Genus Zero Gopakumar-Vafa invariants of multi-Banana configurations

Let $X$ be a quasi-projective Calabi-Yau threefold over $\mathbf{C}$, so that $X$ is smooth and $K_{X}\cong\mathcal{O}_{X}$. Fix a curve class $\beta\in H_{2}(X)$. Let $M=M^{X}_{\beta}$ be the moduli space of Simpson semistable [9], pure, 1-dimensional sheaves $\mathcal{F}$ with proper support on $X$ with $\text{ch}_{2}(\mathcal{F})=\beta^{\vee}$ and $\chi(\mathcal{F})=1$. The genus 0 Gopakumar-Vafa invariants $n^{0}_{\beta}(X)$ are defined mathematically by Katz [6]:

In our previous paper [8], we computed the genus 0 Gopakumar-Vafa invariants of the Banana manifold, $X_{\textnormal{{Ban}}}$, a special kind of Schoen threefold, defined as the conifold resolution given by blowing up along the diagonal of the fiber product of a generic rational elliptic surface $S\to\mathbf{P}^{1}$ with itself :

These results were consistent with the computation of the Donaldson-Thomas invariants of $X_{\textnormal{{Ban}}}$ obtained via topological vertex methods by Bryan [3].

In this paper, we use similar methods as before to obtain the genus 0 Gopakumar-Vafa invariants of certain fiber classes of related local Calabi-Yau threefolds, which we call multi-Banana configurations, and denote by $\widehat{F}_{\textnormal{{mb}}}$. Our motivation is to study the fiberwise contribution of these configurations, 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 by other methods at present. Additionaly, the example configurations we study yield partition functions with modular properties that can be expressed succinctly. Our results appear to be compatible with results that appear in the physics literature [4, Section 3.3].

The twelve singular fibers ${F_{\textnormal{{ban}}}}$ of the regular Banana manifold $X_{\textnormal{{Ban}}}$ are normalizations of the product of $I_{1}$ singular fibers with themselves,

Let ${\widehat{F}_{\textnormal{{ban}}}}$ be the formal completion of $X_{\textnormal{{Ban}}}$ along ${F_{\textnormal{{ban}}}}$. Each ${F_{\textnormal{{ban}}}}$ is isomorphic to a non-normal toric variety whose normalization is isomorphic to $\mathbf{P}^{1}\times\mathbf{P}^{1}$ blown up at two points on the diagonal. We have $\pi_{1}({\widehat{F}_{\textnormal{{ban}}}})=\mathbf{Z}\,\times\mathbf{Z}\,$. See [8, Section 3.1] for details.

We define the local Calabi-Yau threefold $\widehat{F}_{\textnormal{{mb}}}$ as follows:

We sometimes suppress the decoration and write ${F_{\textnormal{{mb}}}}$ and $\widehat{F}_{\textnormal{{mb}}}$ instead. Observe that ${F_{\textnormal{{ban}}}}=F_{\textnormal{{mb}}}^{11}$ and ${\widehat{F}_{\textnormal{{ban}}}}={\widehat{F}_{\textnormal{{mb}}}}^{11}$.

The geometry of multi-Banana configurations was studied by Kanazawa and Lau [5]. In particular, ${\widehat{F}_{\textnormal{{mb}}}}^{vw}$ has $vw+2$ curve classes, generated by three families of curves, $\{A_{i}\}$, $\{B_{j}\}$, and $\{C_{k}\}$, see section 2.2:

In some cases of small $v$ and $w$, the GV invariants have nice formulas. We can express the partition function in terms of $\phi_{\scriptscriptstyle Q}(p)$, the unique weak Jacobi form of weight -2 and index 1,

and $\mathrm{Ell}_{Q,p}(\mathbf{C}^{2},t)$, the equivariant elliptic genus of $\mathbf{C}^{2}$:

When $v=1$, we have the following.

In the case of $v=w=2$, the curve classes are naturally labelled as $A_{0},A_{1},B_{0},B_{1},C_{0},C_{1}$. We have the following result:

We recall the method we used in [8] to compute the genus 0 GV invariants of $X_{\textnormal{{Ban}}}$. The argument carries over largely unchanged for the local multi-Banana configurations $X=\widehat{F}_{\textnormal{{mb}}}$, apart from the final combinatorics computation, so we refer the reader to our previous paper for the details of the proofs of the statements in this summary of our method.

We have a $T\coloneqq\mathbf{C}^{*}\times\mathbf{C}^{*}$ torus action on ${F_{\textnormal{{mb}}}}$, given by translation on the smooth locus, and which extends to an action on all of ${\widehat{F}_{\textnormal{{mb}}}}$. This gives us an action on its coherent sheaves $\operatorname{Coh}({\widehat{F}_{\textnormal{{mb}}}})$ and thus on the moduli space $M_{\beta}^{{\widehat{F}_{\textnormal{{mb}}}}}$. This action preserves the canonical class and is compatible with the symmetric obstruction theory. We can use the motivic nature of the Behrend function weighted Euler characteristic to stratify the moduli space under this group action [1, 2]. The nontrivial torus orbits make no contribution to $e(M_{\beta}^{{\widehat{F}_{\textnormal{{mb}}}}},\nu)$, and we can reduce to considering only the $T$-fixed points of the moduli space $(M_{\beta}^{{\widehat{F}_{\textnormal{{mb}}}}})^{T}$.

We first count the fixed points of the moduli space. This gives us the naive Euler characteristic, $\widetilde{n}^{0}_{\beta}(\widehat{F}_{\textnormal{{mb}}})$, which we define as the Euler characteristic of the moduli space without the Behrend function weighting:

Using stability arguments we show that the sheaves in our moduli space have scheme-theoretic support on the multi-Banana surface ${F_{\textnormal{{mb}}}}$ [8, Proposition 12]. Thus, for computing $\widetilde{n}^{0}_{\beta}(\widehat{F}_{\textnormal{{mb}}})$, it suffices to count $T$-invariant sheaves of ${F_{\textnormal{{mb}}}}$.

We would like to work on the universal cover of a multi-Banana, $U({F_{\textnormal{{mb}}}})$, to make the computations easier. This is an infinite type toric surface, whose irreducible components are isomorphic to the blow-up of $\mathbf{P}^{1}\times\mathbf{P}^{1}$ at two torus fixed points. We give further details of the local geometry in Section 2.2. The universal cover $U({F_{\textnormal{{mb}}}})$ is the same as that of the regular Banana fiber $U({F_{\textnormal{{ban}}}})$ considered in [8].

In order to relate the sheaves of $U({F_{\textnormal{{mb}}}})$ with those of ${F_{\textnormal{{mb}}}}$, we introduce another torus action, which we denote by $P\coloneqq\mathbf{C}^{*}\times\mathbf{C}^{*}$. This $P$ action on $\operatorname{Coh}({F_{\textnormal{{mb}}}})$ is defined by tensoring with degree 0 line bundles of ${F_{\textnormal{{mb}}}}$, as $\operatorname{Pic}^{0}({F_{\textnormal{{mb}}}})\cong\mathbf{C}^{*}\times\mathbf{C}^{*}$ [8, Section 4]. Again, the Euler characteristic contribution can be computed on orbits of the action, and it then suffices to consider only sheaves invariant under the two $\mathbf{C}^{*}\times\mathbf{C}^{*}$ actions, $T$ and $P$. The $T$ torus action also lifts to give an action on the universal cover and its sheaves.

Any sheaf fixed under the $P$ action pulls back to an equivariant sheaf on $U({F_{\textnormal{{mb}}}})$ [8, Proposition 22]. This equivariant sheaf contains a distinguished subsheaf which pushes forward to the original sheaf, and is unique up to deck transformations. Moreover stability, Euler characteristic 1, and invariance under the $T$ torus action is preserved in this correspondence.

The requirements of stability and Euler characteristic equal to 1 then puts restrictions on the allowed invariant stable sheaves. If we further specify that the curve class $\beta$ has degree exactly 1 in one of the curve families of $F_{\textnormal{{mb}}}^{vw}$, then all the $T$ and $P$ fixed sheaves in our moduli space correspond to structure sheaves of possibly non-reduced curves on $U({F_{\textnormal{{mb}}}})$ [8, Proposition 23]. The multiplicity of each component is constrained [8, Proposition 31] by a condition, which is equivalent to requiring that the partition given by multiplicities of successive rational components from the fixed central degree 1 curve has a conjugate partition with odd parts that are distinct. We give the specific details of this condition in Section 4.

This count of the number of fixed points of the moduli space gives the naive Euler characteristic, $\widetilde{n}^{0}_{\beta}(\widehat{F}_{\textnormal{{mb}}})$. However, the Behrend function weighting amounts to a sign $(-1)^{\deg\beta}$, which depends on the total degree of the curve [8, Proposition 39], and this can be incorporated into the partition function.

We note that our technique is limited to computing invariants associated to fiber class curves such that the degree of one of these families is fixed to be 1. We do not yet know how to extend the technique to arbitrary degrees.

Our method allows us to calculate the partition function for ${\widehat{F}_{\textnormal{{mb}}}}^{vw}$ in the general case, for arbitrary $v$ and $w$. However, as $v$ and $w$ increase, there will be unavoidable linear relations among the curve classes, even after fixing the degree of one curve type to be 1. We will only present in detail the $2\times 2$ case (Section 4) and the $1\times w$ case (Section 5) as they illustrate the ideas sufficiently without the notation becoming burdensome.

Instead of taking the fiber product of $S$ with itself, we can also do the following construction.

We now examine the local geometry of the multi-Banana configurations in more detail and establish some notation we will need later.

We recall the construction from [5] and relate it to our discussion.

Let $L$ be a tiling of the plane $(x,y,1)\subset\mathbf{R}^{3}$ given by:

Let $\mathcal{A}$ be the non-finite type toric threefold whose fan consists of all the cones over the proper faces of $L$ (Figure 1). Let $U({F_{\textnormal{{mb}}}})$ be the universal cover of ${F_{\textnormal{{mb}}}}$, and $U({\widehat{F}_{\textnormal{{mb}}}})$ the universal cover of ${\widehat{F}_{\textnormal{{mb}}}}$,

Then $U({F_{\textnormal{{mb}}}})\subset\mathcal{A}$ is the union of the toric divisors of $\mathcal{A}$, and $U({\widehat{F}_{\textnormal{{mb}}}})$ is the formal completion of $\mathcal{A}$ along $U({F_{\textnormal{{mb}}}})$.

We have an action of $G=v\mathbf{Z}\,\times w\mathbf{Z}\,$, $v,w\in\mathbf{Z}\,_{\geq 0}$, on $L\subset\mathbf{R}^{3}$ by translation:

which induces an automorphism $\psi_{G}:\mathcal{A}\to\mathcal{A}$ and also on $U({\widehat{F}_{\textnormal{{mb}}}})$. These are then the deck transformations of the universal cover of the local multi-Banana configuration $\widehat{F}_{\textnormal{{mb}}}$:

We denote by $\Xi$ the irreducible surface which is the momentum polytope of $\mathbf{P}^{1}\times\mathbf{P}^{1}$ blown up at 2 points, and drawn as a hexagon in our diagrams:

Then the momentum polytope of $U({F_{\textnormal{{mb}}}})$ can be represented as a hexagonal tiling of the plane, and the momentum polytope of ${F_{\textnormal{{mb}}}}$ is given by $v\times w$ hexagons glued together along their toric boundary, as depicted in the example of Figure 2.

The irreducible components of the torus fixed curves in ${F_{\textnormal{{mb}}}}$ fall into three families of rational curves. Two of these families, $\{A_{i}\}$ and $\{B_{j}\}$, are proper transforms of the rational curves from the $I_{v}$ and $I_{w}$ singular fibers in ${F_{\textnormal{{mb}}}}$, respectively, and one family, $\{C_{k}\}$, are the exceptional curves of the conifold resolution.

We will draw these curves oriented as shown in Figure 3, so the vertical curves are in the $A$ family, the horizontal curves are $B$ family, and the diagonal curves are from the $C$ family.

When there is no confusion, we will also label irreducible components of the lifts to the universal cover of these torus invariant curves with the curve class of their projection to ${F_{\textnormal{{mb}}}}$. That is, an irreducible component of $pr^{-1}(A_{i})\subset U({F_{\textnormal{{mb}}}})$ will also be referred to as $A_{i}$ in the universal cover.

As each hexagon surface $\Xi$ is the momentum polytope of $\mathbf{P}^{1}\times\mathbf{P}^{1}$ blown up at 2 points, their 6 boundary divisors have 2 relations. For example, in the labeled Figure 3, we have:

from the equivalent ways to express the total transform of the rulings in each $\mathbf{P}^{1}$ factor.

A priori, there are $3vw$ torus invariant irreducible curves in ${F_{\textnormal{{mb}}}}$, but these satisfy standard hexagon relations (Eq. 2), so it is possible to choose a basis of $vw+2$ curves, consisting of $v\times A$ curves, $w\times B$ curves, and $(v-1)(w-1)+1\times C$ curves. For the small examples we consider, we will index the curves in each family in a simple way. It is possible to use a systematic choice of generators and labels for the general case [5, Section 5.1], but it would be notationally cumbersome for these examples, so we do not present that here.

As remarked in the introduction, our technique is limited to considering only curves where we restrict the degree of one family of curves to be exactly 1. For concreteness, we will assume our curves are of class

The hexagonal tiling from $U({F_{\textnormal{{mb}}}})$ possesses a $v\mathbf{Z}\,\times w\mathbf{Z}\,$ periodicity from the deck transformations. In order to get rid of the ambiguity from the deck transformations, we will assume we have fixed a choice of fundamental domain $D$, and any curve we consider has its unique irreducible component covering $B_{0}$ inside $\overline{D}$. In other words, we will require that $T$-torus invariant curves $\mathcal{C}\subset U({F_{\textnormal{{mb}}}})$ with $[pr(\mathcal{C})]=\beta$ also satisfy $\mathcal{C}\cap pr^{-1}(B_{0})\subset\overline{D}$.

From the arguments given in Subsection 1.4, in order to compute the naive Euler characteristic $\widetilde{n}^{0}_{\beta}(\widehat{F}_{\textnormal{{mb}}})$, it suffices to count all configurations of possibly non-reduced $T$-torus invariant curves covering $\beta$ on the universal cover $U({F_{\textnormal{{mb}}}})$, subject to the constraint that the partition given by multiplicities of successive rational components of each tree emanating from $B_{0}$ has a conjugate partition which has odd parts that are distinct. In section 4 and 5, we will illustrate this count in two specific cases, namely when the fundamental domain consists of $2\times 2$ hexagons, and also the case of $1\times w$ hexagons. These configurations exist, for example, in $X_{\textnormal{{22}}}$, and $X_{\textnormal{{15}}}$, respectively, as described in the previous Section 2.1.

We first fix a choice of a fundamental domain $D$ in $U(F_{\textnormal{{22}}})$ so that there is no ambiguity in our counts due to deck transformations.

We label the curves of the $2\times 2$ hexagons of the momentum polytope of the fundamental domain in $U(F_{\textnormal{{22}}})$ with our convention explained in Section 2.2. The vertical curves cover curves in the $A$ family, the horizontal curves those in the $B$ family, and the diagonal curves cover the $C$ family as shown in Figure 4. There is a $2\mathbf{Z}\,\times 2\mathbf{Z}\,$ periodicity of this fundamental domain in the universal cover.

We can choose a basis for the homology classes of curves of $F_{\textnormal{{22}}}$ given by the 6 curves $A_{0},A_{1},B_{0},B_{1},C_{0}$, and $C_{1}$, as labelled in Figure 4. This can be shown 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, as is the sum in each column,

This follows by combining two hexagon relations to the bottom row of hexagons in Fig. 4. For example,

yields $C_{0}+C_{2}=C_{1}+C_{3}$. The other relation is derived similarly.

We can also write the sum of all the diagonal curves in two ways, grouped as rows or as columns. In this case when $I=J=2$, we have:

Together with the previous Eq. 6, this implies that

In a similar fashion, it is easy to deduce that

We can thus choose a basis for the homology classes of curves given by the 6 curves $A_{0},A_{1},B_{0},B_{1},C_{0},C_{1}$.

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 will assume from now on that the support curve $C$ of our sheaf has $\deg B_{0}=1$ and $\deg B_{1}=0$ so that it is in the homology class:

Recall there is a 1-1 correspondence between the $\pi_{1}(F_{\textnormal{{22}}})$-equivariant sheaves on $U({F_{\textnormal{{mb}}}})$ and the $P$-fixed sheaves on ${F_{\textnormal{{mb}}}}$ up to deck transformations [8, Proposition 22]. To remove the ambiguity, we will further require that the corresponding equivariant sheaf in $U({F_{\textnormal{{mb}}}})$ has support curve $\mathcal{C}\subset U({F_{\textnormal{{mb}}}})$, whose reduced irreducible component covering the curve in class $[B_{0}]$ is in our chosen fundamental domain $D$. In this example, notice that there are two possible choices for $B_{0}$, since $[B_{0}]=[B_{2}]$.

The irreducible components of the curve $\mathcal{C}$ may be nonreduced, so we need to keep track of the multiplicity of each component to determine the curve class of $pr(\mathcal{C})$. We let the variable $r_{i}$ track the number of curves which cover $A_{i}$, $i\in\{0,1\}$, and $s_{j}$ track the number of curves which cover $C_{j}$, $j\in\{0,1\}$. A typical curve $\mathcal{C}$ which covers a curve in class Eq. (7) is pictured in Figure 5.

We explain the details of our method of converting the count of invariant stable sheaves into a combinatorics problem.

Recall from the discussion in Section 1.4 the naive Euler characteristic $\widetilde{n}^{0}_{\beta}(\widehat{F}_{\textnormal{{22}}})=e(M^{\widehat{F}_{\textnormal{{22}}}}_{\beta})$ for curves in class $\beta$ of the form Eq. (7) equals a count of $T$-torus invariant structure sheaves of genus 0 curves on the universal cover $\mathcal{C}\subset U({F_{\textnormal{{mb}}}})$ that cover class $\beta$ [8, Proposition 23], subject to the certain conditions [8, Proposition 31] on their multiplicity that we explain below.

First we introduce some terminology. We will refer to irreducible components of $\mathcal{C}$ as edges, and the intersection of two or more edges as vertices. Let $\mathcal{B}_{0}\subset\mathcal{C}$ be the edge that covers class $[B_{0}]$ and lies in the fundamental domain $D$ by assumption.

We will call the union of $\mathcal{B}_{0}$ with any one of the four disjoint subcurves of $\overline{\mathcal{C}\backslash\mathcal{B}_{0}}$ a branch of $\mathcal{C}$. Then the curve counts can be done on each branch separately, and will be the same on each branch, up to relabeling.

The edge $\mathcal{B}_{0}$ is the intersection of two irreducible surface component hexagons isomorphic to $\Xi$ in $U({F_{\textnormal{{mb}}}})$. Let $S$ be one of these and let $g$ be the deck transformation that translates $S$ into the other. The hexagons $g^{m}S$, $m\in\mathbf{Z}\,$, in the orbit of $S$ under the group of deck transformations $\langle g\rangle\cong\mathbf{Z}\,$ will be called inside hexagons. Any other hexagons will be called outside hexagons.

Any edge of $\overline{\mathcal{C}\backslash\mathcal{B}_{0}}$ covers $A_{i}$ or $C_{j}$, $i,j\in\{0,1\}$, and will be the intersection of an inside hexagon and an outside one. These $T$-invariant edges can can have monomial thickening in these two directions. Any thickenings in the direction of the inside hexagon will be called inside thickenings, and those in the direction of the outside hexagon will be called outside thickenings. However, because of stability and the requirement of the Euler characteristic of $\mathcal{O}_{[\mathcal{C}]}$ 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 following properties [8, Proposition 31]:

1. (1)

   Inside thickenings of any edge that intersects $\mathcal{B}_{0}$ is unrestricted.

2. (2)

   All nonzero outside thickenings must be 1.

3. (3)

   Inside thickenings are non-increasing on components along a branch in the direction moving away from $\mathcal{B}_{0}$.

4. (4)

   Inside thickenings for two adjacent edges contained in a common inside hexagon can either be the same or differ by one.

We can interpret the inside multiplicity of each edge as length of a part in a partition. These constraints are independent on each branch, so we examine one branch at a time. Along each branch, the non-increasing length condition says that the allowed multiplicities of edges form a Young diagram. If we examine the conjugate partition of the branch, the fourth condition can be interpreted as saying that any odd parts that appear in the conjugate partition are distinct, with no restriction on the even parts.

The generating function that counts the number of partitions $p(n)$ with odd parts distinct can be written as the product of generating functions for partitions with arbitrary even parts with that of partitions with unique odd parts:

We must further refine the parts, because we have four curve classes $A_{i}$ and $C_{j}$, for $i,j\in\{0,1\}$ to keep track of. In other words, we need to keep track of 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 $e_{1}$ that intersects $\mathcal{B}_{0}$. Then the every odd-numbered edge will contribute to an odd part, and the even-numbered ones to an even part. The first edge $e_{1}$ in this example curve covers $C_{1}$, and we assign the variable $s_{1}$ to track this curve. The second edge $e_{2}$ covers $A_{0}$ and we use the variable $r_{0}$ to track this.

So we refine the generating function above, and replace powers of the variable $q$, by

and for higher powers of $q$, we continue the pattern, so that