Quantum curves and q-deformed Painlevé equations

Abstract

We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve, these are conjectured to be the q-difference Painlevé equations as in Sakai’s classification. More precisely, we propose that the tau functions of q-Painlevé equations are related to the grand canonical topological string partition functions on the corresponding geometry. In the toric cases, we use topological string/spectral theory duality to give a Fredholm determinant representation for the above tau functions in terms of the underlying quantum mirror curve. As a consequence, the zeroes of the tau functions compute the exact spectrum of the associated quantum integrable systems. We provide details of this construction for the local $\mathbb {P}^1\times \mathbb {P}^1$ case, which is related to q-difference Painlevé with affine $A_1$ symmetry, to SU(2) Super Yang–Mills in five dimensions and to relativistic Toda system.

Appendices

The grand potential: definitions

In this section we review the definition of the topological string grand potential ${\mathsf {J}} _X$ associated with a toric CY X with genus one mirror curve. We mainly follow the notation of [22, 56]. As in Sect. 2.1, we denote

$$\begin{aligned} \kappa =\mathrm{e}^{\mu } \end{aligned}$$

(A.1)

the “true” modulus of X, $\mathbf m_X$ the set of mass parameters and ${\varvec{\xi }}$ the rescaled mass parameters. The Kähler parameters of X are denoted by

$$\begin{aligned} t_i , \quad i=1, \ldots s \end{aligned}$$

(A.2)

and can be expressed in terms of the complex moduli through the mirror map. When $\kappa$ is large we have (see [22, 56] and references therein)

$$\begin{aligned} t_i=c_i \mu +\sum _{j=1}^{r_X}a_{ij} \log m_X^{(j)}+\Pi (\kappa ^{-1}, \mathbf{m}_X) \end{aligned}$$

(A.3)

where $\Pi$ is a convergent series in $\kappa ^{-1}$ while $c_i, a_{ij}$ are constants which can be read from the toric data of the CY [43, 44]. For instance for local ${\mathbb {P}}^1\times {\mathbb {P}}^1$ we have $s=2$, $r_X=1$, $c_1=c_2=2$, $a_{11}=0$ and $a_{21}=-1$. Hence,

$$\begin{aligned} t_2=t_1-\log m_{\mathrm{{\mathbb {P}}^1 \times {\mathbb {P}}^1}}. \end{aligned}$$

(A.4)

Geometrically the mirror map can be expressed as the A-period of the underlying mirror curve (2.4). Since this is a genus one curve we denote by ${\mathcal {A}}$ the corresponding A-cycle. Then we have

$$\begin{aligned} t_1\propto \oint _{{\mathcal {A}}} p(x) \mathrm{d}x, \end{aligned}$$

(A.5)

where p(x) is determined by Eq. (2.4). As explained in Sect. 2.1 one can promote the mirror curve to an operator ${{\mathsf {O}}}_{ X}$ [see Eq. (2.8)]. As a consequence, the differential $p(x) \mathrm{d}x$ is promoted to a quantum differential

$$\begin{aligned} p(x,\hbar ) \mathrm{d}x \end{aligned}$$

(A.6)

fulfilling

$$\begin{aligned} {{\mathsf {O}}}_{ X} \left( {1\over \sqrt{ p(x, \hbar )}}\exp \left[ {\mathrm{i}\over \hbar } \int ^x p(y, \hbar )\mathrm{d}y\right] \right) =0. \end{aligned}$$

(A.7)

The quantum mirror map is then defined as [87]

$$\begin{aligned} t_1(\hbar ) \propto \oint _{{\mathcal {A}}} p(x, \hbar ) \mathrm{d}x. \end{aligned}$$

(A.8)

As explained in [87], such map can be computed efficiently at large $\kappa$ and one has

$$\begin{aligned} t_i ( \hbar )=c_i \mu +\sum _{j=1}^{r_X}a_{ij} \log m_X^{(j)}+\Pi (\kappa ^{-1}, \mathbf{m}_X, \hbar ), \end{aligned}$$

(A.9)

where $\Pi$ is a series expansion in $\kappa ^{-1}$ but it is exact in $\hbar$. For instance, when X is the canonical bundle over ${\mathbb {P}}^1\times {\mathbb {P}}^1$, we have

$$\begin{aligned} t_1 ( \hbar )=t(\mu , \xi , \hbar ), \quad t_2 ( \hbar )= t(\mu , \xi , \hbar )-{\hbar \over 2 \pi } \xi , \end{aligned}$$

(A.10)

where the first few terms in the expansion are

$$\begin{aligned} \begin{aligned} t(\mu , \xi ,\hbar )&= 2 \mu -2 (m_{{\mathbb {P}}^1 \times {\mathbb {P}}^1}+1) \kappa ^{-2}\\&\quad +\kappa ^{-4} \left( -3 m_{{\mathbb {P}}^1 \times {\mathbb {P}}^1}^2-\frac{2 m_{{\mathbb {P}}^1 \times {\mathbb {P}}^1} \left( {\mathrm{e}}^{2\mathrm{i}\hbar }+4 {\mathrm{e}}^{\mathrm{i}\hbar }+1\right) }{{\mathrm{e}}^{\mathrm{i}\hbar }}-3\right) +O\left( \kappa ^{-6}\right) , \\&\quad \kappa =\mathrm{e}^{\mu }, \quad m_{{\mathbb {P}}^1 \times {\mathbb {P}}^1}=\mathrm{e}^{{\hbar \over 2 \pi } \xi } . \end{aligned} \end{aligned}$$

(A.11)

Note that in this example we have a series in $\kappa ^{-2}$. For a generic genus one geometry, we would have a series in $\kappa ^{-r}$ where r is determined by the canonical class/intersection data of the geometry. A list of r-parameters for others genus one geometries can be found for instance in Table 1 of [21].

When $\hbar$ is real, it has been checked numerically for many geometries that the series $\Pi$ in (A.9) at large $\kappa$ has a finite radius of convergence (see for instance [55]). For generic complex values of $\hbar$, instead one has to perform a partial resummation, but it is still possible to organize (A.9) into a convergent series by using instanton calculus in 5d gauge theory. Indeed, from a 5d gauge theory viewpoint the quantum mirror map can be computed from the vev of the Wilson loop in the fundamental representation, see for instance [109, 110]. For the example of local ${\mathbb {P}}^1\times {\mathbb {P}}^1$, this is illustrated for instance in [77].

We introduce the topological string free energy

$$\begin{aligned} F^{\mathrm{top}}_{\mathrm{X}}\left( \mathbf{t}, g_s\right) ={1\over 6 g_s^2} a_{ijk} t_i t_j t_k +b_i t_i +F^{\mathrm{GV}}_X\left( \mathbf{t} , g_s\right) \end{aligned}$$

(A.12)

with

$$\begin{aligned} F^{\mathrm{GV}}_X\left( \mathbf{t}, g_s\right) =\sum _{g\ge 0} \sum _\mathbf{d} \sum _{w=1}^\infty {1\over w} n_g^{ \mathbf{d}} \left( 2 \sin { w g_s \over 2} \right) ^{2g-2} \mathrm{e}^{-w \mathbf{d} \cdot \mathbf{t}}, \end{aligned}$$

(A.13)

where $n_g^{ \mathbf{d}}$ are the Gopakumar–Vafa invariants of X and $g_s$ is the string coupling constant. Moreover,

$$\begin{aligned} \mathbf{d}=\{d_1, \ldots ,d_s\} \end{aligned}$$

(A.14)

and the sum over $\mathbf{d}$ in (A.13) runs over all integers

$$\begin{aligned} d_i \in {\mathbb {Z}}, \quad d_i\ge 0 \end{aligned}$$

(A.15)

such that

$$\begin{aligned} \sum _{i=1}^s d_i^2 \ne 0. \end{aligned}$$

(A.16)

The coefficients $a_{ijk}, b_i$ are determined by the classical data of X. In the limit $g_s \rightarrow 0$, we have

$$\begin{aligned} F^{\mathrm{top}}_{\mathrm{X}}\left( \mathbf{t}, g_s\right) \sim \sum _{g\ge 0} F_g(\mathbf{t}) g_s^{2g-2}, \end{aligned}$$

(A.17)

where $F_g(\mathbf{t})$ are called the genus g free energies of topological string. For instance, we have

$$\begin{aligned} \begin{aligned} F_0(\mathbf{t})&={1\over 6} a_{ijk} t_i t_j t_k + \sum _{\mathbf{d}} N_0^{ \mathbf{d}} \mathrm{e}^{-\mathbf{d} \cdot \mathbf{t}},\\ F_1(\mathbf{t})&=b_i t_i + \sum _{\mathbf{d}} N_1^{ \mathbf{d}} \mathrm{e}^{-\mathbf{d} \cdot \mathbf{t}}, \end{aligned} \end{aligned}$$

(A.18)

where $N_g^\mathbf{d}$ are the Gromov–Witten invariants. When X is a toric CY, one has explicit expressions for (A.18) in terms of hypergeometric and standard functions (see for instance [55, 80, 111]).

Let us discuss briefly the convergence properties of (A.13). We first note that (A.13) has poles for $\pi ^{-1}g_s \in {\mathbb {Q}}$ which makes it ill defined on the real $g_s$ axis. If instead $g_s \in {\mathbb {C}}/{\mathbb {R}}$, then (A.13) diverges as a series in $\mathrm{e}^{- \mathbf{t}}$. Nevertheless, by using instanton calculus it is possible to partially resummate it and organize it into a convergent series [24] at least if $g_s \in {\mathbb {C}}/{\mathbb {R}}$.

Similarly, we define the Nekrasov–Shatashvili free energy as

$$\begin{aligned} F^{\mathrm{NS}}(\mathbf{t}, \hbar ) ={1\over 6 \hbar } a_{ijk} t_i t_j t_k +b^{\mathrm{NS}}_i t_i \hbar +F^{\mathrm{NS}}_{\mathrm{inst}}(\mathbf{t}, \hbar ), \end{aligned}$$

(A.19)

where

$$\begin{aligned} F^{\mathrm{NS}}_{\mathrm{inst}}(\mathbf{t}, \hbar ) =\sum _{j_L, j_R} \sum _{w\ge 1 } \sum _{\mathbf{d} } N^{\mathbf{d}}_{j_L, j_R} \frac{\sin \frac{\hbar w}{2}(2j_L+1)\sin \frac{\hbar w}{2}(2j_R+1)}{2 w^2 \sin ^3\frac{\hbar w}{2}} \mathrm{e}^{-w \mathbf{d}\cdot \mathbf{t}} \end{aligned}$$

(A.20)

and $N^{\mathbf{d}}_{j_L, j_R}$ are the refined BPS invariants of X. The sum over $\mathbf{d}$ is as in (A.13). Moreover, there exists a constant vector $\mathbf{B}$, called the B-field, such that

$$\begin{aligned} N^{\mathbf{d}}_{j_L, j_R}\ne 0 \quad \leftrightarrow \quad (-1)^{2j_L + 2 j_R+1}= (-1)^{\mathbf{B} \cdot \mathbf{d}}. \end{aligned}$$

(A.21)

For local ${\mathbb {P}}^1 \times {\mathbb {P}}^1$, this can be set to zero [21, 55]. When $\hbar \rightarrow 0$, we recover the following genus expansion

$$\begin{aligned} F^{\mathrm{NS}}(\mathbf{t}, \hbar ) =\sum _{g\ge 0} F_g^{\mathrm{NS}}(\mathbf{t})\hbar ^{2g-2}. \end{aligned}$$

(A.22)

The convergent properties for the NS free energy are analogous to the ones of (A.13). The topological string grand potential is defined as

$$\begin{aligned} \mathsf {J}_{X}({\mu }, \varvec{\xi },\hbar ) = \mathsf {J}^{\mathrm{WKB}}_X ({\mu }, \varvec{\xi },\hbar )+ \mathsf {J}^{\mathrm{WS}}_X ({\mu }, \varvec{\xi } , \hbar ), \end{aligned}$$

(A.23)

where

$$\begin{aligned} \mathsf {J}^{\mathrm{WS}}_X({\mu }, \varvec{\xi }, \hbar )=F^{\mathrm{GV}}_X\left( {2 \pi \over \hbar }{} \mathbf{t}(\hbar )+ \pi \mathrm{i}\mathbf{B} , {4 \pi ^2 \over \hbar } \right) . \end{aligned}$$

(A.24)

Moreover,

$$\begin{aligned} \mathsf {J}^{\mathrm{WKB}}_X({\mu }, \varvec{\xi }, \hbar )= & {} {t_i(\hbar ) \over 2 \pi } {\partial F^{\mathrm{NS}}(\mathbf{t}(\hbar ), \hbar ) \over \partial t_i} +{\hbar ^2 \over 2 \pi } {\partial \over \partial \hbar } \left( {F^{\mathrm{NS}}(\mathbf{t}(\hbar ), \hbar ) \over \hbar } \right) \nonumber \\&+ {2 \pi \over \hbar } b_i t_i(\hbar ) + A({\varvec{\xi }}, \hbar ), \end{aligned}$$

(A.25)

where $A({\varvec{\xi }}, \hbar )$ denotes the so-called constant map contribution [112]. It is important to notice that, even tough both $\mathsf {J}^{\mathrm{WS}}_X$ and $\mathsf {J}^{\mathrm{WKB}}_X$ have a dense set of poles on the real $\hbar$ axis, their sum (A.23) is free of poles. We usually refer to this mechanism as the HMO cancellation mechanism since it was first discovered in [52] in the context of ABJM theory. Moreover, we have

$$\begin{aligned} \mathsf {J}^{\mathrm{WKB}}_X({\mu }, \varvec{\xi }, \hbar )= & {} {1\over 12 \pi \hbar } a_{ijk} t_i(\hbar ) t_j(\hbar ) t_k(\hbar ) + \left( {2 \pi b_i \over \hbar } + {\hbar b_i^{\mathrm{NS}} \over 2 \pi } \right) t_i(\hbar ) \nonumber \\&+\, {\mathcal {O}}\left( \mathrm{e}^{-t_i(\hbar )} \right) + A({\varvec{\xi }}, \hbar ). \end{aligned}$$

(A.26)

Hence, it is convenient to split

$$\begin{aligned} \mathsf {J}^{\mathrm{WKB}}_X({\mu }, \varvec{\xi }, \hbar )= \mathrm{P}_X({{\mu }, \varvec{\xi }, \hbar })+ \mathsf {J}^{\mathrm{WKB, inst}}_X({\mu }, \varvec{\xi }, \hbar )+ A({\varvec{\xi }}, \hbar ) \end{aligned}$$

(A.27)

where ${\mathrm{P}}_X$ encodes the polynomial part in $t_i$ of $\mathsf {J}^{\mathrm{WKB}}_X$ and

$$\begin{aligned} \mathsf {J}^{{\mathrm{WKB, inst}}}_X({\mu }, \varvec{\xi }, \hbar )= {t_i(\hbar ) \over 2 \pi } {\partial F^{\mathrm{NS}}_{\mathrm{inst}}(\mathbf{t}(\hbar ), \hbar ) \over \partial t_i} +{\hbar ^2 \over 2 \pi } {\partial \over \partial \hbar } \left( {F^{\mathrm{NS}}_{\mathrm{inst}}(\mathbf{t}(\hbar ), \hbar ) \over \hbar } \right) . \end{aligned}$$

(A.28)

For local ${\mathbb {P}}^1 \times {\mathbb {P}}^1$, we have

$$\begin{aligned} {\mathrm{P}}_{{\mathbb {P}}^1 \times {\mathbb {P}}^1}(\mu , \xi ,\hbar ) = -\frac{\xi t(\mu ,\xi ,\hbar )^2}{16 \pi ^2}+\frac{t(\mu ,\xi ,\hbar )^3}{12 \pi \hbar }-\frac{\hbar t(\mu ,\xi ,\hbar )}{24 \pi }+\frac{\pi t(\mu ,\xi ,\hbar )}{6 \hbar }-\frac{\xi }{24}. \end{aligned}$$

(A.29)

The constant map contribution for local ${\mathbb {P}}^1 \times {\mathbb {P}}^1$ reads [113]

$$\begin{aligned} A( \xi ,\hbar )= A_p(\xi ,\hbar )-J_{\mathrm{CS}}\left( {2\pi ^2 \over \hbar },\mathrm{i}\pi +{1\over 2} \xi \right) , \end{aligned}$$

(A.30)

where

$$\begin{aligned} A_p(\xi ,\hbar )={\hbar ^2 \over (4 \pi ^2)^2}\left[ \frac{\xi ^3}{24}+\frac{\pi ^2 \xi }{6 }\right] +A_c\left( {\hbar \over \pi }\right) , \end{aligned}$$

(A.31)

with

$$\begin{aligned} A_{\mathrm{c}}(k)= \frac{2\zeta (3)}{\pi ^2 k}\left( 1-\frac{k^3}{16}\right) +\frac{k^2}{\pi ^2} \int _0^\infty \frac{x}{\mathrm{e}^{k x}-1}\log (1-\mathrm{e}^{-2x})\mathrm{d}x. \end{aligned}$$

(A.32)

Moreover, $J_{\mathrm{CS}}(g_s,t)$ is the non-perturbative Chern–Simons free energy [88, 114]. As explained in [88], this also coincides with the grand potential of the resolved conifold as defined in [55]. More precisely,

$$\begin{aligned} \begin{aligned} J_{\mathrm{CS}}(g_s,T+\mathrm{i}\pi )= J^{\mathrm{pert}}_{{ \mathrm CS}}\left( g_s,T+\mathrm{i}\pi \right) + J^{\mathrm{np}}_{{ \mathrm CS}}\left( g_s,T+\mathrm{i}\pi \right) \end{aligned} \end{aligned}$$

(A.33)

where

$$\begin{aligned} J^{\mathrm{pert}}_{{ \mathrm CS}}\left( g_s,T+\mathrm{i}\pi \right)= & {} -g_s^{-2}\frac{T^3}{12}-g_s^{-2}\frac{\pi ^2 T}{12}-{1\over 24}T+{1\over 2}A_c(4 \pi /g_s)\nonumber \\&+\sum _{n \ge 1}{1\over n}\left( 2 \sin {n g_s \over 2}\right) ^{-2}(-1)^n\mathrm{e}^{-n T}, \end{aligned}$$

(A.34)

$$\begin{aligned} J^{\mathrm{np}}_{{ \mathrm CS}}\left( g_s,T+\mathrm{i}\pi \right)= & {} -\sum _{n\ge 1} {1\over 4 \pi n^2}\csc \left( {2 \pi ^2 n \over g_s}\right) \nonumber \\&\times \left( { 2 \pi n \over g_s} T + {2 \pi ^2 n \over g_s}\cot \left( {2 \pi ^2 n \over g_s}\right) +1\right) \mathrm{e}^{-{2 \pi n \over g_s}T}. \end{aligned}$$

(A.35)

For instance, when $g_s=\pi$ and $T>0$, we have (see also [88], equation (5.11))

$$\begin{aligned} J_{\mathrm{CS}}(\pi ,T+\mathrm{i}\pi )= & {} -(\pi )^{-2}\frac{T^3}{12}-\frac{T}{12}-{1\over 24}T+{1\over 2}A_c(4)\nonumber \\&+{1\over 8 \pi ^2}\mathrm{Li}_3(\mathrm{e}^{-2T})+{T\over 4 \pi ^2}\mathrm{Li}_2(\mathrm{e}^{-2T})\nonumber \\&-\left( {T^2\over 4 \pi ^2}+{1\over 8}\right) \log \left( 1-\mathrm{e}^{-2T}\right) -{1\over 4}{{\mathrm{arctanh}}}(\mathrm{e}^{-T}). \end{aligned}$$

(A.36)

Notice that, even thought $J^{\mathrm{np}}_{{ \mathrm CS}}$ and $J^{\mathrm{pert}}_{{ \mathrm CS}}$ are ill defined at $g_s=\pi$, their sum is perfectly well defined. This is yet another example of the HMO cancellation mechanism.

Nekrasov partition function and topological strings on local ${\mathbb {P}}^1\times {\mathbb {P}}^1$

As explained in [77], when the topological string free energy has a gauge theory interpretation, it is sometimes better to replace the topological string expressions (A.13) and (A.19) with their Nekrasov form since the latter has better convergence properties when $\hbar$ is complex, i.e.  $|q| \ne 1$. We follow the convention of [24] and write the 5d Nekrasov partition function, or q-conformal blocks, as

$$\begin{aligned} \begin{aligned}&{\mathcal {Z}}(u,Z,q_1,q_2)=\sum _{\lambda ,\mu } \left( { Z\over q_1 q_2}\right) ^{|\lambda |+|\mu |}\\&\quad \frac{1}{N_{\lambda ,\lambda }(1,q_1,q_2)N_{\mu ,\mu }(1,q_1,q_2)N_{\lambda ,\mu }(u,q_1,q_2)N_{\mu ,\lambda }(u^{-1},q_1,q_2)} \end{aligned} \end{aligned}$$

(B.1)

where the sum runs over all pairs $(\mu , \lambda )$ of Young diagrams. Moreover, we use

$$\begin{aligned} \begin{aligned} N_{\lambda ,\mu }(u,q_1,q_2) =&\prod _{s\in \lambda } \left( 1-u q_2^{-a_\mu (s)-1}q_1^{\ell _\lambda (s)}\right) \cdot \prod _{s \in \mu } \left( 1-u q_2^{a_\lambda (s)}q_1^{-\ell _\mu (s)-1}\right) \end{aligned} \end{aligned}$$

(B.2)

where $a_{\lambda }(s), l_{\lambda }(s)$ are the arm length and the leg length of the box $s \in \lambda$. For instance, we have

$$\begin{aligned} \begin{aligned}&\log {\mathcal {Z}}(u,Z,q,q^{-1})= \frac{2 q u Z}{(q-1)^2 (u-1)^2} + \mathcal {O}\left( {Z^2}\right) ,\\&\mathrm{i}\lim _{q_2\rightarrow 1} \log \left( q_2\right) \log \left( {\mathcal {Z}}(u,Z,q,q_2)\right) = \frac{i q (q+1) u Z}{(q-1) (q-u) (q u-1)}+ \mathcal {O}\left( {Z^2}\right) . \end{aligned} \end{aligned}$$

(B.3)

As explained in [115,116,117], if we expand ${\mathcal {Z}}(u,Z,q,q^{-1})$ at small u, we recover the partition function (without the one loop contribution) of topological string on local ${\mathbb {P}}^1\times {\mathbb {P}}^1$. Therefore, we identify

$$\begin{aligned} \exp \left[ \mathsf {J}^{\mathrm{WS}}_{{\mathbb {P}}^1 \times {\mathbb {P}}^1}({\mu }, {\xi }, \hbar )\right] \end{aligned}$$

(B.4)

with

$$\begin{aligned} {{1 \over (Q_f q,q,q)_{\infty }^2}} {\mathcal {Z}}\left( Q_f ,{Q_b\over Q_f}, \mathrm{e}^{4\pi ^2 \mathrm{i}/\hbar },\mathrm{e}^{-4\pi ^2 \mathrm{i}/\hbar }\right) . \end{aligned}$$

(B.5)

where

$$\begin{aligned} Q_b=\mathrm{e}^{-{2 \pi \over \hbar }t (\mu , \xi , \hbar )}, \quad Q_f=\mathrm{e}^{\xi } Q_b \end{aligned}$$

(B.6)

and $t (\mu , \xi , \hbar )$ is defined in (A.11).

Likewise for local ${\mathbb {P}}^1 \times {\mathbb {P}}^1$, we identify $F^{\mathrm{NS}}_{\mathrm{inst}}(\mathbf{t}, \hbar )$ given in (A.20) with the Nekrasov–Shatashvili limit of (B.1), see for instance [116]. More precisely, we have

$$\begin{aligned} {- \sum _{w\ge 1}{1\over w^2} \cot \left( {\hbar w\over 2}\right) Q_F^w}+ \mathrm{i}\lim _{q_2\rightarrow 1} \log \left( q_2\right) \log \left( {\mathcal {Z}}\left( Q_F,{Q_B \over Q_F},q_1,q_2\right) \right) , \end{aligned}$$

(B.7)

where

$$\begin{aligned} Q_B=\mathrm{e}^{-t (\mu , \xi , \hbar )}, \quad Q_F=m_{{\mathbb {P}}^1 \times {\mathbb {P}}^1} Q_B, \quad q_1=\mathrm{e}^{\mathrm{i}\hbar }. \end{aligned}$$

(B.8)

As pointed out in the previous section, both quantities in (B.3) are ill defined and have poles at $|q|=1$. However, when we combine them in the grand potential (A.23), these poles cancel.

Some relevant shifts

We define

$$\begin{aligned} V_1(\mu ,\xi ,\hbar )=\mathrm{e}^{ J_{{\mathbb {P}}^1 \times {\mathbb {P}}^1}^{\mathrm{WKB, inst}}(\mu , { \xi },\hbar )}. \end{aligned}$$

(C.1)

Then we have

$$\begin{aligned} \begin{aligned}&{V_1( \mu +\mathrm{i}\pi ,\xi ,\hbar ) V_1( \mu -\mathrm{i}\pi ,\xi ,\hbar )\over V_1( \mu ,\xi ,\hbar )^2}= 1, \\&{V_1( \mu -2 \pi \mathrm{i}, \xi {-4 \mathrm{i}\pi ^2/\hbar },\hbar ) V_1( \mu +2 \pi \mathrm{i},\xi + {4 \mathrm{i}\pi ^2/\hbar },\hbar )\over V_1( \mu ,m,\hbar )^2}= 1,\\&{V_1( \mu -\mathrm{i}\pi , \xi {-4 \mathrm{i}\pi ^2/\hbar },\hbar ) V_1( \mu +\mathrm{i}\pi ,\xi +{4 \mathrm{i}\pi ^2/\hbar },\hbar )\over V_1( \mu ,\xi ,\hbar )^2}= 1 . \end{aligned} \end{aligned}$$

(C.2)

Similarly, we define

$$\begin{aligned} V_2({\mu }, {\xi }, \hbar )= {(Q_f ^{-1}q,q,q)_{\infty } \over (Q_f q,q,q)_{\infty }}. \end{aligned}$$

(C.3)

Then we have

$$\begin{aligned}&{V_2( \mu +\mathrm{i}\pi ,\xi ,\hbar ) V_2( \mu -\mathrm{i}\pi ,\xi ,\hbar )\over V_2( \mu ,\xi ,\hbar )^2}= -{1 \over Q_f } , \nonumber \\&{V_2( \mu -2 \pi \mathrm{i}, \xi {-4 \mathrm{i}\pi ^2/\hbar },\hbar ) V_2( \mu +2 \pi \mathrm{i},\xi +{4 \mathrm{i}\pi ^2/\hbar },\hbar )\over V_2( \mu ,\xi ,\hbar )^2}= -{1 \over Q_f } , \end{aligned}$$

(C.4)

$$\begin{aligned}&{V_2( \mu -\mathrm{i}\pi , \xi {-4 \mathrm{i}\pi ^2/\hbar },\hbar ) V_2( \mu +\mathrm{i}\pi , \xi +{4 \mathrm{i}\pi ^2/\hbar },\hbar )\over V_2(\mu , \hbar ,\hbar )^2}=1, \end{aligned}$$

(C.5)

where we used

$$\begin{aligned} \left( u q;q,q\right) _{\infty } =\prod _{i,j\ge 0}\left( 1-u q q^{i+j}\right) =\exp \left[ - \sum _{s\ge 1}\frac{ u^s}{s \left( q^{s\over 2}-q^{-\frac{s}{2}}\right) ^2} \right] . \end{aligned}$$

(C.6)

Similarly, we define

$$\begin{aligned} V_3( \xi , \hbar )=\exp [A_p( \xi ,\hbar ) ] \end{aligned}$$

(C.7)

and we have

$$\begin{aligned} \begin{aligned}&{V_3( \xi {-4 \mathrm{i}\pi ^2/\hbar },\hbar ) V_3( \xi +{4 \mathrm{i}\pi ^2/\hbar },\hbar )\over V_3( \xi ,\hbar )^2}= \mathrm{e}^{-\xi /4}. \end{aligned} \end{aligned}$$

(C.8)

Also for

$$\begin{aligned} V_4( \mu , \xi , \hbar )=\exp [{\mathrm{P}}_{{\mathbb {P}}^1\times {\mathbb {P}}^1}(\mu , \xi ,\hbar ) ] \end{aligned}$$

(C.9)

we have

$$\begin{aligned} \begin{aligned}&{V_4( \mu +\mathrm{i}\pi ,\xi ,\hbar ) V_4( \mu -\mathrm{i}\pi ,\xi ,\hbar )\over V_4( \mu ,\xi ,\hbar )^2}=\mathrm{e}^{\xi /2} Q_b , \\&{V_4( \mu -2 \pi \mathrm{i}, \xi {-4 \mathrm{i}\pi ^2/\hbar },\hbar ) V_4( \mu +2 \pi \mathrm{i},\xi +{4 \mathrm{i}\pi ^2/\hbar },\hbar )\over V_4( \mu ,\xi ,\hbar )^2}= \mathrm{e}^{2 \xi } Q_b^2,\\&{V_4( \mu -\mathrm{i}\pi , \xi {-4 \mathrm{i}\pi ^2/\hbar },\hbar ) V_4( \mu +\mathrm{i}\pi ,\xi +{4 \mathrm{i}\pi ^2/\hbar },\hbar )\over V_4( \mu ,\xi ,\hbar )^2}=\mathrm{e}^{\xi /2}. \end{aligned} \end{aligned}$$

(C.10)

Moreover, we have

$$\begin{aligned}&J^{\mathrm{np}}_{{ \mathrm CS}} \left( {2\pi ^2 \over \hbar },{1\over 2}\xi + {2 \pi ^2 \mathrm{i}\over \hbar }+\mathrm{i}\pi \right) +J^{\mathrm{np}}_{{ \mathrm CS}} \left( {2\pi ^2 \over \hbar },{1\over 2}\xi - {2 \pi ^2 \mathrm{i}\over \hbar }+\mathrm{i}\pi \right) \nonumber \\&\quad = 2J^{\mathrm{np}}_{{ \mathrm CS}} \left( {2\pi ^2 \over \hbar },{1\over 2}\xi +\mathrm{i}\pi \right) . \end{aligned}$$

(C.11)

Similarly,

$$\begin{aligned} \begin{aligned}&J^{\mathrm{pert}}_{{ \mathrm CS}} \left( {2\pi ^2 \over \hbar },{1\over 2}\xi + {2 \pi ^2 \mathrm{i}\over \hbar }+\mathrm{i}\pi \right) +J^{\mathrm{pert}}_{{ \mathrm CS}} \left( {2\pi ^2 \over \hbar },{1\over 2}\xi - {2 \pi ^2 \mathrm{i}\over \hbar }+\mathrm{i}\pi \right) \\&\qquad - 2J^{\mathrm{pert}}_{{ \mathrm CS}} \left( {2\pi ^2 \over \hbar },{1\over 2}\xi +\mathrm{i}\pi \right) ={1\over 4}\xi +\log \left( \mathrm{e}^{-\xi /2}+1\right) . \end{aligned} \end{aligned}$$

(C.12)

Some identities for $\eta$ function

We denote

$$\begin{aligned} \eta (\tau )=\mathrm{e}^{\mathrm{i}\pi \tau /12}\prod _{n=1}^{\infty } \left( 1-\mathrm{e}^{2 \pi \mathrm{i}n \tau }\right) \end{aligned}$$

(D.1)

the Dedekind $\eta$ function. The Weber modular functions are defined as

$$\begin{aligned} \begin{aligned}&f(\tau )={\eta ^2(\tau )\over \eta (\tau /2)\eta (2\tau )},\\&f_1(\tau )={\eta (\tau /2)\over \eta (\tau )},\\&f_2(\tau )=\sqrt{2}{\eta (2\tau )\over \eta (\tau )}.\\ \end{aligned} \end{aligned}$$

(D.2)

Standard identities of Weber modular functions are

$$\begin{aligned}&f_1(\tau )^8+f_2(\tau )^8=f(\tau )^8, \end{aligned}$$

(D.3)

$$\begin{aligned}&8 j(\tau )=(f_1(\tau )^{16}+f_2(\tau )^{16}+f^{16}(\tau ))^3, \end{aligned}$$

(D.4)

where j is the j-invariant:

$$\begin{aligned} j(\tau )={1\over {{\bar{q}}}}+ 744+196884{{\bar{q}}}+ \mathcal {O}({{{\bar{q}}}^2} ), \quad {{\bar{q}}}=\mathrm{e}^{2 \pi \mathrm{i}\tau } \end{aligned}$$

(D.5)

Summary of conventions

The paper connects different branches of mathematical physics and for this reason has a lot of notations which we summarize here. The complex moduli of local ${\mathbb {P}}_1 \times {\mathbb {P}}_1$ are denoted by

$$\begin{aligned} \kappa , \quad m_{{\mathbb {P}}_1\times {\mathbb {P}}_1}. \end{aligned}$$

(E.1)

The parameter $\hbar$ is such that

$$\begin{aligned}{}[{\mathsf {x}}, {\mathsf {p}}]=\mathrm{i}\hbar , \end{aligned}$$

(E.2)

where ${\mathsf {x}}, {\mathsf {p}}$ are the operators appearing in the quantum mirror curve (3.1). It is sometimes useful to use $(\mu , \xi )$ which are related to (E.1) as

$$\begin{aligned} \kappa =\mathrm{e}^{\mu }, \quad \xi = {2 \pi \over \hbar }\log {m_{{\mathbb {P}}^1\times {\mathbb {P}}^1}}. \end{aligned}$$

(E.3)

The quantum mirror map of local ${\mathbb {P}}_1 \times {\mathbb {P}}_1$ is denoted by

$$\begin{aligned} t (\mu , \xi , \hbar ) . \end{aligned}$$

(E.4)

We also use

$$\begin{aligned} t_1 ( \hbar )=t(\mu , \xi , \hbar ), \quad t_2 ( \hbar )= t(\mu , \xi , \hbar )-{\hbar \over 2 \pi } \xi . \end{aligned}$$

(E.5)

To connect with the topological string/spectral theory correspondence, it is useful to introduce

$$\begin{aligned} Q_b=\mathrm{e}^{-{2 \pi \over \hbar }t (\mu , \xi , \hbar )}, \quad Q_f=\mathrm{e}^{\xi } Q_b , \quad Q_B=\mathrm{e}^{-t (\mu , \xi , \hbar )}, \quad Q_F=m_{{\mathbb {P}}_1\times {\mathbb {P}}_1} Q_B. \end{aligned}$$

(E.6)

Notice that

$$\begin{aligned} Q_b=Q_B^{2\pi /\hbar }, \quad Q_f=Q_F^{2\pi /\hbar }. \end{aligned}$$

(E.7)

In the q-Painlevé literature, one typically uses the variables

$$\begin{aligned} Z, u, q . \end{aligned}$$

(E.8)

These are related to the variables appearing in the topological string/spectral theory correspondence via

$$\begin{aligned} Z^{-1}=\mathrm{e}^{\xi }, \quad q=\mathrm{e}^{\mathrm{i}4 \pi ^2/\hbar }, \quad \quad u=\mathrm{e}^{\xi }Q_b= Q_f. \end{aligned}$$

(E.9)

In the ABJ language, the natural variables are the rank of the gauge group M and the Chern–Simons level k. These are related to the topological string variables via

$$\begin{aligned} \log m_{{\mathbb {P}}^1 \times {\mathbb {P}}^1}=\mathrm{i}\hbar -2 \pi \mathrm{i}M, \quad \hbar = \pi k. \end{aligned}$$

(E.10)