# The Donaldson-Thomas partition function of the Banana Manifold

Jim Bryan

###### Abstract.

A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian
surfaces, whose singular fibers have a singular locus given by a “banana
configuration of curves”. A basic example is given by
$X_{\mathsf{ban}}:=\operatorname{Bl}_{\Delta}(S\times_{{\mathbb{P}}^{1}}S)$,
the blowup along the diagonal of the fibered product of a generic rational
elliptic surface $S\to{\mathbb{P}}^{1}$ with itself.

In this paper we give a closed formula for the Donaldson-Thomas partition
function of the banana manifold $X_{\mathsf{ban}}$ restricted to the
3-dimensional lattice $\Gamma$ of curve classes supported in the fibers of
$X_{\mathsf{ban}}\to{\mathbb{P}}^{1}$. It is given by

$Z^{\,}_{\Gamma}(X_{\mathsf{ban}})=\prod_{d_{1},d_{2},d_{3}\geq
0}\prod_{k}\left(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}\right)^{-12c(||\mathbf{\underline{d}}||,k)}$

where
$||\mathbf{\underline{d}}||=2d_{1}d_{2}+2d_{2}d_{3}+2d_{3}d_{1}-d_{1}^{2}-d_{2}^{2}-d_{3}^{2}$,
and the coefficients $c(a,k)$ have a generating function given by an explicit
ratio of theta functions. This formula has interesting properties and is
closely related to the equivariant elliptic genera of
$\operatorname{Hilb}({\mathbb{C}}^{2})$. In an appendix with S. Pietromonaco,
it is shown that the corresponding genus $g$ Gromov-Witten potential $F_{g}$
is a genus 2 Siegel modular form of weight $2g-2$ for $g\geq 2$, namely it is
the Skoruppa-Maass lift of a multiple of an Eisenstein series, namely
$\frac{6|B_{2g}|}{g(2g-2)!}E_{2g}(\tau)$.

This material is based upon work supported by the National Science Foundation
under Grant No. 1440140, while the author was in residence at the Mathematical
Sciences Research Institute in Berkeley, California, during the Spring
semester 2018. The author was also supported by NSERC.

###### Contents

  1. 1 Introduction
    1. 1.1 Donaldson-Thomas invariants.
    2. 1.2 Banana manifolds and their partition functions
  2. 2 The coefficients $c(a,k)$ and the elliptic genera of $\operatorname{Hilb}({\mathbb{C}}^{2})$.
    1. 2.1 Proof of Proposition 12
  3. 3 Computing the partition function
    1. 3.1 Overview
    2. 3.2 Preliminaries on Notation and Euler Characteristics
    3. 3.3 Pushing forward the $\nu$-weighted Euler characteristic measure.
    4. 3.4 Subschemes supported in an infinitesimal neighborhood of a fiber.
    5. 3.5 Mordell-Weil groups and actions on $\operatorname{Hilb}(\widehat{F}_{y})$.
    6. 3.6 Reduction of the computation to the singular fibers.
    7. 3.7 Reduction to ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$-fixed subschemes.
    8. 3.8 Analysis of ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$-fixed subschemes.
  4. 4 The vertex calculation
    1. 4.1 Overview of computation
    2. 4.2 Notation and Schur function identities
    3. 4.3 The main derivation
  5. 5 Geometry of the Banana manifold
    1. 5.1 Relation to the Schoen threefold and the Hodge numbers
    2. 5.2 Proof of Proposition 24.
  6. 6 BPS invariants from a Donaldson-Thomas partition function in product form.
  7. A The Gromov-Witten potentials are Siegel modular forms (with Stephen Pietromonaco)
    1. A.1 Overview.
    2. A.2 Modular forms and lifts
    3. A.3 The $\lambda$ expansion of $\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t)$.
    4. A.4 Gromov-Witten potentials via the GW/DT correspondence
    5. A.5 Gopakumar-Vafa invariants

##  1\. Introduction

###  1.1. Donaldson-Thomas invariants.

The Donaldson-Thomas invariants of a Calabi-Yau threefold $X$ encode subtle
information about the enumerative geometry of $X$. They are a mathematical
incarnation of counts of BPS states in B-model topological string theory
compactified on $X$.

Let $X$ be a Calabi-Yau threefold, that is a smooth complex threefold with
trivial canonical class. Let $\beta\in H_{2}(X,{\mathbb{Z}})$ be a curve
class, let $n\in{\mathbb{Z}}$, and let

$\operatorname{Hilb}^{\beta,n}(X)=\left\\{Z\subset
X:\quad[Z]=\beta,\,\chi(\mathcal{O}_{Z})=n\right\\}$

be the Hilbert scheme parameterizing dimension one subschemes in the class
$\beta$ and having holomorphic Euler characteristic $n$. The Donaldson-Thomas
invariant $\mathsf{DT}_{\beta,n}(X)$ can be defined [3] as a weighted Euler
characteristic of the Hilbert scheme:

$\mathsf{DT}_{\beta,n}(X)=e\left(\operatorname{Hilb}^{\beta,n}(X),\nu\right):=\sum_{k\in{\mathbb{Z}}}k\cdot
e\left(\nu^{-1}(k)\right)$

where $e(\cdot)$ is topological Euler characteristic and

$\nu:\operatorname{Hilb}^{\beta,n}(X)\to{\mathbb{Z}}$

is Behrend’s constructible function. One can regard $\mathsf{DT}_{\beta,n}(X)$
as a virtual count of the number of curves in the class $\beta$ with Euler
characteristic $n$.

The Donaldson-Thomas partition function is a generating function for the
invariants which we define111Our insertion of a minus sign is somewhat non-
standard. as

$Z^{\,}(X)=\sum_{\beta\in
H_{2}(X,{\mathbb{Z}})}\sum_{n\in{\mathbb{Z}}}\mathsf{DT}_{\beta,n}(X)\,Q^{\beta}(-p)^{n}.$

Here $Q^{\beta}=Q_{1}^{d_{1}}\dotsb Q_{r}^{d_{r}}$ where
$\beta=d_{1}C_{1}+\cdots+d_{r}C_{r}$ and $\\{C_{1},\dots,C_{r}\\}$ is a basis
for $H_{2}(X,{\mathbb{Z}})$ chosen so that $d_{i}\geq 0$ for all effective
curve classes. $Z^{\,}(X)$ is then a formal power series in
$Q_{1},\dots,Q_{r}$ whose coefficients are formal Laurent series in $p$.

The Donaldson-Thomas partition function is very hard to compute:

Currently, there is not a single compact Calabi-Yau threefold $X$ for which
$Z^{\,}(X)$ is completely known, not even conjecturally222Strictly speaking,
we do know (for elementary reasons) that $Z^{\,}(X)=1$ when $X$ is an Abelian
threefold, a product of a $K3$ surface and an elliptic curve, or any other
threefold with a free action by an Abelian variety. There is a modification of
the Donaldson-Thomas invariants which is non-trivial in those cases, and for
Abelian threefolds, we do have a complete, but conjectural, answer for the
partiton function of the modified invariants [12]..

Given some sublattice $\Gamma\subset H_{2}(X,{\mathbb{Z}})$, we can define a
restricted partition function:

$Z^{\,}_{\Gamma}(X)=\sum_{\beta\in\Gamma}\sum_{n\in{\mathbb{Z}}}\mathsf{DT}_{\beta,n}(X)\,Q^{\beta}(-p)^{n}.$

Even for restricted partition functions, there are very few results for
compact $X$. For elliptic or $K3$ fibrations $\pi:X\to B$, the restricted
partition functions $Z^{\,}_{\Gamma}(X)$ have been computed for
$\Gamma=\operatorname{Ker}(\pi_{*}:H_{2}(X)\to H_{2}(B))$, i.e. fiber classes,
by Toda [37, Thm 6.9] in the case of elliptic fibrations and Maulik,
Pandharipande, and Thomas [29, 33] in the case of $K3$ fibrations. In the case
of $K3$ fibrations, the partition functions exhibit modularity properties.

In this paper we compute $Z_{\Gamma}(X)$ where $X$ is a certain kind of
Calabi-Yau threefold (a banana manifold) and $\Gamma$ is a rank 3 lattice. We
give an explicit product formula for $Z_{\Gamma}(X)$, we derive a generating
function for the corresponding Gopakumar-Vafa invariants, and we show
(assuming the GW/DT correspondence) that the associated Gromov-Witten
potentials are Siegel modular forms.

###  1.2. Banana manifolds and their partition functions

We study a class of compact Calabi-Yau threefolds which we call banana
manifolds. The basic example, which we denote $X_{\mathsf{ban}}$, is defined
as follows333 Essentially this construction of the banana manifold is
mentioned briefly in [23, End of section 5.2]; we learned about this from
Georg Oberdieck. .

######  Definition 1.

Let $S\to{\mathbb{P}}^{1}$ be a generic rational elliptic surface. Let

$X_{\mathsf{ban}}=\operatorname{Bl}_{\Delta}\left(S\times_{{\mathbb{P}}^{1}}S\right)$

be the fibered product of $S$ with itself blown up along the diagonal
$\Delta$. See Figure 1.

$n_{j}$$S$$B$$\Delta$ Figure 1. The banana manifold
$X_{\mathsf{ban}}=\operatorname{Bl}(S\times_{{\mathbb{P}}^{1}}S)$.

The map $S\to{\mathbb{P}}^{1}$ is singular at 12 points which gives rise to 12
conifold singularities in the fibered product which all lie on the divisor
$\Delta$. Consequently, $X_{\mathsf{ban}}$ is a conifold resolution of
$S\times_{{\mathbb{P}}^{1}}S$. $X_{\mathsf{ban}}$ is a simply connected,
compact Calabi-Yau threefold with Hodge numbers $h^{1,1}(X_{\mathsf{ban}})=20$
and $h^{1,2}(X_{\mathsf{ban}})=8$ (see § 5).

The generic fiber of $\pi:X_{\mathsf{ban}}\to{\mathbb{P}}^{1}$ is $E\times E$,
the product of an elliptic curve with itself and $\pi$ has 12 singular fibers
which are non-normal toric surfaces, $F_{\mathsf{sing}}$, each of which is a
compactification of ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ by a _banana
configuration_ of ${\mathbb{P}}^{1}$’s.

######  Definition 2.

A _banana configuration_ 444The SYZ mirror of this configuration was studied
by Abouzaid, Auroux, and Katzarkov in [1]. The banana configuration also
appears in the Landau-Ginzburg mirror of a genus two curve as studied by
Gross, Katzarkov, and Ruddat in [18] and Ruddat in [36].  in a Calabi-Yau
threefold $X$ is a union $C=C_{1}\cup C_{2}\cup C_{3}$ of three curves
$C_{i}\cong{\mathbb{P}}^{1}$ with
$N_{C_{i}/X}\cong\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ and such that
$C_{1}\cap C_{2}=C_{2}\cap C_{3}=C_{3}\cap C_{1}=\\{p,q\\}$ where $p,q\in X$
are distinct points. Moreover, there exist coordinates on formal neighborhoods
of $p$ and $q$ such that the curves $C_{i}$ are given by the coordinate axes
in those coordinates. See figure 2; the meaning of the picture on the right is
discussed in section 3.5.

$p$$q$$C_{2}$$C_{1}$$C_{3}$$||$$||$$-$$-$$C_{3}$$C_{2}$$C_{1}$$p$$q$ Figure 2.
The banana configuration of ${\mathbb{P}}^{1}$’s.

The banana curves $C_{1},C_{2},C_{3}$ of any of the singular fibers generate

$\Gamma=\operatorname{Ker}\pi_{*}\subset
H_{2}(X_{\mathsf{ban}},{\mathbb{Z}}),$

the lattice of the fiber classes (Lemma 29). Let

$\beta_{\mathbf{\underline{d}}}=d_{1}C_{1}+d_{2}C_{2}+d_{3}C_{3}$

and define

$||\mathbf{\underline{d}}||=2d_{1}d_{2}+2d_{2}d_{3}+2d_{3}d_{1}-d_{1}^{2}-d_{2}^{2}-d_{3}^{2}.$

This quadratic form is twice the intersection form of a smoth fiber, by which
we mean that if $\beta_{\mathbf{\underline{d}}}$ is represented by a cycle $C$
supported on a smooth fiber, then $||\mathbf{\underline{d}}||=2C\cdot C$ where
$C\cdot C$ is the self-intersection of $C$ in the smooth fiber.

We define banana manifolds in general as follows.

######  Definition 3.

We say that a compact Calabi-Yau threefold $X$ is a _banana manifold_ with $N$
banana fibers if there is an Abelian surface fibration
$\pi:X\to{\mathbb{P}}^{1}$ where the singular locus of $\pi$ consists of $N$
disjoint banana configurations and the smooth locus of $\pi$ is an Abelian
group scheme over ${\mathbb{P}}^{1}$ whose natural action on itself extends to
an action on $X$ (see [13] for examples of rigid Banana manifolds).

Our main theorem is the following:

######  Theorem 4.

Let $X_{\mathsf{ban}}$ be the basic banana manifold and let
$\Gamma=\operatorname{Ker}\pi_{*}\cong{\mathbb{Z}}^{3}$ where
$\pi:X_{\mathsf{ban}}\to{\mathbb{P}}^{1}$. Then the Donaldson-Thomas partition
function of $X_{\mathsf{ban}}$, restricted to $\Gamma$ is given by

$Z^{\,}_{\Gamma}(X_{\mathsf{ban}})=\prod_{d_{1},d_{2},d_{3}\geq
0}\prod_{k}\left(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}\right)^{-12c(||\mathbf{\underline{d}}||,k)}$

where the second product is over all $k\in{\mathbb{Z}}$ unless
$(d_{1},d_{2},d_{3})=(0,0,0)$ in which case $k>0$, and where the
$c(||\mathbf{\underline{d}}||,k)$ are positive integers given by

$\sum_{a=-1}^{\infty}\,\,\sum_{k\in{\mathbb{Z}}}c(a,k)Q^{a}y^{k}=\frac{\sum_{k\in{\mathbb{Z}}}Q^{k^{2}}(-y)^{k}}{\left(\sum_{k\in{\mathbb{Z}}+\frac{1}{2}}Q^{2k^{2}}(-y)^{k}\right)^{2}}=\frac{\vartheta_{4}(2\tau,z)}{\vartheta_{1}(4\tau,z)^{2}}$

where $\vartheta_{4}$ and $\vartheta_{1}$ are the usual theta
functions555Following the conventions from Wolfram Alpha’s Jacobi Theta
Function page. with $Q=e^{2\pi i\tau}$ and $y=e^{2\pi iz}$.

######  Remark 5.

While we have formulated the Theorem for the basic banana manifold, our proof
will work for banana manifolds in general, where the 12 in the exponent on the
right hand side will be replaced by $N$, the number of banana configurations.

######  Remark 6.

The coefficients $c(a,k)$ are also the coefficients of the equivariant
elliptic genus of the plane ${\mathbb{C}}^{2}$, see § 2. We also note that the
right hand side of the above equation, is a meromorphic Jacobi form of weight
$-\frac{1}{2}$ and index -1 for the group $\Gamma(4)$.

######  Corollary 7 (Pietromonaco).

Assuming the Gromov-Witten/Donaldson-Thomas correspondence holds for a banana
manifold $X$, the genus $g$ Gromov-Witten potential function
$F_{g}(Q_{1},Q_{2},Q_{3})$ is a meromorphic Siegel modular form of weight
$2g-2$ for all $g\geq 2$ where $Q_{1}=e^{2\pi iz}$, $Q_{2}=e^{2\pi
i(\tau-z)}$, $Q_{3}=e^{2\pi i(\sigma-z)}$, and where
$\left(\begin{smallmatrix}\tau&z\\\
z&\sigma\end{smallmatrix}\right)\in\mathbb{H}_{2}$ is in the genus 2 Siegel
upper half plane. Namely, $F_{g}$ is given by the Skoruppa-Maass lift of
$a_{g}E_{2g}(\tau)$, the $2g$-th Eisenstein series multiplied by the constant
$a_{g}=\frac{6|B_{2g}|}{g(2g-2)!}$. See Appendix A for full definitions of the
terms.

This corollary has a natural interpretation in terms of mirror symmetry, see
Remark 35.

Our Theorem 4 also completely determines the Gopakumar-Vafa invariants of the
banana manifold. One corollary is:

######  Corollary 8.

The Gopakumar-Vafa invariants of $X_{\mathsf{ban}}$ in the class
$\beta_{\mathbf{\underline{d}}}\,$ only depend on
$||\mathbf{\underline{d}}||$. We thus streamline the notation by writing

$n^{g}_{\beta_{\mathbf{\underline{d}}}}(X_{\mathsf{ban}})=n^{g}_{a}(X_{\mathsf{ban}})\quad\text{where
$a=||\mathbf{\underline{d}}||$.}$

The property that $n^{g}_{\beta_{\mathbf{\underline{d}}}}(X_{\mathsf{ban}})$
only depends on $||\mathbf{\underline{d}}||$ and so in particular the
invariants are independent of the divisibility of
$\beta_{\mathbf{\underline{d}}}$, is an unusual property of $X_{\mathsf{ban}}$
which is also shared by the local $K3$ surface by a deep result of
Pandharipande-Thomas [33].

We can reformulate our main result in terms of the Gopakumar-Vafa invariants.
After some manipulation of generating functions (see § 2 and § 6) we can
deduce:

######  Theorem 9.

The Gopakumar-Vafa invariants
$n^{g}_{\beta_{\mathbf{\underline{d}}}}(X_{\mathsf{ban}})=n^{g}_{a}(X_{\mathsf{ban}})$
in the classes $\beta_{\mathbf{\underline{d}}}\,$ with
$||\mathbf{\underline{d}}||=a$ are given by

$\sum_{a=-1}^{\infty}\sum_{g\geq
0}n_{a}^{g}(X_{\mathsf{ban}})\left(y^{\frac{1}{2}}+y^{-\frac{1}{2}}\right)^{2g}Q^{a+1}=12\prod_{n=1}^{\infty}\frac{(1+yQ^{2n-1})(1+y^{-1}Q^{2n-1})(1-Q^{2n})}{(1+yQ^{4n})^{2}(1+y^{-1}Q^{4n})^{2}(1-Q^{4n})^{2}}.$

It is interesting to compare the above formula to the analogous result for the
local $K3$ surface, namely the Katz-Klemm-Vafa formula:

######  Theorem 10 (Pandharipande-Thomas[33]).

The Gopakumar-Vafa invariants $n^{g}_{\beta}(K3)=n^{g}_{a}(K3)$ in the class
$\beta$ with $\beta^{2}/2=a$ are given by

$\sum_{a=-1}^{\infty}\sum_{g\geq
0}n_{a}^{g}(K3)\left(y^{\frac{1}{2}}+y^{-\frac{1}{2}}\right)^{2g}Q^{a+1}=\prod_{n=1}^{\infty}\frac{1}{(1+yQ^{n})^{2}(1+y^{-1}Q^{n})^{2}(1-Q^{n})^{20}}.$

Note that the right hand side of the above equation, after the substitution
$Q=-yq$, is equal to $\sum_{n}\chi_{y}(\operatorname{Hilb}^{n}(K3))q^{n}$, the
generating function for the $\chi_{y}$-genus of the Hilbert schemes of points
on $K3$. It would be interesting to find an analogous interpretation of the
right hand side of the equation in Theorem 9.

See § A for a table of $n_{a}^{g}(X_{\mathsf{ban}})$ for small values of $a$
and $g$.

##  2\. The coefficients $c(a,k)$ and the elliptic genera of
$\operatorname{Hilb}({\mathbb{C}}^{2})$.

Let $M$ be a compact complex manifold of dimension $d$ and let
$x_{1},\dotsc,x_{d}$ be the Chern roots of $TM$. Then the elliptic genus is
defined by

$\operatorname{Ell}_{q,y}(M)=\int_{M}\,\,\prod_{j=1}^{d}x_{j}y^{-\frac{1}{2}}\,\,\prod_{n=1}^{\infty}\frac{\left(1-ye^{-x_{j}}q^{n-1}\right)\left(1-y^{-1}e^{x_{j}}q^{n}\right)}{\left(1-e^{-x_{j}}q^{n-1}\right)\left(1-e^{x_{j}}q^{n}\right)}.$

If $M$ has a ${\mathbb{C}}^{*}$ action with isolated fixed points, then by
Atiyah-Bott localization we get

(1) $\operatorname{Ell}_{q,y}(M)=\sum_{p\in
M^{{\mathbb{C}}^{*}}}\,\,\prod_{j=1}^{d}y^{-\frac{1}{2}}\prod_{n=1}^{\infty}\frac{\left(1-yt^{-k_{j}(p)}q^{n-1}\right)\left(1-y^{-1}t^{k_{j}(p)}q^{n}\right)}{\left(1-t^{-k_{j}(p)}q^{n-1}\right)\left(1-t^{k_{j}(p)}q^{n}\right)}$

where $k_{1}(p),\dotsc,k_{d}(p)\in{\mathbb{Z}}$ are the weights of the
${\mathbb{C}}^{*}$ action on $T_{p}M$.

If $M$ is non-compact, we take equation (1) to be the definition of the
elliptic genus $\operatorname{Ell}_{q,y}(M,t)$ which then may depend on $t$,
the equivariant parameter666The parameter $t\in H^{*}_{{\mathbb{C}}^{*}}(pt)$
is the Chern character of the universal line bundle so $t=e^{c_{1}}$ where
$c_{1}\in H^{2}_{{\mathbb{C}}^{*}}(pt)$ is an integral generator. .

It is convenient to rewrite this expression in terms of the Fourier expansion
of the theta function $\theta_{1}$. Let $q=\exp\left(2\pi i\tau\right)$ and
$y=\exp\left(2\pi iz\right)$, then $\theta_{1}(q,y)$ is given by

$\displaystyle\theta_{1}(q,y)=$
$\displaystyle-\sum_{k\in{\mathbb{Z}}+\frac{1}{2}}q^{\frac{k^{2}}{2}}(-y)^{k}$
$\displaystyle=$ $\displaystyle-
iq^{\frac{1}{8}}y^{-\frac{1}{2}}\prod_{n=1}^{\infty}(1-q^{n})(1-yq^{n-1})(1-y^{-1}q^{n}).$

Then equation (1) becomes

$\operatorname{Ell}_{q,y}(M,t)=\sum_{p\in
M^{{\mathbb{C}}^{*}}}\prod_{j=1}^{d}\frac{\theta_{1}(q,yt^{-k_{j}(p)})}{\theta_{1}(q,t^{-k_{j}(p)})}$

For example, if we let ${\mathbb{C}}^{*}$ act on ${\mathbb{C}}^{2}$ with
weights $\pm 1$, then the induced action of ${\mathbb{C}}^{*}$ on
$\operatorname{Hilb}^{m}({\mathbb{C}}^{2})$ has isolated fixed points
corresponding to monomial ideals which are in bijective correspondence with
integer partitions of $m$. The tangent weights associated to a partition $R$
are given by $\\{\pm h_{j,k}\\}$ where $h_{j,k}=h_{j,k}(R)$ is the hook length
of the box in position $(j,k)$ in the diagram of $R$ (see § 4.2 for this
notation). Thus

(2)
$\sum_{m=0}^{\infty}\operatorname{Ell}_{q,y}(\operatorname{Hilb}^{m}({\mathbb{C}}^{2}),t)Q^{m}=\sum_{R}Q^{|R|}\prod_{(j,k)\in
R}\frac{\theta_{1}(q,yt^{-h_{j,k}})\theta_{1}(q,yt^{h_{j,k}})}{\theta_{1}(q,t^{-h_{j,k}})\theta_{1}(q,t^{h_{j,k}})}.$

Dijkgraaf-Moore-Verlinde-Verlinde conjectured [15] that

$\sum_{m=0}^{\infty}\operatorname{Ell}_{q,y}(\operatorname{Hilb}^{m}({\mathbb{C}}^{2}),t)\,Q^{m}=\sum_{m=0}^{\infty}\operatorname{Ell}^{orb}_{q,y}(\operatorname{Sym}^{m}({\mathbb{C}}^{2}),t)\,Q^{m}$

where
$\operatorname{Ell}^{orb}_{q,y}(\operatorname{Sym}^{m}({\mathbb{C}}^{2}),t)$
is the orbifold elliptic genera of $\operatorname{Sym}^{m}({\mathbb{C}}^{2})$.
This is a special case of the crepant resolution conjecture for equivariant
elliptic genera which was proven by Waelder [38, Thm 12] based on the non-
equivariant case proven by Borisov-Libgobner [7]. The right hand side of the
DMVV conjecture is easy to compute; consequently, Waelder’s result leads to
the following formula:

######  Theorem 11 (Waelder).

$\sum_{m=0}^{\infty}\operatorname{Ell}_{q,y}(\operatorname{Hilb}^{m}({\mathbb{C}}^{2}),t)Q^{m}=\prod_{m=1}^{\infty}\prod_{n=0}^{\infty}\prod_{l,k\in{\mathbb{Z}}}\left(1-t^{k}q^{n}y^{l}Q^{m}\right)^{-c(nm,l,k)}$

where the coefficients $c(n,l,k)$ are defined by

$\displaystyle\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t)$
$\displaystyle=\frac{\theta_{1}(q,yt)\theta_{1}(q,yt^{-1})}{\theta_{1}(q,t)\theta_{1}(q,t^{-1})}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k,l\in{\mathbb{Z}}}c(n,l,k)q^{n}y^{l}t^{k}.$

We prove the following result about the coefficients $c(n,l,k)$.

######  Proposition 12.

Let

$\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t)=\sum_{n=0}^{\infty}\sum_{l,k\in{\mathbb{Z}}}c(n,l,k)q^{n}y^{l}t^{k}.$

Then $c(n,l,k)$ only depends on the pair $(4n-l^{2},k)$. Writing

$c(n,l,k)=c(4n-l^{2},k)$

we have $c(a,k)=0$ if $a<-1$ and

$\sum_{a=-1}^{\infty}\sum_{k\in{\mathbb{Z}}}c(a,k)Q^{a}t^{k}=\frac{\sum_{k\in{\mathbb{Z}}}Q^{k^{2}}(-t)^{k}}{\left(\sum_{k\in{\mathbb{Z}}+\frac{1}{2}}Q^{2k^{2}}(-t)^{k}\right)^{2}}=\frac{\theta_{4}(Q^{2},t)}{\theta_{1}(Q^{4},t)^{2}}.$

Examining the first two terms in the $Q$ expansion we get the following easy
corollary:

######  Corollary 13.

The coefficients $c(a,k)$ for $a=-1,0$ are given by

$c(-1,k)=\begin{cases}0,&-k\leq 0\\\ -k,&k>0\end{cases}\quad\quad
c(0,k)=\begin{cases}0,&k<0\\\ 1,&k=0\\\ 2k,&k>0\end{cases}$

Applying the Jacobi triple product identity, we also get the following
corollary.

######  Corollary 14.

$\sum_{a=-1}^{\infty}\sum_{k\in{\mathbb{Z}}}c(a,k)Q^{a+1}t^{k}=\frac{-t}{(1-t)^{2}}\prod_{n=1}^{\infty}\frac{(1-Q^{2n})(1-tQ^{2n-1})(1-t^{-1}Q^{2n-1})}{(1-Q^{4n})^{2}(1-tQ^{4n})^{2}(1-t^{-1}Q^{4n})^{2}}$

###  2.1. Proof of Proposition 12

From the product formula for $\theta_{1}(q,t)$ and the fact that
$\theta_{1}(q,t^{-1})=-\theta_{1}(q,t)$, we see we may write

$\theta_{1}(q,t)^{-1}\theta_{1}(q,t^{-1})^{-1}=-\theta_{1}(q,t)^{-2}=q^{-\frac{1}{4}}\frac{t}{(1-t)^{2}}\sum_{i=0}^{\infty}\delta_{i}(t)q^{i}$

where $\delta_{0}=1$ and $\delta_{i}(t)\in{\mathbb{Z}}[t,t^{-1}]$. Then

$\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t)=q^{-\frac{1}{4}}\frac{t}{(1-t)^{2}}\sum_{i=0}^{\infty}\delta_{i}(t)q^{i}\sum_{n,m\in{\mathbb{Z}}}\,\,q^{\frac{1}{2}(n+\frac{1}{2})^{2}+\frac{1}{2}(m+\frac{1}{2})^{2}}(-yt)^{n+\frac{1}{2}}(-yt^{-1})^{m+\frac{1}{2}}.$

If we let $l=m+n+1$ and $b=n-m$ and we note that then $l\equiv b+1$ mod 2 and
$\frac{1}{2}(n+\frac{1}{2})^{2}+\frac{1}{2}(m+\frac{1}{2})^{2}=\frac{1}{4}(l^{2}+b^{2})$
we get

$\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t)=\frac{-t}{(1-t)^{2}}\sum_{i=0}^{\infty}\delta_{i}(t)q^{i}\sum_{\begin{smallmatrix}l,b\in{\mathbb{Z}}\\\
l\equiv b+1\mod
2\end{smallmatrix}}q^{\frac{1}{4}(l^{2}+b^{2}-1)}y^{l}(-t)^{b}$

The terms with $q^{n}y^{l}$ in the above sum occur when
$n=i+\frac{1}{4}(l^{2}+b^{2}-1)$ and thus when $4n-l^{2}=4i+b^{2}-1\geq-1$.
Therefore the $q^{n}y^{l}$ coefficient of
$\operatorname{Ell}_{(q,y)}({\mathbb{C}}^{2},t)$ only depends on $a=4n-l^{2}$
and is zero if $a<-1$:

$\displaystyle\operatorname{Coef}_{q^{n}y^{l}}\left[\operatorname{Ell}_{(q,y)}({\mathbb{C}}^{2},t)\right]$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}c(n,l,k)t^{k}$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}c(4n-l^{2},k)t^{k}$

where

$\displaystyle\sum_{a=-1}^{\infty}\sum_{k\in{\mathbb{Z}}}c(a,k)Q^{a}t^{k}$
$\displaystyle=\frac{-t}{(1-t)^{2}}\sum_{i=0}^{\infty}\sum_{b\in{\mathbb{Z}}}Q^{4i+b^{2}-1}(-t)^{b}\delta_{i}(t)$
$\displaystyle=\frac{-tQ^{-1}}{(1-t)^{2}}\sum_{i=0}^{\infty}\delta_{i}(t)Q^{4i}\sum_{b\in{\mathbb{Z}}}Q^{b^{2}}(-t)^{b}$
$\displaystyle=\theta_{1}(Q^{4},t)^{-2}\sum_{b\in{\mathbb{Z}}}Q^{b^{2}}(-t)^{b}$
$\displaystyle=\frac{\sum_{k\in{\mathbb{Z}}}Q^{k^{2}}(-t)^{k}}{\left(\sum_{k\in{\mathbb{Z}}+\frac{1}{2}}Q^{2k^{2}}(-t)^{k}\right)^{2}}\quad.$

∎

##  3\. Computing the partition function

###  3.1. Overview

Our basic strategy for computing the partition function
$Z^{\,}_{\Gamma}(X_{\mathsf{ban}})$ is to stratify the Hilbert scheme, where
each strata parameterizes subschemes supported on a prescribed set of fibers
of the map $\pi:X\to{\mathbb{P}}^{1}$. Each stratum admits an action by the
Mordell-Weil group of sections on an infinitesimal neighborhood of a fiber.

We use these group actions to reduce the $\nu$-weighted Euler characteristic
computation to the fixed point set of the group actions. The fixed points
correspond to subschemes which are supported on an infinitesimal neighborhood
of the banana configurations and which are formally locally given by monomial
ideals. We can count these fixed points (weighted by $\nu$) using a technique
adapted from [28] to our setting. The outcome is an expression for the
partition function in terms of the topological vertex.

The vertex expression we get has essentially been computed in the physics
literature. The bulk of the computation is done by Hollowood, Iqbal, and Vafa
[19] using geometric engineering. Their derivation requires a certain
geometrically motivated combinatorial conjecture. The conjecture was a special
case of the Equivariant Crepant Resolution Conjecture for Elliptic Genera
(also called the equivariant DMVV conjecture after Dijkgraaf-Moore-Verlinde-
Verlinde [15]), which was subsequently proven by Borisov-Libgobner and Waelder
[7, 38] using motivic integration. We present a mathematically self-contained
version of the Hollowood-Iqbal-Vafa derivation in § 4.

###  3.2. Preliminaries on Notation and Euler Characteristics

For any scheme $Y$ over ${\mathbb{C}}$, let $e(Y)$ be the topological Euler
characteristic of $Y$ in the complex analytic topology. Note that Euler
characteristic is independent of any nilpotent structures, i.e.
$e(Y)=e(Y_{red})$.

For any constructible function $\mu:Y\to{\mathbb{Z}}$, let

$e(Y,\mu)=\sum_{k\in{\mathbb{Z}}}k\cdot e(\mu^{-1}(k))$

be the $\mu$-weighted Euler characteristic of $Y$.

We will need the following standard facts about Euler characterisitics.

  * •

Euler characteristic defines a ring homomorphism
$e:K_{0}(\operatorname{Var}_{{\mathbb{C}}})\to{\mathbb{Z}}$, i.e. it is
additive under the decomposition of a scheme into an open set and its
complement, and it is multiplicative on Cartesian products.

  * •

For any constructible morphism777A constructible morphism is a map which is
regular on each piece of a decomposition of its domain into locally closed
subsets. $f:Y\to Z$ we have (see [27])

(3) $e(Y,\mu)=e(Z,f_{*}\mu)$

where $f_{*}\mu$ is the constructible function given by

$(f_{*}\mu)(x)=e(f^{-1}(x),\mu).$

  * •

If ${\mathbb{C}}^{*}$ acts on a scheme $Y$ with fixed point locus
$Y^{{\mathbb{C}}^{*}}\subset Y$, then (see [5])

$e(Y)=e(Y^{{\mathbb{C}}^{*}}).$

We will also use the following

######  Lemma 15.

Let $G$ be an algebraic group acting on $V$, a scheme (of finite type over
${\mathbb{C}}$). Let $\mu$ be a $G$-invariant constructible function on $V$.
Suppose that each $G$-orbit has zero Euler characteristic, i.e. $e(O_{x})=0$
for all $x\in V$. Then $e(V,\mu)=0$.

###### Proof.

By a theorem of Rosenlicht [35, Thm 2], we have that for the action of any
algebraic group $G$ on any variety $V$, there is a dense open set $U\subset V$
and a morphism $\tau:U\to W$ to a variety $W$, such that for all $x\in U$,
$\tau^{-1}(x)$ is a $G$-orbit. By iteratively applying the same theorem to
$V\setminus U$, we obtain a locally closed $G$-equivariant stratification
$V=\cup_{\alpha}U_{\alpha}$ such that the $G$-action on each strata has a
geometric quotient

$\tau_{\alpha}:U_{\alpha}\to W_{\alpha}\,\,\,.$

Suppose that every $G$-orbit has zero Euler characterisic, $e(O_{x})=0$. Let
$\mu$ be a $G$-invariant constructible function on $V$, then

$\displaystyle e(V,\mu)$ $\displaystyle=\sum_{\alpha}e(U_{\alpha},\mu)$
$\displaystyle=\sum_{\alpha}e(W_{\alpha},(\tau_{\alpha})_{*}(\mu))$
$\displaystyle=\sum_{\alpha}e(W_{\alpha},0)$ $\displaystyle=0.$

This proves the lemma in the case where $V$ is a variety. For $V$ a general
scheme of finite type over ${\mathbb{C}}$, we can easily construct a
$G$-equivariant stratification of the reduced space of $V$

$V_{red}=V_{1}\cup\dotsb\cup V_{N}$

such that each strata $V_{i}$ is a variety. Then

$e(V,\mu)=e(V_{red},\mu)=\sum_{i=1}^{N}e(V_{i},\mu)=0.$

∎

We will use the notation

$\operatorname{Hilb}^{\bullet}(X_{\mathsf{ban}})=\sum_{\mathbf{\underline{d}},n}\operatorname{Hilb}^{\beta_{\mathbf{\underline{d}}}\,\,,n}(X_{\mathsf{ban}})\,Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}(-p)^{n}$

where we regard the right hand side as a formal series in $Q_{1},Q_{2},Q_{3}$,
Laurent in $p$, having coefficients in
$K_{0}(\operatorname{Var}_{{\mathbb{C}}})$, the Grothendieck group of
varieties888We note that $\operatorname{Hilb}^{0,0}(X_{\mathsf{ban}}$) is a
single point corresponding to the empty subscheme. Thus the constant term of
the series $\operatorname{Hilb}^{\bullet}(X_{\mathsf{ban}})$ is 1.. We extend
the operations in the Grothendieck group (addition, multiplication, and Euler
characteristic) to the series in the obvious way. So for example, in this
notation, the partition function is given by:

$Z^{\,}_{\Gamma}(X_{\mathsf{ban}})=e\left(\operatorname{Hilb}^{\bullet}(X_{\mathsf{ban}}),\nu\right).$

We will apply the above notation more generally. Whenever we have any
collection $\\{A^{\mathbf{\underline{d}},n}\\}$ elements of a set (for example
${\mathbb{Z}}$ or $K_{0}(\operatorname{Var}_{{\mathbb{C}}})$) indexed by
$(\mathbf{\underline{d}},n)$, we will write:

$A^{\bullet}=\sum_{\mathbf{\underline{d}},n}A^{\mathbf{\underline{d}},n}\,Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}(-p)^{n}.$

###  3.3. Pushing forward the $\nu$-weighted Euler characteristic measure.

Let

$\operatorname{Conf}^{k}{\mathbb{P}}^{1}=\operatorname{Sym}^{k}{\mathbb{P}}^{1}\setminus\Delta$

be the configuration space of $k$ unordered, distinct points in
${\mathbb{P}}^{1}$, i.e. the $k$th symmetric product of ${\mathbb{P}}^{1}$
with the big diagonal deleted. Let

$\operatorname{Conf}{\mathbb{P}}^{1}=\bigcup_{k}\operatorname{Conf}^{k}{\mathbb{P}}^{1}.$

We define a constructible morphism by

$\displaystyle\rho^{\mathbf{\underline{d}},n}:\operatorname{Hilb}^{\beta_{\mathbf{\underline{d}}}\,,n}(X_{\mathsf{ban}})$
$\displaystyle\to\operatorname{Conf}{\mathbb{P}}^{1}$ $\displaystyle Z$
$\displaystyle\mapsto\operatorname{Supp}(\pi_{*}\mathcal{O}_{Z})$

Then using equation (3) and our bullet notation, we have

$\displaystyle Z_{\Gamma}^{\,}(X_{\mathsf{ban}})$
$\displaystyle=e\left(\operatorname{Hilb}^{\bullet}(X_{\mathsf{ban}}),\nu\right)$
$\displaystyle=e\left(\operatorname{Conf}{\mathbb{P}}^{1},\rho^{\bullet}_{*}\nu\right).$

Note that the constructible function $\rho^{\bullet}_{*}\nu$ takes values in
${\mathbb{Z}}[[Q_{1},Q_{2},Q_{3}]]((p)).$

###  3.4. Subschemes supported in an infinitesimal neighborhood of a fiber.

From here on out, we will write $X$ for $X_{\mathsf{ban}}$ for the sake of
brevity when there is no ambiguity. We will also suppress the superscripts
$\beta,n$ when they are understood and/or unimportant.

######  Definition 16.

Let $F_{y}$ be the fiber of $\pi:X\to{\mathbb{P}}^{1}$ over
$y\in{\mathbb{P}}^{1}$. Let $F_{y}^{(k)}$ be the $k$-order infinitesimal
neighborhood of $F_{y}$ in $X$ and let $\widehat{F}_{y}$ be the formal
neighborhood of $F_{y}$ in $X$. We define

$\operatorname{Hilb}(\widehat{F}_{y})\subset\operatorname{Hilb}(X)$

to be the locally closed subscheme of $\operatorname{Hilb}(X)$ parameterizing
subschemes $Z\subset X$ supported on $\widehat{F}_{y}$, i.e. subschemes which
are set theoretically, but necessarily scheme theoretically, supported in
$F_{y}$. Finally, we define

$\widehat{\operatorname{Hilb}}(\widehat{F}_{y})$

to be the formal neighborhood of $\operatorname{Hilb}(\widehat{F}_{y})$ in
$\operatorname{Hilb}(X)$.

The value of the Behrend function $\nu:\operatorname{Hilb}(X)\to{\mathbb{Z}}$
at a point only depends on the formal neighborhood of that point [22] inside
the Hilbert scheme. Therefore $\nu$ restricted to
$\operatorname{Hilb}(\widehat{F}_{y})$ is completely determined by the formal
scheme $\widehat{\operatorname{Hilb}}(\widehat{F}_{y})$.

######  Remark 17.

The closed points of $\operatorname{Hilb}(\widehat{F}_{y})$ and
$\widehat{\operatorname{Hilb}}(\widehat{F}_{y})$ are of course the same; they
correspond to subschemes $Z\subset X$ such that $Z_{red}\subset F_{y}$.
However, $\operatorname{Hilb}(\widehat{F}_{y})$ and
$\widehat{\operatorname{Hilb}}(\widehat{F}_{y})$ classify different families
of subschemes. For example, the infinitesimal deformations of $Z$
parameterized by $\operatorname{Hilb}(\widehat{F}_{y})$ must preserve the
closed condition $Z_{red}\subset F_{y}$, whereas
$\widehat{\operatorname{Hilb}}(\widehat{F}_{y})$ includes _all_ infinitesimal
deformations of $Z$, even those which (infinitesimally) violate the condition
$Z_{red}\subset X$.

###  3.5. Mordell-Weil groups and actions on
$\operatorname{Hilb}(\widehat{F}_{y})$.

Let

$X^{\circ}=\left\\{x\in X\text{ such that $\pi:X\to{\mathbb{P}}^{1}$ is smooth
at $x$}\right\\}.$

In other words, $X^{\circ}$ is $X$ with the 12 banana configurations removed.
Then, after fixing a section $s_{0}:{\mathbb{P}}^{1}\to X^{\circ}$,

$\pi^{\circ}:X^{\circ}\to{\mathbb{P}}^{1}$

has the structure of an Abelian group scheme over ${\mathbb{P}}^{1}$. Let
$F_{y}^{\circ}=X^{\circ}\cap F_{y}$ be the group of the fiber over $y$. Let

$\\{x_{1},\dotsc,x_{12}\\}\subset{\mathbb{P}}^{1}$

be the points with singular fiber. Then for $y\not\in\\{x_{1},\dotsc
x_{12}\\}$, $F_{y}^{\circ}\cong E\times E$, where $E$ is an elliptic curve,
and for $y\in\\{x_{1},\dotsc,x_{12}\\}$,
$F_{y}^{\circ}\cong{\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$.

######  Definition 18.

Let $\pi^{\circ}:X_{B}^{\circ}\to B$ and $\pi:X_{B}\to B$ denote the
$B$-schemes obtained from $X^{\circ}$ and $X$ by some base change
$B\to{\mathbb{P}}^{1}$. We define $\mathsf{MW}(B)$ be the Mordell-Weil group
of sections of $\pi^{0}$ over $B$.

If $B\to\operatorname{Spec}{\mathbb{C}}$ is finite, then $\mathsf{MW}(B)$ is
the Weil restriction of $X^{\circ}_{B}\to B$ with respect to
$B\to\operatorname{Spec}{\mathbb{C}}$ and is thus an algebraic group over
${\mathbb{C}}$, see [8, §7 Thm 4].

We get an action of $\mathsf{MW}(B)$ on $X_{B}$ defined as follows (c.f. [8,
§7 Thm 6]). The group scheme structure morphism
$X^{\circ}_{B}\times_{B}X^{\circ}_{B}\to X^{\circ}_{B}$ extends (see [14]) to
a morphism

${X_{B}^{\circ}\times_{B}X_{B}}$${X_{B}.}$$\scriptstyle{+}$

We define the $\mathsf{MW}(B)$ action on $X_{B}$ by the composition

${\mathsf{MW}(B)\times
X_{B}}$${X_{B}^{\circ}\times_{B}X_{B}}$${X_{B}}$$\scriptstyle{\phi}$$\scriptstyle{+}$

where $\phi={(ev\circ(Id\times\pi),pr_{X_{B}})}$,

$ev:\mathsf{MW}(B)\times B\to X_{B}$

is the tautological evaluation map $(s,y)\mapsto s(y)$, and

$pr_{X_{B}}:\mathsf{MW}(B)\times X_{B}\to X_{B}$

is projection. $\phi$ is well defined because $\pi\circ
ev\circ(Id\times\pi)=\pi\circ pr_{X_{B}}$.

Concretely, if $x\in X_{B}$ is a point then the action of $s$ on $x$ is given
by translation: $(s,x)\mapsto s(\pi(x))+x$.

Let $\Delta_{y}^{(k)}\subset{\mathbb{P}}^{1}$ be the $k$th order thickening of
$y\in{\mathbb{P}}^{1}$. Since

$\Delta_{y}^{(k)}\cong\operatorname{Spec}{\mathbb{C}}[\epsilon]/\epsilon^{k+1}\to\operatorname{Spec}{\mathbb{C}}$

is finite, $\mathsf{MW}(\Delta_{y}^{(k)})$ is an algebraic group. Restriction
of a section to the closed fiber expresses $\mathsf{MW}(\Delta_{y}^{(k)})$ as
an extension of $F^{\circ}_{y}$ by a vector group of some dimension $D=D(k)$:

(4)
${0}$${{\mathbb{C}}^{D}}$${\mathsf{MW}(\Delta_{y}^{(k)})}$${F^{\circ}_{y}}$${0}$$\scriptstyle{r}$

We write $F_{k}^{(k)}$ for $X_{\Delta_{y}^{(k)}}$, i.e. the $k$th order
thickening of the fiber $F_{y}$. The action of $\mathsf{MW}(\Delta_{y}^{(k)})$
on $F_{y}^{(k)}$ is compatible with the restriction homomorphisms

$\mathsf{MW}(\Delta^{(k+1)}_{y})\to\mathsf{MW}(\Delta^{(k)}_{y})$

and the inclusions

$F_{y}^{(k)}\subset F_{y}^{(k+1)}.$

Let

$\mathsf{MW}^{(\infty)}_{y}=\lim_{\stackrel{{\scriptstyle\leftarrow}}{{k}}}\mathsf{MW}(\Delta^{(k)}_{y})$

be the inverse limit group. Then by construction, $\mathsf{MW}^{(\infty)}_{y}$
acts on $\widehat{F}_{y}$, and this induces an action of
$\mathsf{MW}^{(\infty)}_{y}$ on $\operatorname{Hilb}(\widehat{F}_{y})$ and on
$\widehat{\operatorname{Hilb}}(\widehat{F}_{y})$. Note that

$\operatorname{Hilb}^{\beta,n}(\widehat{F}_{y})\subset\operatorname{Hilb}^{\beta,n}(F_{y}^{(N)})$

for some large $N=N(\beta,n)$ which depends on $\beta$ and $n$. Therefore,
$\operatorname{Hilb}^{\beta,n}(\widehat{F}_{y})$ is acted on by the algebraic
group $\mathsf{MW}(\Delta^{(N)}_{y})$. However,
$\mathsf{MW}(\Delta^{(N)}_{y})$ does not act on
$\widehat{\operatorname{Hilb}}^{\beta,n}(\widehat{F}_{y})$ since this includes
all infinitesimal deformations of any $Z\subset\widehat{F}_{y}$, which can
involve finite neighborhoods of arbitrarily big orders. Thus only the pro-
algebraic group $\mathsf{MW}^{(\infty)}_{y}$ acts on
$\widehat{\operatorname{Hilb}}^{\beta,n}(\widehat{F}_{y})$.

Since the Behrend function
$\nu:\operatorname{Hilb}(\widehat{F}_{y})\to{\mathbb{Z}}$ is determined by
$\widehat{\operatorname{Hilb}}(\widehat{F}_{y})$, the action of
$\mathsf{MW}^{(\infty)}_{y}$ must preserve $\nu$. Consequently we have:

######  Lemma 19.

The Behrend function $\nu$ is invariant under the action of
$\mathsf{MW}(\Delta^{(N)}_{y})$ on
$\operatorname{Hilb}^{\beta,n}(\widehat{F}_{y})$.

In general, we do not know if the sequence (4) splits. However, if $F_{y}$ is
a singular fiber so that
$F^{\circ}_{y}\cong{\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$, then (4) must
split:

${0}$${{\mathbb{C}}^{D}}$${\mathsf{MW}(\Delta_{y}^{(k)})}$${{\mathbb{C}}^{*}\times{\mathbb{C}}^{*}}$${1}$$\scriptstyle{r}$

because then $\mathsf{MW}(\Delta_{y}^{(k)})$ is an affine commutative
algebraic group over ${\mathbb{C}}$ and hence a product of a torus and a
vector space group. Consequently, we have the following

######  Lemma 20.

The group ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ acts on
$\operatorname{Hilb}(\widehat{F}_{\mathsf{sing}})$ where
$\widehat{F}_{\mathsf{sing}}$ is any singular fiber. Moreover, the action
extends to $\widehat{\operatorname{Hilb}}(\widehat{F}_{\mathsf{sing}})$ and
thus it preserves the Behrend function $\nu$.

###  3.6. Reduction of the computation to the singular fibers.

######  Lemma 21.

Let $F_{y}$ be a smooth fiber, then

$e(\operatorname{Hilb}(\widehat{F}_{y}),\nu)=0.$

###### Proof.

By Lemma 19, $\mathsf{MW}=\mathsf{MW}(\Delta_{y}^{N(\beta,n)})$ acts on
$\operatorname{Hilb}^{\beta_{\mathbf{\underline{d}}}\,,n}(\widehat{F}_{y})$
preserving $\nu$. Moreover, the action is fixed point free since the group
$\mathsf{MW}$ acts transitively on $\widehat{F}_{y}$. Let
$\operatorname{Stab}_{x}$ be the $\mathsf{MW}$-stabilizer of
$x\in\operatorname{Hilb}(\widehat{F}_{y})$ and let
$Z_{x}\subset\widehat{F}_{y}$ be the subscheme corresponding to $x$. Then the
image $r(\operatorname{Stab}_{x})$ is a proper subgroup of $F_{y}$ since there
is some element in $F_{y}$ which does not preserve
$\operatorname{Supp}(Z_{x})$. Then the orbit of $x$,
$O_{x}=\mathsf{MW}/\operatorname{Stab}_{x}$ is an Abelian group fiting into
the following exact sequence

$0\longrightarrow\frac{{\mathbb{C}}^{D}}{\operatorname{Stab}_{x}\cap\,\,{\mathbb{C}}^{D}}\longrightarrow
O_{x}\longrightarrow\frac{F_{y}}{r(\operatorname{Stab}_{x})}\longrightarrow
0.$

Therefore, $O_{x}$ is given as the total space of a smooth fibration with base
$F_{y}/r(\operatorname{Stab}_{x})$, a positive dimensional Abelian variety. It
follows that $e(O_{x})=0$. We then can apply Lemma 15 with $G=\mathsf{MW}$,
$V=\operatorname{Hilb}(\widehat{F}_{y})$, and $\mu=\nu$ to complete the proof.

∎

Recall that
$Z_{\Gamma}^{\,}(X)=e(\operatorname{Conf}{\mathbb{P}}^{1},\rho^{\bullet}_{*}\nu)$.
To compute $\rho^{\bullet}_{*}\nu$, we note that the preimage

$\displaystyle(\rho^{\bullet})^{-1}(\\{y_{1},\dotsc,y_{k}\\})$
$\displaystyle=\operatorname{Hilb}^{\bullet}(\widehat{F}_{y_{1}}\cup\dotsb\cup\widehat{F}_{y_{k}})$
$\displaystyle=\prod_{i=1}^{k}\operatorname{Hilb}^{\bullet}(\widehat{F}_{y_{i}})$

where the product takes place in the ring
$K_{0}(\operatorname{Var}_{{\mathbb{C}}})[[Q_{1},Q_{2},Q_{3}]]((p))$.

Lemma 21 implies that

$(\rho^{\bullet}_{*}\nu)(\\{y_{1},\dotsc,y_{k}\\})=0\quad\text{ if
}\quad\\{y_{1},\dotsc,y_{k}\\}\not\subset\\{x_{1},\dotsc,x_{12}\\}$

where $F_{x_{1}},\dotsc,F_{x_{12}}$ are the 12 singular fibers.

In other words, we have shown that the constructible function
$\rho^{*}_{\bullet}\nu$ is supported on
$\operatorname{Conf}(\\{x_{1},\dotsc,x_{12}\\})\subset\operatorname{Conf}{\mathbb{P}}^{1}$.
Therefore

$\displaystyle Z^{\,}_{\Gamma}(X)$
$\displaystyle=e(\operatorname{Conf}{\mathbb{P}}^{1},\nu)$
$\displaystyle=e(\operatorname{Conf}(\\{x_{1},\dotsc,x_{12}\\}),\rho^{*}_{\bullet})$
$\displaystyle=e(\operatorname{Hilb}^{\bullet}(\widehat{F}_{x_{1}}\cup\dotsb\cup\widehat{F}_{x_{12}}),\nu)$
$\displaystyle=\prod_{i=1}^{12}e(\operatorname{Hilb}^{\bullet}(\widehat{F}_{x_{i}}),\nu).$

Since all the formal neighborhoods $\widehat{F}_{x_{i}}$ are isomorphic, we
will write $\widehat{F}_{\mathsf{ban}}$ for this formal scheme, i.e. the
formal neighborhood of $F_{\mathsf{ban}}$, a fiber containing a banana
configuration. In conclusion we have

(5)
$Z^{\,}_{\Gamma}(X_{\mathsf{ban}})=e\left(\widehat{\operatorname{Hilb}}^{\bullet}(\widehat{F}_{\mathsf{ban}}),\nu\right)^{12}.$

Note that here $\nu$ is the restriction of the Behrend function on
$\operatorname{Hilb}(X)$ to $\operatorname{Hilb}(\widehat{F}_{\mathsf{ban}})$,
but since this is determined by
$\widehat{\operatorname{Hilb}}(\widehat{F}_{\mathsf{ban}})$, we may regard
$\nu$ as the Behrend function of the formal scheme
$\widehat{\operatorname{Hilb}}(\widehat{F}_{\mathsf{ban}})$. We will write

$Z^{\,}(\widehat{F}_{\mathsf{ban}})=e\left(\widehat{\operatorname{Hilb}}^{\bullet}(\widehat{F}_{\mathsf{ban}}),\nu\right).$

######  Remark 22.

For more general banana manifolds, the same proof shows that equation 5 holds
with the 12 replaced by $N$, the number of singular fibers (i.e. the number of
banana configurations).

###  3.7. Reduction to ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$-fixed
subschemes.

The ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ action on
$\widehat{F}_{\mathsf{ban}}$ preserves the canonical class by construction. In
particular, the action of ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ is
compatible with the symmetric obstruction theory [4] of
$\operatorname{Hilb}(X)$, restricted to $\widehat{\operatorname{Hilb}}(X)$. We
note that in order to work in this formal setting, we must use the symmetric
obstruction theory associated to the formal moduli space (defined by Jiang in
[22]). The result of Behrend and Fantechi [4] is that for an isolated fixed
point $P\in\widehat{\operatorname{Hilb}}(\widehat{F}_{\mathsf{ban}})$, the
value of the Behrend function is given by

$\nu(P)=(-1)^{\dim T_{P}}$

where

$T_{P}=T_{P}\widehat{\operatorname{Hilb}}(\widehat{F}_{\mathsf{ban}})$

is the Zariski tangent space of
$P\in\widehat{\operatorname{Hilb}}(\widehat{F}_{\mathsf{ban}})$. We note that

$T_{P}=\operatorname{Ext}^{1}_{0}(I_{Z_{P}},I_{Z_{P}})$

where $Z_{P}$ is the subscheme associated to $P$ (see [28]).

We will see that the fixed points of the action are isolated. It then follows
that

$\displaystyle Z^{\,}(\widehat{F}_{\mathsf{ban}})$
$\displaystyle=e\left(\widehat{\operatorname{Hilb}}^{\bullet}(\widehat{F}_{\mathsf{ban}}),\nu\right)$
$\displaystyle=e\left(\widehat{\operatorname{Hilb}}^{\bullet}(\widehat{F}_{\mathsf{ban}})^{{\mathbb{C}}^{*}\times{\mathbb{C}}^{*}},\nu\right)$
$\displaystyle=\sum_{P\in\widehat{\operatorname{Hilb}}(\widehat{F}_{\mathsf{ban}})^{{\mathbb{C}}^{*}\times{\mathbb{C}}^{*}}}(-1)^{\dim
T_{P}}Q_{1}^{d_{1}(P)}Q_{2}^{d_{2}(P)}Q_{3}^{d_{3}(P)}(-p)^{\chi(\mathcal{O}_{Z_{P}})}$

where $d_{i}(P)$ is defined by:

$[Z_{P}]=d_{1}(P)C_{1}+d_{2}(P)C_{2}+d_{3}(P)C_{3}.$

######  Proposition 23.

The ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ fixed points
$P\in\widehat{\operatorname{Hilb}}(\widehat{F}_{\mathsf{ban}})^{{\mathbb{C}}^{*}\times{\mathbb{C}}^{*}}$
are isolated and the fixed point set is in bijective correspondence with the
set $\\{R_{1},R_{2},R_{3},\pi_{1},\pi_{2}\\}$ where $R_{1},R_{2},R_{3}$ is a
triple of 2D partitions and $\pi_{1},\pi_{2}$ is a pair of 3D partitions
asymptotic to $(R_{1},R_{2},R_{3})$ and
$(R^{\prime}_{1},R^{\prime}_{2},R^{\prime}_{3})$ respectively999see for
example [11, Def. 1] for a definition of 3D partitions asymptotic to a triple
of 2D partitions. Also, $A^{\prime}$ denotes the partition conjugate to $A$,
see § 4.. Moreover, the discrete invariants of the corresponding subscheme are
given by

$\displaystyle d_{i}(P)$ $\displaystyle=|R_{i}|$
$\displaystyle\chi(\mathcal{O}_{Z_{P}})$
$\displaystyle=|\pi_{1}|+|\pi_{2}|+\frac{1}{2}\sum_{i=1}^{3}||R_{i}||^{2}+||R^{\prime}_{i}||^{2}$
$\displaystyle(-1)^{\dim T_{P}}$
$\displaystyle=(-1)^{\chi(\mathcal{O}_{Z_{P}})+|R_{1}|+|R_{2}|+|R_{3}|}$

where $|R_{i}|$ is the size of a partition, $|\pi_{i}|$ is the normalized
volume [11, 31], and $||R_{i}||^{2}$ denotes the sum of the squares of the
parts (c.f. § 4).

This proposition will be proved in the next section, but we first use it to
finish the computation $Z(\widehat{F}_{\mathsf{ban}})$. First recall that the
topological vertex is defined to be the generating function

$\mathsf{V}_{R_{1}R_{2}R_{3}}(p)=\sum_{\pi}p^{|\pi|}$

where the sum is over all 3D-partitions asymptotic to $(R_{1},R_{2},R_{3})$
[11, Def. 2], [31]. Then by the proposition we obtain

$\displaystyle Z(\widehat{F}_{\mathsf{ban}})=$
$\displaystyle\sum_{R_{1}R_{2}R_{3}}(-Q_{1})^{|R_{1}|}(-Q_{2})^{|R_{2}|}(-Q_{3})^{|R_{3}|}p^{\frac{1}{2}\sum_{i=1}^{3}||R_{i}||^{2}+||R^{\prime}_{i}||^{2}}\mathsf{V}_{R_{1}R_{2}R_{3}}(p)\mathsf{V}_{R^{\prime}_{1}R^{\prime}_{2}R^{\prime}_{3}}(p)$
$\displaystyle=$ $\displaystyle
M(p)^{2}\sum_{R_{1}R_{2}R_{3}}(-Q_{1})^{|R_{1}|}(-Q_{2})^{|R_{2}|}(-Q_{3})^{|R_{3}|}\widetilde{\mathsf{V}}_{R_{1}R_{2}R_{3}}(p)\widetilde{\mathsf{V}}_{R^{\prime}_{1}R^{\prime}_{2}R^{\prime}_{3}}(p)$

where

$\widetilde{\mathsf{V}}_{R_{1}R_{2}R_{3}}=M(p)^{-1}p^{\frac{1}{2}(||R_{1}||^{2}+||R^{\prime}_{2}||^{2}+||R_{3}||^{2})}V_{R_{1}R_{2}R_{3}}$

is the normalized vertex. The main result of § 4 (Theorem 25) then asserts:

$Z(\widehat{F}_{\mathsf{ban}})=\prod_{a_{i}\geq
0}\prod_{\begin{smallmatrix}k\in{\mathbb{Z}},\\\ k>0\text{ if
}\mathbf{\underline{a}}=\mathbf{\underline{0}}\end{smallmatrix}}\left(1-Q_{1}^{a_{1}}Q_{2}^{a_{2}}Q_{3}^{a_{3}}p^{k}\right)^{-c(||\mathbf{\underline{a}}||,k)}.$

This formula, along with equation (5) then proves Theorem 4.

###  3.8. Analysis of ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$-fixed
subschemes.

The goal of this section is to prove Propostion 23.

To analyze the action of ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ on
$\widehat{F}_{\mathsf{ban}}$ and determine the
${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ invariant subschemes, we obtain an
explicit toric description of $\widehat{F}_{\mathsf{ban}}$ using the following
proposition. Informally, the proposition says that

  1. (1)

$F_{\mathsf{ban}}$ is obtained from
$\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})$, the blow up of
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ at the points
$(0,0),(\infty,\infty)$, by gluing the three pairs of disjoint boundary
divisors to each other in normal crossings.

  2. (2)

$\widehat{F}_{\mathsf{ban}}$ is obtained from
$\widehat{\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})}$, the
formal neighborhood of
$\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})$ inside the total
space of its canonical bundle, by an etalé gluing (which restricts to the
normal crossing gluing of
$\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})$).

  3. (3)

The gluing is ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ equivariant.

Let $F_{\mathsf{sing}}\subset X_{\mathsf{sing}}$ be the image of
$F_{\mathsf{ban}}$ under the blowup $X_{\mathsf{ban}}\to X_{\mathsf{sing}}$.
Let

$\sigma:F^{\mathsf{norm}}_{\mathsf{sing}}\to
F_{\mathsf{sing}},\quad\tau:F^{\mathsf{norm}}_{\mathsf{ban}}\to
F_{\mathsf{ban}}$

be the normalizations. Define $\widehat{F}^{\mathsf{norm}}_{\mathsf{ban}}$ to
be the formal scheme given by the ringed space
$(F^{\mathsf{norm}}_{\mathsf{ban}},\tau^{*}\widehat{\mathcal{O}}_{\widehat{F}_{\mathsf{ban}}})$.

######  Proposition 24.

Then the following hold

  1. (1)

$F^{\mathsf{norm}}_{\mathsf{sing}}\cong{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}.$

  2. (2)

$F^{\mathsf{norm}}_{\mathsf{ban}}\cong\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})$,
the blowup of $F^{\mathsf{norm}}_{\mathsf{sing}}$ at the two points $(0,0)$
and $(\infty,\infty)$ (c.f. [1, Fig 4]).

  3. (3)

$\widehat{F}^{\mathsf{norm}}_{\mathsf{ban}}\cong\widehat{\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})}$,
the formal neighborhood of
$\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})$ viewed as the zero
section in
$\operatorname{Tot}(K_{\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})})$,
the total space of its canonical bundle.

  4. (4)

The induced map
$\widehat{F}^{\mathsf{norm}}_{\mathsf{ban}}\xrightarrow{\widetilde{\tau}}\widehat{F}_{\mathsf{ban}}$
is etalé.

  5. (5)

All of the above maps are ${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$
equivariant.

These results are summarized in the following diagram:

${\widehat{F}_{\mathsf{ban}}}$${\widehat{F}^{\mathsf{norm}}_{\mathsf{ban}}}$${\widehat{\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})}}$${X_{\mathsf{ban}}}$${F_{\mathsf{ban}}}$${F^{\mathsf{norm}}_{\mathsf{ban}}}$${\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})}$${\operatorname{Tot}(K_{\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})})}$${X_{\mathsf{sing}}}$${F_{\mathsf{sing}}}$${F^{\mathsf{norm}}_{\mathsf{sing}}}$${{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}}$$\scriptstyle{\overset{\text{\tiny
formal}}{\text{\tiny neighborhood}}}$$\scriptstyle{\widehat{\tau}}$
etalé${\cong}$$\scriptstyle{\overset{\text{\tiny formal}}{\text{\tiny
neighborhood}}}$$\scriptstyle{\tau}$normalization${\cong}$Blow up
$\scriptstyle(0,0)$ and
$\scriptstyle(\infty,\infty)$$\scriptstyle{\sigma}$normalization${\cong}$

###### Proof.

The proof of this proposition is primarily based on computations in formal
local coordinates. We give the details in Section 5.2. ∎

Using the equivariant etalé morphism

$\hat{\tau}:\widehat{F}_{\mathsf{ban}}^{\mathsf{norm}}\to\widehat{F}_{\mathsf{ban}}$

provided by the lemma, we may study
${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$-invariant subschemes of
$\widehat{F}_{\mathsf{ban}}$ by studying
${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$-invariant of
$\widehat{F}^{\mathsf{norm}}_{\mathsf{ban}}$ satisfying the decent condition.
Since the only proper, invariant subschemes of

$K_{\operatorname{Bl}}:=\operatorname{Tot}\left(K_{\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})}\right)$

are supported on $\widehat{F}^{\mathsf{norm}}_{\mathsf{ban}}\subset
K_{\operatorname{Bl}}$, we may consider torus invariant subschemes of the
toric threefold $K_{\operatorname{Bl}}$ and use the method of MNOP [28] to
count such subschemes — subject to the condition that such subschemes descend
under the etalé relation given by $\hat{\tau}$.

A torus invariant subscheme of a toric Calabi-Yau threefold is determined by
combinatorial data attached to its web diagram, a trivalent planar graph
(which includes non-compact edges). Namely, each edge is labelled by a 2D
partition and each vertex is labelled by a 3D partition which is asymptotic to
the three 2D partitions given by the incident edges (see [28] or also [10, §3
and §B]). The web diagram of $K_{\operatorname{Bl}}$ is a hexagon with 6
additional non-compact edges (the graph on the right in Figure 5.2). The etalé
relation identifies the six vertices to two distinct vertices corresponding to
the points $\\{p,q\\}$ in the banana configuration. The relation also
identifies the edges to three distinct edges corresponding to the curves
$C_{1},C_{2},C_{3}$ in the banana configuration. Each
${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ invariant subscheme is thus
determined by three 2D partitions $R_{1},R_{2},R_{3}$ and two 3D partitions
$\pi_{1},\pi_{2}$. The quantaties $d_{i}(P)$, $\chi(\mathcal{O}_{Z_{P}})$, and
$(-1)^{\dim T_{P}}$ are computed by MNOP in terms of the combinatorial data
associated to the fixed point $P$ for toric Calabi-Yau threefolds. Their
computation applies in our setting (of a formal Calabi-Yau threefold) as well.
Their proofs are based on computing with the Cech open cover given by the
${\mathbb{C}}^{3}$ coordinate charts, but we can equally well use the Cech
cover obtained by intersecting the toric open cover on $K_{\operatorname{Bl}}$
with the formal neighborhood of the banana configuration, and then identifying
via the etalé relation. The proof that fixed points are isolated [28, Lemma
6], the computation of the degree and Euler characteristic [28, §4.4], and the
computation of the parity of the dimensional of the tangent space [28, Thm 2]
all work with this more general Cech cover.

Applying the formulas of MNOP is then straightforward. The formulas in our
Proposition 23 follow directly from the formulas in [28, §4.4] (in particular
Lemma 5), [28, Theorem 2], and the fact that

$\sum_{(i,j)\in R}(i+j+1)\\\ =\frac{1}{2}(||R||^{2}+||R^{\prime}||^{2}).$

This completes the proof of Proposition 23.

##  4\. The vertex calculation

Let

$\widetilde{\mathsf{V}}_{R_{1}R_{2}R_{3}}(p)=s_{R^{\prime}_{3}}(p^{-\rho})\sum_{A}s_{R_{1}^{\prime}/A}(p^{-R_{3}-\rho})s_{R_{2}/A}(p^{-R^{\prime}_{3}-\rho})$

be the normalized vertex and let (see section 3.7)

$Z(\widehat{F}_{\mathsf{ban}})=M(p)^{2}\sum_{R_{1},R_{2},R_{3}}(-Q_{1})^{R_{1}}(-Q_{2})^{R_{2}}(-Q_{3})^{R_{3}}\widetilde{\mathsf{V}}_{R_{1}R_{2}R_{3}}(p)\widetilde{\mathsf{V}}_{R^{\prime}_{1}R^{\prime}_{2}R^{\prime}_{3}}(p)$

where $M(p)=\prod_{m=1}^{\infty}(1-p^{m})^{-m}$.

The purpose of this section is to prove the following.

######  Theorem 25.

$Z(\widehat{F}_{\mathsf{ban}})=\prod_{a_{i}\geq
0}\prod_{\begin{smallmatrix}k\in{\mathbb{Z}},\\\ k>0\text{ if
}\mathbf{\underline{a}}=\mathbf{\underline{0}}\end{smallmatrix}}\left(1-Q_{1}^{a_{1}}Q_{2}^{a_{2}}Q_{3}^{a_{3}}p^{k}\right)^{-c(||\mathbf{\underline{a}}||,k)}$

where $c(a,k)$ is given by Proposition 12.

Most of this computation has previously appeared in the physics literature
under the guise of geometric engineering (in this case a duality between a
certain six dimensional $U(1)$ gauge theory and a certain topological string
theory). The main reference is Hollowood-Iqbal-Vafa [19]. This calculation was
also studied in [25]. These computations assumed an equality between the
generating function for the equivariant elliptic genera of
$\operatorname{Hilb}^{n}({\mathbb{C}}^{2})$, and the orbifold equivariant
elliptic genera of $\operatorname{Sym}^{n}({\mathbb{C}}^{2})$ which they call
the DMVV conjecture (after [15]). This is an instance of the crepant
resolution conjecture for elliptic genera, proven in the compact case by
Borisov-Libgober [7] and the equivariant case by Waelder [38].

We give the derivation here in full detail. We have filled in some minor
details that are missing from the previous accounts and have collected all the
needed results in one place.

###  4.1. Overview of computation

After collecting some standard Schur function identities and proving a few new
ones in the next subsection, we proceed to the main computation. The basic
structure is as follows.

  1. (1)

Writing

$Z^{\prime}_{\mathsf{ban}}=M(p)^{-2}Z(\widehat{F}_{\mathsf{ban}})=\sum_{R}(-Q_{3})^{|R|}Z_{R}(Q_{1},Q_{2},p)$

we use a series of Schur function identities to simplify $Z_{R}$ and write it
as a product

$Z_{R}=Z_{\mathsf{prod}}\cdot Z_{\mathsf{hook},R}$

where $Z_{\mathsf{prod}}$ is a product of terms which do not depend on $R$ and
$Z_{\mathsf{hook},R}$ is also a product of terms which depend on the hook-
lengths of $R$.

  2. (2)

We observe that after the change of variables

(6) $Q_{1}=y,\quad Q_{2}=y^{-1}q,\quad Q_{3}=y^{-1}Q,\quad p=t$

the product $Z_{\mathsf{hook},R}$ is exactly the contribution of a
${\mathbb{C}}^{*}$-fixed point to the computation of

$\sum_{n=0}^{\infty}\operatorname{Ell}_{q,y}(\operatorname{Hilb}({\mathbb{C}}^{2}),t)\,Q^{n},$

the elliptic genera of the Hilbert schemes, via Atiyah-Bott localization. Here
$\lambda\in{\mathbb{C}}^{*}$ acts on the factors of ${\mathbb{C}}^{2}$ with
opposite weights and $t$ is the equivariant parameter.

  3. (3)

An product formula for
$\sum_{n=0}^{\infty}\operatorname{Ell}_{q,y}(\operatorname{Hilb}({\mathbb{C}}^{2}),t)\,Q^{n}$
was conjectured by Dijkgraaf-Moore-Verlinde-Verlinde [15], and proven by
Borisov-Libgobner and Waelder [7, 38]. Using that formula, substituting back
to the $Q_{1},Q_{2},Q_{3},p$ variables, and performing a few easy
manipulations, we arrive at Theorem 4.

###  4.2. Notation and Schur function identities

We will use capital letters $R,A,B,C$, etc. to denote partitions. Via its
diagram, we regard a partition $A$ as a finite subset of
${\mathbb{N}}\times{\mathbb{N}}$ where if $(i,j)\in A$ then $(i-1,j)\in A$ and
$(i,j-1)\in A$. The _rows_ or _parts_ of $A$ are the integers
$A_{j}=\max\\{i|(i,j)\in A\\}$. We use ′ to denote the conjugate partition

$A^{\prime}=\\{(i,j):(j,i)\in A\\},$

and we write

$|A|=\sum_{j}A_{j},\quad||A||^{2}=\sum_{j}A_{j}^{2},$

For each $(i,j)\in A$ we define the _hook length_ :

$h_{ij}(A)=A_{i}+A_{j}^{\prime}-i-j+1.$

We write $\square$ for the unique partition of size 1.

We also use the notation

$M(u,p)=\prod_{m=1}^{\infty}(1-up^{m})^{-m}$

and the short hand $M(p)=M(1,p)$.

For a collection of variables $x=(x_{1},x_{2},\dotsc)$ and two partitions $A$
and $B$ let $s_{A/B}(x)=s_{A/B}(x_{1},x_{2},\dotsc)$ denote the skew Schur
function (see for example [26, § 5]). Let

$\rho=\left(-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\dotsc\right)$

so that for example $p^{-R-\rho}$ is notation for the variable list

$\left(p^{-R_{1}+\frac{1}{2}},p^{-R_{2}+\frac{3}{2}},\dotsc\right).$

Okounkov-Reshetikhin-Vafa derived a formula for the topological vertex in
terms of skew Schur functions. Translating their formulas [31, 3.20& 3.21]
into our notation, we get:

(7)
$\mathsf{V}_{R_{1}R_{2}R_{3}}(p)=M(p)\,p^{-\frac{1}{2}(\|R_{1}\|^{2}+\|R_{2}^{\prime}\|^{2}+\|R_{3}\|^{2})}\,\widetilde{\mathsf{V}}_{R_{1}R_{2}R_{3}}(p)$

where we’ve defined

(8)
$\widetilde{\mathsf{V}}_{R_{1}R_{2}R_{3}}(p)=s_{R_{3}^{\prime}}(p^{-\rho})\,\sum_{A}\,s_{R_{1}^{\prime}/A}(p^{-R_{3}-\rho})\cdot
s_{R_{2}/A}(p^{-R_{3}^{\prime}-\rho}).$

We will need the following Schur function identities. We remark that the Schur
functions which appear are all Laurent expansions in $p$ or in $p^{-1}$ of
rational functions in $p$ and many of the identities should be understood as
equalities of rational functions.

From [31, 3.10] we have101010There is a typo in equation 3.10 in [31] — the
exponent on the right hand side should be $-\nu^{\prime}-\rho$.

(9)
$s_{A/B}(p^{C+\rho})=(-1)^{|A|-|B|}\,s_{A^{\prime}/B^{\prime}}(p^{-C^{\prime}-\rho}).$

From [26, pg 45]:

$s_{R}(1,p,p^{2},\dotsc)=p^{n(R)}\prod_{(i,j)\in R}\frac{1}{1-p^{h_{ij}(R)}}$

where [26, pg 3] $n(R)=\frac{1}{2}||R^{\prime}||^{2}-\frac{1}{2}|R|$ and so
using the homogeneity of $s_{R}$ we see that

$s_{R}(p^{-\rho})=p^{\frac{1}{2}||R^{\prime}||^{2}}\prod_{i,j\in
R}\frac{1}{(1-p^{h_{ij}(R)})}$

and so we arrive at

(10) $s_{R}(p^{-\rho})s_{R^{\prime}}(p^{-\rho})=(-1)^{|R|}\prod_{i,j\in
R}\frac{1}{(1-p^{h_{ij}(R)})(1-p^{-h_{ij}(R)})}.$

We will also need the following identity [26, pg 93(2)]

(11)
$\sum_{R_{1}}\,s_{R_{1}/B}(x)\,s_{R_{1}^{\prime}/A}(y)=\prod_{i,j=1}^{\infty}(1+x_{i}y_{j})\sum_{C}s_{A^{\prime}/C}(x)s_{B^{\prime}/C}(y).$

Finally we will need the following lemma.

######  Lemma 26.

The following equalities hold as formal power series in $u$ whose coefficients
are rational functions of $p$.

(12)
$\displaystyle\prod_{j,k=1}^{\infty}\left(1-up^{R_{j}+R^{\prime}_{k}-j-k+1}\right)$
$\displaystyle=M(u,p)^{-1}\prod_{j,k\in
R}\left(1-up^{h_{jk}(R)}\right)\left(1-up^{-h_{jk}(R)}\right)$ (13)
$\displaystyle\prod_{j,k=1}^{\infty}\left(1-up^{-R_{j}+R_{k}+j-k}\right)$
$\displaystyle=M(u,p)\prod_{j,k\in
R}\left(1-up^{h_{jk}(R)}\right)^{-1}\left(1-up^{-h_{jk}(R)}\right)^{-1}.$

Note that the right hand side of the equations are invariant under
$p\leftrightarrow p^{-1}$ and so we get two more identities by replacing $p$
by $p^{-1}$ on the left hand side of the above equations.

###### Proof.

Moving the $M(u,p)$ to the left hand side of the first equation and then
taking the log, we get

$\displaystyle\quad\log\left(M(u,p)\prod_{j,k=1}^{\infty}\left(1-up^{R_{j}+R^{\prime}_{k}-j-k+1}\right)\right)$
$\displaystyle=$
$\displaystyle\quad\log\left(\prod_{j,k=1}^{\infty}\left(1-up^{j+k-1}\right)^{-1}\right)+\log\left(\prod_{j,k=1}^{\infty}\left(1-up^{R_{j}+R^{\prime}_{k}-j-k+1}\right)\right)$
$\displaystyle=$ $\displaystyle\sum_{n\geq 1}\frac{u^{n}}{n}\sum_{j,k\geq
1}\left(p^{n(j+k-1)}-p^{n(R_{j}+R^{\prime}_{k}-j-k+1)}\right)$
$\displaystyle=$ $\displaystyle\sum_{n\geq 1}-\frac{u^{n}}{n}P_{R}(p^{n})$

where we’ve defined

$\displaystyle P_{R}(x)$ $\displaystyle=\sum_{j,k\geq
1}(x^{R_{j}+R^{\prime}_{k}-j-k+1}-x^{j+k-1})$ $\displaystyle=\left(\sum_{j\geq
1}x^{R_{j}-j+\frac{1}{2}}\sum_{k\geq
1}x^{R_{k}-k+\frac{1}{2}}\right)-\frac{x}{(1-x)^{2}}$
$\displaystyle=s_{\square}(x^{R+\rho})s_{\square}(x^{R^{\prime}+\rho})-\frac{x}{(1-x)^{2}}$

Since by equation (10), we have

$s_{\square}\left(x^{R+\rho}\right)=-s_{\square}\left(x^{-R^{\prime}-\rho}\right),$

we see from the above equation that $P_{R}(x)$ is a sum of two rational
functions in $x$, each invariant under $x\leftrightarrow x^{-1}$. Moreover,
all but a finite number of terms in the sum

$\sum_{j,k\geq
1}\left(x^{R_{j}+R_{k}^{\prime}-j-k+1}-x^{-j-i+1}\right)=P_{R}(x)$

cancel and so we deduce that $P_{R}(x)$ is a Laurent _polynomial_ which is
invariant under $x\leftrightarrow 1/x$. Consequently, $P_{R}(x)$ is uniquely
determined by its terms with positive exponents. Since

$R_{j}+R^{\prime}_{k}-j-k+1=\begin{cases}h_{jk}(R)&\text{if $(j,k)\in R$}\\\
\text{negative}&\text{if $(j,k)\notin R$}\end{cases}$

we see that the terms with positive exponent in the above expression for
$P_{R}$ are precisely $x^{h_{jk}(R)}$ where $(j,k)\in R$. Therefore

$P_{R}(x)=\sum_{j,k\in R}\left(x^{h_{jk}(R)}+x^{-h_{jk}(R)}\right).$

Substituting back, we find

$\displaystyle\log\left(M(u,p)\prod_{j,k=1}^{\infty}\left(1-up^{R_{j}+R^{\prime}_{k}-j-k+1}\right)\right)$
$\displaystyle=$ $\displaystyle\sum_{n\geq 1}-\frac{u^{n}}{n}\sum_{j,k\in
R}\left(x^{h_{jk}(R)}+x^{-h_{jk}(R)}\right)$ $\displaystyle=$
$\displaystyle\sum_{j,k\in
R}\log\left(1-up^{h_{jk}(R)}\right)+\log\left(1-up^{-h_{jk}(R)}\right)$
$\displaystyle=$ $\displaystyle\log\prod_{j,k\in
R}\left(1-up^{h_{jk}(R)}\right)\left(1-up^{-h_{jk}(R)}\right)$

which proves equation (12). To prove equation (13), we observe that since by
equation (10)

$\sum_{k\geq 1}p^{R^{\prime}_{k}-k+\frac{1}{2}}=-\sum_{k\geq
1}p^{-R_{k}+k-\frac{1}{2}}$

so we have

$-\sum_{j,k\geq 1}\left(x^{-R_{j}+R_{k}+j-k}+x^{j+k-1}\right)=\sum_{j,k\in
R}\left(x^{h_{jk}(R)}+x^{-h_{jk}(R)}\right).$

Equation (13) then follows from a similar logarithm argument as we did for
equation (12). ∎

###  4.3. The main derivation

Recall that

$Z^{\prime}_{\mathsf{ban}}=M(p)^{-2}Z(\widehat{F}_{\mathsf{ban}})$

so that

$Z^{\prime}_{\mathsf{ban}}=\sum_{R_{1}R_{2}R_{3}}(-Q_{1})^{R_{1}}(-Q_{2})^{R_{2}}(-Q_{3})^{R_{3}}\,\widetilde{\mathsf{V}}_{R_{1}R_{2}R_{3}}\,\widetilde{\mathsf{V}}_{R^{\prime}_{1}R^{\prime}_{2}R^{\prime}_{3}}.$

We write

$Z^{\prime}_{\mathsf{ban}}=\sum_{R}(-Q_{3})^{R}Z_{R}$

where

$\displaystyle Z_{R}=$ $\displaystyle\hskip 91.04872pt\sum_{R_{1}R_{2}}$
$\displaystyle(-Q_{1})^{R_{1}}(-Q_{2})^{R_{2}}\,\widetilde{\mathsf{V}}_{R_{1}R_{2}R}\,\widetilde{\mathsf{V}}_{R^{\prime}_{1}R_{2}^{\prime}R}$
$\displaystyle=$
$\displaystyle\,\,s_{R}(p^{-\rho})s_{R^{\prime}}(p^{-\rho})\sum_{A\,B\,R_{1}R_{2}}$
$\displaystyle(-Q_{1})^{R_{1}}(-Q_{2})^{R_{2}}\cdot
s_{R^{\prime}_{1}/A}(p^{-R-\rho})\cdot s_{R_{2}/A}(p^{-R^{\prime}-\rho})$
$\displaystyle\hskip 85.35826pt\cdot s_{R_{1}/B}(p^{-R^{\prime}-\rho})\cdot
s_{R^{\prime}_{2}/B}(p^{-R-\rho})$ $\displaystyle=$
$\displaystyle\,\,s_{R}(p^{-\rho})s_{R^{\prime}}(p^{-\rho})\cdot\sum_{A\,B\,R_{2}}$
$\displaystyle(-Q_{2})^{R_{2}}(-Q_{1})^{B}\cdot
s_{R_{2}/A}(p^{-R^{\prime}-\rho})\cdot s_{R_{2}^{\prime}/B}(p^{-R-\rho})$
$\displaystyle\hskip 91.04872pt\cdot\sum_{R_{1}}$ $\displaystyle
s_{R_{1}/B}(-Q_{1}p^{-R^{\prime}-\rho})\cdot
s_{R_{1}^{\prime}/A}(p^{-R-\rho})$

From [26, (2),page 93] we have

$\sum_{R_{1}}s_{R_{1}/B}(x)s_{R_{1}^{\prime}/A}(y)=\prod_{i,j=1}^{\infty}(1+x_{i}y_{j})\sum_{C}s_{A^{\prime}/C}(x)s_{B^{\prime}/C^{\prime}}(y).$

Using the above and equation (10) we get

$\displaystyle Z_{R}=$ $\displaystyle(-1)^{R}\prod_{i,j\in
R}(1-p^{h_{ij}(R)})^{-1}(1-p^{-h_{ij}(R)})^{-1}\cdot\prod_{i,j=1}^{\infty}(1-Q_{1}p^{i+j-1-R_{i}^{\prime}-R_{j}})$
$\displaystyle\hskip
14.22636pt\cdot\sum_{A\,B\,R_{2}\,C}(-Q_{2})^{R_{2}}(-Q_{1})^{B}\cdot
s_{R_{2}/A}(p^{-R^{\prime}-\rho})\cdot s_{R_{2}^{\prime}/B}(p^{-R-\rho})$
$\displaystyle\hskip 136.5733pt\cdot
s_{A^{\prime}/C}(-Q_{1}p^{-R^{\prime}-\rho})\cdot
s_{B^{\prime}/C^{\prime}}(p^{-R-\rho}).$

Using equation (12) from Lemma 26 we get

$\displaystyle Z_{R}=H_{R}\sum_{C\,R_{2}}(-Q_{2})^{R_{2}}(-Q_{1})^{C}$
$\displaystyle\cdot\sum_{A}s_{R_{2}/A}(p^{-R^{\prime}-\rho})\cdot
s_{A^{\prime}/C}(-Q_{1}p^{-R^{\prime}-\rho})$ (14)
$\displaystyle\cdot\sum_{B}s_{R_{2}^{\prime}/B}(p^{-R-\rho})\cdot
s_{B^{\prime}/C^{\prime}}(-Q_{1}p^{-R-\rho})$

where

$H_{R}=(-1)^{R}M(Q_{1},p)^{-1}\prod_{i,j\in
R}\frac{(1-Q_{1}p^{h_{ij}(R)})(1-Q_{1}p^{-h_{ij}(R)})}{(1-p^{h_{ij}(R)})(1-p^{-h_{ij}(R)})}$

Now using (see [26, 5.10 page 72]

$\sum_{\nu}s_{\lambda/\nu}(x)s_{\nu/\mu}(y)=s_{\lambda/\mu}(x,y)$

and equation (9), we can rewrite the second and third sums in equation 4.3 as

$\displaystyle\sum_{A}s_{R_{2}/A}(p^{-R^{\prime}-\rho})\cdot
s_{A^{\prime}/C}(-Q_{1}p^{-R^{\prime}-\rho})$
$\displaystyle=s_{R_{2}/C^{\prime}}(p^{-R^{\prime}-\rho},Q_{1}p^{R+\rho})$
$\displaystyle\sum_{B}s_{R_{2}^{\prime}/B}(p^{-R-\rho})\cdot
s_{B^{\prime}/C^{\prime}}(-Q_{1}p^{-R-\rho})$
$\displaystyle=s_{R_{2}^{\prime}/C}(p^{-R-\rho},Q_{1}p^{R^{\prime}+\rho}).$

Substituting back into equation (4.3) we get

$\displaystyle Z_{R}=$ $\displaystyle
H_{R}\sum_{C,R_{2}}(-Q_{2})^{R_{2}}(-Q_{1})^{C}\cdot
s_{R_{2}/C^{\prime}}(p^{-R^{\prime}-\rho},Q_{1}p^{R+\rho})\cdot
s_{R^{\prime}_{2}/C}(p^{-R-\rho}Q_{1}p^{R^{\prime}+\rho})$ $\displaystyle=$
$\displaystyle H_{R}\sum_{C,R_{2}}(Q_{1}Q_{2})^{R_{2}}\cdot
s_{R_{2}/C^{\prime}}(\mathbf{y},\mathbf{y^{\prime}})\cdot
s_{R_{2}^{\prime}/C}(\mathbf{x},\mathbf{x}^{\prime})$

where

$\mathbf{y}=\\{-Q_{1}^{-1}p^{-R^{\prime}-\rho}\\},\quad\mathbf{y^{\prime}}=\\{-p^{R+\rho}\\},\quad\mathbf{x}=\\{p^{-R-\rho}\\},\quad\mathbf{x}^{\prime}=\\{Q_{1}p^{R^{\prime}+\rho}\\}.$

We now use [26, page 94, equation (b)] to obtain

$\displaystyle Z_{R}=H_{R}$
$\displaystyle\prod_{i=1}^{\infty}(1-Q_{1}Q_{2})^{-1}$ $\displaystyle\cdot$
$\displaystyle\prod_{j,k}(1+Q_{1}^{i}Q_{2}^{i}x_{j}y_{k})\cdot(1+Q_{1}^{i}Q_{2}^{i}x^{\prime}_{j}y_{k})\cdot(1+Q_{1}^{i}Q_{2}^{i}x_{j}y^{\prime}_{k})\cdot(1+Q_{1}^{i}Q_{2}^{i}x^{\prime}_{j}y^{\prime}_{k})$

We deal with each of four factors in the product over $j$ and $k$ using Lemma
26:

$\displaystyle\prod_{j,k}(1+Q_{1}^{i}Q_{2}^{i}x_{j}y_{k})$
$\displaystyle=\prod_{j,k}(1-Q_{1}^{i-1}Q_{2}^{i}\,p^{j+k-1-R_{k}^{\prime}-R_{j}})$
$\displaystyle=M(Q_{1}^{i-1}Q_{2}^{i},p)^{-1}\prod_{j,k\in
R}\left(1-Q_{1}^{i-1}Q_{2}^{i}\,p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i-1}Q_{2}^{i}\,p^{-h_{jk}(R)}\right)$
$\displaystyle\prod_{j,k}(1+Q_{1}^{i}Q_{2}^{i}x^{\prime}_{j}y_{k})$
$\displaystyle=\prod_{j,k}(1-Q_{1}^{i}Q_{2}^{i}\,p^{-j+k-R_{k}^{\prime}+R_{j}^{\prime}})$
$\displaystyle=M(Q_{1}^{i}Q_{2}^{i},p)\prod_{j,k\in
R}\left(1-Q_{1}^{i}Q_{2}^{i}\,p^{h_{jk}(R)}\right)^{-1}\left(1-Q_{1}^{i}Q_{2}^{i}\,p^{-h_{jk}(R)}\right)^{-1}$
$\displaystyle\prod_{j,k}(1+Q_{1}^{i}Q_{2}^{i}x_{j}y^{\prime}_{k})$
$\displaystyle=\prod_{j,k}(1-Q_{1}^{i}Q_{2}^{i}\,p^{j-k-1+R_{k}-R_{j}})$
$\displaystyle=M(Q_{1}^{i}Q_{2}^{i},p)\prod_{j,k\in
R}\left(1-Q_{1}^{i}Q_{2}^{i}\,p^{h_{jk}(R)}\right)^{-1}\left(1-Q_{1}^{i}Q_{2}^{i}\,p^{-h_{jk}(R)}\right)^{-1}$
$\displaystyle\prod_{j,k}(1+Q_{1}^{i}Q_{2}^{i}x^{\prime}_{j}y^{\prime}_{k})$
$\displaystyle=\prod_{j,k}(1-Q_{1}^{i+1}Q_{2}^{i}\,p^{-j-k+1+R_{k}+R_{j}^{\prime}})$
$\displaystyle=M(Q_{1}^{i+1}Q_{2}^{i},p)^{-1}\prod_{j,k\in
R}\left(1-Q_{1}^{i+1}Q_{2}^{i}\,p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i+1}Q_{2}^{i}\,p^{-h_{jk}(R)}\right)$

Substituting back we see

$\displaystyle
Z_{R}=H_{R}\cdot\prod_{i=1}^{\infty}\frac{M(Q_{1}^{i}Q_{2}^{i},p)^{2}}{(1-Q_{1}^{i}Q_{2}^{i})M(Q_{1}^{i-1}Q_{2}^{i},p)M(Q_{1}^{i+1}Q_{2}^{i},p)}$
$\displaystyle\cdot\prod_{j,k\in
R}\frac{\left(1-Q_{1}^{i-1}Q_{2}^{i}p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i-1}Q_{2}^{i}p^{-h_{jk}(R)}\right)\left(1-Q_{1}^{i+1}Q_{2}^{i}p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i+1}Q_{2}^{i}p^{-h_{jk}(R)}\right)}{\left(1-Q_{1}^{i}Q_{2}^{i}p^{h_{jk}(R)}\right)^{2}\left(1-Q_{1}^{i}Q_{2}^{i}p^{-h_{jk}(R)}\right)^{2}}$

Putting $H_{R}$ back in we get

$\displaystyle Z_{R}=Z_{\mathsf{prod}}\cdot(-1)^{R}\cdot\prod_{j,k\in
R}\frac{\left(1-Q_{1}p^{h_{jk}(R)}\right)\left(1-Q_{1}p^{-h_{jk}(R)}\right)}{\left(1-p^{h_{jk}(R)}\right)\left(1-p^{-h_{jk}(R)}\right)}$
$\displaystyle\cdot\prod_{i=1}^{\infty}\frac{\left(1-Q_{1}^{i+1}Q_{2}^{i}p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i+1}Q_{2}^{i}p^{-h_{jk}(R)}\right)\left(1-Q_{1}^{i-1}Q_{2}^{i}p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i-1}Q_{2}^{i}p^{-h_{jk}(R)}\right)}{\left(1-Q_{1}^{i}Q_{2}^{i}p^{h_{jk}(R)}\right)^{2}\left(1-Q_{1}^{i}Q_{2}^{i}p^{-h_{jk}(R)}\right)^{2}}$
$\displaystyle=Z_{\mathsf{prod}}\cdot(-1)^{R}\cdot\prod_{j,k\in R}$
$\displaystyle\cdot\prod_{i=1}^{\infty}\frac{\left(1-Q_{1}^{i-1}Q_{2}^{i}p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i-1}Q_{2}^{i}p^{-h_{jk}(R)}\right)\left(1-Q_{1}^{i-1}Q_{2}^{i}p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i-1}Q_{2}^{i}p^{-h_{jk}(R)}\right)}{\left(1-Q_{1}^{i-1}Q_{2}^{i-1}p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i-1}Q_{2}^{i-1}p^{-h_{jk}(R)}\right)\left(1-Q_{1}^{i}Q_{2}^{i}p^{h_{jk}(R)}\right)\left(1-Q_{1}^{i}Q_{2}^{i}p^{-h_{jk}(R)}\right)}$

where we’ve defined

$\displaystyle Z_{\mathsf{prod}}$
$\displaystyle=M(Q_{1},p)^{-1}\prod_{i=1}^{\infty}\frac{M(Q_{1}^{i}Q_{2}^{i},p)^{2}}{(1-Q_{1}^{i}Q_{2}^{i})M(Q_{1}^{i-1}Q_{2}^{i},p)M(Q_{1}^{i+1}Q_{2}^{i},p)}$
$\displaystyle=\prod_{i=1}^{\infty}\frac{M(Q_{1}^{i}Q_{2}^{i},p)^{2}}{(1-Q_{1}^{i}Q_{2}^{i})M(Q_{1}^{i-1}Q_{2}^{i},p)M(Q_{1}^{i}Q_{2}^{i-1},p)}$

We now make the variable change given by equation (6), sum over the remaining
partition, and write the result in terms of elliptic genera:

$\displaystyle Z^{\prime}_{\mathsf{ban}}=$ $\displaystyle
Z_{\mathsf{prod}}\cdot\sum_{R}(y^{-1}Q)^{R}$
$\displaystyle\cdot\prod_{i=1}^{\infty}\prod_{j,k\in
R}\frac{(1-t^{h_{jk}}yq^{i-1})(1-t^{-h_{jk}}yq^{i-1})(1-t^{h_{jk}}y^{-1}q^{i})(1-t^{-h_{jk}}y^{-1}q^{i})}{(1-t^{h_{jk}}q^{i-1})(1-t^{-h_{jk}}q^{i-1})(1-t^{h_{jk}}q^{i})(1-t^{-h_{jk}}q^{i})}$
$\displaystyle=$ $\displaystyle
Z_{\mathsf{prod}}\cdot\sum_{k=0}^{\infty}Q^{k}\operatorname{Ell}_{q,y}(({\mathbb{C}}^{2})^{[k]},t)$
$\displaystyle=$ $\displaystyle
Z_{\mathsf{prod}}\cdot\prod_{n=0}^{\infty}\prod_{m=1}^{\infty}\prod_{l,k\in{\mathbb{Z}}}(1-t^{k}q^{n}y^{l}Q^{m})^{-c(nm,l,k)}$
$\displaystyle=$ $\displaystyle
Z_{\mathsf{prod}}\cdot\prod_{n=0}^{\infty}\prod_{m=1}^{\infty}\prod_{l,k\in{\mathbb{Z}}}(1-t^{k}q^{n}y^{l}Q^{m})^{-c(4nm-l^{2},k)}$

where the last three equalities come from equation (1), Theorem 11, and
Proposition 12 respectively.

Returning to the DT variables via

$t=p,\quad q=Q_{1}Q_{2},\quad y=Q_{1},\quad Q=Q_{1}Q_{3},$

reindexing by

$d_{1}=n+l+m,\quad d_{2}=n,\quad d_{3}=m,$

and observing that

$||d||=2d_{1}d_{2}+2d_{2}d_{3}+2d_{3}d_{1}-d_{1}^{2}-d_{2}^{2}-d_{3}^{2}=4mn-l^{2}$

we find

$Z^{\prime}_{\mathsf{ban}}=Z_{\mathsf{prod}}\cdot\prod_{d_{3}=1}^{\infty}\prod_{d_{2}=0}^{\infty}\prod_{d_{1},k\in{\mathbb{Z}}}(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}})^{-c(||d||,k)}$

Observing further that when $d_{1}<0$ and $d_{3}>0$,
$||d||=4d_{3}d_{1}-(d_{2}-d_{1}-d_{3})^{2}<-1$ and so $c(||d||,k)=0$, we get

$\displaystyle Z(\widehat{F}_{\mathsf{ban}})$ $\displaystyle=M(p)^{2}\cdot
Z^{\prime}_{\mathsf{ban}}$ $\displaystyle=M(p)^{2}\cdot
Z_{\mathsf{prod}}\cdot\prod_{d_{3}=1}^{\infty}\prod_{d_{2},d_{3}=0}^{\infty}\prod_{k\in{\mathbb{Z}}}(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}})^{-c(||d||,k)}$

Finally we claim that

$M(p)^{2}\cdot
Z_{\mathsf{prod}}=\prod_{(*)}(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}})^{-c(||d||,k)}$

where the product is over

$(*)\quad\quad\quad d_{3}=0,\quad d_{1},d_{2}\geq 0,\quad\text{and
}\begin{cases}k\in{\mathbb{Z}}&(d_{1},d_{2})\neq(0,0),\\\
k>0&(d_{1},d_{2})=(0,0).\end{cases}$

Indeed, if $d_{3}=0$, then $||d||=-(d_{2}-d_{1})^{2}$ and by Corollary 13 the
product over $(*)$ reduces to the terms where

$k\in{\mathbb{Z}},\quad(d_{1},d_{2})=(d,d-1),(d-1,d),(d,d),\quad d>1$

and the special case $d_{1}=d_{2}=0,k>0$. Applying Corollary 13 we easily
deduce the above claim.

Substituting the claimed equation back into the previous equation for
$Z(\widehat{F}_{\mathsf{ban}})$ then finishes the proof of Theorem 4. ∎

##  5\. Geometry of the Banana manifold

In this section we compute the Hodge numbers of the banana manifold
$X_{\mathsf{ban}}$, we show that the fiber classes are spanned by the banana
curves $C_{1},C_{2},C_{3}$, and we prove Proposition 24 which describes the
formal neighborhood of the singular fibers.

Let $p:S\to{\mathbb{P}}^{1}$ be a rational elliptic surface with 12 singular
fibers, each having one node. Let
${p}^{\scriptscriptstyle{\\#}}:{S}^{\scriptscriptstyle{\\#}}\to{\mathbb{P}}^{1}$
be an isomorphic copy of $S$ and consider the fibered product

$X_{\mathsf{sing}}=S\times_{{\mathbb{P}}^{1}}{S}^{\scriptscriptstyle{\\#}}.$

$X_{\mathsf{sing}}$ is singular at the 12 points where both $p$ and
${p}^{\scriptscriptstyle{\\#}}$ are not smooth, namely at the product of the
nodes. To see that these points are conifold singularities, note that for each
node $n\in S$, and corresponding node
${n}^{\scriptscriptstyle{\\#}}\in{S}^{\scriptscriptstyle{\\#}}$, there exists
formal local coordinates
$(x,y),({x}^{\scriptscriptstyle{\\#}},{y}^{\scriptscriptstyle{\\#}}),t$ about
$n,$ ${n}^{\scriptscriptstyle{\\#}}$, and
$p(n)={p}^{\scriptscriptstyle{\\#}}({n}^{\scriptscriptstyle{\\#}})\in{\mathbb{P}}^{1}$
respectively such that the maps $p$ and ${p}^{\scriptscriptstyle{\\#}}$ are
given by $xy=t$ and
${x}^{\scriptscriptstyle{\\#}}{y}^{\scriptscriptstyle{\\#}}=t$. Consequently,
$xy={x}^{\scriptscriptstyle{\\#}}{y}^{\scriptscriptstyle{\\#}}$ is the local
equation of $X_{\mathsf{sing}}$ at $(n,{n}^{\scriptscriptstyle{\\#}})$.

Let $\Delta\subset X_{\mathsf{sing}}$ be the divisor given by the diagonal in
the fibered product. From the above local description of the singularities,
all of which lie on $\Delta$, we see that

$X_{\mathsf{ban}}=\operatorname{Bl}_{\Delta}(X_{\mathsf{sing}}),$

the blowup of $X_{\mathsf{sing}}$ along the diagonal is smooth and

$X_{\mathsf{ban}}\to X_{\mathsf{sing}}$

is a conifold resolution.

###  5.1. Relation to the Schoen threefold and the Hodge numbers

######  Lemma 27.

$e(X_{\mathsf{ban}})=24$.

###### Proof.

Since the map $X_{\mathsf{ban}}\to X_{\mathsf{sing}}$ contracts 12
${\mathbb{P}}^{1}$s to 12 singular points, we can write the following equation
in the Grothendieck group of varieties,

$[X_{\mathsf{ban}}]=[X^{o}_{\mathsf{sing}}]+12[{\mathbb{P}}^{1}]$

where $X_{\mathsf{sing}}^{o}\subset X_{\mathsf{sing}}$ is the non-singular
locus. Since $e(-)$ is a homomorphism from the Grothendieck group to the
integers, we see that $e(X_{\mathsf{ban}})=e(X_{\mathsf{sing}}^{o})+24$. We
claim the Euler characteristic of the fibers of
$X_{\mathsf{sing}}^{o}\to{\mathbb{P}}^{1}$ are all zero. The smooth fibers are
$E\times E$, a product of smooth elliptic curves and hence have zero Euler
characterisitic. The singular fibers have a
${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ action constructed in § 3.5. The
fixed points of the action on the fibers of
$X_{\mathsf{sing}}\to{\mathbb{P}}^{1}$ are exactly the conifold points and so
the action on the fibers of $X^{o}_{\mathsf{sing}}\to{\mathbb{P}}^{1}$ is
free. Consequently, the Euler characteristics of these fibers are zero. The
lemma follows. ∎

The banana manifold is related to the Schoen Calabi-Yau threefold by a
conifold transition. This will allow us to compute the Hodge numbers of the
banana manifold in terms of the (well-known) Hodge numbers of the Schoen
threefold. The Schoen threefold can be defined as

$X_{\mathsf{Sch}}\subset{\mathbb{P}}^{2}\times{\mathbb{P}}^{2}\times{\mathbb{P}}^{1},$

the intersection of two generic hypersurfaces of multi-degree $(3,0,1)$ and
$(0,3,1)$. It is a simply connected Calabi-Yau threefold with
$h^{1,1}(X_{\mathsf{Sch}})=19$ and $h^{2,1}(X_{\mathsf{Sch}})=19$.

######  Proposition 28.

$X_{\mathsf{ban}}$ is a simply connected Calabi-Yau with
$h^{2,1}(X_{\mathsf{ban}})=8$ and $h^{1,1}(X_{\mathsf{ban}})=20$.

###### Proof.

We first show that there is a conifold transition from $X_{\mathsf{Sch}}$ to
$X_{\mathsf{ban}}$. The generic hypersurface in
${\mathbb{P}}^{2}\times{\mathbb{P}}^{1}$ of degree $(3,1)$ is a rational
elliptic surface $S\to{\mathbb{P}}^{1}$ and consequently the projections of
$X_{\mathsf{Sch}}\subset{\mathbb{P}}^{2}\times{\mathbb{P}}^{2}\times{\mathbb{P}}^{1}$
onto various subfactors realize $X_{\mathsf{Sch}}$ as a fibered product

$X_{\mathsf{Sch}}=S_{0}\times_{{\mathbb{P}}^{1}}S_{1}$

where $S_{i}\to{\mathbb{P}}^{1}$ are distinct generic rational surfaces. We
may choose a 1-parameter family of elliptic surfaces $S_{t}$ which
interpolates between $S_{0}$ and $S_{1}$. Then the family of threefolds

$X_{t}=S_{0}\times_{{\mathbb{P}}^{1}}S_{t}$

are all Schoen threefolds for $t\neq 0$ and $X_{0}=X_{\mathsf{sing}}$ and
since $X_{\mathsf{ban}}\to X_{\mathsf{sing}}$ is a conifold resolution, we get
a conifold transition $X_{\mathsf{Sch}}\rightsquigarrow X_{\mathsf{ban}}$.
Since conifold transitions preserve simple connectivity and the Calabi-Yau
condition, we see that $X_{\mathsf{ban}}$ is a simply connected Calabi-Yau
threefold. The change in $h^{1,1}$ through a conifold transition is determined
by the codimension of the family of singular threefolds inside the full
deformation space of the threefold (see [30, § 3.1]). The family of Schoen
threefolds is 19 dimensional: there are the two 8 dimensional families of
rational elliptic surfaces $S_{i}\to{\mathbb{P}}^{1}$ and the three
dimensional family of isomorphisms between the bases of
$S_{i}\to{\mathbb{P}}^{1}$ required to form the fibered product
$S_{0}\times_{{\mathbb{P}}^{1}}S_{1}$. The locus of such fibered products with

$\delta=12$

conifold singularities is 8 dimensional: the 12 nodes of $S_{0}$ and $S_{1}$
must occur over the same fibers which implies that $S_{0}\cong S_{1}$ and the
isomorphism of the base is the identity. The codimension $\sigma$ is thus

$\sigma=19-8=11.$

Following [30, § 3.1], we compute:

$\displaystyle h^{1,1}(X_{\mathsf{ban}})$
$\displaystyle=h^{1,1}(X_{\mathsf{Sch}})+\delta-\sigma$
$\displaystyle=19+12-11=20.$

Then since
$e(X_{\mathsf{ban}})=24=2(h^{1,1}(X_{\mathsf{ban}})-h^{2,1}(X_{\mathsf{ban}}))$
we see that $h^{2,1}(X_{\mathsf{ban}})=8$ (in particular, the 8 dimensional
space of banana manifolds constructed above is the whole deformation space). ∎

######  Lemma 29.

Let $C_{1}$, $C_{2}$, $C_{3}$ be the banana curves in a singular fiber
$F_{\mathsf{sing}}$. Then
$\Gamma=\operatorname{Ker}(\pi_{*}:H_{2}(X_{\mathsf{ban}},{\mathbb{Z}})\to
H_{2}({\mathbb{P}}^{1},{\mathbb{Z}}))$ is spanned by the classes of $C_{1}$,
$C_{2}$, and $C_{3}$.

###### Proof.

By the previous discussion of the conifold transition, we have that

$H_{2}(X_{\mathsf{ban}},{\mathbb{Z}})\cong
H_{2}(X_{\mathsf{Sch}},{\mathbb{Z}})\oplus{\mathbb{Z}}$

where the ${\mathbb{Z}}$ factor is spanned by the exceptional curves of the
conifold resolution, in particular, the exceptional curves are all homologous.
Let

$p_{1},p_{2}:X_{\mathsf{ban}}\to S$

be the projections on to the first and second factors of the fibered product.
Then

$\Gamma=\operatorname{Ker}(p_{1})_{*}\cup\operatorname{Ker}(p_{2})_{*}.$

This follows since if a connected algebraic 1-cycle in $X_{\mathsf{ban}}$ maps
to a point in ${\mathbb{P}}^{1}$, then by properties of the fiber product, it
either maps to a point under $p_{1}$ or $p_{2}$. Moreover, it is enough to
consider algebraic cycles since $H_{2}(X_{\mathsf{ban}},{\mathbb{Z}})\cong
H^{4}(X_{\mathsf{ban}},{\mathbb{Z}})\subset H^{2,2}(X_{\mathsf{ban}})$ has no
torsion and the Hodge conjecture holds for threefolds.

Let $x_{1},\dotsc,x_{12}\in{\mathbb{P}}^{1}$ be points corresponding to
singular fibers and we label the banana curves $C_{1}(i),C_{2}(i),C_{3}(i)$ in
the singular fiber over $x_{i}$ such that $C_{3}(i)$ is an exceptional curve
for the conifold resolution and $C_{1}(i)$ and $C_{2}(i)$ are such that

$p_{j}^{-1}(p_{j}(C_{3}(i)))=C_{j}(i)\cup C_{3}(i),\quad j=1,2.$

The fibers of $p_{j}:X_{\mathsf{ban}}\to S$ are all irreducible except for the
fibers $C_{j}(i)\cup C_{3}(i)$ for $i=1,\dotsc,12$. Since the exceptional
curves are all homologous, we have a single class $C_{3}=C_{3}(i)$ for all $i$
and since the fibers of $p_{j}$ are all homologous we get

$C_{1}(i)+C_{3}=C_{1}(i^{\prime})+C_{3},\text{ and
}C_{2}(i)+C_{3}=C_{2}(i^{\prime})+C_{3}.$

Hence we all $C_{1}(i)$ are homologous to a single class $C_{1}$ and all
$C_{2}(i)$ are homologous to a single class $C_{2}$ and $\Gamma$ is spanned by
$C_{1},C_{2},C_{3}$.

∎

###  5.2. Proof of Proposition 24.

Let $C_{\mathsf{sing}}\subset S$ be a singular fiber of $S\to{\mathbb{P}}^{1}$
and let
${C}^{\scriptscriptstyle{\\#}}_{\mathsf{sing}}\subset{S}^{\scriptscriptstyle{\\#}}$
be an isomorphic copy so that

$X_{\mathsf{sing}}=S\times_{{\mathbb{P}}^{1}}{S}^{\scriptscriptstyle{\\#}}\text{
and
}F_{\mathsf{sing}}=C_{\mathsf{sing}}\times{C}^{\scriptscriptstyle{\\#}}_{\mathsf{sing}}.$

$C_{\mathsf{sing}}$ is a nodal rational curve whose normalization
${\mathbb{P}}^{1}\to C_{\mathsf{sing}}$ identifies the points
$0,\infty\in{\mathbb{P}}^{1}$ to the nodal point $n\in C_{\mathsf{sing}}$.
Thus $F^{\mathsf{norm}}_{\mathsf{sing}}$ is isomorphic to
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ and the map
$F^{\mathsf{norm}}_{\mathsf{sing}}\to F_{\mathsf{sing}}$ identifies
$0\times{\mathbb{P}}^{1}$ and $\infty\times{\mathbb{P}}^{1}$ to
$n\times{C}^{\scriptscriptstyle{\\#}}_{\mathsf{sing}}$ and identifies
${\mathbb{P}}^{1}\times 0$ and ${\mathbb{P}}^{1}\times\infty$ to
$C_{\mathsf{sing}}\times{n}^{\scriptscriptstyle{\\#}}$. Note that this map is
${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$ equivariant with respect to the usual
toric action on ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ and our constructed
action on $F_{\mathsf{sing}}$.

The formal neighborhood $\widehat{F}_{\mathsf{ban}}$ is obtained from
$\widehat{F}_{\mathsf{sing}}$ by blowing up the diagonal
$\Delta\subset\widehat{F}_{\mathsf{sing}}$. The blow down
$\widehat{F}_{\mathsf{ban}}\to\widehat{F}_{\mathsf{sing}}$ contracts the
exceptional ${\mathbb{P}}^{1}$ to the conifold point
$n\times{n}^{\scriptscriptstyle{\\#}}$ and is an isomorphism elsewhere. We
study this blowup and the normalizations in formal local coordinates about
$n\times{n}^{\scriptscriptstyle{\\#}}$:

Let $(x,y)$ and
$({x}^{\scriptscriptstyle{\\#}},{y}^{\scriptscriptstyle{\\#}})$ be formal
local coordinates on $S$ and ${S}^{\scriptscriptstyle{\\#}}$ about the points
$n$ and ${n}^{\scriptscriptstyle{\\#}}$ such that the maps
$S\to{\mathbb{P}}^{1}$ and ${S}^{\scriptscriptstyle{\\#}}\to{\mathbb{P}}^{1}$
are given by $t=xy$ and
$t={x}^{\scriptscriptstyle{\\#}}{y}^{\scriptscriptstyle{\\#}}$ where $t$ is a
formal local coordinate on ${\mathbb{P}}^{1}$. Moreover, we may choose the
coordinates so that the action of
$(\lambda,{\lambda}^{\scriptscriptstyle{\\#}})\in{\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$
on $\widehat{F}_{\mathsf{sing}}$ is given by

$(x,y,{x}^{\scriptscriptstyle{\\#}},{y}^{\scriptscriptstyle{\\#}})\mapsto(\lambda
x,\lambda^{-1}y,{\lambda}^{\scriptscriptstyle{\\#}}{x}^{\scriptscriptstyle{\\#}},({\lambda}^{\scriptscriptstyle{\\#}})^{-1}{y}^{\scriptscriptstyle{\\#}}).$

Then the formal neighborhood of
$n\times{n}^{\scriptscriptstyle{\\#}}\in\widehat{F}_{sing}$ is isomorphic to

$\left\\{xy={x}^{\scriptscriptstyle{\\#}}{y}^{\scriptscriptstyle{\\#}}\right\\}\subset\operatorname{Spec}{\mathbb{C}}[[x,y,{x}^{\scriptscriptstyle{\\#}},{y}^{\scriptscriptstyle{\\#}}]]$

where the closed fiber $F_{\mathsf{sing}}\subset\widehat{F}_{\mathsf{sing}}$
is given by

$\left\\{xy={x}^{\scriptscriptstyle{\\#}}{y}^{\scriptscriptstyle{\\#}}=0\right\\}.$

The blowup of $\widehat{F}_{\mathsf{sing}}$ along the diagonal
$\Delta=\\{x={x}^{\scriptscriptstyle{\\#}},y={y}^{\scriptscriptstyle{\\#}}\\}$
is canonically isomorphic to the blowup along any of the planes111111There are
canonically two small resolutions of the conifold singularity
$xy={x}^{\scriptscriptstyle{\\#}}{y}^{\scriptscriptstyle{\\#}}$. These two are
given by blowing up any plane given by affine cone over a line in one of the
two rulings of the quadric surface
$\\{xy={x}^{\scriptscriptstyle{\\#}}{y}^{\scriptscriptstyle{\\#}}\\}\subset{\mathbb{P}}^{3}$.

$\left\\{ax={a}^{\scriptscriptstyle{\\#}}{x}^{\scriptscriptstyle{\\#}},{a}^{\scriptscriptstyle{\\#}}y=a{y}^{\scriptscriptstyle{\\#}}\right\\},\quad\quad(a:{a}^{\scriptscriptstyle{\\#}})\in{\mathbb{P}}^{1}.$

Choosing $(a:{a}^{\scriptscriptstyle{\\#}})=(1:0)$, we get two affine toric
charts for the blow up with coordinate rings given by

$\displaystyle{\mathbb{C}}[[x,{x}^{\scriptscriptstyle{\\#}},y,{y}^{\scriptscriptstyle{\\#}}]][u]/(x-u{y}^{\scriptscriptstyle{\\#}},{x}^{\scriptscriptstyle{\\#}}-uy)$
$\displaystyle\cong{\mathbb{C}}[[y,{y}^{\scriptscriptstyle{\\#}}]][u],$
$\displaystyle{\mathbb{C}}[[x,{x}^{\scriptscriptstyle{\\#}},y,{y}^{\scriptscriptstyle{\\#}}]][v]/(y-v{x}^{\scriptscriptstyle{\\#}},{y}^{\scriptscriptstyle{\\#}}-vx)$
$\displaystyle\cong{\mathbb{C}}[[x,{x}^{\scriptscriptstyle{\\#}}]][v].$

The coordinate change between the charts is given by

$u=v^{-1},\quad
y=v{x}^{\scriptscriptstyle{\\#}},\quad{y}^{\scriptscriptstyle{\\#}}=vx$

and the induced $(\lambda,{\lambda}^{\scriptscriptstyle{\\#}})$ action is
given by

$(\lambda
x,{\lambda}^{\scriptscriptstyle{\\#}}{x}^{\scriptscriptstyle{\\#}},(\lambda{\lambda}^{\scriptscriptstyle{\\#}})^{-1}v),\quad(\lambda^{-1}y,({\lambda}^{\scriptscriptstyle{\\#}})^{-1}{y}^{\scriptscriptstyle{\\#}},\lambda{\lambda}^{\scriptscriptstyle{\\#}}u).$

The coordinates on the blowup of the formal neighborhood of
$n\times{n}^{\scriptscriptstyle{\\#}}\in\widehat{F}_{\mathsf{sing}}$ and the
corresponding blowdown are encoded in the following momentum “polytopes” where
the coordinate lines are labelled by their corresponding variables:

$x$${x}^{\scriptscriptstyle{\\#}}$$v$${y}^{\scriptscriptstyle{\\#}}$$y$$u$$x$$y$${x}^{\scriptscriptstyle{\\#}}$${y}^{\scriptscriptstyle{\\#}}$

The $x$ and $y$ coordinates about $n\in S$ (which correspond to the two
branches of the node in $C_{\mathsf{sing}}$) become local coordinates near $0$
and $\infty$ in ${\mathbb{P}}^{1}$, the normalization of $C_{\mathsf{sing}}$.

Thus $F^{\mathsf{norm}}_{\mathsf{sing}}$ and
$F^{\mathsf{norm}}_{\mathsf{ban}}$, when endowed with local coordinate rings
given by $\sigma^{*}\widehat{\mathcal{O}}_{\widehat{F}_{\mathsf{sing}}}$ and
$\tau^{*}\widehat{\mathcal{O}}_{\widehat{F}_{\mathsf{ban}}}$ respectively,
have momentum polytopes given by:

$x$${x}^{\scriptscriptstyle{\\#}}$${y}^{\scriptscriptstyle{\\#}}$$y$$x$${x}^{\scriptscriptstyle{\\#}}$${y}^{\scriptscriptstyle{\\#}}$$y$$y$${x}^{\scriptscriptstyle{\\#}}$${y}^{\scriptscriptstyle{\\#}}$$x$${y}^{\scriptscriptstyle{\\#}}$$x$${x}^{\scriptscriptstyle{\\#}}$$y$$v$$x$${x}^{\scriptscriptstyle{\\#}}$$y$$u$${y}^{\scriptscriptstyle{\\#}}$${x}^{\scriptscriptstyle{\\#}}$$x$$v$$u$${y}^{\scriptscriptstyle{\\#}}$$y$$x$${x}^{\scriptscriptstyle{\\#}}$$v$${y}^{\scriptscriptstyle{\\#}}$$u$$y$

We see that

$F^{\mathsf{norm}}_{\mathsf{ban}}\cong\operatorname{Bl}_{(0,0)(\infty,\infty)}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1}),\quad\quad\widehat{F}^{\mathsf{norm}}_{\mathsf{ban}}\cong\widehat{\operatorname{Bl}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})}$

as asserted.

##  6\. BPS invariants from a Donaldson-Thomas partition function in product
form.

The Gopakumar-Vafa invariants (a.k.a. BPS invariants) $n_{\beta}^{g}(X)$ can
be defined in terms of Gromov-Witten or Donaldson-Thomas invariants and
(conjecturally) have better finiteness properties. Unlike the Donaldson-Thomas
invariants, it is expected that there are only a finite number of non-zero
Gopakumar-Vafa invariants for each curve class $\beta$.

One definition of Gopakumar-Vafa invariants, which is equivalent to the usual
one given in terms of Gromov-Witten invariants via the Gopakumar-Vafa formula,
is the following:

######  Definition-Theorem 30.

Let $X$ be any Calabi-Yau threefold and suppose that the Donaldson-Thomas
partition function is given by

(15) $Z^{\,}(X)=\prod_{\beta\in
H_{2}(X)}\prod_{k\in{\mathbb{Z}}}\left(1-p^{k}Q^{\beta}\right)^{-a(\beta,k)}$

for some integers $a(\beta,k)\in{\mathbb{Z}}$. Then for $\beta\neq 0\in
H_{2}(X)$ and $g\in{\mathbb{Z}}$ the Gopakumar-Vafa invariants
$n^{g}_{\beta}(X)$ can be defined by the formula

$\sum_{g}n^{g}_{\beta}(X)\,\left(y^{\frac{1}{2}}+y^{-\frac{1}{2}}\right)^{2g}=\left(y^{\frac{1}{2}}+y^{-\frac{1}{2}}\right)^{2}\,\sum_{k\in{\mathbb{Z}}}a(\beta,k)\,(-y)^{k}.$

This is equivalent, assuming the Gromov-Witten/Donaldson-Thomas
correspondence, to defining $n^{g}_{\beta}(X)$ in terms of the Gromov-Witten
invariants via the Gopakumar-Vafa formula.

######  Remark 31.

It is expected that $n^{g}_{\beta}(X)$ is zero unless $0\leq g\leq
g_{\mathsf{max}}$ where $g_{\mathsf{max}}$ is the maximal arithmetic genus of
curves in the class $\beta$. Note that the left hand side of the above formula
is a palandromic polynomial in $y$ (i.e. invariant under $y\leftrightarrow
y^{-1}$). One can see that the invariants $n^{g}_{\beta}(X)$ are well-defined
by the above definition as follows. Any power series
$Z\in{\mathbb{Z}}((p))[[Q]]$ can be uniquely written in the form of equation
(15) for some collection of integers $a(\beta,k)$. By a theorem of Bridgeland
[9], the right hand side of equation (15) is a rational function of $y$,
invariant under $y\leftrightarrow y^{-1}$, possibly having poles at $y=-1$.
Such functions have a basis given by the functions

$\left(y^{\frac{1}{2}}+y^{-\frac{1}{2}}\right)^{2g}=y^{-g}(1+y)^{2g},\quad
g\in{\mathbb{Z}}$

and so the left hand side of (15) is well defined and uniquely determines
$n^{g}_{\beta}(X)$. The conjecture that $n_{\beta}^{g}(X)=0$ if $g<0$ is
equivalent to the right hand side of (15) not having a pole.

###### Proof.

The Gopakumar-Vafa formula [17] expresses the reduced Gromov-Witten potential
function $F^{\prime}(X)$ of a Calabi-Yau threefold $X$ in terms of
conjecturally integer invariants $n_{\beta}^{g}(X)$, which are commonly called
Gopakumar-Vafa invariants, or BPS invariants:

$F^{\prime}(X)=\sum_{g\geq 0}\sum_{\beta\neq
0}GW^{g}_{\beta}(X)\lambda^{2g-2}Q^{\beta}=\sum_{g\geq 0}\sum_{\beta\neq
0}\sum_{m>0}n_{\beta}^{g}(X)\frac{1}{m}\left(2\sin\frac{m\lambda}{2}\right)^{2g-2}Q^{m\beta}.$

One can re-express the above formula in terms of the partition function
$Z^{\prime}(X)=\exp\left(F^{\prime}(X)\right)$ as follows121212This expression
essentially appears in [24, eqn 18], although our convention for
$\binom{n}{k}$ allows us to write the formula more uniformly. :

$Z^{\prime}(X)=\prod_{\beta\neq
0}\prod_{m\in{\mathbb{Z}}}\left(1-Q^{\beta}p^{m}\right)^{-\sum_{g\geq
0}n^{g}_{\beta}(X)\binom{2g-2}{g-1-m}(-1)^{m}}$

where $p=\exp\left(i\lambda\right)$ and
$\binom{n}{k}=\frac{n(n-1)\dotsb(n-k+1)}{k!}$ is defined for all $k\geq 0$ and
$n\in{\mathbb{Z}}$.

The Donaldson-Thomas partition function can always be written in the following
form

$Z^{\,}(X)=\prod_{\beta}\prod_{m\in{\mathbb{Z}}}\left(1-Q^{\beta}p^{m}\right)^{-c(\beta,m)}$

for some $c(\beta,m)\in{\mathbb{Z}}$. Then for $\beta\neq 0$,

$c(\beta,m)=\sum_{n=0}^{\beta}n_{\beta}^{g}(X)\binom{2g-2}{g-1-m}(-1)^{m}.$

The Definition-Theorem is then an easy consequence of the binomial theorem.

∎

##  Appendix A The Gromov-Witten potentials are Siegel modular forms (with
Stephen Pietromonaco)

###  A.1. Overview.

Let $F_{g}(Q_{1},Q_{2},Q_{3})$ be the genus $g\geq 2$ Gromov-Witten potential
for fiber classes in $X_{\mathsf{ban}}$. Namely, let

$F_{g}(Q_{1},Q_{2},Q_{3})=\sum_{d_{1},d_{2},d_{3}\geq
0}GW^{g}_{\mathbf{\underline{d}}}(X_{\mathsf{ban}})\,Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}$

where $GW^{g}_{\mathbf{\underline{d}}}(X_{\mathsf{ban}})$ is the genus $g$
Gromov-Witten invariant of $X_{\mathsf{ban}}$ in the class
$\beta_{\mathbf{\underline{d}}}=d_{1}C_{1}+d_{2}C_{2}+d_{3}C_{3}$.

Assuming that the Gromov-Witten/Donaldson-Thomas correspondence conjectured in
[28] holds, we may compute $F_{g}(Q_{1},Q_{2},Q_{3})$ using the formula for
the Donaldson-Thomas partition function derived in the main text. The main
result of this appendix is that $F_{g}$ is an explicit genus 2 Siegel modular
form of weight $2g-2$.

######  Definition 32.

Let $\Omega=\left(\begin{smallmatrix}\tau&z\\\
z&\sigma\end{smallmatrix}\right)\in\mathbb{H}_{2}$ be the standard coordinates
on the genus 2 Siegel upper half plane. A holomorphic (resp. meromorphic)
genus 2 Siegel modular form of weight $k$ is a holomorphic (resp. meromorphic)
function $F(\Omega)$ satisfying

$F\left((A\Omega+B)(C\Omega+D)^{-1}\right)=\operatorname{det}\left((C\Omega+D)^{k}\right)F(\Omega)$

for all $\left(\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\right)\in
Sp_{4}({\mathbb{Z}})$ (c.f. [16]). We denote the space of meromorphic genus 2
Siegel forms of weight $k$ by $\mathsf{Siegel}_{k}$.

As we will detail in § A.2, a standard way to construct a genus 2 Siegel
modular form of weight $2g-2$ is to take the so-called Maass lift of a Jacobi
form of weight $2g-2$ and index 1. Moreover, such Jacobi forms are easily
obtained by taking a modular form of weight $2g$ and multiplying it by
$\phi_{-2,1}$, the unique weak Jacobi form of weight $-2$ and index 1\. Some
authors call this the Skoruppa lift of the modular form. Schematically we have

${\mathsf{Mod}_{2g}}$${\mathsf{Jac}_{2g-2,1}}$${\mathsf{Siegel}_{2g-2}}$SkoruppaMaass

We call the composition the Skoruppa-Maass lift. It takes weight $2g$ modular
forms to genus 2, weight $2g-2$, Siegel modular forms131313This lift is
different from the famous Saito-Kurokawa lift which also takes
$\mathsf{Mod}_{2g}$ to $\mathsf{Siegel}_{2g-2}$. While both use the Maass
lift, the Saito-Kurokawa lift uses a combination of the Shimura correspondence
and a lift studied by Eichler-Zagier to go from $\mathsf{Mod}_{2g}$ to
$\mathsf{Jac}_{2g-2,1}$ [16].. Our main result is:

######  Theorem 33.

Assume that the Gromov-Witten/Donaldson-Thomas correspondence holds for
$X_{\mathsf{ban}}$. Then for $g\geq 2$, the genus $g$ Gromov-Witten potentials
$F_{g}(Q_{1},Q_{2},Q_{3})$ of $X_{\mathsf{ban}}$ are meromorphic genus 2
Siegel modular forms of weight $2g-2$ where

$Q_{1}=e^{2\pi i\,z},\quad Q_{2}=e^{2\pi i(\tau-z)},\quad Q_{3}=e^{2\pi
i(\sigma-z)}.$

Specifically, $F_{g}$ is the Skoruppa-Maass lift of $a_{g}E_{2g}(\tau)$, the
$2g$-th Eisenstein series times the constant
$a_{g}=\frac{6|B_{2g}|}{g(2g-2)!}$ where $B_{2g}$ is the $2g$-th Bernoulli
number.

The ring of holomorphic, even weight, genus 2 Siegel modular forms is a
polynomial ring generated by the Igusa cusp forms $\chi_{10}$ and $\chi_{12}$
of weight 10 and 12, and the Siegel Eisenstein series $\mathcal{E}_{4}$ and
$\mathcal{E}_{6}$ of weight 4 and 6 [20, 21]. Although $F_{g}$ are
meromorphic, the denominators can be determined explicitly:

######  Corollary 34.

For $g\geq 2$, the product $\chi_{10}^{g-1}\cdot F_{g}$ is a holomorphic
Siegel form of weight $12g-12$.

This corollary follows from Aoki’s proof of [2, Thm 14]. In terms of the
generators $\chi_{10},\chi_{12},\mathcal{E}_{4},\mathcal{E}_{6}$, the first
few potentials are given explicitly as

$\displaystyle F_{2}$
$\displaystyle=\frac{1}{240}\left(\frac{\chi_{12}}{\chi_{10}}\right),$
$\displaystyle F_{3}$
$\displaystyle=\frac{-1}{60480}\left(6\,\mathcal{E}_{4}-5\left(\frac{\chi_{12}}{\chi_{10}}\right)^{2}\right),$
$\displaystyle F_{4}$
$\displaystyle=\frac{1}{3628800}\left(\frac{35}{2}\left(\frac{\chi_{12}}{\chi_{10}}\right)^{3}-\frac{63}{2}\left(\frac{\chi_{12}}{\chi_{10}}\right)\mathcal{E}_{4}+15\,\mathcal{E}_{6}\right),$
$\displaystyle F_{5}$
$\displaystyle=\frac{-1}{106444800}\left(\frac{-175}{3}\left(\frac{\chi_{12}}{\chi_{10}}\right)^{4}+140\left(\frac{\chi_{12}}{\chi_{10}}\right)^{2}\mathcal{E}_{4}-\frac{200}{3}\left(\frac{\chi_{12}}{\chi_{10}}\right)\mathcal{E}_{6}-14\,\mathcal{E}_{4}^{2}\right).$

Note that the prefactor is given by

$F_{g}(0,0,0)=\frac{12B_{2g-2}|B_{2g}|}{g(4g-4)(2g-2)!},$

the degree 0 genus $g$ Gromov-Witten invariant of $X_{\mathsf{ban}}$. See [34]
for details of this computation.

######  Remark 35.

Theorem 33 has a nice interpretation in terms of mirror symmetry. The Gromov-
Witten potentials are functions of local coordinates on the Kähler moduli
space. Under mirror symmetry, these become coordinates on the complex moduli
space of the mirror. Since the arguments of a genus 2 Siegel modular form are
coordinates on the moduli space of genus 2 curves (or Abelian surfaces), we
expect the complex moduli space of $\check{X}_{\mathsf{ban}}$, the mirror of
the banana manifold, to contain a subspace isomorphic to the moduli space of
genus 2 curves. Indeed, it has already been observed that the mirror of a
_local_ banana configuration should be a genus 2 curve [1, 18, 36].

######  Remark 36.

We also determine the genus 0 and genus 1 potentials. Up to degree 0 terms
(which are unstable for $g=0$ or $g=1$), $F_{0}$ is the Skoruppa-Maass lift of
the constant $12$ (viewed as a weight 0 modular form) and $F_{1}$ is the Maass
lift of $12\wp\cdot\phi_{-2,1}$ where $\wp$ is the Weierstrass
$\wp$-function141414Interestingly, the Jacobi form
$\phi_{0,1}=12\wp\cdot\phi_{-2,1}$ is also equal to
$\frac{1}{2}\operatorname{Ell}(K3)$. .

###  A.2. Modular forms and lifts

######  Definition 37.

A weight $k$ _modular form_ is a holomorphic function $f(\tau)$ on
${\mathbb{H}}=\\{\tau\in{\mathbb{C}},\operatorname{Im}\tau>0\\}$ satisfying

  * •

For all $\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in
SL_{2}({\mathbb{Z}})$

$f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau),$

  * •

$f(\tau)$ admits a Fourier series of the form

$f(\tau)=\sum_{n=0}^{\infty}a_{n}q^{n},\quad\quad q=e^{2\pi i\tau}.$

We denote the space of weight $k$ modular forms by $\mathsf{Mod}_{k}$. We
sometimes abuse notation by writing $f(q)$ for the Fourier expansion of
$f(\tau)$.

The $2g$-th Eisenstein series is given by

$E_{2g}(q)=1-\frac{4g}{B_{2g}}\sum_{n=1}^{\infty}\sum_{d|n}d^{2g-1}q^{n}$

where $B_{2g}$ is the $2g$-th Bernoulli number. $E_{2g}$ is a modular form of
weight $2g$ for all $g\geq 2$.

######  Definition 38.

A _weak Jacobi form_ of weight $k$ and index $m$ is a holomorphic function
$\phi(\tau,z)$ on ${\mathbb{H}}\times{\mathbb{C}}$ satisfying

  * •

For all $\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in
SL_{2}({\mathbb{Z}})$

$\phi\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right)=(c\tau+d)^{k}e^{\frac{2\pi
icmz^{2}}{c\tau+d}}\phi(\tau,z),$

  * •

for all $u,v\in{\mathbb{Z}}$,

$\phi(\tau,z+u\tau+v)=e^{-2\pi im(\tau u^{2}+2zu)}\phi(\tau,z),$

  * •

and $\phi$ admits a Fourier expansion of the form

$\phi(\tau,z)=\sum_{n=0}^{\infty}\sum_{l\in{\mathbb{Z}}}c_{\phi}(n,l)q^{n}y^{l}$

where $q=e^{2\pi i\tau}$ and $y=e^{2\pi iz}$.

In the case of index $m=1$, the Fourier coefficients $c_{\phi}(n,l)$ only
depend on $4n-l^{2}$ and so we will sometimes in this case write

$c_{\phi}(4n-l^{2})=c_{\phi}(n,l).$

We denote the space of weak Jacobi forms of weight $k$ and index $m$ by
$\mathsf{Jac}_{k,m}$. We sometimes abuse notation by writing $\phi(q,y)$ for
the Fourier expansion of $\phi(\tau,z)$. A basic example is given by
$\phi_{-2,1}$ whose Fourier expansion is given by

$\phi_{-2,1}(q,y)=y^{-1}(1-y)^{2}\prod_{n=1}^{\infty}\frac{(1-yq^{n})^{2}(1-y^{-1}q^{n})^{2}}{(1-q^{n})^{4}}.$

Up to a multiplicative constant, $\phi_{-2,1}$ is the unique weak Jacobi form
of weight -2 and index 1. We also will use the Weierstrass $\wp$-function:

$\wp(q,y)=\frac{1}{12}+\frac{y}{(1-y)^{2}}+\sum_{n=1}^{\infty}\sum_{d|n}d(y^{d}+y^{-d}-2)q^{n}$

and we note that up to a multiplicative constant

$\phi_{0,1}=12\cdot\phi_{-2,1}\cdot\wp$

is the unique weak Jacobi form of weight 0 and index 1.

The product of a weak Jacobi form of weight $k$ and index $m$ with a modular
form of weight $n$ is a weak Jacobi form of $k+n$ and index $m$. In
particular, multiplication by $\phi_{-2,1}$ defines a map which we call the
_Skoruppa lift_ :

${\mathsf{Mod}_{2g}}$${\mathsf{Jac}_{2g-2,1}.}$Skoruppa

######  Definition 39.

Let $k$ be even. For $m$ a non-negative integer, the _$m$ -th Hecke operator_

$V_{m}:\mathsf{Jac}_{k,1}\to\mathsf{Jac}_{k,m}$

is given by taking

$\phi(q,y)=\sum_{n=0}^{\infty}\sum_{l\in{\mathbb{Z}}}c_{\phi}(n,l)q^{n}y^{l}$

to

$(\phi|V_{m})=\sum_{n=0}^{\infty}\sum_{r\in{\mathbb{Z}}}\sum_{d|(n,r,m)}d^{k-1}c_{\phi}\left(\frac{nm}{d^{2}},\frac{r}{d}\right)q^{n}y^{r}$

for $m>0$ and

$(\phi|V_{0})=c_{\phi}(0,0)\frac{-B_{k}}{2k}+\sum_{n=0}^{\infty}\sum_{\begin{smallmatrix}r\in{\mathbb{Z}}\\\
r>0\text{ if
}n=0\end{smallmatrix}}\sum_{d|(n,r)}d^{k-1}c_{\phi}\left(0,\frac{r}{d}\right)q^{n}y^{r}$

for $m=0$.

######  Definition 40.

Let $\phi\in\mathsf{Jac}_{k,1}$ with $k$ even. The _Maass lift_ of $\phi$ is
given by

$ML(\phi)=\sum_{m=0}^{\infty}(\phi|V_{m})Q^{m}.$

The following is due to Eichler-Zagier [16] in the case of holomorphic Jacobi
forms, and Borcherds [6, Thm 9.3] and Aoki [2] in the case of weak Jacobi
forms151515We thank H. Aoki and G. Oberdieck for discussion on this point.:

######  Theorem 41.

The Maass lift of a weak Jacobi form of weight $k>0$ and index 1 is a
meromorphic genus 2 Siegel modular form of weight $k$. If the Jacobi form is
holomorphic, then the Maass lift is also holomorphic. Here $Q=e^{2\pi
i\sigma}$, $q=e^{2\pi i\tau}$, and $y=e^{2\pi iz}$ where
$\left(\begin{smallmatrix}\tau&z\\\
z&\sigma\end{smallmatrix}\right)\in{\mathbb{H}}_{2}$.

We may reformulate the Maass lift in terms of polylogarithms. Let

$\operatorname{Li}_{a}(x)=\sum_{n=1}^{\infty}n^{-a}x^{n}.$

Then a straightforward computation yields the following

######  Lemma 42.

Let
$\phi=\sum_{n=0}^{\infty}\sum_{l\in{\mathbb{Z}}}c_{\phi}(4n-l^{2})q^{n}y^{l}\in\mathsf{Jac}_{k,1}$
with $k$ even. Then

$ML(\phi)=c_{\phi}(0)\frac{-B_{k}}{2k}+\sum_{n,m\geq
0}\sum_{\begin{smallmatrix}l\in{\mathbb{Z}}\\\ l>0\text{ if
}(n,m)=(0,0)\end{smallmatrix}}c_{\phi}(4nm-l^{2})\operatorname{Li}_{1-k}(Q^{m}q^{n}y^{l}).$

###  A.3. The $\lambda$ expansion of
$\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t)$.

Recall from § 2 that the coefficients $c(d,k)$ are defined by the expansion of
the equivariant elliptic genus of ${\mathbb{C}}^{2}$:

$\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t)=\sum_{n=0}^{\infty}\sum_{l,k\in{\mathbb{Z}}}\,c(4n-l^{2},k)\,q^{n}y^{l}t^{k}.$

Let $t=e^{i\lambda}$. Theorem 4.4 in Zhou [39] gives the expansion of
$\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t)$ as a Laurent series in
$\lambda$. His result is161616 We use the $r=1$ case of Zhou’s theorem. Our
$\lambda$ is $2\pi t$ in Zhou’s notation (his $t$ is not our $t$). There is a
typo in his formula: the $\eta(\tau)$ should be $\eta(\tau)^{3}$.

$\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},e^{i\lambda})=\lambda^{-2}\cdot\phi_{-2,1}(q,y)\cdot\left(1+\wp(q,y)\lambda^{2}+\sum_{g=2}^{\infty}\frac{|B_{2g}|}{2g(2g-2)!}E_{2g}(q)\lambda^{2g}\right).$

Let

$\psi_{2g-2}(q,y)=\operatorname{Coef}_{\lambda^{2g-2}}\left[\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},e^{i\lambda})\right]$

so that Zhou’s result can be expressed as

$\psi_{2g-2}(q,y)=\phi_{-2,1}(q,y)\cdot\begin{cases}1&g=0\\\ \wp(q,y)&g=1\\\
\frac{|B_{2g}|}{2g(2g-2)!}E_{2g}(q)&g>1\end{cases}$

We observe that $\psi_{2g-2}$ is a weak Jacobi form of weight $2g-2$ and index
1 and consequently has an expansion

$\psi_{2g-2}(q,y)=\sum_{n=0}^{\infty}\sum_{l\in{\mathbb{Z}}}c_{2g-2}(4n-l^{2})q^{n}y^{l},$

which defines the coefficients $c_{2g-2}(d)$.

Comparing coefficients in the $\lambda$ and the $t$ expansions of
$\operatorname{Ell}_{q,y}({\mathbb{C}}^{2},t=e^{i\lambda})$ we get the
following fundamental relationship between the coefficients $c_{2g-2}(d)$ and
$c(d,l)$:

(16)
$\sum_{g=0}^{\infty}c_{2g-2}(d)\lambda^{2g-2}=\sum_{l\in{\mathbb{Z}}}c(d,l)e^{il\lambda}.$

###  A.4. Gromov-Witten potentials via the GW/DT correspondence

The GW/DT correspondence is a conjectural equivalence between the Gromov-
Witten and the Donaldson-Thomas invariants of a Calabi-Yau threefold [28]. It
has been proven for a broad class of Calabi-Yau threefolds including complete
intersections in products of projective spaces [32], which unfortunately does
not include $X_{\mathsf{ban}}$.

However, if we assume that the GW/DT correspondence holds for
$X_{\mathsf{ban}}$, we may compute the genus $g$ Gromov-Witten potentials from
our formula for the Donaldson-Thomas partition function (Theorem 4). We define
the reduced genus $g$ Gromov-Witten potential (for banana curve classes) of
$X_{\mathsf{ban}}$ by

$F^{\prime}_{g}(Q_{1},Q_{2},Q_{3})=\sum_{\mathbf{\underline{d}}>0}GW^{g}_{\mathbf{\underline{d}}}(X_{\mathsf{ban}})Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}.$

Here $GW^{g}_{\mathbf{\underline{d}}}(X_{\mathsf{ban}})$ denotes the genus $g$
Gromov-Witten invariant in the class
$\beta_{\mathbf{\underline{d}}}=d_{1}C_{1}+d_{2}C_{2}+d_{3}C_{3}$ and
$\mathbf{\underline{d}}>0$ means $d_{i}\geq 0$ and
$(d_{1},d_{2},d_{3})\neq(0,0,0)$.

The GW/DT correspondence asserts that

$\sum_{g=0}^{\infty}F^{\prime}_{g}(Q_{1},Q_{2},Q_{3})\lambda^{2g-2}=\log\left(\frac{Z_{\Gamma}(X_{\mathsf{ban}})}{Z_{\Gamma}(X_{\mathsf{ban}})|_{Q_{i}=0}}\right)$

under the change of variables $p=e^{i\lambda}$.

We now prove Theorem 33. Applying Theorem 4 and using Equation (16), we get

$\displaystyle\sum_{g=0}^{\infty}F^{\prime}_{g}(Q_{1},Q_{2},Q_{3})\lambda^{2g-2}$
$\displaystyle=\log\left(\prod_{\mathbf{\underline{d}}>0}\,\prod_{k\in{\mathbb{Z}}}(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}})^{-12c(||\mathbf{\underline{d}}||,k)}\right)$
$\displaystyle=\sum_{\mathbf{\underline{d}}>0}\,\sum_{k\in{\mathbb{Z}}}12c(||\mathbf{\underline{d}}||,k)\sum_{n=1}^{\infty}\frac{1}{n}p^{nk}Q_{1}^{nd_{1}}Q_{2}^{nd_{2}}Q_{3}^{nd_{3}}$
$\displaystyle=12\sum_{\mathbf{\underline{d}}>0}\sum_{n=1}^{\infty}\frac{1}{n}Q_{1}^{nd_{1}}Q_{2}^{nd_{2}}Q_{3}^{nd_{3}}\sum_{g=0}^{\infty}c_{2g-2}(||\mathbf{\underline{d}}||)n^{2g-2}\lambda^{2g-2}.$

Thus we find that

$\displaystyle F^{\prime}_{g}(Q_{1},Q_{2},Q_{3})$
$\displaystyle=12\sum_{\mathbf{\underline{d}}>0}c_{2g-2}(||\mathbf{\underline{d}}||)\sum_{n=1}^{\infty}n^{2g-3}Q_{1}^{nd_{1}}Q_{2}^{nd_{2}}Q_{3}^{nd_{3}}$
$\displaystyle=12\sum_{\mathbf{\underline{d}}>0}c_{2g-2}(||\mathbf{\underline{d}}||)\operatorname{Li}_{3-2g}(Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}).$

Substituting

$Q_{1}=e^{2\pi iz}=y,\quad\quad Q_{2}=e^{2\pi i(\tau-z)}=qy^{-1},\quad\quad
Q_{3}=e^{2\pi i(\sigma-z)}=Qy^{-1},$

reindexing by

$d_{1}=l+n+m,\quad d_{2}=n,\quad d_{3}=m,$

so that $||\mathbf{\underline{d}}||=4nm-l^{2}$, and noting that
$\mathbf{\underline{d}}>0$ is equivalent to $n,m\geq 0,l\in{\mathbb{Z}}$ and
$l>0$ if $n=m=0$, we find that

$F_{g}^{\prime}(q,y,Q)=12\sum_{n,m\geq
0}\sum_{\begin{smallmatrix}l\in{\mathbb{Z}}\\\ l>0\text{ if
}(n,m)=(0,0)\end{smallmatrix}}c_{2g-2}(4nm-l^{2})\operatorname{Li}_{3-2g}(Q^{m}q^{n}y^{l}).$

For $g>1$, the full genus $g$ Gromov-Witten potential is the reduced potential
plus the constant term:

$F_{g}=GW^{g}_{\mathbf{\underline{0}}}(X_{\mathsf{ban}})+F^{\prime}_{g}.$

Using for example the formulas in [28, § 2.1] we know

$\displaystyle GW^{g}_{\mathbf{\underline{0}}}(X_{\mathsf{ban}})$
$\displaystyle=(-1)^{g}\frac{1}{2}e(X_{\mathsf{ban}})\frac{|B_{2g}|\cdot|B_{2g-2}|}{2g(2g-2)(2g-2)!}$
$\displaystyle=12\cdot\left(\frac{-B_{2g-2}}{4g-4}\right)\cdot\left(\frac{-|B_{2g}|}{g(2g-2)!}\right).$

Examining the $y^{0}q^{0}$ term of
$\psi_{2g-2}(q,y)=\phi_{-2,1}\cdot\frac{|B_{2g}|}{2g(2g-2)!}E_{2g}$ for $g>1$
yields

$c_{2g-2}(0)=\frac{-|B_{2g}|}{g(2g-2)!}$

and hence we find

$F_{g}=12\cdot\left(c_{2g-2}(0)\frac{-B_{2g-2}}{4g-4}+\sum_{n,m\geq
0}\sum_{\begin{smallmatrix}l\in{\mathbb{Z}}\\\ l>0\text{ if
}(n,m)=(0,0)\end{smallmatrix}}c_{2g-2}(4nm-l^{2})\operatorname{Li}_{3-2g}(Q^{m}q^{n}y^{l})\right).$

By Lemma 42, the above is exactly the Maass lift of $12\psi_{2g-2}$, and hence
we find that $F_{g}$ is the Skoruppa-Maass lift of $a_{g}E_{2g}$ where

$a_{g}=\frac{6|B_{2g}|}{g(2g-2)!}.$

This completes the proof of Theorem 33.

######  Remark 43.

Note that the proof of the theorem also shows that up to constant terms, the
genus 0 and the genus 1 Gromov-Witten potentials are given by the Maass lifts
of $12\phi_{-2,1}$ and $12\phi_{-2,1}\wp=\phi_{0,1}$ respectively. Although
these are weak Jacobi forms of index 1, they are not of positive weight, and
hence their Maass lifts are not guarenteed to be Siegel forms. See [34] for a
further discussion.

###  A.5. Gopakumar-Vafa invariants

In this section we give tables of values of the Gopakumar-Vafa invariants
$n^{g}_{a}(X_{\mathsf{ban}})$ for small values of $g$ and $a$. Since all
values are divisible by 12, we list
$\frac{1}{12}\,n^{g}_{a}(X_{\mathsf{ban}})$ (which can also be regarded as the
Gopakumar-Vafa invariants of a local banana configuration). We note that the
non-zero values have $a$ congruent to $0$ or $-1$ modulo 4 and so we organize
the tables as such.

$\frac{1}{12}\,n^{g}_{4n-1}(X_{\mathsf{ban}})$ | $g=0$ | $g=1$ | $g=2$ | $g=3$ | $g=4$ | $g=5$ | $g=6$
---|---|---|---|---|---|---|---
$n=0$ | 1 | 0 | 0 | 0 | 0 | 0 | 0
$n=1$ | 8 | -6 | 1 | 0 | 0 | 0 | 0
$n=2$ | 39 | -46 | 17 | -2 | 0 | 0 | 0
$n=3$ | 152 | -242 | 139 | -34 | 3 | 0 | 0
$n=4$ | 513 | -1024 | 800 | -304 | 56 | -4 | 0
$n=5$ | 1560 | -3730 | 3683 | -1912 | 548 | -82 | 5

$\frac{1}{12}\,n^{g}_{4n}(X_{\mathsf{ban}})$ | $g=0$ | $g=1$ | $g=2$ | $g=3$ | $g=4$ | $g=5$ | $g=6$
---|---|---|---|---|---|---|---
$n=0$ | -2 | 1 | 0 | 0 | 0 | 0 | 0
$n=1$ | -12 | 10 | -2 | 0 | 0 | 0 | 0
$n=2$ | -56 | 72 | -30 | 4 | 0 | 0 | 0
$n=3$ | -208 | 352 | -220 | 60 | -6 | 0 | 0
$n=4$ | -684 | 1434 | -1194 | 492 | -100 | 8 | 0
$n=5$ | -2032 | 5056 | -5252 | 2908 | -902 | 148 | -10

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