Matrix Resolvent and the Discrete KdV Hierarchy

Abstract

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are obtained, and applications to enumeration of ribbon graphs with even valencies and to certain special cubic Hodge integrals are considered.

On Consequence of the Hodge–GUE Correspondence

In this appendix, we derive a consequence of the Hodge–GUE correspondence that has a similar flavour to formula (121). Note that

$$\begin{aligned}&{\mathcal {H}}\bigl (\mathbf {t}(x,\mathbf {s}); \sqrt{2}\epsilon \bigr ) \\ {}&~ \; = \; \sum _{g\ge 0} 2^{g-1} \epsilon ^{2g-2} \sum _{k\ge 0} \frac{1}{k!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \prod _{m=1}^k \biggl (\sum _{i_m\ge 0} t_{i_m}(x,\mathbf {s}) \psi _m^{i_m}\biggr )\\ {}&~ \; = \; \sum _{g\ge 0} 2^{g-1} \epsilon ^{2g-2} \sum _{k\ge 0} \frac{1}{k!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \sum _{l=0}^k\left( {\begin{array}{c}k\\ l\end{array}}\right) \prod _{m=l+1}^k \biggl ((x -1) - \frac{\psi _m^2}{1-\psi _m} \biggr )\\ {}&\qquad \cdot \sum _{ p_1,\dots ,p_l} \prod _{m=1}^{l}\frac{p_m{\bar{s}}_{p_m}}{1- p_m\psi _m}. \end{aligned}$$

Then by comparing the coefficients of $s_{p_1} \dots s_{p_l}$ of the both sides of (118) we get

$$\begin{aligned}&\langle \sigma _{p_1}\dots \sigma _{p_l}\rangle _g(x) \nonumber \\ {}&\quad = \sum _{k\ge l} \frac{1}{(k-l)!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \prod _{m={l+1}}^k \biggl ((x -1) - \frac{\psi _m^2}{1-\psi _m} \biggr ) \prod _{m=1}^{l}\frac{p_m \left( {\begin{array}{c}2p_m\\ p_m\end{array}}\right) }{1- p_m\psi _{m}} \nonumber \\ {}&\quad \; + \delta _{g,0} \delta _{l,2} \, \frac{p_1 \, p_2}{p_1+p_2} \left( {\begin{array}{c}2p_1\\ p_1\end{array}}\right) \left( {\begin{array}{c}2p_2\\ p_2\end{array}}\right) \, + \, \frac{1}{2}\delta _{g,0} \delta _{l,1} \left( {\begin{array}{c}2p_1\\ p_2\end{array}}\right) \biggl ( \frac{p_1}{1+p_1} - x \biggr ) . \end{aligned}$$

(124)

Here $\langle \sigma _{p_1}\dots \sigma _{p_l}\rangle (x;\epsilon ) =: \sum _{g\ge 0} \epsilon ^{2g-2} \langle \sigma _{p_1}\dots \sigma _{p_l}\rangle _g(x)$, and $\langle \sigma _{p_1}\dots \sigma _{p_l}\rangle (x;\epsilon )$ are the modified GUE correlators with even couplings defined in (113). Taking $x=1$ we find

$$\begin{aligned} \langle \sigma _{p_1}\dots \sigma _{p_l}\rangle _g|_{x=1} \; = \; \sum _{k\ge l} \frac{1}{(k-l)!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \prod _{m={l+1}}^k \biggl (- \frac{\psi _m^2}{1-\psi _m} \biggr ) \prod _{m=1}^{l}\frac{p_m \left( {\begin{array}{c}2p_m\\ p_m\end{array}}\right) }{1- p_m\psi _{m}} . \end{aligned}$$

(125)

A further consideration to (125) was given in [5].

Combining (124) with (115) we find for any fixed $l\ge 1$, $p_1,\dots ,p_l\ge 1$ the following identities:

$$\begin{aligned}&l! \, x^{2-2g -l+|j|} \sum _{\begin{array}{c} g_1,r\ge 0 \\ g_1+r=g \end{array}} \left( {\begin{array}{c}2-2g_1-l+|j|\\ 2r\end{array}}\right) \, \frac{E_{2r}}{2^{2r}} \,a_{g_1}(2p_1, \dots , 2p_l) \nonumber \\&\quad \; = \; \sum _{k\ge l} \frac{1}{(k-l)!} \int _{\overline{{\mathcal {M}} }_{g,k}} \Omega _{g,k} \prod _{m={l+1}}^k \biggl ((x -1) - \frac{\psi _m^2}{1-\psi _m} \biggr ) \prod _{m=1}^{l}\frac{p_m \left( {\begin{array}{c}2p_m\\ p_m\end{array}}\right) }{1- p_m\psi _{m}} \nonumber \\&\qquad \, + \, \delta _{g,0} \delta _{l,2} \, \frac{p_1 \, p_2}{p_1+p_2} \left( {\begin{array}{c}2p_1\\ p_1\end{array}}\right) \left( {\begin{array}{c}2p_2\\ p_2\end{array}}\right) \, + \, \frac{1}{2}\delta _{g,0} \delta _{l,1} \left( {\begin{array}{c}2p_1\\ p_2\end{array}}\right) \biggl ( \frac{p_1}{1+p_1} - x \biggr ), \qquad g\ge 0. \end{aligned}$$

(126)

Note that for any $g\ge 0$, the RHS is a priori a power series of $x-1$, but the LHS shows that it is actually a monomial of x and so is also a polynomial of $x-1$. This subset of the identities deserve a further investigation. Moreover, the LHS vanishes when g is sufficiently large, an so is the RHS; this provides another subset of the identities for the cubic Hodge integrals.