Genus zero Gopakumar-Vafa invariants of Multi-Banana configurations
aa r X i v : . [ m a t h . AG ] F e b GENUS ZERO GOPAKUMAR-VAFA INVARIANTS OFMULTI-BANANA CONFIGURATIONS
NINA MORISHIGE
Abstract.
The multi-Banana configuration b F mb is a local Calabi-Yauthreefold of Schoen type. Namely, b F mb is a conifold resolution of b I v × D b I w ,where b I v → D is an elliptic surface over a formal disc D with an I v singulararity on the central fiber. We generalize the technique developedin our earlier paper to compute genus 0 Gopakumar-Vafa invariants ofcertain fiber curve classes. We illustrate the computation explicitly for v = 1 and v = w = 2. The resulting partition function can be expressedin terms of elliptic genera of C , or classical theta functions, respectively. Introduction
Background.
Let X be a quasi-projective Calabi-Yau threefold over C , so that X is smooth and K X ∼ = O X . Fix a curve class β ∈ H ( X ). Let M = M Xβ be the moduli space of Simpson semistable [9], pure, 1-dimensionalsheaves F with proper support on X with ch ( F ) = β ∨ and χ ( F ) = 1. Thegenus 0 Gopakumar-Vafa invariants n β ( X ) are defined mathematically byKatz [6]: Definition 1.
The genus 0 Gopakumar-Vafa (GV) invariants n β ( X ) of X in curve class β are defined as the Behrend function weighted Euler charac-teristics of the moduli space M Xβ .(1) n β ( X ) = e ( M Xβ , ν ) := X k ∈ Z k · e top ( ν − ( k ))where e top is topological Euler characteristic and ν : M Xβ → Z is Behrend’sconstructible function [1].In our previous paper [8], we computed the genus 0 Gopakumar-Vafainvariants of the Banana manifold, X Ban , a special kind of Schoen threefold,defined as the conifold resolution given by blowing up along the diagonal ofthe fiber product of a generic rational elliptic surface S → P with itself : X Ban := Bl ∆ ( S × P S ) . These results were consistent with the computation of the Donaldson-Thomasinvariants of X Ban obtained via topological vertex methods by Bryan [3].In this paper, we use similar methods as before to obtain the genus 0Gopakumar-Vafa invariants of certain fiber classes of related local Calabi-Yau threefolds, which we call multi-Banana configurations, and denote by
Date : February 17, 2021. b F mb . Our motivation is to study the fiberwise contribution of these con-figurations, which exist as formal subschemes in special Schoen manifolds,(Section 2.1). Unlike in our previous paper, even the genus 0 Gopakumar-Vafa invariants associated to these configurations cannot be obtained byother methods at present. Additionaly, the example configurations we studyyield partition functions with modular properties that can be expressed suc-cinctly. Our results appear to be compatible with results that appear in thephysics literature [4, Section 3.3].1.2. The multi-Banana configuration b F mb . The twelve singular fibers F ban of the regular Banana manifold X Ban are normalizations of the productof I singular fibers with themselves, F ban ∼ = Bl ∆ ( I × I ) ⊂ X Ban . Let b F ban be the formal completion of X Ban along F ban . Each F ban is iso-morphic to a non-normal toric variety whose normalization is isomorphic to P × P blown up at two points on the diagonal. We have π ( b F ban ) = Z × Z .See [8, Section 3.1] for details.We define the local Calabi-Yau threefold b F mb as follows: Definition 2.
The multi-Banana F vw mb and the local multi-Banana configu-ration b F vw mb are the ´etale covers of F ban and b F ban , respectively, F vw mb → F ban , b F vw mb → b F ban , associated to the subgroup v Z × w Z ⊂ Z × Z .We sometimes suppress the decoration and write F mb and b F mb instead.Observe that F ban = F mb and b F ban = b F mb .The geometry of multi-Banana configurations was studied by Kanazawaand Lau [5]. In particular, b F vw mb has vw + 2 curve classes, generated by threefamilies of curves, { A i } , { B j } , and { C k } , see section 2.2: β ∈ w − X i =0 Z [ A i ] ⊕ v − X j =0 Z [ B j ] ⊕ ( v − w − X k =0 Z [ C k ] , β ∈ H ( b F vw mb ) . Main results.
In some cases of small v and w , the GV invariants havenice formulas. We can express the partition function in terms of φ Q ( p ), theunique weak Jacobi form of weight -2 and index 1, φ Q ( p ) = p − (1 − p ) ∞ Y m =1 (1 − Q m p − ) (1 − Q m p ) (1 − Q m ) ,Q = exp(2 πiτ ) , p = exp(2 πiz ) , ( τ, z ) ∈ H × C . and Ell Q,p ( C , t ), the equivariant elliptic genus of C : V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 3
Ell
Q,p ( C , t ) = p φ Q ( pt ) φ Q ( p − t ) φ Q ( t ) . When v = 1, we have the following. Theorem 3. (See Theorem 10 for details and notation.) Fix a curve class β ( a , c ) in the local multi-Banana b F mb = b F w mb : β ( a , c ) = w − X i =0 a i [ A i ] + c [ C ] + [ B ] , a = ( a , . . . , a w − ) ∈ Z w ≥ , c ∈ Z ≥ . Then the genus 0 Gopakumar-Vafa invariants n β ( a ,c ) ( b F w mb ) can be ex-pressed as: X a ,c n β ( a , c ) ( b F mb ) r a s c = s · φ Q ( s ) w − X i =0 i + w − Y k = i Ell
Q,s ( C , R i ; k ) , where Q := w − Y i =0 ( r i s ) ,R a ; b := r a · r a +1 · r a +2 · · · r b · s b − a +1 , a ≤ b,r k + w := r k . In the case of v = w = 2, the curve classes are naturally labelled as A , A , B , B , C , C . We have the following result: Theorem 4. (See Theorem 9 for details and notation.) Let v = w = 2 , andfix a curve class β ( a , c ) in the local multi-Banana b F mb = b F mb : β ( a , c ) = a [ A ] + a [ A ] + c [ C ] + c [ C ] + [ B ] , a = ( a , a ) , c = ( c , c ) ∈ Z ≥ . Then the genus 0 Gopakumar-Vafa invariants n β ( a , c ) ( b F mb ) are given by thefollowing: X a ,a ,c ,c n β ( a , c ) ( b F mb ) r a r a s c s c = 2 (cid:26) φ Q ( r ) φ Q ( s ) φ Q ( r ) φ Q ( s ) φ Q ( r s ) φ Q ( r s ) (cid:27) / , where Q := r r s s φ Q ( p ) := φ − , ( Q, p ) . Remark . We note that the appearance of the elliptic genera in the parti-tion function of the multi-Banana suggests a correspondence via geometricengineering [7] to partition functions of Yang-Mills gauge theories on sur-faces. This viewpoint is discussed further in the previously cited physicsliterature [4].
NINA MORISHIGE
Outline of method.
We recall the method we used in [8] to computethe genus 0 GV invariants of X Ban . The argument carries over largelyunchanged for the local multi-Banana configurations X = b F mb , apart fromthe final combinatorics computation, so we refer the reader to our previouspaper for the details of the proofs of the statements in this summary of ourmethod.We have a T := C ∗ × C ∗ torus action on F mb , given by translation on thesmooth locus, and which extends to an action on all of b F mb . This gives usan action on its coherent sheaves Coh( b F mb ) and thus on the moduli space M b F mb β . This action preserves the canonical class and is compatible with thesymmetric obstruction theory. We can use the motivic nature of the Behrendfunction weighted Euler characteristic to stratify the moduli space underthis group action [1, 2]. The nontrivial torus orbits make no contribution to e ( M b F mb β , ν ), and we can reduce to considering only the T -fixed points of themoduli space ( M b F mb β ) T .We first count the fixed points of the moduli space. This gives us the naiveEuler characteristic, e n β ( b F mb ), which we define as the Euler characteristic ofthe moduli space without the Behrend function weighting: e n β ( b F mb ) := e ( M b F mb β ) . Using stability arguments we show that the sheaves in our moduli space havescheme-theoretic support on the multi-Banana surface F mb [8, Proposition12]. Thus, for computing e n β ( b F mb ), it suffices to count T -invariant sheavesof F mb .We would like to work on the universal cover of a multi-Banana, U ( F mb ),to make the computations easier. This is an infinite type toric surface,whose irreducible components are isomorphic to the blow-up of P × P at two torus fixed points. We give further details of the local geometry inSection 2.2. The universal cover U ( F mb ) is the same as that of the regularBanana fiber U ( F ban ) considered in [8].In order to relate the sheaves of U ( F mb ) with those of F mb , we introduceanother torus action, which we denote by P := C ∗ × C ∗ . This P actionon Coh( F mb ) is defined by tensoring with degree 0 line bundles of F mb ,as Pic ( F mb ) ∼ = C ∗ × C ∗ [8, Section 4]. Again, the Euler characteristiccontribution can be computed on orbits of the action, and it then sufficesto consider only sheaves invariant under the two C ∗ × C ∗ actions, T and P .The T torus action also lifts to give an action on the universal cover and itssheaves.Any sheaf fixed under the P action pulls back to an equivariant sheaf on U ( F mb ) [8, Proposition 22]. This equivariant sheaf contains a distinguishedsubsheaf which pushes forward to the original sheaf, and is unique up to decktransformations. Moreover stability, Euler characteristic 1, and invarianceunder the T torus action is preserved in this correspondence.The requirements of stability and Euler characteristic equal to 1 then putsrestrictions on the allowed invariant stable sheaves. If we further specify that V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 5 the curve class β has degree exactly 1 in one of the curve families of F vw mb ,then all the T and P fixed sheaves in our moduli space correspond to struc-ture sheaves of possibly non-reduced curves on U ( F mb ) [8, Proposition 23].The multiplicity of each component is constrained [8, Proposition 31] by acondition, which is equivalent to requiring that the partition given by mul-tiplicities of successive rational components from the fixed central degree 1curve has a conjugate partition with odd parts that are distinct. We givethe specific details of this condition in Section 4.This count of the number of fixed points of the moduli space gives thenaive Euler characteristic, e n β ( b F mb ). However, the Behrend function weight-ing amounts to a sign ( − deg β , which depends on the total degree of thecurve [8, Proposition 39], and this can be incorporated into the partitionfunction.We note that our technique is limited to computing invariants associatedto fiber class curves such that the degree of one of these families is fixed tobe 1. We do not yet know how to extend the technique to arbitrary degrees.Our method allows us to calculate the partition function for b F vw mb in thegeneral case, for arbitrary v and w . However, as v and w increase, there willbe unavoidable linear relations among the curve classes, even after fixing thedegree of one curve type to be 1. We will only present in detail the 2 × × w case (Section 5) as they illustrate the ideassufficiently without the notation becoming burdensome.2. Geometry
In this section, we give two examples of multi-Banana configurations b F mb that exist as formal neighborhoods of surfaces inside compact Calabi-Yauthreefolds. We then discuss some of the local geometry of multi-Bananaconfigurations needed for the following sections.2.1. Global geometry.Definition 6.
A multi-Banana manifold X mb is a smooth Calabi-Yau three-fold which is a conifold resolution of the fiber product of two rational ellipticsurfaces, and such that the formal neighborhood of each singular fiber is amulti-Banana configuration. Example . Let S π → P be a rational elliptic surface with singular fibersconsisting of four I and four I singular fibers. Suppose S has a 2-torsionsection. This induces an order 2 automorphism φ that interchanges thenodes of each of the I fibers.We can then form the fiber product of S with itself, S × P S . In order toget a conifold resolution, we blow up the generalized diagonal e ∆, consistingof the diagonal ∆, as well as all its translates by iterations of φ , e ∆ := ( φ ( i )2 × φ ( j )2 )∆ , ≤ i, j < . NINA MORISHIGE
We will call this multi-Banana manifold X . X := Bl e ∆ ( S × P S )In this case, the multi-Banana contains four F mb configurations, and fourordinary Banana fibers F ban .Instead of taking the fiber product of S with itself, we can also do thefollowing construction. Example . Let S π → P be a rational elliptic surface with two I and two I singular fibers. Then S has a 5-torsion section, which induces an order 5automorphism φ which acts on each I fiber by cycling the nodes.Now, let us take the quotient of S by the action of φ , and let S ′ be theresolution of the quotient: S ′ := Res ( S/φ ) . Notice that by construction, S ′ π ′ → P is another rational elliptic surfacewith singular fibers over the same base points: P sing := { p ∈ P | π − ( p ) ⊂ S singular } = { p ∈ P | π ′− ( p ) ⊂ S ′ singular } . We also have that the smooth fibers of S and S ′ are isogenous: φ | p : π − ( p ) → π ′ − ( p ) ,φ | p is an isogeny, ∀ p ∈ P \ P sing . In this case, there is a conifold resolution of the fiber product, S × P S ′ .From the construction, we have a rational map of schemes over P , S S ′ = Res ( S/φ k ) , so we get a graph Γ ⊂ S × P S ′ . Then the conifold resolution is given by blowing up this graph Γ. We willcall this multi-Banana threefold X : X := Bl Γ ( S × P S ′ ) . The multi-Banana manifold X is a rigid Calabi-Yau threefold and containstwo F mb multi-Banana configurations, and two F mb multi-Banana configu-rations.2.2. Local geometry.
We now examine the local geometry of the multi-Banana configurations in more detail and establish some notation we willneed later.We recall the construction from [5] and relate it to our discussion.Let L be a tiling of the plane ( x, y, ⊂ R given by: { x = m, z = 1 } ∪ { y = n, z = 1 } ∪ { y − x = r, z = 1 } ,m, n, r ∈ Z , ( x, y, z ) ∈ R . V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 7
Let A be the non-finite type toric threefold whose fan consists of all thecones over the proper faces of L (Figure 1). Let U ( F mb ) be the universalcover of F mb , and U ( b F mb ) the universal cover of b F mb , U ( F mb ) pr −−→ F mb U ( b F mb ) pr −−→ b F mb . Then U ( F mb ) ⊂ A is the union of the toric divisors of A , and U ( b F mb ) is theformal completion of A along U ( F mb ). Figure 1.
The fan of A .We have an action of G = v Z × w Z , v, w ∈ Z ≥ , on L ⊂ R by transla-tion: ( v, w ) · ( x, y,
1) = ( x + v, y + w, . which induces an automorphism ψ G : A → A and also on U ( b F mb ). Theseare then the deck transformations of the universal cover of the local multi-Banana configuration b F mb : U ( b F mb ) → U ( b F mb ) /ψ G ∼ = b F mb . We denote by Ξ the irreducible surface which is the momentum polytopeof P × P blown up at 2 points, and drawn as a hexagon in our diagrams:Ξ := Bl p ,p ( P × P ) , p , p ∈ P × P . Then the momentum polytope of U ( F mb ) can be represented as a hexagonaltiling of the plane, and the momentum polytope of F mb is given by v × w hexagons glued together along their toric boundary, as depicted in theexample of Figure 2.The irreducible components of the torus fixed curves in F mb fall into threefamilies of rational curves. Two of these families, { A i } and { B j } , are propertransforms of the rational curves from the I v and I w singular fibers in F mb ,respectively, and one family, { C k } , are the exceptional curves of the conifoldresolution.We will draw these curves oriented as shown in Figure 3, so the verticalcurves are in the A family, the horizontal curves are B family, and thediagonal curves are from the C family. NINA MORISHIGE A A A A A A B B B B B B B B C C C C C C C C C C C C Figure 2. F mb in the case v = 3 and w = 4. Here, the topboundary curves are identified with those along the bottom,and also the left edge with the right edge. A i B j C k A l B m C n Figure 3.
Curve labels for hexagon ΞWhen there is no confusion, we will also label irreducible componentsof the lifts to the universal cover of these torus invariant curves with thecurve class of their projection to F mb . That is, an irreducible component of pr − ( A i ) ⊂ U ( F mb ) will also be referred to as A i in the universal cover.As each hexagon surface Ξ is the momentum polytope of P × P blownup at 2 points, their 6 boundary divisors have 2 relations. For example, inthe labeled Figure 3, we have:(2) A i + C k = A l + C n ,B m + C k = B j + C n . from the equivalent ways to express the total transform of the rulings ineach P factor.A priori, there are 3 vw torus invariant irreducible curves in F mb , butthese satisfy standard hexagon relations (Eq. 2), so it is possible to choosea basis of vw + 2 curves, consisting of v × A curves, w × B curves, and( v − w −
1) + 1 × C curves. For the small examples we consider, wewill index the curves in each family in a simple way. It is possible to use asystematic choice of generators and labels for the general case [5, Section 5.1], V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 9 but it would be notationally cumbersome for these examples, so we do notpresent that here.As remarked in the introduction, our technique is limited to consideringonly curves where we restrict the degree of one family of curves to be exactly1. For concreteness, we will assume our curves are of class(3) β = X a i [ A i ] + [ B ] + X c k [ C k ] . The hexagonal tiling from U ( F mb ) possesses a v Z × w Z periodicity fromthe deck transformations. In order to get rid of the ambiguity from thedeck transformations, we will assume we have fixed a choice of fundamentaldomain D , and any curve we consider has its unique irreducible componentcovering B inside D . In other words, we will require that T -torus invariantcurves C ⊂ U ( F mb ) with [ pr ( C )] = β also satisfy C ∩ pr − ( B ) ⊂ D .From the arguments given in Subsection 1.4, in order to compute thenaive Euler characteristic e n β ( b F mb ), it suffices to count all configurations ofpossibly non-reduced T -torus invariant curves covering β on the universalcover U ( F mb ), subject to the constraint that the partition given by multi-plicities of successive rational components of each tree emanating from B has a conjugate partition which has odd parts that are distinct. In sec-tion 4 and 5, we will illustrate this count in two specific cases, namely whenthe fundamental domain consists of 2 × × w hexagons. These configurations exist, for example, in X , and X ,respectively, as described in the previous Section 2.1.3. Notation and conventions
We gather in this section the conventions we use for product and sumexpansions for Jacobi forms and elliptic genera.Recall, the weak Jacobi form φ − , ( q, p ) of weight -2 index 1 is defined as φ − , ( q, p ) = p − (1 − p ) ∞ Y m =1 (1 − q m p − ) (1 − q m p ) (1 − q m ) , The Jacobi theta function θ ( q, p ) function is given as θ ( q, p ) = − X q k ( − p ) k = − iq p − ∞ Y m =1 (1 − q m )(1 − q m − p )(1 − q m p − )and the Dedkind η function is η ( q ) = q ∞ Y m =1 (1 − q m ) . Here, q = exp(2 πiτ ) , p = exp(2 πiz ) , ( τ, z ) ∈ H × C . Since the first variable will be constant within our partition functions, wewill use the shortened notation, φ Q ( p ) := φ − , ( Q, p ) θ Q ( p ) := θ ( Q, p ) η Q := η ( Q )Treating these expressions as formal power series, it is easy to verify theidentities :(4) p φ Q ( p ) = iθ Q ( p ) η Q , p p φ Q ( p ) = s Qp φ Q ( Qp ) , p φ Q ( p ) = − q φ Q ( p − ) . Suppose M is a non-compact complex manifold of dimension d with a C ∗ action with isolated fixed points { x } of tangent weights k i . We define theequivariant elliptic genus of M to be:Ell q,y ( M, t ) = X x ∈ M C ∗ d Y j =1 y − ∞ Y m =1 (1 − q m − yt − k j ( x ) )(1 − q m p − t k j ( x ) )(1 − q m − t − k j ( x ) )(1 − q m t k j ( x ) ) . In particular ([10, Theorem 12]), we have:Ell q,y ( C , t ) = θ ( q, yt ) θ ( q, yt − ) θ ( q, t ) θ ( q, t − )= p φ − , ( q, yt ) φ − , ( q, y − t ) φ − , ( q, t )= p φ Q ( yt ) φ Q ( y − t ) φ Q ( t ) . (5) 4. Case × b F := b F mb , and we use this exampleto illustrate in detail how the method described in the Section 1.4 leads tothe computation of the Gopakumar-Vafa invariants.4.1. T -Torus fixed curves on b F . We first fix a choice of a fundamentaldomain D in U ( F ) so that there is no ambiguity in our counts due to decktransformations.We label the curves of the 2 × U ( F ) with our convention explained in Section 2.2.The vertical curves cover curves in the A family, the horizontal curves thosein the B family, and the diagonal curves cover the C family as shown inFigure 4. There is a 2 Z × Z periodicity of this fundamental domain in theuniversal cover. V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 11 A A A A A A B B B B B B C C C C C C C Figure 4. × U ( F ).We can choose a basis for the homology classes of curves of F given bythe 6 curves A , A , B , B , C , and C , as labelled in Figure 4. This can beshown using simple applications of the standard hexagon relations (Eq. 2)as follows.First observe that the sum of the the C curves in each row is constant, asis the sum in each column,(6) C + C = C + C ,C + C = C + C . This follows by combining two hexagon relations to the bottom row ofhexagons in Fig. 4. For example, A + C = A + C ,A + C = A + C . yields C + C = C + C . The other relation is derived similarly.We can also write the sum of all the diagonal curves in two ways, groupedas rows or as columns. In this case when I = J = 2, we have:( C + C ) + ( C + C ) = ( C + C ) + ( C + C ) , Together with the previous Eq. 6, this implies that C = C ,C = C . In a similar fashion, it is easy to deduce that A = A ,A = A ,B = B ,B = B . We can thus choose a basis for the homology classes of curves given by the6 curves A , A , B , B , C , C .We are interested in sheaves invariant under the action of the torus T ,so their support must be contained in the T -torus fixed curves. We willassume from now on that the support curve C of our sheaf has deg B = 1and deg B = 0 so that it is in the homology class:(7) [ C ] = β ∈ X i =0 , a i [ A i ] + [ B ] + X j =0 , c j [ C j ] . Recall there is a 1-1 correspondence between the π ( F )-equivariant sheaveson U ( F mb ) and the P -fixed sheaves on F mb up to deck transformations [8,Proposition 22]. To remove the ambiguity, we will further require that thecorresponding equivariant sheaf in U ( F mb ) has support curve C ⊂ U ( F mb ),whose reduced irreducible component covering the curve in class [ B ] is inour chosen fundamental domain D . In this example, notice that there aretwo possible choices for B , since [ B ] = [ B ].The irreducible components of the curve C may be nonreduced, so we needto keep track of the multiplicity of each component to determine the curveclass of pr ( C ). We let the variable r i track the number of curves which cover A i , i ∈ { , } , and s j track the number of curves which cover C j , j ∈ { , } .A typical curve C which covers a curve in class Eq. (7) is pictured in Figure 5.4.2. Translating to combinatorics.
We explain the details of our methodof converting the count of invariant stable sheaves into a combinatorics prob-lem.Recall from the discussion in Section 1.4 the naive Euler characteristic e n β ( b F ) = e ( M b F β ) for curves in class β of the form Eq. (7) equals a countof T -torus invariant structure sheaves of genus 0 curves on the universalcover C ⊂ U ( F mb ) that cover class β [8, Proposition 23], subject to thecertain conditions [8, Proposition 31] on their multiplicity that we explainbelow.First we introduce some terminology. We will refer to irreducible compo-nents of C as edges, and the intersection of two or more edges as vertices.Let B ⊂ C be the edge that covers class [ B ] and lies in the fundamentaldomain D by assumption.We will call the union of B with any one of the four disjoint subcurvesof C\B a branch of C . Then the curve counts can be done on each branchseparately, and will be the same on each branch, up to relabeling.The edge B is the intersection of two irreducible surface componenthexagons isomorphic to Ξ in U ( F mb ). Let S be one of these and let g be thedeck transformation that translates S into the other. The hexagons g m S , m ∈ Z , in the orbit of S under the group of deck transformations h g i ∼ = Z will be called inside hexagons. Any other hexagons will be called outsidehexagons. V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 13 B r s r s r s r s r s r s r s r s r s r s r s Figure 5.
A typical torus fixed curve in U ( F ) which cov-ers a curve with deg[ B ] = deg[ B ] = 1. Here r i and s j areused to track the multiplicity of curves which cover A i and C j curves, respectively.Any edge of C\B covers A i or C j , i, j ∈ { , } , and will be the intersectionof an inside hexagon and an outside one. These T -invariant edges can canhave monomial thickening in these two directions. Any thickenings in thedirection of the inside hexagon will be called inside thickenings, and thosein the direction of the outside hexagon will be called outside thickenings.However, because of stability and the requirement of the Euler characteristicof O [ C ] to be 1, the possible thickenings can only be of a particular form.Thickenings on the edges that cover A i or C j are subject to the followingproperties [8, Proposition 31]:(1) Inside thickenings of any edge that intersects B is unrestricted.(2) All nonzero outside thickenings must be 1.(3) Inside thickenings are non-increasing on components along a branchin the direction moving away from B .(4) Inside thickenings for two adjacent edges contained in a commoninside hexagon can either be the same or differ by one.We can interpret the inside multiplicity of each edge as length of a partin a partition. These constraints are independent on each branch, so we examine one branch at a time. Along each branch, the non-increasing lengthcondition says that the allowed multiplicities of edges form a Young diagram.If we examine the conjugate partition of the branch, the fourth conditioncan be interpreted as saying that any odd parts that appear in the conjugatepartition are distinct, with no restriction on the even parts.The generating function that counts the number of partitions p ( n ) withodd parts distinct can be written as the product of generating functions forpartitions with arbitrary even parts with that of partitions with unique oddparts: X p ( n ) q n = Y − q n Y (1 + q n − )We must further refine the parts, because we have four curve classes A i and C j , for i, j ∈ { , } to keep track of. In other words, we need to keep trackof the residue classes mod 4 in our partition.Consider for example the northeast branch of the curve shown above,which we reproduce in Figure 6. Suppose we number the edges consecutively,starting with the first edge e that intersects B . Then the every odd-numbered edge will contribute to an odd part, and the even-numbered onesto an even part. The first edge e in this example curve covers C , and weassign the variable s to track this curve. The second edge e covers A andwe use the variable r to track this. B s r s r s r s Figure 6.
Detail of the northeast branch of the curve shownin Figure 5.So we refine the generating function above, and replace powers of thevariable q , by q s q s r q s r s q s r s r , V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 15 and for higher powers of q , we continue the pattern, so that q m + i = q m q i ( s r s r ) m q i We also use the notation Q := r s r s . Now, the generating function that counts the number of partitions withodd parts distinct can be expressed for the northeast branch as:(1 + s )(1 + s r s )(1 + Qs )(1 + Qs r s ) · · · (1 − r s )(1 − Q )(1 − Qr s )(1 − Q ) · · · = (1 + s ) ∞ Q m =1 (1 + Q m s )(1 + Q m r − )(1 − r s ) ∞ Q m =1 (1 − Q m )(1 − r s Q m )We do this for each branch and multiply the contribution from all fourbranches. Notice also that there are two distinct possible locations for thecurve B . However, they give the same contribution to the generating func-tion, since the four branches in either location consist of the same sequenceof curves, up to renaming of the branches.Hence, using the identities (4), the partition function can be expressed interms of the theta function θ Q ( p ) as:2 iη − Q θ Q ( − r ) θ Q ( − s ) θ Q ( − r ) θ Q ( − s ) θ Q ( r s ) θ Q ( r s ) . (8)or in terms of the weak Jacobi form φ Q ( p ) as:2 (cid:26) φ Q ( − r ) φ Q ( − s ) φ Q ( − r ) φ Q ( − s ) φ Q ( r s ) φ Q ( r s ) (cid:27) / (9)As we explained in the introduction, this count of the fixed points corre-sponds to the naive Euler characteristic contribution, e n β ( b F mb ), X a ,a c ,c e n β ( a , c ) ( b F mb ) r a r a s c s c = 2 (cid:26) φ Q ( − r ) φ Q ( − s ) φ Q ( − r ) φ Q ( − s ) φ Q ( r s ) φ Q ( r s ) (cid:27) / . (10)However, the Behrend function weighting amounts to a sign that dependson the degree of the curve class [8, Remark 33]: e n β ( a , c ) ( b F mb ) = ( − a + a + c + c n β ( a , c ) ( b F mb ) . We can incorporate this sign by replacing our tracking variables by theirnegatives.Hence, we have shown the following.
Theorem 9.
Fix a curve class β ( a , c ) in the local multi-Banana b F mb = b F mb , β ( a , c ) = a [ A ] + a [ A ] + c [ C ] + c [ C ] + [ B ] , a = ( a , a ) , c = ( c , c ) ∈ Z ≥ . Then the genus 0 Gopakumar-Vafa invariants n β ( a , c ) ( b F mb ) are given by thefollowing: X a ,a c ,c n β ( a , c ) ( b F mb ) r a r a s c s c = 2 (cid:26) φ Q ( r ) φ Q ( s ) φ Q ( r ) φ Q ( s ) φ Q ( r s ) φ Q ( r s ) (cid:27) / , where we use the notation Q := r r s s φ Q ( p ) := φ − , ( Q, p ) . and φ Q ( p ) = φ − , ( Q, p ) is the unique weak Jacobi form of weight -2 andindex 1: φ Q ( p ) = p − (1 − p ) ∞ Y m =1 (1 − Q m p − ) (1 − Q m p ) (1 − Q m ) ,Q = exp(2 πiτ ) , p = exp(2 πiz ) , ( τ, z ) ∈ H × C . Case × w In this section, we look in detail at the case of F w mb , when v = 1 and w ≥ U ( F w mb ) has a Z × w Z periodicityin the universal cover. Its momentum polytope is given by 1 × w hexagons.Using the hexagon relations (Eq. 2) as in the previous section, it is easyto see there is only one horizontal curve class, which we call B , and onediagonal curve class, which we call C . There are w distinct vertical curveclasses, A i , ≤ i ≤ w −
1, (Figure 7).Let us assume that the support curve C of our sheaf has deg B = 1 sothat [ C ] ∈ w X i =0 a i [ A i ] + [ B ] + c [ C ] . Let r i track the number of A i curves and s track the number of C curves.For this case, we define the variable Q to be:(11) Q := w − Y i =0 ( r i s ) . We will also use the following multi product notation:(12) R a ; b := r a · r a +1 · r a +2 · · · r b · s b − a +1 , a ≤ b, where the subscript of r is interpreted mod w : r k + w := r [ k ] , [ k ] ∈ Z /w Z . V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 17 A A A A w − A A A A w − B B B B BC C C CC C C C
Figure 7. × w hexagon momentum polytope of the fun-damental domain in U ( F w mb ).In particular, R b := b Y i =0 ( r i s ) . First, suppose the single B curve is located connected to an A curve. Then,we can count the number of partitions with odd parts distinct in the sameway as before, and the generating function for these configurations is ex-pressed as follows:(1 + s ) ∞ Y m =1 (1 + sQ m ) (1 + s − Q m ) (1 − Q m ) × w − Y k =0 (cid:26) (1 + sR k )(1 + s − R k )(1 − R k ) × ∞ Y m =1 (1 + sR k Q m )(1 + s − R k Q m )(1 + sR − k Q m )(1 + s − R − k Q m )(1 − R k Q m ) (1 − R − k Q m ) ) We can write this more succinctly using the weak Jacobi form φ Q ( p ) as:(13) ( − s ) φ Q ( − s ) w − Y k =0 ( p φ Q ( − sR k ) p φ Q ( − sR k ) φ Q ( R k ) ) ;or alternatively, in terms of the theta function θ Q ( p ) function as:(14) ( − s ) φ Q ( − s ) w − Y k =0 ( θ Q ( − sR k ) θ Q ( − sR − k ) θ Q ( R k ) θ Q ( R − k ) ) . Notice that, from Eq. (5), the product can be expressed in terms of theequivariant elliptic genus of C ,(15) ( − s ) φ Q ( − s ) w − Y k =0 Ell Q, − s ( C , R k ) . There are w different locations possible for the B curve in the 1 × w hexagon, characterized by the choice of which A i curve, 0 ≤ i ≤ w − B curve is connected to. Although the generating function for thepartitions with distinct odd parts associated to these other configurationsdepends on the particular location of B , it is easy to see that it differs fromthe previous formula only by a cyclic shift of indices in the R k variable.The total partition function counts the contribution from all possible lo-cations of the B curve, and is thus expressed as a sum over the generatingfunctions from each location,(16) ( − s ) φ Q ( − s ) w − X i =0 i + w − Y k = i Ell Q, − s ( C , R i ; k ) . As in the previous section, this count of fixed points corresponds to thenaive Euler characteristic. To take account of the Behrend function weight-ing, we can incorporate a sign based on the degree of the curve class bysimply replacing our tracking variables by their negatives.Thus, for the case of b F w mb , we have the following partition function. Theorem 10.
Fix a curve class β ( a , c ) in the local multi-Banana b F mb = b F w mb : β ( a , c ) = w − X i =0 a i [ A i ] + c [ C ] + [ B ] , a = ( a , . . . , a w − ) ∈ Z w ≥ , c ∈ Z ≥ . Then the genus 0 Gopakumar-Vafa invariants n β ( a ,c ) ( b F w mb ) can be ex-pressed as: X a ,c n β ( a , c ) ( b F mb ) r a s c = s · φ Q ( s ) w − X i =0 i + w − Y k = i Ell
Q,s ( C , R i ; k ) . where Ell q,y ( C , t ) is the equivariant elliptic genus of C , and we use thenotation: r a := r a r a . . . r a w − w − ,Q := w − Y i =0 ( r i s ) ,R a ; b := r a · r a +1 · r a +2 · · · r b · s b − a +1 , a ≤ b,r k + w := r [ k ] , [ k ] ∈ Z /w Z . V INVARIANTS OF MULTI-BANANA CONFIGURATIONS 19
We also mention that it is possible to choose to fix the degree of the A family of curve classes or the C class to be 1 instead of the B curve. However,in this 1 × w case, doing so reduces to the ordinary Banana configuration F ban case and yields the same formula as in the earlier paper [8]. References [1] Kai Behrend. Donaldson-Thomas type invariants via microlocal geometry.
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Nina Morishige, Department of Mathematics, The University of BritishColumbia, Vancouver, BC, V6T 1Z2 Canada
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