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Generating series for GUE correlators

Abstract

We extend to the Toda lattice hierarchy the approach of Bertola et al. (Phys D Nonlinear Phenom 327:30–57, 2016; IMRN, 2016) to computation of logarithmic derivatives of tau-functions in terms of the so-called matrix resolvents of the corresponding difference Lax operator. As a particular application we obtain explicit generating series for connected GUE correlators. On this basis an efficient recursive procedure for computing the correlators in full genera is developed.

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Notes

  1. They are closely related to the multi-loop generating functions

    $$\begin{aligned} \left\langle \mathrm{tr}\frac{1}{\lambda _1-M} \cdots \mathrm{tr}\frac{1}{\lambda _k-M}\right\rangle _c=\sum _{i_1, \ldots , i_k=0}^\infty \frac{\left\langle \mathrm{tr}\, M^{i_1}\cdots \mathrm{tr}\, M^{i_k} \right\rangle _c}{\lambda _1^{i_1+1}\cdots \lambda _k^{i_k+1}} \end{aligned}$$

    often considered in the literature on random matrices, see, e.g., [11].

  2. For more details about the Toda lattice hierarchy (and about its extension), one can see, for example, ref. [10].

  3. In this calculation we omit the index n of \(\tau _n\) and \(R_n\).

  4. Under certain assumptions for the polynomial V(M) it can be rigorously justified [12, 20, 21] that the formal series considered below are asymptotic expansions at \(N\rightarrow \infty \) of convergent matrix integrals.

  5. The rational numbers \(a_g(i_1,\ldots ,i_k)\) have the following alternative expression

    $$\begin{aligned} a_g(i_1,\ldots ,i_k)=\prod _{j=1}^k i_j \, \sum _G \frac{1}{\#\, \mathrm{Sym} \, G} \end{aligned}$$
    (A.3.5)

    where the summation is taken over connected oriented ribbon graphs G of genus g with unlabelled half-edges and unlabelled vertices of valencies \(i_1\), ..., \(i_k\).

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Acknowledgements

The work is partially supported by the Russian Science Foundation Grant No. 16-11-10260 “Geometry and Mathematical Physics of Integrable Systems” and by PRIN 2010-11 Grant “Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions” of Italian Ministry of Universities and Researches. The authors thank the anonymous referees for several valuable constructive comments that helped to improve the presentation of the results of the paper.

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Correspondence to Boris Dubrovin.

A Appendix: GUE, Toda lattice, and enumeration of ribbon graphs

A Appendix: GUE, Toda lattice, and enumeration of ribbon graphs

A.1 GUE partition function and orthogonal polynomials

Consider the GUE partition function represented as an integral over the space \({\mathcal H}(N)\) of \(N\times N\) Hermitian matrices \(M=\left( M_{ij}\right) \)

$$\begin{aligned} Z_N(\mathbf{s}; \epsilon )=\frac{(2\pi )^{-{N}} \epsilon ^{-\frac{1}{12}}}{\mathrm{Vol}(N)} \int _{{\mathcal H}(N)} e^{-\frac{1}{\epsilon } \mathrm{tr}\,V(M)} \mathrm{d}M. \end{aligned}$$
(A.1.1)

Here the formal series V depending on the parameters \(\mathbf{s}=(s_1,s_2,s_3, \ldots )\) has the form

$$\begin{aligned} V(M)=\frac{1}{2} M^2 - \sum _{j\ge 1} s_j M^j. \end{aligned}$$
(A.1.2)

The integral with respect to the measure

$$\begin{aligned} \mathrm{d}M=\prod _{i=1}^N \mathrm{d}M_{ii} \prod _{i<j} \mathrm{d}\mathrm{Re}( M_{ij})\, \mathrm{d}\mathrm{Im}(M_{ij}) \end{aligned}$$

will be understood as a formal seriesFootnote 4 with respect to the parameters \(s_j\). The prefactor \(\mathrm{Vol}(N)^{-1}\) corresponds to the volume, with respect to the Haar measure, of the quotient of the unitary group over the maximal torus \(\left[ U(1)\right] ^N\)

$$\begin{aligned} \mathrm{Vol}(N)=\mathrm{Vol} \left( U(N)/\left[ U(1)\right] ^N\right) =\frac{\pi ^{\frac{N(N-1)}{2}} }{G(N+1)} \end{aligned}$$
(A.1.3)

Here G is the Barnes G-function taking the value

$$\begin{aligned} G(N+1)=\prod _{n=1}^{N-1} n! \end{aligned}$$
(A.1.4)

at positive integers. Formula (A.1.3) will be re-derived below.

Denote \({\mathcal D}_N\) the set of diagonal \(N\times N\) matrices \(\Lambda =\mathrm{diag}(\lambda _1, \lambda _2, \ldots , \lambda _N)\) with real ordered eigenvalues \(\lambda _1\le \lambda _2\le \cdots \le \lambda _N\). The map

$$\begin{aligned}&U(N)/\left[ U(1)\right] ^N\times {\mathcal D}_N\rightarrow {\mathcal H}(N) \nonumber \\&\left( U, \Lambda \right) \mapsto U\,\Lambda \, U^* \end{aligned}$$
(A.1.5)

is a local diffeomorphism away from a subset of codimension three in \({\mathcal H}(N)\). Because of invariance of the measure w.r.t. to the action of unitary group one obtains

$$\begin{aligned} \int _{{\mathcal H}(N)} e^{-\frac{1}{\epsilon } \mathrm{tr}\,V(M)} \mathrm{d}M=\mathrm{Vol}\left( U(N)/\left[ U(1)\right] ^N\right) \int _{{\mathcal D}_N}\!\varDelta ^2(\lambda ) \, e^{-\frac{1}{\epsilon } \sum _{k=1}^N V(\lambda _k)}\mathrm{d}\lambda _1\cdots \mathrm{d}\lambda _N. \end{aligned}$$

Here

$$\begin{aligned} \varDelta (\lambda )=\prod _{i<j} (\lambda _i-\lambda _j) \end{aligned}$$

is the Vandermonde determinant. Due to symmetry of the integrand one can rewrite the last formula as

$$\begin{aligned} \int _{{\mathcal H}(N)} e^{-\frac{1}{\epsilon } \mathrm{tr}\,V(M)} \mathrm{d}M=\frac{1}{N!} \mathrm{Vol} \left( U(N)/\left[ U(1)\right] ^N\right) \int _{\mathbb R^N} \varDelta ^2(\lambda ) \, e^{-\frac{1}{\epsilon } \sum _{k=1}^N V(\lambda _k)}\mathrm{d}\lambda _1\cdots \mathrm{d}\lambda _N. \end{aligned}$$

Denote

$$\begin{aligned} p_n(\lambda )=\lambda ^n+a_{1n} \lambda ^{n-1}+\cdots +a_{nn}, \quad n=0,1, 2, \ldots \end{aligned}$$
(A.1.6)

a system of monic polynomials orthogonal w.r.t. to the exponential weight

$$\begin{aligned} \int _{-\infty }^\infty p_n(\lambda ) \, p_m(\lambda ) \, e^{-\frac{1}{\epsilon } V(\lambda )}\mathrm{d}\lambda =: h_n \, \delta _{mn}. \end{aligned}$$
(A.1.7)

Representing the Vandermonde as

$$\begin{aligned} \varDelta (\lambda )=\det \left( \begin{array}{cccc} p_0(\lambda _1) &{} p_0(\lambda _2) &{} \cdots &{} p_0(\lambda _N)\\ p_1(\lambda _1) &{} p_1(\lambda _2) &{} \cdots &{} p_1(\lambda _N)\\ \cdot &{} \cdot &{} \cdots &{} \cdot \\ \cdot &{} \cdot &{} \cdots &{} \cdot \\ \cdot &{} \cdot &{} \cdots &{} \cdot \\ p_{N-1}(\lambda _1) &{} p_{N-1}(\lambda _2) &{} \cdots &{} p_{N-1}(\lambda _N)\end{array}\right) \end{aligned}$$

one obtains an expression of the last integral via the normalizing factors of the orthogonal polynomials

$$\begin{aligned} \int _{\mathbb R^N} \varDelta ^2(\lambda ) e^{-\frac{1}{\epsilon } \sum _{k=1}^N V(\lambda _k)}\mathrm{d}\lambda _1\cdots \mathrm{d}\lambda _N=N! \,h_0 \, h_1\cdots h_{N-1}. \end{aligned}$$

We conclude that

$$\begin{aligned} \int _{{\mathcal H}(N)} e^{-\frac{1}{\epsilon } \mathrm{tr}\,V(M)} \mathrm{d}M=\mathrm{Vol} \left( U(N)/\left[ U(1)\right] ^N\right) \, h_0 \, h_1\cdots h_{N-1}. \end{aligned}$$
(A.1.8)

Formula (A.1.3) for the volume \(\mathrm{Vol} \left( U(N)/\left[ U(1)\right] ^N\right) \) can be easily derived from the last equation. Indeed, evaluating the LHS of Eq. (A.1.8) at the Gaussian point \(\mathbf{s}=0\) one obtains

$$\begin{aligned} \int _{{\mathcal H}(N)} e^{-\frac{1}{2\epsilon } \mathrm{tr}\, M^2} \mathrm{d}M=2^{\frac{N}{2}} \left( \pi \,\epsilon \right) ^{\frac{N^2}{2}}. \end{aligned}$$

At \(\mathbf{s}=0\) the orthogonal polynomials (A.1.6)–(A.1.7) are expressed via Hermite polynomials

$$\begin{aligned} p_n(\lambda )=\epsilon ^{\frac{n}{2}}\, \mathrm{He}_n(x), \quad \lambda =\epsilon ^{\frac{1}{2}}x. \end{aligned}$$

From

$$\begin{aligned} \int _{-\infty }^\infty \mathrm{He}_n(x) \, \mathrm{He}_m(x) \, e^{-\frac{1}{2} x^2} \mathrm{d}x=\sqrt{2\pi } \,n! \,\delta _{mn} \end{aligned}$$

it follows that

$$\begin{aligned} h_n(\mathbf{s}=0)= \epsilon ^{n+\frac{1}{2}} \sqrt{2\pi } \,n! \end{aligned}$$

So, Eq. (A.1.8) at \(\mathbf{s}=0\) takes the form

$$\begin{aligned} 2^{\frac{N}{2}} \left( \pi \,\epsilon \right) ^{\frac{N^2}{2}}= \mathrm{Vol} \left( U(N)/\left[ U(1)\right] ^N\right) \cdot \left( 2\pi \right) ^{\frac{N}{2}} \epsilon ^{\frac{N^2}{2}} \prod _{n=1}^{N-1} n! \end{aligned}$$

This implies (A.1.3).

We conclude this section with the following expression for the GUE partition function

$$\begin{aligned} Z_N(\mathbf{s}; \epsilon ) = h_0 \, h_1 \cdots h_{N-1}. \end{aligned}$$
(A.1.9)

Our nearest goal is to prove that this partition function is the tau-function of a particular solution of Toda hierarchy.

A.2 GUE and Toda

Denote \(v_n\), \(w_n\) the coefficients of the three-term recursion relation for the orthogonal polynomials \(p_n(\lambda )\)

$$\begin{aligned} \lambda \, p_n(\lambda ) =p_{n+1}(\lambda ) +v_n \, p_n(\lambda ) + w_n \, p_{n-1}(\lambda ), \quad n\ge 0 \end{aligned}$$
(A.2.1)

\(p_{-1}=0\). That is, the orthogonal polynomials are eigenvectors of the second-order difference operator

$$\begin{aligned} \left( L\, \psi \right) _n = \psi _{n+1} +v_n \, \psi _n + w_n \, \psi _{n-1}. \end{aligned}$$
(A.2.2)

The corresponding tri-diagonal matrix will also be denoted \(L=\left( L_{ij}\right) \).

Denote

$$\begin{aligned} (f, g)=\int _{-\infty }^\infty f(\lambda ) \, g(\lambda ) \, e^{-\frac{1}{\epsilon } V(\lambda )}\mathrm{d}\lambda \end{aligned}$$
(A.2.3)

an inner product on the space of polynomials. Recall that all integrals are understood as formal series in the \(s_j\)-variables. The symmetry

$$\begin{aligned} \left( \lambda \, p_n, p_m\right) =\left( p_n, \lambda \, p_m\right) ~\Leftrightarrow ~ L_{mn} h_m=L_{nm} h_n \end{aligned}$$

implies

$$\begin{aligned} w_n=\frac{h_n}{h_{n-1}}=\frac{Z_{n+1} Z_{n-1}}{Z_{n}^2}. \end{aligned}$$
(A.2.4)

Here \(h_n=(p_n, p_n)\) [see Eq. (A.1.7)].

For an arbitrary square matrix \(X=(X_{ij})\) denote \(X_-\) and \(X_+\) its upper- and lower-triangular parts

$$\begin{aligned} X_-=\left( X_{ij}\right) _{i<j}, \quad X_+=\left( X_{ij}\right) _{i\ge j}, \quad X=X_+ + X_-. \end{aligned}$$

Lemma A.2.1

The orthogonal polynomials \(p_n=p_n(\lambda )\) satisfy

$$\begin{aligned} \epsilon \,\frac{\partial p_n}{\partial s_j} +(A_j \, p)_n=0, \quad A_j = -\left( L^{j} \right) _-, \quad j\ge 1. \end{aligned}$$
(A.2.5)

Proof

Write

$$\begin{aligned} \frac{\partial p_n(\lambda )}{\partial s_j} =\sum _{i=0}^{n-1} A_{i\, n}^{(j)} \, p_i(\lambda ), \quad n\ge 1 \end{aligned}$$

for some coefficients \(A_{in}^{(j)}\). Differentiating in \(s_j\) the equation \((p_n, p_m)=0\) for \(m<n\) we obtain

$$\begin{aligned} A_{mn}^{(j)}h_m +\frac{1}{\epsilon } \left( \lambda ^{j} p_n, p_m\right) =0. \end{aligned}$$

Introduce matrices of multiplication by powers of \(\lambda \)

$$\begin{aligned} \lambda ^{j}p_n(\lambda )=\sum _{i=0}^{n+j} \left( L^{j}\right) _{in} p_i(\lambda ). \end{aligned}$$
(A.2.6)

We have

$$\begin{aligned} \left( \lambda ^{j}p_n, p_m\right) =\left( L^{j}\right) _{mn} h_m, \end{aligned}$$

hence

$$\begin{aligned} \epsilon \, A_{mn}^{(j)} =- \left( L^{j}\right) _{mn}, \quad m<n \end{aligned}$$
(A.2.7)

that is,

$$\begin{aligned} \epsilon \, A^{(j)}=- \left( L^{j}\right) _-. \end{aligned}$$

\(\square \)

Repeating a similar calculation for \(m=n\) we obtain

$$\begin{aligned} \frac{\partial }{\partial s_j} \log \frac{Z_{n+1}}{Z_n} \equiv \frac{\partial }{\partial s_j}\log h_n = \left( L^{j} \right) _{nn}. \end{aligned}$$
(A.2.8)

Corollary A.2.2

The difference operator L satisfies

$$\begin{aligned} \epsilon \, \frac{\partial L}{\partial s_j} =\left[ A_j, L\right] , \quad A_j=\left( L^{j}\right) _+. \end{aligned}$$
(A.2.9)

Proof

Differentiating equation

$$\begin{aligned} \lambda \, p_n =\sum _{i=0}^{n+1} L_{in} \, p_i \end{aligned}$$

in \(s_j\) and using Eq. (A.2.5) we obtain

$$\begin{aligned} \epsilon \, \frac{\partial L}{\partial s_j} =\left[ L, \left( L^{j}\right) _-\right] . \end{aligned}$$

Since the operators L and \(L^{j}\) commute we arrive at (A.2.9). \(\square \)

Proposition A.2.3

The GUE partition function \(Z_n\) is a tau-function, in the sense of Definition 1.2.4 of the Toda lattice hierarchy, where the time variables are defined by \(t_j=s_{j+1}/\epsilon , ~j=0,1,2,\ldots \).

Proof

Cor. A.2.2 tells that \(w_n,v_n\) is a particular solution to the Toda lattice hierarchy. It then follows from (A.2.4) and (A.2.8) that \(Z_n\) satisfies Eqs. (1.2.9) and (1.2.10). Equation (A.2.8) implies that

$$\begin{aligned} \frac{\partial }{\partial t_{j}} \log \frac{Z_{n+1}}{Z_n} = (j+1)\, h_{j-1}(n), \quad j\ge 0 \end{aligned}$$

where \(h_{j-1}(n):=\frac{1}{j+1} (L^{j+1})_{nn}\). Define

$$\begin{aligned} \gamma _n(\lambda )= \frac{1}{\lambda } + \sum _{j\ge 0} \frac{ (j+1)\, h_{j-1}(n-1)}{\lambda ^{j+2}}. \end{aligned}$$

We have

$$\begin{aligned} \sum _{j\ge 0}\frac{1}{\lambda ^{j+2}} \frac{\partial }{\partial t_{j}} \log \frac{Z_{n+1}}{Z_n} = \sum _{j\ge 0}\frac{1}{\lambda ^{j+2}} (j+1)\, h_{j-1}(n) = \gamma _{n+1}(\lambda )-\frac{1}{\lambda }. \end{aligned}$$

So

$$\begin{aligned} \sum _{i,j\ge 0}\frac{1}{\mu ^{i+2}\lambda ^{j+2}} \frac{\partial ^2}{\partial t_{j}\partial t_i} \log \frac{Z_{n+1}}{Z_n} = \nabla (\mu )\, \gamma _{n+1}(\lambda ). \end{aligned}$$

Here, \(\nabla (\mu )\) is defined in (2.2.9). Noting that

$$\begin{aligned} \nabla (\mu )\, \gamma _{n+1}(\lambda )= & {} \frac{\gamma _{n+1}(\lambda )(1+2\alpha _{n+1}(\mu ))-\gamma _{n+1}(\mu )(1+2\alpha _{n+1}(\lambda ))}{\lambda -\mu }\\&+\gamma _{n+1}(\lambda )\gamma _{n+1}(\mu )\\= & {} \frac{\mathrm{tr}\, R_{n+1}(\lambda ) R_{n+1}(\mu ) - \mathrm{tr}\, R_n(\lambda ) R_n(\mu )}{(\lambda -\mu )^2} \end{aligned}$$

we obtain

$$\begin{aligned} \sum _{i,j\ge 0}\frac{1}{\mu ^{i+2}\lambda ^{j+2}} \frac{\partial ^2}{\partial t_{j}\partial t_i} \log Z_n = \frac{\mathrm{tr}\, R_n(\lambda ) R_n(\mu ) -1}{(\lambda -\mu )^2} . \end{aligned}$$

In the above formulae, \(R_n\) is the matrix resolvent of L. The proposition is proved. \(\square \)

A.3 GUE and enumeration of ribbon graphs

Expanding in powers of \(s_1\), \(s_2\), \(s_3\), ...

$$\begin{aligned} \int _{{\mathcal H}(N)} e^{-\frac{1}{\epsilon } \mathrm{tr}\,V(M)} \mathrm{d}M=\sum _{k\ge 0}\frac{\epsilon ^{-k}}{k!}\int _{{\mathcal H}(N)} e^{-\frac{1}{2\epsilon } \mathrm{tr}\, M^2}\sum _{i_1, \ldots , i_k} s_{i_1}\cdots s_{i_k} \mathrm{tr}\, M^{i_1} \cdots \mathrm{tr}\, M^{i_k} \,\mathrm{d}M \end{aligned}$$

and using an obvious formula

$$\begin{aligned} \frac{\int _{{\mathcal H}(N)} e^{-\frac{1}{2\epsilon } \mathrm{tr}\, M^2} \mathrm{tr}\, M^{i_1} \cdots \mathrm{tr}\, M^{i_k} \,\mathrm{d}M}{\int _{{\mathcal H}(N)} e^{-\frac{1}{2\epsilon } \mathrm{tr}\,M^2} \mathrm{d}M}=\epsilon ^{\frac{i_1+\cdots +i_k}{2}}\frac{\int _{{\mathcal H}(N)} e^{-\frac{1}{2} \mathrm{tr}\, M^2} \mathrm{tr}\, M^{i_1} \cdots \mathrm{tr}\, M^{i_k} \,\mathrm{d}M}{\int _{{\mathcal H}(N)} e^{-\frac{1}{2} \mathrm{tr}\,M^2} \mathrm{d}M} \end{aligned}$$

(note that both sides of this equation vanish if \(i_1+\cdots +i_k=\,\)odd) one obtains

$$\begin{aligned} \frac{Z_N(\mathbf{s}; \epsilon )}{Z_N(0; \epsilon )}= \sum _{m\in \mathbb Z} \epsilon ^m \sum _{ k\ge 0} \frac{1}{k!} \sum _{ i_1+\cdots +i_k= 2k+2 m} s_{i_1}\cdots s_{i_k} \left\langle \mathrm{tr}\, M^{i_1}\cdots \mathrm{tr}\, M^{i_k} \right\rangle \qquad \end{aligned}$$
(A.3.1)

where, as above

$$\begin{aligned} \left\langle \mathrm{tr}\, M^{i_1}\cdots \mathrm{tr}\, M^{i_k} \right\rangle :=\frac{\int \mathrm{tr}\, M^{i_1}\cdots \mathrm{tr}\, M^{i_k}\,e^{-\frac{1}{2} \mathrm{tr}\, M^2} \, \mathrm{d}M}{\int e^{-\frac{1}{2} \mathrm{tr}\, M^2}\mathrm{d}M}. \end{aligned}$$
(A.3.2)

The coefficients (A.3.2) of the perturbative expansion (A.3.1) are polynomials in N that can be computed by applying the Wick rule. For example,

$$\begin{aligned} \left\langle \mathrm{tr}\, M^4\right\rangle =2N^3+N, \quad \left\langle \left( \mathrm{tr}\, M^3\right) ^2\right\rangle =12 N^3 + 3 N, \quad \left\langle \mathrm{tr}\, M^6\right\rangle =5N^4 + 10 N^2 \end{aligned}$$

etc. Terms of polynomial (A.3.2) correspond to oriented ribbon graphs with k vertices. Expansion of the logarithm of the partition function has a similar structure keeping connected graphs only

$$\begin{aligned} \log \frac{Z_N(\mathbf{s}; \epsilon )}{Z_N(0; \epsilon )}= \sum _{m} \epsilon ^m \sum _{ k\ge 0} \frac{1}{k!} \, \sum _{ i_1+\cdots +i_k=2k+2 m} s_{i_1}\cdots s_{i_k} \left\langle \mathrm{tr}\, M^{i_1}\cdots \mathrm{tr}\, M^{i_k} \right\rangle _c.\nonumber \\ \end{aligned}$$
(A.3.3)

Introduce the ’t Hooft coupling parameter

$$\begin{aligned} x=N\, \epsilon . \end{aligned}$$

Re-expanding in \(\epsilon \) the logarithm of the partition function we arrive at the main statement of this section, see [7].

Theorem A.3.1

Logarithm of the tau-function of the solution to the Toda hierarchy given by the GUE partition function has the following expansion

(A.3.4)

where \(h=2-2g -\left( k-\frac{|i|}{2}\right) , |i|=i_1+\cdots + i_k\), and the last summation is taken over all connected ribbon graphs \(\Gamma \) (with labelled half-edges and unlabelled vertices) of genus g with k vertices of valencies \(i_1\), ..., \(i_k\), and \(\#\,\mathrm{Sym}\, \Gamma \) is the order of the symmetry group of \(\Gamma \) generated by permuting the vertices.Footnote 5

In the particular case \(i_1=i_2=\cdots =i_k=3\) the dual to the ribbon graph is a triangulation of the surface of genus g consisting of k triangles. Thus, \(a_g\!\left( 3^k\right) :=a_g(3, \ldots , 3)\) (k times) is equal to the weighted number of triangulations of genus g with k triangles. In a similar way, \(a_g\!\left( 4^k\right) \) is the weighted number of quadrangulations of a surface of genus g with k squares, etc.

Since

$$\begin{aligned} Z_N(0; \epsilon )=(2\pi )^{-\frac{N}{2}} \epsilon ^{\frac{N^2}{2}-\frac{1}{12}}\, G(N+1) \end{aligned}$$

we can also expand the first term in (A.3.4) with the help of the asymptotic expansion of the Barnes G-function (cf. [2, 23, 37])

This yields the following genus expansion of the logarithm of the tau-function of the interpolated Toda hierarchy where the shift operator \(\psi _n\mapsto \psi _{n+1}\) acting on functions on a lattice is replaced with the translation \(\psi (x) \mapsto \psi (x+\epsilon )\) acting on smooth functions on the real line,

$$\begin{aligned} \log \tau (x; \mathbf{s};\epsilon )&= \frac{x^2}{2\epsilon ^2} \left( \log x -\frac{3}{2}\right) -\frac{1}{12} \log x +\zeta '(-1)\nonumber \\&\quad +\sum _{g\ge 2} \epsilon ^{2g-2} \frac{B_{2g}}{4g(g-1)x^{2g-2}} \nonumber \\&\quad +\sum _{g\ge 0} \epsilon ^{2g-2} \sum _{k\ge 0} \sum _{ i_1, \ldots , i_k} a_g(i_1, \ldots , i_k) \, s_{i_1} \cdots s_{i_k} x^{2-2g -\left( k-\frac{|i|}{2}\right) }. \end{aligned}$$
(A.3.6)

Remark A.3.2

The genus expansion (A.3.6) is often written as 1 / N expansion, setting \(x=1\), so \(\epsilon =1/N\),

$$\begin{aligned} \log Z_N(\mathbf{s}; N^{-1})&= \sum _{g\ge 0} N^{2-2g} \mathcal{F}_g (\mathbf{s}) \\ \mathcal{F}_g(\mathbf{s})&= - \delta _{g,0} \, \frac{3}{4} + \delta _{g,1} \, \zeta '(-1) +\delta _{g\ge 2} \, \frac{B_{2g}}{4g(g-1)}\\&\quad + \sum _{k\ge 0} \sum _{i_1, \ldots , i_k} a_g(i_1, \ldots , i_k) \, s_{i_1} \cdots s_{i_k}. \end{aligned}$$

The coefficients \(a_g(i_1, \ldots , i_k)\) are the same as in (A.3.4).

Observe that the coefficients of the connected correlators as polynomials in N can be expressed via the numbers \(a_g(i_1, \ldots , i_k)\) enumerating ribbon graphs. Namely, \(\forall \,k\ge 1\) and \(\forall \, i_1,\ldots ,i_k\) such that |i| is even, we have

$$\begin{aligned} \langle \mathrm{tr}\, M^{i_1} \, \mathrm{tr}\, M^{i_2} \, \cdots \, \mathrm{tr}\, M^{i_k} \rangle _c\,=\,k!\,\sum _{0\le g \le \frac{|i|}{4}-\frac{k}{2}+\frac{1}{2}} \, a_g(i_1,\ldots ,i_k) \, N^{2-2g-k+\frac{|i|}{2}}.\nonumber \\ \end{aligned}$$
(A.3.7)

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Dubrovin, B., Yang, D. Generating series for GUE correlators. Lett Math Phys 107, 1971–2012 (2017). https://doi.org/10.1007/s11005-017-0975-6

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Keywords

  • Toda lattice
  • Matrix resolvent
  • Tau-function
  • GUE correlator
  • Enumeration of ribbon graphs

Mathematics Subject Classification

  • Primary 37K10
  • Secondary 15A52