Reconstructing GKZ via Topological Recursion

Abstract

In this article, a novel description of the hypergeometric differential equation found from Gel’fand–Kapranov–Zelevinsky’s system (referred to as GKZ equation) for Givental’s J-function in the Gromov–Witten theory will be proposed. The GKZ equation involves a parameter $\hbar$, and we will reconstruct it as a quantum curve from the classical limit $\hbar \rightarrow 0$ via the topological recursion. In this analysis, the spectral curve (referred to as GKZ curve) plays a central role, and it can be described by the critical point set of the mirror Landau–Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes phenomenon for the equivariant ${\mathbb {C}}\mathbf{P }^{1}$ model, and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis, we will study Dubrovin’s conjecture for this equivariant model.

Appendices

Appendix A. GKZ curve from the J-function

In this appendix we will discuss a heuristic derivation of the GKZ curves from the J-functions for the projective space and complete intersections. For this purpose, we will introduce the on-shell equivariant J-function.¹⁹ Let X be the projective space ${\mathbb {C}}\mathbf{P }_{\varvec{w}}^{N-1}$ or the complete intersection $X=X_{\varvec{l};\varvec{w},\varvec{\lambda }}$ in ${\mathbb {C}}\mathbf{P }^{N-1}$. Let $\ell :H^*_T({\mathbb {C}}\mathbf{P }^{N-1}) \rightarrow {\mathbb {C}}$ be a ${\mathbb {C}}$-linear map. For the equivariant J-function $J_{X}(x)$, the composite map $(\ell \circ J_{X})(x)$ is a ${\mathbb {C}}$-valued function satisfying GKZ equation. We assume that $\ell$ is a ${\mathbb {C}}$-algebra homomorphism. Then by

$$\begin{aligned} H^*_T({\mathbb {C}}\mathbf{P }^{N-1}) \cong {\mathbb {C}}[p]/(\prod _{i=0}^{N-1}(p-w_i)), \end{aligned}$$

(A.1)

$\ell (p)$ must be one of equivariant parameters $w_i$$(i=0,\ldots ,N-1)$.

Definition A.1

The function ${\mathcal {J}}_X(x)$ is named on-shell equivariant J-function, if the second equivariant cohomology element $p\in H_T^{2}(X)$ in Proposition 2.3 is replaced with one of equivariant parameters $w_i$$(i=0,\ldots ,N-1)$:

$$\begin{aligned} {\mathcal {J}}_X(x)=J_{X}(x)|_{p=w_i}. \end{aligned}$$

(A.2)

By the construction, we have the following lemma:

Lemma A.2

The on-shell equivariant J-function obeys the GKZ equation.

$$\begin{aligned} {\widehat{A}}_X({\widehat{x}},{\widehat{y}}){\mathcal {J}}_X(x)=0. \end{aligned}$$

(A.3)

We remark that if $w_i \ne w_j$ for $i\ne j$, then ${\mathbb {C}}[p]/(\prod _{i=0}^{N-1}(p-w_i)) \cong ({\mathbb {C}}[p]/(p-w_0)) \times \cdots \times ({\mathbb {C}}[p]/(p-w_{N-1}))$ is a product of N-copies of the ${\mathbb {C}}$-algebra ${\mathbb {C}}$. Then the ${\mathbb {C}}$-algebra homomorphisms $H^*_T({\mathbb {C}}\mathbf{P }^{N-1}) \rightarrow {\mathbb {C}}, p \mapsto w_i$ give a ${\mathbb {C}}$-basis of the space of ${\mathbb {C}}$-linear maps: $H^*_T({\mathbb {C}}\mathbf{P }^{N-1}) \rightarrow {\mathbb {C}}$. Thus the on-shell J-functions give basis of the solution of GKZ equation.

Now we will consider the asymptotic expansion of the on-shell equivariant J-function.

$$\begin{aligned} {\mathcal {J}}_X(x)\sim \mathrm {exp}\left( \sum _{m=0}^{\infty } \hbar ^{m-1}S_m(x)\right) . \end{aligned}$$

(A.4)

Using (2.28) and this asymptotic expansion we find the defining equation of the GKZ equation $A_X(x,y)=0$ in $(x,y)\in {\mathbb {C}}^*\times {\mathbb {C}}$ from the relation:

$$\begin{aligned} y=x\frac{dS_0(x)}{dx}\in {\mathbb {C}}. \end{aligned}$$

(A.5)

In the following we will evaluate the saddle point value $S_0(x)$ of the on-shell equivariant J-function ${\mathcal {J}}_X(x)$ for the projective space $X={\mathbb {C}}\mathbf{P }_{\varvec{w}}^{N-1}$ and the smooth Fano complete intersection $X_{\varvec{l};\varvec{w},\varvec{\lambda }}$ in a heuristic way. As a consequence we will show that the defining equation $A_X(x,y)=0$ of the GKZ curve is obtained for these two cases of X.

(1) Projective space ${\mathbb {C}}\mathbf{P }_{\varvec{w}}^{N-1}$:

Let p denote one of equivariant parameters $w_i$$(i=0,\ldots ,N-1)$. We focus on the factor $\prod _{m=1}^{d}(p-w+m\hbar )$ to find the saddle point value of the on-shell equivariant J-function ${\mathcal {J}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x)$ in (2.7). In the $\hbar \rightarrow 0$ limit while keeping $d\hbar =z$ finite, we use the Riemann integral as follows:

$$\begin{aligned}&\prod _{m=1}^{d}(p-w+m\hbar )= \mathrm {exp}\left[ \sum _{m=1}^d\log (p-w+m\hbar )\right] \nonumber \\&\underset{\begin{array}{c} \hbar \rightarrow 0\\ d\hbar =u:\mathrm {finite} \end{array}}{\sim } \mathrm {exp}\left[ \frac{1}{\hbar }\int ^u_0 du'\,\log (p-w+u')\right] \nonumber \\&\quad =\mathrm {exp}\left[ \frac{1}{\hbar } \left( (u+p-w)\log (u+p-w)-u-(p-w)\log (p-w)\right) \right] . \end{aligned}$$

(A.6)

Adopting this factor we can approximate ${\mathcal {J}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x)$ by the integral on z as

$$\begin{aligned}&{\mathcal {J}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x) \underset{\begin{array}{c} \hbar \rightarrow 0 \end{array}}{\sim } \int _{\gamma } du \,\exp \left[ \frac{1}{\hbar }{\mathcal {W}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(u;x) \right] , \nonumber \\&\quad {\mathcal {W}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(u;x) =(u+p)\log x\nonumber \\&\quad -\sum _{i=0}^{N-1}\bigl ((u+p-w_i)\log (u+p-w_i)-u-(p-w_i)\log (p-w_i)\bigr ), \end{aligned}$$

(A.7)

where we interpret the term $(p-w_i)\log (p-w_i)$ as 0 when $p=w_i$. Here we call ${\mathcal {W}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(u;x)$effective superpotential, and an analytical continuation can be performed by deforming the integration path $\gamma$ on the complex u plane.²⁰ Assuming such analytical continuation, we can approximate the integral (A.7) by the saddle point value in $\hbar \rightarrow 0$ limit:

$$\begin{aligned} S_0(x)={\mathcal {W}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(u_c;x),\qquad \frac{\partial {\mathcal {W}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(u;x)}{\partial u}\Bigg |_{u=u_c} =0. \end{aligned}$$

The saddle point condition is then given by

$$\begin{aligned} \frac{\partial {\mathcal {W}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(u;x)}{\partial u}\Bigg |_{u=u_c} =\log x-\sum _{i=0}^{N-1}\log (u_c+p-w_i) =0, \end{aligned}$$

(A.8)

and by (A.5) one has

$$\begin{aligned} y=x\frac{\partial {\mathcal {W}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(u;x)}{\partial x}\Bigg |_{u=u_c} =u_c+p. \end{aligned}$$

(A.9)

By eliminating the variable $u_c$ from these relations (A.8) and (A.9), we find a constraint equation on $(x,y)\in {\mathbb {C}}^*\times {\mathbb {C}}$:

$$\begin{aligned} \prod _{i=0}^{N-1}(y-w_i)-x=0, \end{aligned}$$

(A.10)

which agrees with the defining equation $A_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x,y) = 0$ of the GKZ curve (2.22).

(2) Complete intersection $X_{\varvec{l};\varvec{w},\varvec{\lambda }}$ in ${\mathbb {C}}\mathbf{P }_{\varvec{w}}^{N-1}$:

Adopting the similar approximation for the factor (A.6) in ${\mathcal {J}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x)$, we obtain the effective superpotential ${\mathcal {W}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(u;x)$ for the on-shell equivariant J-function ${\mathcal {J}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(x)$ in (2.8) as

$$\begin{aligned}&{\mathcal {J}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(x) \underset{\begin{array}{c} \hbar \rightarrow 0 \end{array}}{\sim } \int _{\gamma }du\,\exp \left[ {\mathcal {W}}_{X_{\varvec{l}; \varvec{w},\varvec{\lambda }}}(u;x)\right] ,\nonumber \\&{\mathcal {W}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(u;x)=(u+p)\log x\nonumber \\&\quad -\sum _{i=0}^{N-1}\bigl ((u+p-w_i)\log (u+p-w_i)-u-(p-w_i)\log (p-w_i)\bigr ) \nonumber \\&\quad +\sum _{a=1}^n\bigl ((l_a u+l_a p-\lambda _a) \log (l_a u+l_a p-\lambda _a)-l_a u - (l_ap-\lambda _a)\log (l_ap-\lambda _a)\bigr ), \end{aligned}$$

(A.11)

where p denotes one of equivariant parameters $w_i$$(i=0,\ldots ,N-1)$ and we interpret the term $(p-w_i)\log (p-w_i)$ as 0 when $p=w_i$. The saddle point condition is then given by

$$\begin{aligned} \frac{\partial {\mathcal {W}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(u;x)}{\partial u}\Bigg |_{u=u_c}&=\log x-\sum _{i=0}^{N-1}\log (u_c+p-w_i) \nonumber \\&\quad -\sum _{a=1}^{n}\log (l_a u_c+l_ap-\lambda _a)^{-l_a} =0, \end{aligned}$$

(A.12)

and by (A.5) one has

$$\begin{aligned} y=x\frac{\partial {\mathcal {W}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(u;x)}{\partial x}\Bigg |_{u=u_c} =u_c+p. \end{aligned}$$

(A.13)

As a result of the elimination of the variable $u_c$ from these relations (A.12) and (A.13), we find a constraint equation on $(x,y)\in {\mathbb {C}}^*\times {\mathbb {C}}$:

$$\begin{aligned} \prod _{i=0}^{N-1}(y-w_i)-x\prod _{a=1}^n(l_ay-\lambda _a)^{l_a}=0, \end{aligned}$$

(A.14)

which agrees with the defining equation $A_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(x,y) = 0$ of the GKZ curve (2.24).

Appendix B. GKZ equations for oscillatory integrals

In this appendix we will give a proof of Propositions 2.9 and 2.12.

1.1 B.1 Proof of Propositions 2.9

We will show that the oscillatory integral ${\mathcal {I}}_X(x)$ satisfies the GKZ equation for the projective space $X={\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}$ and the Fano complete intersection $X_{\varvec{l};\varvec{w},\varvec{\lambda }}$ separately.

Proposition B.1

The oscillatory integral ${\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x)$ in (2.17) satisfies the GKZ equation (2.9).

Proof

Act a differential operator $\left( \hbar x\frac{d}{dx}-w_i\right) \left( \hbar x\frac{d}{dx}-w_0\right)$ on the oscillatory integral ${\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x)$,

$$\begin{aligned}&\left( \hbar x\frac{d}{dx}-w_i\right) \left( \hbar x\frac{d}{dx}-w_0\right) {\mathcal {I}}_{{\mathbb {C}} \mathbf{P }^{N-1}_{\varvec{w}}}(x)\\&\quad =\left( \hbar x\frac{d}{dx}-w_i\right) \int _{\Gamma }\prod _{i=1}^{N-1} du_i\,\frac{x}{(u_1\cdots u_{N-1})^2} \,\mathrm {e}^{\frac{1}{\hbar }W_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}} (u_1,\ldots ,u_{N-1};x)}\\&\quad =\int _{\Gamma }\prod _{i=1}^{N-1}du_i\,\frac{x}{(u_1\cdots u_{N-1})^2} \left( \frac{x}{u_1\cdots u_{N-1}}+w_0-w_i+\hbar \right) \mathrm {e}^{\frac{1}{\hbar }W_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}} (u_1,\ldots ,u_{N-1};x)}, \end{aligned}$$

where $W_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}$ is the Landau–Ginzburg potential given in (2.14). To manipulate further we will use the following integration by parts:

$$\begin{aligned} 0&=\int _{\Gamma }\prod _{i=1}^{N-1}du_i\, \hbar \frac{d}{du_i}\left( \frac{1}{u_i} \,\mathrm {e}^{\frac{1}{\hbar }W_{{\mathbb {C}} \mathbf{P }^{N-1}_{\varvec{w}}}(u_1,\ldots ,u_{N-1};x)}\right) \\&=\int _{\Gamma }\prod _{i=1}^{N-1}du_i\,\frac{1}{u_i^2}\Biggl [ \left( u_i+w_i-w_0-\hbar -\frac{x}{u_1\cdots u_{N-1}}\right) \Biggr ] \mathrm {e}^{\frac{1}{\hbar }W_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}} (u_1,\ldots ,u_{N-1};x)}. \end{aligned}$$

In this computation, any boundary contributions do not appear, because the image of the Lefschetz thimble $\Gamma$ is a relative cycle starting from a non-degenerate critical point $p_{\mathrm {crit}}$ of the Landau–Ginzburg potential $W_X$ to the infinity. Then one finds that

$$\begin{aligned}&\left( \hbar x\frac{d}{dx}-w_i\right) \left( \hbar x\frac{d}{dx}-w_0\right) {\mathcal {I}}_{{\mathbb {C}} \mathbf{P }^{N-1}_{\varvec{w}}}(x)\\&\quad =\int _{\Gamma }\prod _{i=1}^{N-1}du_i\,\frac{xu_i}{(u_1\cdots u_{N-1})^2} \,\mathrm {e}^{\frac{1}{\hbar }W_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}} (u_1,\ldots ,u_{N-1};x)}. \end{aligned}$$

Repeating the above manipulations for $\left[ \prod _{i=1}^{N-1}\left( \hbar x\frac{d}{dx}-w_i\right) \right] \left( \hbar x\frac{d}{dx}-w_0\right)$, the following relation is obtained:

$$\begin{aligned} \begin{aligned}&\left[ \prod _{i=1}^{N-1}\left( \hbar x\frac{d}{dx}-w_i\right) \right] \left( \hbar x\frac{d}{dx}-w_0\right) {\mathcal {I}}_{{\mathbb {C}} \mathbf{P }^{N-1}_{\varvec{w}}}(x)\\&\quad =\int _{\Gamma }\prod _{i=1}^{N-1}du_i\,\frac{xu_1\cdots u_{N-1}}{(u_1\cdots u_{N-1})^2} \,\mathrm {e}^{\frac{1}{\hbar }W_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}} (u_1,\ldots ,u_{N-1};x)} =x{\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x). \end{aligned} \end{aligned}$$

(B.1)

This differential equation is the same as the GKZ equation (2.9) for the J-function $J_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}$. $\quad \square$

Proposition B.2

The oscillatory integral ${\mathcal {I}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(x)$ in (2.18) satisfies the GKZ equation (2.9).

Proof

Consider the GKZ equation (B.1) for the oscillatory integral ${\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x)$ for the mirror Landau–Ginzburg model of the projective space $X={\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}$ denoted by

$$\begin{aligned} 0&={\widehat{A}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}} \left( {\widehat{x}},{\widehat{y}}\right) {\mathcal {I}}_{{\mathbb {C}} \mathbf{P }^{N-1}_{\varvec{w}}}(x) =\prod _{i=0}^{N-1}\left( {\widehat{y}}-w_i\right) {\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x) -{\widehat{x}}{\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x), \end{aligned}$$

where ${\widehat{x}}$ (resp. ${\widehat{y}}$) acts on ${\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(x)$ as x (resp. $\hbar x d/dx$). Perform the Laplace transformation of this differential equation:

$$\begin{aligned} 0&=\int _0^{\infty }dv_1\cdots \int _0^{\infty }dv_n\, \mathrm {e}^{-\frac{\sum _{i=1}^n(v_i+\lambda _i\log v_i)}{\hbar }} \\ {}&\quad {\widehat{A}}_{{\mathbb {C}}\mathbf {P }^{N-1}_{\varvec{w}}}\left( v_1^{l_1}\cdots v_n^{l_n}{\widehat{x}},{\widehat{y}}\right) {\mathcal {I}}_{{\mathbb {C}} \mathbf {P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x)\\ {}&=\int _0^{\infty }dv_1\cdots \int _0^{\infty }dv_n\, \mathrm {e}^{-\frac{\sum _{a=1}^n(v_a+\lambda _a\log v_a)}{\hbar }}\\ {}&\quad \prod _{i=0}^{N-1}\left( \hbar x\frac{d}{dx}-w_i\right) {\mathcal {I}}_{{\mathbb {C}}\mathbf {P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x)\\ {}&\quad -\int _0^{\infty }dv_1\cdots \int _0^{\infty }dv_n\, \mathrm {e}^{-\frac{\sum _{a=1}^n(v_a+\lambda _a\log v_a)}{\hbar }} v_1^{l_1}\cdots v_n^{l_n}x{\mathcal {I}}_{{\mathbb {C}}\mathbf {P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x)\\&=\prod _{i=0}^{N-1}\left( \hbar x\frac{d}{dx}-w_i\right) {\mathcal {I}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(x)\\ {}&\quad -x\int _0^{\infty }dv_1\cdots \int _0^{\infty }dv_n\, \mathrm {e}^{-\frac{\sum _{a=1}^n(v_a+\lambda _a\log v_a)}{\hbar }} v_1^{l_1}\cdots v_n^{l_n} {\mathcal {I}}_{{\mathbb {C}}\mathbf {P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x). \end{aligned}$$

To manipulate further we will use the following integration by parts repeatedly for each $v_i$’s:

$$\begin{aligned} 0&=\int _0^{\infty }dv_a\,\hbar \frac{d}{dv_a}\left( v_a^{m} \mathrm {e}^{-\frac{v_a+\lambda _a\log v_a}{\hbar }} {\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x) \right) \\&=\int _0^{\infty }dv_a\left( -v_a-\lambda _a+m\hbar \right) v_a^{m-1}\mathrm {e}^{-\frac{v_a+\lambda _a\log v_a}{\hbar }} {\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x) \\&\quad +\int _0^{\infty }dv_av_a^{m}\mathrm {e}^{-\frac{v_a+\lambda _a\log v_a}{\hbar }} \hbar l_a\frac{x}{v_a}\frac{d}{dx}{\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x) \\&=-\int _0^{\infty }dv_a\,v_a^{m}\mathrm {e}^{-\frac{v_a+\lambda _a\log v_a}{\hbar }} {\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x) \\&\quad +\left( l_a\hbar x\frac{d}{dx}-\lambda _a+m\hbar \right) \int _0^{\infty } dv_a\,v_a^{m-1}\mathrm {e}^{-\frac{v_a+\lambda _a\log v_a}{\hbar }} {\mathcal {I}}_{{\mathbb {C}}\mathbf{P }^{N-1}_{\varvec{w}}}(v_1^{l_1}\cdots v_n^{l_n}x). \end{aligned}$$

Then one finds the GKZ equation (2.11) for the J-function $J_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(x)$:

$$\begin{aligned} \left[ \prod _{i=0}^{N-1}\left( \hbar x\frac{d}{dx}-w_i\right) -x\prod _{a=1}^{n}\prod _{m=1}^{l_a}\left( l_a\hbar x\frac{d}{dx}-\lambda _a+m\hbar \right) \right] {\mathcal {I}}_{X_{\varvec{l};\varvec{w},\varvec{\lambda }}}(x)=0. \end{aligned}$$

(B.2)

$\square$

1.2 B.2 Proof of Proposition 2.12

Here we investigate the behavior of coefficients when $x \rightarrow \infty$ in the saddle point approximation (2.34) of the oscillatory integral for

$$\begin{aligned} W_X= & {} \sum _{i=1}^{N-1} (u_i + w_i \log u_i) - \sum _{a=1}^{n} (v_a + \lambda _a \log v_a) + \frac{v_1 \cdots v_n}{u_1 \cdots u_{N-1}} x \\&+w_0 \log \left( \frac{v_1 \cdots v_n}{u_1 \cdots u_{N-1}} x \right) , \end{aligned}$$

which is mirror to $X = X_{\varvec{w},\varvec{\lambda }}$. In this subsection we write $W = W_X$ for simplicity.

To prove (i) in Proposition 2.12, it is enough to find an asymptotic behavior of second derivatives of W:

$$\begin{aligned} \frac{\partial ^{2} W}{\partial u_{i}^2}= & {} - \frac{w_i - w_0}{u_i^2} + \frac{2}{u_i^2} \, \frac{v_1 \cdots v_n}{u_1 \cdots u_{N-1}} x \\ \frac{\partial ^{2} W}{\partial u_{i} \partial u_j}= & {} \frac{1}{u_i u_j} \, \frac{v_1 \cdots v_n}{u_1 \cdots u_{N-1}} x \quad (i \ne j)\\ \frac{\partial ^{2} W}{\partial u_{i} \partial v_a}= & {} - \frac{1}{u_i v_a} \, \frac{v_1 \cdots v_n}{u_1 \cdots u_{N-1}} x \\ \frac{\partial ^{2} W}{\partial v_{a} \partial v_b}= & {} \frac{1}{v_a v_b} \, \frac{v_1 \cdots v_n}{u_1 \cdots u_{N-1}} x \quad (a \ne b)\\ \frac{\partial ^{2} W}{\partial v_{a}^2}= & {} \frac{\lambda _a - w_0}{v_a^2}. \end{aligned}$$

At a critical point $({\varvec{u}}^{\mathrm{(c)}},{\varvec{v}}^{\mathrm{(c)}})$, we can use

$$\begin{aligned} \left( \frac{v_1 \cdots v_n}{u_1 \cdots u_{N-1}} x \right) \biggl |_{({\varvec{u}},{\varvec{v}}) = ({\varvec{u}}^{\mathrm{(c)}},{\varvec{v}}^{\mathrm{(c)}})} = u_i^{\mathrm{(c)}} + w_i - w_0 = v_a^{\mathrm{(c)}} + \lambda _a - w_0. \end{aligned}$$

This behaves as $O(x^{\frac{1}{N-n}})$ in the case of (2.32), and as $\lambda _b - w_0 + O(x^{-1})$ in the case of (2.33). Therefore, we can find the behavior

$$\begin{aligned} \mathrm{Hess}({\varvec{u}}^{\mathrm{(c)}},{\varvec{v}}^{\mathrm{(c)}}) = {\left\{ \begin{array}{ll} O(x^{-\frac{N+n-1}{N-n}}) &{} \text {in the case of } (2.32), \\ O(x^{2}) &{} \text {in the case of }(2.33) \end{array}\right. } \end{aligned}$$

of the Hessian when $x \rightarrow \infty$. The claim of (i) in Proposition 2.12 follows immediately from this computation.

Let us prove (ii) in Proposition 2.12. We take a coordinate at a critical point $({\varvec{u}}^{\mathrm{(c)}}, {\varvec{v}}^\mathrm{(c)})$ of W:

$$\begin{aligned} {\varvec{\xi }} = (\xi _1, \dots , \xi _{N+n-1}) = (u_1 - u_1^\mathrm{(c)},\dots ,u_{N-1} - u_{N-1}^{\mathrm{(c)}}, v_1 - v_1^{\mathrm{(c)}}, \dots , v_n - v_n^{\mathrm{(c)}}), \end{aligned}$$

and consider the Taylor expansion of W at the critical point ${\varvec{\xi }}^{\mathrm{(c)}} = {\varvec{0}}$:

$$\begin{aligned} W({\varvec{\xi }};x)= & {} W({\varvec{\xi }}^{\mathrm{(c)}};x) + \frac{1}{2!}\sum _{i,j} W_{ij}({\varvec{\xi }}^{\mathrm{(c)}};x) \, \xi _i \xi _j \nonumber \\&+ \sum _{m \ge 3} \frac{1}{m!} \sum _{i_1, \dots , i_m} W_{i_1 \dots i_m}({\varvec{\xi }}^\mathrm{(c)};x) \, \xi _{i_1} \cdots \xi _{i_m}, \end{aligned}$$

(B.3)

where $W_{i_1 \dots i_m} = (\partial ^m W)/(\partial \xi _{i_1} \cdots \partial \xi _{i_m})$. Since we have chosen generic $w_i$ and $\lambda _a$ so that the critical points are non-degenerate, the Hesse matrix H has non-zero determinant at ${\varvec{\xi }}^{\mathrm{(c)}}$. We further take a linear transformation $\xi _i = (- \hbar )^{1/2}\sum _j C_{i}^j(x) s_j$ of the coordinates which transforms the quadratic part of W as

$$\begin{aligned} \frac{1}{2 \hbar }\sum _{i,j} W_{ij}({\varvec{\xi }}^{\mathrm{(c)}};x)\, \xi _i \xi _j = - \frac{1}{2} \sum _{i=1}^{N+n-1} s_i^2. \end{aligned}$$

(B.4)

We can find these coefficients $C_{i}^{j}(x)$ by applying the simultaneous completing the square to the quadratic form in the left hand side of (B.4). Eventually we can find the behavior of the coefficients $C_{i}^{j}(x)$ for large x as follows:

• Let us consider the case when the critical point $({\varvec{u}}^{\mathrm{(c)}},{\varvec{v}}^{\mathrm{(c)}})$ behaves as (2.32) in Lemma 2.11. Since $W_{ij}({\varvec{\xi }}^\mathrm{(c)};x)$ behaves as $O(x^{- \frac{1}{N-n}})$ in this case, we can show that

  $$\begin{aligned} C_{i}^{j}(x) = O(x^{\frac{1}{2(N-n)}}) \end{aligned}$$

  (B.5)

  holds for all $i,j=1,\dots , N+n-1$, when $x \rightarrow \infty$.

• Let us consider the case when the critical point $({\varvec{u}}^{\mathrm{(c)}},{\varvec{v}}^{\mathrm{(c)}})$ behaves as (2.33) in Lemma 2.11. Let $i_b \in \{N,\dots ,N+n-1 \}$ be the label of $v_b$; that is, $\xi _{i_b} = v_b - v_b^{\mathrm{(c)}}$. Since we can arrange the quadratic part of W as

  $$\begin{aligned}&W_{i_b i_b}({\varvec{\xi }}^{\mathrm{(c)}};x) \xi _{i_b}^2 + 2\sum _{i \ne i_b} W_{i \, i_b}({\varvec{\xi }}^{\mathrm{(c)}};x) \xi _{i_b} \xi _i + \sum _{i,j \ne i_b} W_{ij}({\varvec{\xi }}^{\mathrm{(c)}};x) \xi _{i} \xi _j \\&\quad =W_{i_b i_b}({\varvec{\xi }}^{\mathrm{(c)}};x) \left( \xi _{i_b} + \sum _{i \ne i_b} \frac{W_{i\, i_b}({\varvec{\xi }}^{\mathrm{(c)}};x)}{W_{i_b i_b}({\varvec{\xi }}^{\mathrm{(c)}};x)} \xi _i \right) ^2 \\&\qquad - \frac{1}{W_{i_b i_b}({\varvec{\xi }}^{\mathrm{(c)}};x)} \left( \sum _{i \ne i_b} W_{i\, i_b}({\varvec{\xi }}^{\mathrm{(c)}};x) \xi _i\right) ^2 + \sum _{i,j \ne i_b} W_{ij}({\varvec{\xi }}^{\mathrm{(c)}};x) \xi _{i} \xi _j. \end{aligned}$$

  Thus we can choose

  $$\begin{aligned} s_{i_b} = W_{i_b i_b}^{1/2}({\varvec{\xi }}^{\mathrm{(c)}};x) \left( \xi _{i_b} + \sum _{i \ne i_b} \frac{W_{i\, i_b}({\varvec{\xi }}^{\mathrm{(c)}};x)}{W_{i_b i_b}({\varvec{\xi }}^{\mathrm{(c)}};x)} \xi _i \right) \end{aligned}$$

  as one of new coordinates, and other $s_i$’s are written in terms of $\xi _i$’s except for $\xi _{i_b}$. Since $W_{i_b i_b}({\varvec{\xi }}^\mathrm{(c)};x) = O(x^2)$, $W_{i\, i_b}({\varvec{\xi }}^{\mathrm{(c)}};x) = O(x)$ if $i \ne i_b$ and $W_{ij}({\varvec{\xi }}^{\mathrm{(c)}};x)=O(1)$ for $i,j \ne i_b$, we can conclude

  $$\begin{aligned} C_{i}^{j}(x) = {\left\{ \begin{array}{ll} O(x^{-1}) &{} \text {if }i = i_b \\ O(1) &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

  (B.6)

  hold when $x \rightarrow \infty$.

Let us proceed the computation of saddle point expansion. The above change of the coordinate yields

$$\begin{aligned} d\xi _1 \cdots d\xi _{N+n-1} = \frac{(-\hbar )^{\frac{N+n-1}{2}}}{\sqrt{\mathrm{Hess}({\varvec{u}}^\mathrm{(c)},{\varvec{v}}^{\mathrm{(c)}})}} \, ds_1\cdots ds_{N+n-1}. \end{aligned}$$

Then, by the standard argument of the saddle point method, the asymptotic expansion of the oscillatory integral is computed by term-wise integration:

$$\begin{aligned} {{\mathcal {I}}}_{X}(x)\sim & {} {} \exp \left( \frac{1}{\hbar } W({\varvec{\xi }}^\mathrm {(c)};x) \right) \, \frac{(-\hbar )^{\frac{N+n-1}{2}}}{u^{\mathrm {(c)}}_1 \cdots u^{\mathrm {(c)}}_{N-1} \,\sqrt{\mathrm {Hess}({\varvec{\xi }}^{\mathrm {(c)}})}}\nonumber \\&\times \left( 1 + \sum _{m = 1}^{\infty } \hbar ^{\frac{m}{2}} \sum _{i_1, \dots , i_{m}} f_{i_1 \dots i_m} \int _{{{\mathbb {R}}}^{N+n-1}} ds_1 \cdots ds_{N+n-1} \right. \nonumber \\&\left. \qquad \exp \left( - \frac{1}{2} \sum _{i=1}^{N+n-1} s_i^2 \right) s_{i_1} \cdots s_{i_m} \right) , \end{aligned}$$

(B.7)

where $f_{i_1 \dots i_m}$ is the Taylor coefficient given by

$$\begin{aligned}&\frac{u^{\mathrm{(c)}}_1 \cdots u^{\mathrm{(c)}}_{N-1}}{u_1 \cdots u_{N-1}}\exp \left( \frac{1}{\hbar } \sum _{m \ge 3} \frac{1}{m!} \sum _{i_1, \dots , i_m} W_{i_1 \dots i_m}({\varvec{\xi }}^{\mathrm{(c)}};x) \xi _{i_1} \cdots \xi _{i_m} \right) \Biggl |_{\xi _i = (- \hbar )^{1/2}\sum _j C_{i}^j(x) s_j} \\&\quad =1 + \sum _{m = 1}^{\infty } \hbar ^{\frac{m}{2}} \sum _{i_1, \dots , i_{m}} f_{i_1 \dots i_m} s_{i_1} \cdots s_{i_m}. \end{aligned}$$

• If $({\varvec{u}}^{\mathrm{(c)}},{\varvec{v}}^{\mathrm{(c)}})$ behaves as (2.32), then $W_{i_1 \dots i_m}({\varvec{\xi }}^{\mathrm{(c)}};x) = O(x^{- \frac{m-1}{N-n}})$, and hence,

  $$\begin{aligned} W_{i_1 \dots i_m}({\varvec{\xi }}^{\mathrm{(c)}};x) \, C_{i_1}^{j_1} \cdots C_{i_m}^{j_m} = O(x^{- \frac{m-2}{2(N-n)}}) \quad \text {for any }j_1, \dots , j_m. \end{aligned}$$

  For $m \ge 3$, this tends to 0 when $x \rightarrow \infty$.

• If $({\varvec{u}}^{\mathrm{(c)}},{\varvec{v}}^{\mathrm{(c)}})$ behaves as (2.33), then $W_{i_1 \dots i_m}({\varvec{\xi }}^{\mathrm{(c)}};x) = O(x^{\ell _b})$, where $\ell _b$ is the number of $i_b$ in the indices $i_1, \dots , i_m$. Therefore,

  $$\begin{aligned} W_{i_1 \dots i_m}({\varvec{\xi }}^{\mathrm{(c)}};x) \, C_{i_1}^{j_1} \cdots C_{i_m}^{j_m} = O(1) \quad \text {for any }j_1, \dots , j_m. \end{aligned}$$

We can also verify that

$$\begin{aligned} \frac{u_i^{\mathrm{(c)}}}{u_i} = \left( 1 + (-\hbar )^{1/2} \sum _{i} \frac{C_i^{j}}{u_i^{\mathrm{(c)}}} s_j \right) ^{-1}, \end{aligned}$$

and the coefficient satisfies

$$\begin{aligned} \frac{C_i^{j}}{u_i^{\mathrm{(c)}}} = {\left\{ \begin{array}{ll} O(x^{- \frac{1}{2(N-n)}}) &{} \text {for the case of }(2.32) \\ O(1) &{} \text {for the case of }(2.33) \end{array}\right. } \end{aligned}$$

when $x \rightarrow \infty$. Therefore, we can conclude that

$$\begin{aligned} f_{i_1 \dots i_m} = {\left\{ \begin{array}{ll} O(x^{- \frac{1}{2(N-n)}}) &{} \text {for the case of } (2.32) \\ O(1) &{} \text {for the case of }(2.33) \end{array}\right. } \end{aligned}$$

(B.8)

for $m \ge 1$. After evaluating the Gaussian integrals in (B.7) by using

$$\begin{aligned} \int _{{\mathbb {R}}} ds_i\, e^{- \frac{1}{2} s_i^2} \, s_i^{k} = {\left\{ \begin{array}{ll} 0 &{} \text {if }k\text { is odd}, \\ \displaystyle \sqrt{2\pi } \, {(k-1)!!} &{} \text {if }k\text { is even}, \end{array}\right. } \end{aligned}$$

we obtain the saddle point approximation (2.34). In particular, (B.8) proves the claim (ii) in Proposition 2.12.

Appendix C. Computational results by iteration and topological recursion

In this appendix we will firstly give some explicit computational results of the WKB solutions to the GKZ equations. In Appendix C.2 we will explicitly perform the WKB reconstruction (3.13) for the equivariant ${{\mathbb {C}}}\mathbf{P }^1$ model, and see agreements with the results in Appendix C.1.

1.1 C.1. Some iterative computations for the GKZ equation

Assume the saddle point approximation of the oscillatory integral

$$\begin{aligned} {\mathcal {I}}_X(x)\sim \exp \left( \sum _{m=0}^{\infty }\hbar ^{m-1}S_m(x)\right) , \end{aligned}$$

one finds a set of the first order differential equations for $S_m$’s by expanding the GKZ equation around $\hbar =0$.

$\underline{{\mathbb {C}}\mathbf{P }^{N-1}\hbox { model}}$

The GKZ equation for the (non-equivariant) ${\mathbb {C}}\mathbf{P }^{N-1}$ model is

$$\begin{aligned} \left[ \left( \hbar x\frac{d}{dx}\right) ^{N}-x\right] {\mathcal {I}}_{{\mathbb {C}} \mathbf{P }^{N-1}}(x)=0. \end{aligned}$$

(C.1)

Some computational results of $S_m$’s for $N=2, 3, 4$ are listed in Table 2.

Table 2 $S_m(x)$ for the ${\mathbb {C}}\mathbf{P }^{N-1}$ models ($N=2,3,4$) obtained from the GKZ equation (C.1)

$\underline{\hbox {Equivariant }{\mathbb {C}}\mathbf{P }^{1}\hbox { model}}$

The GKZ equation for the equivariant ${\mathbb {C}}\mathbf{P }^{1}$ model is

$$\begin{aligned} \left[ \left( \hbar x\frac{d}{dx}-w_0\right) \left( \hbar x\frac{d}{dx}-w_1\right) -x\right] {\mathcal {I}}_{{\mathbb {C}} \mathbf{P }^{1}_{\varvec{w}}}(x)=0. \end{aligned}$$

(C.2)

For this model there are two solutions which have the formal power series expansion:

$$\begin{aligned} {\mathcal {I}}^{(\pm )}_{{\mathbb {C}}\mathbf{P }^{1}_{\varvec{w}}}(x)\sim \exp \left( \sum _{m=0}^{\infty }\hbar ^{m-1}S_m^{(\pm )}(x) \right) . \end{aligned}$$

Computational results of $S^{(\pm )}_m$ for $m=0,1,2,3,4$ are listed in Table 3 modulo constant shifts.

Table 3 $S_m^{(\pm )}(x)$ for the equivariant ${\mathbb {C}}\mathbf{P }^{1}$ model obtained from the GKZ equation (C.2)

$\underline{\hbox {Degree 1 hypersurface in }{\mathbb {C}}\mathbf{P }^{1}}$

The GKZ equation for the degree 1 hypersurface $X_{\varvec{w},\lambda }=X_{l=1;\varvec{w},\lambda }$ in ${\mathbb {C}}\mathbf{P }^{1}$ is

$$\begin{aligned} \left[ \left( \hbar x\frac{d}{dx}-w_0\right) \left( \hbar x\frac{d}{dx}-w_1\right) -x\left( \hbar x\frac{d}{dx}-\lambda +\hbar \right) \right] {\mathcal {I}}_{X_{\varvec{w}, \lambda }}(x)=0. \end{aligned}$$

(C.3)

For this model we also find two solutions which have the formal power series expansion:

$$\begin{aligned} {\mathcal {I}}^{(\pm )}_{X_{\varvec{w},\lambda }}(x) \sim \exp \left( \sum _{m=0}^{\infty }\hbar ^{m-1}S_m^{(\pm )}(x) \right) . \end{aligned}$$

Computational results of $S^{(\pm )}_m$ for $m=0,1,2,3$ are listed in Table 4 modulo constant shifts.

Table 4 $S_m^{(\pm )}(x)$ for degree 1 hypersurface in ${\mathbb {C}}\mathbf{P }^{1}$ obtained from the GKZ equation (C.3)

Especially, focusing on the $S^{(\pm )}_1(x)$ we find the following expansion around $x=\infty$:

$$\begin{aligned} \begin{aligned} \mathrm {e}^{S^{(+)}_1(x)}&=1+\frac{(w_0-\lambda )(w_1-\lambda )}{2x^2}+O(x^{-3}),\\ \mathrm {e}^{S^{(-)}_1(x)}&=\frac{1}{x}-\frac{w_0+w_1 -2\lambda }{x^2}+O(x^{-3}), \end{aligned} \end{aligned}$$

and these asymptotic expansions are consistent with Proposition 2.12 (i).

1.2 C.2 Topological recursion for the equivariant ${\mathbb {C}}\mathbf{P }^{1}$ model

In the following, for the equivariant ${\mathbb {C}}\mathbf{P }^{1}$ model, we will explicitly recover the computational result in Table 3 by applying the topological recursion (3.9) in [43]. The GKZ curve

$$\begin{aligned} \Sigma _{{\mathbb {C}}\mathbf{P }^{1}_{\varvec{w}}}=\Big \{\; (x,y)\in {\mathbb {C}}^*\times {\mathbb {C}}\; \Big |\; (y-w_0)(y-w_1)-x=0\; \Big \} \end{aligned}$$

is parametrized by a local coordinate z as follows:

$$\begin{aligned} x(z)=z^2-\Lambda ,\qquad y(z)=z+\frac{1}{2}(w_0+w_1),\qquad \Lambda =\frac{1}{4}(w_0-w_1)^2. \end{aligned}$$

The spectral curve $\Sigma _{{\mathbb {C}}\mathbf{P }^{1}_{\varvec{w}}}$ has only one simple ramification point at $z=0$ in this local coordinate. Starting from

$$\begin{aligned} \omega _{1}^{(0)}(z)=0,\qquad \omega _{2}^{(0)}(z_1,z_2)=B(z_1,z_2)=\frac{dz_1dz_2}{(z_1-z_2)^2}, \end{aligned}$$

the differentials $\omega _{n}^{(g)}$ for $(g,n) \ne (0,1), (0,2)$ are defined by the topological recursion (3.9)

$$\begin{aligned} \begin{aligned} \omega _{n+1}^{(g)}(z,\varvec{z}_N)&= \mathop {\mathrm {Res}}_{w=0}\, \frac{\int _{-w}^{w}B(\cdot , z)}{2\left( y(w)-y(-w)\right) dx(w)/x(w)}\bigg (\omega _{n+2}^{(g-1)} (w,-w,\varvec{z}_N)\\&\quad +\sum _{\ell =0}^{g}\sum _{\emptyset =J\subseteq N}\omega _{|J|+1}^{(g-\ell )}(w,\varvec{z}_J)\omega _{|N|-|J|+1}^{(\ell )} (-w,\varvec{z}_{N \backslash J}) \bigg ), \end{aligned} \end{aligned}$$

where $N=\{1,2,\ldots ,n\}\supset J=\{i_1,i_2,\ldots ,i_j\}$, and $N\backslash J=\{i_{j+1},i_{j+2},\ldots ,i_n\}$. Integrating these multi-differentials, one finds the free energies

$$\begin{aligned} \begin{aligned} F_{1}^{(0)}(x)&=\int ^z_{z_*} y(z')\frac{dx(z')}{x(z')},\qquad F_{2}^{(0)}(x)=\int ^z_{z_*}\int ^z_{z_*}\left( B(z_1',z_2') -\frac{dx(z_1')dx(z_2')}{(x(z_1')-x(z_2'))^2}\right) ,\\ F_{n}^{(g)}(x)&=\int ^z_{z_*} \cdots \int ^z_{z_*} \omega ^{(g)}_{n} (z_1',\ldots ,z_n'),\quad (g,n) \ne (0,1), (0,2), \end{aligned} \end{aligned}$$

(C.4)

where $z_*$ denotes a reference point.

The WKB reconstruction (3.13) of wave function is defined by $F_{n}^{(g)}(x)$’s as

$$\begin{aligned} \psi _{{\mathbb {C}}\mathbf{P }^{1}_{\varvec{w}}}(x)= \exp \left( \sum _{g=0,n=1}^{\infty }\frac{\hbar ^{2g-2+n}}{n!} F^{(g)}_{n}(x)\right) . \end{aligned}$$

(C.5)

We fix the reference point by $z_*=\infty$ which corresponds to $x(z_*)=\infty$. Here note that $F_{1}^{(0)}$ and $F_{2}^{(0)}$ need to be regularized by certain constant shifts so as to depend on $z_*$. Some explicit computational results of the free energies $F_n^{(g)}(x)$ are listed in Table 5.

Table 5 Free energies for the equivariant ${\mathbb {C}}\mathbf{P }^{1}$ model. Here we have two types free energies corresponding to the branches $z=\pm \sqrt{x+\Lambda }$

The wave function (C.5) is reorganized by

$$\begin{aligned} \psi _{{\mathbb {C}}\mathbf{P }^{1}_{\varvec{w}}}(x)=\exp \left( \sum _{m=0}^{\infty }\hbar ^{m-1}F_m(x)\right) ,\qquad F_m(x)=\sum _{\begin{array}{c} g\ge 0,\;n\ge 1,\\ 2g+n-1=m \end{array}}\frac{1}{n!}F_n^{(g)}(x), \end{aligned}$$

where corresponding to two branches $z=\pm \sqrt{x+\Lambda }$ we find two types of free energies $F_m(x)=F_m^{(\pm )}(x)$. Since the leading term $S_0(x)$ of the asymptotic expansion of the J-function obeys

$$\begin{aligned} x\frac{dS_0(x)}{dx}=y(x), \end{aligned}$$

the free energy $F_1^{(0)}(x)$ in (C.4) agrees with $S_0^{(\pm )}(x)$ up to a constant shift. Using the computational results in Table 5, $F_m^{(\pm )}(x)$’s ($m\ge 1$) are computed immediately and summarized in Table 6.

Table 6 $F_m(x)=F_m^{(\pm )}(x)$ for the equivariant ${\mathbb {C}}\mathbf{P }^{1}$ model

Comparing the computational results in Tables 3 and 6, one finds the agreement

$$\begin{aligned} F_m^{(\pm )}(x)=S_m^{(\pm )}(x), \end{aligned}$$

for $m=1,2,3,4$ up to a constant shift of $F_1^{(\pm )}(x)$.