1 Introduction and statements of the main results↩︎

Enumerating ribbon graphs, also known as (combinatorial) maps, is a fundamental problem at the interface of mathematics and mathematical physics, attracting interest from combinatorics (cf. [1]–[5]), and revealing deep relations to quantum field theory [1], [6]–[10] and geometry [4], [11]–[15]. Let \(\mathcal{R}^{\rm conn}_{g}(b_1,\dots,b_k)\) be the set of connected oriented labelled¹ ribbon graphs of genus \(g\) with \(k\) vertices of valencies \(b_1,\dots,b_k\), and let \(n_{g}(b_1,\dots,b_k)\mathrel{\vcenter{:}}=|\mathcal{R}^{\rm conn}_{g}(b_1,\dots,b_k)|\). Here, \(g\geq 0\) and \(b_1,\dots,b_k\geq 1\). By the Euler formula, the number \(n_g(b_1,\dots,b_k)\) vanishes unless \(2-2g-k+\frac{|b|}{2}\) is a positive integer. By looking at the dual graphs, one can also understand \(n_g(b_1,\dots,b_k)\) as the number of polygon-angulations with \(k\) polygons of sizes \(b_1,\dots,b_k\) on a genus \(g\) closed oriented surface.

Following [13], [16], define a power series of infinitely many variables \({\boldsymbol{s}}=(s_1,s_2,\dots)\) by \[\begin{align} \mathcal{F}(x,{\boldsymbol{s}};\epsilon) = & \, \frac{x^2}{2\epsilon^2}\Bigl(\log x-\frac{3}{2}\Bigr) - \frac{\log x}{12} + \zeta'(-1) + \sum_{g\geq2} \frac{\epsilon^{2g-2} B_{2g}}{4g(g-1)x^{2g-2}}\nonumber\\ &+\sum_{g\geq 0}\epsilon^{2g-2}\sum_{k\geq 1}\frac{1}{k!}\sum_{b_1,\dots,b_k\ge1}n_{g}(b_1,\dots,b_k) \, s_{b_1} \cdots s_{b_k} x^{2-2g - k + \frac{|{\boldsymbol{b}}|}{2}}\,, \label{Fgue1x39} \end{align}\tag{1}\] called the free energy, and define \(Z(x,{\boldsymbol{s}};\epsilon)\mathrel{\vcenter{:}}= e^{\mathcal{F}(x,{\boldsymbol{s}};\epsilon)}\), called the partition function. Here, \(x\) is a formal variable, \(\zeta(s)\) denotes the Riemann zeta function, and \(B_m\) denotes the \(m\)th Bernoulli number. It is known from e.g. [1], [4], [15]–[17] that \(Z(x,{\boldsymbol{s}};\epsilon)\) can be understood as the following integral \[\label{matrixintegral} 2^{-n}\pi^{-\frac{n(n+1)}{2}} \epsilon^{-\frac{1}{12}} G(n+1) \int_{{\mathcal{H}}(n)} \exp\Bigl(-\frac{1}{\epsilon}{\rm tr}\Bigl(\frac{1}{2} M^2-\sum_{b\geq 1}s_b M^b \Bigr)\Bigr) dM\,,\tag{2}\] where \(x=n\epsilon\), \(G\) denotes Barnes’s \(G\)-function, and \[dM = \prod_{1\leq i\leq n} d M_{ii} \prod_{1\leq i<j\leq n} d{\rm Re} M_{ij}\, d{\rm Im}M_{ij}\,.\]

Define \(\mathcal{F}_g(x,{\boldsymbol{s}})\mathrel{\vcenter{:}}=[\epsilon^{2g-2}]\mathcal{F}(x,{\boldsymbol{s}};\epsilon)\), \(g\geq 0\), the genus \(g\) free energy. Structures for \(\mathcal{F}_g(x,{\boldsymbol{s}})\) have been studied in [13]–[16], [18]–[20]. By definition, \(\mathcal{F}_g(x,{\boldsymbol{s}})\) encodes the enumeration of arbitrary tilings on a genus \(g\) surface (throughout this paper, all surfaces are assumed to be closed and oriented). The special case of enumeration of \(b\)-angulations, with a fixed value of \(b\), has attracted a lot of interest [16], [21]–[24] and will be the main focus of this paper. This corresponds to restricting \(\mathcal{F}(x,{\boldsymbol{s}};\epsilon)\) to \({\boldsymbol{s}}=(0,\dots,0,s_b=s,0,\dots)\), and by rescaling we can assume \(s=1\), so we will consider \[\begin{align} \label{Fgexplicit} \mathcal{F}_g^{\{b\}}(x)\mathrel{\vcenter{:}}=&\,\mathcal{F}_g(x,{\boldsymbol{s}}={\boldsymbol{1}}_b) = \delta_{g,0}\frac{x^2}{2}\Bigl(\log x-\frac{3}{2}\Bigr) +\delta_{g,1}\Bigl(-\frac{\log x}{12} + \zeta'(-1)\Bigr)\nonumber\\ &\,+\, \delta_{g\geq 2}\frac{B_{2g}}{4g(g-1)x^{2g-2}} +\sum_{k\geq 1}\frac{n_{g}(b^k)}{k!}x^{2-2g+(\frac{b}{2}-1)k}\,, \end{align}\tag{3}\] where \({\boldsymbol{1}}_b\) denotes the infinite vector with the \(b\)th component equal to 1 and all other components equal to 0.

According to e.g. [15]–[17], [25], \(Z(x,{\boldsymbol{s}};\epsilon)\) is a tau-function of the Toda lattice hierarchy. The latter will be one of the main tools for this paper. Following [13], [15], [16], [26], define \[u=\frac{\partial^2 \mathcal{F}^{\{b\}}_0(x)}{\partial x^2}\,,\quad w=e^{u}\,.\]

The case \(b=3\). Following [16], [27], introduce a power series \(v=v(x)=6 x+324 x^2+31104 x^3+\cdots\) as the unique solution to the following cubic equation \[\label{triveqn} 6 \, x = (1-9v+18v^2) \, v\,.\tag{4}\] According to [16] we have the identity \[\label{triweqn} w = \frac{x}{1-6v}\,.\tag{5}\] (We will give a new proof of 5 in Section 3.) It follows from 4 , 5 that \(w=x+36x^2+3240x^3+\cdots\) is the unique solution to the cubic equation \[\label{triweqn2} x^2=w^2-72w^3\,.\tag{6}\] Equation 6 was also obtained in [27]. Building on, for example,[16], [17], [25] (see also [27]) and using a method from [28], we will prove in Section 3 the following theorem.

For \(g\ge2\), the coefficient \(a_{g,5g-5}\) has the expression \[\label{tritopcoef} a_{g,5g-5}=\frac{162^g}{3888}\frac{C_g}{(5g-3)(5g-5)},\tag{9}\] which can be straightforwardly deduced using 8 and the well-known result (see e.g. [27]) \[\label{tricorrasymp} n_g(3^{2j})\sim \frac{16}{\sqrt{3}} \, \frac{\bigl(108\sqrt{3}\bigr)^j}{\bigl(256\sqrt{3}\bigr)^{g}} \frac{(2j)!\,(2j)^{\frac{5g-7}{2}}}{\Gamma({\frac{5g-1}{2}})}\,C_g \quad (j\to \infty) \,.\tag{10}\] Here \(C_g\), with \(C_0=-1, \ldots\), are constants determined (cf. e.g. [3], [27]–[32]) by requiring that the formal series \(U=\sum_{g\geq 0}C_g X^{\frac{1-5g}{2}}\) satisfies the Painlevé I equation \[\label{painleveI} \frac{d ^2 U}{d X^2}+\frac{1}{16}U^2-\frac{1}{16}X=0\,.\tag{11}\]

Table 1 consists of \(n_g(3^{4g-4+2d})\) for \(g=0,\dots,4\) and \(d=1,\dots,5\). For the reader’s convenience, we also provide \[\begin{align} \mathcal{F}^{\{3\}}_2&=\frac{\partial_x^2 (\log w)}{240}-\frac{351}{8 (1-108 w)^3}+\frac{27}{8 (1-108w)^4}+\frac{189}{10 (1-108 w)^5}\,, \tag{12}\\ \mathcal{F}^{\{3\}}_3&=-\frac{\partial_x^4 (\log w)}{6048}+\frac{589761}{4 (1-108 w)^6}-\frac{8203437}{28 (1-108 w)^7} -\frac{448335}{2 (1-108 w)^8}+\frac{324405}{(1-108 w)^9}\nonumber\\ & \quad\quad +\frac{178605}{(1-108 w)^{10}}\,.\tag{13} \end{align}\]

The case \(b=4\). According to [1], [20], [23], \(w=x+12x^2+288x^3+\cdots\) is the unique solution to \[\label{quadwgenus0eqn} x =w - 12 \, w^2\,.\tag{14}\] Explicitly, \(w=\frac{1-\sqrt{1-48x}}{24}\).

The proof, which is based on [16], [17], [25] and uses a method from [28], is given in Section 4.

It is easy to deduce using 15 and a result of [22] that the number \(\tilde{a}_{g,5g-5}\) equals to \(\frac{48^g}{576}\frac{C_g}{(5g-3)(5g-5)}\) (this statement can also be proved using just the Toda lattice theory), where \(C_g\) are the universal constants introduced in 10 .

The case \(b=2\nu\). In this case, the following theorem (using our notations) was originally conjectured in [23] and confirmed recently in the third arXiv version of [23].

([23]) For \(\nu\geq 2\), define \(q=q(x)\) as the power-series-in-\(x\) solution to the equation \[\label{qeqn} 1-q+\frac{(2\nu)!}{\nu!(\nu-1)!} \; x^{\nu-1} q^{\nu}=0\,.\qquad{(1)}\] Then for \(g\geq 2\), \[\label{evenFform4} x^{2g-2}\mathcal{F}_g^{\{2\nu\}}(x) =\sum_{\ell=2g-2}^{5g-5}\frac{r_{g,\ell}(\nu)}{(\nu-(\nu-1)q)^{\ell}}\,,\qquad{(2)}\] where for each \(\nu\geq 2\), \(r_{g,2g-2}(\nu),\dots,r_{g,5g-5}(\nu)\) are rational numbers.

We mention here that for the case when \(b=4\) one can use \(g\) parameters to express the genus \(g\) free energy with \(g\ge2\) (see 15 ) instead of \(3g-2\) parameters (see ?? ). For the case when \(b=6\) we have similar observations in which \(2g-1\) parameters are sufficient, whose proof will be given elsewhere.

We will also give several proofs of Theorem A, which were completed before noticing the updated version of [23].

In Section 4, a proof of Theorem A will be achieved based on Theorem 2. By using a formula of [22], we will give a second proof, which also corresponds to the proof in [23].

In [14], [20] a relationship between \(n_g(2\nu_1,\dots,2\nu_k)\) and certain cubic Hodge integrals on the Deligne–Mumford moduli spaces, called the Hodge–GUE correspondence, was established. As an application of the Hodge–GUE correspondence, we will give a third proof of Theorem A.

Motivated by a study of Gharakhloo–Latimer [24], we give in the following theorem a fine structure for \(\mathcal{F}_g^{\{2\nu\}}(x)\) as another application of the Hodge–GUE correspondence.

The proof is in Section 5.

Following [24], define \(S_{g,k}(\nu)\) by \[\label{evenexplicit} n_{g}((2\nu)^k) =:\biggl(\frac{(2\nu)!}{(\nu+1)!\nu!}\biggr)^k S_{g,k}(\nu)\,.\tag{16}\] By ?? , ?? we have \[\begin{align} \label{Sgjdef} S_{g,k}(\nu)=k!(\nu(\nu+1))^k \sum_{m=0}^{k} \binom{\nu k-1}{k-m}\frac{(\nu-1)^m}{m!} \sum_{\ell=2g-2}^{5g-5}r_{g,\ell}(\nu)(\ell-1)_m\,. \end{align}\tag{17}\] Theorem 3 then implies a conjectural statement given by Gharakhloo–Latimer in [24]: for \(g\geq 0\) and \(k\geq 1\), \(S_{g,k}(\nu)\) is a polynomial in \(\nu\) of degree \(3g-3+3k\). A further conjectural statement on distributions of zeros of \(S_{g,k}(\nu)\) was also proposed in [24]. We recall that, when \(g=0\), the expressions of \(n_0((2\nu)^k)\), \(\nu\geq 2\), were given in [33] (the \(\nu=2\) case also given in [34]); when \(g=1\), the expressions of \(n_1((2\nu)^k)\) were obtained in [34] for \(\nu=2\) and in [35] for \(\nu\ge2\); when \(g=2\), the expressions of \(n_2((2\nu)^k)\), \(\nu\geq 2\), \(k=1,2,3\), were given in [24]; the expressions of \(n_{g}(4^k)\), \(g=2,\dots,7\), \(k\geq 1\), were given in [35], and the expressions of \(n_{g}(6^k)\), \(g=2,\dots,5\), \(k\geq 1\), were given in [24]. We also note that, the above-proved polynomiality of \(S_{g,k}(\nu)\) can alternatively be deduced from the celebrated quasi-polynomiality of [36], [37] (cf. also [38], [39]), which is deeply related to topological recursion.

In Section 2 we review earlier works on the free energy \(\mathcal{F}(x,{\boldsymbol{s}};\epsilon)\). In Section 3 we prove Theorem 1. In Section 4 we prove Theorem 2. In Section 5 we prove Theorem 3.

We thank Paul Norbury for pointing out the reference [19]. The work of E.G-F. is supported by the Ramón y Cajal Fellowship RYC2023-045188-I, funded by MCIN/AEI/10.13039/501100011033 and by the FSE+. She also acknowledges support by the project PID2024-155686NB-I00 of the Spanish Ministry of Science and Innovation, the ANR CarteEtPlus ANR-23-CE48-0018, a Tremplin grant from Sorbonne Université, a PEPS grant from the CNRS and the ERC Synergy Grant ReNewQuantum. The work of D.Y. and J.X. is supported by NSFC 12371254 and CAS YSBR-032.

2 Review of the GUE free energy \(\mathcal{F}(x,{\boldsymbol{s}};\epsilon)\)↩︎

In this section we review several properties of the GUE free energy \(\mathcal{F}(x,{\boldsymbol{s}};\epsilon)\).

It is known (cf. e.g. [4], [25], [40]–[42]) that the partition function \(Z=Z(x,{\boldsymbol{s}};\epsilon)\) satisfies the following Virasoro constraints: \[\label{virasoro} L_{k}(Z(x,{\boldsymbol{s}};\epsilon))=0\,,\quad k\geq -1\,,\tag{18}\] where \(L_k\) are linear operators given by \[\begin{align} &L_{-1}=\sum_{j\geq 2}j s_j \frac{\partial}{\partial s_{j-1}}-\frac{\partial}{\partial s_1}+\frac{x s_1}{\epsilon^2}\,,\tag{19}\\ &L_0=\sum_{j\geq 1}j s_j \frac{\partial}{\partial s_j}-\frac{\partial}{\partial s_2}+\frac{x^2}{\epsilon^2}\,,\tag{20}\\ &L_k=\epsilon^2\sum_{j=1}^{k-1}\frac{\partial^2 }{\partial s_j \partial s_{k-j}}+2x\frac{\partial}{\partial s_k}+\sum_{j\geq 1}j s_j\frac{\partial}{\partial s_{j+k}}-\frac{\partial}{\partial s_{k+2}}\,,\quad k\geq 1\,, \end{align}\] which satisfy the Virasoro commutation relations: \[[L_k,L_\ell]=(k-\ell)L_{k+\ell}\,,\quad k,\ell\geq -1\,.\] The \(k=-1\) equation in 18 is also known as the string equation. It also follows from 2 that \(Z\) satisfies the following dilaton equation: \[\sum_{j\geq1} s_j \, \frac{\partial Z}{\partial s_j} + \epsilon\, \frac{\partial Z}{\partial\epsilon} + x \, \frac{\partial Z}{\partial x} + \frac{1}{12} \, Z = \frac{1}{2} \, \frac{\partial Z}{\partial s_2}\,. \label{dilaton}\tag{21}\]

Denote by \(\Lambda=e^{\epsilon\partial_x}\) the shift operator. It is known (cf. [16], [17], [25], [41]) that \((V^{\rm GUE}=V^{\rm GUE}(x,{\boldsymbol{s}};\epsilon),W^{\rm GUE}=W^{\rm GUE}(x,{\boldsymbol{s}};\epsilon))\) defined by \[\begin{align} \label{defVWintro} V^{\rm GUE}(x,{\boldsymbol{s}};\epsilon) &= \epsilon(\Lambda-1) \frac{\partial\mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_1}\,,\\ W^{\rm GUE}(x,{\boldsymbol{s}};\epsilon) &=\epsilon^2\frac{\partial^2 \mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_1 \partial s_1} \end{align}\tag{22}\] is a solution to the Toda lattice hierarchy [43], [44]: \[\epsilon\frac{\partial L}{\partial s_j}=[A_j, L]\,,\quad j\geq 1\,,\] where \[L=\Lambda+V+W \Lambda^{-1}\,,\quad A_j\mathrel{\vcenter{:}}=(L^j)_{+}\,.\] Here and below, for a difference operator \(P=\sum_{k\in\mathbb{Z}} P_k \Lambda^k\), \(P_{+}\mathrel{\vcenter{:}}=\sum_{k\geq 0} P_k \Lambda^k\) and \({\rm res} \, P\mathrel{\vcenter{:}}= P_0\). Denote \[V^{\{b\}}(x,\epsilon)\mathrel{\vcenter{:}}= V^{\rm GUE}(x,{\boldsymbol{1}}_b;\epsilon)\,, \quad W^{\{b\}}(x,\epsilon)\mathrel{\vcenter{:}}= W^{\rm GUE}(x,{\boldsymbol{1}}_b;\epsilon)\,.\] By definition, we know that \(W^{\{b\}}(x,\epsilon)\) has the following genus expansion: \[\label{topoWbexpand} W^{\{b\}}(x,\epsilon) = \sum_{g\geq0} \epsilon^{2g} W^{\{b\}}_{g}(x)\,.\tag{23}\]

Dividing the \(k=-1\) equation in 18 by \(Z(x,{\boldsymbol{s}};\epsilon)\), we have \[\label{string2} \sum_{j\geq 2}j s_j \;\frac{\partial\mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_{j-1}}+\frac{x s_1}{\epsilon^2} =\frac{\partial\mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_1}\,.\tag{24}\] Applying \(\epsilon(\Lambda-1)\) on both sides of 24 yields \[\label{string21} \epsilon\sum_{j\geq2}j s_j (\Lambda-1)\frac{\partial\mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_{j-1}}+s_1=\epsilon(\Lambda-1)\frac{\partial\mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_1}\,.\tag{25}\] Following [16], define \(h_j\mathrel{\vcenter{:}}=\frac{1}{j+2}{\rm res}L^{j+2}\in\mathbb{Q}[V,W,\Lambda^{\pm 1}V,\Lambda^{\pm 1}W,\dots]\). In particular, \(h_{-1}=V\). From [16] we know \[\epsilon(\Lambda-1)\frac{\partial\mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_j}=j h_{j-2}\bigl|\bigr._{\Lambda^i V=\Lambda^i V^{\rm GUE},\,\Lambda^i W=\Lambda^i W^{\rm GUE},\, i\in\mathbb{Z}} \, ,\] then 25 becomes \[\label{Ptype1} \sum_{j\geq 2}j(j-1) s_j \; h_{j-3}\bigl|\bigr._{\Lambda^i V=\Lambda^i V^{\rm GUE},\,\Lambda^i W=\Lambda^i W^{\rm GUE},\, i\in\mathbb{Z}} +s_1 =V^{\rm GUE}\,.\tag{26}\] Applying \(\epsilon^2\frac{\partial}{\partial{s_1}}\) on both sides of 24 , we get \[\label{string22} \epsilon^2\sum_{j\geq2}j s_j \frac{\partial^2 \mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_1\partial s_{j-1}}+x=\epsilon^2\frac{\partial^2 \mathcal{F}(x,{\boldsymbol{s}};\epsilon)}{\partial s_1^2}\,.\tag{27}\] Then according to [16] and [45], we obtain \[\label{Ptype2} \sum_{j\geq2}j s_{j}(\Lambda+1)^{-1}(j h_{j-2}-(j-1)V h_{j-3}) |_{\Lambda^i V=\Lambda^i V^{\rm GUE},\,\Lambda^i W=\Lambda^i W^{\rm GUE},\, i\in\mathbb{Z}}+x =W^{\rm GUE}\,.\tag{28}\] We note that special cases of 26 and 28 were given in e.g. [7], [27], [34], [46].

Define \({\boldsymbol{v}}^{\rm GUE}(x,{\boldsymbol{s}})=(v^{\rm GUE}(x,{\boldsymbol{s}}),u^{\rm GUE}(x,{\boldsymbol{s}}))\mathrel{\vcenter{:}}=(V^{\rm GUE}(x,{\boldsymbol{s}};0), \log W^{\rm GUE}(x,{\boldsymbol{s}};0))\). Based on the Dubrovin–Zhang theory, the following formulas were obtained in [13] (cf. also [3], [15], [18], [47]):

\[\begin{align} \mathcal{F}_0(x,{\boldsymbol{s}}) = & \frac{1}{2}\sum_{p,q\geq0} (p+1)! (q+1)! \bigl(s_{p+1}-\frac{1}{2}\delta_{p,1}\bigr) \bigl(s_{q+1}-\frac{1}{2}\delta_{q,1}\bigr) \Omega^{{\mathbb{P}^1},[0]}_{2,p;2,q}(v^{\rm GUE}(x,{\boldsymbol{s}}),u^{\rm GUE}(x,{\boldsymbol{s}})) \nonumber\\ & + x\sum_{p\geq0} (p+1)! \bigl(s_{p+1}-\frac{1}{2}\delta_{p,1}\bigr) \theta^{\mathbb{P}^1}_{2,p}(v^{\rm GUE}(x,{\boldsymbol{s}}),u^{\rm GUE}(x,{\boldsymbol{s}})) + \frac{1}{2} x^2 u^{\rm GUE}(x,{\boldsymbol{s}})\,, \tag{29}\\ \mathcal{F}_g(x,{\boldsymbol{s}}) = &F_g^{\mathbb{P}^1}\biggl(v^{\rm GUE}(x,{\boldsymbol{s}})\,, u^{\rm GUE}(x,{\boldsymbol{s}}), \dots,\frac{\partial^{3g-2} v^{\rm GUE}(x,{\boldsymbol{s}})}{\partial x^{3g-2}},\frac{\partial^{3g-2} u^{\rm GUE}(x,{\boldsymbol{s}})}{\partial x^{3g-2}}\biggr) \nonumber\\ & + \bigl(\zeta'(-1)-\frac{1}{24} \log(-1)\bigr)\delta_{g,1}\,,\quad g\geq1\,.\tag{30} \end{align}\] Here, \(\Omega^{{\mathbb{P}^1},[0]}_{2,p;2,q}(v,u)\) and \(\theta^{{\mathbb{P}^1}}_{2,p}(v,u)\) are certain genus \(0\) two-point correlation functions of the \(\mathbb{P}^1\)-Frobenius manifold, and \(F_g^{\mathbb{P}^1}(v,u,v_1,u_1,\dots,v_{3g-2},u_{3g-2})\) denotes the genus \(g\) free energy in jet variables. For \(g=1\), \[\label{jetF1} F_1^{\mathbb{P}^1}(v,u,v_1,u_1)=\frac{1}{24}\log(v_1^2-e^{u}u_1^2)-\frac{1}{24}u\,.\tag{31}\] For \(g\geq 2\), \(F_g^{\mathbb{P}^1}(v,u,v_1,u_1,\dots,v_{3g-2},u_{3g-2})\) depends polynomially on \(v_2,u_2,\dots,v_{3g-2},u_{3g-2}\) and rationally on \(v_1,u_1\) with coefficients being smooth functions of \(v\) and \(u\). Moreover, \[\deg F^{\mathbb{P}^1}_{g}(v,u,v_1,u_1,\dots,v_{3g-2},u_{3g-2})=2g-2\,,\] where \(\deg v_k=\deg u_k\mathrel{\vcenter{:}}= k\). The reader is referred to e.g. [12], [15], [26] for details on the \(\mathbb{P}^1\)-Frobenius manifold, \(\theta^{{\mathbb{P}^1}}_{2,p}(v,u)\), \(\Omega^{{\mathbb{P}^1},[0]}_{2,p;2,q}(v,u)\), and \(F_g^{\mathbb{P}^1}(v,u,v_1,u_1,\dots,v_{3g-2},u_{3g-2})\).

Following for example [10], consider the even GUE free energy \(\mathcal{F}^{\rm even}\), which is the GUE free energy restricted to even couplings, i.e., \[\mathcal{F}^{\rm even} = \mathcal{F}^{\rm even}(x,s_2,s_4,\dots;\epsilon)\mathrel{\vcenter{:}}=\mathcal{F}(x,{\boldsymbol{s}};\epsilon)|_{s_1=s_3=\cdots=0}\] and plays an important role in two-dimensional quantum gravity [10]. Denote by \(\mathcal{F}^{{\rm even}}_g\mathrel{\vcenter{:}}=[\epsilon^{2g-2}]\mathcal{F}^{\rm even}\) the genus \(g\) part of \(\mathcal{F}^{\rm even}\), and denote \[v_{\rm even}=v^{\rm GUE}(x,{\boldsymbol{s}})|_{s_1=s_3=\dots=0}\,,\; u_{\rm even}=u^{\rm GUE}(x,{\boldsymbol{s}})|_{s_1=s_3=\dots=0}\,.\] It was proved in [20] that \(v_{\rm even}\equiv 0\) and \(w_{\rm even}\mathrel{\vcenter{:}}= e^{u_{\rm even}}\) satisifies \[\label{evengenus0eqn} x=w_{\rm even}-\sum_{\nu\geq 1}\frac{(2\nu)!}{\nu!(\nu-1)!}s_{2\nu} w_{\rm even}^{\nu}\,.\tag{32}\]

The Hodge–GUE correspondence, which gives an explicit relationship between a certain cubic Hodge free energy and \(\mathcal{F}^{\rm even}\), was established in [14], [20]. Let \(\overline{\mathcal{M}}_{g,n}\) denote the Deligne–Mumford moduli space of stable genus-\(g\) curves with \(n\) marked points. Denote by \(\psi_i\) the first Chern class of the \(i\)th tautological line bundle over \(\overline{\mathcal{M}}_{g,n}\), and by \(\lambda_i\) the \(i\)th Chern class of the rank-\(g\) Hodge bundle on \(\overline{\mathcal{M}}_{g,n}\).

We define the Chern polynomial \(\Lambda_g(z)\mathrel{\vcenter{:}}=\sum_{i=0}^g \lambda_i z^i\), and set \[\mathcal{H}({\boldsymbol{t}};\epsilon)\mathrel{\vcenter{:}}=\sum_{g\geq 0}\epsilon^{2g-2}\sum_{n\geq 0}\frac{1}{n!}\sum_{i_1,\cdots,i_n\geq 0} \prod_{m=1}^{n}t_{i_m}\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_g(-1)\Lambda_g(-1)\Lambda_g(\tfrac12) \psi_1^{i_1}\cdots\psi_{n}^{i_n}\,.\] The Hodge–GUE correspondence says that \[\mathcal{F}^{\rm even}(x,s_2,s_4,\dots)= \epsilon^{-2}A(x,s_2,s_4,\dots)+\zeta'(-1) +(\Lambda^{\frac{1}{2}}+\Lambda^{-\frac{1}{2}})\mathcal{H}({\boldsymbol{t}}(x,s_2,s_4,\dots);\epsilon)\,,\] where \(A=A(x,s_2,s_4,\dots)\) is defined by \[A=\frac{1}{2}\sum_{k_1,k_2\geq 1}\frac{k_1 k_2}{k_1+k_2} \binom{2k_1}{k_1}\binom{2k_2}{k_2}s_{2k_1}s_{2k_2} -\sum_{k\geq 1}\frac{k}{1+k}\binom{2k}{k}s_{2k}+x\sum_{k\geq1}\binom{2k}{k}s_{2k}+\frac{1}{4}-x\,,\] and \[t_i(x,s_2,s_4,\dots)\mathrel{\vcenter{:}}=\sum_{k\geq 1} k^{i+1}\binom{2k}{k}s_{2k}-1+\delta_{i,1}+x \delta_{i,0}\,,\quad i\geq 0\,.\] This correspondence yields the only known ELSV-type formula for strictly monotone Hurwitz numbers [48], to the best of our knowledge. Explicit expressions for \(\mathcal{F}^{{\rm even}}_g\) for \(g=1,\dots,5\) in terms of \(u_{\rm even}\) and its \(x\)-derivatives were given in [20] and its arXiv preprint version. For example, [20] shows that \[\label{Feven1general} \mathcal{F}^{{\rm even}}_1= \frac{1}{12} \log \frac{\partial u_{\rm even}}{\partial x} +\zeta'(-1).\tag{33}\]

3 On triangulations↩︎

In this section we prove Theorem 1.

Taking \({\boldsymbol{s}}={\boldsymbol{1}}_3\) in 26 and 28 , we have \[\begin{align} 3 \, \bigl(V^{\{3\}}(x,\epsilon)^2 + W^{\{3\}}(x,\epsilon) + W^{\{3\}}(x+\epsilon,\epsilon)\bigr) & = V^{\{3\}}(x,\epsilon) \,, \tag{34}\\ x + 3 \, W^{\{3\}}(x,\epsilon) \, \bigl(V^{\{3\}}(x,\epsilon)+V^{\{3\}}(x-\epsilon,\epsilon)\bigr) & = W^{\{3\}}(x,\epsilon) \tag{35} \end{align}\] (cf. also e.g. [27], [34]). Introduce \[\widetilde{V}^{\{3\}}(x,\epsilon) \mathrel{\vcenter{:}}= V^{\{3\}}\bigl(x-\frac{\epsilon}{2},\epsilon\bigr)\,,\] then equations 34 –35 become \[\begin{align} 3 \, \Bigl(\widetilde{V}^{\{3\}}(x,\epsilon)^2 + W^{\{3\}}\bigl(x-\frac{\epsilon}{2},\epsilon\bigr) \, +\, W^{\{3\}}\bigl(x+\frac{\epsilon}{2},\epsilon\bigr)\Bigr) & = \widetilde{V}^{\{3\}}(x,\epsilon) \,, \tag{36}\\ x + 3 W^{\{3\}}(x,\epsilon) \Bigl(\widetilde{V}^{\{3\}}\bigl(x-\frac{\epsilon}{2},\epsilon\bigr)+\widetilde{V}^{\{3\}}\bigl(x+\frac{\epsilon}{2},\epsilon\bigr)\Bigr) & = W^{\{3\}}(x,\epsilon) \,. \tag{37} \end{align}\]

From the definition of \(V^{\rm GUE}(x,{\boldsymbol{s}};\epsilon)\), we know that \(\widetilde{V}^{\{3\}}(x,\epsilon)\) has the following genus expansion: \[\begin{align} \widetilde{V}^{\{3\}}(x,\epsilon) = \sum_{g\geq0} \epsilon^{2g} \widetilde{V}^{\{3\}}_g(x). \label{topoVt} \end{align}\tag{38}\] Substituting 23 and 38 in equations 36 –37 , we obtain \[\begin{align} & 3 \, \sum_{g_1,g_2\geq0} \epsilon^{2(g_1+g_2)} \, \widetilde{V}^{\{3\}}_{g_1} \, \widetilde{V}^{\{3\}}_{g_2} + 6\, \sum_{g,k} \frac{\epsilon^{2g+2k}}{2^{2k} (2k)!} \frac{\partial^{2k} W^{\{3\}}_{g}}{\partial x^{2k}} = \sum_{g\geq0} \epsilon^{2g} \, \widetilde{V}^{\{3\}}_{g}(x) \,, \tag{39}\\ & x + 6 \, \sum_{g_1,g_2,k\geq0} \frac{\epsilon^{2g_1+2g_2+2k}}{2^{2k} (2k)!} W^{\{3\}}_{g_1}(x) \, \frac{\partial^{2k} \widetilde{V}^{\{3\}}_{g_2}}{\partial x^{2k}} = \sum_{g\geq0} \epsilon^{2g} \, W^{\{3\}}_{g}(x) \,. \tag{40} \end{align}\] Comparing coefficients of \(\epsilon^{2g}\) (\(g \geq 0\)) on both sides of 39 –40 , we find \[\begin{align} 3 \, \widetilde{V}^{\{3\}}_{0}(x)^2 + 6 \, W^{\{3\}}_{0}(x) & = \widetilde{V}^{\{3\}}_{0}(x) \,, \tag{41}\\ x + 6 \, W^{\{3\}}_{0}(x) \, \widetilde{V}^{\{3\}}_{0}(x) & = W^{\{3\}}_{0}(x) \,, \tag{42} \end{align}\] and for \(g\geq1\), \[\begin{align} & \bigl(1 - 6 \widetilde{V}^{\{3\}}_{0} \bigr) \, \widetilde{V}^{\{3\}}_{g} \,-\, 6 \, W^{\{3\}}_{g} = 3 \, \sum_{g_1=1}^{g-1} \widetilde{V}^{\{3\}}_{g_1} \, \widetilde{V}^{\{3\}}_{g-g_1} + 6 \, \sum_{k=1}^g \frac{1}{2^{2k}(2k)!} \frac{\partial^{2k} W^{\{3\}}_{g-k}}{\partial x^{2k}} \,, \tag{43}\\ & - \, 6 \, W^{\{3\}}_{0} \, \widetilde{V}^{\{3\}}_{g} + (1- 6 \, \widetilde{V}^{\{3\}}_{0}) \, W^{\{3\}}_{g} = 6 \, \sum_{g_1,g_2\leq g-1 \atop g_1+g_2+k=g} \frac{1}{2^{2k} (2k)!} W^{\{3\}}_{g_1} \, \frac{\partial^{2k} \widetilde{V}^{\{3\}}_{g_2}}{\partial x^{2k}} \,. \tag{44} \end{align}\]

Solving 41 –42 we find that \(v=v(x)\mathrel{\vcenter{:}}=\widetilde{V}^{\{3\}}_{0}(x)\) and \(w=w(x)\mathrel{\vcenter{:}}= W^{\{3\}}_{0}(x)\) satisfy 4 and 5 , which imply 6 as well as \[\label{vwithw} v=\frac{1-\sqrt{1-72w}}{6}\,.\tag{45}\] From 6 we get \[\label{tridw} \partial_x = \frac{\sqrt{1-72\,w}}{1-108\,w} \, \partial_w\,.\tag{46}\]

From Proposition 1 we know that \(\widetilde{V}^{\{3\}}(x,\epsilon)\), \(W^{\{3\}}(x,\epsilon)\) can be written as \[\label{trigenusinpart1} W^{\{3\}}(x,\epsilon)=w+\sum_{g\geq 1}\epsilon^{2g}\sum_{\ell=3g-1}^{5g-1}\frac{A_{g,\ell}}{(1-108w)^{\ell}}\,, \quad \widetilde{V}^{\{3\}}(x,\epsilon)=v+\sum_{g\geq 1}\epsilon^{2g}\sum_{\ell=-\infty}^{5g-1}\frac{\widetilde{B}_{g,\ell}}{(1-108w)^{\ell}}\,,\tag{50}\] for some real numbers \(A_{g,5g-1},\ldots,A_{g,3g-1},\widetilde{B}_{g,5g-1},\ldots\). Substituting 50 in 36 –37 , one can obtain that \(U=2^{\frac{9}{5}}3^{\frac{11}{5}}\sum_{g\geq 0}A_{g,5g-1}(2^{\frac{2}{5}}3^{\frac{8}{5}}X)^{\frac{1-5g}{2}}\) satisfies the Painlevé I equation 11 . Therefore, \[\label{univ95const} A_{g,5g-1}=\frac{162^g}{108}C_g\,,\tag{51}\] with \(C_g\) the constants introduced in 10 .

We proceed to calculate \(\mathcal{F}^{\{3\}}(x,\epsilon)\). Taking \({\boldsymbol{s}}={\boldsymbol{1}}_3\) in 22 we have \[\log W^{\{3\}}(x,\epsilon) = \mathcal{F}^{\{3\}}(x+\epsilon,\epsilon) \,+\, \mathcal{F}^{\{3\}}(x-\epsilon,\epsilon) \,-\, 2 \, \mathcal{F}^{\{3\}}(x,\epsilon) \,. \label{WF}\tag{52}\] Therefore, \[\begin{align} \label{wf} \mathcal{F}^{\{3\}}(x,\epsilon) & = \frac{1}{\epsilon^2 \partial_x^2} \, \frac{\epsilon^2 \partial_x^2}{e^{\epsilon\partial_x} + e^{-\epsilon\partial_x} -2} \bigl( \log W^{\{3\}}(x,\epsilon) \bigr)\nonumber\\ &= \frac{1}{\epsilon^2 \partial_x^2} \, \sum_{k\geq 0} \frac{(1-2k) B_{2k}}{(2k)!} \, \epsilon^{2k} \partial_x^{2k} \bigl( \log W^{\{3\}}(x,\epsilon) \bigr) \,. \end{align}\tag{53}\]

It is helpful to introduce the following lemma for computing 53 :

Using Proposition 1, we have \[\begin{align} \label{triWstep2} \log \, W^{\{3\}}(x,\epsilon) &=\log \, w \,+\, \log \, \biggl(1\,+\,\sum_{g\geq1} \, \frac{\epsilon^{2g} Q_g(w)}{(1-108w)^{5g-1}}\biggr) \,. \end{align}\tag{56}\] By substituting 56 in 53 , and using Lemma 1 and Lemma 2, we obtain for \(g\geq 2\), \[\begin{align} \mathcal{F}^{\{3\}}_g =& \frac{1-2g}{(2g)!}B_{2g}\partial_x^{2g-2}(\log w) \nonumber\\ &+\sum_{k=1}^{g-1} \frac{1-2k}{(2k)!}B_{2k}\partial_x^{2k-2} \biggl(\sum_{\ell=1}^{g-k}\frac{(-1)^{\ell-1}}{\ell (1-108w)^{5g-5k-\ell}} \sum_{m_1,\dots,m_{\ell}\geq 1\atop m_1+\cdots+m_{\ell}=g-k}\prod_{i=1}^{\ell}Q_{m_i}(w)\biggr)\nonumber\\ &+\partial_x^{-2} \biggl(\sum_{\ell=1}^{g}\frac{(-1)^{\ell-1}}{\ell (1-108w)^{5g-\ell}} \sum_{m_1,\dots,m_{\ell}\geq 1\atop m_1+\cdots+m_{\ell}=g}\prod_{i=1}^{\ell}Q_{m_i}(w)\biggr)\tag{57} \\ =&\frac{1-2g}{(2g)!}B_{2g}\partial_x^{2g-2}(\log w) +\sum_{\ell=3g-3}^{5g-5}\frac{a_{g,\ell}}{(1-108w)^{\ell}} +\alpha_g w\sqrt{1-72w}+\beta_g\nonumber\\ &+\gamma_g \cdot \biggl(72w \sqrt{3-216w}\, {\rm arctanh}(\sqrt{3-216w})\nonumber\\ &-72z+\sum_{i=1}^{3g-4}\frac{1}{(1-108w)^i}\frac{(-2)^{i+2}}{i(i+2)}\frac{(i+2)!^2}{(2i+4)!}\biggr)\,.\tag{58} \end{align}\] Here \(\alpha_g\) and \(\beta_g\) are integration constants, and \(a_{g,3g-3},\dots, a_{g,5g-5},\gamma_g\) are rational numbers.

For further simplification of 58 , introduce \[\label{Gdef} \mathcal{G}^{\{3\}}(x,\epsilon) \mathrel{\vcenter{:}}= 6 \, \Bigl(\frac{1}{4} \,-\, \frac{x^2}{\epsilon^2} + 3 \, (\epsilon\partial_\epsilon+ x \partial_x) \, (\mathcal{F}^{\{3\}}(x,\epsilon))\Bigr)\,.\tag{59}\] It has the genus expansion \(\mathcal{G}^{\{3\}}(x,\epsilon)=\sum_{g\geq 0}\epsilon^{2g-2}\mathcal{G}^{\{3\}}_g(x)\), and for \(g\geq 0\), \[\label{GandF} \mathcal{G}^{\{3\}}_g=-6x^2 \delta_{g,0}+\frac{3}{2}\delta_{g,1}+18(2g-2)\mathcal{F}^{\{3\}}_g+18x\partial_{x}(\mathcal{F}^{\{3\}}_g)\,.\tag{60}\]

By taking \({\boldsymbol{s}}={\boldsymbol{1}}_3\) in equations 19 , 20 and 21 , we find \(\mathcal{G}^{\{3\}}(x,\epsilon) = \frac{\partial Z}{\partial s_1}\), so from 22 we know \[\frac{1}{\epsilon}\, \widetilde{V}^{\{3\}}(x,\epsilon) = \mathcal{G}^{\{3\}}(x+\epsilon/2,\epsilon) - \mathcal{G}^{\{3\}}(x-\epsilon/2,\epsilon) \,. \label{VF}\tag{61}\] Therefore \[\mathcal{G}^{\{3\}}(x,\epsilon) = \frac{1}{\epsilon^2\partial_x} \frac{\epsilon\partial_x}{(\Lambda^{1/2}-\Lambda^{-1/2})} (\widetilde{V}^{\{3\}}(x,\epsilon)) = \partial_x^{-1} \sum_{g\geq0} \epsilon^{2g-2} \sum_{k+g_1=g} \frac{(-1)^k B_{2k}(\frac{1}{2})}{(2k)!}\partial_x^{2k} (\widetilde{V}^{\{3\}}_{g_1} (x))\,, \label{triGstructure}\tag{62}\] and \[\label{triGg} \mathcal{G}_g^{\{3\}}(x,\epsilon)=\partial_x^{-1}(\widetilde{V}^{\{3\}}_{g}(x))+\sum_{k=1}^g \frac{(-1)^k B_{2k}(\frac{1}{2})}{(2k)!}\partial_x^{2k-1} (\widetilde{V}^{\{3\}}_{g-k} (x))\,,\quad g\geq 0\,.\tag{63}\] Here \(B_k(p)\) denotes the \(k\)th Bernoulli polynomial.

We are ready to prove Theorem 1.

We end this section by presenting several consequences of Theorem 1. First of all, note that, using Lemmas 1, 2, 50 , 57 , 58 and Theorem 1, we can achieve a new self-contained proof of the expression 9 of \(a_{g,5g-5}\), which in turn implies 10 .

Multiplying both sides of 8 by \(dx/x^{3-2g+j}\) and taking the residue at \(x=0\), we obtain \[\begin{align} n_{g}(3^{2j})&=\sigma_{g,j}+72^{j-2g+2}(2j)!\sum_{m=0}^{j-2g+2} \binom{\frac{3j}{2}-3g+3-m}{\frac{j}{2}-g+1} \sum_{\ell=3g-3}^{5g-5}(\ell-1)_{m}\,\frac{3^m}{2^m m!}a_{g,\ell}\,, \end{align}\] where \(\sigma_{g,j}=(2j)!\mathop{\rm res}\limits_{w=0}\frac{(1-108w)dw}{w^{j-2g+3}(1-72w)^{\frac{j}{2}-g+2}}\frac{(1-2g)B_{2g}}{2g!}\partial_x^{2g-2}(\log w)\) and \((a)_m\mathrel{\vcenter{:}}= a(a+1)\cdots (a+m-1)\) is the Pochhammer symbol.

Following [21] (cf. [3]), introduce \[\label{triqchange} q=\frac{1}{1-72w}\,.\tag{64}\] Formulas 7 , 12 are then translated to \[\begin{align} \mathcal{F}^{\{3\}}_1 =-\frac{1}{12}&\log x-\frac{1}{24} \log \biggl(\frac{3-q}{2}\biggr)+\zeta'(-1)\, ,\tag{65}\\ \mathcal{F}^{\{3\}}_2 =\frac{1}{x^2}\biggl(&\frac{3}{64 \left(3-q\right)}-\frac{29}{32 \left(3-q\right)^2}+\frac{191}{48 \left(3-q\right)^3}-\frac{55}{8 \left(3-q\right)^4}+\frac{21}{5 \left(3-q\right)^5}\,\biggr)\,,\tag{66}\\ \mathcal{F}^{\{3\}}_3 =\frac{1}{x^4}\biggl(&\frac{63}{256 \left(3-q\right)^2}-\frac{22765}{1152 \left(3-q\right)^3}+\frac{7925}{24 \left(3-q\right)^4}-\frac{39311}{16 \left(3-q\right)^5}\nonumber\\ &+\frac{1443995}{144 \left(3-q\right)^6} -\frac{4055053}{168\left(3-q\right)^7}+\frac{68625}{2\left(3-q\right)^8}-\frac{26730}{\left(3-q\right)^9}+\frac{8820}{\left(3-q\right)^{10}}\biggr) \,. \tag{67} \end{align}\] Formula 66 was given in [22] (cf. [3]), and 67 was given in [3]. It was proved by Eynard [3] (using a slightly different notation) that for \(g\geq2\), \[\label{triFform4} x^{2g-2}\mathcal{F}^{\{3\}}_g(x) =\sum_{k=g-1}^{5g-5}\frac{r_{g,k}}{(3-q)^{k}}\,,\tag{68}\] where \(r_{g,g-1},\dots,r_{g,5g-5}\) are rational numbers. Noting that \[\label{pxlogform3} x^{2k} \partial_x^{2k} (\log w)=\sum_{\ell=0}^{3k-1}\frac{H_{k,\ell}}{(3-q)^{k+\ell}}\, , \quad k\ge1\,,\, H_{k,0},\dots,H_{k,3k-1}\in\mathbb{Q}\,,\tag{69}\] and using Theorem 1 and formula 6 , we can achieve a new proof of 68 . Multiplying 68 by \(dx/x^{j+1}\) and taking the residue at \(x=0\), we obtain

\[\begin{align} n_{g}(3^{2j}) =72^j (2j)! \sum_{m=0}^j\sum_{\ell=g-1}^{5g-5} \frac{(\ell-1)_m}{2^{m+\ell} m!} \binom{\frac{3j}{2}-1}{j-m} r_{g,\ell}\,.\label{triformula3} \end{align}\tag{70}\]

Another direct consequence of Theorem 1 is that for \(g\geq2\), the genus \(g\) free energy \(\mathcal{F}^{\{3\}}_g(x)\) can be written in the form \[\label{triFform2} \mathcal{F}^{\{3\}}_g(x)=\sum_{k=1}^{5g-5}\frac{b_{g,k}}{(1-108w)^{k}} +\sum_{\ell=1}^{2g-2}\frac{b'_{g,k}}{w^{k}}\,,\tag{71}\] where \(b_{g,1},\dots,b_{g,5g-5},b'_{g,1},\dots,b'_{g,2g-2}\) are rational numbers.

Finally, introduce \(p=p(x)\) by \[\label{tripchange} p=\frac{108w}{1-108w}\,.\tag{72}\]

Explicitly, \[\begin{align} &\mathcal{F}^{\{3\}}_2 =\frac{1}{x^2}\biggl(-\frac{1}{240}+\frac{7 p^5}{12960}+\frac{29 p^4}{10368}+\frac{35 p^3}{10368}\,\biggr)\,, \nonumber\\ &\mathcal{F}^{\{3\}}_3 =\frac{1}{x^4}\biggl(\frac{1}{1008}+\frac{245 p^{10}}{1679616}+\frac{965 p^9}{559872}+\frac{2945 p^8}{373248}+\frac{813587 p^7}{47029248}+\frac{29969 p^6}{1679616}+\frac{5005 p^5}{746496}\biggr) \,. \nonumber \end{align}\] Multiplying ?? by \(dx/x^{j+1}\) and taking the residue at \(x=0\), we obtain for \(g\ge2\) \[\begin{align} n_{g}(3^{2j})=108^j (2j)!\sum_{\ell=2g-1}^{j} c_{g,\ell}\sum_{m=0}^{j-\ell} \frac{(\frac{3j}{2}-m)_m }{m!}\frac{(\frac{j}{2}+1)_{j-\ell-m}}{(j-\ell-m)!}(-3)^{-j+\ell+m} \label{triformula2} \end{align}\tag{74}\] (\(c_{g,\ell}\) are defined as 0 if \(\ell>5g-5\)).

4 On quadrangulations and \((2\nu)\)-angulations↩︎

In this section we prove Theorem 2.

Taking \({\boldsymbol{s}}={\boldsymbol{1}}_4\) in 26 and 28 , we have \[\begin{align} &V^{\{4\}}(x,\epsilon)\equiv 0 \,,\\ & 4 \, W^{\{4\}}(x,\epsilon)(W^{\{4\}}(x,\epsilon)+W^{\{4\}}(x+\epsilon,\epsilon)+W^{\{4\}}(x-\epsilon,\epsilon)) \,-\, W^{\{4\}}(x,\epsilon) + x = 0 \label{differencequad} \end{align}\tag{75}\] (cf. also e.g. [7], [34], [46]). Substituting 23 in 75 , we find \(w=W^{\{4\}}_0(x)\) satisfies 14 , and for \(g\geq1\), \[\label{wgrec} W^{\{4\}}_{g} = \frac{4}{1-24 w} \Biggl(\sum_{g_2=1}^{g-1} W^{\{4\}}_{g_2} W^{\{4\}}_{g-g_2}+2 \sum_{0\leq g_1,g_2\leq g-1, \, j\geq0 \atop g_1+g_2+j=g} \frac{1}{(2j)!} W^{\{4\}}_{g_2} \partial_x^{2j}(W^{\{4\}}_{g_1})\Biggr)\,.\tag{76}\] From equation 14 we get \[\label{quaddw} \partial_x=\frac{1}{1-24 w}\partial_{w}\,.\tag{77}\]

Proposition 3 says that \[\label{quadW} W^{\{4\}}(x,\epsilon)=w+\sum_{g\geq 1}\epsilon^{2g}\sum_{\ell=4g-1}^{5g-1}\frac{\widetilde{A}_{g,\ell}}{(1-24w)^{\ell}}\,,\tag{79}\] for some rational numbers \(\widetilde{A}_{g,4g-2}, \dots, \widetilde{A}_{g,5g-1}\). Substituting 79 in 75 , we find that \(U=2^{\frac{11}{5}}3^{\frac{4}{5}}\sum_{g\geq 0}\widetilde{A}_{g,5g-1}(2^{\frac{8}{5}}3^{\frac{2}{5}}X)^{\frac{1-5g}{2}}\) again satisfies the Painlevé I equation 11 . Therefore, \[\widetilde{A}_{g,5g-1} =\frac{48^g}{24}C_g\,, \quad g\geq 1\,,\] where \(C_g\) is again the same universal constant as we found for triangulations 51 .

Taking \({\boldsymbol{s}}={\boldsymbol{1}}_4\) in 22 , we have \[\log W^{\{4\}}(x,\epsilon) = \mathcal{F}^{\{4\}}(x+\epsilon,\epsilon) + \mathcal{F}^{\{4\}}(x-\epsilon,\epsilon) \,-\, 2 \, \mathcal{F}^{\{4\}}(x,\epsilon) \,. \label{quadWF}\tag{80}\] Similar to 53 , we have \[\begin{align} \label{quadwf} &\mathcal{F}^{\{4\}}(x,\epsilon) = \frac{1}{\epsilon^2 \, \partial_x^2} \, \sum_{k\geq 0} \frac{(1-2k) B_{2k}}{(2k)!} \, \epsilon^{2k}\partial_x^{2k} \bigl( \log W^{\{4\}}(x,\epsilon) \bigr) \,. \end{align}\tag{81}\]

Multiplying both sides of 15 by \(dx/x^{3-2g+k}\) and taking the residue at \(x=0\), we obtain \[\begin{align} n_{g}(4^k)&=\tilde{\sigma}_{g,k}+12^{k-2g+2}k! \sum_{m=0}^{k-2g+2}\binom{2k-4g+4-m}{k-2g+2-m}\Bigl(\sum_{\ell=4g-4}^{5g-5}\tilde{a}_{g,\ell}(\ell-1)_m\Bigr)\frac{2^m}{m!}\,, \end{align}\] where \(\tilde{\sigma}_{g,k}\mathrel{\vcenter{:}}= k!\mathop{\rm res}\limits_{w=0}\frac{(1-24w)dw}{w^{k-2g+3}(1-12w)^{k-2g+3}}\frac{(1-2g)B_{2g}}{(2g)!}\partial_x^{2g-2}(\log w)\).

Another direct consequence of Theorem 2 is that for \(g\geq2\), \(\mathcal{F}^{\{4\}}_g(x)\) admits the form \[\label{quadFform2} \mathcal{F}^{\{4\}}_g(x) =\sum_{k=1}^{5g-5}\frac{\tilde{b}_{g,k}}{(1-24w)^k} +\sum_{k=1}^{2g-2}\frac{\tilde{b}'_{g,k}}{w^{k}}\,,\tag{83}\] where \(\tilde{b}_{g,1},\dots,\tilde{b}_{g,5g-5},\tilde{b}'_{g,1},\dots,\tilde{b}'_{g,2g-2}\) are rational numbers satisfying \(\tilde{b}_{g,2}=\tilde{b}_{g,4}=\cdots=\tilde{b}_{g,4g-6}=0\).

Setting \[\label{quadpchange} p=\frac{24w}{1-24w}\tag{84}\] leads to the following corollary.

Multiplying both sides of ?? by \(dx/x^{k+1}\) and taking the residue at \(x=0\), we obtain \[\begin{align} n_{g}(4^{k})=24^k k!\sum_{\ell=2g-1}^{k} \tilde{c}_{g,\ell}\sum_{m=0}^{k-\ell} \frac{(2k-m)_m }{m!}\frac{(k+1)_{k-\ell-m}}{(k-\ell-m)!}\,(-2)^{-k+\ell+m} \end{align}\] (\(\tilde{c}_{g,\ell}\) are defined as 0 if \(\ell>5g-5\)).

Following [21], introduce \(q\) by \[\label{quadqchange} w=\frac{q-1}{12q}\,.\tag{85}\]

The proof is again omitted.

It was proved in [22] that, for any fixed \(\nu\geq 2\), the generating function \(e_g(s)\mathrel{\vcenter{:}}=\sum_{k\geq 1}\frac{n_g((2\nu)^k)}{k!}s^k\) admits the expression \[\label{Eregexpression} e_g(s)=C^{(g)}+\sum_{\ell=2g-2}^{5g-5}\frac{\tilde{r}_{g,\ell}(\nu)}{(\nu-(\nu-1)z_0)^\ell}\,, \quad g\geq 2\,.\tag{86}\] Here, \(\tilde{r}_{g,2g-2}(\nu),\dots,\tilde{r}_{g,5g-5}(\nu)\) are some rational numbers, \(C^{(g)}\) is a certain constant independent of \(\nu\), and \(z_0=z_0(s)=1+\frac{(2\nu)!}{\nu!(\nu-1)!}s+\cdots\) is the unique solution to \[\label{z0eqn} z_0=1+\frac{(2\nu)!}{\nu!(\nu-1)!}s\, z_0^{\nu}\,,\tag{87}\] Here, \(z_0(x^{\nu-1})=q(x)\). For the case when \(\nu=2\), Eynard [3] gave an expression of \(e_g(s)\), which is equivalent to 86 . Ercolani–Lega–Tippings [23] conjectured that \[\label{ELTconj} C^{(g)}=-\frac{B_{2g}}{4g(g-1)},\quad g\geq 2.\tag{88}\] By using 3 , ?? , 87 and based on 86 , we find that Ercolani–Lega–Tippings’s conjecture 88 is equivalent to Conjecture A.

We note that the topological recursion (see e.g. [3]) should also lead to a proof of Theorem A.

Now we use a formula from [22] to give a second proof of Theorem A. It was already shown in [22] that the \(C^{(g)}\) for any \(\nu\) satisfy the following recursion: \[\label{ErcolaniRecursive} \frac{(2g-1)!}{(2g+2)!}-\frac{(2g-1)!}{12(2g)!} +\sum_{k=2}^{g}\frac{(1-2g)_{2g-2k+2}}{(2g-2k+2)!}C^{(k)}=0\,, \quad g\geq 2\,.\tag{89}\]

5 More on the \(b=2\nu\) case↩︎

In this section, we prove Theorem 3 and give some more discussions on the \(b=2\nu\) case.

As a particular example of equation 32 , we know that \(w\) satisfies \[\label{2nugenus0eqn} x=w-\frac{(2\nu)!}{\nu!(\nu-1)!}w^{\nu}\,.\tag{93}\] Obviously the variable \(q\) introduced in ?? is related to \(w\) by \(q=w/x\).

Using 30 , 31 and Lemma 4, we obtain \[\mathcal{F}_1^{\{2\nu\}}(x)=-\frac{1}{12}\log x-\frac{1}{12}\log (\nu-(\nu-1)q)+\zeta'(-1)\,.\]

Similar to [28], by using Theorem A we can prove that, for \(2m-2+j>0\), \[\label{evendxFm} \partial_x^j (\mathcal{F}^{\{2\nu\}}_m(x))=\frac{1}{x^{2m-2+j}}\sum_{\ell=2m-2+j}^{5m-5+2j}\frac{r_{m,\ell,j}(\nu)}{(\nu-(\nu-1)q)^{\ell}}\,.\tag{97}\] By 22 we have \[\label{evenWstep1} W^{\{2\nu\}}_{g}(x)=w\sum_{n= 1}^{g}\frac{2^n}{n!}\sum_{k_1+m_1,\dots,k_n+m_n\geq 1 \atop |k|+|m|=g} \prod_{i=1}^{n}\frac{\partial_x^{2k_i+2}(\mathcal{F}_{m_i}^{\{2\nu\}}(x))}{(2k_i+2)!}\,,\quad g\geq 1\,.\tag{98}\] Using 97 and 98 , we obtain \[\label{evenWg} x^{2g-1} W^{\{2\nu\}}_{g}(x)=q\sum_{\ell=2g}^{5g-1}\frac{t_{g,\ell}(\nu)}{(\nu-(\nu-1)q)^{\ell}}\,, \quad g\ge1\,.\tag{99}\] Using Theorem 3, we show that \(r_{m,\ell,j}(\nu)\) are polynomials in \(\nu\) of degree at most \(3m-3+j\). Hence, we obtain:

We note that part of the motivation of Corollary 4 comes from [24]. Indeed, following [24], define \[Q_{g,k}(\nu) := k![x^{1-2g+(\nu-1)k}]W^{\{2\nu\}}_{g}(x)/\Bigl(\frac{(2\nu)!}{\nu!(\nu-1)!}\Bigr)^k\,.\] (As shown in [33], \(k![x^{1-2g+(\nu-1)k}]W^{\{2\nu\}}_{g}(x)\) counts the number of two-legged \(2\nu\)-valent genus-\(g\) maps with \(k\) vertices.) It then follows from ?? and 99 that \[\label{Qgk} Q_{g,k}(\nu)=k! \sum_{m=0}^k \binom{\nu k}{k-m}\frac{(\nu-1)^m}{m!}\sum_{\ell=2g}^{5g-1} t_{g,\ell}(\nu) (\ell-1)_m\,,\tag{100}\] which, together with Corollary 4, implies validity of a conjectural statement in [24] that \(Q_{g,k}(\nu)\) are polynomials in \(\nu\) of degree \(3g-1+k\).

Finally, we note that it can be deduced using Theorem A and a result of [22] that \[\label{rgCg} r_{g,5g-5}(\nu)=\frac{\nu^{3g-3}}{12^{g}}\frac{C_g}{(5g-3)(5g-5)}\,, \quad g\ge2\,,\tag{101}\] which can also be proved using the Hodge–GUE correspondence (cf. [14] and [15]) and [31]. Here \(C_g\) are defined by 11 . By 97 –99 and 101 we have \[t_{g,5g-1}(\nu)=\frac{\nu^{3g-1}}{12^g}C_g\,.\]

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