χprime) = 1|CG(a)||CG(b)|summationdisplayg∈G(a,b)χ(gbg−1)∗χprime(g−1ag)∗= 1|G|summationdisplayg∈Ka,h∈Kb∩CG(g)χ(xhx−1)∗χprime(ygy−1)∗,T(a,χ)(aprime,χprime) = δa,aprimeδχ,χprime χ(a)χ(e)where G(a,b) = {g ∈ G|agbg−1 = gbg−1a} and where x and y are anysolutions to g = x−1ax and h = y−1by. If G(a,b) is empty then the sum isequal to zero.The matrix S is symmetric and unitary, and gives rise (via the Verlindeformula) to non-negative integer fusion coefficients N(c,χ3)(a,χ1)(b,χ2), whereNcab =summationdisplayd∈ΦSadSbdS∗cdS0d .One way to generalize the group data is by introducing some twisting.The twisting of the modular data is described in [20], where the explicit34Chapter 3. The Fusion Algebra of an Extraspecial p-groupexpressions for the modular matrix S appear.The primary fields Φα in the model twisted by a given 3-cocycle α consistof all pairs (g,χ) where g ∈ R and χ is a βg twisted irreducible character ofCG(g), where βg is defined byβg(a,b) = α(g,a,b)α(a,a−1ga,b)−1α(a,b,(ab)−1gab)from a normalized element α of H3(G,U(1)). The S matrix is calculated asSα(a,χ)(b,χprime) = 1|G|summationdisplayg∈Ka,gprime∈Kb∩CG(g)parenleftbiggβg(gprime,x−1)βgprime(g,y−1)βg(x−1,h)βgprime(y−1,hprime)parenrightbigg∗χ(h)∗χprime(hprime)∗,= 1|G|summationdisplayg∈Ka,gprime∈Kb∩CG(g)parenleftbiggβa(x,gprime)βa(xgprime,x−1)βb(y,g)βb(yg,y−1)βa(x,x−1)βb(y,y−1)parenrightbigg∗χ(h)∗χprime(hprime)∗,where g = x−1ax = y−1hprimey, gprime = y−1by = x−1hx, h ∈ CG(a), hprime ∈ CG(b).In order to compute the T matrix, in [20]the 1-cochain epsilon1a : CG(a) → U(1)is introduced. It is determined by the following equalities:epsilon1a(e) = 1,βa(h,g) = epsilon1a(h)epsilon1a(g)epsilon1a(hg)−1,epsilon1x−1ax(x−1hx) = βa(x,x−1hx)βa(h,x) epsilon1a(h),for all g, h ∈ CG(a) and x ∈ G. Then the T matrix isTα(a,χ)(aprime,χprime) = δa,aprimeδχ,χprime χ(a)χ(e)epsilon1a(a).35Chapter 3. The Fusion Algebra of an Extraspecial p-group3.3 Twisted example G = (Z/p)3The simplest non-cyclic group is G = (Z/n)2 for a positive integer n, but itdoes not lead to something new (see [8]). The cohomology groupH3((Z/n)2,U(1)) = (Z/n)3has three generators, but all 3-cocycles give β’s which are all coboundaries.Thus all twistings are cohomologically trivial.The more interesting case of G = (Z/n)3 is given in [20]. Here, we recallthat work in order to compare it with our calculation in the next section:The cohomology H3(G,U(1)) = (Z/n)7 is generated by the followingcocycles:α(j)I (a,b,c) = expbraceleftBig2ipiaj(bj +cj −bj +cj)/n2bracerightBig, 1 ≤ j ≤ 3,α(jk)II (a,b,c) = expbraceleftBig2ipiaj(bk +ck −bk +ck)/n2bracerightBig, 1 ≤ j < k ≤ 3,αIII(a,b,c) = expbraceleftBig2ipia1b2c3/nbracerightBig,where the group elements are the triplets a = (a1,a2,a3). The cocycleswhich involve a non-trivial power of αIII define non-trivial twistings. The2-cocycles are of the formβa(b,c) = expbraceleftBig2ipiq(a1b2c3 −b1a2c3 +b1c2a3)/nbracerightBig.Given a, we wish to count the number of classes b (elements here)which are βa-regular, i.e., which satisfy βa(b,c) = βa(c,b) for all c. Tak-36Chapter 3. The Fusion Algebra of an Extraspecial p-grouping c = (1,0,0),(0,1,0) and (0,0,1), the βa-regular elements b are thosewhich satisfya2b3 −a3b2 ≡ a1b3 −a3b1 ≡ a1b2 −a2b1 ≡ 0 (mod f),where f = n/gcd(q,n). The number of solutions (b1,b2,b3) ∈ (Z/n)3 to thissystem is equal to n3[gcd(a1,a2,a3,f)/f]2.It remains to sum those numbers for all a to obtain the number of pri-maries:|Φα| = n6f3productdisplayp|fp primebracketleftBig(pkp −1)(1+p−1 +p−2)+1bracketrightBig.We now consider the case when n is an odd prime number p, and whenthe 3-cocycle is αIII.When G is abelian, all factors in the formula for Sα that involve thecocycles drop out, and one is left with the simple expression:Sα(a,˜χ),(b,˜χprime) = 1|G|χ∗(b)χprime∗(a),where χ and χprime are respectively βa and βb-projective characters, for thecocycles given above with q = 1.It remains to compute the projective characters. One then finds p dis-tinct, irreducible βa-projective representations of dimension p if a is not theidentity, while there are of course p3 representations of dimension 1 if a = e.Depending on the value of a = (a1,a2,a3), the characters are given in thefollowing table, where it is implicit that the element g = (g1,g2,g3) mustbe βa-regular for the character not to vanish. In the first three cases, thecharacter label u runs over Z/p, and in the last column, vectoru takes all values37Chapter 3. The Fusion Algebra of an Extraspecial p-groupin (Z/p)3.a1 negationslash= 0 a1 = 0, a2 negationslash= 0 a1 = a2 = 0, a3 negationslash= 0 a1 = a2 = a3 = 0χ(g) pξa−11 ug1−a−11 a2a3g21/2p pξa−12 ug2p pξa−13 ug3p ξvectoru·vectorgpIf for instance a2 is also invertible, then a−11 g1 = a−12 g2, so that the firstcharacter value is also equal to ˜χ(g) = pξa−12 ug2−a1a−12 a3g22/2p . The primaryfields are thus (e,χvectoru) and (a, ˜χu), for a total of|Φα| = p3 +(p3 −1)p = p4 +p3 −p.The matrices Sα and Tα are now straightforward to establish. Takingthe condition of β-regularity into account, one finds that Sα(a,χu)(b,χuprime)isalmost block-diagonal:1p1p21pξ−u1b1−u2b2−u3b3p 1pξ−u2b2−u3b3p 1pξ−u3b3p1pξ−uprime1a1−uprime2a2−uprime3a3pξ−ua−11 b1−uprimeb−11 a1+(a2a3b1+b2b3a1)/2p×δ(b2−a−11 a2b1)δ(b3−a−11 a3b1) 0 01pξ−uprime2a2−uprime3a3p 0ξ−ua−12 b2−uprimeb−12 a2p×δ(b3−a−12 a3b2) 01pξ−uprime3a3p 0 0 ξ−ua−13 b3−uprimeb−13 a3pwhere the blocks correspond to the subsets {a = e}, {a1 negationslash= 0} , {a1 =0, a2 negationslash= 0}, and {a1 = a2 = 0, a3 negationslash= 0}.The entries Tα(a,χ)(a,χ) are 1 for a = e = (0,0,0) and ξu−a1a2a3/2p in anyother case.In the next section, we observe that the modular data of the extraspecialp-group gives the same fusion coefficients with this twisted case. Hence wecan conclude that these two fusion algebras are the same.38Chapter 3. The Fusion Algebra of an Extraspecial p-group3.4 Modular data for extraspecial p-groupWhen discussing what he called electric/magnetic duality in [35], Propitiusobserved that the modular data for (Z/2)3 with a particular twist equalsthat of the untwisted dihedral group D8 (for an appropriate identificationof primary fields). In the more recent work of [20] it is claimed that thereare many more such examples; when calculating the modular data for anappropriately twisted (Z/p)3, it is noted that the quantum dimensions andnumber of primaries suggest that this twisted data would yield the modulardata for the extraspecialp-groupH of orderp3, which is the central extensionof the cyclic subgroup Z/p×Z/p. This group may be represented as〈A,B,C|Ap = Bp = Cp = [A,C] = [B,C] = 1,[A,B] = C〉.In this section we prove that this is indeed the case.It is convenient to use an isomorphic realization of H via (r,s,t) ∈ (Z/p)3with the following defining relations.(r,s,t)−1 = (−r,−s,−t−rs),(r,s,t)(rprime,sprime,tprime) = (r +rprime,s+sprime,t+tprime −srprime).Theirreduciblecharactersarecalculatedin[29]. Forfixedx, y, z in{1,2,...,p−1}, we define the character asχ(r,s,t) =0, if pd notbar s∨(pd notbar r∧z negationslash= 0);pξtz+rx+syp , otherwise;where d = gcd(z,p). It may be rewritten in a more compact form using the39Chapter 3. The Fusion Algebra of an Extraspecial p-groupKronecker delta function:χ(r,s,t) =ξrx+syp , if z=0;δ(p|r)δ(p|s)pξrx+sy+tzp , if z negationslash= 0;where δ(p|r) = 1 if p = r and 0 otherwise. Using this triple realization, theconjugacy class of the element (a,b,c) can be given as C(a,b,c) = {(a,b,c +bx−ay) | x,y ∈ Z/p}. Thus two elements, (a,b,c) and (aprime,bprime,cprime), belong tothe same conjugacy class if and only if a = aprime, b = bprime and there exist integersx, y and z such that c = cprime −ay +bx+pz. This equation is solvable if andonly if c ≡ cprime ( mod gcd(a,b,p)). Therefore every conjugacy class containsexactly one element of the setL = {(a,b,c) ∈ {0....,p−1}3|c < gcd(a,b,p)}.If neither a nor b is equal to 0 then gcd(a,b,p) = 0. Hence c should beequal to 0. If a = b = 0 then c can be any number in {0,...,p − 1} asgcd(0,0,p) = p in this case. So the representative of the conjugacy classesare either in the form of (a,b,0) negationslash= (0,0,0) or (0,0,c). We now calculatethe centralizer for each element. The elements (r,s,t) of CG(a,b,0) shouldsatisfy(r,s,t)(a,b,0) = (a,b,0)(r,s,t),(a+r,s+b,t−as) = (a+r,s+b,t−rb).Therefore as = rb and (r,s,t) = (a,asr−1,c). CG(a,b,0) is of order p2.From the basic results of finite group theory there are two groups of orderp2 up to isomorphism. CG(a,b,0) cannot be the cyclic one as it has at leasttwo elements of order p, namely (a,0,0) and (0,0,c). Thus, CG(a,b,0) is40Chapter 3. The Fusion Algebra of an Extraspecial p-groupisomorphic to Z/p×Z/p.The elements (r,s,t) of CG(0,0,c) should satisfy(r,s,t)(0,0,c) = (0,0,c)(r,s,t),(r,s,c+t) = (r,s,c+t).Hence CG(0,0,c) = G.Let us denote the irreducible characters by Γi|p2i=1 and χi|p2+p+1i=1 for Z/p×Z/p and G, respectively. The primary fields of the modular data may bewritten as parenleftbig(0,0,c),χiparenrightbig for all c ∈ G and parenleftbig(a,b,0),Γiparenrightbig where (a,b,0) negationslash=(0,0,0). We find that the number of the primary fields is p(p2 + p − 1) +(p2 −1)p2 = p4 + p3 −p which is equal to the number of primary fields oftwisted modular data of (Z/p)3.We next calculate the entries of the S matrix. We first consider theprimary fields (a,χi) and (b,χj) where a = (0,0,r) and b = (0,0,t). We usethe formula given in the second section:S(a,χi)(b,χj) = 1|CG(a)||CG(b)|summationtextg∈G(a,b) χi(gbg−1)∗χj(g−1ag)∗.We know |CG(a)||CG(b)| = p6, G(a,b) = {g ∈ G|agbg−1 = gbg−1a} ={g ∈ G|(0,0,r + t) = (0,0,r + t)} = G as well as χi(gbg−1) = χi(b) forg ∈ G. HenceS(a,χi)(b,χj) = 1p3χi(b)∗χj(a)∗.By the above formula for χ, if z in the formula is equal to 0 then there arep2 characters such that χ(0,0,t) = 0. We denote these characters by χi,jwhere i and j are in {0,...,p−1}. If z negationslash= 0 then there are p−1 charactersof the form as z ∈ {1,...,p−1}. We denote these characters by χzi afterindexing the (p−1) z’s. Hence, by the formula for the S matrix, we have41Chapter 3. The Fusion Algebra of an Extraspecial p-groupS((0,0,r),χi,j)((0,0,t),χiprime,jprime) = 1/p3,S((0,0,r),χzi)((0,0,t),χzj) = 1/pξ−rzj−tzip ,S((0,0,r),χi,j)((0,0,t),χz) = 1/p2ξ−rzp .One may compare these results with the Sα matrix of the twisted mod-ular data for (Z/p)3. In order to get the same entries as Sα, we change((0,0,t),χzi) to ((0,0,z−1i t),χzi), and notice that zi corresponds to a3 inthe notation of the twisted matrix. We basically proved that the blocksat the corners of each matrix are the same. It is trivial to see that thedimension of these blocks is also the same.We now need to check the other blocks. For the other blocks we need toconsider the primary fields ((r,s,0),Γi) where s or t are not equal to 0. Wehave the following formulaS((r,s,0),Γi)((0,0,t),χz) = 1|CG(a)||CG(b)|summationdisplayg∈G(a,b)Γi(gbg−1)∗χz(g−1ag)∗where a = (0,0,t) and b = (r,s,0). By the Kronecker delta functions δ(p|r)and δ(p|s) in the definition of χ, the character χz(g−1ag) should be equalto zero as at least one of r or s is nonzero. These zeros corresponds to the(4,2), (4,3), (2,4) and (3,4) block entries of the Sα matrix of the twistedcase if we index the blocks starting from the left upper corner which is the(1,1) block.Another kind of entry that we need to consider isS((0,0,t),χi,j)((r,s,0),Γi) = 1|CG(a)||CG(b)|summationdisplayg∈G(a,b)χi,j(gbg−1)∗Γi(g−1ag)∗42Chapter 3. The Fusion Algebra of an Extraspecial p-groupwhere a = (0,0,t) and b = (r,s,0). Here, we know |CG(a)| = p3, |CG(b)| =p2 andG(a,b) = {g ∈ G|agbg−1 = gbg−1a}= {g ∈ G|(r,s,t−as+rb) = (r,s,t−as+rb)} = G.Moreover, we have g−1ag = a for g ∈ G andgbg−1 = (g1,g2,g3)(r,s,0)(−g1,−g2,−g3 −g1g2) = (r,s,g2r−g1s)by the formula for χ, χi,j(gbg−1) = ξrx+syp where x, y ∈ G are fixed for eachcharacter χi,j. We can change the notation x and y to u1 to u2. On the otherhand, Γi(0,0,t) = ξa3tp where we can define Γi(r,s,t) = ξvsp ξa3tp where v anda3 are fixed for each character Γi. We recall that, for (x,y,z) ∈ CG(r,s,0),x and y are the same up to a constant while z is independent than the othercoordinates. For the character Γi of Z/p×Z/p we thus consider the secondand third coordinates without loss of generality. We use this fact again inthe next calculation. As a result, we obtainS((0,0,t),χi,j)((r,s,0),Γi) = 1/p2ξ−ru1−su2−ta3p .Hence if r negationslash= 0 this block coincides with (1,2) and (2,1) blocks of the Sαmatrix. If r = 0 it coincides with the (1,3) and (3,1) blocks if we change r,s and t to a1, a2 and u3, respectively.The remaining blocks that we need to check are the four blocks atthe center. Let us first check S((0,s,0),Γi)((rprime,sprime,0),Γj) where rprime negationslash= 0. In thiscase we see that G(a,b) = {g ∈ G|agbg−1 = gbg−1a} = {g ∈ G|(rprime,s +sprime,g2rprime − g1sprime) = (rprime,s + sprime,g2rprime − g1sprime − srprime) = ∅ because srprime negationslash= 0. Hence43Chapter 3. The Fusion Algebra of an Extraspecial p-groupS((0,s,0),Γi)((rprime,sprime,0),Γj) = 0 corresponds to the zero blocks ((2,3) and (3,2)blocks) at the center.We continue with the block corresponding to S((0,s,0),Γi)((0,sprime,0),Γj). Herewe have G(a,b) = {g ∈ G|agbg−1 = gbg−1a} = {g ∈ G|(0,s + sprime,−g1sprime) =(0,s + sprime,−g1sprime)} = G, gbg−1 = (0,sprime,−sprimeg1) and g−1ag = (0,s,sg1) whereg = (g1,g2,g3), and recall that we defined Γi and Γj as Γi(r,s,t) = ξvsp ξa3tpand Γj(r,s,t) = ξvprimesp ξb3tp . If a3sprime = b3s thenΓi(gbg−1)∗Γj(g−1ag)∗ = ξ−vsprimep ξa3sprimeg1p ξ−vprimesp ξ−b3sg1p= ξ−vsprime−vprimesp ξg1(sprimea3−b3s)p = ξ−vsprime−vprimesp .If a3sprime negationslash= b3s which means δ(a3sprime −b3s) = 0, thenΓi(gbg−1)∗Γj(g−1ag)∗ = ξ−vsprime−vprimesp ξg1(sprimea3−b3s)p .We now consider the S-matrix formulaS((0,s,0),Γi)((0,sprime,0),Γj) = 1/p4summationdisplayg∈Gξ−vsprime−vprimesp ξg1(sprimea3−b3s)p .If a3sprime negationslash= b3s, then ξg1(sprimea3−b3s)p is summed while g1 is covering all Z/p p2times. As the sum of all the roots of unity is 0, the entry S((0,s,0),Γi)((0,sprime,0),Γj)must be zero. Otherwise S((0,s,0),Γi)((0,sprime,0),Γj) = 1/pξ−vsprime−vprimesp . Any non-zeropower of ξp is also a pth root unity. Hence we can choose v = ua−12 andvprime = uprimeb−12 . If we change the notation s = a2 and sprime = b2, we obtain the(3,3) block of the twisted S matrix of (Z/p)3. Note that this assignmentdoes not change the order of the entries in the first column or in the firstrow as they are determined by second factor ξa3tp of the character Γi.44Chapter 3. The Fusion Algebra of an Extraspecial p-groupThe (2,2) block is the only block left in the twisted Sα of (Z/p)3. Weneed to calculate S((r,s,0),Γi)((rprime,sprime,0),Γj). Here G(a,b) = {g ∈ G|agag−1 =gbg−1a} = {g ∈ G|(r+rprime,s+sprime,g2r−g1sprime−rprimes) = (r+rprime,s+sprime,−g1sprime−sprimer)is G if and only if rprimes = sprimer. Otherwise G(a,b) is an empty set, a case whichis covered by examining the Kronecker delta functionδ(sprime−r−1srprime), as theseterms are in the abelian group Z/p.We now assume that rprimes = sprimer. We also have gbg−1 = (rprime,sprime,rprimeg2 −sprimeg1) and g−1ag = (r,s,sg1 − rg1) where g = (g1,g2,g3). By the formulaeΓi(r,s,t) = ξvsp ξa3tp and Γj(r,s,t) = ξvprimesp ξb3tp we get the multiplicationΓi(gbg−1)∗Γj(g−1ag)∗ = ξ−vsprime−vprimesp ξg1(sprimea3−b3s)+g2(rprimea3−rb3)p .We note that sprimea3 −b3s = 0 if and only if rprimea3 −rb3 as rprimes = sprimer. If theseexponents are non-zero, the entry in our S is zero because gi is coveringthe set G by resolving the power sum of the ξp which is equal to zero.Hence another Kronecker delta function appears in our expression which isδ(rprimea3 −rb3). If δ(rprimea3 −rb3) = 1 then S((r,s,0),Γi)((rprime,sprime,0),Γj) = 1/pξ−vsprime−vprimesp .We now make the conventional choice of v and vprime as ua−12 − a1a3/2 anduprimeb2−1 −b1b3/2. HenceS((r,s,0),Γi)((rprime,sprime,0),Γj) = 1/pξ−(ua−12 −a1a3/2)sprime−(uprimeb2−1−b1b3/2)sp .We also change the notation r = a1, s = a2, rprime = b1 and sprime = b2 to obtainξ−ua−12 b2−uprimeb−12 a2+(a1a3b2+b1b3b2)/2p = ξ−ua−11 b1−uprimeb−11 a1+(a2a3b1+b2b3a1)/2,pwhere the equality follows from b1a2 = a1b2 and b2a3−b3a2 = 0. The choiceof v and vprime is the same as the last part we have calculated because r = a1 = 045Chapter 3. The Fusion Algebra of an Extraspecial p-groupin that case. We note that the change in variables does not change the orderof the entries in first column or row as they are determined by second factorξa3tp of the character Γi. This completes our calculation of the S matrix of Hwhich as a result is the same as the twisted Sα matrix of the group (Z/p)3We can write the change of variables as a map from the primary field((r,s,0),Γi) to ((a1,a2,a3),χu) in the twisted fusion algebra where r and sare mapped to a1 and a2, respectively. The two factors in the character Γiare mapped to a3 and u.We next calculate the T matrix. We again start with the primary fieldof type ((0,0,t),χi,j) for whichT((0,0,t),χi,j)((0,0,t),χi,j) = χi,j(0,0,t)/χi,j(0,0,0) = 1.If the field is ((0,0,t),χzi) thenT((0,0,t),χzi)((0,0,t),χzi) = χzi(0,0,t)/χzi(0,0,0) = ξtzip .Remember that we replace (0,0,t) by (0,0,tz−1i ). Hence T((0,0,t),χzi)((0,0,t),χzi)is ξtp.If the field is of type ((0,s,0),Γi) thenT((0,s,0),Γi)((0,s,0),Γi) = Γi(0,s,0)/Γi(0,0,0) = ξsvp .By our previous change of v and s to ua−12 and a2, respectively, we obtainξupfor this primary field.The last primary field to check is the field of type ((r,s,0),Γi) for whichT((r,s,0),Γi)((r,s,0),Γi) = Γi(r,s,0)/Γi(0,0,0) = ξsvp ,46Chapter 3. The Fusion Algebra of an Extraspecial p-groupwhere we replace s and v by a2 and ua−12 −a1a3/2, respectively. Thereforewe obtain the term ξu−a1a2a3/2p . We conclude that the matrix Tα of the group(Z/p)3 for a particular twist α and the T matrix of the extraspecial p-groupH of order p3 with exponent p coincide.As a result, we have proved the following theorem.Theorem 3.4.1. The twisted modular data of the group (Z/p)3 are the sameas the modular data of the extraspecial p-group of order p3 with exponent p.We also have the following corollary.Corollary 3.4.2. The twisted fusion algebra of (Z/p)3, which is isomor-phic to the twisted orbifold K-theory αKorb(∧[∗/G]) in the sense of [8], isisomorphic to the fusion algebra of the extraspecial p-group of order p3 withexponent p, which is isomorphic to the orbifold K-theory Korb(∧[∗/H]).47Chapter 4Cohomology of ToroidalOrbifolds4.1 IntroductionLet G be a finite group and ϕ : G → GLn(Z) an integral representationof G. In this way G acts linearly on Rn preserving the integral lattice Zn,thus inducing a G-action on the torus Xϕ = X := Rn/Zn. The quotientX → X/G naturally has the structure of an orbifold as a global quotient,and these kind of orbifolds are usually referred to as toroidal orbifolds. Thegoal of this chapter is to compute the cohomology groups H∗(X/G;Z) forthe particular case where G = Z/p for a prime number p.The quotients of the form X/G appear naturally in different contexts.For example, given a topological space Y, the m-th cyclic product of Y isdefined to be the quotientCPm(Y) := Y m/Z/m,where Z/m acts by cyclically permuting the product Y m. In the particularcase where the representation ϕ : G → GLn(Z) induces the ZG-module(ZG)n, the associated torus X is (S1)p)n, where G acts cyclically on each48Chapter 4. Cohomology of Toroidal Orbifolds(S1)p and diagonally on the product ((S1)p)n. In this caseX/G = (((S1)p)n)/Z/p ∼= ((S1)n)p/Z/pwhere now Z/p acts cyclically on the ((S1)n)p. ThereforeX/G ∼= CPp((S1)n).The homology groups of quotient spaces of the form Xm/K, where K ⊂ Σm,have long been studied. In particular, in [38] Swan formulated a method forthe computation of homology of cyclic products of topological spaces.4.2 PreliminariesLet G be a finite group and ϕ : G → GLn(Z) an integral representation ofG. Consider X = Xϕ the standard torus with the action of G induced fromϕ. Then there is a fibration sequenceX → X ×G EG → BG. (4.1)Using the long exact sequence in homotopy groups associated to this fi-bration, it follows that X ×G EG is an Eilenberg-Maclane space of typeK(Γ,1), where Γ := pi1(X×GEG). The G-action on X makes L := pi1(X) ∼=H1(X;Z) ∼= Zn into a ZG-module. Moreover, since [0] ∈ Rn/Zn is alwaysa fixed point for the action of G on X, it follows that the fibration (4.1)has a section. The existence of such a section implies that the short exactsequence1 → pi1(X) → pi1(X ×G EG) → pi1(BG) → 149Chapter 4. Cohomology of Toroidal Orbifoldshas a section and therefore Γ ∼= LmulticloserightGis a semi-direct product. For example,when the representation ϕ is injective, the group Γ is a crystallographicgroup.The cohomology groups of the groups of the form Γ ∼= L multicloseright G, whenG = Z/p for a prime number p, were computed in [4, Theorem 1.1] whereit is shown that a certain special free resolution epsilon1 : F → Z of Z as aZ[L]-module admits an action of G compatible with ϕ. This implies thatthe Lyndon-Hochshild-Serre spectral sequence associated to the short exactsequence1 → L → Γ → G → 1collapses on the E2-term without extension problems, thus for any k ≥ 0Hk(Γ;Z) ∼=circleplusdisplayi+j=kHi(G;jlogicalanddisplay(L∗)),where L∗ denotes the dual module Hom(L,Z).One application of this, by [4, Theorem 1.2] is when G = Z/p acts on Xvia a representation ϕ : G → GLn(Z), thenHk(X ×G EG;Z) ∼=circleplusdisplayi+j=kHi(G;Hj(X;Z)).for each k ≥ 0. The strategy we will use to compute the cohomology groupsH∗(X/G;Z) is as follows. Let F denote the subspace of fixed points underthe G-action. In general, F will be a disjoint union of product of circles.Then the long exact sequence in cohomology associated to the pair (X/G,F)can be used to compute the cohomology groups H∗(X/G;Z). To make this50Chapter 4. Cohomology of Toroidal Orbifoldswork, the relative cohomology groups H∗(X/G,F;Z) need to be determined.For this, we consider the equivariant projection pi1 : X ×EG → X. On thelevel of orbit spaces pi1 induces a mapφ : X ×G EG → X/G.By [17, Proposition V 1.1], φ induces the isomorphismφ∗ : H∗(X/G,F;Z) → H∗G(X,F;Z).This reduces the problem to one of determining the cohomology groupsH∗G(X,F;Z). These groups will be computed using representation theoryand the fact that the Lyndon-Hochshild-Serre spectral sequence associatedto the fibration sequenceX → X ×G EG → BGcollapses on the E2-term without extension problems.Let L := H1(X;Z) ∼= pi1(X). As explained, L has the structure of aZG-lattice; this structure determines the cohomology groups of X/G. LetR = Z(p) be the ring of integers localized at the prime p. Then (see [21])there are only three distinct isomorphism classes of RG-lattices, namelythe trivial module R, the augmentation ideal IG and the group ring RG.Moreover, if L is any finitely generated ZG-lattice, then there is a ZG-latticeLprime ∼= Zr ⊕ZGs ⊕IGt and ZG homomorphism f : Lprime → L such that f is anisomorphism after tensoring with R. In general, a ZG-module of the formZr ⊕ZGs ⊕IGt is called a module of type (r,s,t).51Chapter 4. Cohomology of Toroidal OrbifoldsA fundamental tool in the computation of the cohomology groups of thetoroidal orbifolds is the next lemma.Lemma 4.2.1. Suppose that p is a prime number. Let G = Z/p act on afinite dimensional space X with fixed point set F. If there is an integer Nsuch that Hk(X,F;Z) = 0 for k > N, then HkG(X,F;Z) = 0 for k > N.Proof. This follows by applying [17, Exercise III.9] and [17, PropositionVII 1.1].From now on G denotes the group Z/p for a prime number p. Givena representation ϕ : G → GLn(Z), X denotes the G-space Rn/Zn withthe G-action induced by ϕ and F denotes the fixed point set under thisaction. The representation ϕ makes L : H1(X;Z) into a ZG-module whosestructure completely determines the cohomology groups H∗(X/G;Z) as itwill be shown in the next section in the case of L = IGm.4.3 The case L = IGmIn this section the particular case where the ZG-lattice L equals IGm isconsidered. Let ρm := ⊕mρ : G → GLN(Z) be the integral representationinducing the ZG-module L, where N = m(p − 1). The fixed point set ofsuch an action is easily identified by the following straightforward lemma.Lemma 4.3.1. If X = X⊕mρ is the G-space induced by the representation⊕mρ then the fixed point set F under this G-action is a discrete set with pmpoints.Proof. Consider the short exact sequence of G-modules defining the G-space X0 → L → LmulticloserightR → (LmulticloserightR)/L = X → 0.52Chapter 4. Cohomology of Toroidal OrbifoldsThis short exact sequence induces a long exact sequence on the level of groupcohomology0 → H0(G,L) → H0(G,LmulticloserightR) → H0(G,X) → H1(G,L) → H1(G,LmulticloserightR) → ...Note that H1(G,L multicloseright R) = 0 and H0(G,L) = H0(G,L multicloseright R) = 0, thusthere is an isomorphism F = H0(G,X) ∼= H1(G,L) ∼= (Z/p)m.We are interested in computing the cohomology groups H∗G(X,F). Thesecan be computed using the Serre spectral sequence for the pair (X,F)Ei,j2 = Hi(G,Hj(X,F)) =⇒ Hi+jG (X,F).Therefore the structure of Hj(X,F) as a G-module needs to be studied. Tothisend, weconsiderthelongexactsequenceincohomologyassociatedtothepair(X,F). SinceHj(F) = 0whenj ≥ 1, itfollows thatHj(X,F) ∼= Hj(X)for j ≥ 2 and there is a short exact sequence0 → Zpm−1 → H1(X,F) → (IG)m → 0. (4.2)By the classification theorem for ZG-modules in [21], there are integers aj,bjand cj ≥ 0, ideals A1,...,Aaj of ZG rank p−1 and projective indecomposablemodules P1,...,Pbj such thatHj(X,F) ∼=parenleftBigg ajcircleplusdisplay1AiparenrightBigg⊕bjcircleplusdisplay1Pj⊕parenleftBigg cjcircleplusdisplay1ZparenrightBigg.53Chapter 4. Cohomology of Toroidal OrbifoldsIn particular,Hi(G,Hj(X,F)) ∼=Zbj+cj if i = 0,(Z/p)aj if i is odd and i > 0,(Z/p)cj if i is even and i > 0.On the other hand, for j ≥ 2 we obtainHj(X,F) ∼= Hj(X) ∼=jlogicalanddisplay(IG)m.Hence the rank of logicalandtextj(IG)m is equal to the rank of Hj(X,F) ∼= (circleplustextaj1 Ai)⊕(circleplustextbj1 Pi)⊕(circleplustextcj1 Z). This implies the following equation when j ≥ 2:parenleftBigm(p−1)jparenrightBig= aj(p−1)+bjp+cj. (4.3)For each m,j ≥ 0, we define pm(j) to be the number of all possiblesequences of integers l1,...,lm such that 0 ≤ lr ≤ p−1 and l1 +···lm = j.This forces pm(j) = 0 for j > N = (p−1)m, pm(0) = 1, pm(1) = m and byinduction it is easy to see thatm(p−1)summationdisplayj=0pm(j) = pm.When j ≥ 2, it follows from [3, Proposition 1.10] thatHi(G,Hj(X,F)) ∼=(Z/p)pm(j) if i+j is even and i > 0,0 if i+j is odd and i > 0.54Chapter 4. Cohomology of Toroidal OrbifoldsThis result impliesaj =pm(j) if j is odd and j ≥ 2,0 if j is even and j ≥ 2,andcj =0 if j is odd and j ≥ 2,pm(j) if j is even and j ≥ 2We also need to find H0(G,Hj(X,F)) ∼= Zbj+cj. When j ≥ 2 by equa-tion (4.3) we obtainbj =1pbracketleftBigparenleftBigm(p−1)jparenrightBig−(p−1)pm(j)bracketrightBigif j is odd,1pbracketleftBigparenleftBigm(p−1)jparenrightBig−pm(j)bracketrightBigif j is even.The Ei,j2 of the this spectral sequence is described belowEi,j2 =0 if j = 0,Zbj+cj if i = 0 and j > 0,(Z/p)pm(j) if i+j is even and i > 0, j ≥ 2,0 if i+j is odd and i > 0, j ≥ 2,(Z/p)a1 if i is odd and j = 1,(Z/p)c1 if i is even and j = 1.Consider the Serre spectral sequence˜Ei,j2 = Hi(G,Hj(X)) =⇒ Hi+jG (X)55Chapter 4. Cohomology of Toroidal Orbifoldsassociated to the fibration sequence X → X ×G EG → BG. As it waspointed out before this sequence collapses on the E2-term. The inclusionX → (X,F) defines a map of spectral sequences fi,jr : Ei,jr → ˜Ei,jr . Noticethat fi,j2 is an isomorphism when j ≥ 2. This implies that the only possiblenontrivial differentials in the spectral sequence {Ei,jr } must land in Ei,1r . Onthe other hand, by Lemma 4.2.1 it follows that HkG(X,F) = 0 for k > N; thismeans that there are no permanent cocycles of total degree k with k > N.We notice that all the differentials that end at E2k+1,12 ∼= (Z/p)a1 start attrivial groups. This implies that if 2k + 1 > N then E2k+1,12 must be thetrivial group and thus a1 = 0. On the other hand, when 2 ≤ j ≤ N and i+jeven, the only possible nonzero differential starting at Ei,j2 ∼= (Z/p)pm(j) isdj : Ei,jj → Ei+j,1j .If i + j > N, there are no permanent cocycles in Ei,jj and thus dj must beinjective. We note that the spectral sequence Ei,jr is a spectral sequence ofH∗(BG)-modules and we can find t ∈ H2(BG) such that multiplication bytk is an isomorphism tk : Ei,jj → Ei+2k,jj . Since dj is a map of H∗(BG)modules, we are able to choose k big enough such that i + j + 2k > N; itfollows that dj : Ei,jj → Ei+j,1j is injective for all i+j > 0 and its image hasrank cj = pm(j) when i = 0 and j is even. Moreover Zbj ⊂ (Hj(X,F))Glies inside the image of the norm map.We next consider the transfer map associated with the trivial subgroup{1} → G. This map preserves the filtrations that induce the Serre spectralsequence and thus it induces a map of the corresponding spectral sequencesτG1 : Hi({1},Hj(X,F)) → Hi(G,Hj(X,F)).56Chapter 4. Cohomology of Toroidal OrbifoldsSince the image of the transfer mapτG1 : H0({1},Hj(X,F)) → H0(G,Hj(X,F))consists of elements in the image of the norm map, it follows that all thedifferentials in the Serre spectral sequence vanish on the elements that arein the image of the norm map; in particular, dj is trivial on the summandZbj.We conclude that when 2k > N, E2k,12 ∼= (Z/p)c1, E2k,13 ∼= (Z/p)c1−pm(2),... ,E2k,1N+1 ∼= (Z/p)c1−(PNj=2 pm(j)). Also, if 2k+1 > N, the group E2k,1N+1 mustvanish because its elements are not permanent cocycles and there are nonontrivial differentials with target E2k,1N+1 as Ei,jN+1 = 0 for j ≥ 2. This showsthat c1 = (summationtextNj=2 pm(j)) = pm −1−m. By the short exact sequence (4.2),we obtain b1 = m.Therefore E0,1∞ ∼= Z(PNj=1 pm(j)) = Zpm−1, Ek,1∞ ∼= (Z/p)(PNj=k+1 pm(j)) for0 < k ≤ N even, E0,k∞ ∼= Zbk+ck for 1 < k ≤ N and the other groups aretrivial.By studying the Serre spectral sequence for the pair (X,F) with coeffi-cients in Z/q with q a prime number different from p and also with rationalcoefficients, we can conclude that there are no extension problems and thefollowing theorem is obtained.Theorem 4.3.2. Suppose that G = Z/p acts on X ∼= ((S1)p−1)m via the57Chapter 4. Cohomology of Toroidal Orbifoldsrepresentation ⊕mρ. If F denotes the fixed point set under this action, thenHkG(X,F) ∼=0 if k = 0,Zpm−1 if k = 1,Zγk if k is even and k > 0,Zγk ⊕(Z/p)(Pm(p−1)j=k pm(j)) if k is odd and k > 0.where γk = bk +ck = 1/p[parenleftBigm(p−1)kparenrightBig+(−1)k(p−1)pm(k).Consider now the long exact sequence in cohomology associated to thepair (X/G,F). Since F is a discrete set in this case, it follows at once thatHk(X/G,F) ∼= Hk(X/G) whenever k ≥ 2. On the other hand, for k = 1it is easy to see directly that H1(X/G) = 0. This fact, together with [17,Proposition VII 1.1] proves the following corollary.Corollary 4.3.3. Suppose that G = Z/p acts on X ∼= ((S1)p−1)m via therepresentation ρm, thenHk(X/G;Z) ∼=Z if k = 0,0 if k = 1,Zγk if k is even and k > 0,Zγk ⊕(Z/p)(Pm(p−1)j=k pm(j)) if k is odd and k > 0.where γk = bk +ck = 1pbracketleftBigparenleftBigm(p−1)kparenrightBig+(−1)k(p−1)pm(k)bracketrightBig.58Bibliography[1] J. F. Adams. Stable homotopy and generalised homology. The Universityof Chicago Press, 1974.[2] A.Adem. Automorphisms and cohomology of discrete groups. Journalof Algebra, 182(3): 721-737, 1996.[3] A.Adem. Z/p actions on (Sn)k. Trans. Amer. Math. Soc., 300(2): 791-809, 1987.[4] A. Adem, J. Ge, J. Pan and N. Petrosyan. Compatible actions andcohomology of crystallographic groups. J. 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