March 16, 2023
In [1] Okuyama and Sakai gave a conjectural equality for the higher genus generalized Brézin–Gross–Witten (BGW) free energies. In a recent work [2] we established the Hodge-BGW correspondence on the relationship between certain special cubic Hodge integrals and the generalized BGW correlators, and a proof of the Okuyama–Sakai conjecture was also given ibid. In this paper, we give a new proof of the Okuyama–Sakai conjecture by a further application of the Dubrovin–Zhang theory for the KdV hierarchy.
The Brézin–Gross–Witten (BGW) model was introduced in [3], [4]: \[Z_{\rm BGW}(A,A^{\dagger};\hbar)\sim \int\,[dU]\,e^{\frac{1}{\hbar^2}\,{\rm tr}(A^{\dagger}U\,+\,AU^{\dagger})}\, ,\] where the integration is over \(M\times M\) unitary matrices with Haar measure \([dU]\) and \(A\) is an \(M\times M\) complex matrix. In [5], a one parameter deformation of \(Z_{\rm BGW}\) was given via a generalized Kontsevich model [6]–[9]: \[Z_{\rm gBGW}(N,{\boldsymbol{T}};\hbar) \sim \int\, [d\Phi]\, e^{\frac{1}{\hbar^2}\, {\rm tr}\big(\Lambda^2\,\Phi\,+\,\frac{1}{\Phi}\,+\,(N\,-\,M)\,\hbar^2\,\log\,\Phi\big)}\, ,\] where \({\boldsymbol{T}}=(T_1,T_3,T_5,\dots)\), \(T_{2k+1}:= (2k-1)!!\, {\rm tr}\Lambda^{-2k-1}\), \(k\ge0\), and \(N\) is an indeterminate. The normalization constant for \(Z_{\rm gBGW}(N,{\boldsymbol{T}};\hbar)\) is often made so that \[\label{Zgbgw1} Z_{\rm gBGW}(N, {\boldsymbol{0}}; \hbar) \;\equiv\; 1 \,.\tag{1}\] We call \(Z_{\rm gBGW}(N,{\boldsymbol{T}};\hbar)\) the normalized generalized BGW partition function with the parameter \(N\). The logarithm \(\log Z_{\rm gBGW}(N, {\boldsymbol{T}}; \hbar) =: \mathcal{F}_{\rm gBGW}(N, {\boldsymbol{T}}; \hbar)\), belonging [6] to \(\mathbb{C}[\hbar][[N,{\boldsymbol{T}}]]\), is called the normalized generalized BGW free energy with the parameter \(N\).
Following Alexandrov [6], introduce \[x \; = \; N\, \hbar \,\sqrt{-2}\,.\] Then the free energy \(\mathcal{F}_{\rm gBGW}(N, {\boldsymbol{T}}; \hbar)\) has the genus expansion [6]: \[\label{genusbgsalex} \mathcal{F}_{\rm gBGW}\biggl(\frac{x}{\hbar\,\sqrt{-2}}, {\boldsymbol{T}}; \hbar\biggr) \;=:\; \sum_{g\geq 0}\hbar^{2g-2}\,\mathcal{F}^{\rm gBGW}_{g}(x, {\boldsymbol{T}}).\tag{2}\] We call \(\mathcal{F}_{\rm gBGW}\bigl(\frac{x}{\hbar\,\sqrt{-2}}, {\boldsymbol{T}}; \hbar\bigr)\) the normalized generalized BGW free energy, \(\mathcal{F}^{\rm gBGW}_{g}(x, {\boldsymbol{T}})\) its genus \(g\) part, and \(Z_{\rm gBGW}\bigl(\frac{x}{\hbar\,\sqrt{-2}}, {\boldsymbol{T}}; \hbar\bigr)\) the normalized generalized BGW partition function.
According to [6] (see also [5], [7], [10]), the normalized generalized BGW partition function \(Z_{\rm gBGW}\bigl(\frac{x}{\hbar\,\sqrt{-2}}, {\boldsymbol{T}}; \hbar\bigr)\) satisfies the Virasoro constraints, leading to the topological recursion of the Chekhov–Eynard–Orantin type [6], [11]–[13] for the computation of the corresponding connected correlators. It is also known that the normalized generalized BGW partition function is a particular tau-function for the Korteweg–de Vries (KdV) hierarchy (see e.g. [5], [7]). This enables one to apply theories of tau-functions for the KdV hierarchy to the study of \(Z_{\rm gBGW}\bigl(\frac{x}{\hbar\,\sqrt{-2}}, {\boldsymbol{T}}; \hbar\bigr)\). Recall that the matrix-resolvent method [14], [15] gives the explicit formulae for the generating series of the logarithmic derivatives of an arbitrary KdV tau-function; using this method, explicit formulae for the generating series of the \(n\)-point generalized BGW correlators were obtained [15] (see also [7] for other proofs for the explicit formulae). Recalling also that the KdV hierarchy is a reduction of the Kadomtsev–Petviashvili (KP) hierarchy, one can interprete \(Z_{\rm gBGW}\bigl(\frac{x}{\hbar\,\sqrt{-2}}, {\boldsymbol{T}}; \hbar\bigr)\) as a point in the Sato Grassmannian for the KP hierarchy; in particular, the corresponding affine coordinates were calculated out [15]–[17]. The KdV hierarchy can also be viewed as a reduction of the BKP hierarchy [18], and the BKP affine coordinates for \(Z_{\rm gBGW}\bigl(\frac{x}{\hbar\,\sqrt{-2}}, {\boldsymbol{T}}; \hbar\bigr)\) were given in [13], [19]. A new formula for \(Z_{\rm gBGW}\) based on Virasoro constraints and the KdV/BKP theory was recently obtained in [20]–[22].
Another important theory of tau-functions for the KdV hierarchy was partially motivated from the quantum gravity and topological field theories [23]–[26], and was systematically developed by Dubrovin and Zhang [27] in the framework normal forms of evolutionary PDEs. In our previous work [2], we applied this theory from viewpoints of Virasoro constraints, and found the Hodge-BGW correspondence (for details about the Hodge-BGW correspondence see Section 2 below). In particular, by using the Hodge-BGW correspondence and by deriving the loop equations we proved [2] a conjecture of Okuyama and Sakai [1].
([1]). Define a power series \(y(x,{\boldsymbol{T}})\in\mathbb{C}[[x+2]][[{\boldsymbol{T}}]]\) by \[y(x,{\boldsymbol{T}})\; = \;\frac{\partial^2 \mathcal{F}^{\rm gBGW}_0(x,{\boldsymbol{T}})}{\partial T_1^2}.\] For every \(g\geq 1\), the genus g part of the generalized BGW free energy satisfy the identity: for \(g=1\), \[\begin{align} & \mathcal{F}^{\rm gBGW}_{1}(x,{\boldsymbol{T}}) \; = \; \frac{1}{24} \log \bigg(\frac{\partial y(x,{\boldsymbol{T}})}{\partial T_1}\bigg) \,-\, \frac{\log 2}{24} \,-\, \frac{1}{12} \, \log\Bigl(-\frac{x}{2}\Bigr) \,, \label{eqn:bgwf1-wkjet} \end{align}\qquad{(1)}\] and for \(g\geq 2\), \[\label{eqn:kw-bgwjet} \mathcal{F}^{\rm gBGW}_{g}(x,{\boldsymbol{T}}) \; = \; F^{\rm WK}_g\bigg(\frac{\partial y(x,{\boldsymbol{T}})}{\partial T_1},\dots,\frac{\partial^{3g-2} y(x,{\boldsymbol{T}})}{\partial T_1^{3g-2}}\bigg)\, -\,\frac{1}{x^{2g-2}}\frac{(-1)^g \, 2^{g-1} \, B_{2g}}{2g \, (2g-2)}\,.\qquad{(2)}\] Here, \(B_k\) denotes the \(k\)th Bernoulli number, and \(F^{\rm WK}_g(z_1,\dots,z_{3g-2})\), \(g\ge2\), are certain specific functions of \((3g-2)\) variables (see 19 , 20 , 21 for the definitions).
We recall that in [1] the above conjectural identities ?? , ?? were verified for \(g=1,2\), and were also checked [1] for special evaluations up to \(g=20\).
The goal of this paper is to give a new proof of the Okuyama–Sakai conjecture by using the Hodge-BGW correspondence, and by considering a further application of the Dubrovin–Zhang theory [27] (see also [28], [29]) to the KdV hierarchy which can be interpreted like in [29] for the Laguerre Unitary Ensemble case (see also [30]) as a universality class of criticality in the renormalization theory of quantum field theories.
Theorem 1. The Okuyama–Sakai conjecture holds.
The rest of the paper is organized as follows. In Section 2 we give a review of the Hodge-BGW correspondence. In Section 3 we prove Theorem 1.
The work was partially supported by the CAS Project for Young Scientists in Basic Research No. YSBR-032 and by the NSFC No. 12061131014.
Let \(\overline{\mathcal{M}}_{g,n}\) denote the Deligne–Mumford moduli space of stable algebraic curves of genus \(g\) with \(n\) distinct marked points [31]. Let \(\mathcal{L}_p\) be the \(p\)th tautological line bundle on \(\overline{\mathcal{M}}_{g,n}\), and \(\mathbb{E}_{g,n}\) the Hodge bundle. Denote by \(\psi_p:=c_1(\mathcal{L}_p)\), \(p=1,\dots,n\), the first Chern class of \(\mathcal{L}_p\), and by \(\lambda_j:= c_j(\mathbb{E}_{g,n})\), \(j=0, \dots, g\), the \(j\)th Chern class of \(\mathbb{E}_{g,n}\). The Hodge integrals are defined as the following intersection numbers of mixed \(\psi\)-, \(\lambda\)-classes on \(\overline{\mathcal{M}}_{g,n}\): \[\label{hodgekappaint} \int_{\overline{\mathcal{M}}_{g,n}} \, \psi_1^{i_1}\cdots \psi_n^{i_n} \, \lambda_1^{j_1} \cdots \lambda_g^{j_g} \,,\tag{3}\] where \(i_1, \dots, i_n, j_1, \dots, j_g \geq 0\). These integrals vanish unless \[\label{ddhodge} (i_1\; + \;\cdots\; + \; i_n) \; + \; (j_1\; + \;2 \, j_2\; + \;\cdots\; + \;g \, j_g)\; = \; 3 \, g \,-\, 3 \; + \; n \,.\tag{4}\] For the case when all \(j\)’s are taken to be zero, the Hodge integrals 3 are famously known as the intersection numbers of \(\psi\)-classes, which, according to Witten’s conjecture [26] and Kontsevich’s proof [9], have an explicit connection to the KdV hierarchy (cf. e.g. [9], [15], [26], [27], [32]).
Let \({\rm ch}_k(\mathbb{E}_{g,n})\), \(k\ge0\), be the components of the Chern character of \(\mathbb{E}_{g,n}\). According to Mumford [33], the odd components of the Chern character of \(\mathbb{E}_{g,n}\) vanish. Denote by \[\label{hodgeint} \mathcal{H}({\boldsymbol{t}};{\boldsymbol{\sigma}}; \epsilon) \; := \; \sum_{g,n\geq0} \,\epsilon^{2g-2}\, \sum_{i_1,\dots,i_n\geq0}\, \frac{t_{i_1}\cdots t_{i_n}}{n!} \, \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{i_1} \cdots \psi_n^{i_n} \cdot \exp\biggl(\sum_{j\ge1} \sigma_{2j-1} {\rm ch}_{2j-1}(\mathbb{E}_{g,n})\biggr)\tag{5}\] the generating series of Hodge integrals, called the Hodge free energy, and by \(\mathcal{H}_g({\boldsymbol{t}}; {\boldsymbol{\sigma}})\) its genus \(g\) part, i.e., \[\mathcal{H}({\boldsymbol{t}}; {\boldsymbol{\sigma}}; \epsilon) \; = \; \sum_{g\geq0} \, \epsilon^{2g-2} \, \mathcal{H}_g({\boldsymbol{t}}; {\boldsymbol{\sigma}})\,.\] Denote also \[\label{hodgepart} Z_{\rm H}({\boldsymbol{t}}; {\boldsymbol{\sigma}}; \epsilon) \; := \; e^{\mathcal{H}({\boldsymbol{t}}; {\boldsymbol{\sigma}}; \epsilon)}.\tag{6}\] We call \(Z_{\rm H}({\boldsymbol{t}}; {\boldsymbol{\sigma}}; \epsilon)\) the Hodge partition function. The specialization \[\label{wkpart} Z_{\rm H}({\boldsymbol{t}}; {\boldsymbol{0}}; \epsilon)=:Z_{\rm WK} ({\boldsymbol{t}};\epsilon), \quad \mathcal{H}({\boldsymbol{t}}; {\boldsymbol{0}}; \epsilon) =: \mathcal{F}^{\rm WK} ({\boldsymbol{t}};\epsilon) =: \sum_{g\ge0} \epsilon^{2g-2} \mathcal{F}^{\rm WK}_g ({\boldsymbol{t}})\tag{7}\] are called the Witten–Kontsevich partition function, the Witten–Kontsevich free energy, respectively.
We will be particularly interested in the Hodge integrals of the following form: \[\label{chbcy} \int_{\overline{\mathcal{M}}_{g,n}} \, \Lambda_g(-1)^2 \, \Lambda_g\Bigl(\frac{1}{2}\Bigr) \, \psi_1^{i_1} \cdots \psi_n^{i_n} \,,\tag{8}\] where \(\Lambda_g(z) := \sum_{j=0}^g \lambda_j z^j\) denotes the Chern polynomial of \(\mathbb{E}_{g,n}\). Significance of the Hodge integrals in 8 was manifested by the Hodge-GUE correspondence [34], [35] (see also [36]), by the Gopakumar–Mariño–Vafa conjecture on the string/Chern–Simons duality [37]–[41], by the Hodge-BGW correspondence [2], and etc. Generating series of these Hodge integrals can be obtained from 5 by specializing the parameters \(\sigma\)’s as follows: \[\sigma_{2j-1} \; = \; - 2\, (1-4^{-j}) \, (2j-2)! , \quad j\geq1\] (see e.g. [36]). Denote \[\begin{align} & \mathcal{H}_{\rm special}({\boldsymbol{t}};\epsilon) := \mathcal{H}\bigl({\boldsymbol{t}}; \{- 2\, (1-4^{-j}) \, (2j-2)! \} ; \epsilon\bigr), \label{cubichodgeint} \\ & Z_{\rm special}({\boldsymbol{t}};\epsilon) \; := \; Z_H\bigl({\boldsymbol{t}}; \{- 2\, (1-4^{-j}) \, (2j-2)! \};\epsilon\bigr). \end{align}\tag{9}\] We call \(\mathcal{H}_{\rm special}({\boldsymbol{t}};\epsilon)\) (and \(Z_{\rm special}({\boldsymbol{t}};\epsilon)\)) the Hodge free energy (and respectively Hodge partition function) associated to \(\Lambda_g(-1)^2\Lambda_g(1/2)\). We also denote by \(\mathcal{H}^{\rm special}_{g}({\boldsymbol{t}})\) the genus \(g\) part of the Hodge free energy associated to \(\Lambda_g(-1)^2\Lambda_g(1/2)\), i.e., \[\mathcal{H}_{\rm special}({\boldsymbol{t}};\epsilon) \; = \; \sum_{g\geq0} \, \epsilon^{2g-2} \, \mathcal{H}^{\rm special}_{g}({\boldsymbol{t}})\,.\]
In [2] an explicit relationship (see the following theorem) between the Hodge partition function associated to \(\Lambda(-1)^2\Lambda(1/2)\) and the generalized BGW partition function, called the Hodge-BGW correspondence, was established. As in [2], define \[\label{def:gbgw-part} \mathcal{F}(x, {\boldsymbol{T}}; \hbar) \; := \; \mathcal{F}_{\rm gBGW}\Bigl(\frac{x}{\hbar \,\sqrt{-2}}, {\boldsymbol{T}}; \hbar \Bigr) \; + \; B(x,\hbar)\,,\tag{10}\] where \[\label{eqn:bernoulli} B(x, \hbar) \; = \; \frac{1}{\hbar^2} \biggl(\frac{x^2}{4} \, \log \Bigl(-\frac{x}{2}\Bigr) - \frac{3}{8} \, x^2\biggr) \; + \; \frac{1}{12} \, \log\Bigl(-\frac{x}{2}\Bigr) \; + \; \sum_{g\geq 2} \frac{\hbar^{2g-2}}{x^{2g-2}}\frac{(-1)^g \, 2^{g-1} \, B_{2g}}{2g \, (2g-2)}\tag{11}\] with \(B_k\) denoting the \(k\)th Bernoulli number. We call \(\mathcal{F}(x, {\boldsymbol{T}}; \hbar)\) the generalized BGW free energy, and its exponential \[\label{def:Z} \exp(\mathcal{F}(x, {\boldsymbol{T}}; \hbar)) =: Z(x, {\boldsymbol{T}}; \hbar)\tag{12}\] the generalized BGW partition function. The genus expansion 2 implies the genus expansion \[\label{eqn:genus-expan} \mathcal{F}(x, {\boldsymbol{T}};\hbar) \;=:\; \sum_{g\ge0} \hbar^{2g-2}\, \mathcal{F}_g(x, {\boldsymbol{T}}),\tag{13}\] and we call \(\mathcal{F}_g(x, {\boldsymbol{T}})\) the genus \(g\) part of the generalized BGW free energy, for short the genus \(g\) generalized BGW free energy. We are ready to state the Hodge-BGW correspondence.
Theorem 2 (Hodge-BGW correspondence [2]). The following identity \[\begin{align} \label{mainid} e^{\frac{A( x,{\boldsymbol{T}})}{\hbar^2}} Z_{\rm special}\bigl({\boldsymbol{t}}( x, {\boldsymbol{T}}); \hbar\,\sqrt{-4} \bigr) \; = \; Z( x, {\boldsymbol{T}}; \hbar) \end{align}\tag{14}\] holds true in \(\mathbb{C}((\hbar^2))[[x+2]][[{\boldsymbol{T}}]]\). Here, \[\begin{align} \label{deftT} & t_i( x, {\boldsymbol{T}}) \; = \; \delta_{i,0} \, x \; + \;\delta_{i,1}\, -\, \Bigl(-\frac{1}{2}\Bigr)^{i-1}\,-\, 2\, \sum_{a\geq0} \,\Bigl(-\frac{2a+1}{2}\Bigr)^i \; \frac{ {T}_{2a+1} }{a!} \, , \quad i\geq 0\,, \end{align}\tag{15}\] and \(A(x, {\boldsymbol{T}})\) is a quadratic series given by \[\label{defA1215} A(x,{\boldsymbol{T}}) \; = \; \frac{1}{2} \, \sum_{a,b\geq0} \, \frac{\widetilde{ T}_{2a+1} \,\widetilde{ T}_{2b+1}}{a! \, b! \, (a+b+1)} \, - \, \sum_{b\geq0} \, \frac{ x \, \widetilde{ T}_{2b+1}}{b!\, (2b+1)} \,,\tag{16}\] with \(\widetilde{T}_{2a+1} \; = \; T_{2a+1} \,-\, \delta_{a,0}\), \(a\geq0\).
We note that Norbury [42] also gave a conjectural topological interpretation for \(\log Z_{\rm BGW}\) (proved recently by Chidambaram, Garcia-Failde and Giacchetto [11]), and that Kazarian and Norbury [43] gave another new topological interpretation (which is conjectured to be equivalent to the construction of [42]).
It is known that (cf. [24], [36], [44]–[48]) for any \(g\ge1\), the genus \(g\) part of the Hodge free energy \(\mathcal{H}_g({\boldsymbol{t}}; {\boldsymbol{\sigma}})\) admits the jet-variable representation, i.e., for \(g=1\), \[\mathcal{H}_1({\boldsymbol{t}}; {\boldsymbol{\sigma}}) = H_1\biggl(v({\boldsymbol{t}}), \frac{\partial v({\boldsymbol{t}})}{\partial t_0}; {\boldsymbol{\sigma}}\biggr), \quad H_1(z,z_1) := \frac{1}{24} \log z_1 +\frac{\sigma_1}{24}z \label{jetgenus1}\tag{17}\] and for \(g\ge2\), there exists a unique function \(H_g(z_1, \dots, z_{3g-2}; {\boldsymbol{\sigma}})\) of \((3g-2)\) variables, such that \[\label{jetgenusg} \mathcal{H}_g({\boldsymbol{t}}; {\boldsymbol{\sigma}}) = H_g\biggl(\frac{\partial v({\boldsymbol{t}})}{\partial t_0}, \dots, \frac{\partial^{3g-2} v({\boldsymbol{t}})}{\partial t_0^{3g-2}}; {\boldsymbol{\sigma}}\biggr), \quad g\geq2.\tag{18}\] Here \(v({\boldsymbol{t}})=\partial_{t_0}^2 \mathcal{F}^{\rm WK}_0(\boldsymbol{t})\). For the special case when all \({\boldsymbol{\sigma}}={\boldsymbol{0}}\), we know that \[\begin{align} &\mathcal{F}^{\rm WK}_1({\boldsymbol{t}}) = F^{\rm WK}_1\biggl(\frac{\partial v({\boldsymbol{t}})}{\partial t_0}\biggr), \tag{19}\\ &F^{\rm WK}_1(z_1) := \frac{1}{24} \log z_1\,, \tag{20} \end{align}\] and that for \(g\ge2\), there exists a unique function \(F^{\rm WK}_g(z_1, \dots, z_{3g-2})\) of \((3g-2)\) variables, such that \[\label{jstwk2} \mathcal{F}^{\rm WK}_g({\boldsymbol{t}}) = F^{\rm WK}_g\biggl(\frac{\partial v({\boldsymbol{t}})}{\partial t_0}, \dots, \frac{\partial^{3g-2} v({\boldsymbol{t}})}{\partial t_0^{3g-2}}\biggr), \quad g\geq2.\tag{21}\] The unique functions \(F^{\rm WK}_g(z_1, \dots, z_{3g-2})\) are the ones that appear in the Okuyama–Sakai conjecture.
It follows from the jet-variable representation of \(\mathcal{H}^{\rm special}_{g}({\boldsymbol{t}})\) (cf. 17 –18 ) and Theorem 2 that the genus \(g\) generalized BGW free energy \(\mathcal{F}_g(x, {\boldsymbol{T}})\) for \(g\ge1\) has the jet-variable representation: there exist functions \(F_g(z_0, z_1,\dots,z_{3g-2})\), \(g\ge1\), which for \(g\ge2\) belong to \(\mathbb{C}[z_0^{\pm},z_1^{\pm1}][z_2,\dots,z_{3g-2}]\), such that \[\label{jetfgx} \mathcal{F}_g(x, {\boldsymbol{T}}) \; = \; F_g \biggl( u(x,{\boldsymbol{T}}), \frac{\partial u(x,{\boldsymbol{T}})}{\partial x}, \dots, \frac{\partial^{3g-2} u(x,{\boldsymbol{T}})}{\partial x^{3g-2}} \biggr)\,, \quad g\ge1\,,\tag{22}\] where \[u(x,{\boldsymbol{T}}) \; := \; - 4 \, \frac{\partial^2 \mathcal{F}_0(x,{\boldsymbol{T}})}{\partial x^2} \, .\] (This representation was given in the Proposition 5 of [2]). Recall from [2] that the power series \(u(x,{\boldsymbol{T}})\) is related to \(v({\boldsymbol{t}})\) by \[\label{uvequal} u(x,{\boldsymbol{T}}) \; = \; v({\boldsymbol{t}}(x,{\boldsymbol{T}}))\,.\tag{23}\] Now let \[\label{def:y} y=y(x,{\boldsymbol{T}}) \; := \; \frac{\partial^2 \mathcal{F}_0(x,{\boldsymbol{T}})}{\partial T_1^2} \; = \;\frac{\partial^2 \mathcal{F}^{\rm gBGW}_0(x,{\boldsymbol{T}})}{\partial T_1^2}\,.\tag{24}\] According to [2] we know that \[\label{relationsyu} y\; = \; e^{-u} \,, \quad u_x \; = \; -\frac{y_x}{y} \,, \quad y_x \; = \; - \frac{y_{T_1}}{2y^{1/2}} \,.\tag{25}\] Using 25 and 22 , we find that there exist functions \(\bar F_g(z_0,z_1,\dots,z_{3g-2})\), \(g\ge1\), such that \[\label{jetfgT1} \mathcal{F}_{g}(x,{\boldsymbol{T}}) \; = \; \bar F_g\bigg(y(x,{\boldsymbol{T}}), \frac{\partial y(x,{\boldsymbol{T}})}{\partial T_1},\dots,\frac{\partial^{3g-2} y(x,{\boldsymbol{T}})}{\partial T_1^{3g-2}}\bigg) \,,\quad g\ge1\,.\tag{26}\] These representations will be used in the next section.
In this section we prove the Okuyama–Sakai conjecture by applying the Dubrovin–Zhang theory [27] to the KdV hierarchy.
Before proving Theorem 1, let us first recall the following well-known lemma.
Lemma 1. For \(g\geq 1\) the function \(F^{\rm WK}_{g}=F^{\rm WK}_{g}(z_1,\dots,z_{3g-2})\) satisfies the following equations: \[\begin{align} \sum_{k\geq 1} \frac{k+2}{2} z_{k}\frac{\partial F^{\rm WK}_{g}}{\partial z_k} \; = \;&\frac{\delta_{g,1}}{16},\\ \sum_{k\geq 1}k z_k \frac{\partial F^{\rm WK}_{g}}{\partial z_k} \; = \;&(2g-2)F_{g}\; + \;\frac{\delta_{g,1}}{24}. \end{align}\]
Proof. It is well known that \(Z_{\rm WK}({\boldsymbol{t}};\epsilon)\) satisfies the following two equations: \[\begin{align} & \sum_{i\geq 0} \frac{2i+1}{2}t_{i} \frac{\partial Z_{\rm WK}({\boldsymbol{t}}; \epsilon)}{\partial t_i}\; + \; \frac{1}{16} Z_{\rm WK}({\boldsymbol{t}}; \epsilon) \; = \; \frac{3}{2}\frac{\partial Z_{\rm WK}({\boldsymbol{t}}; \epsilon)}{\partial t_1} \, . \tag{27}\\ & \sum_{i\geq 0} t_i \, \frac{\partial Z_{\rm WK}({\boldsymbol{t}}; \epsilon)}{\partial t_i} \; + \; \epsilon\, \frac{\partial Z_{\rm WK}}{\partial\epsilon} \; + \; \frac{1}{24} \, Z_{\rm WK}({\boldsymbol{t}}; \epsilon) \; = \; \frac{\partial Z_{\rm WK}({\boldsymbol{t}}; \epsilon)}{\partial t_1} \,, \tag{28} \end{align}\] The lemma is proved by using 7 , 19 , 20 , 21 , 27 , 28 , and \[\frac{\partial v({\boldsymbol{t}})}{\partial t_i}\; = \;\frac{v({\boldsymbol{t}})^i}{i!}\frac{\partial v({\boldsymbol{t}})}{\partial t_0}\,,\quad i\geq 0\,.\] ◻
Proof of Theorem 1. Let us first recall some properties of the genus \(0\) part of the generalized BGW free energy. Following [2] (see (112) therein), introduce \[Q\; = \;Q(x,{\boldsymbol{T}})\; = \;\exp\Big({-\frac{u(x,{\boldsymbol{T}})}{2}}\Big)\,.\] Then \(Q\) has [2] the following properties: \[\begin{align} &\frac{\partial Q}{\partial T_{2a+1}} \; = \; - \, \frac{2}{a!} \, Q^{2a+1} \, \frac{\partial Q}{\partial x} \,, \quad a\geq 0\,,\tag{29}\\ &Q(x,{\boldsymbol{0}}) \; = \; - \, \frac{x}{2}\,. \tag{30} \end{align}\] Alternatively, the power series \(Q\) can be uniquely determined by the following equation [2]: \[\label{eqn:euler-lagrange-Q} Q \; = \; - \, \frac{x}{2} \; + \; \sum_{a\geq0} \, {T}_{2a+1} \, \frac{Q^{2a+1}}{a!} \,.\tag{31}\] Moreover, the genus \(0\) free energy \(\mathcal{F}_0(x,\boldsymbol{T})\) of generalized BGW model satisfies the following equations [2]: \[\begin{align} \frac{\partial^2 \mathcal{F}_0(x,{\boldsymbol{T}}) }{\partial T_{2a+1} \partial T_{2b+1} } & \; = \; \frac{Q^{2a+2b+2}}{a!\, b!\,(a+b+1)}\,, \tag{32} \\ \frac{\partial^2\mathcal{F}_0(x,{\boldsymbol{T}})}{\partial x \partial T_{2b+1} } & \; = \; - \, \frac{Q^{2b+1}}{b! \, (2 b+1)}\,, \tag{33} \\ \frac{\partial^2\mathcal{F}_0(x,{\boldsymbol{T}})}{\partial x \partial x} & \; = \; \frac{1}{2}\, \log Q \,. \tag{34} \end{align}\] Here \(a,b\geq 0\). It has also been proved in [2] that the genus zero part of the generalized BGW free energy \(\mathcal{F}_0(x, {\boldsymbol{T}})\) has the expression \[\begin{align} \label{dubrovin-f0} \mathcal{F}_0(x, {\boldsymbol{T}}) \; = \; & \frac{1}{2} \, \sum_{a,b\geq0} \, \widetilde{T}_{2a+1} \, \widetilde{T}_{2b+1} \, \frac{Q^{2a+2b+2}}{a!\,b!\,(a+b+1)} \,-\, x \, \sum_{b\geq0} \, \widetilde{T}_{2b+1} \, \frac{Q^{2b+1}}{b!\,(2b+1)} \; + \; \frac{x^2}{4} \, \log Q \,. \end{align}\tag{35}\]
By using 24 and 32 , we find \(y=Q^2\). Then by using 29 and 31 , we have \[\begin{align} \frac{\partial y}{\partial T_{2a+1}} &\; = \;\frac{y^a}{a!}\,\frac{\partial y}{\partial T_1}\,,\quad a\geq 0\,, \label{eqn:flow-v} \end{align}\tag{36}\] and \[y|_{T_3=T_5=\cdots=0}\; = \;\frac{x^2}{4\,(1-T_1)^2}\,.\] From 32 (again noticing that \(y=Q^2\)), we know that \(e^{\hbar^{-2}\mathcal{F}_{0}(x,{\boldsymbol{T}})}\) is the tau-function of the solution \(y\) to the dispersionless KdV hierarchy 36 .
Define \[U(x, {\boldsymbol{T}};\hbar) \; := \; y(x, {\boldsymbol{T}}) \; + \; \sum_{g\geq 1} \hbar^{2g} \,\frac{\partial^2 F_g^{\rm WK}\bigl(\frac{\partial y(x, {\boldsymbol{T}})}{\partial T_1},\dots,\frac{\partial^{3g-2}y(x, {\boldsymbol{T}})}{\partial T_1^{3g-2}}\bigr)}{\partial T_1^2}.\] According to Dubrovin and Zhang [27], the power series \(U(x, {\boldsymbol{T}};\hbar)\) is a particular solution to the KdV hierarchy: \[\label{eqn:KdV} \frac{\partial U}{\partial T_{2i+1}}\; = \;\frac{1}{(2i+1)!!}\Big[\big(L^{\frac{2i+1}{2}}\big)_{+},U\Big],\quad i\geq0\,,\tag{37}\] where \(L=\hbar^2\partial^2_{T_1}+2U\) is the Lax operator of the KdV hierarchy (cf. e.g. [32]). Moreover, the power series \(\tau (x,{\boldsymbol{T}};\hbar)\), defined by \[\label{defitau10} \tau (x,{\boldsymbol{T}};\hbar) \; := \; \exp\biggl(\hbar^{-2}\mathcal{F}_0(x, {\boldsymbol{T}})+\sum_{g\geq 1}\hbar^{2g-2}F_g^{\rm KW}\biggl(\frac{\partial y(x, {\boldsymbol{T}})}{\partial T_1},\dots,\frac{\partial^{3g-2}y(x, {\boldsymbol{T}})}{\partial T_1^{3g-2}}\biggr)\biggr) \,,\tag{38}\] is the tau-function of the solution \(U(x, {\boldsymbol{T}};\hbar)\) to the KdV hierarchy [27]. The conjectural identies ?? , ?? are now equivalent to \[Z(x,{\boldsymbol{T}};\hbar)\; = \; 2^{-\frac{1}{24}} \, \tau(x,{\boldsymbol{T}};\hbar)\,.\]
It is known ([2], [6]) that \(Z(x, {\boldsymbol{T}}; \hbar)\) satisfies the following equation: \[\begin{align} \label{eqn:string-gBGW} L_0 \bigl(Z(x, {\boldsymbol{T}}; \hbar)\bigr) \; = \; 0\,, \end{align}\tag{39}\] where \[\begin{align} L_0\; = \;&\sum_{a\geq 0}\frac{2a+1}{2}\widetilde{T}_{2a+1}\frac{\partial}{\partial T_{2a+1}} \; + \;\frac{1}{16}+\frac{x^2}{8\hbar^2}\,. \end{align}\]
By equation 39 and 12 , 13 , \(\mathcal{F}_{0}(x,{\boldsymbol{T}})\) satisfies the following equation \[\label{eqn:string-gBGW-F0} \sum_{a\geq 0}\frac{2a+1}{2}\widetilde{T}_{2a+1}\frac{\partial\mathcal{F}_{0}(x,{\boldsymbol{T}})}{\partial T_{2a+1}} \; + \;\frac{x^2}{8}\; = \;0.\tag{40}\] By repeatedly taking derivatives of the above equation with respect to \(T_1\) and by using the commutation relation \[\Big[\frac{\partial}{\partial T_1},L_0\Big]\; = \;\frac{1}{2}\,\frac{\partial}{\partial T_1}\,,\] we get \[\sum_{a\geq 0}\frac{2a+1}{2}\widetilde{T}_{2a+1}\frac{\partial}{\partial T_{2a+1}}\bigg(\frac{\partial^k y}{\partial T_1^k}\bigg)\; = \;-\frac{k+2}{2}\frac{\partial^k y}{\partial T_1^k}\,,\quad k\geq 0\,.\] Therefore, \[\sum_{a\geq 0}\frac{2a+1}{2}\widetilde{T}_{2a+1}\frac{\partial F_g^{\rm KW}\bigl(\frac{\partial y}{\partial T_1},\dots,\frac{\partial^{3g-2}y}{\partial T_1^{3g-2}}\bigr)}{\partial T_{2a+1}} \; = \; \Biggl(-\sum_{k=1}^{3g-2}\frac{k+2}{2} z_k\frac{\partial F_g^{\rm WK}}{\partial z_k}\Biggr)\biggl(\frac{\partial y}{\partial T_1},\dots,\frac{\partial^{3g-2}y}{\partial T_1^{3g-2}}\biggr)\,.\] Together with Lemma 1, we arrive at \[\label{eqn:string-gBGW-Fg} \sum_{a\geq 0}\frac{2a+1}{2}\widetilde{T}_{2a+1}\frac{\partial F_g^{\rm KW}\bigl(\frac{\partial y}{\partial T_1},\dots,\frac{\partial^{3g-2}y}{\partial T_1^{3g-2}}\bigr)}{\partial T_{2a+1}} \; + \;\frac{\delta_{g,1}}{16} \; = \;0\,\quad g\geq 1\,.\tag{41}\] Hence \[\begin{align} L_0 \big(\tau(x,{\boldsymbol{T}};\hbar)\big)\; = \;0\,.\label{eqn:string-gBGW-jet} \end{align}\tag{42}\]
It is also known ([2], [6]) that \(Z(x, {\boldsymbol{T}}; \hbar)\) satisfies the following dilaton equation: \[\begin{align} \label{eqn:dilaton-gBGW} L_{\rm dilaton}\bigl(Z(x, {\boldsymbol{T}}; \hbar)\bigr) \; = \; 0\,, \end{align}\tag{43}\] where \[\begin{align} L_{\rm dilaton} \; = \; &\sum_{a\geq 0}\,\widetilde{T}_{2a+1}\,\frac{\partial}{\partial T_{2a+1}}\; + \;x\,\frac{\partial}{\partial x}\; + \;\hbar\,\frac{\partial}{\partial\hbar}\; + \;\frac{1}{24} \,. \end{align}\]
By equation 43 and 12 , 13 , \(\mathcal{F}_{0}(x,{\boldsymbol{T}})\) satisfies the following equation \[\label{eqn:dilaton-gBGW-F0} \sum_{a\geq 0}\widetilde{T}_{2a+1}\frac{\partial\mathcal{F}_{0}(x,{\boldsymbol{T}})}{\partial T_{2a+1}} \; + \;x\,\frac{\partial\mathcal{F}_{0}(x,{\boldsymbol{T}})}{\partial x}\; = \;2\, \mathcal{F}_{0}(x,{\boldsymbol{T}}).\tag{44}\] Like the above, using the commutation relation \[\Big[\frac{\partial}{\partial T_1},L_{\rm dilaton}\Big]\; = \;\frac{\partial}{\partial T_1}\,,\] we get \[\sum_{a\geq 0}\widetilde{T}_{2a+1}\frac{\partial}{\partial T_{2a+1}}\bigg(\frac{\partial^k y}{\partial T_1^k}\bigg) \; + \;x\frac{\partial}{\partial x}\bigg(\frac{\partial^k y}{\partial T_1^k}\bigg) \; = \;-k\,\frac{\partial^k y}{\partial T_1^k}\,,\quad k\geq 0\,.\] Together with Lemma 1, we obtain \[\begin{align} L_{\rm dilaton}\big(\tau(x,{\boldsymbol{T}};\hbar)\big)\; = \;0\,.\label{eqn:dilaton-gBGW-jet} \end{align}\tag{45}\]
Denote \[\label{defiwtf310} \log(\tau (x,{\boldsymbol{T}};\hbar)) \,=:\, \widetilde{\mathcal{F}}(x,{\boldsymbol{T}};\hbar) \,=:\, \sum_{g\geq 0}\hbar^{2g-2}\, \widetilde{\mathcal{F}}_g(x,{\boldsymbol{T}}) \, .\tag{46}\] By equation 39 and equation 42 , set \(T_{2a+1}=0\), \(a=1,2,\cdots\), we have \[\label{eqn:tildeF-pT} \hbar^2\frac{\partial\widetilde{\mathcal{F}}}{\partial T_1}(x,T_1, 0, 0, \dots;\hbar) \; = \;\hbar^2\frac{\partial{\mathcal{F}}}{\partial T_1}(x,T_1, 0, 0, \dots;\hbar) \; = \;\frac{1}{1-T_1} \biggl(\frac{\hbar^2}{8}+\frac{x^2}{4}\biggr)\,.\tag{47}\] Taking the derivative with respective to \(T_1\) in the above equation, we get \[U(x,T_1, 0, 0, \dots;\hbar) \; = \;U_{\rm gBGW}(x,T_1, 0, 0, \dots;\hbar) \; = \;\frac{1}{(1-T_1)^2} \biggl(\frac{\hbar^2}{8}+\frac{x^2}{4}\biggr)\,,\] where \(U_{\rm gBGW}(x,{\boldsymbol{T}};\hbar):=\hbar^2\frac{\partial^2 {\mathcal{F}}}{\partial T_1^2}(x,{\boldsymbol{T}};\hbar)\). So we have proved that these two power series, \(U(x,{\boldsymbol{T}};\hbar)\) and \(U_{\rm gBGW}(x,{\boldsymbol{T}};\hbar)\), both satisfy the KdV hierarchy and have the same initial value, thus by uniqueness of the solution to the KdV hierarchy, we have \[U(x,{\boldsymbol{T}};\hbar)=U_{\rm gBGW}(x,{\boldsymbol{T}};\hbar).\]
It follows that \(\mathcal{F}(x,{\boldsymbol{T}}; \epsilon)\) and \(\widetilde{\mathcal{F}}(x,{\boldsymbol{T}};\epsilon)\) could only differ by an affine function of \(T_1, T_2, \dots\) (the coefficients can depend on \(x\) and \(\epsilon\)). From 39 and 42 we know that this affine function could only be a constant with respect to \(T_1, T_2, \dots\), that is a function of \(x\), \(\epsilon\). Combining with the genus expansions 13 and 46 , we can now write \[\mathcal{F}(x,{\boldsymbol{T}}; \epsilon) - \widetilde{\mathcal{F}}(x,{\boldsymbol{T}};\epsilon) \, = : \, \sum_{g\geq0} \epsilon^{2g-2} \mathcal{K}_g(x) \,.\] We are left to show that the functions \(\mathcal{K}_g(x)\), \(g\ge0\), all vanish. Using 38 , 46 , 17 , 23 , 13 , 20 , we find that this is true for \(g=0,1\). Using equation 43 and equation 45 , we know that \(\mathcal{K}_g(x)\) for \(g\ge2\) must have the form \[\mathcal{K}_g(x) \; = \; \frac{c_g}{x^{2g-2}}\,, \quad g\ge2\,.\] Because of 26 and 38 , 46 , we know that for each \(g\ge2\) there exists a function \(K_g(z_0, z_1,\dots,z_{3g-2})\in\mathbb{C}[z_0^{\pm1},z_1^{\pm1}][z_2,\dots,z_{3g-2}]\) such that \[\label{kkequal} \mathcal{K}_g(x) \; = \; K_g\biggl(y(x,{\boldsymbol{T}}), \frac{\partial y(x, {\boldsymbol{T}})}{\partial T_1},\dots,\frac{\partial^{3g-2}y(x, {\boldsymbol{T}})}{\partial T_1^{3g-2}}\biggr) \,,\tag{48}\] where \(y(x,{\boldsymbol{T}})\) is given by 24 . Like in the first arXiv preprint version of [34], taking derivatives with respect to \(T_{2m+1}\), \(m\ge0\), on both sides of 48 , using 36 , and dividing both sides by \(y(x,{\boldsymbol{T}})^m\), we find that the right-hand side becomes a polynomial of \(m\), which vanishes identically in \(m\). Comparing the coefficients of powers of \(m\) from the highest degree to the lowest degree we obtain \[\label{vanishingfirstderiv} \frac{\partial K_g}{\partial z_k}\biggl(y(x,{\boldsymbol{T}}), \frac{\partial y(x, {\boldsymbol{T}})}{\partial T_1},\dots,\frac{\partial^{3g-2}y(x, {\boldsymbol{T}})}{\partial T_1^{3g-2}}\biggr) \; = \; 0 \,, \quad k=0,\dots,3g-2\,.\tag{49}\] This, by an elementary exercise, leads to the conclusion that for each \(g\ge2\) the Laurent polynomial \(K_g(z_0, z_1,\dots,z_{3g-2})\) must be a pure constant. Therefore, \(c_g=0\). The theorem is proved. ◻
We note that it is shown in [2] that Theorem 1 implies Kazarian–Norbury’s conjectural identity [43] for kappa class integrals.
After the first version of this paper appeared on arXiv, Chekhov kindly communicated to us the paper [49], where the so-called Born–Infeld (NBI) matrix model was considered which is very similar to the generalized BGW model. It is shown in [49] that the partition function of the NBI model can be identified with the Witten–Kontsevich tau-function by shifting times (by constants) in the power-series ring, therefore with the partition function of certain kappa class integrals (see e.g. [14] and the references therein). However, the corrected/normalized partition function of the generalized BGW model is a power series of times which cannot be obtained by shifting times by constants from the Witten–Kontsevich tau-function. A simple way to see this is that the Virasoro constraints for the NBI model start from \(L_{-1}\) while for the generalized BGW model they start from \(L_0\). Nevertheless, it seems to us that the tau-function given from the viewpoint of [15] for the KdV hierarchy can unify the two models (see the right part of (115) in [15]). We study the precise relation of these two models in subsequent publications.
Di Yang
School of Mathematical Sciences, University of Science and Technology of China,
Hefei 230026, P.R. China
diyang@ustc.edu.cn
Qingsheng Zhang
School of Mathematical Sciences, Peking University,
Beijing 100871, P.R. China
zqs@math.pku.edu.cn