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Computing motivic Donaldson-Thomas invariants Morrison, Andrew James 2012

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Computing Motivic Donaldson–Thomas Invariants by Andrew James Morrison B.A. Math., University of Cambridge, 2006 M.A. Cantab., University of Cambridge, 2007 M.Math., University of Cambridge, 2010 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2012 c© Andrew James Morrison 2012 Abstract This thesis develops a method (dimensional reduction) to compute motivic Donaldson–Thomas invariants. The method is then employed to compute these invariants in several different cases. Given a moduli scheme with a symmetric obstruction theory a Donaldson– Thomas type invariant can be defined by integrating Behrend’s function over the scheme. Motivic Donaldson–Thomas theory aims to provide a more re- fined invariant associated to each such moduli space - a virtual motive. From the modern point of view motivic Donaldson–Thomas invariants should be defined for a three dimensional Calabi–Yau category. These categories often arise in a geometric context as the derived category of representations of a quiver with potential. Provided the potential has a linear factor we are able to reduce the problem of computing the corresponding virtual motives to a much simpler one. This includes geometric examples coming from local curves which we compute ex- plicitly. ii Preface This thesis is a compendium of three papers. The Chapters 2, 3, and 4 are essentially three separate articles. Chapter 2 is a version of independent work published in [38]; Chapter 3 provides an application of the reduction theorem proven in Chap- ter 2 to the resolved conifold singularity. It is joint work with Prof. Sergey Mozgovoy, Prof. Kentaro Nagao and Prof. Balazs Szendrői [39], my indi- vidual contribution to this project was Section 3.3.2; Chapter 4 generalizes the results of Chapter 3 and comprises joint work with Prof. Kentaro Nagao [40]. Sections 4.5 and 4.9 are my main individual contribution to the work; [38] A. Morrison, Motivic invariants of quivers via dimensional reduc- tion, Selecta Mathematica, published online January 11th 2012. [39] A. Morrison, S. Mozgovoy, K. Nagao, B. Szendrői. Motivic Donaldson–Thomas invariants of the conifold and the refined topological ver- tex. submitted to Advances in Math., July 2011. [40] A. Morrison, K. Nagao. Motivic Donaldson–Thomas invariants of small crepant resolutions, submitted to Algebra & Number Theory, Novem- ber 2011. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Quiver with Potential . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Stability for Quiver Representations . . . . . . . . . . . . . . 8 2.4 Grothendieck Rings . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Virtual Motives . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Motivic Quantum Torus . . . . . . . . . . . . . . . . . . . . . 15 2.7 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 The Conifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.1 Motives . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2 λ-Rings and power structures . . . . . . . . . . . . . 32 iv 3.2.3 Quivers and moduli spaces . . . . . . . . . . . . . . . 34 3.2.4 Motivic DT invariants . . . . . . . . . . . . . . . . . . 36 3.2.5 Twisted algebra and central charge . . . . . . . . . . 38 3.2.6 Generating series of motivic DT invariants . . . . . . 40 3.3 The Universal DT Series of the Conifold Quiver . . . . . . . 41 3.3.1 Motivic DT invariants for the conifold quiver . . . . . 41 3.3.2 First proof . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.3 Second proof: another dimensional reduction . . . . . 48 3.3.4 Decomposing the universal series . . . . . . . . . . . . 53 3.4 Motivic DT with Framing . . . . . . . . . . . . . . . . . . . . 55 3.4.1 Framed quiver . . . . . . . . . . . . . . . . . . . . . . 55 3.4.2 Stability for framed representations . . . . . . . . . . 55 3.4.3 Motivic DT invariants with framing . . . . . . . . . . 56 3.4.4 Relating the universal and framed series . . . . . . . 57 3.4.5 Application to the conifold . . . . . . . . . . . . . . . 59 3.5 DT/PT Series . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5.1 Chambers and the moduli spaces for the conifold . . 60 3.5.2 Motivic PT and DT invariants . . . . . . . . . . . . . 61 3.5.3 Connection with the refined topological vertex . . . . 63 3.5.4 Connection to the cohomological Hall algebra . . . . 65 4 Local Toric Examples . . . . . . . . . . . . . . . . . . . . . . . 66 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.1.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1.2 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3 Non-Commutative Crepant Resolutions . . . . . . . . . . . . 74 4.3.1 Quivers with potential . . . . . . . . . . . . . . . . . 74 4.3.2 NCCR and derived equivalence . . . . . . . . . . . . . 77 4.3.3 Mutation and derived equivalence . . . . . . . . . . . 77 4.3.4 Cut and mutation . . . . . . . . . . . . . . . . . . . . 80 4.4 Motivic Donaldson–Thomas Invariants . . . . . . . . . . . . 81 4.4.1 Motives . . . . . . . . . . . . . . . . . . . . . . . . . . 81 v 4.4.2 Quivers and moduli spaces . . . . . . . . . . . . . . . 82 4.4.3 Motivic DT invariants . . . . . . . . . . . . . . . . . . 84 4.4.4 Generating series of motivic DT invariants . . . . . . 84 4.5 The Universal DT Series: Special Case . . . . . . . . . . . . 85 4.5.1 Step One: The invertible case Iσ(y) . . . . . . . . . . 90 4.5.2 Step Two: The nilpotent case Nσ(y) . . . . . . . . . 91 4.6 The Universal DT Series: General Case . . . . . . . . . . . . 103 4.6.1 Mutation and the root system . . . . . . . . . . . . . 103 4.6.2 Wall-crossing formula . . . . . . . . . . . . . . . . . . 103 4.6.3 Factorization of the universal series . . . . . . . . . . 104 4.7 Motivic DT with Framing . . . . . . . . . . . . . . . . . . . . 105 4.8 DT/PT Series . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.8.1 Chambers in the moduli spaces . . . . . . . . . . . . 106 4.8.2 Motivic PT and DT invariants . . . . . . . . . . . . . 106 4.8.3 Connection with the refined topological vertex . . . . 107 4.9 Linear Algebra Computation . . . . . . . . . . . . . . . . . . 107 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 126 5.2 Connections to String Theory . . . . . . . . . . . . . . . . . 127 5.3 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.3.1 Localization . . . . . . . . . . . . . . . . . . . . . . . 127 5.3.2 Motivic zeta functions . . . . . . . . . . . . . . . . . . 129 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 vi Acknowledgements I would like to thank all the staff and faculty at the UBC mathematics de- partment for creating the brilliant environment in which this work was done. First and foremost thanks is due to my supervisor Prof. Dr. Jim Bryan for all his mathematical support, and for providing such exciting and fruitful mathematical problems to think about. Secondly, I am grateful to Prof. Dr. Kai Behrend for his mathematical help and especially for his courses taught on foundational algebraic geometry. Then a general word of thanks goes to all my other friends, Prof. Dr. Patrick Brosnan, Prof. Dr. Kalle Karu, Prof. Dr. Zinovy Reichstein and all the postdocs and graduate students who have been a part of the algebraic geometry group here in Vancouver. As always I thank my family; Jim, Jeannie, Phil and Annette for their love and kindness to me. Since coming to the west coast of Canada I have been immensely blessed by the hospitality of the Fleming family; Johnnie, Carolyn, Ross, Rachel, Graham and Coen. I am especially grateful to my aunt Carolyn for her kindness, generosity and encouragement over these years. I will miss you all greatly. vii Dedication For Rachel and Coen Sanchez. viii Chapter 1 Introduction This thesis contains work collected from three research papers [38], [39], [40] found in Chapters 2, 3 and 4 respectively. Each of the chapters begins with an introduction to the results it contains. Chapter 2 - contains the dimensional reduction theorem used throughout the thesis in the computation of motivic Donaldson–Thomas invariants. The first six sections introduce all the basic material on quivers, moduli spaces, motives and vanishing cycles. The last section contains the proof of our dimensional reduction theorem. Chapter 3 - is our first application of the reduction theorem to the resolved conifold. Section 3.2 contains the preliminaries. Section 3.3 computes the re- duced series. Then section 3.4 computes the motivic DT invariants (for any choice of stability parameter). The final section identifies the DT/PT mod- uli spaces for the resolved conifold, giving a motivic version of the DT/PT correspondence. Chapter 4 - is a generalization of the results in Chapter 3 to a collection of local toric Calabi-Yau threefolds. Sections 4.2-4.4 contain preliminaries on root systems, non-commutative crepant resolutions and motivic DT invari- ants. Section 4.5 computes the reduced series in a special case (modulo the linear algebra calculations of 4.9). Then in section 4.6 this is generalized using mutations of the quiver and potential. Section 4.7 is almost identi- cal to 3.4 and computes the motivic Donaldson–Thomas invariants (for any choice of stability parameter). Section 4.8 gives another motivic DT/PT type correspondence. 1 Chapter 5 - is our conclusion. It recapitulates the main results of the thesis with some added analysis. We mention some related work in theoretical physics and end with proposals for future research. 2 Chapter 2 Dimensional Reduction This chapter contains a version of independent work published in [38]. It provides a reduction formula for the motivic Donaldson–Thomas invariants associated to a quiver with potential. The method is valid provided the superpotential has a linear factor, it allows us to compute virtual motives in terms of ordinary motivic classes of simpler quiver varieties. 2.1 Introduction In [60] Thomas defines integer valued invariants associated to compact mod- uli spaces of stable sheaves on a Calabi–Yau threefold. These numbers con- tain geometric information about the underlying manifold. In particular they provide a virtual count for the number of curves in each homology class, conjecturally equivalent [37] to the invariants of Gromov–Witten. Recently there have been several extensions of the integer valued Donaldson– Thomas invariant [33], [32], [4], [27]. In [33] Kontsevich and Soibelman pro- pose to work in a general three dimensional Calabi–Yau category. For a choice of stability condition they get moduli spaces of objects in this cate- gory each of which has a motivic DT invariant. The Euler numbers of these motives specialize to the classical invariant. Quivers with potential provide concrete examples of such CY3 categories. Although quiver categories are of independent interest in representation the- ory [54] they often contain geometric content, in particular the derived cat- egory of many local CY3 folds can be realized in this way. We provide a reduction theorem for motivic DT invariants of a general class of quivers with potential. The theorem expresses the motivic DT in- variants in terms of the ordinary motivic classes of certain reduced quiver 3 representations. In this way the theorem can be seen as a generalization of [4] where the quiver with potential models the Hilbert scheme of C3. Or from another point of view, in the cohomological Hall algebra, the reduction statement of [32] section 4.8 is an analog of our motivic result. The theorem is used here to compute the motivic DT invariants for crepant resolutions of toric singularities. We finish by mentioning some joint work with co-authors on the wall crossing behavior of these motives. Let Q be a finite quiver with vertex set Q0, that is a finite directed graph. The representations of Q are indexed by a dimension vector α ∈ NQ0 . At each vertex i we have a vector space Vi of dimension αi, and for each arrow a : i → j we have a linear map Ma : Vi → Vj . Such representations are considered unique up to the linear action of GL(Vi) on each vector space Vi. We produce a fine moduli space of framed quiver representations by taking a G.I.T quotient. The framing comes from the extra data of a vector f ∈ NQ0 \ {0}, and the stability condition is determined by a linearization of the group action χ; NQ(α, f) = ∏ a:i→j Hom(Vi, Vj)× ∏ i∈Q0 (Vi) fi//χ ∏ i∈Q0 GL(Vi). It is a smooth quasi-projective variety (see Section 2.3). The path algebra CQ of the quiver Q is the vector space over C with basis given by the paths in the quiver, the multiplication is then defined by concatenation of paths. An element W ∈ CQ of the path algebra is called a potential if it is the sum of cycles, i.e. paths that form loops. We are specifically interested in potentials that have a linear factor; W = L ·R here L is linear, that is a sum of length one paths. 4 The potential defines a function on the moduli space of representations of Q. Given a representation (Ma)a:i→j insert the matrix Ma in place of each occurrence of a in W then take the trace; TrWα,f : N Q(α, f)→ C the definition is invariant under the action of the gauge group and so gives a well defined regular function on the moduli space of framed quiver represen- tations. For a specific pair of vectors α, f ∈ NQ0 we define the DT moduli space of Q,W to be the scheme theoretic degeneracy locus of TrWα,f ; DTQ,W (α, f) = {dTrWα,f = 0} ⊂ NQ,W (α, f). Behrend, Bryan and Szendrői [4] define a virtual motive associated to each such a locus, essentially given by the motivic Milnor fibre of the map TrWα,f . Using the same definition we have a motivic DT invariant; [DTQ,W (α, f)]vir ∈ K0(V arC)[L− 1 2 ] taking values in the Grothendieck ring of varieties, adjoined with a for- mal inverse square root of the Lefschetz motive L = [A1C]. The conditions described in Section 2.5 allow us to neglect the usual µ̂-action on these in- variants. When the potential has a linear factor W = L · R, there exists a reduced space of quiver representations; RQ,W (α) = {(Ma)a:i→j | R((Ma)a:i→j) = 0} ⊂ ∏ a:i→j Hom(Vi, Vj). Here the space is a variety and also we have no G.I.T quotient. Our result expresses the virtual motives of the DT moduli spaces in terms of the ordi- nary motives of the reduced spaces. It is best phrased in the context of a motivic quantum torus. Let MStC be the Grothendieck ring of varieties where we formally invert the general linear groups and a square root of the Lefschetz motive. Now 5 consider the formal power series over this ring, as aMStC -module it is defined TQ :=  ∑ α∈NQ0 mαt α | mα ∈MStC  . The Euler form 〈, 〉Q, a pairing associated to the adjacency matrix of Q, defines a non-commutative product ∗ on this set given by, tα ∗ tβ = L 12(〈α,β〉Q−〈β,α〉Q)tα+β. TQ is the motivic quantum torus of the quiver Q. The virtual motives of the DT moduli spaces give an element; ZQ,W,f (t) := ∑ α∈NQ0 [DTQ,W (α, f)]virt α ∈ TQ. Similarly the reduced spaces can be assembled into a series; RQ,W (t) := ∑ α∈NQ0 [RQ,W (α)] [GL(α)] L 1 2 〈α,α〉Qtα ∈ TQ here GL(α) = ∏ i∈Q0 GLαi . In this context our main result reads simply, RQ,W (L f 2 t) = ZQ,W,f (t) ∗ RQ,W (L− f2 t) where (Lβt)α = Lβ·αtα. This describes all the motivic DT invariants in terms of the ordinary motivic classes of the reduced spaces. 2.2 Quiver with Potential Let Q be a finite quiver with vertex set Q0 and arrows a : i→ j. A path in the quiver is a sequence of arrows a1 ·a2 · · · an so that the head of the arrow ai coincides with the tail of the arrow ai+1, each path has a length given by the number of arrows of which it is composed. The path algebra of the quiver CQ is the C vector space with basis the 6 set of all paths and with multiplication given by concatenation of paths. An element W ∈ CQ is called a potential if all the monomials in W are cycles. Then W̃ ∈ CQ/[CQ,CQ] is the class of W up to equivalence of cyclically permuting terms in any monomial. Ginzburg [20], provides a detailed account of such algebras and potentials. Our main interest is in their representations. For each dimension vector α ∈ NQ0 we define a linear space of representations of Q, MQ(α) = ∏ a:i→j Hom(Cαi ,Cαj ) the group of automorphisms GL(α) = ∏ i∈Q0 GLαi(C) acts on this space by change of basis. The potential W defines a function on the representations of Q. So that given a representation (Ma)a:i→j , we define W ((Ma)a:i→j) to be the endomorphism obtained by inserting Ma in the place of each occurrence of a in W . Taking the trace of this linear map gives a complex number TrW ((Ma)a:i→j) well defined under cyclic reordering of each monomial and invariant under the action of GL(α). In particular this means that the class W̃ ∈ CQ/[CQ,CQ] defines a GL(α) equivariant Chern–Simons functional, Tr W̃α : M Q(α)→ C. The potentials we consider in this paper satisfy the following special condi- tion. Definition 2.1. A potential W̃ ∈ CQ/[CQ,CQ] has a linear factor if for some lift W ∈ CQ there exists a factorization W = L ·R ∈ CQ where L is a linear combination of arrows, so that for each pair of vertices i, j ∈ Q0 there is at most one such arrow a : i→ j between them. Moreover we have that no arrow in L occurs in R. 7 For such a potential we define MQ,W1 (α) := Tr W̃ −1 α (1), MQ,W0 (α) := Tr W̃ −1 α (0), RQ,W (α) := {(Ma)a:i→j ∈MQ(α) | R((Ma)a:i→j) = 0} the first two being fibres of the Chern–Simons functional and the latter the zero locus of the reduced equations R = 0. The corresponding moduli stacks are defined, MQ,W1 (α) := [M Q,W 1 (α)/GL(α)], MQ,W0 (α) := [M Q,W 0 (α)/GL(α)], RQ,W (α) := [RQ,W (α)/GL(α)]. There exists a partial order on NQ0 given by α ≥ β if αi ≥ βi for all i ∈ Q0 and we say that α > β if α ≥ β and α 6= β. As a final piece of notation we introduce two pairings on the semi-group NQ0 to be used later. Let α, β ∈ ZQ0 then α · β = ∑ i αiβi 〈α, β〉Q = ∑ i αiβi − ∑ a:i→j αiβj . 2.3 Stability for Quiver Representations We fix a dimension vector α ∈ NQ0 . The stability condition depends upon a choice of framing vector f ∈ NQ0 \ {0}, introducing such will produce a 8 fine moduli space of framed quiver representations. Let UQ(α, f) = {(Ma,ml) | dimC〈Ma〉{ml} = α} ⊂MQ(α)× ∏ i∈Q0 (Cαi)fi where C〈Ma〉{ml}, is defined to be the span of the collection of vectors ml under the successive action of the matrices Ma. Now the group GL(α) acts freely on the UQ(α, f) giving a fine moduli space of quiver representations. Proposition 2.1. [30, King] There exists a character χ : GL(α) → C∗ such that the open subset UQ(α, f) is precisely the subset of stable points for the action of GL(α) linearized by χ. In particular the action of GL(α) on UQ(α, f) is free, and the quotient NQ(α, f) := MQ(α)× ∏ i∈Q0 (Cαi)fi//χ GL(α) = UQ(α, f)/GL(α) is a smooth quasi-projective G.I.T. quotient. The Chern–Simons functional can be defined on UQ(α, f) as the pullback of the one on MQ(α) via the natural projection. This functional is GL(α) equivariant and consequently descends to a regular function on the smooth quotient Tr W̃α,f : N Q(α, f)→ C. Our main object of interest is the scheme theoretic degeneracy locus of this functional, which we define; DTQ,W (α, f) := {dTr W̃α,f = 0} ⊂ NQ(α, f). Example 2.1. Let Q be the quiver with one vertex {?} and three arrows {x, y, z}, potential W = x(yz − zy) then it is well known [4, Prop 2.1], DTQ,W (n, 1) = Hilbn(C3). Remark 2.1. Details of more general quiver stability conditions can be found in section 3.2.3. 9 2.4 Grothendieck Rings Here is an account of Grothendieck rings following Bridgeland [8] (c.f. Toën [59]). We are interested in spaces that are varieties or stacks over C. Denote by V arC ⊂ StC the inclusion of the category of varieties in the category of Artin stacks of finite type. Our first definition is the Grothendieck ring of varieties. Definition 2.2. The Grothendieck ring of varieties K0(V arC) is the free abelian group on isomorphism classes of varieties over C, modulo the scissor relations [X] = [Z] + [X \ Z] for Z a subvariety ofX. Multiplication inK0(V arC) is given by fibre product [X] · [Y ] = [X ×C Y ]. For a variety X we will call [X] the motivic class of X. For example consider a Zariski fibration pi : E → B with fibre F , then stratifying the base by trivializing neighborhoods and using the scissor relations we get [E] = [F ] · [B] ∈ K0(V arC). We can use this fact to give a nice formula for the motivic class of the general linear group. Lemma 2.2. We have [GLn] = n−1∏ k=0 (Ln − Lk) ∈ K0(V arC) where L = [A1C]. Proof. The map pi : GLn → Cn \ {0}, 10 sending a matrix to its first column, is a Zariski fibration with fibre equal to GLn−1×Cn−1 so by what was said above [GLn] = (Ln − 1)Ln−1[GLn−1], and the result follows by induction.  Recall that the Grassmannian can be defined as the free quotient of the general linear group by the subgroup of matrices fixing a hyperplane of dimension k, so as an immediate corollary of the above lemma we also have a formula for the motivic class of the Grassmannian [Gr(k, n)] = [GLn] [GLk][GLn−k][Hom(Ck,Cn−k)] ∈ K0(V arC). To define the Grothendieck ring of stacks we need the notion of geometric bijection, we say that a representable morphism f : X → Y is a geometric bijection if it induces an isomorphism f : X(C)→ Y (C) between the groupoids of C-valued points. In our definition we shall also need to consider stacks with affine stabilizers, that is stacksX locally of finite type over C such that for all x ∈ X(C), IsomC(x, x) is an affine algebraic group. Definition 2.3. The Grothendieck ring of stacks K0(StC) is the free abelian group of isomorphism classes of finite type Artin stacks of over C with affine stabilizers, modulo the relations 1. [X unionsq Y ] = [X] + [Y ]. 11 2. [X] = [Y ] for every geometric bijection f : X → Y . 3. [X] = [Y ] for every pair of Zariski fibrations with the same base and fibre. Again multiplication comes from the fibre product. The inclusion of the category of varieties in the category of stacks gives an obvious homomorphism of rings K0(V arC) ϕ−→ K0(StC). The following lemma shows that the image of the general linear group is invertible. Lemma 2.3. Every principal GLn bundle pi : Y → X is a Zariski fibration. Hence [Y ] = [X] · [GLn] ∈ K0(StC) and in particular taking Y to be a point we have that [GLn] is invertible with inverse [BGLn] ∈ K0(StC). Proof. The fibration pi : Y → X is a principal GLn bundle if its pullback to any scheme S is Y pi  S f // X. So f∗pi : f∗Y → S is a principal GLn bundle, therefore locally trivial in the étale topology. As the group GLn is special [56] this bundle is a Zariski fibration.  This lemma provides an extension of the map ϕ above. We have K0(V arC)[[GLn] −1 : n ≥ 1] ϕ̃−→ K0(StC). Kresch proves [34] that stacks with affine stabilizers are in geometric bijec- tion with quotients Y/GLn where Y is a variety and n is sufficiently large. 12 This shows that the map ϕ̃ above is surjective. In fact ϕ̃ is actually an isomorphism of rings. Lemma 2.4. [8, c.f. Bridgeland, Lemma 3.9] The ring homomorphism ϕ̃ : K0(V arC)[[GLn] −1 : n ≥ 1]→ K0(StC) is an isomorphism. Since we need them for our definition of virtual motives we hereby define two extended rings of motivic classes MC := K0(V arC)[L− 1 2 ] and MStC := K0(StC)[L− 1 2 ]. The topological Euler characteristic of classes in K0(V arC) extends to MC by letting χ(L− 1 2 ) = −1, giving a ring homomorphism χ :MC → Z. 2.5 Virtual Motives Let f : M → C be a regular function on a smooth quasi-projective variety with Z the scheme theoretic degeneracy locus of f . Using arc spaces Denef and Loeser [14] define the motivic vanishing cycle [ϕf ]Z ∈ K µ̂0 (V arZ). This motivic class is relative over the the degeneracy locus Z and has an action of the profinite group µ̂ = limd µd of all roots of unity. From this one can recover the cohomology with its monodromy action of the Milnor fibre at each point z ∈ Z (see [14] Theorem 3.5.5.). This relative motive can be pushed forward to a point giving a coarser class [ϕf ] ∈ K µ̂0 (V arC) sometimes described as the integral over Z and this class can be used directly to give a definition of the virtual motive in the following case. 13 Definition 2.4. [4] Let f : M → C be a regular function on the a smooth quasi-projective variety, with Z = {df = 0} ⊂ M the scheme theoretic degeneracy locus of f . The virtual motive of Z is defined, [Z]vir = −L−dimM2 [ϕf ] ∈Mµ̂C. This really depends not only on the scheme Z but on the function f and space M . However taking the Euler characteristic will give a number intrinsic to Z. In particular if Z is a moduli space of stable sheaves on a Calabi–Yau threefold then χ([Z]vir) will equal the Donaldson-Thomas in- variant, by Behrend’s description of the constructible function ν in terms of the Milnor fibre [2]. In the case where f is equivariant with respect to a C∗-action satisfying the properties below, the virtual motive lives in the subring MC ⊂ Mµ̂C of motivic classes with trivial monodromy action. Moreover, it is given by an easy formula. Definition 2.5. (Property (?).) Let f : M → C be a regular function on a smooth quasi-projective variety M . Then f satisfies Property (?) if there exists a C∗ acting on M , so that for all m ∈M and t ∈ C∗ we have f(t ·m) = t · f(m) and secondly we have that for all m ∈M the limit lim t→0 t ·m exists. Property (?) means that the map f : M → C is a trivial fibration over C∗. This is easy to see, letting M1 = f −1(1) and M0 = f−1(0) the trivialization is C∗ ×M1 t·m //M \M0 sending a point m in the fibre over 1 to the point t ·m ∈M \M0. 14 Proposition 2.5. [4, Behrend, Bryan, Szendrői] Let f : M → C be a regu- lar function on a smooth quasi-projective variety M with C∗-action satisfying Property (?). The motivic class of vanishing cycles [ϕf ] can be expressed as the motivic difference of the general and central fibres [ϕf ] = [f −1(1)]− [f−1(0)] ∈ K0(V arC) in particular [Z]vir = L− dimM 2 ([f−1(0)]− [f−1(1)]) ∈MC. In [4] the above theorem is stated under the additional assumption that the C∗-action be circle-compact this implies that in addition to Property (?) the C∗-fixed locus be compact in the analytic topology. As they them- selves mention [4, Section 1.7] this requirement is stronger than needed and Property (?) will suffice. Remark 2.2. More details on equivariant motivic classes and vanishing cycles can be found in the excellent expository article [14]. 2.6 Motivic Quantum Torus The reduction theorem is most easily expressed as a multiplicative relation in a certain quantum torus. Kontsevich and Soibelman see this arising naturally in their work on motivic DT invariants of a three dimensional Calabi–Yau categories [33] and again in the quiver context [32]. Definition 2.6. Let Q be a finite quiver with pairing 〈, 〉Q. The algebra TQ, is an MStC -module TQ := ∏ α∈NQ0 MStC tα, with a non-commutative product, tα ∗ tβ = L 12(〈α,β〉Q−〈β,α〉Q)tα+β. 15 Roughly speaking this is a non-commutative ring of power series with motivic classes as coefficients. All the motivic classes we are interested in can be neatly packaged in two such series. The DT series, ZQ,W,f (t) := ∑ α∈NQ0 [DTQ,W (α, f)]virt α ∈ TQ, and the reduced series, RQ,W (t) := ∑ α∈NQ0 [RQ,W (α)] [GL(α)] L 1 2 〈α,α〉Qtα ∈ TQ. Remark 2.3. Notice that in the special case when the quiverQ is symmetric the algebra TQ is commutative. 2.7 Main Theorem Here we extend the results of [4, §2.7] to a general quiver and potential with a linear factor. First we work without stability and consider the Chern–Simons functional TrW̃α : M Q(α)→ C. One immediate consequence of the linearity of the potential is that the above map satisfies Property (?), hence is a trivial fibration over C∗. To see this just consider C∗ acting diagonally on the left of the matrices appearing in the linear factor L, one of which will be non-zero if the trace map is. Recall the definitions of Section 2.2 MQ,W1 (α) := Tr W̃ −1 α (1), MQ,W0 (α) := Tr W̃ −1 α (0), RQ,W (α) := {(Ma) ∈MQ(α) | R(Ma) = 0}, 16 the first space here is the general fibre, the second the central fibre and the space RQ,W (α) we call the reduced space, since it is the locus of the reduced equation R = 0. The corresponding stacks defined in Section 2.2 were denoted MQ,W1 (α),M Q,W 0 (α) and R Q,W (α). The following proposition expresses the difference of the general and central fibres in terms of the reduced space. Proposition 2.6. Let Q be a finite quiver with potential W . Suppose W has a linear factor. For any dimension vector α ∈ NQ0 let MQ,W1 (α), MQ,W1 (α), RQ,W (α) be the stacks defined above. Then [MQ,W0 (α)]− [MQ,W1 (α)] = [RQ,W (α)] in the Grothendieck ring of stacks K0(StC). Proof. The idea is to use the triviality of the fibration and the linearity of the potential to obtain two simple relations between the motivic classes. Fix a dimension vector α ∈ NQ0 . Consider the map Tr W̃α : MQ(α) → C, we stratify the base C = C∗ unionsq {0}. The fibre over {0} we call MQ,W0 (α), the central fibre. Since the map is a C∗-equivariant family with Property (?), then over C∗ it is a trivial fibration, with general fibre MQ,W1 (α). This decomposition of MQ(α) into two pieces gives our first motivic relation, [MQ(α)] = [MQ,W0 (α)] + (L− 1)[MQ,W1 (α)]. (2.1) Now split the arrows of the quiver into two sets, if and only if they occur in the linear factor of the potential; A := {a : i→ j | a ∈ L} and B := {b : i→ j | b 6∈ L} Let m = (Ma)a:i→j be a Q representation of dimension vector α, using the splitting we decompose m into parts (mA,mB) ∈ ⊕ a:i→j∈A Hom(Cαi ,Cαj )× ⊕ b:i→j∈B Hom(Cαi ,Cαj ). 17 Then L(m) = L(mA) ∈ ⊕ a:i→j∈A Hom(Cαi ,Cαj ), and since W = L ·R is a sum of cycles R(m) = R(mB) ∈ ⊕ a:i→j∈A Hom(Cαj ,Cαi). The trace map gives a non-degenerate pairing between these spaces Tr : ⊕ a:i→j∈A Hom(Cαi ,Cαj )× ⊕ a:i→j∈A Hom(Cαj ,Cαi)→ C defined as follows. Let X ∈ ⊕ a:i→j∈A Hom(Cαi ,Cαj ) and Y ∈ ⊕ a:i→j∈A Hom(Cαj ,Cαi) take the product of these linear maps to get X · Y ∈ ⊕ a:i→j∈A Hom(Cαj ,Cαj ) and then take the trace of this endomorphism Tr : (X,Y ) 7→ Tr(X · Y ). We use this pairing to compute MQ,W0 (α) in order to get a second relation. Let m = (mA,mB) ∈ MQ,W0 (α), so that Tr(L(mA) · R(mB)) = 0. There are two cases to consider firstly when R(mB) = 0 and secondly when R(mB) 6= 0. By definition the locus where R = 0 is equal to RQ,W (α). On the compliment of this set consider the projection pi : MQ(α) \RQ,W (α)→ {mB | R(mB) 6= 0} : (mA,mB) 7→ mB. 18 This map is a trivial vector bundle. For fixed mB the condition Tr(L(mA) · R(mB)) = 0 is a single linear condition on the matrices mA in the fibre, and so in the second case the locus of m ∈ MQ,W0 (α) such that R 6= 0, is a vector bundle of rank one lower. Both cases considered we have a formula for the class of MQ,W0 (α), [MQ,W0 (α)] = [R Q,W (α)] + ([MQ(α)]− [RQ,W (α)])L−1. (2.2) Finally relations (2.1),(2.2) together imply [MQ,W0 (α)]− [MQ,W1 (α)] = [RQ,W (α)] Dividing by the motivic classes of the automorphism groups GL(α) gives the corresponding result for stacks.  In Section 2.3 we showed that for a choice of framing vector f ∈ NQ0\{0} there exists a fine moduli space of quiver representations with Chern–Simons functional TrW̃α,f : N Q(α, f)→ C. Again we can use the C∗-action given by acting diagonally on the left of all matrices occurring in the linear factor L to produce a C∗-action satisfying Property (?). Lemma 2.7. There exists a C∗-action on the moduli space NQ(α, f) satis- fying Property (?). Proof. Let (mA,mB, v) ∈ ⊕ a:i→j∈A Hom(Cαi ,Cαj )× ⊕ β:i→j∈B Hom(Cαi ,Cαj )× ⊕ i∈Q0 Cαi as above we consider the C∗-action given by acting on the left of the matrices in the linear factor t : (mA,mB, v) 7→ (t ·mA,mB, v). 19 This action is GL(α) equivariant and descends to a C∗-action on NQ(α, f) = ∏ a:i→j∈Q1 Hom(Cαi ,Cαj )× ∏ i∈Q0 Cαi//χ GL(α). Moreover by the linearity of the potential the function Tr W̃α,f is C∗-equivariant. All that remains is to check that the limits exist as t → 0 (c.f. Lemma 2.4 [4]). We consider the affine quotient NQ0 (α, f) = ∏ a:i→j∈Q1 Hom(Cαi ,Cαj )× ∏ i∈Q0 Cαi//0 GL(α). given by the GIT quotient at zero stability. Now NQ(α, f) is projective over NQ0 (α, f). For any m = (mA,mB, v) ∈ NQ0 (α, f) the limit limt→0 t ·m exists, indeed it is given by the image of (0,mB, v) in the affine quotient. Since limits exist in NQ0 (α, f) and N Q(α, f) is projective over NQ0 (α, f) the result follows.  So by Proposition 2.5 the virtual motive of DTQ,W (α, f) is the difference of the general and central fibres of Tr W̃α,f [DTQ,W (α, f)]vir = L− dimNQ(α,f) 2 ([Tr W̃−1α,f (0)]− [Tr W̃−1α,f (1)]). In Proposition 2.6 we ignored stability and expressed the difference of the general and central fibres as equal to the reduced space. Now on adding the stability condition we will get a recursion relation for the virtual motives in terms of the reduced spaces. Theorem 2.8. Let Q be a finite quiver with potential W . Suppose that W has a linear factor. For any dimension vector α ∈ NQ0 and framing f ∈ NQ0 \ {0}, the virtual motives of the moduli spaces DTQ,W (α, f) satisfy 20 a recursion, [RQ,W (α)]L f ·α 2 = ∑ β≤α ( L−〈α−β,β〉Q−1/2〈β,β〉Q · [DTQ,W (β, f)]vir ·[RQ,W (α− β)]L− f ·(α−β)2 ) . in the ring MStC . Or equivalently rephrased in the language of Section 2.6 RQ,W (L f 2 t) = ZQ,W,f (t) ∗ RQ,W (L− f2 t). Proof. Fix a dimension vector α ∈ NQ0 and a framing vector f ∈ NQ0 \ {0}. First define all the objects without stability XQ(α, f) := MQ(α)× ∏ i∈Q0 (Cαi)fi , Y Q,W (α, f) := MQ,W0 (α)× ∏ i∈Q0 (Cαi)fi , ZQ,W (α, f) := MQ,W1 (α)× ∏ i∈Q0 (Cαi)fi . As shown in Proposition 2.6 the two spaces above are related to the reduced space, [RQ,W (α)] · Lf ·α = [Y Q,W (α, f)]− [ZQ,W (α, f)]. The stability condition introduced in Section 2.3 depends upon the span of the vectors {ml} under the matrices (Ma)a:i→j . This vector space was earlier defined as C〈Ma〉{ml}. For a given dimension vector β ∈ NQ0 set, XQ(α, β, f) := {(Ma,ml) | dimC〈Ma〉{ml} = β} ⊂ XQ(α, f), Y Q,W (α, β, f) := XQ(α, β, f) ∩ Y Q,W (α, f) ⊂ Y Q,W (α, f), ZQ,W (α, β, f) := XQ(α, β, f) ∩ ZQ,W (α, f) ⊂ ZQ,W (α, f). 21 Since the stability condition required the vectors {ml} generate the entire representation, then in the notation of Section 2.3 UQ(α, f) = XQ(α, α, f) and NQ(α, f) = XQ(α, α, f)/GL(α). So the virtual motive of DTQ,W (α, f) is the difference of the general and central fibres [DTQ,W (α, f)]vir = L− dimNQ(α,f) 2 ( [Y Q,W (α, α, f)] [GL(α)] − [Z Q,W (α, α, f)] [GL(α)] ) . The dimension of NQ(α, f) is dimNQ(α, f) = dimMQ(α)− dim GL(α) + dim ∏ i∈Q0 (Cαi)fi = −〈α, α〉Q + α · f. The remaining task is to compute the difference [Y Q,W (α, α, f)]−[ZQ,W (α, α, f)]. Let us start with Y Q,W (α, β, f) and ZQ,W (α, β, f). Let Gr(β, α) be the Grassmannian of β dimensional subspaces in α dimen- sional space, that isGr(β, α) = ∏ i∈Q0 Gr(βi, αi). An element of Y Q,W (α, β, f) defines a subspace C〈Ma〉{ml} with dimension vector β ∈ NQ0 , the associ- ated map Y Q,W (α, β, f)→ Gr(β, α) is a Zariski fibration. To compute the motivic class of the fibre we fix a basis so that Ma = ( M ′a M∗a 0 M ′′a ) with  M ′a ∈ Hom(Cβi ,Cβj ) M∗a ∈ Hom(Cβi ,Cαj−βj ) M ′′a ∈ Hom(Cαi−βi ,Cαj−βj ) when a : i→ j, and vectors ml = ( m′l 0 ) with { m′l ∈ Hom(1,Cβi) 0 ∈ Hom(1,Cαi−βi) 22 when ml is at vertex i ∈ Q0. The image of the vectors m′l under the matrices M ′a is now the entire β-dimensional subspace. The Chern–Simons functional also splits with respect to this basis, Tr W̃α(Ma) = Tr W̃β(M ′ a) + Tr W̃α−β(M ′′ a ) = 0, in particular there is no restriction on the M∗a , and they factor out an affine space of dimension −〈α−β, β〉Q+ (α−β) ·β. The two cases to consider are {Tr W̃β(M ′a) = Tr W̃α−β(M ′′a ) = 0}, and {Tr W̃β(M ′a) = −Tr W̃α−β(M ′′a ) 6= 0}. In the first case we get an element of Y Q,W (β, β, f) and an element of MQ,W0 (α − β). The other stratum is a trivial C∗-bundle, by the nonzero value of the Chern–Simons functional. Looking at the fibre over 1 gives an element of ZQ,W (β, β, f) and an element of MQ,W1 (α−β). The total motivic class of the fibre is then, [Y Q,W (β, β, f)] · L−〈α−β,β〉Q+(α−β)·β · [MQ,W0 (α− β, f)] +(L− 1)[ZQ,W (β, β, f)] · L−〈α−β,β〉Q+〈α−β,β〉 · [MQ,W1 (α− β, f)]. The space ZQ,W (α, β, f) also fibres over the Grassmannian Gr(β, α), the motivic class of the fibre is computed similarly, as [Y Q,W (β, β, f)] · L−〈α−β,β〉Q+(α−β)·β · [MQ,W1 (α− β, f)] +(L− 2)[ZQ,W (β, β, f)] · L−〈α−β,β〉Q+(α−β)·β · [MQ,W1 (α− β, f)] +[ZQ,W (β, β, f)] · L−〈α−β,β〉Q+(α−β)·β · [MQ,W0 (α− β, f)]. We are now ready to deduce the recursion. Stratifying Y Q,W (α, f) and Y Q,W (α, f) by the dimension of C〈Ma〉{ml} we have Y Q,W (α, f) = ∐ β≤α Y Q,W (α, β, f) and ZQ,W (α, f) = ∐ β≤α ZQ,W (α, β, f). 23 As mentioned the motivic difference of these two spaces was equal to the class of the reduced space (Proposition 2.6) [RQ,W (α)]Lf ·α = ∑ β≤α [Y Q,W (α, β, f)]− [ZQ,W (α, β, f)]. Substituting in our formulas for Y Q,W (α, β, f) and ZQ,W (α, β, f) above gives [RQ,W (α)]Lf ·α = ∑ β≤α [Gr(β, α)]L−〈α−β,β〉Q+(α−β)·β · ([Y Q,W (β, β, f)]− [ZQ,W (β, β, f)]) · ( [MQ,W0 (α− β, f)]− [MQ,W1 (α− β, f)] ) = ∑ β≤α [GL(α)] [GL(β)][GL(α− β)]L −〈α−β,β〉Q · ([Y Q,W (β, β, f)]− [ZQ,W (β, β, f)]) · [RQ,W (α− β)] = ∑ β≤α [GL(α)] [GL(α− β)]L −〈α−β,β〉Q−1/2〈β,β〉Q+1/2f ·β ·[DTQ,W (β, f)]vir · [RQ,W (α− β)]. This recursion formula is a relation in the ring MC. Dividing out by the motivic classes of the automorphism groups GL(α) and GL(α−β) gives the corresponding result in MStC the Grothendieck ring of stacks [RQ,W (α)]L f ·α 2 = ∑ β≤α ( L−〈α−β,β〉Q−1/2〈β,β〉Q · [DTQ,W (β, f)]vir ·[RQ,W (α− β)]L− f ·(α−β)2 ) .  Remark 2.4. As seen in the proof of Theorem 2.8 our result is essentially a relation in the ring MC. On the last line however we divide by the motivic classes of some automorphism groups to get a corresponding statement in MStC the Grothendieck ring of stacks. We make this remark as it is not know 24 whether or not the homomorphism MC →MC[(Ln − 1)−1 : n ≥ 1] ∼=MStC is injective. 25 Chapter 3 The Conifold This chapter is joint work with Prof. Sergey Mozgozoy, Prof. Kentaro Nagao and Prof. Balazs Szendrői [39]. We compute the motivic Donaldson– Thomas theory of the resolved conifold, in all chambers of the space of stability conditions of the corresponding quiver. The answer is a product formula whose terms depend on the position of the stability vector, generaliz- ing known results for the corresponding numerical invariants. Our formulae imply in particular a motivic form of the DT/PT correspondence for the resolved conifold. The answer for the motivic PT series is in full agreement with the prediction of the refined topological vertex formalism. 3.1 Introduction A Donaldson-Thomas (DT) invariant of a Calabi-Yau 3-fold Y is a counting invariant of coherent sheaves on Y , introduced in [60] as a holomorphic analogue of the Casson invariant of a real 3-manifold. A component of the moduli space of (say stable) coherent sheaves on Y carries a symmetric obstruction theory and a virtual fundamental cycle [6, 7]. A DT invariant of a compact Y is then defined as the integral of the constant function 1 over the virtual fundamental cycle of the moduli space. It is known that the moduli space of coherent sheaves on Y can be locally described as the critical locus of a function, the holomorphic Chern–Simons functional (see [27]). Behrend provided a description of DT invariants in terms of the Euler characteristic of the Milnor fiber of the CS functional [2]. Inspired by this result, the proposal of [33, 4] was to study the motivic Milnor fiber of the CS functional as a motivic refinement of the DT invariant. Such a refinement had been expected in string theory [26, 17]. 26 The purpose of this chapter is to show how the ideas of Szendrői [58] and Nagao and Nakajima [46] can be used to study the motivic refinement of DT theory and related enumerative theories associated to the local conifold Y = OP1(−1,−1), the threefold total space over P1 of the rank two bundle OP1(−1)⊕OP1(−1). In [58], it was realized that a counting problem closely related to the original DT counting on Y can be formulated algebraically, in terms of counting representations of a certain quiver with potential (see below), the so-called conifold quiver. It was also conjectured there that the algebraic and geometric counting problems are related by wall crossing. The paper [46] realized this, by • describing the natural chamber structure on the space of stability pa- rameters of the conifold quiver, • finding chambers which correspond to geometric DT and stable pair (PT), as well as algebraic non-commutative DT invariants, and • computing the generating function of Donaldson-Thomas type invari- ants for each chamber. In this chapter, we consider motivic refinements of these formulae. The motivic refinement is given by the motivic class of vanishing cycles of the conifold potential. This virtual motive “motivates” DT theory and its vari- ants (PT, NCDT) in the sense that its Euler characteristic specialization is the corresponding enumerative invariant of the moduli space. The main result of this chapter is the computation of the generating series of these virtual motives in all chambers of the space of stability con- ditions. We constantly use the torus action on Y , together with a result of [4]. We use the factorization property of [33, 32, 44, 51]. We also need one explicit evaluation, Theorem 3.8; we give two proofs of that result, one using an explicit calculation of the generating series of motives of a certain space of matrices, another relying on a further “dimensional reduction” to a problem on a tame quiver. At large volume, our result agrees (up to a subtlety involving the Hilbert scheme of points) with the refined topological vertex formulae of [26], also 27 discussed in [17] in this context. The motives considered here exist globally over the moduli spaces. Thus our point of view is slightly different from that of [33], whose general frame- work involves building motivic invariants from local data. The results here are fully compatible with theirs, but the proofs do not depend on the par- tially conjectural setup of [33], in particular their integration map. As well as a motivic refinement, there is also a “categorification” given by the mixed Hodge module of vanishing cycles of the potential; compare [32]. Our results can also be interpreted as computing the generating series of E-polynomials of this categorification. 3.1.1 Main result Let J = J(Q,W ) be the non-commutative crepant resolution of the conifold, a quiver algebra with relations coming from the Klebanov–Witten potential W (see Section 3.3.1 for details). Let J̃ = J(Q̃,W ) be the framed algebra given by adding the new vertex ∞ to the quiver of J . In [46], the authors introduce a notion of ζ-(semi)stability of J̃-modules Ṽ with dim Ṽ∞ ≤ 1 for a stability parameter ζ = (ζ0, ζ1) ∈ R2. Let α ∈ N2 and let Mζ(J̃ , α) be the moduli space of ζ-stable J̃-modules Ṽ , with dim Ṽ = (α, 1). We want to compute the motivic generating series Zζ(y0, y1) = ∑ α∈N2 [ Mζ ( J̃ , α )] vir · yα00 yα11 ∈MC[[y0, y1]]. Here [•]vir denotes the virtual motive (see Section 3.2.1), an element of a suitable ring of motives MC. As proved in [46], the stability parameter space R2 is a countable union of chambers, within which the moduli spaces and therefore the generating series Zζ remain unchanged. The chambers are separated by a set of walls, defined by a set of positive roots ∆+ = ∆ re + unionsq∆im+ , 28 where ∆re+ = {(i, i− 1) | i ≥ 1} ∪ {(i− 1, i) | i ≥ 1}, ∆im+ = {(i, i) | i ≥ 1}. To each element α = (α0, α1) ∈ ∆+, we associate a finite product as follows: for real roots α ∈ ∆re+ , put Zα(−y0, y1) = α0−1∏ j=0 ( 1− L−α02 + 12+jyα00 yα11 ) , whereas for imaginary roots α ∈ ∆im+ , put Zα(−y0, y1) = α0−1∏ j=0 ( 1− L−α02 +1+jyα00 yα11 )−1 ( 1− L−α02 +2+jyα00 yα11 )−1 . Our main result is the following product formula: Theorem 3.1. For ζ ∈ R2 not orthogonal to any root, Zζ(y0, y1) = ∏ α∈∆+ ζ·α<0 Zα(y0, y1). By [2, 4], the specialization Zζ(y0, y1)|L 12→1 is the DT-type series at the generic stability parameter ζ, computed in special cases in [3, 37, 58, 62] and in general in [13, 46]. Previously, all these results have been obtained by torus localization; we obtain new proofs of all these formulae. Since [46] identifies the DT and PT chambers for Y , we in particular get motivic results for these two chambers. Corollary 3.2. The refined DT and PT series of the resolved conifold are given by the formulae ZPT(−s, T ) = ∏ m≥1 m−1∏ j=0 ( 1− L−m2 + 12+jsmT ) 29 and ZDT(−s, T ) = ZPT(−s, T )· ∏ m≥1 m−1∏ j=0 ( 1− L−m2 +1+jsm )−1 ( 1− L−m2 +2+jsm )−1 , written in the geometric variables s, T , with s representing the point class and T representing the curve class as usual. Thus in particular we compute the first instance of a motivic DT parti- tion function for the original geometric problem of rank-1 invariants of ideal sheaves of points and curves [37] where a curve is present. Corollary 3.2 also proves the motivic version of the DT/PT wall crossing formula. These re- sults are compared to the expectation from the refined topological vertex [26] in Section 3.5.3. 3.2 Preliminaries 3.2.1 Motives We are working in a version of the ring of motivic weights: let MC denote the K-group of the category of effective Chow motives over C, extended by L− 1 2 , where L is the Lefschetz motive. It has a natural structure of a λ-ring [19, 23] (see Section 3.2.2 for the definition of a λ-ring) with σ-operations defined by σn([X]) = [X n/Sn] and σn(L 1 2 ) = L n 2 . There is a dimensional completion [5] M̃C =MC[[L−1]], which is also a λ-ring. Note that in this latter ring, the elements (1− Ln), and therefore the motives of general linear groups, are invertible. The rings MC ⊂ M̃C sit in larger ringsMµ̂C ⊂ M̃µ̂C of equivariant motives, where µ̂ is the group of all roots of unity [35]. The map that sends a smooth projective variety X to its E-polynomial E(X,u, v) = ∑ p,q≥0 (−1)p+q dimHp,q(X,C)upvq 30 can be extended to the ring homomorphism E : M̃C → Q[u, v][[(uv)− 12 ]]. This map is a λ-ring homomorphism, where the λ-ring structure onQ[u, v][[(uv)− 1 2 ]] is given by Adams operations (see Section 3.2.2) ψn(f(u, v)) = f(u n, vn). The map E :MC → Q[u, v, (uv)− 12 ] can be further specialized to the Euler number e :MC → Q by u 7→ 1, v 7→ 1, (uv)− 12 7→ 1. Remark 3.1. Note that the Euler number specialization of L 1 2 is L 1 2 7→ 1. This differs from the conventions of [4], where the specialization is L 1 2 7→ −1. This difference results from the fact that [4] uses the λ-ring structure onMC with σn(−L 12 ) = (−L 12 )n [4, Remark 1.7]. Let f : X → C be a regular function on a smooth variety X. Using arc spaces, Denef and Loeser [14, 35] define the motivic nearby cycle [ψf ] ∈Mµ̂C and the motivic vanishing cycle [ϕf ] = [ψf ]− [f−1(0)] ∈Mµ̂C of f . Note that if f = 0, then [ϕ0] = −[X]. The following result was proved in [4, Prop. 1.11]. Theorem 3.3. Let f : X → C be a regular function on a smooth variety X. Assume that X admits a C∗-action such that f is C∗-equivariant i.e. f(tx) = tf(x) for t ∈ C∗, x ∈ X, and such that there exist limits limt→0 tx for all x ∈ X. Then [ϕf ] = [f −1(1)]− [f−1(0)] ∈MC ⊂Mµ̂C. Following [4], we define the virtual motive of crit(f) to be [crit(f)]vir = −(−L 12 )− dimX [ϕf ] ∈Mµ̂C. Thus for a smooth variety X with f = 0, [X]vir = [crit(0X)]vir = (−L 12 )− dimX · [X]. 31 Remark 3.2. The ring MC is known to be a homomorphic image of the naive motivic ring K0(VarC)[L− 1 2 ]. Some of the works cited above work in this ring; the quoted constructions and results carry over toMC under this ring homomorphism. We prefer to work in MC since that is known to be a λ-ring. 3.2.2 λ-Rings and power structures Let Λ = lim←−Z[[x1, . . . , xn]] Sn be the ring of symmetric functions [36]. It is well-known that Λ is generated as an algebra over Z by elementary symmetric functions en = ∑ i1<···<in xi1 . . . xin as well as by complete symmetric functions hn = ∑ i1≤···≤in xi1 . . . xin . Moreover ΛQ = Λ⊗Q is generated over Q by power sums pn = ∑ xni . A Q-algebra R is called a λ-ring if it is endowed with a map ◦ : Λ×R→ R called plethysm, such that (− ◦ r) : Λ → R is a ring homomorphism for any r ∈ R and the maps ψn = (pn ◦ −), called Adams operations, are ring homomorphisms satisfying ψ1 = IdR and ψmψn = ψmn for m,n ≥ 1. Note that plethysm is uniquely determined by Adams operations. It is also uniquely determined by maps λn = (en ◦ −) : R → R called λ-operations and by maps σn = (hn ◦ −) : R→ R called σ-operations. Given a λ-ring R, we endow the ring A = R[[x1, . . . , xm]] with a λ-ring structure by the rule ψn(rx α) = ψn(r)x nα, r ∈ R,α ∈ Nm. Let A+ ⊂ A be an ideal generated by x1, . . . , xm. We define a map Exp : 32 A+ → 1 +A+, called plethystic exponential, by the rule [19, 43] Exp(f) = ∑ n≥0 σn(f) = exp (∑ n≥1 1 n ψn(f) ) . This map has an inverse Log : 1 +A+ → A+, called plethystic logarithm, Log(f) = ∑ n≥1 µ(n) n ψn log(f), where µ is a Möbius function. We define a power structure map Pow : (1 + A+)× A→ 1 + A+ by the rule [43] Pow(f, g) = Exp(g Log(f)). In the case when R is a ring of motives, the power structure map has the following geometric interpretation [22]. Let f = 1 + ∑ α>0 [Aα]x α, where Aα are algebraic varieties. Then Pow(f, [X]) = ∑ k:Nm→N [( F|k|X × ∏ α∈Nm Ak(α)α ) / ∏ α∈Nm Sk(α) ] x ∑ k(α)α, where the sum runs over maps k : Nm → N with finite support, |k| =∑ α∈Nm k(α), the configuration space FnX is given by FnX = {(x1, . . . , xn) ∈ Xn | xi 6= xj for i 6= j}, and the product of symmetric groups ∏ α∈Nm Sk(α) acts on both factors in the obvious way. The quotient in square brackets parametrizes elements in⋃ ψ:X→Nm ∏ x∈X Aψ(x) with ψ : X → Nm satisfying #{x ∈ X | ψ(x) = α} = k(α) for any α ∈ 33 Nm\{0} (see [41]). Therefore we can also write Pow(f, [X]) = ∑ ψ:X→Nm ∏ x∈X [Aψ(x)]x ψ(x), where the sum runs over maps ψ : X → Nm with finite support. 3.2.3 Quivers and moduli spaces Let Q be a quiver, with vertex set Q0 and edge set Q1. For an arrow a ∈ Q1, we denote by s(a) ∈ Q0 (resp. t(a) ∈ Q0) the vertex at which a starts (resp. ends). We define the Euler-Ringel form χ on ZQ0 by the rule χ(α, β) = ∑ i∈Q0 αiβi − ∑ a∈Q1 αs(a)βt(a), α, β ∈ ZQ0 . We define the skew-symmetric bilinear form 〈•, •〉 of the quiver Q to be 〈α, β〉 = χ(α, β)− χ(β, α), α, β ∈ ZQ0 . Given a Q-representation M , we define its dimension vector dimM ∈ NQ0 by dimM = (dimMi)i∈Q0 . Let α ∈ NQ0 be a dimension vector and let Vi = Cαi , i ∈ Q0. We define R(Q,α) = ⊕ a∈Q1 Hom(Vs(a), Vt(a)) and Gα = ∏ i∈Q0 GL(Vi). Note that Gα naturally acts on R(Q,α) and the quotient stack M(Q,α) = [R(Q,α)/Gα] gives the moduli stack of representations of Q with dimension vector α. Let W be a potential on Q, a finite linear combination of cyclic paths in Q. Denote by J = JQ,W the Jacobian algebra, the quotient of the path 34 algebra CQ by the two-sided ideal generated by formal partial derivatives of the potential W . Let fα : R(Q,α)→ C be the Gα-invariant function defined by taking the trace of the map asso- ciated to the potential W . As it is now well known [55, Proposition 3.8], a point in the critical locus crit(fα) corresponds to a J-module. The quotient stack M(J, α) = [ crit(fα)/Gα ] gives the moduli stack of J-modules with dimension vector α. Definition 3.1. A central charge is a group homomorphism Z : ZQ0 → C such that Z(α) ∈ H+ = {reipiϕ | r > 0, 0 < ϕ ≤ 1} for any α ∈ NQ0\{0}. Given α ∈ NQ0\{0}, the number ϕ(α) = ϕ ∈ (0, 1] such that Z(α) = reipiϕ, for some r > 0, is called the phase of α. Definition 3.2. For any nonzero Q-representation (resp. J-module) V , we define ϕ(V ) = ϕ(dimV ). A Q-representation (resp. J-module) V is said to be Z-(semi)stable if for any proper nonzero Q-subrepresentation (resp. J-submodule) U ⊂ V we have ϕ(U)(≤)ϕ(V ). Definition 3.3. Given ζ ∈ RQ0 , define the central charge Z : ZQ0 → C by the rule Z(α) = −ζ · α+ i|α|, where |α| = ∑i∈Q0 αi. We say that a Q-representation (resp. J-module) is ζ-(semi)stable if it is Z-(semi)stable. Remark 3.3. Let the central charge Z be as in Definition 3.3. Define the slope function µ : NQ0\{0} → R by µ(α) = ζ·α|α| . If l ⊂ H = H+∪{0} is a ray such that Z(α) ∈ l then l = R≥0(−µ(α), 1). This implies that ϕ(α) < ϕ(β) if and only if µ(α) < µ(β). 35 We say that ζ ∈ RQ0 is α-generic if for any 0 < β < α we have ϕ(β) 6= ϕ(α). This condition implies that any ζ-semistable Q-representation (resp. J-module) is automatically ζ-stable. LetRζ(Q,α) denote the open subset ofR(Q,α) consisting of ζ-semistable representations. Let fζ,α denote the restriction of fα to Rζ(Q,α). The quotient stacks Mζ(Q,α) = [ Rζ(Q,α)/Gα ] , Mζ(J, α) = [ crit(fζ,α)/Gα ] (3.1) give the moduli stacks of Q-representations and J-modules with dimension vector α. 3.2.4 Motivic DT invariants Let (Q,W ) be a quiver with a potential and let J = JQ,W be its Jacobian algebra. Recall that the degeneracy locus of the function fα : R(Q,α)→ C defines the locus of J-modules, so that the quotient stack M(J, α) = [crit(fα)/Gα] is the stack of J-modules with dimension vector α. We define motivic Donaldson-Thomas invariants [M(J, α)]vir = [crit(fα)]vir [Gα]vir , where [Gα]vir refers to the virtual motive of the pair (Gα, 0). Definition 3.4. A subset I ⊂ Q1 is called a cut of (Q,W ) if in the associated grading gI on Q given by gI(a) = 1 a ∈ I,0 a /∈ I, the potential W is homogeneous of degree 1. Throughout this section we assume that (Q,W ) admits a cut. Then the 36 space R(Q,α) admits a C∗-action satisfying the conditions of Theorem 3.3 for the function fα : R(Q,α)→ C. This implies [M(J, α)]vir = (−L 12 )−dimR(Q,α) [f −1 α (0)]− [f−1α (1)] [Gα]vir = (−L 12 )χ(α,α) [f −1 α (0)]− [f−1α (1)] [Gα] . (3.2) Generally, for an arbitrary stability parameter ζ, we define [Mζ(J, α)]vir = (−L 1 2 )χ(α,α) [f−1ζ,α(0)]− [f−1ζ,α(1)] [Gα] , (3.3) where, as before, fζ,α denote the restriction of fα : R(Q,α)→ C to Rζ(Q,α). Lemma 3.4. Let α ∈ NQ0 be such that αi = 1 for some i ∈ Q0 (this will be the case for framed representations studied later) and let ζ ∈ RQ0 be α-generic. Then [Mζ(J, α)]vir = [crit(fζ,α)]vir [Gα]vir . Proof. Let Mζ(Q,α) = Rζ(Q,α)/Gα be the smooth moduli space of ζ-semistable Q-representations having di- mension vector α, and let f ′ζ,α : Mζ(Q,α) → C be the map induced by fζ,α : Rζ(Q,α)→ C. Note that Rζ(Q,α)→ Mζ(Q,α) is a principal bundle with the structure group PGα = Gα/C∗. The group PGα is a product of general linear groups (here we use our assumption that there exists i ∈ Q0 with αi = 1). Therefore Rζ(Q,α) → Mζ(Q,α) is locally trivial in Zariski topology. This implies [crit(fζ,α)]vir [Gα]vir = [crit(f ′ζ,α)]vir [GL1]vir . As (Q,W ) admits a cut, the space Mζ(Q,α) admits a C∗-action satisfying the conditions of Theorem 3.3 for the function f ′ζ,α : Mζ(Q,α) → C (one 37 uses the fact that Mζ(Q,α) is projective over R(Q,α)//Gα). This implies [crit(f ′ζ,α)]vir [GL1]vir = −(−L 12 )− dimMζ(Q,α)[ϕf ′ζ,α ] (−L 12 )−1(L− 1) = (−L 12 )dimGα−dimR(Q,α) L− 1 ([f ′−1 ζ,α (0)]− [f ′−1ζ,α (1)]) = (−L 12 )χ(α,α) [f −1 ζ,α(0)]− [f−1ζ,α(1)] [Gα] . (3.4)  3.2.5 Twisted algebra and central charge Definition 3.5. The twisted motivic algebra associated to the quiver Q is the associative M̃C-algebra TQ = ∏ α∈NQ0 M̃C · yα generated by formal variables yα that satisfy the relation yα · yβ = (−L 12 )〈α,β〉yα+β, with 〈•, •〉 the skew-symmetric form of the quiver Q. Note that if the quiver Q is symmetric, i.e. its skew-symmetric form is identically zero, then TQ is commutative. Remark 3.4. This algebra, which is all we are going to need, is (a com- pletion of) the “positive half” of the motivic quantum torus of Kontsevich– Soibelman [33]. A ray in the upper half plane H = H+ ∪ {0} is a half line which has the origin as its end. For a ray l ⊂ H and a central charge Z, we put TZ,l = ∏ α∈Z−1(l)∩NQ0 M̃C · yα, 38 a subalgebra of the twisted algebra TQ. Lemma 3.5 (Kontsevich–Soibelman [31, Theorem 6]). For any element A = ∑ α∈NQ0 Aαy α ∈ TQ with A0 = 1, there is a unique factorization A = y∏ l⊂H AZ,l (3.5) with AZ,l ∈ TZ,l, where the product is taken in the clockwise order over all rays. Proof. For a positive real number r, we put T (r)Q = ∏ α∈NQ0 ,|α|<r M̃C · yα, T (r)Z,l = ∏ α∈Z−1(l)∩NQ0 ,|α|<r M̃C · yα which we consider as factor algebras of TQ and TZ,l respectively. Let A(r) ∈ T (r)Q denote the image of A under the canonical projection TQ  T (r)Q . It is enough to show that for any r there is a unique factorization A(r) = y∏ l⊂H A (r) Z,l with A (r) Z,l ∈ T (r)Z,l . Note that the set of rays l ∈ H such that {α ∈ Z−1(l) | |α| < r} 6= ∅ is finite. We order this set (l1, . . . , lN ) so that argZ(l1) < · · · < argZ(lN ). First we put A (r) Z,l1 to be the summand of A(r) contained in T (r)Z,l1 . For 1 < i ≤ N we define A(r)Z,li to be the summand of A(r) · (A(r)Z,l1)−1 · · · · · (A(r)Z,li−1)−1 contained in T (r)Z,li . Uniqueness is clear from the construction.  39 3.2.6 Generating series of motivic DT invariants Let (Q,W ) be a quiver with a potential admitting a cut, and let J = JQ,W be its Jacobian algebra. Definition 3.6. We define the generating series of the motivic Donaldson- Thomas invariants of (Q,W ) by AU = ∑ α∈NQ0 [M(J, α)]vir · yα = ∑ α∈NQ0 [crit(fα)]vir [Gα]vir · yα ∈ TQ, the subscript referring to the fact that we think of this series as the universal series. Given a cut I of (Q,W ), we define a new quiver QI = (Q0, Q1\I). Let JW,I be the quotient of CQI by the ideal (∂IW ) = (∂W/∂a, a ∈ I). Proposition 3.6. If (Q,W ) admits a cut I, then AU = ∑ α∈NQ0 (−L 12 )χ(α,α)+2dI(α) [R(JW,I , α)] [Gα] yα, where dI(α) = ∑ (a:i→j)∈I αiαj for any α ∈ ZQ0. Proof. Let f = fα : R(Q,α)→ C. According to (3.2) we have [M(J, α)]vir = (−L 12 )χ(α,α) [f −1(0)]− [f−1(1)] Gα . It is proved in [51, Theorem 4.1] and [38, Prop. 7.1] that [f−1(1)]− [f−1(0)] = −LdI(α)[R(JW,I , α)]. Therefore [M(J, α)]vir = (−L 12 )χ(α,α)+2dI(α) [R(JW,I , α)] [Gα] .  40 Let ζ ∈ RQ0 be some stability parameter and let Z : ZQ0 → C be the central charge determined by ζ as in Definition 3.3. Definition 3.7. Let l = R≥0(−µ, 1) ⊂ H be a ray (see Remark 3.3). We put AZ,l = Aζ,µ = ∑ α∈NQ0 Z(α)∈l [Mζ(J, α)]vir · yα ∈ TQ. The Harder-Narashimhan filtrations provide a filtration onR(Q,α). This filtration induces the following factorization property. Theorem 3.7. Assume that (Q,W ) has a cut. Then we have AU = y∏ l AZ,l, where the product is taken in the clockwise order over all rays. Proof. This is originally a result of Kontsevich–Soibelman [33], though their proof depends on a conjectural integral identity. Assuming the existence of a cut, Theorem 3.3 leads to a simplified proof, written out in [51] and [44].  3.3 The Universal DT Series of the Conifold Quiver 3.3.1 Motivic DT invariants for the conifold quiver Let (Q,W ) be the conifold quiver with potential. Recall that Q has vertices 0, 1 and arrows ai : 0→ 1, bi : 1→ 0 for i = 1, 2. The potential is given by W = a1b1a2b2 − a1b2a2b1. We have χ(α, α) = α20 + α 2 1 − 4α0α1. Let JW = CQ/∂W be the Jacobian algebra of (Q,W ). Then I = {a1} is easily seen to be a cut for (Q,W ). Let QI = (Q0, Q1\I) be the quiver 41 defined by the cut, and JW,I the quotient of CQI by the ideal (∂IW ) = (∂W/∂a, a ∈ I). It follows from Proposition 3.6 that the coefficients of the universal Donaldson- Thomas series AU = ∑ α∈NQ0 Aαy α are given by Aα = (−L 12 )χ(α,α)+2α0α1 [R(JW,I , α)] [Gα] = (−L 12 )(α0−α1)2 [R(JW,I , α)] [Gα] . The goal of this section is to prove the following result. Theorem 3.8. We have AU (y0, y1) = Exp ( (L+ L2)y0y1 − L 12 (y0 + y1) L− 1 ∑ n≥0 (y0y1) n ) . (3.6) Equivalently, AU (y0, y1) = ∏ α∈∆+ Aα(y0, y1), (3.7) where for roots α ∈ ∆+, we put Aα(y0, y1) =  Exp ( −L− 12 1− L−1 y α ) = ∏ j≥0 ( 1− L−j− 12 yα ) α ∈ ∆re+ , Exp ( 1 + L 1− L−1 y α ) = ∏ j≥0 ( 1− L−jyα)−1 (1− L−j+1yα)−1 α ∈ ∆im+ . The equivalence of the exponential and product forms (3.6)–(3.7) follows from formal manipulations. In the following two subsections, we give two proofs of Theorem 3.8. The first one develops the method of [18] (c.f. [12]). The second proof uses another “dimensional reduction” to reduce the prob- lem to that of representations of the tame quiver of affine type A (1) 1 . 3.3.2 First proof The goal is to compute the generating function of motives of moduli of representations of JW,I -modules. Up to a group action, these moduli spaces 42 are given concretely as spaces of triples of matrices: R(JW,I , α) = {(A2, B1, B2) ∈ Hom(V0, V1)×Hom(V1, V0)×2 | B1A2B2 = B2A2B1}. In this section, for simplicity, we will denote this space by R(α). The proof begins by reducing the problem to two simpler ones via a stratification of R(α). For (A2, B1, B2) ∈ R(α), consider the linear map A2 ⊕B2 : V0 ⊕ V1 → V0 ⊕ V1. For any such endomorphism, the vector space V = V0 ⊕ V1 has a second decomposition V = V I⊕V N on which A2⊕B2 decomposes into an invertible map and a nilpotent map (c.f. [18, Lemma 1]). Namely, AI2 ⊕BI2 : V I → V I and AN2 ⊕BN2 : V N → V N . Define V Ii (resp. V N i ) to be the intersection of Vi and V I (resp. V N ), so that we have decompositions V0 = V I 0 ⊕ V N0 and V1 = V I1 ⊕ V N1 , with A2 = A I 2 ⊕AN2 ∈ Hom(V I0 , V I1 )⊕Hom(V N0 , V N1 ), B2 = B I 2 ⊕BN2 ∈ Hom(V I1 , V I0 )⊕Hom(V N1 , V N0 ). Notice that AI2, B I 2 are invertible, in particular we have dim(V I0 ) = dim(V I 1 ) = 1 2 dim(V I). A little bit of linear algebra shows that a matrix B1 satisfying B1A2B2 = B2A2B1 has a similar block decomposition with respect to the splitting V = V I ⊕ V N ; B1 = B I 1 ⊕BN1 ∈ Hom(V I1 , V I0 )⊕Hom(V N1 , V N0 ). 43 The relation B1A2B2 = B2A2B1 now becomes two independent sets of equa- tions, BI1A I 2B I 2 = B I 2A I 2B I 1 and B N 1 A N 2 B N 2 = B N 2 A N 2 B N 1 . Define RIa = {(A2, B1, B2) ∈ R((a, a)) | A2 ⊕B2 is invertible}; RNα = {(A2, B1, B2) ∈ R(α) | A2 ⊕B2 is nilpotent}. Over the stratum of R(α) where dim(V I) = 2a, we have a Zariski locally trivial fibre bundle RIa ×RNα // {(A2, B1, B2) ∈ R(α) | dim(VI) = 2a}  M(a, α) where M(a, α) is the space of direct sum decompositions V0 ∼= V I0 ⊕ V N0 , V1 ∼= V I1 ⊕ V N1 . Hence stratifying R(α) by dim(V I) gives the following relation in the Grothendieck ring of varieties: [R(α)] = min(α0,α1)∑ a=0 [RIa] · [RN(α0−a,α1−a)] · [GLα0 ] [GLa][GLα0−a] · [GLα1 ] [GLa][GLα1−a] . We collect the above motives into two generating series I(y) = ∑ a≥0 [RIa] [GLa]2 ya and N(y0, y1) = ∑ α∈NQ [RNα ] [GLα0 ][GLα1 ] (−L1/2)(α0−α1)2yα00 yα11 . Multiplying the above relation by (−L1/2)(α0−α1)2tα00 tα11 and summing gives an equality of power series AU = I(y0y1) ·N(y0, y1). It remains to compute I(y) and N(y0, y1). 44 First consider I(y). If pi is a partition of a we will write pi ` a, and denote its length l(pi), and size |pi|. Then we have two spaces RIa = {(A2, B1, B2) ∈ Iso ( V I0 , V I 1 )×Hom(V I1 , V I0 )×Iso(V I1 , V I0 ) | B1A2B2 = B2A2B1} and CIa = {(C1, C2) ∈ End(V I1 )×GL(V I1 ) | C−12 C1C2 = C1}, together with a map β : RIa → CIa given by β(A2, B1, B2) = (A2B1, A2B2). The map β is a GL(V I0 )-torsor associated to a global gauge fixing, g · (A2, B1, B2) = (A2g −1, gB1, gB2). Since the general linear group is a special group, the map β is a locally trivial GL(V I0 ) bundle in the Zariski topology. The base of the fibration CIa is a commuting variety whose motivic class is known [12] to equal [GLa] ∑ pi`a Ll(pi). Therefore I(y) = ∑ a≥0 [RIa] [GLa]2 ya = ∑ a≥0 [CIa ] [GLa] ya = ∑ pi Ll(pi)y|pi| = ∞∏ i=1 1 1− Lyi = Exp ( L ∑ n≥1 yn ) . All that remains is to compute the series N(y0, y1). Given now that the matrix A2 ⊕ B2 is nilpotent, there exists a basis of V N , {ai1s1 , bi2s2 , ci3s3 , di4s4}, 45 1 ≤ ij ≤ kj , 1 ≤ sj ≤ rji , such that V N0 3 ai1r1i A27→ ai1 r1i−1 B27→ ai1 r1i−2 A27→ · · · A27→ ai11 ∈ V N1 \ 0 and B2(ai11 ) = 0, V N1 3 bi2r2i B27→ bi2 r2i−1 A27→ bi2 r2i−2 B27→ · · · B27→ bi21 ∈ V N0 \ 0 and A2(bi21 ) = 0, V N0 3 ci3r3i A27→ ci3 r3i−1 B27→ ci3 r3i−2 A27→ · · · B27→ ci31 ∈ V N0 \ 0 and A2(ci31 ) = 0, V N1 3 di4r4i B27→ di4 r4i−1 A27→ di4 r4i−2 B27→ · · · A27→ di41 ∈ V N1 \ 0 and B2(di41 ) = 0. As the numbers r1i , r 2 i are always even and r 3 i , r 4 i always odd, it is com- binatorially convenient to define r̂1i = r 1 i /2, r̂ 2 i = r 2 i /2, r̂ 3 i = (r 3 i + 1)/2, r̂ 4 i = (r4i + 1)/2. Also after reordering we may assume that r̂ j i ≥ r̂ji+1. Up to a choice of the above basis, the matrix A2⊕B2 is determined by four partitions pi1 : |pi1| = r̂11 + r̂12 + r̂13 + . . .+ r̂1k1 pi2 : |pi2| = r̂21 + r̂22 + r̂23 + . . .+ r̂2k2 pi3 : |pi3| = r̂31 + r̂32 + r̂33 + . . .+ r̂3k3 pi4 : |pi4| = r̂41 + r̂42 + r̂43 + . . .+ r̂4k4 with α0 = |pi1|+ |pi2|+ |pi3|+ |pi4| − l(pi4) α1 = |pi1|+ |pi2|+ |pi3|+ |pi4| − l(pi3). With respect to the above basis, denote the normal form of A2 ⊕ B2 by A {pij} 2 ⊕B{pij}2 . The space RNα can be stratified by this data, giving [RNα ] = ∑ pi1,pi2,pi3,pi4 [R(pi1, pi2, pi3, pi4)], where R(pi1, pi2, pi3, pi4) is the stratum of R N α , where A2⊕B2 has normal form A {pij} 2 ⊕B{pij}2 . The space R(pi1, pi2, pi3, pi4) is a vector bundle p : R(pi1, pi2, pi3, pi4)→ {(A2, B2) | A2 ⊕B2 ∼ A{pij}2 ⊕B{pij}2 } 46 over the space of all matrices with this normal form, with fibre the linear space of matrices {B1 | B1A{pij}2 B{pij}2 = B{pij}2 A{pij}2 B1}. We compute the fibre and base by a linear algebra calculation to deduce [R(pi1, pi2, pi3, pi4)] = [GLα0 ]·[GLα1 ]f(pi1)f(pi2)g(pi3)g(pi4)(−L1/2)−(l(pi3)−l(pi4)) 2 , where we are given f(pi) = ∏ i≥1 Lb 2 i /[GLbi ] for pi = (1 b12b23b3 · · · ), and g(pi) = ∏ i≥1 (−L1/2)b2i /[GLbi ] for pi = (1b12b23b3 · · · ). Substituting this into the generating series gives N(y0, y1) = ∑ α0,α1≥0 ∑ pi1,pi2,pi3,pi4 α0=|pi1|+|pi2|+|pi3|+|pi4|−l(pi4) α1=|pi1|+|pi2|+|pi3|+|pi4|−l(pi3) f(pi1)f(pi2)g(pi3)g(pi4)y α0 0 y α1 1 = ∑ pi1 f(pi1)(y0y1) |pi1| ∑ pi2 f(pi2)(y0y1) |pi2| ∑ pi3 g(pi3)(y0y1) |pi3|y−l(pi3)1 · ∑ pi4 g(pi4)(y0y1) |pi4|y−l(pi4)0 . The series for f and g have well known formulas [36] ∑ pi f(pi)y|pi| = ∞∏ i,j=1 (1− L1−jyi)−1 = Exp ( L L− 1 ∑ n≥1 yn ) , and ∑ pi g(pi)y|pi|a−l(pi) = ∞∏ i,j=1 (1+(−L1/2)−2j+1yia−1) = Exp ( L1/2 1− L ∑ n≥1 yna−1 ) . 47 Hence N(y0, y1) = Exp ( 2L L− 1 ∑ n≥1 (y0y1) n ) · Exp (−L1/2 L− 1 ∑ n≥1 yn0 y n−1 1 + y n−1 0 y n 1 ) . Multiplying the series I and N gives AU (y0, y1) = Exp ( (L+ L2)y0y1 − L1/2(y0 + y1) L− 1 ∑ n≥0 (y0y1) n ) . 3.3.3 Second proof: another dimensional reduction Recall that representations of the cut algebra JW,I are given by triples (A2, B1, B2), where A2 : V0 → V1, B1, B2 : V1 → V0 are linear maps satisfy- ing B1A2B2 = B2A2B1. (3.8) The pair (A2, B2) gives a representation of the quiver C2 = (0, 1; a : 0→ 1, b : 1→ 0). Given the dimension vector α ∈ N2, let R(JW,I , α) be the space of repre- sentations of JW,I having dimension vector α. Let R(C 2, α) be the space of representations of C2 having dimension vector α. There is a forgetful map g : R(JW,I , α)→ R(C2, α), (A2, B1, B2)→ (A2, B2). Its fibers are linear vector spaces. This map is equivariant with respect to the natural action of Gα = GLα0 ×GLα1 on both sides. Given a C2- representation M = (M0,M1;Ma,Mb), let ρ(M) be the dimension of the fiber of g over M . Let M0 = (M0;MbMa) and M 1 = (M1;MaMb) be representations of the Jordan quiver C1 (one vertex and one loop). Then it follows from (3.8) that ρ(M) = dim HomC1(M 1,M0). 48 More generally, for any two representations of C2 M = (M0,M1;Ma,Mb), N = (N0, N1;Na, Nb) we define ρ(M,N) = dim HomC1(M 1, N0). If M is some representation of C2 having dimension vector α, then the contribution of its Gα-orbit (i.e. isomorphism class) to [R(C 2, α)]/[Gα] is 1/[AutM ]. The contribution of the preimage of itsGα-orbit to [R(JW,I , α)]/[Gα] is Lρ(M)/[AutM ]. Let M = ⊕Mnii be a decomposition of a C2-representation M into the sum of indecomposable representations. Then by [41, Theorem 1.1] [AutM ] = [End(M)] · ∏ i (L−1)ni , where (q)n = (q; q)n = ∏n k=1(1 − qk) is the q-Pochhammer symbol. Thus the contribution of the preimage of the Gα-orbit of M to [R(JW,I , α)]/[Gα] is Lρ(M) [AutM ] = Lρ(M,M)−h(M,M)∏ i(L−1)ni = ∏ i,j Lninj(ρ(Mi,Mj)−h(Mi,Mj))∏ i(L−1)ni , (3.9) where h(M,N) = dim Hom(M,N) for any C2-representations M,N . We will compute the numbers h(M,N) and ρ(M,N) for indecomposable repre- sentations M,N of C2. As is well known, the indecomposable representa- tions of C2 are the following: 1. Representations In of dimension (n, n− 1), n ≥ 1. 2. Representations Pn of dimension (n− 1, n), n ≥ 1. 3. Representations Rt,n = (Idn, Jt,n), n ≥ 1, t ∈ C, of dimension (n, n). There are also representations R∞,n = (J0,n, Idn), n ≥ 1, of dimension (n, n). Here Jt,n denotes the Jordan block of size n with value t on the diagonal. 49 Remark 3.5. Define duality on representations of C2 by D(M0,M1;M12,M21) = (M ∨ 0 ,M ∨ 1 ;M ∨ 21,M ∨ 12). Then Hom(DM,DN) = Hom(N,M)∨ and D(In) = In, D(Pn) = Pn, D(Rt,n) = Rt−1,n. Define equivalence (cyclic shift) by C(M0,M1;M12,M21) = (M1,M0;M21,M12). Then Hom(CM,CN) = Hom(M,N) and C(In) = Pn, C(Pn) = In, C(Rt,n) = Rt−1,n. The proofs of the following two propositions are easy exercises. Proposition 3.9. We have 1. h(Rs,m, Rt,n) = min{m,n} if s = t or s, t ∈ {0,∞}. It is 0 otherwise. 2. h(Im, Rt,n) = h(Rt,n, Pm) =  min{m− 1, n} t = 0; min{m,n} t =∞; 0 t ∈ C∗. 3. h(Rt,n, Im) = h(Pm, Rt,n) =  min{m− 1, n} t =∞; min{m,n} t = 0; 0 t ∈ C∗. 4. h(Im, In) = h(Pm, Pn) = min{m,n}. 5. h(Im, Pn) = h(Pn, Im) = min{m,n} − 1. Proposition 3.10. We have 1. ρ(Rs,m, Rt,n) = min{m,n} if s = t or s, t ∈ {0,∞}. It is 0 otherwise. 50 2. ρ(Im, Rt,n) = ρ(Rt,n, Pm) = min{m− 1, n} t = 0,∞;0 t ∈ C∗. 3. ρ(Rt,n, Im) = ρ(Pm, Rt,n) = min{m,n} t = 0,∞;0 t ∈ C∗. 4. ρ(Im, In) = ρ(Pn, Pm) = min{m− 1, n}. 5. ρ(Im, Pn) = min{m,n} − 1. 6. ρ(Pn, Im) = min{m,n}. Corollary 3.11. For any C2-representations M,N , let d(M,N) = ρ(M,N)− h(M,N). If M,N are indecomposable, then d(M,N)+d(N,M) =  1 M = Im, N = Pn; −1− δm,n M = Im, N = In or M = Pm, N = Pn; 0 otherwise. Proof of Theorem 3.8. We can decompose any C2-representation as M = I ⊕ P ⊕ ⊕ t∈P1 Rt = ⊕ i≥1 Imii ⊕ ⊕ i≥1 Pnii ⊕ ⊕ t∈P1 ⊕ i≥1 R ri(t) t,i . With the representation M we associate partitions µ, η ∈ P and λ(t) ∈ P, for t ∈ P1, in the following way: µk = ∑ i≥k mi, ηk = ∑ i≥k ni, λk(t) = ∑ i≥k ri(t). 51 Applying Corollary 3.11, we obtain ρ(M,M)− h(M,M) = ∑ i,j≥1 minj − ∑ i≥j≥1 (mimj + ninj) = −1 2 ((∑ i≥1 (mi − ni) )2 + ∑ i≥1 m2i + ∑ i≥1 n2i ) , (3.10) an expression that we are going to denote by d(µ, η). The dimension vectors of the summands of M are given by dim I = (∑ i≥1 imi, ∑ i≥1 (i− 1)mi ) = (|µ|, |µ| − µ1), dimP = (∑ i≥1 (i− 1)ni, ∑ i≥1 ini) ) = (|η| − η1, |η|), dimRt = (∑ i≥1 iri(t), ∑ i≥1 iri(t) ) = (|λ(t)|, |λ(t)|). Applying equation (3.9) we obtain AU = ∑ α∈N2 (−L 12 )(α0−α1)2 [R(JW,I , α)] [Gα] yα = ∑ µ,η∈P (−L 12 )(µ1−η1)2 y |µ|+|η|−η1 0 y |µ|+|η|−µ1 1 Ld(µ,η)∏ i≥1(L−1)µi−µi+1(L−1)ηi−ηi+1 ∑ λ:P1→P ∏ t∈P1 fλ(t), (3.11) where fλ = (y0y1) |λ|∏ i≥1(L−1)λi−λi+1 . By Hua formula’s (see [24] or [43, Theorem 6]) applied to the quiver with one loop, we obtain f = ∑ λ∈P fλ = Exp ( L L− 1 ∑ n≥1 (y0y1) n ) . Therefore, using the geometric description of power structures (Section 3.2.2), 52 we obtain∑ λ:P1→P ∏ t∈P1 fλ(t) = Pow(f, [P1]) = Exp ( (L+ 1)L L− 1 ∑ n≥1 (y0y1) n ) . On the other hand it follows from (3.10) that (µ1 − η1)2 + 2d(µ, η) = − ∑ i≥1 m2i − ∑ i≥1 n2i and therefore AU = Exp ( L+ L2 L− 1 ∑ n≥1 (y0y1) n ) ∑ µ,η∈P y |µ|+|η|−η1 0 y |µ|+|η|−µ1 1 (−L 1 2 )− ∑ i(m 2 i+n 2 i )∏ i≥1(L−1)mi(L−1)ni , (3.12) where we denote mi = µi − µi+1, ni = ηi − ηi+1. Define H(x, q 1 2 ) = ∑ n≥0 (−q 12 )−n2xn (q−1)n = ∑ n≥1 (xq 1 2 )n (q)n = Exp ( xq 1 2 1− q ) , where the last equality follows from the Heine formula [28, 42]. Then the sum in (3.12) can be written in the form ∑ (mi)i≥1,(ni)i≥1 ∏ i≥1 (yi0y i−1 1 ) mi(yi−10 y i 1) ni(−L 12 )−m2i−n2i (L−1)mi(L−1)ni = ∏ i≥1 (∑ m≥0 (yi0y i−1 1 ) m(−L 12 )−m2 (L−1)m ∑ n≥0 (yi−10 y i 1) n(−L 12 )−n2 (L−1)n ) = ∏ i≥1 H(yi0y i−1 1 ,L 1 2 )H(yi−10 y i 1,L 1 2 ) = Exp ( L 1 2 1− L ∑ i≥1 (yi0y i−1 1 + y i−1 0 y i 1) ) . The second proof of Theorem 3.8 is complete.  3.3.4 Decomposing the universal series In this section, we decompose the product from Theorem 3.8. We will say that a stability parameter ζ is generic, if for any stable J-module V , we 53 have ζ · dimV 6= 0. For generic stability parameter ζ, let M+ζ (J, α) (resp. M−ζ (J, α)) denote the moduli stacks of J-modules V such that dimV = α and such that all the HN factors F of V with respect to the stability parameter ζ satisfy ζ · dimF > 0 (resp. < 0). We put A±ζ = ∑ α∈NQ0 [M±ζ (J, α)]vir · yα. Lemma 3.12. The generating series A±ζ are given by A±ζ = ∏ α∈∆+ ±ζ·α<0 Aα, where Aα = Aα(y0, y1) were defined in Theorem 3.8. We have AU = A + ζ A − ζ . Proof. By Theorem 3.7, we have a factorization in TQ (note that TQ is commutative and we don’t need to take the ordered product) AU = ∏ µ∈R Aζ,µ, whereAζ,µ were defined in Definition 3.7. Similarly we haveA ± ζ = ∏ ±µ>0Aζ,µ. By Theorem 3.8, we have AU = ∏ α∈∆+ Aα, where Aα contain only powers ykα, k ≥ 0. By the uniqueness of the factor- izations from Lemma 3.5, we obtain Aζ,µ = ∏ α∈∆+ µ(α)=µ Aα and the statement of the lemma follows.  54 3.4 Motivic DT with Framing 3.4.1 Framed quiver Let Q be a quiver with a distinguished vertex 0 ∈ Q0 and let W be a potential. We denote by Q̃ the corresponding framed quiver, the new quiver obtained from Q by adding a new vertex ∞ and a single new arrow ∞→ 0. Let J̃ = J Q̃,W be the Jacobian algebra corresponding to the quiver with potential (Q̃,W ), where we view W as a potential for Q̃ in the obvious way. Any Q̃-representation (resp. J̃-module) Ṽ can be written as a triple (V, Ṽ∞, s), where V is a Q-representation (resp. J-module), Ṽ∞ is a vector space, and s : Ṽ∞ → V0 is a linear map. We will always do this identification without mentioning. The twisted motivic algebra TQ of the original quiver sits as a subalgebra inside the algebra T Q̃ associated to the framed quiver Q̃. Note that in T Q̃ we have y∞ · y(α,0) = (−L 12 )−α0 · y(α,1) = L−α0 · y(α,0) · y∞, (3.13) where we put y∞ = y(0,1). In particular, T Q̃ is never commutative. 3.4.2 Stability for framed representations Let ζ ∈ RQ0 be a vector, which we will refer to as the stability parameter. Definition 3.8. A Q̃-representation (resp. J̃-module) Ṽ with dim Ṽ∞ = 1 is said to be ζ-(semi)stable, if it is (semi)stable with respect to (ζ, ζ∞) ∈ RQ̃ (see Definition 3.2), where ζ∞ = −ζ · dimV . Equivalently, the following conditions should be satisfied: • for any Q̃-subrepresentation (resp. J̃-submodule) 0 6= Ṽ ′ ⊂ Ṽ with Ṽ ′∞ = 0, we have ζ · dimV ′ (≤) 0; 55 • for any Q̃-quotient representation (resp. J̃-quotient module) Ṽ  Ṽ ′′ 6= 0 with Ṽ ′′∞ = 0, we have ζ · dimV ′′ (≥) 0. As in Section 3.3.4, a stability parameter ζ ∈ RQ0 is said to be generic, if for any stable J-module V we have ζ · dimV 6= 0. 3.4.3 Motivic DT invariants with framing For a stability parameter ζ ∈ RQ0 and a dimension vector α ∈ NQ0 , let as before ζ∞ = −ζ · α, α̃ = (α, 1), and let Mζ(Q̃, α) = [R(ζ,ζ∞)(Q̃, α̃)/Gα], Mζ(J̃ , α) = [R(ζ,ζ∞)(J̃ , α̃)/Gα] denote the moduli stack of ζ-stable Q̃-representations (resp. J̃-modules) with dimension vector α̃. The corresponding stacks for the trivial stability ζ = 0 will be denoted by M(Q̃, α) and M(J̃ , α). Remark 3.6. Note that the stack Mζ(Q̃, α) is slightly different from the stack M(ζ,ζ∞)(Q̃, α̃) which was defined in (3.1) to be [R(ζ,ζ∞)(Q̃, α̃)/Gα̃]. The same applies to the stacks of J̃-modules. Definition 3.9. Let ÃU = ∑ α∈NQ0 [ M ( J̃ , α )] vir · yα̃ ∈ T Q̃ , where [ M ( J̃ , α )] vir is defined similarly to (3.2). For any stability parameter ζ ∈ RQ0 let Ãζ = ∑ α∈NQ0 [ Mζ ( J̃ , α )] vir · yα̃ ∈ T Q̃ , where [ Mζ ( J̃ , α̃ )] vir is defined similarly to (3.3). Let also, as in the Intro- duction, Zζ = ∑ α∈NQ0 [ Mζ ( J̃ , α )] vir · yα ∈ TQ. 56 3.4.4 Relating the universal and framed series In this subsection we assume that Q is a symmetric quiver and therefore TQ is commutative. The following theorem relates results of the previous section on the universal series to the framed invariants of this section. Theorem 3.13. For generic stability parameter ζ, we have Zζ = A−ζ (−L 1 2 y0, y1, . . . ) A−ζ (−L− 1 2 y0, y1, . . . ) , (3.14) where A−ζ were defined in Section 3.3.4. This result is [45, Corollary 4.17]. In the rest of this subsection, we provide an alternative approach to this theorem. The main difference is that here we study just two stability parameters, while the result in [45] was obtained by studying an infinite sequence of parameters between these two. Proposition 3.14. Let Ṽ be a Q̃-representation (resp. a J̃-module) with dim Ṽ∞ = 1. Then there exists the unique filtration 0 = Ũ0 ⊂ Ũ1 ⊂ Ũ2 ⊂ Ũ3 = Ṽ such that with Ṽ i = Ũ i/Ũ i−1 we have 1. Ṽ 1∞ = 0 and all the HN factors F of V 1 with respect to the stability parameter ζ satisfy ζ · dimF > 0, 2. Ṽ 2∞ = 1 and Ṽ 2 is ζ-semistable, 3. Ṽ 3∞ = 0 and all the HN factors F of V 3 with respect to the stability parameter ζ satisfy ζ · dimF < 0. Proof. We will work only with Q̃-representations. We take sufficiently small ε > 0 and define the central charge Zζ,ε(α̃) = −ζ · α+ (ε|α|+ α∞) √−1, α̃ = (α, α∞). 57 Let W̃ be a Q̃-representation with dim W̃∞ = 1. For any submodule W̃ ′ = W ′ of W̃ with W̃ ′∞ = 0, we have ζ · dimW ′ ≷ 0 ⇐⇒ argZζ,ε(dim W̃ ′) ≷ argZζ,ε(dim W̃ ). Hence W̃ is Zζ,ε-stable if and only if it is ζ-stable. Then, the Harder- Narashimhan filtration for Zζ,ε-stability is the required filtration.  The filtration from Proposition 3.14 induces the following factorization in the same way as Theorem 3.7: Proposition 3.15. We have ÃU = A + ζ · Ãζ ·A−ζ in the motivic algebra T Q̃ . Proposition 3.16. ÃU = AU · y∞. Proof. Any Q̃-module (resp. J̃-module) Ṽ with dim Ṽ∞ = 1 and dimV = α has a unique filtration 0 ⊂ V ⊂ Ṽ with Ṽ /V ' S∞, the simple module concentrated at the vertex ∞. Thus the factorization follows.  Proof of Theorem 3.13. We have Ãζ = (A + ζ ) −1 · ÃU · (A−ζ )−1 (Proposition 3.15) = (A+ζ ) −1 · (A+ζ ·A−ζ · y∞) · (A−ζ )−1 (Prop. 3.16 and Lemma 3.12) = y∞ · A−ζ (Ly0, y1, . . . ) A−ζ (y0, y1, . . . ) . (Equation (3.13)) (3.15) 58 It follows from (3.13) that y∞ · Zζ(−L 12 y0, . . . ) = Ãζ . Combining this with (3.15) we get the statement of Theorem 4.21.  3.4.5 Application to the conifold The following theorem is the main result of this section, announced as The- orem 3.1. Let (Q,W ) be the conifold quiver with potential. Theorem 3.17. For generic ζ ∈ R2, Zζ(y0, y1) = ∏ α∈∆+ ζ·α<0 Zα(y0, y1), (3.16) with Zα(−y0, y1)=  α0−1∏ j=0 ( 1− L−α02 + 12+jyα ) α ∈ ∆re+ α0−1∏ j=0 ( 1− L−α02 +1+jyα )−1 ( 1− L−α02 +2+jyα )−1 α ∈ ∆im+ Proof. Substituting the result of Lemma 3.12 into Theorem 3.13, we get the product form (3.16), with Zα(y0, y1) = A α(−L 12 y0, y1)/Aα(−L− 12 y0, y1). Now use the expression for Aα from Theorem 3.8.  59 3.5 DT/PT Series 3.5.1 Chambers and the moduli spaces for the conifold Let (Q,W ) be the conifold quiver with potential. In the space R2 of stability parameters, consider the lines L+(m) = {(ζ0, ζ1) | mζ0 + (m− 1)ζ1 = 0} (m ≥ 1), L∞ = {(ζ0, ζ1) | ζ0 + ζ1 = 0}, L−(m) = {(ζ0, ζ1) | mζ0 + (m+ 1)ζ1 = 0} (m ≥ 0). It is immediately seen that these are exactly the lines orthogonal to the roots in ∆+ with respect to the standard inner product. Let L ⊂ R2 denote the union of this countable set of lines. The complement of L in R2 is a countable union of open cones. Denote by Y + the flop of Y along the embedded rational curve. Theorem 3.18. [46, Lemma 3.1 and Propositions 2.10-2.13] The set of generic parameters in R2 is the complement of the union L of the lines defined above. 1. For ζ with ζ0 < 0 and ζ1 < 0, the moduli spaces Mζ(J̃ , α) are the NCDT moduli spaces, the moduli spaces of cyclic J-modules from [58]. 2. For ζ near the line L∞ with ζ0 < ζ1 and ζ0 + ζ1 < 0, the moduli spaces Mζ(J̃ , α) are the commutative DT moduli spaces of Y from [37], the moduli spaces of subschemes on Y with support in dimension at most 1. 3. For ζ near the line L∞ with ζ0 < ζ1 and ζ0 + ζ1 > 0, the moduli spaces Mζ(J̃ , α) are the PT moduli spaces of Y introduced in [53]; these are moduli spaces of stable rank-1 coherent systems. 4. For ζ near the line L∞ with ζ0 > ζ1 and ζ0 + ζ1 < 0, the moduli spaces Mζ(J̃ , α) are the commutative DT moduli spaces of the flop Y +. 60 5. For ζ near the line L∞ with ζ0 < ζ1 and ζ0 + ζ1 > 0, the moduli spaces Mζ(J̃ , α) are the PT moduli spaces of the flop Y +. 6. For ζ with ζ0 > 0 and ζ1 > 0, the moduli space Mζ(J̃ , α) consists of a point for α = 0 and is otherwise empty. Remark 3.7. Note that “near” in the above statements means sufficiently near depending on the dimension vector (α, 1). 3.5.2 Motivic PT and DT invariants Proposition 3.19. The refined partition functions of the resolved conifold Y for the DT and PT chambers are given by ZPT(−y0, y1) = ∏ m≥1 m−1∏ j=0 ( 1− L−m2 + 12+jym0 ym−11 ) (3.17) and ZDT(−y0, y1) = ZPT(−y0, y1) · ∏ m≥1 m−1∏ j=0 ( 1− L−m2 +1+jym0 ym1 )−1 ( 1− L−m2 +2+jym0 ym1 )−1 . (3.18) Proof. Let ζ = (−1 + ε, 1), 0 < ε  1, be some stability corresponding to PT moduli spaces. Then {α ∈ ∆+ | ζ · α < 0} = {(m,m− 1) | m ≥ 1}. Applying Theorem 3.17 we obtain ZPT (−y0, y1) = ∏ ζ·α<0 Zα(−y0, y1) = ∏ m≥1 m−1∏ j=0 ( 1− L−m2 + 12+jym0 ym−11 ) . The proof of the second formula is similar.  Let us re-write these formulae in the perhaps more familiar large radius 61 parameters T = y−11 , s = y0y1, corresponding to the cohomology class of a point and a curve on the geometry Y . We obtain ZPT(−s, T ) = ∏ m≥1 m−1∏ j=0 ( 1− L−m2 + 12+jsmT ) (3.19) and ZDT(−s, T ) = ZPT(−s, T )· ∏ m≥1 m−1∏ j=0 ( 1− L−m2 +1+jsm )−1 ( 1− L−m2 +2+jsm )−1 . (3.20) The specializations at L 1 2 = 1 are the PT and DT series of the resolved conifold respectively, given by the standard expressions ZPT(−s, T ) = ∏ m≥1 (1− Tsm)m = Exp ( −T (s 1 2 − s− 12 )2 ) and ZDT(−s, T ) = M(s)2 ∏ m≥1 (1− Tsm)m , with M(s) = ∏ m≥1(1 − sm)−m the MacMahon function, and denoting generating series of numerical (as opposed to motivic) invariants. Wall crossing at the special wall L∞ is the PT/DT wall crossing of [53]. On the PT side, the coefficient of the T 0 term is just 1, since if there is no curve present, the only possible PT pair consists of the structure sheaf of Y (with zero map). On the DT side, the moduli space with zero curve class is the moduli space of ideal sheaves of point clusters on Y , in other words the Hilbert scheme of points of Y . Hence the ratio of the T 0 terms gives the generating function of virtual motives of the Hilbert scheme of points of Y [4]: ∞∑ n=0 [Y [n]]vir(−s)n = ∏ m≥1 m−1∏ j=0 ( 1− L1+j−m2 sm )−1 ( 1− L2+j−m2 sm )−1 . 62 At L 1 2 = 1, we obtain the MNOP result M(s)2. Note in particular that, as proved in [4] but contrary to the speculations of [17], the motivic refinement is not a square, though both products are combinatorial refinements of the usual MacMahon series. Remark 3.8. Note that our results in fact imply a full factorization ZDT(s, T ) = ( ∞∑ n=0 [Y [n]]virs n ) ZPT(s, T ), (3.21) with the middle sum being a product of refined MacMahon series as in [4]. This is a motivic analogue of the factorization ZX,DT(s, T ) = M(s) e(X)ZX,PT(s, T ) (3.22) conjectured for a quasi-projective Calabi–Yau threefold X in [37, 53], proved in [9] following earlier proofs of a version of this statement in [57, 61]. In gen- eral, the only definition we have of the motivic ZX,DT and ZX,PT is through the partially conjectural setup of [33]. Assuming that relevant parts of [33] are put on a firm footing, including the integration ring homomorphism from the motivic Hall algebra to the motivic quantum torus, it seems likely that the proof of [9] can be adapted to prove the motivic version (3.21) in general. 3.5.3 Connection with the refined topological vertex The standard way to compute the unrefined PT series of the resolved conifold Y is via the topological vertex [1]. From the toric combinatorics, we obtain the formula Z vertex PT (−s, T ) = ∑ λ Cλ∅∅(s)Cλt∅∅(s)(−T )|λ|, see e.g. [26, (63)]. On the right hand side, the sum runs over all partitions; for a partition λ, λt denotes the conjugate partition, and Cλµν(s) is the topological vertex expression of [1]. In the case when µ = ν = ∅, Cλ∅∅(s) can be expressed as a simple Schur function, and then Cauchy’s identity 63 immediately gives Z vertex PT (−s, T ) = ∏ m≥1 (1− Tsm)m = ZPT(−s, T ). In mathematical terms [37], this equality (or rather its DT version) ex- presses torus localization, the combined expression M(s)Cλµν(s) being the generating function of 3-dimensional partitions with given 2-dimensional asymptotics along the coordinate axes. The refined PT partition function as computed by the refined topological vertex is [26, (67)] ZvertexPT (t, q, T ) = ∑ λ Cλ∅∅(q, t)Cλt∅∅(t, q)(−T )|λ|, where Cλµν(q, t) is now the refined topological vertex expression. Using the Cauchy identity again, this sum reduces to [26, (67)] ZvertexPT (t, q, T ) = ∏ i,j≥1 ( 1− Tqi− 12 tj− 12 ) (3.23) = exp (∑ n≥1 −Tn n(q n 2 − q−n2 )(tn2 − t−n2 ) ) = Exp ( −T (q 1 2 − q− 12 )(t 12 − t− 12 ) ) . Proposition 3.20. We have ZvertexPT (t, q, T ) = ZPT (−s, T ), when we make the change of variables q = L 1 2 s, t = L− 1 2 s, with (qt) 1 2 = s. Proof. This is immediate when we compare (3.19) with (3.23).  Thus we obtain a proof of the “motivic=refined” correspondence [17] in this example. Note however that the situation is very different from the unrefined story: here we are not giving a full mathematical interpretation of the refined topological vertex expression Cλµν(q, t); indeed, it remains a 64 very interesting problem to find one. We are only checking that the results agree in the particular case of the resolved conifold. 3.5.4 Connection to the cohomological Hall algebra An alternative to considering the refined motivic invariants is to consider the mixed Hodge modules of vanishing cycles [16, 32]. The description as the vanishing locus of the trace of the potential endows the moduli spaces Mζ ( J̃ , α̃ ) with the mixed Hodge modules of vanishing cycles of the trace function. Recently, the cohomologies of the moduli spaces with coefficients in these mixed Hodge modules have been organized into an algebra in [32], the (critical) cohomological Hall algebra. Replacing L by q in all our formulae, we obtain generating series of E-polynomials of these mixed Hodge modules, the analogues of the formulae of [16] in our situation. 65 Chapter 4 Local Toric Examples This chapter is joint work with Prof. Kentaro Nagao [40]. Here we generalize the work of the previous chapter to compute the motivic Donaldson–Thomas theory of small crepant resolutions of a toric Calabi–Yau 3-folds. 4.1 Introduction As mentioned this Chapter is a continuation of [39]. We recall that a Donaldson–Thomas (DT) invariant of a Calabi–Yau 3-fold Y is a count- ing invariant of coherent sheaves on Y , introduced in [60] as a holomorphic analogue of the Casson invariant of a real 3-manifold. A component of the moduli space of stable coherent sheaves on Y carries a symmetric obstruc- tion theory and a virtual fundamental cycle [6, 7]. A DT invariant of a compact Y is then defined as the integral of the constant function 1 over the virtual fundamental cycle of the moduli space. It is known that the moduli space of coherent sheaves on Y can be locally described as the critical locus of a function, the holomorphic Chern–Simons functional (see [27]). Behrend provided a description of DT invariants in terms of the Euler characteristic of the Milnor fiber of the CS functional [2]. Inspired by this result, the proposal of [33, 4] was to study the motivic Milnor fiber of the CS functional as a motivic refinement of the DT invariant. Such a refinement had been expected in string theory [26, 17]. On the other hand, in [58], it was proposed to study counting invariants for the non-commutative crepant resolution (NCCR) of the conifold, which are called non-commutative Donaldson–Thomas (ncDT) invariants. It was also conjectured there that ncDT and DT invariants are related by wall crossing. The paper [46] realized this, by 66 • describing the chamber structure on the space of stability parameters for the NCCR, • finding chambers which correspond to geometric DT and stable pair (PT), as well as ncDT invariants, and • computing the generating function of DT type invariants for each chamber. For the conifold, the dimension of the fiber of the crepant resolution is less than 2 (we say that the resolution is small). This condition plays an important role in many places of the paper. Affine toric Calabi–Yau 3-folds which have small crepant resolutions are classified as follows: 1. X = XN0,N1 := {XY − ZN0WN1} for N0 > 0 and N1 ≥ 0, or 2. X = X(Z/2Z)2 := C3/(Z/2Z)2 where (Z/2Z)2 acts on C3 with weights (1, 0), (0, 1) and (1, 1). Figure 4.1: Polygons for XN0,N1 and X(Z/2Z)2 In [47], counting invariants for non-commutative and commutative crepant resolutions of {XY − ZN0WN1} were studied. First, we provided descrip- tions of NCCRs of {XY − ZN0WN1} in terms of a quiver with potential. Given N0 and N1, the quivers with potential are not unique. However it was also shown that any such quivers with potential are related by a sequence of mutations. Finally, generalizations of the results in [46] are given. In [39], we provided motivic refinements of formulae in [46]. For the proof, we needed one explicit evaluation of the “universal” series ([39, §2]) and a wall-crossing argument ([39, §3]). 67 In this chapter, we will show similar formulae for {XY −ZN0WN1}, that is, motivic refinements of the formulae in [47]. The wall-crossing argument works without modifications (§4.7, 4.8), while the evaluation part is more involved (Theorem 4.1). Our strategy is as follows: • First, in §4.5, we evaluate the universal series for a specific NCCR using a generalization of the calculation [39, §2.2]. • Then, in §4.6, we evaluate the universal series for a general NCCR. In [51], Nagao has provided a formula which describes how the universal series changes under mutation (§4.7, 4.8). Although we assume that the quiver has no loops and 2-cycles in [51], we can apply a parallel argument in our setting as well. Since any two NCCRs are related by a sequence of mutations, the evaluation is done. 4.1.1 Main result Let Γ be the quadrilateral (or the triangle in case N1 = 0) as in Figure 4.1 and σ be a partition Γ, that is, a division of Γ into N -tuples of triangles with area 1/2. We will associate σ with a quiver with potential (Qσ, ωσ). The set of vertices of the quiver Qσ is Î := Z/NZ, which is identified with {0, . . . , N − 1}. A vertex has a loop if and only if it is in the subset Îr ⊂ Î (see (4.1) for the definition). It is shown in [47, §1] that the Jacobian algebra Jσ := J(Qσ, ωσ) is an NCCR of X := Spec (C[X,Y, Z,W ]/(XY − ZN0WN1)) . Let ∆ be the set of roots of type ÂN and ∆σ,+ (resp. ∆ re σ,+, ∆ im σ,+) denote the set of positive (resp. positive real, positive imaginary) roots. 1 For α ∈ NÎ , let M(Jσ, α) be the moduli stack of Jσ-modules V with dimV = α. We define the generating series of the motivic DT invariants of 1From the view point of the root system, a choice of a partition σ corresponds to a choice of a set of simple roots. 68 (Qσ,Wσ) by AσU (y) = A σ U (y0, . . . , yN−1) := ∑ α∈NQ0 [M(Jσ, α)]vir ·yα ∈MC[[y0, . . . , yN−1]].2 Here yα := ∏ (yi) αi and [•]vir denotes the virtual motive (see Section 4.4.1), an element of a suitable ring of motivesMC. The subscript referring to the fact that we think of this series as the universal series. To each root α ∈ ∆σ,+, we associate an infinite product as follows: • for a real root α ∈ ∆reσ,+ such that ∑ k/∈Îr αk is odd, put Aα(y) := Exp ( −L−1/2 1− L−1 y α ) = ∏ j≥0 ( 1− L−j−1/2yα ) • for a real root α ∈ ∆reσ,+ such that ∑ k/∈Îr αk is even, put Aα(y) := Exp ( 1 1− L−1 y α ) = ∏ j≥0 ( 1− L−jyα)−1 • for an imaginary root α ∈ ∆imσ,+, put Aα(y) := Exp ( N − 1 + L 1− L−1 y α ) = ∏ j≥0 ( 1− L−jyα)1−N · (1− L−j+1yα)−1 . The main result of this paper is the following formula: 2For the wall-crossing of motivic DT theory, a twisted product on yα’s twisted by the Euler form plays a crucial role. In this case, the twisted product coincides with the usual commutative product since the Euler form is trivial. 69 Theorem 4.1. AσU (y) = ∏ α∈∆σ,+ Aα(y). This is proved in §4.5 and §4.6.2. 4.1.2 Corollaries Let J̃σ = J(Q̃σ,Wσ) be the framed algebra given by adding the new vertex ∞ and the new arrow from ∞ to 0 to the quiver of Jσ. In [46], the authors introduce a notion of ζ-(semi)stability of J̃-modules Ṽ with dim Ṽ∞ ≤ 1 for a stability parameter ζ ∈ RÎ . For α ∈ NÎ , let Mζ(J̃ , α) be the moduli space of ζ-stable J̃-modules Ṽ with dim Ṽ = (α, 1). We want to compute the motivic generating series Zζ(y) = Zζ(y0, . . . , yN−1) := ∑ α∈NÎ [ Mζ ( J̃ , α )] vir · yα ∈MC[[y0, . . . , yN−1]]. To each root α ∈ ∆σ,+, we put Zα(y0, . . . , yN−1) := Aα(−L1/2y0, y1, . . . , yN−1) Aα(−L−1/2y0, y1, . . . , yN−1) . They are given as follows: • for a real root α ∈ ∆reσ,+ such that ∑ k/∈Îr αk is odd, we have Zα(−y0, . . . , yN−1) = α0−1∏ i=0 ( 1− L−α02 + 12+iyα ) , • for a real root α ∈ ∆reσ,+ such that ∑ k/∈Îr αk is even, we have Zα(−y0, . . . , yN−1) = α0−1∏ i=0 ( 1− L−α02 +1+iyα )−1 , 70 • for an imaginary root α ∈ ∆imσ,+, we have Zα(−y0, . . . , yN−1) = α0−1∏ i=0 ( 1− L−α02 +1+iyα )1−N ·(1− L−α02 +2+jyα)−1 Applying the same argument as [39, §3], we get the following formula (§4.7): Corollary 4.2. For ζ ∈ RĨ not orthogonal to any root, we have Zζ(y) = ∏ α∈∆σ,+ ζ·α<0 Zα(y0, . . . , yN−1). By [2, 4], the specialization Zζ(y)|L 12→1 is the DT-type series at the generic stability parameter ζ, computed in [47]. Let Yσ → X be the crepant resolution corresponding to σ. The non- commutative crepant resolution Jσ is derived equivalent to Yσ. In [46, §3], we find a stability parameter ζDT (resp. ζPT) such that the moduli space coincides with the Hilbert scheme (resp. the stable pair moduli space) for Yσ. Let ZσDT(s, T1, . . . , TN−1) (resp. Z σ DT(s, T1, . . . , TN−1)) be the generating function of DT (resp. PT) invariants of Yσ. Here s is the variable for the homology class of a point and Ti is the variable for the homology class of the i-th component Ci of the exceptional curve. The variable change induced by the derived equivalence is given as follows: s := y0 · y1 · · · · · yN−1, Ti = yi. For 1 ≤ a ≤ b ≤ N − 1, we put C[a,b] := [Ca] + · · ·+ [Cb] ∈ H2(Yσ,Z), where Ci is a component of the exceptional curve and let T[a,b] = Ta · · · · · Tb 71 be the corresponding monomial. Let c(a, b) denote the number of (−1,−1)- curves in {Ci | a ≤ i ≤ b}. We define infinite products as follows: • If c(a, b) is odd, we put Z[a,b] = Z[a,b](s, T[a,b]) := ∞∏ n=1 ( n−1∏ i=0 ( 1− L−n2 + 12+i · (−s)n · T[a,b] )) . • If c(a, b) is even, we put Z[a,b] = Z[a,b](s, T[a,b]) := ∞∏ n=1 ( n−1∏ i=0 ( 1− L−n2 +1+i · (−s)n · T[a,b] )−1) . • For imaginary roots, we put Zim = Zim(s) := ∞∏ n=1 ( n−1∏ i=0 ( 1− L−n2 +1+i(−s)n )1−N ( 1− L−n2 +2+i(−s)n )−1) . Corollary 4.3. (1) The refined DT and PT series of Yσ are given by the formulae : ZDT(s, T1, . . . , TN−1) = Zim(s) · ∏ 1≤a≤b≤N−1 Z[a,b](s, T[a,b]) and ZPT(s, T1, . . . , TN−1) = ∏ 1≤a≤b≤N−1 Z[a,b](s, T[a,b]) (2) The generating function of virtual motives of the Hilbert scheme of points on Yσ is given by the formula : Z0-dim(s) := ∞∑ n=0 [ (Yσ)[n] ] vir · sn = Zim. (3) The refined version of DT-PT correspondence for Yσ holds : ZDT(s, T1, . . . , TN−1) = Z0-dim(s)× ZPT(s, T1, . . . , TN−1). 72 Remark. The formula in (2) is a direct consequence of the formula for ZDT in (1), since the polynomial in the T[a,b] variables does not contribute. 4.2 Root Systems Let N0 > 0 and N1 ≥ 0 be integers such that N0 ≥ N1 and set N = N0 +N1. We set I = {1, . . . , N − 1} , Î = {0, 1, . . . , N − 1} , Ĩ = { 1 2 , 3 2 , . . . , N − 1 2 } , Z̃ = { n+ 1 2 ∣∣∣n ∈ Z} . For l ∈ Z and j ∈ Z̃, let l ∈ Î and j ∈ Ĩ be the elements such that l − l ≡ j − j ≡ 0 modulo N . Let ZÎ be the free Abelian group with basis {αi | i ∈ Î}, where αi is called a simple root. We put ∆fin+ := {α[a,b] := αa + · · ·αb | 1 ≤ a ≤ b ≤ N − 1} ∆re,++ := {α[a,b] + n · δ | α[a,b] ∈ ∆fin+ , n ∈ Z≥0} ∆re,−+ := {−α[a,b] + n · δ | α[a,b] ∈ ∆fin+ , n ∈ Z>0} and ∆re+ := ∆ re,+ + unionsq∆re,−+ , ∆im+ := {n · δ | n ∈ Z>0} where δ := α0 + · · ·+ αN−1 is the (positive minimal) imaginary root. For k ∈ Î, the simple reflection at k is the group homomorphism given by ZÎ → ZÎ αi 7→ αi − Cik · αk where C is the Cartan matrix of type ÂN . This gives a self-bijection of 73 ∆re,++ \{αk}. 4.3 Non-Commutative Crepant Resolutions 4.3.1 Quivers with potential We denote by Γ the quadrilateral (or the triangle in case N1 = 0) with vertices (0, 0), (0, 1), (N0, 0) and (N1, 1). Note that the affine toric Calabi– Yau 3-fold corresponding to Γ is X = {XY − ZN0WN1}. A partition σ of Γ is a pair of functions σx : Ĩ → Z̃ and σy : Ĩ → {0, 1} such that • σ(i) := (σx(i), σy(i)) gives a bijection between Ĩ and the following set:{( 1 2 , 0 ) , ( 3 2 , 0 ) , . . . , ( N0 − 1 2 , 0 ) , ( 1 2 , 1 ) , ( 3 2 , 1 ) , . . . , ( N1 − 1 2 , 1 )} , • if i < j and σy(i) = σy(j) then σx(i) > σx(j). Giving a partition σ of Γ is equivalent to dividing Γ into N -tuples of triangles {Ti}i∈Ĩ with area 1/2 so that Ti has (σx(i)± 1/2, σy(i)) as its vertices. Let Γσ be the corresponding diagram, ∆σ be the fan and fσ : Yσ → X be the crepant resolution of X . We put Îr := { k ∈ Î ∣∣∣σy(k − 1 2 ) = σy(k + 1 2 ) } . (4.1) Example 4.1. Let us consider as an example the case N0 = 4, N1 = 2 and (σ(i))i∈Ĩ = (( 7 2 , 0 ) , ( 3 2 , 1 ) , ( 5 2 , 0 ) , ( 3 2 , 0 ) , ( 1 2 , 1 ) , ( 1 2 , 0 )) . We show the corresponding diagram Γσ in Figure 4.2. Let S be the union of an infinite number of rhombi with edge length 1 as in Figure 4.3 which is located so that the centers of the rhombi are on a line parallel to the x-axis in R2 and H be the union of infinite number of 74 Figure 4.2: Γσ hexagons with edge length 1 as in Figure 4.4 which is located so that the centers of the hexagons are in a line parallel to the x-axis in R2. We make Figure 4.3: S Figure 4.4: H the sequence τ = τσ : Z→ {S,H} which maps l to S (resp. H) if l modulo N is not in Îr (resp. is in Îr) and cover the whole plane R2 by arranging S’s and H’s according to this sequence (see Figure 4.5). We regard this as a graph on the 2-dimensional torus R2/Λ, where Λ is the lattice generated by ( √ 3, 0) and (N0 −N1, (N0 −N1) √ 3 +N1). We can color the vertices of this graph black or white so that each edge connects a black vertex and a white one. Let Pσ denote this bipartite graph on the torus. For each edge 75 Figure 4.5: Pσ in case Example 4.1 h∨ in Pσ, we make its dual edge h directed so that we see the black end of h∨ on our right hand side when we cross h∨ along h in the given direction. Let Qσ denote the resulting quiver. The set of vertices of the quiver Qσ is Î, which is identified with Z/NZ. The set of edges of the quiver Qσ is given by H := ∐ i∈Ĩ h+i  unionsq ∐ i∈Ĩ h−i  unionsq ∐ k∈Îr rk  . Here h+i (resp. h − i ) is an edge from i− 12 to i+ 12 (resp. from i+ 12 to i− 12), rk is an edge from k to itself. For each vertex q of Pσ, let ωq be the potential 3 which is the composition of all arrows in Qσ corresponding to edges in Pσ with q as their ends. We define ωσ := ∑ q : black ωq − ∑ q : white ωq. The relations of the Jacobian algebra are as follows: • h+i ◦ ri− 1 2 = ri+ 1 2 ◦ h+i and ri− 1 2 ◦ h−i = h−i ◦ ri+ 1 2 for i ∈ Ĩ such that 3A potential of a quiver Q is an element in CQ/[CQ,CQ], i.e. a linear combination of equivalence classes of cyclic paths in Q where two paths are equivalent if they coincide after a cyclic rotation. 76 i− 12 , i+ 12 ∈ Îr. • h+i ◦ ri− 1 2 = h−i+1 ◦ h+i+1 ◦ h+i and ri− 1 2 ◦ h−i = h−i ◦ h−i+1 ◦ h+i+1 for i ∈ Ĩ such that i− 12 ∈ Îr, i+ 12 /∈ Îr. • h+i ◦ h+i−1 ◦ h−i−1 = ri+ 1 2 ◦ h+i and h+i−1 ◦ h−i−1 ◦ h−i = h−i ◦ ri+ 1 2 for i ∈ Ĩ such that i− 12 /∈ Îr, i+ 12 ∈ Îr. • h+i ◦h+i−1◦h−i−1 = h−i+1◦h+i+1◦h+i and h+i−1◦h−i−1◦h−i = h−i ◦h−i+1◦h+i+1 for i ∈ Ĩ such that i− 12 , i+ 12 /∈ Îr. • h+ i− 1 2 ◦ h− i− 1 2 = h− i+ 1 2 ◦ h+ i+ 1 2 for k ∈ Îr. Remark 4.1. Note these quivers were previously considered in ??. Another detailed account of their definition can be found there. 4.3.2 NCCR and derived equivalence Let pi : Yσ → X be the crepant resolution corresponding to σ. Theorem 4.4. [47, Theorem 1.15 and Theorem 1.20] Db(modJσ) ' Db(CohYσ) The equivalence is given by an explicit tilting vector bundle which is a direct sum of line bundles [47, Theorem 1.10]. In particular, the following map is compatible with the derived equivalence: H0(Yσ,Z) ⊕ H2(Yσ,Z) → ZI [pt] 7→ δ [Ci] 7→ αi where αi is the i-th fundamental vector and δ := α0 + α1 + · · ·αN−1. 4.3.3 Mutation and derived equivalence Derksen–Weyman–Zelevinsky’s mutation ([15]) of a quiver with a potential induces a derived equivalence of the derived categories of Ginzburg’s dgas 77 ([29]). Moreover, the relation between the module categories of Jacobian algebras has a description in terms of torsion pair and tilting, which plays a crucial role for the wall-crossing formulae ([33, 48]). In this paper, we can not apply [15] and [29] since we have loops and oriented 2-cycles in the quiver. In this subsection, we see derived equivalences and descriptions of module categories using the explicit computations given in [47, §3]. Let k be an edge of the partition σ which is a diagonal of a parallelogram. Note that such k corresponds to a vertex without loops. Let σ′ denote the partition which is obtained by a “flip” of the edge k. Let Pi be the indecomposable projective Jσ-module associated to a ver- tex i. Note that as a vector space Pi is the space of linear combinations of path ending at the vertex i. We define P ′k := coker(Pk → Pk−1 ⊕ Pk+1). and put P ′i = Pi for i 6= k. Here the map Pk → Pk±1 above is induced by the arrow from k to k ± 1. Theorem 4.5. [47, Proposition 3.1] (1) End(⊕P ′i )op ' Jσ′ . (2) Φk := RHom(⊕P ′i , •) : Db(modJσ)→ Db(modJσ′) provides an equivalence. For a Jσ-module V = ⊕i∈ÎVi, we have ( HjmodJσ′ (Φk(V )) ) i =  Vi i 6= k, j = 0, ker (Vk−1 ⊕ Vk+1 → Vk) i = k, j = 0, coker (Vk−1 ⊕ Vk+1 → Vk) i = k, j = 1, 0 otherwise. 78 As for dimension vectors, the simple reflection is compatible with the derived equivalence. By the description above, we have modJσ ∩ Φ−1k (modJσ′) = {V ∈ modJσ | coker (Vk−1 ⊕ Vk+1 → Vk) = 0} = {V ∈ modJσ | Hom(V, sk) = 0} =: (modJσ) k, modJσ ∩ Φ−1k (modJσ′)[1] = {V ∈ modJσ | Vi = 0 (i 6= k)} =: Sk. In other-words, ((modJσ) k,Sk) is a torsion pair of modJσ and Φ−1k (modJσ′) is obtained from modJσ by tilting with respect to this torsion pair (see [48, §3.1]). Then we have modJσ′ ∩ Φk(modJσ) = {V ∈ modJσ′ | Hom(s′k, V ) = 0} =: (modJσ′)k. In summary, we have the following: Proposition 4.6. The equivalence Φk induces an equivalence of (modJσ) k and (modJσ′)k. In the proof of [47, Proposition 3.1], the author provides the isomorphism in Proposition 4.5 (1) explicitly. For V ∈ modJσ ∩ Φ−1k (modJσ′), the map( H0modJσ′ (Φk(V )) ) k−1 → ( H0modJσ′ (Φk(V )) ) k is induced by the following morphism: Rk−1 ⊕Rk−1,k+1 : Vk−1 → Vk−1 ⊕ Vk+1 where Rk−1 := rk−1 k − 1 ∈ Îr,h+ k− 3 2 ◦ h− k− 3 2 k − 1 /∈ Îr 79 and Rk−1,k+1 := h−k+ 1 2 ◦ h+ k+ 1 2 ◦ h+ k− 1 2 . 4.3.4 Cut and mutation Let (Q,W ) be a quiver with potential. To each subset C ⊂ Q1 we associate a grading gC on Q by gC(a) = 1 a ∈ C,0 a ∈ C. A subset C ⊂ Q1 is called a cut if W is homogeneous of degree 1 with respect to gC . Denote by QC , the subquiver of Q with the vertex set Q0 and the arrow set Q1\C. We define the truncated Jacobian algebra by J(Q,W )C := J(Q,W )/〈C〉 Let k be a vertex of Qσ without loops and C be a cut of (Qσ, wσ) such that gC(h + k+ 1 2 ) = 14. We define a cut C ′ of (Qσ′ , wσ′) by the following conditions: • gC′(h+k− 1 2 ) = 1, and • gC′(h±i ) = gC(h±i ) if i 6= k − 12 , k + 12 . Proposition 4.7. (See [51, Proposition 4.12]) The equivalence Φk induces an equivalence of (modJσ,C)k and (modJσ′,C′) k. Proof. It is enough to show that if h+ k+ 1 2 vanishes on V then h+ k− 1 2 vanished on Φk(V ). Since gC(h ± k− 1 2 ) = 0, we have • gC(rk−1) = 1 if k − 1 ∈ Îr, and • gC(h+k− 3 2 ) = 1 or gC(h − k− 3 2 ) = 1 if k − 1 /∈ Îr 4We can construct a cut of (Qσ, wσ) as follows: First, by coupling h + i and h − i for each i, we group the arrows in Qσ into N + |Îr| groups. Note that N + |Îr| is even. These groups have the natural cyclic order and we label each of them by odd or even. Choose (any) one arrow from each odd (or even) labelled group, then we get a cut. 80 and so Rk−1 vanishes. Since gC(h+k+ 1 2 ) = 1, we see that Rk−1,k+1 vanishes.  4.4 Motivic Donaldson–Thomas Invariants 4.4.1 Motives We are working in a version of the ring of motivic weights: let MC denote the K-group of the category of effective Chow motives over C, extended by L− 1 2 , where L is the Lefschetz motive. It has a natural structure of a λ-ring [19, 23] with σ-operations defined by σn([X]) = [X n/Sn] and σn(L 1 2 ) = L n 2 . We put M̃C =MC[[L−1]], which is also a λ-ring. Note that in this latter ring, the elements (1− Ln), and therefore the motives of general linear groups, are invertible. The rings MC ⊂ M̃C sit in larger ringsMµ̂C ⊂ M̃µ̂C of equivariant motives, where µ̂ is the group of all roots of unity [35]. Let f : X → C be a regular function on a smooth variety X. Using arc spaces, Denef and Loeser [14, 35] define the motivic nearby cycle [ψf ] ∈Mµ̂C and the motivic vanishing cycle [ϕf ] := [ψf ]− [f−1(0)] ∈Mµ̂C of f . Note that if f = 0, then [ϕ0] = −[X]. The following result was proved in [4, Prop. 1.11]. Theorem 4.8. Let f : X → C be a regular function on a smooth variety X. Assume that X admits a C∗-action such that f is C∗-equivariant i.e. f(tx) = tf(x) for t ∈ C∗, x ∈ X, and such that there exist limits limt→0 tx for all x ∈ X. Then [ϕf ] = [f −1(1)]− [f−1(0)] ∈MC ⊂Mµ̂C. 81 Following [4], we define the virtual motive of crit(f) to be [crit(f)]vir := −(−L 12 )− dimX [ϕf ] ∈Mµ̂C. For a smooth variety X, we put [X]vir := [crit(0X)]vir = (−L 12 )− dimX · [X]. 4.4.2 Quivers and moduli spaces Let Q be a quiver, with vertex set Q0 and edge set Q1. For an arrow a ∈ Q1, we denote by s(a) ∈ Q0 (resp. t(a) ∈ Q0) the vertex at which a starts (resp. ends). We define the Euler–Ringel form χ on ZQ0 by the rule χ(α, β) = ∑ i∈Q0 αiβi − ∑ a∈Q1 αs(a)βt(a), α, β ∈ ZQ0 . Given a Q-representation M , we define its dimension vector dimM ∈ NQ0 by dimM = (dimMi)i∈Q0 . Let α ∈ NQ0 be a dimension vector and let Vi = Cαi , i ∈ Q0. We define R(Q,α) := ⊕ a∈Q1 Hom(Vs(a), Vt(a)) and Gα := ∏ i∈Q0 GL(Vi). Note that Gα naturally acts on R(Q,α) and the quotient stack M(Q,α) := [R(Q,α)/Gα] gives the moduli stack of representations of Q with dimension vector α. Let W be a potential on Q, a finite linear combination of cyclic paths in Q. Denote by J = JQ,W the Jacobian algebra, the quotient of the path algebra CQ by the two-sided ideal generated by formal partial derivatives 82 of the potential W . Let fα : R(Q,α)→ C be the Gα-invariant function defined by taking the trace of the map asso- ciated to the potential W . As it is now well known [55, Proposition 3.8], a point in the critical locus crit(fα) corresponds to a J-module. The quotient stack M(J, α) := [ crit(fα)/Gα ] gives the moduli stack of J-modules with dimension vector α. Definition 4.1. A central charge is a group homomorphism Z : ZQ0 → C such that Z(α) ∈ H+ = {reipiϕ | r > 0, 0 < ϕ ≤ 1} for any α ∈ NQ0\{0}. Given α ∈ NQ0\{0}, the number ϕ(α) = ϕ ∈ (0, 1] such that Z(α) = reipiϕ, for some r > 0, is called the phase of α. Definition 4.2. For any nonzero Q-representation (resp. J-module) V , we define ϕ(V ) = ϕ(dimV ). A Q-representation (resp. J-module) V is said to be Z-(semi)stable if for any proper nonzero Q-subrepresentation (resp. J-submodule) U ⊂ V we have ϕ(U)(≤)ϕ(V ). Definition 4.3. Given ζ ∈ RQ0 , define the central charge Z : ZQ0 → C by the rule Z(α) = −ζ · α+ i|α|, where |α| = ∑i∈Q0 αi. We say that a Q-representation (resp. J-module) is ζ-(semi)stable if it is Z-(semi)stable. Remark 4.2. Let the central charge Z be as in Definition 4.3. Define the slope function µ : NQ0\{0} → R by µ(α) = ζ·α|α| . If l ⊂ H = H+∪{0} is a ray such that Z(α) ∈ l then l = R≥0(−µ(α), 1). This implies that ϕ(α) < ϕ(β) if and only if µ(α) < µ(β). 83 We say that ζ ∈ RQ0 is α-generic if for any 0 < β < α we have ϕ(β) 6= ϕ(α). This condition implies that any ζ-semistable Q-representation (resp. J-module) is automatically ζ-stable. LetRζ(Q,α) denote the open subset ofR(Q,α) consisting of ζ-semistable representations. Let fζ,α denote the restriction of fα to Rζ(Q,α). The quotient stacks Mζ(Q,α) := [ Rζ(Q,α)/Gα ] , Mζ(J, α) := [ crit(fζ,α)/Gα ] (4.2) give the moduli stacks of ζ-semistable Q-representations and J-modules with dimension vector α. 4.4.3 Motivic DT invariants Let (Q,W ) be a quiver with a potential and let J = JQ,W be its Jacobian algebra. Recall that the degeneracy locus of the function fα : R(Q,α)→ C defines the locus of J-modules, so that the quotient stack M(J, α) := [crit(fα)/Gα] is the stack of J-modules with dimension vector α. We define motivic Donaldson–Thomas invariants by [M(J, α)]vir := [crit(fα)]vir [Gα]vir . For a stability parameter ζ, we define [Mζ(J, α)]vir = [crit(fζ,α)]vir [Gα]vir . (4.3) where, as before, fζ,α denote the restriction of fα : R(Q,α)→ C to Rζ(Q,α). 4.4.4 Generating series of motivic DT invariants Let (Q,W ) be a quiver with a potential admitting a cut, and let J = JQ,W be its Jacobian algebra. 84 Definition 4.4. We define the generating series of the motivic Donaldson– Thomas invariants of (Q,W ) by AU (y) = ∑ α∈NQ0 [M(J, α)]vir · yα = ∑ α∈NQ0 [crit(fα)]vir [Gα]vir · yα ∈ TQ, the subscript referring to the fact that we think of this series as the universal series. Given a cut C of (Q,W ), we define a new quiver QC = (Q0, Q1\C). Let JC be the quotient of CQC by the ideal (∂CW ) = (∂W/∂a, a ∈ C). Proposition 4.9. [39, Proposition 1.14] If (Q,W ) admits a cut C, then AU (y) = ∑ α∈NQ0 (−L 12 )χ(α,α)+2dI(α) [R(JC , α)] [Gα] yα, where dC(α) = ∑ (a:i→j)∈C αiαj for any α ∈ ZQ0. The quiver with potential (Qσ, wσ) introduced in §4.3 admits a cut (see §4.3.4) and Proposition 4.9 can be applied. In the next section we use this to compute the universal series in a specific case. 4.5 The Universal DT Series: Special Case Let us fix an integer N ′ with 0 ≤ N ′ ≤ N . Throughout this section we fix σ to be the unique partition defined such that Îr = {0, 1, 2, 3, . . . N ′ − 1}. In other words the partition such that the quiver with potential (Qσ, wσ) has loops at the first N ′ vertices only. The aim of this section is to prove Theorem 4.1 for this quiver with potential. 85 We define three fixed subsets of the vertices I1 := {0, 1, . . . , N ′ − 1} ⊂ Z/N, I2 := {N ′, N ′ + 2, N ′ + 4, . . . , N − 2} ⊂ Z/N, I3 := {N ′ + 1, N ′ + 3, N ′ + 5, . . . , N − 1} ⊂ Z/N. Then there exists a cut C given by the collection of arrows C = { h−i | i− 1 2 6∈ I2 } . By Proposition 4.9 the coefficients of the universal DT series AσU (y) =∑ α∈NQ Aαy α are given by Aα = ( −L 12 )χ(α,α)+2dC(α) [R(Jσ,C , α)] [Gα] yα where dC(α) = ∑ (a:i→j)∈C αiαj . To begin we find a simple expression for the term χ(α, α) + 2dC(α) in the exponent. We know by definition that χ(α, α) = ∑ i∈I1∪I2∪I3 α2i − ∑ i∈I1 α2i − ∑ i∈I1∪I2∪I3 αiαi+1 − ∑ i∈I1∪I2∪I3 αi+1αi, dI(α) = ∑ i∈I1 αiαi+1 + ∑ i∈I3 αi+1αi, so it follows χ(α, α) + 2dC(α) = ∑ i∈I2∪I3 α2i − 2 · ∑ i∈I2 αiαi+1, = ∑ i∈I2 (αi+1 − αi)2. Our next goal is to factorize AσU (y) into two simpler series. This proceeds by analyzing the motivic classes [R(Jσ,C , α)]. 86 Given a dimension vector α ∈ NQ0 and a representation of a Jσ,C-module V = ⊕ i∈I1∪I2∪I3 Vi we focus on a specific element H := h+1/2 + h + 3/2 + · · ·+ h+N−1/2 ∈ ⊕ i∈I1∪I2∪I3 Hom(Vi, Vi+1). This map H acts as an endomorphism of the vector space V . Given any such linear map H : V → V there exists a unique splitting V = V I ⊕ V N with maps HI : V I → V I invertible HN : V N → V N nilpotent so that H = HI ⊕HN . Moreover in our case the above splitting respects the grading by i ∈ I1 ∪ I2 ∪ I3. To be explicit we have that V I = ⊕ i∈I1∪I2∪I3 V Ii where V Ii := Vi∩V I (similarly V N = ⊕ i∈I1∪I2∪I3 V N i with V N i := Vi∩V N ). One immediate consequence of this is that dim ( V Ii ) = dim ( V Ii+1 ) for all i ∈ I1 ∪ I2 ∪ I3, indeed this is clear since the block form of HI demands that it map V Ii to V Ii+1 via an isomorphism. We are now ready to decompose the computation of AσU (y) into two simpler subproblems. 87 Definition 4.5. (Invertible series) We define RI(a) := {r ∈ R(Jσ,C , α) | H is invertible, αi = a ∀i} and the series Iσ(y) := ∑ a≥0 [RI(a)] [GL(a)]N ya. Definition 4.6. (Nilpotent series) We define RN (α) := {r ∈ R(Jσ,C , α) | H is nilpotent} and the series Nσ(y) := ∑ α∈NQ0 (−L1/2) ∑ i∈I2 (αi+1−αi) 2 [RN (α)] [Gα] yα. The following lemma shows that the series AσU (y) factorizes into the product of the two just defined. Lemma 4.10. We have AσU (y) = I σ(y0 · · · yN−1) ·Nσ(y). Proof. This formula follows directly from a stratification of the varietyR(Jσ,C , α) by the dimension of V Ii . Fix α ∈ NQ0 , we stratify R(Jσ,C , α) by dim(V Ii ) = a. Let a := (a, a, . . . , a) ∈ NQ0 , and α′ such that α = a+ α′ ∈ NQ0 . There is a Zariski locally trivial fibration 88 RI(a)×RN (α′) → {r ∈ R(Jσ,C , α) | dim(V Ii ) = a for H ∈ r} ↓ M(a, α). Here M(a, α) is the space parameterizing splittings Vi = V Ii ⊕ V Ni . To see this one checks that the arrows ri, h − i+1/2 in the representation also preserve the splitting, so the entire representation splits into V I⊕V N . This follows easily from the relations and some basic linear algebra. Splittings of the vector space Vi = V I i ⊕ V Ni are parameterized by GL(αi)/ ( GL(a)×GL(α′i) ) and hence the motivic class of the base is [M(a, α)] = [Gα] [GL(a)]N · [Gα′ ] . Summing over each stratum with dim(V Ii ) = a we get [R(Jσ,C , α)] = [Gα] · mini{αi}∑ a=0 [RI(a)] [GL(a)]N · [R N (α′)] [Gα′ ] . Multiplying both sides of this expression by (−L1/2) ∑ i∈I2 (αi+1−αi) 2 yα and summing gives AσU (y) = ∑ a≥0 [RI(a)] [GL(a)]N N−1∏ i=0 yai ·  ∑ α′∈NQ0 (−L1/2) ∑ i∈I2(α ′ i+1−α′i) 2 [RN (α′)] [Gα′ ] yα ′  proving the result.  In the next two sections we compute formulas for Iσ(y) and Nσ(y). 89 4.5.1 Step One: The invertible case Iσ(y) Proposition 4.11. We have Iσ(y) = Exp ( L y 1− y ) . Proof. A Jσ,C-module r ∈ R(Jσ,C , α) is given by a vector space V = ⊕ i∈I1∪I2∪I3 Vi of dimension α ∈ NQ0 and a collection of linear maps ri : Vi → Vi for i ∈ I1 h−i+1/2 : Vi+1 → Vi for i ∈ I2 h+i+1/2 : Vi → Vi+1 for i ∈ I1 ∪ I2 ∪ I3 satisfying the relations coming from cyclic differentiation of the potential rih + i−1/2 = h + i−1/2ri−1 for i ∈ [1, N ′ − 1] ∩ I1 r0h + N−1/2 = h + N−1/2h + N−3/2h − N−3/2 h−N ′+1/2h + N ′+1/2h + N ′−1/2 = h + N ′−1/2rN ′−1 h−i+3/2h + i+3/2h + i+1/2 = h + i+1/2h + i−1/2h − i−1/2 for i = [N ′ + 1, N − 3] ∩ I3. assuming moreover that r ∈ RI(a) then h+i+1/2 : Vi → Vi+1 is invertible ∀i ∈ I1 ∪ I2 ∪ I3. This allows us to express RI(a) as a ∏N−1 i=1 GL(Vi) torsor over a commuting variety pi : RI(a) → C(a) : (ri, h + i+1/2, h − i+1/2) 7→ (r0, h+N−1/2h+N−3/2 · · ·h+3/2h+1/2) where C(a) = {(A,B) ∈ End(V0)×GL(V0) | AB = BA}. 90 The free action of ∏N−1 i=1 GL(Vi) on R I(a) is given by (g1, . . . , gN−1) : ri 7→ girig−1i for i ∈ [1, N ′ − 1] : h+1/2 7→ g1h+1/2 : h+N−1/2 7→ h+N−1/2g−1N−1 : h+i+1/2 7→ gi+1h+i+1/2g−1i for i ∈ [1, N − 2] : h−i+1/2 7→ gih−i+1/2g−1i+1 for i ∈ I2. As GL(a) is a special group the torsor splits in the Zariski topology, motivi- cally we have [RI(a)] = [GL(a)]N−1 · [C(a)]. Thus Iσ(y) = ∑ a≥0 [C(a)] [GL(a)] ya. The generating series for the commuting variety is obtained in [12] giving the result.  4.5.2 Step Two: The nilpotent case Nσ(y) This section is the final step in the calculation. Here we compute Nσ(y) and obtain the formula of AσU (y). We fix a dimension vector α ∈ NQ0 . As before a Jσ,C-module is given by a vector space V = ⊕ i∈I1∪I2∪I3 Vi of dimension α and a collection of linear maps ri : Vi → Vi for i ∈ I1 h−i+1/2 : Vi+1 → Vi for i ∈ I2 h+i+1/2 : Vi → Vi+1 for i ∈ I1 ∪ I2 ∪ I3 satisfying the relations of the potential (see Proposition 4.11). Throughout 91 this section we insist that the map H = h+1/2 + h + 3/2 + · · ·h+N−1/2 ∈ ⊕ i∈I1∪I2∪I3 Hom(Vi, Vi+1) is nilpotent. In fact RN (α) is exactly the collection of all such representa- tions (see Definition 4.6). In particular if we let |α| := dim(V ) then we know that H |α| = 0. This gives a filtration of the vector space, V = V |α| ⊃ V |α|−1 ⊃ · · · ⊃ V 1 ⊃ V 0 = {0} where V j = {v ∈ V | Hj(v) = 0}. Moreover the filtration respects the grading by i ∈ I1 ∪ I2 ∪ I3, by which we mean that V j = ⊕ i∈I1∪I2∪I3 ( V j ∩ Vi ) where Vi is the summand at the ith vertex of the quiver. By considering the vector space V as a representation of the nilpotent matrix H we can identify V with a C[x]-module supported at the origin. Modules for a principal ideal domain have a simple structure. In particular we have V ∼= d⊕ j=1 ( C[x]/(xj) )⊕bj as a C[x]-module. The next proposition provides a more refined version of this statement where each factor in this decomposition is generated by a vector from a vector space Vi. Proposition 4.12. For each i ∈ I1∪I2∪I3 there exists collection of integers bij so that V ∼= ⊕ i∈I1∪I2∪I3 d⊕ j=1 ( C[x]/(xj) )⊕bij where the factor ( C[x]/(xj) )⊕bij is generated as a C[x]-module by vectors in 92 Vi. Moreover the numbers b i j are uniquely determined by the above condi- tions. Proof. We will argue by induction on d, the largest integer such that bd 6= 0. As such we can assume that for each j ≤ d − 1 the factor C[x]/(xj) is generated by a vector in some Vi. Now let e1, . . . , ebd be a generating set for the factor ( C[x]/(xd) )⊕bd , and define W := span{e1, . . . , ebd}. We consider the projection operators pi : V → Vi/Vi ∩ V d−1 and set Wi := pi(W ) and b i d = dimWi. We claim that p0 ⊕ · · · ⊕ pN−1 : W →W0 ⊕ · · · ⊕WN−1 is an isomorphism. The map is clearly onto and an injection since any vector in the kernel must lie in V d−1. Now consider a lifting of the vector space Vi ⊃W ′i Wi ⊂ Vi/Vi ∩ V d−1 then W ′i ⊕HW ′i ⊕ · · · ⊕Hd−1W ′i ⊂ V is a submodule of V isomorphic to ( C[x]/(xd) )⊕bid . Summing over all i we have that ( C[x]/(xd) )∑ i b i d is a submodule of V , hence it follows that∑ i dimWi = ∑ i b i d ≤ bd = dimW and so for dimension reasons we get V ∼= ( N−1⊕ i=0 ( C[x]/(xd) )⊕bid)⊕d−1⊕ j=1 ( C[x]/(xj) )⊕bj . Here each factor ( C[x]/(xd) )⊕bid is generated by vectors in Vi, so by our inductive hypothesis the entire module is generated by vectors in Vi. Finally we prove the uniqueness statement. Assume we have two distinct such decompositions V ∼= N−1⊕ i=0 d⊕ j=1 (C[x]/(xj))⊕b i j ∼= N−1⊕ i=0 d⊕ j=1 (C[x]/(xj))⊕c i j . By restricting to subrepresentations if necessary we can assume that bid 6= cid 93 for some i. However in this case bid = dim ( ker(Hd : Vi → Vi+d)/Vi ∩ V d−1 ) = cid is a contradiction. This proves the last part of the lemma.  Next we organize this data in the way most helpful to our cause. Definition 4.7. Let 0 ≤ a, b ≤ N − 1. We define |b− a| = min{r ∈ {0, 1, . . . , N − 1} | b = a+ r mod N}. Intuitively this is the distance from a to b in the cyclic direction i→ i+1 corresponding to the map H. Definition 4.8. Suppose we have a decomposition of V as a C[x]-module as in Proposition 4.12 . Define V a,b to be the vector subspace corresponding to summand ⊕ l≥1 ( C[x]/(xN(l−1)+|b−a|+1) )ba N(l−1)+|b−a|+1 . and relabel the integers ba,bl := b a N(l−1)+|b−a|+1, to define partitions pi[a,b] := (1b a,b 1 2b a,b 2 3b a,b 3 · · · ). Notice that the above definition depends on the choice of the decompo- sition in Proposition 4.12. However all such vector spaces are isomorphism abstractly as C[x]-modules. We can think of these vector spaces as being generated by the nilpotent vectors that start at the ath vertex and are an- nihilated at the b+ 1th vertex under the action of the map H. The next lemma makes explicit how to recover the dimension vector of a representation from the datum of the N2 partitions {pi[a,b] | 0 ≤ a, b ≤ N − 1}. 94 Lemma 4.13. Given a representation r ∈ RN (α) so that the endomorphism H has type {pi[a,b]} the dimension vector of the representation r is given by αi = ∑ a,b |pi[a,b]| − ∑ a,b:i 6∈[a,b] l(pi[a,b]) where |pi[a,b]| and l(pi[a,b]) are the size and length of the partition pi[a,b]. Proof. This is clear since V = ⊕ a,b V a,b and dim ( V a,b ∩ Vi ) = { |pi[a,b]| if i ∈ [a, b] |pi[a,b]| − l(pi[a,b]) if i 6∈ [a, b].  We can use this to give a simple reformulation of the term χ(α, α) + 2dC(α) appearing in the series N σ. Corollary 4.14. We have χ(α, α) + 2dC(α) = ∑ i∈I2 ∑ b6=i l(pi[i+1,b])− ∑ c 6=i+1 l(pi[c,i]) 2 . Proof. In our initial analysis of these terms we saw that χ(α, α) + 2dC(α) = ∑ i∈I2 (αi+1 − αi)2 and now by lemma 4.13 we have αi+1 − αi = ∑ b 6=i l(pi[i+1,b])− ∑ c 6=i+1 l(pi[c,i]).  The above classification has been for the purpose of breaking the variety RN (α) down into simpler pieces. 95 Definition 4.9. Given N2 partitions {pi[a,b] | 0 ≤ a, b ≤ N − 1} and a dimension vector α as in lemma 4.13 we define R({pi[a,b]}) = {r ∈ RN (α) | H has type {pi[a,b]}}. This provides a stratification of RN (α) into strata where the normal form of H has a fixed type. We will proceed to compute the motivic classes of each of these stratum. A representation in R({pi[a,b]}) is given explicitly by a vector space V =⊕ i∈I1∪I2∪I3 Vi and a collection of linear maps corresponding to the arrows ri with i ∈ I1 ,h−i+1/2 with i ∈ I2 and h+i+1/2 with i ∈ I1∪ I2∪ I3. In addition the linear maps satisfy relations rih + i−1/2 = h + i−1/2ri−1 for i ∈ [1, N ′ − 1] ∩ I1 r0h + N−1/2 = h + N−1/2h + N−3/2h − N−3/2 h−N ′+1/2h + N ′+1/2h + N ′−1/2 = h + N ′−1/2rN ′−1 h−i+3/2h + i+3/2h + i+1/2 = h + i+1/2h + i−1/2h − i−1/2 for i = [N ′ + 1, N − 3] ∩ I3. and we require that the map H = h+1/2 + h + 3/2 + · · ·h+N−1/2 ∈ ⊕ i∈I1∪I2∪I3 Hom(Vi, Vi+1) has a type given by the partitions {pi[a,b] | 0 ≤ a, b ≤ N−1}. The linear map H contains all the information of the maps h+i+1/2. For brevity we make the following definition packaging all the remaining linear maps into one. Definition 4.10. Given a representation as above, we define the linear map L := r0+r1+· · · rN ′−1+h−N ′+1/2+· · ·h−N−3/2 ∈ ⊕ i∈I1 Hom(Vi, Vi) ⊕ i∈I2 Hom(Vi+1, Vi). From now on in order to compute the motivic class of R({pi[a,b]}) we will work with a choice of coordinates. Let va,bl (k) ∈ Va 96 be such that va,bl (k) generates the kth summand of C[x]/(x N(l−1)+|b−a|+1)⊕b a,b l in the decomposition of Proposition 4.12. Then we have that B := {Hpva,bl (k) | 1 ≤ k ≤ ba,bl , 0 ≤ a, b ≤ N−1, 0 ≤ p ≤ N(l−1)+|b−a|+1} forms a basis of V . Definition 4.11. We define H(pi[a,b]) to be the matrix representation of the map H with respect to the basis B. Also define F ({pi[a,b]}) := {L | (L,H(pi[a,b])) ∈ R({pi[a,b]})} N({pi[a,b]}) := {H | H has type {pi[a,b]}}. Then R({pi[a,b]}) has a decomposition as a vector bundle. Lemma 4.15. R({pi[a,b]}) has the structure of a vector bundle F ({pi[a,b]}) → R({pi[a,b]}) ↓ N({pi[a,b]}). In particular we have that [R({pi[a,b]})] = [F ({pi[a,b]})] · [N({pi[a,b]})] in the Grothendieck ring of varieties. Proof. The projection map p : R({pi[a,b]}) → N({pi[a,b]}) : (L,H) 7→ H. defines the bundle structure with zero section H 7→ (0, H). The fibre is the linear space of all such L.  Here the base of the vector bundle is the space of all matrices of type 97 {pi[a,b]} these are all conjugate to H(pi[a,b]), therefore we have a torsor, pi : Gα → N(pi[a,b]) : P 7→ PH(pi[a,b])P−1. This is a torsor for the group S′({pi[a,b]}) := StabGα(H(pia,b)). This group is given as the group of units in an algebra. Definition 4.12. We identify S′({pi[a,b]}) with the group of multiplicative units in the following algebra S({pi[a,b]}) := { N ∈ N−1∏ i=0 End(αi) ∣∣∣ NH(pi[a,b]) = H(pi[a,b])N} . Since S′({pi[a,b]}) is the group of units of an algebra it is a special group and so the above torsor splits in the Zariski topology. The next lemma gives a formula of the motivic class of the group S′({pi[a,b,]}) and via the splitting of the above torsor we deduce a formula for the class of N({pi[a,b]}). Before stating the lemma we create some notation. Definition 4.13. The following linear spaces have dimension T ({pi[a,b]}) := dimF ({pi[a,b]}) B({pi[a,b]}) := dimS({pi[a,b]}). Lemma 4.16. We have [S′({pi[a,b]})] = [S({pi[a,b]})] · ∏ 0≤a,b≤N−1 1 f ( pi[a,b] ) where f ( pi[a,b] ) := ∏ l≥1 [End(ba,bl )] [GL(ba,bl )] . So as a consequence [R({pia,b})] = [Gα] · LT ({pi[a,b]})−B({pi[a,b]}) · ∏ 0≤a,b≤N−1 f ( pi[a,b] ) . 98 Proof. Let W a,bl := spanC{va,bl (k) | 1 ≤ k ≤ ba,bl } be the span of the basis elements va,bl (k) for 1 ≤ k ≤ ba,bl . We have both inclusion and projection W a,bl ↪→ V W a,bl . This gives a map of algebras pi : S({pi[a,b]})→ ∏ a,b,l End(W a,bl ) : N 7→ ⊕a,b,lN |Wa,bl . This splits as a trivial vector bundle, whose rank is the dimension of the total space minus the dimension of the base. Since we have that the group S′({pi[a,b]}) is the group of units in S({pi[a,b]}), we can identify S′({pi[a,b]}) as the inverse image of the units on the right hand side. This is a trivial vector bundle of rank equal to dimS({pi[a,b]}) − dim∏a,b,l End(W a,bl ). We have an isomorphism of varieties S′({pi[a,b]}) ≡ S({pi [a,b]})∏ a,b,l End(W a,b l ) × ∏ a,b,l GL(W a,bl ) so motivically we have [S′({pi[a,b]})] = [S({pi[a,b]})] · ∏ 0≤a,b≤N−1 1 f ( pi[a,b] ) . In lemma 4.15 we saw that [R({pi[a,b]})] = [F ({pi[a,b]})] · [N({pi[a,b]})] Now we know that N({pi[a,b]}) is a torsor for the group S′({pi[a,b]}) whose 99 motive we have just computed we deduce [R({pi[a,b]})] = [F ({pi[a,b]})] · [Gα] [S′({pi[a,b]})] = [F ({pi[a,b]})] · [Gα] [S({pi[a,b]})] · ∏ 0≤a,b≤N−1 f ( pi[a,b] ) = [Gα] · LT ({pi[a,b]})−B({pi[a,b]}) · ∏ 0≤a,b≤N−1 f ( pi[a,b] ) .  The next proposition computes the difference T ({pi[a,b]}) − B({pi[a,b]}). Its proof is found in Section 4.9. Proposition 4.17. We have T ({pi[a,b]})−B({pi[a,b]}) equals to −1 2 ∑ i∈I2 ∑ b6=i l(pi[i+1,b])− ∑ c 6=i+1 l(pi[c,i]) 2 −1 2 ∑ a∈I3,b 6∈I2 ∑ i≥1 (ba,bi ) 2 − 1 2 ∑ a6∈I3,b∈I2 ∑ i≥1 (ba,bi ) 2. Proof. The proof is a linear algebra calculation. See Section 4.9.  As a corollary we deduce the formula for Nσ(y). Proposition 4.18. Let S = {[a, b] | a ∈ I3, b 6∈ I2 or a 6∈ I3, b ∈ I2}, y[a,b] = ya · ya+1 · · · yb, y′ = y0 · y1 · · · yN−1, then we have Nσ(y) = Exp  L L− 1 1 1− y′  ∑ [a,b] 6∈S y[a,b] − L− 1 2 ∑ [a,b]∈S y[a,b]  . 100 Proof. Recall our initial definition of Nσ(y) Nσ(y) = ∑ α∈NQ0 (−L1/2)χ(α,α)+2dC(α) [R N (α)] [Gα] yα In Proposition 4.12 we saw that it was possible to stratify each of the varieties RN (α) by the type {pi[a,b]} of the cycle H. This gives Nσ(y) = ∑ α∈NQ0 (−L1/2)χ(α,α)+2dC(α)[Gα]−1  ∑ {pi[a,b]}`α [R({pi[a,b]})]  yα. The motivic class of R({pi[a,b]}) was computed in lemma 4.16 substituting this into the above formula gives ∑ α∈NQ0 (−L1/2)χ(α,α)+2dC(α)  ∑ {pi[a,b]}`α LT ({pi [a,b]})−B({pi[a,b]}) · ∏ 0≤a,b≤N−1 f ( pi[a,b] ) yα. Lemma 4.13 showed how the dimension vector depended on the partitions we had αi = ∑ 0≤a,b≤N−1 |pi[a,b]| − ∑ [a,b]63i l(pi[a,b]) and an immediate corollary was that χ(α, α) + 2dC(α) = ∑ i∈I2 ∑ b 6=i l(pi[i+1,b])− ∑ c 6=i+1 l(pi[c,i]) 2 . Combining this with the formula for the difference T ({pi[a,b]})−B({pi[a,b]}) (Proposition 4.17) gives Nσ(y) = ∑ {pi[a,b]}  ∏ [a,b] 6∈S f ( pi[a,b] ) ·  ∏ [a,b]∈S f ( pi[a,b] )∏ l≥1 (−L 12 )−(ba,bl )2  · N−1∏ i=0 y ∑ 0≤a,b≤N−1 |pi[a,b]|− ∑ [a,b] 63i l(pi [a,b]) i . 101 To simplify notation set g (pi) := f (pi) ·∏l≥1(−L 12 )−b2l for pi = (1b12b23b3 · · · ) then rearranging the products and summations gives Nσ(y) = ∏ [a,b] 6∈S ∑ pi[a,b] f ( pi[a,b] ) · y′|pi[a,b]|−l(pi[a,b]) · yl(pi[a,b])[a,b] · ∏ [a,b]∈S ∑ pi[a,b] g ( pi[a,b] ) · y′|pi[a,b]|−l(pi[a,b]) · yl(pi[a,b])[a,b] . Both of these series are know to have product expansions [36] f(t, a) = ∑ pi f (pi) a l(pi)t|pi|−l(pi) = Exp ( 1 1−L−1 · a1−t ) g(t, a) = ∑ pi g (pi) a l(pi)t|pi|−l(pi) = Exp ( (−L 12 )−1 1−L−1 · a1−t ) . Now Nσ is a product of such series and multiplying together the correspond- ing exponential generating series gives the desired result Nσ(y) = Exp  L L− 1 1 1− y′  ∑ [a,b] 6∈S y[a,b] − L− 1 2 ∑ [a,b]∈S y[a,b]  .  Now we have computed Iσ, Nσ and so by lemma 4.10 AσU (y) = Exp L y′ 1− y′ + L L− 1 1 1− y′  ∑ [a,b]6∈S y[a,b] − L−1/2 ∑ [a,b]∈S y[a,b]  102 or to reformulate this as a product over the set of roots Exp  11− L−1 (L+N − 1) ∑ α∈∆imσ,+ yα + ∑ α∈∆reσ,+∑ I2∪I3 αi is even yα − L− 12 ∑ α∈∆reσ,+∑ I2∪I3 αi is odd yα   . Thus proving Theorem 4.1 AσU (y) = ∏ α∈∆σ,+ Aα(y) for the special case of the partition σ. 4.6 The Universal DT Series: General Case In this section we will prove Theorem 4.1 for any partition σ. 4.6.1 Mutation and the root system Recall that the simple reflection provides a bijection between ∆σ,+\{αk} and ∆σ′,+\{α′k} (see §4.3.3). The simple root αk maps to −α′k. For α ∈ ∆re+ , let xα be a simple module with dimα. By [47, Proposition 2.14], ∑ i/∈Îr αi is odd (resp. even) if and only if ext 1(x, x) = 0 (resp. = 1). In particular, the parity of ∑ i/∈Îr αi is preserved by the simple reflection. 4.6.2 Wall-crossing formula Theorem 4.19. [51, Theorem 4.9] Aσ ′ U (y) = AσU (y) E(yk) × E(y−1k ) Proof. Step 1 : By the observation in §4.3.3, we have the following factor- ization: AσU = E(yk)×Aσk 103 where E(y) := ∑ n≥0 [pf] [GLn]vir · yn, yk := yαk and Aσk is the generating series of virtual motives of moduli stacks of objects in (modJσ)k. We also have Aσ ′ U = A σ′,k × E(y−1k ) where Aσ ′,k is the generating series of virtual motives of moduli stacks of objects in (modJσ′) k. Step 2 : By Proposition 4.7, we have Aσk = A σ′,k (see [51, Theorem4.7]).  Now Theorem 4.1 for any σ follows from the result in §4.5 combined with Theorem 4.19 and the remark in §4.6.1. Remark 4.3. These mutations were first used in [47] for the case of the classical invariants. 4.6.3 Factorization of the universal series We will say that a stability parameter ζ is generic, if for any stable Jσ- module V , we have ζ · dimV 6= 0. For generic stability parameter ζ, let M+ζ (Jσ, α) (resp.M − ζ (Jσ, α)) denote the moduli stacks of Jσ-modules V such that dimV = α and such that all the HN factors F of V with respect to the stability parameter ζ satisfy ζ · dimF > 0 (resp. < 0). Let [M±ζ (Jσ, α)]vir denote the virtual motive of the moduli stack defined in the same way as (4.3). We put A±ζ (y) = ∑ α∈NÎ [M±ζ (J, α)]vir · yα. Lemma 4.20. [39, Lemma 2.6] The generating series A±ζ are given by A±ζ (y) = ∏ α∈∆σ,+,±ζ·α<0 Aα(y). 104 4.7 Motivic DT with Framing We denote by Q̃σ the new quiver obtained from Qσ by adding a new vertex ∞ and a single new arrow∞→ 0. Let J̃σ = JQ̃σ ,wσ be the Jacobian algebra corresponding to the quiver with potential (Q̃σ, wσ), where we view wσ as a potential for Q̃σ in the obvious way. Let ζ ∈ RÎ be a vector, which we will refer to as the stability parameter. A J̃σ-representation Ṽ with dim Ṽ∞ = 1 is said to be ζ-(semi)stable, if it is (semi)stable with respect to (ζ, ζ∞) ∈ RÎunionsq{∞} (see Definition 4.2), where ζ∞ = −ζ · dimV . As in §4.4.2, a stability parameter ζ ∈ RQ0 is said to be generic, if for any stable J-module V we have ζ · dimV 6= 0. For a stability parameter ζ ∈ RQ0 and a dimension vector α ∈ (Z≥0)Î , let Mζ(J̃σ, α) denote the moduli stack of ζ-semistable J̃σ-representations with dimension vector (α, 1). As in the introduction, we define the generating function: Zζ(y0, . . . , yN−1) = Zζ(y) := ∑ α∈(Z≥0)Î [ Mζ ( J̃σ, α )] vir · yα. Theorem 4.21. [39, Proposition 4.6] For a generic stability parameter ζ, we have Zζ(y) = A−ζ (−L 1 2 y0, y1, . . . , yN−1) A−ζ (−L− 1 2 y0, y1, . . . , yN−1) , (4.4) where A−ζ were defined in §4.6.3. Combined with Lemma 4.20, we get the formula in Corollary 4.2. Remark 4.4. If we cross the wall Wα, we get (or lose) a factor Zα(y) in the generating function. This is compatible with the result in [50]. 105 4.8 DT/PT Series 4.8.1 Chambers in the moduli spaces For a root α ∈ Λ, let Wα denote the hyperplane in the space RÎ of stability parameters which is orthogonal to α. We put W = Wδ ∪ ⋃ α∈∆reσ,+ Wα. A connected component of the complement of W in RÎ is called a chamber. Theorem 4.22. [47, Proposition 2.10], [46, Proposition 3.10, 3.11] The set of generic parameters in RÎ is the compliment of W . 1. For ζ with ζi < 0 (∀i), the moduli spaces Mζ(J̃ , α) are the NCDT moduli spaces, the moduli spaces of cyclic J-modules from [58]. 2. For ζ in the same chamber as (1−N + ε, 1, 1, . . . , 1) (0 < ε 1), the moduli spaces Mζ(J̃ , α) are the DT moduli spaces of Yσ from [37], the moduli spaces of subschemes on Yσ with support in dimension at most 1. 3. For ζ in the same chamber as (1 − N − ε, 1, 1, . . . , 1) (0 < ε  1), the moduli spaces Mζ(J̃ , α) are the PT moduli spaces of Yσ introduced in [53]; these are moduli spaces of stable rank-1 coherent systems. Remark 4.5. In the above statements ε depends on the dimension vector (α, 1). 4.8.2 Motivic PT and DT invariants Let ζDT = (1−N − ε, 1, 1, . . . , 1), ζPT = (1−N + ε, 1, 1, . . . , 1) 106 (0 < ε  1) be stability parameters corresponding to DT and PT moduli spaces. Then we have {α ∈ ∆σ,+ | ζDT · α < 0} = ∆re,++ , {α ∈ ∆σ,+ | ζPT · α < 0} = ∆re,++ unionsq∆im+ . As we mentioned in the introduction the variable change induced by the derived equivalence is given by s := y0 · y1 · · · · · yN−1, Ti = yi. Here s is the variable for the homology class of a point and Ti is the variable for the homology class of Ci. Then we get the formulae in Corollary 4.3. 4.8.3 Connection with the refined topological vertex As studied in [49], we can apply the vertex operator method [52] to get a product expansion of the refined topological vertex for Yσ. Then we see that the PT generating function can be described by the refined topological vertices normalized by the refined MacMahon functions. 5 4.9 Linear Algebra Computation Throughout this computation we will work with a fixed choice of basis B. In §4.5.2 we chose a basis B = {Hpva,bl (k) | 1 ≤ k ≤ ba,bl , 0 ≤ a, b ≤ N−1, 0 ≤ p ≤ N(l−1)+|b−a|+1} and defined linear spaces F ({pi[a,b]} equal toL ∈⊕ i∈I1 Hom(Vi, Vi)⊕ ⊕ i∈I2 Hom(Vi+1, Vi) ∣∣∣ (L,H(pi[a,b])) ∈ R({pi[a,b]})  5Unfortunately, the DT generating function does not coincide with the refined topo- logical vertex. See [39, §4.3] for detail. 107 and S({pi[a,b]}) equal toN ∈ ⊕ i∈I1∪I2∪I3 Hom(Vi, Vi) ∣∣∣ [N,H(pi[a,b])] = 0  with dimensions T ({pi[a,b]}) = dimF ({pi[a,b]}) andB({pi[a,b]}) = dimS({pi[a,b]}). Our goal is to prove Proposition 4.17, that is to show that the difference T ({pi[a,b]})−B({pi[a,b]}) is equal to −1 2 ∑ i∈I2 ∑ b6=i l(pi[i+1,b])− ∑ c 6=i+1 l(pi[c,i]) 2 −1 2 ∑ a∈I3,b 6∈I2 ∑ i≥1 (ba,bi ) 2 − 1 2 ∑ a6∈I3,b∈I2 ∑ i≥1 (ba,bi ) 2. For some early examples it becomes clear that the dimension of F ({pi[a,b]}) and S({pi[a,b]}) are determined by solving a set of linearly independent equa- tions. We will see that these dimensions are quadratic polynomials in the number of parts ba,bl of the partitions {pi[a,b]}. An initial means of simplify- ing the calculation is to break the spaces F ({pi[a,b]}) and S({pi[a,b]}) down into simpler spaces. One easy observation is that not only are the spaces F ({pi[a,b]}) and S({pi[a,b]}) linear but they come with a natural vector space structure, the origin corresponding to the zero matrix in both cases. This means that as vector spaces we have decompositions F ({pi[a,b]}) = ⊕ 0≤a,b,c,d≤N−1 F (pi[a,b], pi[c,d]) S({pi[a,b]}) = ⊕ 0≤a,b,c,d≤N−1 S(pi[a,b], pi[c,d]) whose summands are given by the following definition. 108 Definition 4.14. We define F (pi[a,b], pi[c,d]) = F ({pi[a,b]}) ∩ ⊕ i∈I1∪I2 Hom(V a,b, V c,d) S(pi[a,b], pi[c,d]) = S(pi[a,b], pi[c,d]) ∩ ⊕ i∈I1∪I2∪I3 Hom(V a,b, V c,d). These subspaces are essentially given by the block matrices for the de- composition V = ⊕ 0≤a,b≤N−1 V a,b. Definition 4.15. We define T (pi[a,b], pi[c,d]) = dimF (pi[a,b], pi[c,d]) B(pi[a,b], pi[c,d]) = dimS(pi[a,b], pi[c,d]). Both T (pi[a,b], pi[c,d]) and B(pi[a,b], pi[c,d]) can be written as quadratic ex- pressions in the number of parts of pi[a,b] and pi[c,d]. To do this we introduce a quadratic form on the space of all partitions and a combinatorial operation that removes a box from each column of the the partition. Definition 4.16. We define M : P ⊗ P → Z≥0 : (1b12b23b3 · · · )⊗ (1c12c23c3 · · · ) 7→ ∑ i≥1 ∑ j≥i bj ∑ j≥i cj  ′ : P → P : pi = (1b12b23b3 · · · ) 7→ pi′ = (1b22b33b4 · · · ). Let us begin with the easier case. We compute dimensions B(pi[a,b], pi[c,d]) of the spaces S(pi[a,b], pi[c,d]). Lemma 4.23. Let N ∈ S(pi[a,b], pi[c,d]) then the matrix N is uniquely deter- mined by its value on the vectors va,bl (k). Moreover the only restriction on the image of such a vector is that it lie in the linear subspace N(va,bl ) ∈ Va ∩ V c,d ∩ V N ·(l−1)+|b−a|+1. 109 Proof. To define the linear map N on the space V a,b it suffices to define its value at each of the basis vectors {Hrva,bl (k) | 0 ≤ r ≤ N · (l − 1) + |b− a|, 1 ≤ k ≤ ba,bl }. However for N ∈ S(pi[a,b], pi[c,d]) we have N(Hrva,bl (k)) = H r(Nva,bl (k)), therefore the value of N at each Hrva,bl (k) is determined by Nv a,b l (k). This proves the first part of the lemma. Now we know that the matrix N maps the vector space at the ath vertex to itself Va → Va, also since N ∈ S(pi[a,b], pi[c,d]) we insist that its image be in V c,d. The only additional condition on the image of the vector va,bl (k) is HN ·(l−1)+|b−a|+1(Nva,bl (k)) = N(H N ·(l−1)+|b−a|+1va,bl (k)) = 0. Combining these three conditions above we have, N(va,bl (k)) ∈ Va ∩ V c,d ∩ V N ·(l−1)+|b−a|+1.  Corollary 4.24. We have B(pi[a,b], pi[c,d]) =  M(pi[a,b], pi[c,d]) if a ∈ [c, d] and |d− a| ≤ |b− a| M((pi[a,b])′, pi[c,d]) if a ∈ [c, d] and |d− a| > |b− a| M(pi[a,b], (pi[c,d])′) if a 6∈ [c, d] and |d− a| ≤ |b− a| M((pi[a,b])′, (pi[c,d])′) if a 6∈ [c, d] and |d− a| > |b− a|. Proof. Let N ∈ S(pi[a,b], pi[c,d]). Each vector va,bl (k) with 1 ≤ k ≤ ba,bl can take any value in the vector space Va ∩ V c,d ∩ V N ·(l−1)+|b−a|+1 and so the dimension of S(pia,b, pi[c,d]) is given by B(pi[a,b], pi[c,d]) = ∑ l≥0 ba,bl · dim ( Va ∩ V c,d ∩ V N ·(l−1)+|b−a|+1 ) . 110 Counting the number of basis vectors of V c,d that lie in Va we see there are four possibilities for dim ( Va ∩ V c,d ∩ V N ·(l−1)+|b−a|+1 ) : ∑l i=1 ib c,d i + l ∑ i≥l b c,d i if a ∈ [c, d] and |d− a| ≤ |b− a|∑l−1 i=1 ib c,d i + (l − 1) ∑ i≥l b c,d i if a ∈ [c, d] and |d− a| > |b− a|∑l i=1 ib c,d i+1 + l ∑ i≥l b c,d i+1 if a 6∈ [c, d] and |d− a| ≤ |b− a|∑l−1 i=1 ib c,d i+1 + (l − 1) ∑ i≥l b c,d i+1 if a 6∈ [c, d] and |d− a| > |b− a|. Consider the first case a ∈ [c, d] and |d− a| ≤ |b− a| then B(pi[a,b], pi[c,d]) = ∑ l≥1 ba,bl ·  l∑ i=1 ibc,di + l ∑ i≥l bc,di  = ∑ i≥1 ∑ l≥i ba,bl  · ∑ l≥i bc,dl  = M(pi[a,b], pi[c,d]). The other three cases are identical. The relabeling of the partitions in these cases is encoded by the operation pi 7→ pi′.  Now we turn to computing the dimensions T (pi[a,b], pi[c,d]) of the spaces F (pi[a,b], pi[c,d]). This will be more intricate. Lemma 4.25. Suppose a ∈ I1 ∪ I3 and L ∈ F (pi[a,b], pi[c,d]) then the map L is uniquely determined by its value on the vectors va,bl (k). Moreover the only restriction on the image of such a vector is that it lie in a linear subspace Lva,bl (k) ∈  Va ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d if a ∈ I1 and b 6∈ I2 Va ∩ V N ·(l−1)+|b−a| ∩ V c,d if a ∈ I1 and b ∈ I2 Va−1 ∩ V N ·(l−1)+|b−a|+2 ∩ V c,d if a ∈ I3 and b 6∈ I2 Va−1 ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d if a ∈ I3 and b ∈ I2. Proof. To define the linear map L on the space V a,b it suffices to define its 111 value at each of the basis vectors {Hrva,bl (k) | 0 ≤ r ≤ N · (l − 1) + |b− a|, 1 ≤ k ≤ ba,bl }. However for L ∈ F (pi[a,b], pi[c,d]) we know that the pair (L,H(pi[a,b])) ∈ R({pi[a,b]}) satisfy the relations coming from the potential: rih + i−1/2 = h + i−1/2ri−1 for i ∈ [1, N ′ − 1] ∩ I1 r0h + N−1/2 = h + N−1/2h + N−3/2h − N−3/2 h−N ′+1/2h + N ′+1/2h + N ′−1/2 = h + N ′−1/2rN ′−1 h−i+3/2h + i+3/2h + i+1/2 = h + i+1/2h + i−1/2h − i−1/2 for i = [N ′ + 1, N − 3] ∩ I3. As in Lemma 4.23 once the value of L is determined for va,bl (k) it is uniquely determined for all Hrva,bl (k) by the condition that the above relations be satisfied for the pair (L,H(pi[a,b]). To be precise if a ∈ I1 we have L : Hr(va,bl (k)) 7→  HrL(va,bl (k)) if a+ r ∈ I1 0 if a+ r ∈ I2 Hr−1L(va,bl (k)) if a+ r ∈ I3 and if a ∈ I3 then L : Hr(va,bl (k)) 7→  Hr+1L(va,bl (k)) if a+ r ∈ I1 0 if a+ r ∈ I2 HrL(va,bl (k)) if a+ r ∈ I3. Since L ∈ F (pi[a,b], pi[c,d]) by definition its image must lie in the space V c,d, also if a ∈ I1 then L : Va → Va and if a ∈ I3 then L : Va → Va−1. The only further condition on the image of a vector va,bl (k) is that its image be killed by a high enough power of H. It is given that HN ·(l−1)+|b−a|+1va,bl (k) = 0 so then Ht(Lva,bl (k)) = 0 where the exponent t is read off for the defining 112 relations on L above. In the separate cases Lva,bl (k) ∈  Va ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d if a ∈ I1 and b 6∈ I2 Va ∩ V N ·(l−1)+|b−a| ∩ V c,d if a ∈ I1 and b ∈ I2 Va−1 ∩ V N ·(l−1)+|b−a|+2 ∩ V c,d if a ∈ I3 and b 6∈ I2 Va−1 ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d if a ∈ I3 and b ∈ I2. proving the result.  We have a result similar to Lemma 4.25 when a ∈ I2. Lemma 4.26. Suppose a ∈ I2 and L ∈ F (pi[a,b], pi[c,d]) then the map L is uniquely determined by its value on the vectors Hva,bl (k). Moreover the only restriction on the image of such a vector is that it lie in a linear subspace L(Hva,bl (k)) ∈ { Va ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d if b 6∈ I2 Va ∩ V N ·(l−1)+|b−a| ∩ V c,d if b ∈ I2 Proof. Again we know that to define the linear map L on the space V a,b it suffices to define its value at each of the basis vectors {Hrva,bl (k) | 0 ≤ r ≤ N · (l − 1) + |b− a|, 1 ≤ k ≤ ba,bl }. Since by definition if a ∈ I2 then Lva,bl (k) = 0 the map is already trivially determined on these vectors and their image does not suffice to determine the map in general. However if we consider the vectors Hva,bl (k) then once the value of L is determined for Hva,bl (k) it is uniquely determined for all Hrva,bl (k) by the condition that the relations (see Lemma 4.25) be satisfied by the pair (L,H(pi[a,b]). To be precise if a ∈ I2 we have L : Hr(va,bl (k)) 7→  HrL(Hva,bl (k)) if a+ r ∈ I1 0 if a+ r ∈ I2 Hr−1L(Hva,bl (k)) if a+ r ∈ I3. By definition we know that the image of L lies in V c,d and also that for a ∈ I2 we have L : Va+1 → Va. As before the only remaining condition on 113 the image of va,bl (k) is that it be killed by a high enough power of H. From the definition of L above we see that L(Hva,bl (k)) ∈ { Va ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d if b 6∈ I2 Va ∩ V N ·(l−1)+|b−a| ∩ V c,d if b ∈ I2 proving the result.  The following notation collects the dimensions of all the vector spaces encountered in the last two Lemmas. Definition 4.17. We define integers da,b:c,d(l) =  dim(Va ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d) if a ∈ I1 ∪ I2 and b 6∈ I2 dim(Va ∩ V N ·(l−1)+|b−a| ∩ V c,d) if a ∈ I1 ∪ I2 and b ∈ I2 dim(Va−1 ∩ V N ·(l−1)+|b−a|+2 ∩ V c,d) if a ∈ I3 and b 6∈ I2 dim(Va−1 ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d) if a ∈ I3 and b ∈ I2. From Lemmas 4.25 and 4.26 we deduce the dimension of the spaces F (pi[a,b], pi[c,d]). Corollary 4.27. If a ∈ I1 ∪ I2 and b 6∈ I2 then T (pi[a,b], pi[c,d]) equals M(pi[a,b], pi[c,d]) if a ∈ [c, d] and |d− a| ≤ |b− a| M((pi[a,b])′, pi[c,d]) if a ∈ [c, d] and |d− a| > |b− a| M(pi[a,b], (pi[c,d])′) if a 6∈ [c, d] and |d− a| ≤ |b− a| M((pi[a,b])′, (pi[c,d])′) if a 6∈ [c, d] and |d− a| > |b− a|. If a ∈ I1 ∪ I2 and b ∈ I2 then T (pi[a,b], pi[c,d]) equals M(pi[a,b], pi[c,d]) if a ∈ [c, d] and |d− a| ≤ |b− a| − 1 M((pi[a,b])′, pi[c,d]) if a ∈ [c, d] and |d− a| > |b− a| − 1 M(pi[a,b], (pi[c,d])′) if a 6∈ [c, d] and |d− a| ≤ |b− a| − 1 M((pi[a,b])′, (pi[c,d])′) if a 6∈ [c, d] and |d− a| > |b− a| − 1. 114 If a ∈ I3 and b 6∈ I2 then T (pi[a,b], pi[c,d]) equals M(pi[a,b], pi[c,d]) if a− 1 ∈ [c, d] and |d− (a− 1)| ≤ |b− a|+ 1 M((pi[a,b])′, pi[c,d]) if a− 1 ∈ [c, d] and |d− (a− 1)| > |b− a|+ 1 M(pi[a,b], (pi[c,d])′) if a− 1 6∈ [c, d] and |d− (a− 1)| ≤ |b− a|+ 1 M((pi[a,b])′, (pi[c,d])′) if a− 1 6∈ [c, d] and |d− (a− 1)| > |b− a|+ 1. If a ∈ I3 and b ∈ I2 then T (pi[a,b], pi[c,d]) equals M(pi[a,b], pi[c,d]) if a− 1 ∈ [c, d] and |d− (a− 1)| ≤ |b− a| M((pi[a,b])′, pi[c,d]) if a− 1 ∈ [c, d] and |d− (a− 1)| > |b− a| M(pi[a,b], (pi[c,d])′) if a− 1 6∈ [c, d] and |d− (a− 1)| ≤ |b− a| M((pi[a,b])′, (pi[c,d])′) if a− 1 6∈ [c, d] and |d− (a− 1)| > |b− a|. Proof. We know that if a ∈ I1 ∪ I3 (resp. a ∈ I2) then the map L ∈ F (pi[a,b], pi[c,d]) is determined by its value at the vectors va,bl (k) (resp. Hv a,b l (k)) for 1 ≤ k ≤ ba,bl . In the notation of the previous definition such a vector takes values in a space of dimension da,b;c,d(l). So in all cases the total dimension of the space F (pi[a,b], pi[c,d]) equals T (pi[a,b], pi[c,d]) = ∑ l≥1 ba,bl · da,b;c,d(l). In the above definition of da,b;c,d(l) there are four possible forms depending on the value of a and b. Lets consider the first case when a ∈ I1 ∪ I2 and b 6∈ I2. Then we have that da,b;c,d(l) = dim ( Va ∩ V N ·(l−1)+|b−a|+1 ∩ V c,d ) . Counting the number of basis vectors of V c,d that lie in Va we see there are four possibilities for dim ( Va ∩ V c,d ∩ V N ·(l−1)+|b−a|+1 ) : ∑l i=1 ib c,d i + l ∑ i≥l b c,d i if a ∈ [c, d] and |d− a| ≤ |b− a|∑l−1 i=1 ib c,d i + (l − 1) ∑ i≥l b c,d i if a ∈ [c, d] and |d− a| > |b− a|∑l i=1 ib c,d i+1 + l ∑ i≥l b c,d i+1 if a 6∈ [c, d] and |d− a| ≤ |b− a|∑l−1 i=1 ib c,d i+1 + (l − 1) ∑ i≥l b c,d i+1 if a 6∈ [c, d] and |d− a| > |b− a|. 115 In the first case a ∈ [c, d] and |d− a| ≤ |b− a| and T (pi[a,b], pi[c,d]) = ∑ l≥1 ba,bl ·  l∑ i=1 ibc,di + l ∑ i≥l bc,di  = ∑ i≥1 ∑ l≥i ba,bl  · ∑ l≥i bc,dl  = M(pi[a,b], pi[c,d]). In the second case a ∈ [c, d] and |d− a| > |b− a| and T (pi[a,b], pi[c,d]) = ∑ l≥1 ba,bl ·  l−1∑ i=1 ibc,di + (l − 1) ∑ i≥l bc,di  = ∑ i≥1 ∑ l≥i ba,bl+1  · ∑ l≥i bc,dl  = M((pi[a,b])′, pi[c,d]). In the third case a 6∈ [c, d] and |d− a| ≤ |b− a| and T (pi[a,b], pi[c,d]) = ∑ l≥1 ba,bl ·  l∑ i=1 ibc,di+1 + l ∑ i≥l bc,di+1  = ∑ i≥1 ∑ l≥i ba,bl  · ∑ l≥i bc,dl+1  = M(pi[a,b], (pi[c,d])′). 116 Finally in the fourth case a 6∈ [c, d] and |d− a| > |b− a| and we have T (pi[a,b], pi[c,d]) = ∑ l≥1 ba,bl ·  l−1∑ i=1 ibc,di+1 + (l − 1) ∑ i≥l bc,di+1  = ∑ i≥1 ∑ l≥i ba,bl+1  · ∑ l≥i bc,dl+1  = M((pi[a,b])′, (pi[c,d])′). This completes the situation when a ∈ I1 ∪ I2 and b 6∈ I2. In the other situations a ∈ I1 ∪ I2 and b ∈ I2, or a ∈ I3 and b 6∈ I2, or a ∈ I3 and b ∈ I2. All of these cases can be dealt with in a similar manner.  Now we have computed all the dimensions T (pi[a,b], pi[c,d]) andB(pi[a,b], pi[c,d]). The next lemma combines corollaries 4.24 and 4.27 to compute their differ- ence. We see that in most cases there is an exact cancellation. Lemma 4.28. We have T (pi[a,b], pi[c,d]) = B(pi[a,b], pi[c,d]) aside from the following cases, Case 1: a ∈ I1 ∪ I2, b = d ∈ I2 M((pi[a,b])′, pi[c,b])−M(pi[a,b], pi[c,b]) if a ∈ [c, b] M((pi[a,b])′, (pi[c,b])′)−M(pi[a,b], (pi[c,b])′) if a 6∈ [c, b]. Case 2: a ∈ I3, b 6∈ I2, d = a− 1 ∈ I2 M(pi[a,b], pi[a,a−1])−M((pi[a,b])′, pi[a,a−1]) if a = c M(pi[a,b], pi[c,a−1])−M((pi[a,b])′, (pi[c,a−1])′) if a 6= c. Case 3: a ∈ I3, b 6∈ I2, a = c, d 6= a− 1, M(pi[a,b], (pi[a,d])′)−M(pi[a,b], pi[a,d]) if |d− a| ≤ |b− a|, M((pi[a,b])′, (pi[a,d])′)−M((pi[a,b])′, pi[a,d]) if |d− a| > |b− a|. 117 Case 4: a ∈ I3, b ∈ I2, d = a− 1, M(pi[a,b], pi[a,a−1])−M((pi[a,b])′, pi[a,a−1]) if a = c and b 6= a− 1 M(pi[a,a−1], pi[c,a−1])−M(pi[a,a−1], (pi[c,a−1])′) if a 6= c and b = a− 1 M(pi[a,b], pi[c,a−1])−M((pi[a,b])′, (pi[c,a−1])′) if a 6= c and b 6= a− 1. Case 5: a ∈ I3, b ∈ I2, a− 1 ∈ [c, d], d 6= a− 1, b = d M((pi[a,b])′, pi[c,b])−M(pi[a,b], pi[c,b]). Case 6: a ∈ I3, b ∈ I2, a− 1 6∈ [c, d], a = c, |d− a| < |b− a| M(pi[a,b], (pi[a,d])′)−M(pi[a,b], pi[a,d]). Case 7: a ∈ I3, b ∈ I2, a− 1 6∈ [c, d] M((pi[a,b])′, (pi[a,b])′)−M(pi[a,b], pi[a,b]) if a = c and b = d M((pi[a,b])′, (pi[a,d])′)−M((pi[a,b])′, pi[a,d]) if a = c and |d− a| > |b− a| M((pi[a,b])′, (pi[c,b])′)−M(pi[a,b], (pi[c,b])′) if a 6= c and b = d. Proof. Compare corollaries 4.24 and 4.27.  Our aim throughout this appendix has been to prove Proposition 4.17 and deduce that the difference ∑ 0≤a,b,c,d≤N−1 T (pi [a,b], pi[c,d])−B(pi[a,b], pi[c,d]) equals −1 2 ∑ i∈I2 ∑ b6=i l(pi[i+1,b])− ∑ c 6=i+1 l(pi[c,i]) 2−1 2 ∑ a∈I3,b 6∈I2 ∑ i≥1 (ba,bi ) 2−1 2 ∑ a6∈I3,b∈I2 ∑ i≥1 (ba,bi ) 2. So all that remains is to check this sum agrees with the values we computed. First we will transform it into a expression in terms of the M(pi[a,b], pi[c,d]). 118 To do this we need the simple identities M(pi[a,b], pi[c,d])−M((pi[a,b])′, (pi[c,d])′) = ∑ l≥1 ∑ i≥l ba,bi · ∑ i≥l bc,di − ∑ i≥l ba,bi+1 · ∑ i≥l bc,di+1  = ∑ i≥1 ba,bi · ∑ i≥1 bc,di = l(pi[a,b]) · l(pi[c,d]) and M(pi[a,b], pi[a,b])−M((pi[a,b])′, pi[a,b]) = ∑ l≥1 ∑ i≥l ba,bi · ∑ i≥l ba,bi − ∑ i≥l ba,bi+1 · ∑ i≥l ba,bi  = ∑ l≥1 ba,bl · ∑ i≥l bc,di = 1 2 l(pi[a,b])2 + 1 2 ∑ l≥1 (ba,bl ) 2. Using these two identities and some simple algebraic manipulations we can rewrite Proposition 4.17 as the statement that the difference ∑ 0≤a,b,c,d≤N−1 T (pi [a,b], pi[c,d])− B(pi[a,b], pi[c,d]) equals∑ i∈I2 ∑ b6=i,c 6=i+1 M(pi [i+1,b], pi[c,i]) − M((pi[i+1,b])′, (pi[c,i])′) + ∑ i∈I2 ∑ b<d b,d 6=i M((pi[i+1,b])′, (pi[i+1,d])′) − M(pi[i+1,b], pi[i+1,d]) + ∑ i∈I2 ∑ a<c a,c6=i+1 M((pi[a,i])′, (pi[c,i])′) − M(pi[a,i], pi[c,i]) + ∑ a∈I3,b∈I2,b6=a−1 M((pi [a,b])′, (pi[a,b])′) − M(pi[a,b], pi[a,b]) + ∑ [a,b]∈S M((pi [a,b])′, pi[a,b]) − M(pi[a,b], pi[a,b]). We will take a systematic approach, accounting for these terms one by one, all in all we will check nine separate cases. First let us assess the contribution from terms involving partitions pi[r,s] with r, s ∈ I1. Comparing with Lemma 4.28 in all seven cases there is no discrepancy when a, b ∈ I1 or c, d ∈ I1 and therefore there is no contribution from these terms in agreement with the above sum. 119 Secondly we assess the contribution from terms involving partitions pi[r,s] with r ∈ I1 and s ∈ I2. Considering Lemma 4.28 we note the following cases, Case 1: a ∈ I1 ∪ I2, b ∈ I2, b = d M((pi[a,b])′, pi[c,b])−M(pi[a,b], pi[c,b]) if a ∈ [c, b] M((pi[a,b])′, (pi[c,b])′)−M(pi[a,b], (pi[c,b])′) if a 6∈ [c, b]. Case 2: a ∈ I3, b 6∈ I2, c ∈ I1, d = a− 1 ∈ I2 M(pi[a,b], pi[c,a−1])−M((pia,b)′, (pi[c,a−1])′). Case 4: a ∈ I3, b ∈ I2, c ∈ I1, d = a− 1 M(pi[a,a−1], pi[c,a−1])−M(pi[a,a−1], (pi[c,a−1])′) if b = a− 1 M(pi[a,b], pi[c,a−1])−M((pi[a,b])′, (pi[c,a−1])′) if b 6= a− 1. Case 5: a ∈ I3, b ∈ I2, c ∈ I2, b = d, a− 1 ∈ [c, b], b 6= a− 1 M((pi[a,b])′, pi[c,b])−M(pi[a,b], pi[c,b]). Case 7: a ∈ I3, b ∈ I2, c ∈ I1, b = d, a− 1 6∈ [c, b] M((pi[a,b])′, (pi[c,b])′)−M(pi[a,b], (pi[c,b])′). The sum total of these cases gives∑ a∈I1,b∈I2 M((pi[a,b])′, pi[a,b]) − M(pi[a,b], pi[a,b]) + ∑ a<c:a,c6=b+1 a∈I1,b∈I2 or c∈I1,b∈I2 M((pi[a,b])′, (pi[c,b])′) − M(pi[a,b], pi[c,b]) + ∑ b6=i,c∈I1,i∈I2 M(pi[i+1,b]pi[c,i]) − M((pi[i+1,b])′, (pi[c,i])′) this accounts for all the terms involving partitions pi[a,b] with a ∈ I1 and b ∈ I2. Thirdly we assess the contribution from terms involving partitions pi[r,s] with r ∈ I1 and s ∈ I3. Comparing with lemma 4.28 in all seven cases there is no discrepancy when a ∈ I1 and b ∈ I3 or c ∈ I1 and d ∈ I3 120 and therefore there is no contribution from these terms in agreement with Proposition 4.17. We have now observed the correct contributions from all terms involving partitions pi[r,s] where r ∈ I1. Fourthly we will consider contributions from terms involving partitions pi[r,s] where r ∈ I2 and s ∈ I1. As in the first and third cases on comparing with lemma 4.28 in all seven cases there is no discrepancy when a ∈ I2 and b ∈ I1 or c ∈ I2 and d ∈ I1. Again this is in full agreement with Proposition 4.17. Fifthly we consider contributions from terms involving partitions pi[r,s] where r ∈ I2 and s ∈ I2. This time on comparing with lemma 4.28 we ob- serve some nontrivial contributions as desired. This case is almost identical to when r ∈ I1 and s ∈ I2. In the lemma the following cases contribute Case 1: a ∈ I1 ∪ I2, b ∈ I2, b = d M((pi[a,b])′, pi[c,b])−M(pi[a,b], pi[c,b]) if a ∈ [c, b] M((pi[a,b])′, (pi[c,b])′)−M(pi[a,b], (pi[c,b])′) if a 6∈ [c, b]. Case 2: a ∈ I3, b 6∈ I2, c ∈ I2, d = a− 1 ∈ I2 M(pi[a,b], pi[c,a−1])−M((pia,b)′, (pi[c,a−1])′). Case 4: a ∈ I3, b ∈ I2, c ∈ I2, d = a− 1 M(pi[a,a−1], pi[c,a−1])−M(pi[a,a−1], (pi[c,a−1])′) if b = a− 1 M(pi[a,b], pi[c,a−1])−M((pi[a,b])′, (pi[c,a−1])′) if b 6= a− 1. Case 5: a ∈ I3, b ∈ I2, c ∈ I2, b = d, a− 1 ∈ [c, b], b 6= a− 1 M((pi[a,b])′, pi[c,b])−M(pi[a,b], pi[c,b]). Case 7: a ∈ I3, b ∈ I2, c ∈ I2, b = d, a− 1 6∈ [c, b] M((pi[a,b])′, (pi[c,b])′)−M(pi[a,b], (pi[c,b])′). 121 The sum total of these cases gives∑ a∈I2,b∈I2 M((pi[a,b])′, pi[a,b]) − M(pi[a,b], pi[a,b]) + ∑ a<c:a,c6=b+1 a,b∈I2 or c,b∈I2 M((pi[a,b])′, (pi[c,b])′) − M(pi[a,b], pi[c,b]) + ∑ c∈I2i∈I2b6=i M(pi[i+1,b]pi[c,i]) − M((pi[i+1,b])′, (pi[c,i])′). These terms again agree with those of Proposition 4.28. Sixthly we move to consider terms involving partitions pi[r,s] where r ∈ I2 and s ∈ I3. considering the lemma we see that there is no contribution for these terms as desired. We have now observed the correct contributions from all terms involving partitions pi[r,s] where r ∈ I1 ∪ I2 only the cases when r ∈ I3 remain, we may restrict to consider only those differences T (pi[a,b], pi[c,d])−B(pi[a,b], pi[c,d]) where both a, c ∈ I3. Seventhly we consider terms involving partitions pi[r,s] where r ∈ I3 and s ∈ I1. Considering lemma 4.28 we see that the following cases give non- trivial contributions. Case 2: a ∈ I3, b 6∈ I2, d = a− 1 ∈ I2 M(pi[a,b], pi[a,a−1])−M((pi[a,b])′, pi[a,a−1]) if a = c M(pi[a,b], pi[c,a−1])−M((pi[a,b])′, (pi[c,a−1])′) if a 6= c. Case 3: a ∈ I3, b 6∈ I2, a = c, d 6= a− 1 M(pi[a,b], (pi[a,d])′)−M(pi[a,b], pi[a,d]) if |d− a| ≤ |b− a|, M((pi[a,b])′, (pi[a,d])′)−M((pi[a,b])′, pi[a,d]) if |d− a| > |b− a|. Case 6: a ∈ I3, b ∈ I2, a− 1 6∈ [c, d], a = c, |d− a| < |b− a|, d ∈ I1 M(pi[a,b], (pi[a,d])′)−M(pi[a,b], pi[a,d]). Case 7: a ∈ I3, b ∈ I2, a− 1 6∈ [c, d], d ∈ I1 M((pi[a,b])′, (pi[a,d])′)−M((pi[a,b])′, pi[a,d]) if a = c and |d− a| > |b− a|. 122 The sum total of these cases gives∑ a∈I3,b∈I1 M((pi[a,b])′, pi[a,b]) − M(pi[a,b], pi[a,b]) + ∑ b<d:b,d 6=a a∈I2 and b∈I1 or d∈I1 M((pi[a+1,b])′, (pi[a+1,d])′) − M(pi[a+1,b], pi[a+1,d]) + ∑ i∈I2b∈I1 M(pi[i+1,b]pi[c,i]) − M((pi[i+1,b])′, (pi[c,i])′). Eighthly we consider terms involving partitions pi[r,s] where r ∈ I3 and s ∈ I3. Here considering lemma 4.28 we see that the following cases give non-trivial contributions. Case 2: a ∈ I3, b 6∈ I2, d = a− 1 ∈ I2 M(pi[a,b], pi[a,a−1])−M((pi[a,b])′, pi[a,a−1]) if a = c M(pi[a,b], pi[c,a−1])−M((pi[a,b])′, (pi[c,a−1])′) if a 6= c. Case 3: a ∈ I3, b 6∈ I2, a = c, d 6= a− 1 M(pi[a,b], (pi[a,d])′)−M(pi[a,b], pi[a,d]) if |d− a| ≤ |b− a|, M((pi[a,b])′, (pi[a,d])′)−M((pi[a,b])′, pi[a,d]) if |d− a| > |b− a|. Case 6: a ∈ I3, b ∈ I2, a− 1 6∈ [c, d], a = c, |d− a| < |b− a|, d ∈ I3 M(pi[a,b], (pi[a,d])′)−M(pi[a,b], pi[a,d]). Case 7: a ∈ I3, b ∈ I2, a− 1 6∈ [c, d], d ∈ I3 M((pi[a,b])′, (pi[a,d])′)−M((pi[a,b])′, pi[a,d]) if a = c and |d− a| > |b− a|. 123 The sum total of these cases gives∑ a∈I3,b∈I3 M((pi[a,b])′, pi[a,b]) − M(pi[a,b], pi[a,b]) + ∑ b<d:b,d 6=a a∈I2 and b∈I3 or d∈I3 M((pi[a+1,b])′, (pi[a+1,d])′) − M(pi[a+1,b], pi[a+1,d]) + ∑ i∈I2b∈I3 M(pi[i+1,b]pi[c,i]) − M((pi[i+1,b])′, (pi[c,i])′). Now we have accounted for all terms involving partitions other that the case pi[r,s] with r ∈ I3 and s ∈ I2. So we can now restrict to consider terms T (pi[a,b], pi[c,d])−B(pi[a,b], pi[c,d]) with a, c ∈ I3 and b, d ∈ I2 as follows. Ninthly we consider terms involving partitions pi[r,s] where r ∈ I3 and s ∈ I2. Here considering lemma 4.28 we see that the following cases give non-trivial contributions. Case 4: a ∈ I3, b ∈ I2, d = a− 1, M(pi[a,b], pi[a,a−1])−M((pi[a,b])′, pi[a,a−1]) if a = c and b 6= a− 1 M(pi[a,a−1], pi[c,a−1])−M(pi[a,a−1], (pi[c,a−1])′) if a 6= c and b = a− 1 M(pi[a,b], pi[c,a−1])−M((pi[a,b])′, (pi[c,a−1])′) if a 6= c and b 6= a− 1. Case 5: a ∈ I3, b ∈ I2, a− 1 ∈ [c, d], d 6= a− 1, b = d M((pi[a,b])′, pi[c,b])−M(pi[a,b], pi[c,b]). Case 6: a ∈ I3, b ∈ I2, a− 1 6∈ [c, d], a = c, |d− a| < |b− a| M(pi[a,b], (pi[a,d])′)−M(pi[a,b], pi[a,d]). Case 7: a ∈ I3, b ∈ I2, a− 1 6∈ [c, d] M((pi[a,b])′, (pi[a,b])′)−M(pi[a,b], pi[a,b]) if a = c and b = d M((pi[a,b])′, (pi[a,d])′)−M((pi[a,b])′, pi[a,d]) if a = c and |d− a| > |b− a| M((pi[a,b])′, (pi[c,b])′)−M(pi[a,b], (pi[c,b])′) if a 6= c and b = d. 124 The sum total of these cases gives the remaining terms∑ a∈I3,b∈I2,a−16=b M((pi[a,b])′, (pi[a,b])′) − M(pi[a,b], pi[a,b]) + ∑ b<d:b,d 6=a a∈I2 and b∈I2 or d∈I2 M((pi[a+1,b])′, (pi[a+1,d])′) − M(pi[a+1,b], pi[a+1,d]) + ∑ a<c:a,c6=b b∈I2 and a∈I3 or c∈I3 M((pi[a,b])′, (pi[c,b])′) − M(pi[a,b], pi[c,b]) + ∑ b 6=i and a−16=i i∈I2 and b∈I2 or c∈I3 M(pi[i+1,b], pi[c,i]) − M((pi[i+1,b])′, (pi[c,i])′). This completes the proof of Proposition 4.17. 125 Chapter 5 Conclusion 5.1 Summary of Results The central goal of this thesis was to produce a method to compute the motivic Donaldson–Thomas invariants of three dimensional Calabi–Yau cat- egories [33]. Theorem 2.8 now provides a schematic way to determine these invariants. We have applied this to the derived category of coherent sheaves on various Calabi–Yau threefolds [39] [40]. In particular we have produced formulas for the motivic Donaldson–Thomas invariants of crepant resolutions in all chambers of our space of stability parameters, for example Theorem 3.1 and Corollary 4.2. One obvious drawback of the method is that the categories we work with must come from a quiver with potential. In this way it is not possible to study derived categories of sheaves on a projective Calabi–Yau threefold. In- deed this remains a problem in the computation of the classical invariants, which have yet to be computed for the quintic threefold, the main known examples are toric [37]. Secondly, from the quiver point of view we have an affine space of stability conditions. In the cases we have studied the classical DT/PT/ncDT moduli spaces are all present in this picture [46],[47]. However when the threefold contains a divisor class (e.g. local surfaces) this is not known to be true. Hopefully new techniques can be developed in the future to broaden our 126 field of view. Joint work with Jim Bryan is underway in this direction, see future work below. 5.2 Connections to String Theory In type IIB string theory there exists a collection of dynamical objects know as D-branes. The branes which saturate a certain energy bound are called BPS-states. Mathematically these BPS-branes can be described by the cat- egory of coherent sheaves on a Calabi–Yau threefold. Bridgeland defined a stability condition on this category inspired by the notion of Π-stability in the physics literature [10]. From this modern point of view the Donadlson– Thomas invariants should be considered virtual counts of the number of stable BPS-branes. Recently string theorist have been working to understand a so called re- fined BPS state count. It is conjectured that these refined counts are given by the motivic Donaldson–Thomas invariants computed in this thesis. One validation of this comes from comparing our results with those of Dimofte and Gukov [17]. Another interesting recent development of physicists is the formulation of a refined topological vertex [26], this provides a formalism to compute refined BPS invariants on toric Calabi–Yau threefolds. We no- tice that this formalism is in agreement with our motivic calculations (see Section 3.5.3 and Section 4.8.3). Hopefully future work will be fruitful in producing a mathematical version of this refined vertex. 5.3 Future Research 5.3.1 Localization This is work in progress with Jim Bryan. Our aim is to discover a localiza- tion type formula for computing motivic vanishing cycles (and hence motivic Donaldson–Thomas invariants). 127 Let f : X → C be a regular function on a smooth quasi-projective scheme. We assume that f is C∗-equivariant so that f(t · x) = t · f(x) and also that limt→0 t · x exists for all x ∈ X. Then we know (see Theorem 2.5) that the (absolute) motivic vanishing cycle has a trivial monodromy action and equals the difference of the general and central fibers [ϕf ] = [f −1(1)]− [f−1(0)] ∈ K0(V arC). We would like to describe this motivic class as a sum over the fixed points of the C∗-action. We have shown that the only fixed points that have non- trivial contribution are those in the singular locus of the central fibre. As mentioned above a good localization formula would potentially be useful in the calculation of motivic DT invariants. We would like to prove the following: Conjecture 5.1. The motivic DT series for the orbifold C3/Z2×Z2 equals Z(y0, y1, y2, y3) = ∏ m≥1 ( m−1∏ k=0 (1− L−m2 +1+kym)−3(1− L−m2 +2+kym)−1 ) · ( m−1∏ k=0 3∏ i=0 (1 + L− m 2 + 1 2 +kymyi) −1(1 + L− m 2 + 1 2 +kymy−1i ) −1 ) · m−1∏ k=0 ∏ 0≤i<j≤3 (1− L−m2 +1+kymy−1i y−1j )  where y := y0y1y2y3. I have verified this conjecture with a computer in low degrees. In [62], B. Young solved the corresponding enumerative problem by using Graber–Pandharipande localization [21]. Then the classical DT invariant equals the number of torus fixed points on the Hilbert scheme. There is a bijection {piles of 3d - boxes} ←→ {C∗- fixed points of Hilb(C3)}. 128 Young enumerates piles of colored 3d - boxes using a clever trick involving vertex operator algebras. I am optimistic that finding a motivic localization formula will provide a refined combinatorial generating series which can be handled in a similar way. Through solving this problem we are aiming to develop a mathematical framework for the refined topological vertex. In physics the topological vertex [1] provides a powerful formalism to com- pute the A-model topological string partition function. A mathematical counterpart of this was developed via localization [37] [11] giving a way to compute the classical DT invariants of any toric Calabi-Yau threefold solely in terms of its web diagram. Recently, A. Iqbal, C. Kozçaz and C. Vafa [26] have proposed a refined topological vertex as a means to compute the refined topological string par- tition function. This vertex is more subtle giving refined BPS state counts and breaking much of the symmetry enjoyed by the usual vertex. I hope that by providing a localization formula for motivic vanishing cycles we will give a mathematical description of this object and provide a calculus formalizing the computation of motivic DT invariants on toric Calabi-Yau threefolds. However such a localization formula could be useful for computing the co- homology of vanishing cycles in other contexts. 5.3.2 Motivic zeta functions Another interesting avenue for future research comes from the singularities of schemesM = {df = 0} that are the degeneracy locus of a regular function on a smooth variety f : X → C. The motivic zeta function contains much more information about the singularities of M than the motivic vanishing cycle. The theory of local zeta functions also has analytic and p-adic versions. The interaction of these theories is very interesting. In particular the geometry of the singularities of f determines much of the number theoretic behavior. It is conjectured (see [25]) that the poles of the local zeta function are given by the roots of the Bernstein-Sato polynomial bf (s). 129 Bibliography [1] M. Aganagic, A. Klemm, M. Marino, C. Vafa. The Topological Vertex, Commun.Math.Phys. 254 (2005) 425-478. [2] K. Behrend. 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