1 Introduction

In this paper we study the refined topological string partition function of a class of toric elliptically fibered Calabi–Yau threefolds which are dual to a set of parallel M5-branes with a transverse direction compactified to a circle. The size of the circle is related to the Kähler parameter of the elliptic curve class on the Calabi–Yau threefolds. We determine the Gopakumar–Vafa invariants for all curve classes and also show that for these Calabi–Yau threefolds the genus g free energy takes a particularly simple for and is given by the Eisenstein series. We also study the modular properties of the refined partition function and show that it satisfies the non-perturbative modular transformation discovered in [1]. As was shown in [1] for the case of \(N=1\), N being the number of M5-branes, the partition function is not modular invariant but satisfies a more involved transformation which maps the \(g_{s}\) to \(\frac{1}{g_{s}}\) where \(g_{s}\) is the topological string coupling constant—hence the name non-perturbative modular transformation.

This note is organized as follows. In Sect. 2 we discuss the Calabi–Yau geometry and the dual brane configuration. In Sect. 3 we use the refined topological vertex formalism to express the partition function as a trace of an operator acting on the fermionic Fock space. In Sect. 4 we determine the Gopakumar–Vafa invariants for all curve classes and the genus g fee energy. In Sect. 5 we express the partition function in terms of double elliptic gamma functions and show that it satisfies the non-perturbative modular transformation property.

Fig. 1
figure 1

a The (p, q) brane configuration dual to \(X_{N}\). The web lives on \(\mathbb {R}\times S^1\) and the horizontal lines are glued together. b The Newton polygon of \(X_{N}\)

Full size image

2 Elliptic CY3fold and dual brane configuration

The class elliptic Calabi–Yau threefolds we are interested in are dual to certain brane configurations which arose in the study of M-strings [2]. These Calabi–Yau threefolds are birationally equivalent \(\widehat{A}_{N-1}\times _{f}\mathbb {C}\) where \(\widehat{A}_{N-1}\) is affine \(A_{N-1}\) space blown up at N points and it is fibered over \(\mathbb {C}\) to obtain the Calabi–Yau threefold. We will denote these by \(X_{N}\).

The affine \(A_{N-1}\) space has N \(\mathbb {P}^1\)’s corresponding to the simple roots of affine SU(N). We denote these curve classes by \(C_{a}\) with \(a=1,2,\ldots , N\). The blow-up introduces N new curve classes, which we denote by \(M_{a}\) with \(a=1,2,\ldots ,N\). The class of the elliptic curve E is given by

$$\begin{aligned} E=C_{1}+C_{2}+\cdots +C_{N}. \end{aligned}$$
(2.1)

The complexified Kähler parameters associated with these curves are given by the Kähler form \(\omega \):

$$\begin{aligned} t_{a}=\int _{C_{a}}\,\omega ,\quad \rho =\int _{E}\,\omega ,\quad m=\int _{M_{a}}\,\omega ,\quad a=1,2,\ldots ,N.\nonumber \\ \end{aligned}$$
(2.2)

Because of Eq. (2.1) \(\rho \) is given by

$$\begin{aligned} \rho =t_{1}+t_{2}+\cdots +t_{N}. \end{aligned}$$
(2.3)

The (p, q) 5-brane web dual to \(X_{N}\) and the Newton polygon are given by Fig. 1a, b, respectively.

The Calabi–Yau threefolds \(X_{N}\) are dual to certain M5-brane configurations. Consider N M5-brane with worldvolume coordinates \(X^{A},A=0,1,2,3,4,5\). The transverse space is \(\mathbb {R}^{5}\) with coordinates \(X^{I},I=6,7,8,9,10\). We consider the \(X^6\) direction and separate the N M5-branes along this direction and then compactify this direction to a circle so that M5-branes are sprinkled on the \(X^6\) circle. The transverse space to the M5-branes is \(\mathbb {R}^{4}_{\perp }\times S^1\) and one can introduce a deformation by fibering this transverse space over the \(X^6\) circle so that as one goes around the \(X^6\) circle there is an SO(4) action on \(\mathbb {R}^4_{\perp }\) given by [2],

$$\begin{aligned} (w_{1},w_{2})\mapsto (e^{2\pi i m}w_{1},e^{-2\pi im}w_{2}). \end{aligned}$$
(2.4)

The deformation m breaks the maximal supersymmetry of the brane configuration. In the type IIB picture the theory living on the (p, q) brane web is five dimensional quiver gauge theory with eight supercharges. The partition function of this gauge theory can be calculated using Nekrasov’s instanton calculus or by applying the refined topological vertex formalism to the (p, q) brane web of Fig. 1a.

3 The partition function from topological vertex

The partition function of the (p, q) brane web shown in Fig. 1 can be calculated by applying refined topological vertex formalism to the web. The refined vertex requires a choice of preferred direction [3] which determines the form of the partition function as a sum over Young diagrams but the partition function itself is invariant under the change in the preferred direction. In this case the simplest choice for the preferred direction is vertical and, as we will see below, in this case the sum can be carried out exactly and partition function becomes an infinite product. We denote the topological string partition function of this Calabi–Yau threefold by \(Z_{N}\) and it is given by [4]

$$\begin{aligned} Z_{N}&:=\sum _{\vec {\lambda }}\prod _{a=0}^{N-1}\left[ \phantom {\left. \left( \sum _{\mu }(-Q_{m})^{|\mu |}\,C_{\lambda ^{t}_{(a)}\,\mu \emptyset }(t, q)\,C_{\lambda _{(a+1)}\mu ^{t}\emptyset }(q, t)\right) \right] }(-Q_{a})^{|\lambda _{(a)}|}\right. \nonumber \\&\quad \;\,\times \left. \left( \sum _{\mu }(-Q_{m})^{|\mu |}\,C_{\lambda ^{t}_{(a)}\,\mu \emptyset }(t, q)\,C_{\lambda _{(a+1)}\mu ^{t}\emptyset }(q, t)\right) \right] , \end{aligned}$$
(3.1)

where

$$\begin{aligned} C_{\lambda \mu \emptyset }(t,q)&=\Big (\frac{q}{t}\Big )^{\frac{|\lambda |-|\mu |}{2}}\nonumber \\&\quad \;\,\times \sum _{\eta }\Big (\frac{q}{t}\Big )^{\frac{|\eta |}{2}}\,s_{\lambda ^{t}/\eta }(t^{-\rho })\,s_{\mu /\eta }(q^{-\rho }) \end{aligned}$$
(3.2)

is the refined topological vertex. The length of the slanted lines in Fig. 1 are all equal to m and we have defined \(Q_{m}=e^{-m}\), similarly the length of the horizontal lines is \(T_{a}\) and we have defined \(Q_{a}=e^{-T_{a}}\) such that \(t_{a}=T_{a}+m\) is the distance between the two vertical lines. \(\lambda \), \(\mu \) and \(\eta \) are Young diagrams and \(s_{\lambda /\mu }(x_{1},x_{2},\ldots )\) is the skew-Schur polynomial in the variables \(x_{1},x_{2},\ldots \). \(t^{-\rho }\) denotes the following set of variables: \(t^{-\rho }=\{t^{1/2},t^{3/2},\ldots \}\).

Using repeatedly the identity

$$\begin{aligned} \sum _{\lambda }s_{\lambda ^{t}/\eta }(x)s_{\lambda /\sigma }(y)=\prod _{i,j}(1+x_{i}y_{j})\sum _{\tau }s_{\sigma ^{t}/\tau }(x)\,s_{\eta ^{t}/\tau ^{t}}(y) \end{aligned}$$
(3.3)

and the following properties of the skew-Schur functions:

$$\begin{aligned}&s_{\lambda ^t/\sigma ^t}(q^{-\rho })=s_{\lambda /\sigma }(-q^{\rho }),\nonumber \\&\quad \sum _{\eta }s_{\lambda /\eta }(x)\,s_{\eta /\sigma }(y)=s_{\lambda /\sigma }(x,y), \end{aligned}$$
(3.4)

the partition function in Eq. (3.1) can be written as

$$\begin{aligned} Z_{N}&=\Pi (Q_{m})^{N}\sum _{\mathbf {\lambda }, \mathbf {\tau }} Q_{\rho }^{|\lambda _{(0)}|}\nonumber \\&\quad \;\,\times \prod _{a=0}^{N-1}s_{\lambda _{(a)}/\tau _{(a+1)}}(\mathbf{x}_{a+1})\,s_{\lambda _{(a+1)}/\tau _{(a+1)}}(\mathbf{y}_{a+1}), \end{aligned}$$
(3.5)

where

$$\begin{aligned} Q_{\rho }&= \prod \limits _{a=0}^{N-1} (Q_a Q_m),\quad Q_{1,a+1}=(Q_{1}Q_{2}\ldots Q_{a})Q_{m}^{a}\nonumber \\ \mathbf{x}_{a+1}&= Q_m^{-1/2} Q_{1,a+1} \left\{ Q_{m}\sqrt{\frac{q}{t}}t^{\rho },t^{-\rho }\right\} ,\\ \quad \;\,\mathbf{y}_{a+1}&= Q_m^{-1/2} Q_{1,a+1}^{-1}\,\left\{ Q_{m}\sqrt{\frac{t}{q}}q^{-\rho },q^{\rho }\right\} , \end{aligned}$$

and

$$\begin{aligned} \Pi (x)=\prod _{i,j=1}^{\infty }\Big (1-x\,q^{-\rho _{i}}\,t^{-\rho _{j}}\Big ). \end{aligned}$$
(3.6)

In the above equations \(-log(Q_{\rho })\) is the Kähler parameter associated to the elliptic curve which is dual to the circle transverse to the M5-branes. The form of the partition function given in Eq. (3.5) was obtained in [4] where it was related to the periodic Schur process of period N.

3.1 Product representation of partition function using free fermions

In this section we express the partition function given in Eq. (3.5) as an infinite product. To do this, following [5] we introduce the free fermion Fock space spanned by creation and annihilation operators \((\psi _a,\psi _a^*)\) satisfying the relations

$$\begin{aligned}&\{\psi _a,\psi _b\}=\{\psi _a^*,\psi _b^*\}=0,\quad \{\psi _a,\psi _b^*\}=\delta _{ab},\,a,b\in \mathbb {Z}+\tfrac{1}{2}.\nonumber \\ \end{aligned}$$
(3.7)

With these we can construct the operators

$$\begin{aligned} \Gamma _{\pm }(z)&= \text{ exp }\left( \sum _{n=1}^\infty \frac{z^n J_n}{n}\right) ,\quad \text{ where }\nonumber \\&\quad J_n = \sum _{k\in \mathbb {Z}+\frac{1}{2}}\psi _{k+n}\psi ^*_{k},\quad n = \pm 1,\pm 2,\ldots . \end{aligned}$$
(3.8)

These operators satisfy the following commutation relation:

$$\begin{aligned} \Gamma _+(z)\Gamma _-(w)=(1-zw)\Gamma _-(w)\Gamma _+(z). \end{aligned}$$
(3.9)

The fermionic Fock space is spanned by states which are in a one-to-one correspondence with partitions. For a partition \(\lambda =(\lambda _{1},\lambda _{2},\ldots )\) we have the state

$$\begin{aligned} |\lambda \rangle =\psi _{\lambda _{1}-\frac{1}{2}}\psi _{\lambda _{2}-\frac{3}{2}}\psi _{\lambda _{3}-\frac{5}{2}}\ldots \,|0\rangle . \end{aligned}$$
(3.10)

The ground state \(|0\rangle \) is annihilated by \(\psi _{k}\) for \(k<0\). The operators defined in Eq. (3.8) are useful when working with symmetric polynomials due to the following:

$$\begin{aligned} \prod _{i}\Gamma _{+}(x_{i})|\lambda \rangle= & {} \sum _{\mu }s_{\mu /\lambda }(x_{1},x_{2},\ldots )|\mu \rangle \,\nonumber \\ \prod _{i}\Gamma _{-}(x_{i})|\lambda \rangle= & {} \sum _{\mu }s_{\lambda /\mu }(x_{1},x_{2},\ldots )|\mu \rangle . \end{aligned}$$
(3.11)

To simplify the equations we will use the following notation: \(\prod _{i}\Gamma _{\pm }(x_{i})=\Gamma _{\pm }(\mathbf{x})\). Using Eqs. (3.5) and (3.11) we can write \(\widehat{Z}_{N}:=Z_{N}/\Pi (Q_{m})^N\) as a trace, over the free fermionic Fock space, of an infinite product of operators:

$$\begin{aligned} \widehat{Z}_{N}=\text{ Tr }\Big ((Q_{1}Q_{m})^{L_{0}}\mathcal{O}(Q_{2}Q_{m})^{L_{0}}\mathcal{O}\ldots (Q_{N}Q_{m})^{L_{0}}\mathcal{O}\Big ), \end{aligned}$$
(3.12)

with the operator insertions \(\mathcal {O}\) being built from \(\Gamma _\pm \),

$$\begin{aligned} \mathcal{O}&=\prod _{i=1}^{\infty }\Big (\Gamma _{+}\left( Q_{m}^{-1}\,t^{-i+\frac{1}{2}}\right) \Gamma _{+}\left( \sqrt{\frac{q}{t}}t^{i-\frac{1}{2}}\right) \nonumber \\&\quad \times \prod _{j=1}^{\infty }\left( \Gamma _{-}\left( q^{j-\frac{1}{2}}\right) \Gamma _{-}\left( Q_{m}\sqrt{\frac{t}{q}}q^{-j+\frac{1}{2}}\right) \right) . \end{aligned}$$
(3.13)

In the above equations \(L_{0}\) is such that \(L_{0}|\lambda \rangle =|\lambda |\,|\lambda \rangle \). Using the commutation relations of \(\Gamma _{\pm }(\mathbf{x})\), we can write Eq. (3.12) as

$$\begin{aligned} \widehat{Z}_{N}&=\text{ Tr }\left( Q_{\rho }^{L_{0}}\prod _{a=1}^{k}\Gamma _{+}\left( Q_{1}Q_{2}\ldots Q_{a}Q_{m}^{a}{} \mathbf{x}\right) \right. \nonumber \\&\qquad \quad \times \left. \phantom {\left( Q_{\rho }^{L_{0}}\prod _{a=1}^{k}\Gamma _{+}(Q_{1}Q_{2}\ldots Q_{a}Q_{m}^{a}{} \mathbf{x})\right. }\Gamma _{-}\left( Q_{1}^{-1}Q_{2}^{-1}\ldots Q_{a}^{-1}Q_{m}^{-a}{} \mathbf{y}\right) \right) , \end{aligned}$$
(3.14)

where

$$\begin{aligned}&\mathbf{x}=\left\{ Q_{m}^{-1}t^{-i+\frac{1}{2}},\sqrt{\frac{q}{t}}t^{i-\frac{1}{2}}\,|\,i=1,2,\ldots \right\} ,\nonumber \\&\mathbf{y}=\left\{ q^{i-\frac{1}{2}},Q_{m}\sqrt{\frac{t}{q}}q^{-i+\frac{1}{2}}\,|\,i=1,2,\ldots \right\} . \end{aligned}$$
(3.15)

Using the commutation relation of \(\Gamma _{\pm }(x)\) repeatedly we get

$$\begin{aligned} \widehat{Z}_{N}=\left( \prod _{1\le a<b\le k}F_{ab}\right) \text{ Tr }\left( Q_{\rho }^{L_{0}}\Gamma _{+}(\mathbf{X})\Gamma _{-}(\mathbf{Y})\right) , \end{aligned}$$
(3.16)

where we have denoted

$$\begin{aligned} \mathbf{X}&=\left\{ Q_{1}Q_{m}{} \mathbf{x},Q_{1}Q_{2}Q_{m}^{2}{} \mathbf{x},\ldots , Q_{1}Q_{2}\ldots Q_{N}Q_{m}^{N}{} \mathbf{x}\right\} ,\end{aligned}$$
(3.17)
$$\begin{aligned} \mathbf{Y}&=\Big \{Q_{1}^{-1}Q_{m}^{-1}{} \mathbf{y},Q_{1}^{-1}Q_{2}^{-1}Q_{m}^{-2}{} \mathbf{y},\ldots , Q_{1}^{-1}Q_{2}^{-1}\ldots Q_{N}^{-1}Q_{m}^{-N}{} \mathbf{y}\Big \},\end{aligned}$$
(3.18)
$$\begin{aligned} F_{ab}&=\prod _{i,j=1}^{\infty }\frac{\left( 1-Q_{ab}Q_{m}^{-1}t^{i-\frac{1}{2}}q^{j-\frac{1}{2}}\right) \left( 1-Q_{ab}Q_{m}t^{i-\frac{1}{2}}q^{j-\frac{1}{2}}\right) }{(1-Q_{ab}t^{i}q^{j-1})(1-Q_{ab}t^{i-1}q^{j})},\end{aligned}$$
(3.19)
$$\begin{aligned} Q_{ab}&=Q_{a+1}\ldots Q_{b}Q_{m}^{b-a}. \end{aligned}$$
(3.20)

The trace appearing in Eq. (3.16) can be written in the form of a product [6],

$$\begin{aligned} \text{ Tr }\left( Q_{\rho }^{L_{0}}\Gamma _{+}(\mathbf{x})\Gamma _{-}(\mathbf{y})\right) =\prod _{n=1}^{\infty }\left( 1-Q_{\rho }^{n}\right) ^{-1}\prod _{i,j}\left( 1-Q_{\rho }^{n}x_{i}y_{j}\right) ^{-1}. \end{aligned}$$
(3.21)

such that the partition function is given by

$$\begin{aligned} \widehat{Z}_{N}&=\left( \displaystyle \prod \limits _{n=1}^{\infty }\left( 1-Q_{\rho }^{n}\right) ^{-1}\right) \left( \displaystyle \prod \limits _{1\le a<b\le N}F_{ab}\right) \left( \displaystyle \prod \limits _{a,b=1}^{N}H_{ab}\right) ,\nonumber \\ \end{aligned}$$
(3.22)

where

$$\begin{aligned} H_{ab}&=\prod _{n,i,j=1}^{\infty }\frac{\left( 1-Q_{\rho }^{n}\widetilde{Q}_{ab}Q_{m}^{-1}\,t^{i-\frac{1}{2}}q^{j-\frac{1}{2}}\right) \left( 1-Q_{\rho }^{n}\widetilde{Q}_{ab}Q_{m}\,t^{i-\frac{1}{2}}q^{j-\frac{1}{2}}\right) }{\left( 1-Q_{\rho }^{n}\widetilde{Q}_{ab}t^{i}q^{j-1}\right) \left( 1-Q_{\rho }^{n}\widetilde{Q}_{ab}t^{i-1}q^{j}\right) },\nonumber \\ \widetilde{Q}_{ab}&=Q_{1}Q_{2}\ldots Q_{a}Q_{1}^{-1}\ldots Q_{b}^{-1}Q_{m}^{a-b}. \end{aligned}$$
(3.23)

4 Free energy and Gopakumar–Vafa invariants

Using the refined topological string partition function we can calculate the free energy and the Gopakumar–Vafa invariants [7, 8] for the different curve classes. The free energy is given by

$$\begin{aligned} F_{N}(\rho ,t_{1},\ldots ,t_{N-1},m)= & {} \text{ ln }\,Z_{N}\nonumber \\= & {} N\,\text{ ln }\,\Pi (Q_{m})+\text{ ln }\widehat{Z}_{N}, \end{aligned}$$
(4.1)

where \(\Pi (Q_m)\) is given by Eq. (3.6) and \(\widehat{Z}_{N}\) is given by Eq. (3.22). After some simplification \(F_{N}\) can be written as

$$\begin{aligned}&F_{N}(\rho ,t_{1},\ldots ,t_{N-1},m)\nonumber \\&\quad =-N\,\sum _{k=1}^{\infty }\frac{Q_{m}^{k}}{k}\,\left[ \frac{1}{\left( q^{\frac{k}{2}}-q^{-\frac{k}{2}}\right) \left( t^{\frac{k}{2}}-t^{-\frac{k}{2}}\right) }\right] +\sum _{n=1}^{\infty }\sum _{k=1}^{\infty }\frac{Q_{\rho }^{n\,k}}{k}\nonumber \\&\qquad +\sum _{1\le a<b\le N}\sum _{k=1}^{\infty }\frac{Q_{ab}^{k}}{k}\,\left[ \frac{(t/q)^{\frac{k}{2}}+(q/t)^{\frac{k}{2}}-\left( Q_{m}^{k}+Q_{m}^{-k}\right) \,}{\left( q^{\frac{k}{2}}-q^{-\frac{k}{2}}\right) \left( t^{\frac{k}{2}}-t^{-\frac{k}{2}}\right) }\right] \nonumber \\&\qquad +\sum _{n=1}^{\infty }\sum _{a,b=1}^{N}\sum _{k=1}^{\infty }\frac{Q_{\rho }^{n\,k}\widetilde{Q}_{ab}^{k}}{k}\left[ \frac{(t/q)^{\frac{k}{2}}+(q/t)^{\frac{k}{2}}-\left( Q_{m}^{k}+Q_{m}^{-k}\right) \,}{\left( q^{\frac{k}{2}}-q^{-\frac{k}{2}}\right) \left( t^{\frac{k}{2}}-t^{-\frac{k}{2}}\right) }\right] . \end{aligned}$$
(4.2)

From Eq. (4.2) we can see that the \(SU(2)_{L}\times SU(2)_{R}\) spin content of various curve classes is given by

Curve

\(\sum _{j_{L},j_{R}}N_{C}^{j_{L},j_{R}}(j_{L},j_{R})\)

\(n\,E+C_{ab},\quad n\ge 0,\,1\le a<b\le N,\,c=a-1,b+1\)

\((0,\frac{1}{2})\)

\(n\,E+C_{ab}+M_{c},\quad n\ge 0,\,1\le a<b\le N,\,c=a-1,b+1\)

(0, 0)

\(n\,E+C_{ab}-M_{c},\quad n\ge 0,\,1\le a<b\le N,\,c=a+1,b-1\)

(0, 0)

\(n\,E-C_{ab},\quad n\ge 1,\,1\le a<b\le N\)

\((0,\frac{1}{2})\)

\(n\,E-C_{ab}+M_{c},\quad n\ge 1,\,1\le a<b\le N,\,c=a-1,b+1\)

(0, 0)

\(n\,E-C_{ab}-M_{c},\quad n\ge 1,\,1\le a<b\le N,\,c=a+1,b-1\)

(0, 0)

\(n\,E,\quad n\ge 1\)

\((\frac{1}{2},0)\oplus (N-1)(0,\frac{1}{2})\)

Here

$$\begin{aligned} C_{ab}=C_{a}+C_{a+1}+\cdots +C_{b}. \end{aligned}$$

From Eq. (4.2) we can isolate the contribution of the elliptic curve E to \(F_{N}\), which we will denote by \(F^{E}_{N}\), by considering only those terms which depend only on \(Q_{\rho }\) and the other Kähler parameter,

$$\begin{aligned} F^{E}_{N}(\rho ,q,t)= & {} \sum _{n,k=1}^{\infty }\frac{Q_{\rho }^{n\,k}}{k}\left[ 1+N\left( \frac{t^{k}+q^{k}}{(1-q^{k})(1-t^k)}\right) \right] .\nonumber \\ \end{aligned}$$
(4.3)

We will restrict ourselves to the unrefined case so that \(q=t\) and in terms of the topological string coupling constant \(g_{s}\),

$$\begin{aligned} q=t=e^{i\,g_{s}}. \end{aligned}$$
(4.4)

Since

$$\begin{aligned} \frac{2q}{(1-q)^2}= & {} \frac{2}{g_{s}^2}-\frac{1}{6}-2\sum _{g\ge 2}^{\infty }\,g_{s}^{2g-2}\,\frac{B_{2g}}{(2g)(2g-2)!}, \end{aligned}$$
(4.5)

where \(B_{k}\) are the Bernoulli numbers, we have

$$\begin{aligned} F^{E}_{N}(\rho ,q)=\sum _{g=0}^{\infty }g_{s}^{2g-2}F^{E}_{N,g}(\rho )\, \end{aligned}$$
(4.6)

with

$$\begin{aligned} F^{E}_{N, g\ge 2}&= N\frac{B_{2g}B_{2g-2}}{2(2g)\,(2g-2)!}\nonumber \\&\quad \;\,-2\,N\frac{B_{2g}}{(2g)\,(2g-2)!}\,\,\frac{B_{2g-2}}{4}\,E_{2g-2}(\rho ),\nonumber \\ F^{E}_{1}&=-\frac{1}{12}\text{ ln }\left[ \prod _{n=1}^{\infty }\left( 1-Q_{\rho }^n\right) ^{12-2N}\right] ,\nonumber \\&\quad \;\, \frac{\partial ^2\,F^{E}_{0}}{\partial \rho ^2}=2N\,\sum _{n=1}^{\infty }n^2\,\text{ ln }\left( 1-Q_{\rho }^n\right) ^{-1}\end{aligned}$$
(4.7)

where \(E_{2g}(\rho )\) is the Eisenstein series defined by

$$\begin{aligned} E_{2g}(\rho )=1-\frac{4}{B_{2g}}\sum _{n\ge 1}n^{2g-1}\,\left( \frac{Q_{\rho }^n}{1-Q_{\rho }^n}\right) . \end{aligned}$$
(4.8)

5 Non-perturbative modular transformation

For an elliptic Calabi–Yau threefold with modular parameter \(\rho \) one naively would expect the partition function to be invariant under the modular transformation,

$$\begin{aligned} Z_{N}\left( -\frac{1}{\tau },\frac{\epsilon _1}{\tau },\frac{\epsilon _2}{\tau }\right) =Z_{N}\Big (\tau ,\epsilon _1,\epsilon _2\Big ) \end{aligned}$$
(5.1)

where \(q=e^{i\epsilon _{1}}\) and \(t=e^{-i\epsilon _{2}}\). However, quite surprisingly, in [1] it was shown that the refined topological partition function of \(X_{1}\), which is dual to a single M5-brane wrapped on a circle, satisfies a non-perturbative modular transformation,

$$\begin{aligned} Z_{1}\left( -\frac{1}{\tau },\frac{\epsilon _1}{\tau },\frac{\epsilon _2}{\tau }\right) =\frac{Z_{1}(\tau ,\epsilon _1,\epsilon _2)}{Z_{1}\left( \frac{\tau }{\epsilon _1},-\frac{1}{\epsilon _1},\frac{\epsilon _2}{\epsilon _1}\right) Z_{1}\left( \frac{\tau }{\epsilon _2},\frac{\epsilon _1}{\epsilon _2},-\frac{1}{\epsilon _2}\right) }.\nonumber \\ \end{aligned}$$
(5.2)

Since \(\epsilon _1,\epsilon _2\) are related to the topological string coupling constant \(g_{s}\) by

$$\begin{aligned} \epsilon _1\epsilon _2=-g_{s}^2, \end{aligned}$$
(5.3)

Equation (5.2) implies that

$$\begin{aligned} Z_{1}\left( -\frac{1}{\tau },\frac{\epsilon _1}{\tau },\frac{\epsilon _2}{\tau }\right) =Z_{1}(\tau ,\epsilon _1,\epsilon _2)+O\left( e^{-\frac{1}{g_{s}}}\right) , \end{aligned}$$
(5.4)

i.e., modularity holds only up to non-perturbative corrections. For this reason the transformation in Eq. (5.2) was called a non-perturbative modular transformation and it was argued that this gives the non-perturbative completion of the topological string partition function for the case of elliptic Calabi–Yau threefolds.

To show that \(Z_{N}\) also satisfies an equation similar to Eq. (5.2) we write it in terms of double elliptic gamma functions. Recall that the double elliptic Gamma function is defined by

$$\begin{aligned} G_{2}(x;\tau ,\epsilon _{1},\epsilon _{2})&=\prod _{k,i,j=1}^{\infty }\left( 1-Q_{\tau }^{k-1}q^{i-1}t^{-j+1}x\right) \nonumber \\&\quad \times ~\left( 1-Q_{\tau }^{k}q^{i}t^{-j}x^{-1}\right) \end{aligned}$$
(5.5)

and satisfies the following modular transformation:

$$\begin{aligned}&G_{2}(z;\rho ,\epsilon _1,\epsilon _2)\nonumber \\&\quad =G_{2}\left( \frac{z}{\rho };-\frac{1}{\rho },\frac{\epsilon _1}{\rho },\frac{\epsilon _2}{\rho }\right) G_{2}\left( \frac{z}{\epsilon _1};\frac{\rho }{\epsilon _1},-\frac{1}{\epsilon _1},\frac{\epsilon _2}{\epsilon _1}\right) \nonumber \\&\qquad \times ~G_{2}\left( \frac{z}{\epsilon _2};\frac{\rho }{\epsilon _2},\frac{\epsilon _1}{\epsilon _2},-\frac{1}{\epsilon _2}\right) \,\text{ exp }\left( \frac{i\pi }{12}B_{44}\right) , \end{aligned}$$
(5.6)

where \(B_{4,4}\) is given by

$$\begin{aligned} B_{4,4}(z;\rho ,\epsilon _1,\epsilon _2)=\frac{\mathrm{d}^{4}}{\mathrm{d}x^4}\frac{x^4\,e^{z\,x}}{(e^{\rho \,x}-1)(e^{\epsilon _{1}\,x}-1)(e^{\epsilon _{2}\,x}-1)}|_{x=0}. \end{aligned}$$
(5.7)

The triple infinite product in Eq. (3.23) can be written in terms of double elliptic gamma functions so that the full partition function \(Z_N\) is given by

$$\begin{aligned}&Z_{N}=Z_{1}^{N}\left( \prod _{n=1}^{\infty }(1-Q_{\rho }^n)\right) ^{N-1}\nonumber \\&\quad \;\,\times ~\prod _{1\le a<b\le N}\frac{G_{2}\left( Q_{ab}Q_{m}\sqrt{t\,q};\rho ,\epsilon _1,-\epsilon _2\right) G_{2}\left( Q_{ab}\,Q_{m}^{-1}\sqrt{t\,q};\rho ,\epsilon _1,-\epsilon _2\right) }{G_{2}\left( Q_{ab}\,t;\rho ,\epsilon _1,-\epsilon _2\right) G_{2}\left( Q_{ab}\,q;\rho ,\epsilon _1,-\epsilon _2\right) }\nonumber \\&=Z_{1}^{N}\left( \prod _{n=1}^{\infty }(1-Q_{\rho }^n)\right) ^{N-1}\nonumber \\&\quad \;\,\times ~\prod _{1\le a<b\le N}\frac{G_{2}\left( Q_{ab};\rho ,\epsilon _1,\epsilon _2\right) G_{2}\left( Q_{\rho }Q_{ab}^{-1};\tau ,\epsilon _1,\epsilon _2\right) }{G_{2}\left( Q_{ab}Q_{m}\sqrt{t\,q};\rho ,\epsilon _1,\epsilon _2\right) G_{2}\left( Q_{ab}\,Q_{m}^{-1}\sqrt{t\,q};\rho ,\epsilon _1,\epsilon _2\right) } \end{aligned}$$
(5.8)

with the explicit expression

$$\begin{aligned} Z_{1}&=\left( \prod _{n=1}^{\infty }(1-Q_{\rho }^{n})^{-1}\right) \nonumber \\&\quad \times ~\prod _{i,j=1}^{\infty }\frac{\left( 1-Q_{\rho }^{n-1}\,Q_{m}\,q^{i-\frac{1}{2}}\,t^{j-\frac{1}{2}}\right) \left( 1-Q_{\rho }^{n}Q_{m}^{-1}\,q^{i-\frac{1}{2}}t^{j-\frac{1}{2}}\right) }{ \left( 1-Q_{\rho }^{n}q^{i}t^{j-1}\right) \left( 1-Q_{\rho }^{n}q^{i-1}t^{j}\right) }. \end{aligned}$$
(5.9)

Using Eq. (5.8) and the modular transformation satisfied by \(G_{2}(z;\rho ,\epsilon _1,\epsilon _2)\) we see that

$$\begin{aligned}&\displaystyle \frac{Z_{N}(\tau ,\epsilon _1,\epsilon _2)}{Z_{N}\left( \frac{\tau }{\epsilon _1},-\frac{1}{\epsilon _1},\frac{\epsilon _2}{\epsilon _1}\right) Z_{N}\left( \frac{\tau }{\epsilon _2},\frac{\epsilon _1}{\epsilon _2},-\frac{1}{\epsilon _2}\right) }=Z_{N}\left( -\frac{1}{\tau },\frac{\epsilon _1}{\tau },\frac{\epsilon _2}{\tau }\right) . \end{aligned}$$
(5.10)

Thus this class of elliptic Calabi–Yau threefolds do indeed satisfy the non-perturbative modular transformation of Lockhart and Vafa [1].

6 Discussion

In this paper we have shown that there is a simple class of elliptic Calabi–Yau threefolds for which the product representation of the refined topological string partition function can be determined using the relation between Schur functions and free fermionic Fock space. The product representation allows us to study the non-perturbative modular transformations of [1]. The class of elliptic Calabi–Yau threefolds discussed in this paper were simplest in the sense that they had no compact 4-cycles. It would be interesting to see if the non-perturbative modular transformation holds for an elliptic Calabi–Yau threefold with compact 4-cycles. We hope to report on this in the future [9].