Hodge–GUE Correspondence and the Discrete KdV Equation

Abstract

We prove the conjectural relationship recently proposed in Dubrovin and Yang (Commun Number Theory Phys 11:311–336, 2017) between certain special cubic Hodge integrals of the Gopakumar–Mariño–Vafa type (Gopakumar and Vafa in Adv Theor Math Phys 5:1415–1443, 1999, Mariño and Vafa in Contemp Math 310:185–204, 2002) and GUE correlators, and the conjecture proposed in Dubrovin et al. (Adv Math 293:382–435, 2016) that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy.

Appendices

Appendix A: Givental Quantization

Denote by ${{\mathcal {V}}}$ the space of Laurent polynomials in z with coefficients in ${{\mathbb {C}}}$. Define a symplectic bilinear form $\omega$ on ${{\mathcal {V}}}$ by

$$\begin{aligned} \omega (f,g):=- \mathrm{Res}_{z=\infty } f(-z) g(z) \frac{dz}{z^2} = - \omega (g,f) , \quad \forall f,g\in {{\mathcal {H}}}. \end{aligned}$$

The pair $({{\mathcal {V}}},\omega )$ is called a Givental symplectic space. For any $f\in {{\mathcal {V}}}$, write

$$\begin{aligned} f = \sum _{i\ge 0} q_i z^{-i} + \sum _{i\ge 0} p_i (-z)^{i+1} . \end{aligned}$$

Then $\{q_i,\,p_i\}|_{i=0}^\infty$ gives a system of canonical coordinates for $({{\mathcal {V}}},\omega ).$ The canonical quantization in these coordinates yields operators of the form

$$\begin{aligned} \widehat{p_i} =\epsilon \frac{\partial }{\partial q_i} ,\quad \widehat{q_i} = \frac{1}{\epsilon } q_i \end{aligned}$$

on the Fock space of formal power series in $q_i$. For any infinitesimal symplectic transformation A on $({{\mathcal {V}}},\omega )$ which satisfies

$$\begin{aligned} \omega (A f,g)+\omega (f, A g) = 0 , \quad \forall f,g\in {{\mathcal {V}}}, \end{aligned}$$

the Hamiltonian associated to A is defined by

$$\begin{aligned} H_{A}(f) = \frac{1}{2} \omega (f, A f ) = - \frac{1}{2} \mathrm{Res}_{z=\infty } f(-z) A f(z) \frac{dz}{z^2} . \end{aligned}$$

This Hamiltonian is a quadratic function on ${{\mathcal {V}}}$, and its quantization is defined via

$$\begin{aligned} \widehat{p_i p_j} = \epsilon ^2\frac{\partial ^2}{\partial q_i \partial q_j} ,\quad \widehat{p_i q_j} = q_j \frac{\partial }{\partial q_i} , \quad \widehat{q_i q_j} = \frac{1}{\epsilon ^2} q_i q_j . \end{aligned}$$

Denote the quantization of $H_{A}$ by ${\widehat{A}}$. We have, for any two infinitesimal symplectic transformations A, B,

$$\begin{aligned} \Bigl [{\widehat{A}} , {\widehat{B}}\Bigr ] = \widehat{[A, B]}+{{\mathcal {C}}}\left( H_A , H_B\right) , \end{aligned}$$

where ${{\mathcal {C}}}$ is the so-called 2-cocycle term satisfying

$$\begin{aligned} {{\mathcal {C}}}(p_ip_j, q_k q_l) = -{{\mathcal {C}}}(q_k q_l, p_i p_j) = \delta _{i,k} \delta _{j,l}+\delta _{i,l} \delta _{j,k} , \end{aligned}$$

and ${{\mathcal {C}}}=0$ for all other pairs of quadratic monomials in p, q.

Define the operators $\ell _k$ by

$$\begin{aligned} \ell _k = (-1)^{k+1} z^{3/2} \partial _z^{k+1} z^{-1/2} ,\quad k\ge -1 . \end{aligned}$$

(A.1)

Then we have

Lemma A.1

([22]). The operators $\ell _k$ are infinitesimal symplectic transformations on ${{\mathcal {V}}}$, and their quantizations yield the Virasoro operators defined in (2.5)–(2.7) as follows:

$$\begin{aligned} L_k \left( \epsilon ^{-1}{\mathbf{t}}, \epsilon \partial /\partial {\mathbf{t}}\right) = \left. \widehat{\ell _k}\right| _{q_i\mapsto t_i, \partial _{q_i} \mapsto \partial _{t_i}, i\ge 0} + \frac{\delta _{k,0}}{16} , \quad k\ge -1. \end{aligned}$$

Lemma A.2

([22]). The multiplication operators $z^{1-2j}, j\ge 1$ are infinitesimal symplectic transformations on ${{\mathcal {V}}}$, and the operators $D_j, j\ge 1$ defined in (2.2) can be represented as

$$\begin{aligned} D_j = \left. \widehat{z^{1-2j}}\right| _{q_i\mapsto t_i, \partial _{q_i} \mapsto \partial _{t_i}, i\ge 0}. \end{aligned}$$

(A.2)

Consider now the quantization ${{\widehat{\Phi }}}$ of the symplectomorphism $f(z)\mapsto \Phi (z) f(z)$, where the function $\Phi (z)$ was defined by equation (2.19). It is defined by

$$\begin{aligned} {\widehat{\Phi }} =\left. e^{\widehat{\log \Phi (z)}}\right| _{q_i\mapsto t_i, \partial _{q_i} \mapsto \partial _{t_i}, i\ge 0} , \end{aligned}$$

where we replace $\log \Phi (z)$ by its asymptotic expansion as $|z|\rightarrow \infty$, $\mathrm{Re}(z)\ne 0$. The latter has the form, up to an inessential piecewise constant term

$$\begin{aligned} \log \Phi (z)\sim \sum _{k=1}^\infty \frac{B_{2k}}{k(2k-1)} \frac{2^{-2k}-1}{z^{2k-1}} . \end{aligned}$$

By using (2.2) we arrive at the following lemma.

Lemma A.3

We have

$$\begin{aligned} {\widehat{\Phi }} = e^{\sum _{k=1}^\infty \frac{B_{2k}}{k(2k-1)} \left( 2^{-2k}-1\right) D_k}. \end{aligned}$$

(A.3)

Remark A.4

The function $\Phi (z)$ is analytic near $z=0$, $\Phi (0)=1$ and

$$\begin{aligned} \log \Phi (z) = -2z \log 2 - 2 \sum _{k=1}^\infty \frac{2^{2k}-1}{2k+1} \zeta (2k+1) \, z^{2k+1} , \quad |z|<\frac{1}{2} . \end{aligned}$$

(A.4)

One can define another quantum operator ${{\hat{\Phi }}}_0$ by quantizing the series (A.4). Geometric interpretation of this quantum operator remains an interesting open question.

Appendix B: List of Symbols

$t_i$ :

indeterminate tracing the $i{\mathrm{th}}$ power of $\psi$-class (aka coupling constant), $i\ge 0$

${{\tilde{t}}}_i$ :

${{\tilde{t}}}_i=t_i-\delta _{i,1}$, $i\ge 0$

$s_j$ :

the coupling constant to the $j{\mathrm{th}}$ power of Hermitian matrix, $j\ge 1$

${{\bar{s}}}_{k}$ :

${{\bar{s}}}_{k} = \left( {\begin{array}{c}2k\\ k\end{array}}\right) s_{2k}$, $k\ge 1$

${{\tilde{s}}}_{2k}$ :

${{\tilde{s}}}_{2k} = s_{2k} - \tfrac{1}{2} \delta _{k,1}$, $k\ge 1$

Z :

the partition function of intersection numbers of $\psi$-classes

$Z_\mathrm{cubic}$ :

the special cubic Hodge partition function

$Z_{\mathrm{GUE}}$ :

the GUE partition function

$Z_\mathrm{even}$ :

the GUE partition function with even couplings

${\widetilde{Z}}$ :

the modified GUE partition function with even couplings

${{\mathcal {H}}}_\mathrm{cubic}$ :

the special cubic Hodge free energy

${{\mathcal {F}}}_{\mathrm{GUE}}$ :

the GUE free energy

${{\mathcal {F}}}_\mathrm{even}$ :

the GUE free energy with even couplings

$\widetilde{{{\mathcal {F}}}}$ :

the modified GUE free energy with even couplings

${{\mathcal {H}}}_g$ :

the genus g part of the the special cubic Hodge free energy

${{\mathcal {F}}}_g$ :

the genus g part of the GUE free energy with even couplings

${{\widetilde{{{\mathcal {F}}}}}}_g$ :

the genus g part of the modified GUE free energy with even couplings

$H_g$ :

the genus g part of the special cubic Hodge free energy in jets, $g\ge 1$

$F_g$ :

the genus g part of the GUE free energy with even couplings in jets, $g\ge 1$

v :

genus zero Witten–Kontsevich solution to the KdV hierarchy

u :

genus zero GUE solution to the discrete KdV hierarchy