1 Introduction

It has recently been shown that the partition function of a certain Hermitian \(\Phi ^4\)-matrix model corresponds to a zero-energy solution of a Schrödinger equation for the Hamiltonian of N-body harmonic oscillator system [7]. This \(\Phi ^4\)-matrix model is obtained by changing the potential of the Kontsevich model [14] from \(\Phi ^3\) to \(\Phi ^4\).Footnote 1 The N-body harmonic oscillator system can be extended to the integrable Calogero–Moser model [4, 15]. It is thus natural to conjecture that there should be matrix models whose partition functions satisfy the Schrödinger equation for the Calogero–Moser model. It is precisely this which we demonstrate in this paper.

Let \(\Phi \) be a real symmetric \(N\times N\) matrix, E be a positive diagonal \(N\times N\) matrix \(E:= \textrm{diag} (E_1, E_2, \cdots ,E_N )\) without degenerate eigenvalues, and \(\eta \) be a positive real number, called coupling constant. We deal in this paper with the following symmetric one-matrix model defined by

$$\begin{aligned} S_E&= N~ \textrm{Tr} \{ E \Phi ^2 + \frac{\eta }{4} \Phi ^4 \} \nonumber \\&= N \left( \sum _{i,j}^N E_{i}\Phi _{ij}\Phi _{ji} + \frac{\eta }{4} \sum _{i,j,k,l}^N \Phi _{ij}\Phi _{jk}\Phi _{kl}\Phi _{li} \right) . \end{aligned}$$
(1.1)

The main theorem of this paper is:

Theorem 1.1

Let \(Z(E, \eta )\) be the partition function defined by

$$\begin{aligned} Z(E, \eta )= \int _{S_N} d\Phi ~e^{-S_E[\Phi ]}, \end{aligned}$$
(1.2)

where \(S_N\) is the space of real symmetric \(N\times N\)-matrices. Let \(\Delta (E)\) be the Vandermonde determinant \(\Delta (E):= \prod _{k<l} (E_l -E_k)\). Then, the function

$$\begin{aligned} \Psi (E, \eta ) := e^{-\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}}\Delta (E)^{\frac{1}{2}} {Z}(E,\eta ) \end{aligned}$$

is a zero-energy solution of the Schrödinger type equation

$$\begin{aligned} {{\mathcal {H}}}_{CM} \Psi (E, \eta ) = 0, \end{aligned}$$

where \({{\mathcal {H}}}_{CM}\) is the Hamiltonian for the Calogero–Moser model:

$$\begin{aligned} {\mathcal H}_{CM}:=\frac{-\eta }{2N}\left( \sum _{i=1}^{N}\frac{\partial ^{2}}{\partial E_{i}^{2}}+\frac{1}{4}\sum _{i\ne j}\frac{1}{(E_{i}-E_{j})^{2}}\right) +2\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}. \end{aligned}$$
(1.3)

Furthermore, since the Calogero–Moser model admits a Virasoro algebra representation, it gives rise to a family of differential equations satisfied by the partition function \({Z}(E,\eta )\). We will see this result in Theorem 4.1.

2 Schwinger–Dyson equation

Let \(\Phi \) be a real symmetric \(N\times N\) matrix. Let H be a real symmetric \(N\times N\) matrix with nondegenerate eigenvalues \(\{E_1, E_2, \cdots ,E_N ~ | ~ E_i \ne E_j ~\text{ for }~ i \ne j \}\). Let \(\eta \) be a real positive number. We consider the following action

$$\begin{aligned} S&= N~ \textrm{Tr} \{ H \Phi ^2 + \frac{\eta }{4} \Phi ^4 \} \nonumber \\&= N \left( \sum _{i,j,k}^N H_{ij}\Phi _{jk}\Phi _{ki} + \frac{\eta }{4} \sum _{i,j,k,l}^N \Phi _{ij}\Phi _{jk}\Phi _{kl}\Phi _{li} \right) . \end{aligned}$$
(2.1)

The partition function is defined by

$$\begin{aligned} Z(E, \eta ) := \int _{S_N} d \Phi ~e^{-S} , \end{aligned}$$
(2.2)

where \({d}\Phi =\prod _{i=1}^{N}d\Phi _{ii} \prod _{1\le i<j\le N}d\Phi _{ij}\) is the Lebesgue measure and \(S_N\) the space of real symmetric \(N\times N\) matrices. We denote expectation values with this action S by \(\langle O \rangle := \int _{S_N} d \Phi ~ O\; e^{-S} \). Note that we do not normalize it here, i.e. \(\langle 1 \rangle = Z(E, \eta ) \ne 1\). Note that the partition function \(Z(E, \eta ) \) depends only on the eigenvalues of H, because the integral measure is O(N) invariant. Indeed, \(Z(E, \eta ) \) is equal to the partition function (1.2) built from the action \(S_E\) in (1.1).

The following discussion in this section runs parallel to [7], so the calculations in [7] will also be helpful.

First, a Schwinger–Dyson equation is derived from

$$\begin{aligned} \int _{S_N} {d}\Phi \frac{\partial }{\partial \Phi _{tt}}\left( \Phi _{tt}e^{-S[\Phi ]}\right) =0, \end{aligned}$$

which is expressed as

$$\begin{aligned} {Z}(E,\eta )-2N\sum _{i=1}^{N}\left<H_{it}\Phi _{tt}\Phi _{ti}\right>-\eta N\sum _{k,l=1}^{N}\left<\Phi _{tk}\Phi _{kl}\Phi _{lt}\Phi _{tt}\right>=0. \end{aligned}$$
(2.3)

Similarly, for \(p\ne s\), from

$$\begin{aligned} \int _{S_N} {d}\Phi \frac{\partial }{\partial \Phi _{ps}}\left( \Phi _{ps}e^{-S[\Phi ]}\right) =0 , \end{aligned}$$
(2.4)

the following is obtained:

$$\begin{aligned} {Z}(E,\eta )-2N\sum _{i=1}^{N}\left( \left<H_{ip}\Phi _{ps}\Phi _{si}\right>+\left<H_{si}\Phi _{ip}\Phi _{ps}\right>\right) -2N\eta \sum _{k,l=1}^{N}\left<\Phi _{sk}\Phi _{kl}\Phi _{lp}\Phi _{ps}\right>=0. \end{aligned}$$
(2.5)

From (2.3) and (2.5), after taking sum over the indices t, p, s, we get the following:

$$\begin{aligned} \frac{N(N+1)}{2}{Z}(E,\eta )-2N\sum _{i,p,s=1}^{N}H_{ip} \left<\Phi _{is}\Phi _{sp}\right>-\eta N\sum _{k,l,s,p=1}^{N}\left<\Phi _{ps}\Phi _{sk}\Phi _{kl}\Phi _{lp}\right>=0. \end{aligned}$$
(2.6)

By using

$$\begin{aligned}&\frac{\partial {Z}(E,\eta )}{\partial H_{ps}}=-2N\sum _{k=1}^{N}\left<\Phi _{pk}\Phi _{ks}\right>\hspace{2mm}\text{ for }\hspace{2mm}\ p\ne s \\&\frac{\partial {Z}(E,\eta )}{\partial H_{pp}}=-N\sum _{k=1}^{N}\left<\Phi _{pk}\Phi _{kp}\right> \\&\frac{\partial ^{2}{Z}(E,\eta )}{\partial H_{ps}\partial H_{tu}}=4N^{2}\sum _{k,l=1}^{N}\left<\Phi _{pk}\Phi _{ks}\Phi _{tl}\Phi _{lu}\right> \text{ for }\hspace{2mm} p\ne s, t\ne u \\&\frac{\partial ^{2}{Z}(E,\eta )}{\partial H_{pp}\partial H_{pp}}=N^{2}\sum _{k,l=1}^{N}\left<\Phi _{pk}\Phi _{kp}\Phi _{pl}\Phi _{lp}\right>, \end{aligned}$$

a partial differential equation is obtained:

$$\begin{aligned}&\frac{N(N+1)}{2}{Z}(E,\eta )+ \sum _{ i\ne p} H_{ip} \frac{\partial }{\partial H_{ip}}{Z}(E,\eta )+2\sum _{p=1}^{N}H_{pp}\frac{\partial }{\partial H_{pp}}{Z}(E,\eta )\nonumber \\&-\frac{\eta }{N}\sum _{s=1}^{N}\frac{\partial ^{2}}{\partial H_{ss}\partial H_{ss}}{Z}(E,\eta )-\frac{\eta }{4N}\sum _{s\ne l} \frac{\partial ^{2}}{\partial H_{sl}\partial H_{ls}}{Z}(E,\eta )=0, \end{aligned}$$
(2.7)

where we denote \( \sum _{p=1}^{N}\sum _{i=1, i\ne p}^{N}\) by \(\sum _{i \ne p} \). We define \(H'_{ij}\) by \(H_{ii}=\sqrt{2}H^{'}_{ii} \) for \(i=1,\cdots ,N\) and \(H_{ij}=H^{'}_{ij}\) for \(i,j=1,\cdots ,N\hspace{2mm}(i\ne j)\), and we use an indices set \(U=\{(p,s)|\quad p\le s, p,s\in \{1,2,\cdots , N\}\}\), for convenience.

Proposition 2.1

The partition function \(Z(E, \eta )\) satisfies the following partial differential equation:

$$\begin{aligned} {{\mathcal {L}}}_{SD}^H Z(E, \eta ) = 0 . \end{aligned}$$
(2.8)

Here, \({{\mathcal {L}}}_{SD}^H \) is a second-order differential operator defined by

$$\begin{aligned} -{{\mathcal {L}}}_{SD}^H:=&\frac{N(N+1)}{2}+2\sum _{(p,s)\in U} H_{ps}\frac{\partial }{\partial H_{ps}} -\frac{\eta }{2N} \sum _{(p,s)\in U} \frac{\partial ^{2}}{\partial H^{'}_{ps}\partial H^{'}_{sp}} . \end{aligned}$$
(2.9)

Next we rewrite this Schwinger–Dyson equation in terms of the eigenvalues \(E_n (n= 1,2, \cdots , N)\) of H. References [11, 13] are helpful in the following calculations. Let P(x) be the characteristic polynomial:

$$\begin{aligned} P(x): = \det (x~ Id_N - H) =\det B=\prod _{i=1}^N (x-E_i), \end{aligned}$$

where \(B(x)=x~ Id_N - H\). Using this P(x), the formula

$$\begin{aligned} \frac{\partial E_{t}}{\partial H_{ij}}=&\frac{2(^T\!\widetilde{B}(E_{t}))_{ij}- (^T\!\widetilde{B}(E_{t}))_{ii}\delta _{ij}}{P'(E_{t})} \end{aligned}$$
(2.10)

for the derivative is obtained, where \(\widetilde{B}\) is the cofactor matrix of B. The proof of (2.10) is given in Appendix A.

At first, let us rewrite the second and the third terms of (2.7) by using (2.10). Since \(\widetilde{B}\) is a symmetric matrix,

$$\begin{aligned}&2\sum _{(p,s)\in U}^{N}H_{ps}\frac{\partial }{\partial H_{ps}}{Z}(E,\eta )=2\sum _{(p,s)\in U}^{N}\sum _{k=1}^{N}H_{ps}\frac{2(\widetilde{B}(E_{k}))_{ps}-(\widetilde{B}(E_{k}))_{pp}\delta _{ps}}{P'(E_{k})}\frac{\partial Z}{\partial E_{k}}\\&=2\sum _{p,k,s=1}^{N}H_{ps}\frac{\widetilde{B}(E_{k})_{ps}}{P'(E_{k})}\frac{\partial Z}{\partial E_{k}}\\&=-2\sum _{p,k,s=1}^{N}(E_{k}\delta _{ps}-H_{ps})\frac{\widetilde{B}(E_{k})_{ps}}{P'(E_{k})}\frac{\partial Z}{\partial E_{k}} +2\sum _{p,k,s=1}^{N}E_{k}\delta _{ps}\frac{\widetilde{B}(E_{k})_{ps}}{P'(E_{k})}\frac{\partial Z}{\partial E_{k}}. \end{aligned}$$

Due to the fact that

$$\begin{aligned} \sum _{s=1}^{N}(E_{k}\delta _{ps}-H_{ps}){\widetilde{B}(E_{k})_{ps}} = \det B(E_{k}) = P(E_k)=0 \end{aligned}$$
(2.11)

and

$$\begin{aligned} \sum _{p,s=1}^{N}\delta _{ps}{\widetilde{B}(E_{k})_{ps}} = \sum _{p=1}^{N}\det \begin{pmatrix} E_{k}-H_{11}&{}-H_{12}&{}\cdots &{}\cdots &{}-H_{1N}\\ \vdots &{}\ddots &{} &{} &{}\vdots \\ 0&{}\cdots &{}\delta _{pp}&{}\cdots &{}0\\ \vdots &{}&{} &{} \ddots &{}\vdots \\ -H_{N1}&{}-H_{N2}&{}\cdots &{}\cdots &{}E_{k}-H_{NN} \end{pmatrix} = P'(E_k), \end{aligned}$$
(2.12)

we finally get

$$\begin{aligned} 2\sum _{(p,s)\in U}^{N}H_{ps}\frac{\partial }{\partial H_{ps}}{Z}(E,\eta ) =&2\sum _{k=1}^{N}E_{k}\frac{\partial Z}{\partial E_{k}} . \end{aligned}$$
(2.13)

As a next step, we rewrite the Laplacian \(\sum _{(p,s)\in U} \frac{\partial ^{2}}{\partial H^{'}_{ps}\partial H^{'}_{sp}}{Z} \) in terms of \(E_p\). It is a well known fact (see e.g. [6, sec. 1.2]) that in terms of the Vandermonde determinant \(\Delta (E):= \prod _{k<l} (E_l -E_k)\), the Jacobian for the change of variables reads

$$\begin{aligned} \prod _{i=1}^{N}dH_{ii}\!\!\!\!\!\! \prod _{1\le i<j\le N} \!\!\!\!\!\! dH_{ij} =\Delta (E)\prod _{i=1}^{N}dE_{i} \!\!\!\!\!\! \prod _{1\le k<l\le N} \!\!\!\!\!\! dO_{lk} =(\sqrt{2})^{N}\prod _{i=1}^{N}dH'_{ii} \!\!\!\!\!\! \prod _{1\le i<j\le N} \!\!\!\!\!\! dH^{'}_{ij}, \end{aligned}$$
(2.14)

where \(\prod _{1\le k<l\le N}dO_{lk}\) is the Haar measure on O(n). Then, the Laplacian is rewritten as

$$\begin{aligned} \sum _{(p,s)\in U} \frac{\partial ^{2}}{\partial H^{'}_{ps}\partial H^{'}_{sp}}{Z}(E,\eta ) =&\frac{(\sqrt{2})^{N}}{\Delta (E)} \sum _{i=1}^{N}\frac{\partial }{\partial E_{i}}\left( \frac{\displaystyle \Delta (E)}{(\sqrt{2})^{N}}\frac{\partial }{\partial E_{i}}\right) {Z}(E,\eta )\nonumber \\ =&\sum _{l\ne i}^{N} \frac{1}{E_{i}-E_{l}}\frac{\partial }{\partial E_{i}}{Z}(E,\eta )+\sum _{i=1}^{N}\frac{\partial ^{2}}{\partial E_{i}^{2}}{Z}(E,\eta ) . \end{aligned}$$
(2.15)

From (2.13), (2.15) and Proposition 2.1, we obtain the following.

Theorem 2.2

The partition function defined by \({Z}(E,\eta ):=\int _{S_N} d \Phi \exp \left( -S[\Phi ]\right) \) satisfies the partial differential equation

$$\begin{aligned} {{\mathcal {L}}}_{SD} Z(E, \eta ) = 0 , \end{aligned}$$
(2.16)

where

$$\begin{aligned} {{\mathcal {L}}}_{SD} := \left\{ \frac{\eta }{2N}\sum _{i=1}^{N}\frac{\partial ^{2}}{\partial E_{i}^{2}} +\frac{\eta }{2N} \sum _{l\ne i}^{N} \frac{1}{E_{i}-E_{l}}\frac{\partial }{\partial E_{i}}-2\sum _{k=1}^{N}E_{k}\frac{\partial }{\partial E_{k}}-\frac{N(N+1)}{2} \right\} . \end{aligned}$$
(2.17)

3 Diagonalization of \({{\mathcal {L}}}_{SD}\)

In this section, we prove the main theorem (Theorem 1.1). The calculations in this section are performed in the similar manner as the calculations in [7]; we refer to [7] for further details.

As the first step, we prove the following proposition.

Proposition 3.1

The differential operator \({{\mathcal {L}}}_{SD} \) defined in (2.17) is transformed as

$$\begin{aligned}&e^{-\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}}\Delta (E)^{\frac{1}{2}} \mathcal {L}_{SD}\Delta (E)^{-\frac{1}{2}}e^{\frac{N}{\eta } \sum _{i=1}^{N}E_{i}^{2}} = - {{\mathcal {H}}}_{CM} . \end{aligned}$$
(3.1)

Here, we denote the Hamiltonian of the Calogero–Moser model by \({{\mathcal {H}}}_{CM}\):

$$\begin{aligned} {\mathcal H}_{CM}:=-\frac{\eta }{2N}\left( \sum _{i=1}^{N}\frac{\partial ^{2}}{\partial E_{i}^{2}}+\frac{1}{4}\sum _{i\ne j}\frac{1}{(E_{i}-E_{j})^{2}}\right) +2\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}. \end{aligned}$$
(3.2)

Proof

By direct calculations, we obtain

$$\begin{aligned}&\Delta (E)^{\frac{1}{2}}\left( \frac{\eta }{2N}\sum _{i=1}^{N}\frac{\partial ^{2}}{\partial E_{i}^{2}} +\frac{\eta }{2N} \sum _{l\ne i}^{N} \frac{1}{E_{i}-E_{l}}\frac{\partial }{\partial E_{i}}\right) \Delta (E)^{-\frac{1}{2}} \nonumber \\&=\frac{\eta }{2N}\sum _{i=1}^{N}\frac{\partial ^{2}}{\partial E_{i}^{2}} +\frac{\eta }{8N} \sum _{l\ne i}^{N} \frac{1}{(E_{i}-E_{l})^{2}}. \end{aligned}$$
(3.3)

Here, we used \( \sum _{i\ne l\ne k \ne i} \frac{1}{(E_i-E_l)(E_i-E_k)}=0\). Next we calculate the following:

$$\begin{aligned}&\Delta (E)^{\frac{1}{2}}\left( -2\sum _{k=1}^{N}E_{k}\frac{\partial }{\partial E_{k}}\right) \Delta (E)^{-\frac{1}{2}} = \sum _{l\ne k}^{N} \frac{E_{k}}{E_{k}-E_{l}}-2\sum _{k=1}^{N}E_{k}\frac{\partial }{\partial E_{k}}\nonumber \\&=\sum _{k>l}1-2\sum _{k=1}^{N}E_{k}\frac{\partial }{\partial E_{k}} =\frac{N(N-1)}{2}-2\sum _{k=1}^{N}E_{k}\frac{\partial }{\partial E_{k}} . \end{aligned}$$
(3.4)

Then, we obtain

$$\begin{aligned}&\Delta (E)^{\frac{1}{2}} {{\mathcal {L}}}_{SD} \Delta (E)^{-\frac{1}{2}} =&\frac{\eta }{2N}\Biggl \{\sum _{i=1}^{N}\frac{\partial ^{2}}{\partial E_{i}^{2}}+\frac{1}{4}\sum _{i\ne j}\frac{1}{(E_{i}-E_{j})^{2}}\Biggl \}-2\sum _{k=1}^{N}E_{k}\frac{\partial }{\partial E_{k}}-N. \end{aligned}$$
(3.5)

Using

$$\begin{aligned}&e^{-\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}}\left( \frac{\eta }{2N}\sum _{i=1}^{N}\left( \frac{\partial }{\partial E_{i}}\right) ^{2}\right) e^{\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}}\nonumber \\&=N+2\sum _{i=1}^{N}E_{i}\frac{\partial }{\partial E_{i}}+\frac{\eta }{2N}\sum _{i=1}^{N}\frac{\partial ^{2}}{\partial E_{i}^{2}}+\frac{2N}{\eta }\sum _{i=1}^{N}E_{i}^{2} \end{aligned}$$
(3.6)

and

$$\begin{aligned} e^{-\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}}\left( -2\sum _{k=1}^{N}E_{k}\frac{\partial }{\partial E_{k}}\right) e^{\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}}=&-4\frac{N}{\eta }\sum _{k=1}^{N}E_{k}^{2}-2\sum _{k=1}^{N}E_{k}\frac{\partial }{\partial E_{k}}, \end{aligned}$$
(3.7)

finally we obtain

$$\begin{aligned}&e^{-\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}}\Delta (E)^{\frac{1}{2}}\mathcal {L}_{SD}\Delta (E)^{-\frac{1}{2}}e^{\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}} = -\mathcal {H}_{CM} . \end{aligned}$$
(3.8)

\(\square \)

We introduce a function \(\Psi (E,\eta ):=e^{-\frac{N}{\eta }\sum _{i=1}^{N}E_{i}^{2}}\Delta (E)^{\frac{1}{2}}{Z}(E,\eta )\); then, we obtain \(\mathcal {H}_{CM}\Psi (E,\eta )=0\) from Proposition 3.1 and Theorem 2.2. Thus, Theorem 1.1 was proved.

The Hamiltonian of the Calogero–Moser model is defined as follows [4, 12]:

$$\begin{aligned} \displaystyle {H}_{C_{\beta }}:=\frac{1}{2}\sum _{j=1}^{N}\left( -\frac{\partial ^{2}}{\partial y_{j}^{2}}+y_{j}^{2}\right) +\sum _{j>k}\frac{\beta (\beta -1)}{(y_{j}-y_{k})^{2}}. \end{aligned}$$
(3.9)

After changing variable \(\sqrt{\frac{2N}{\eta }}E_{i}=y_{i}\), if \(\displaystyle \beta =\frac{1}{2}\), (1.3) is identified with (3.9) up to global factor \(\frac{1}{2}\):

$$\begin{aligned} \displaystyle H_{C_{\beta =\frac{1}{2}}}=\frac{1}{2}\sum _{j=1}^{N}\left( -\frac{\partial ^{2}}{\partial y_{j}^{2}}+y_{j}^{2}\right) -\frac{1}{4}\sum _{j>k}\frac{1}{(y_{j}-y_{k})^{2}}=\frac{1}{2}\mathcal {H}_{CM}. \end{aligned}$$
(3.10)

In the following, we consider only the case \(\displaystyle \beta =\frac{1}{2}\).

4 Virasoro algebra

Bergshoeff and Vasiliev proved in [1] that the Calogero–Moser model is associated with a Virasoro algebra structure. In this section, we discuss the Virasoro algebra representation in our \(\Phi ^4\) real symmetric matrix model.

As a start, a variable transformation is performed so that the Hamiltonian obtained in the previous section coincides with the Hamiltonian of the one in [1].

Using \(y_{i}= \sqrt{\frac{2N}{\eta }}E_{i}\), \({\mathcal L}_{SD}\) is expressed as

$$\begin{aligned} -\frac{1}{2}{{\mathcal {L}}}_{SD} =&\sum _{k=1}^{N}y_{k}\frac{\partial }{\partial y_{k}} -\frac{1}{2}\Biggl \{ \sum _{i=1}^{N}\frac{\partial ^{2}}{\partial y_{i}^{2}}+\frac{1}{2} \sum _{l\ne i}^{N}\frac{1}{y_{i}-y_{l}}\left( \frac{\partial }{\partial y_{i}}-\frac{\partial }{\partial y_{l}}\right) \Biggl \}+\frac{N(N+1)}{4}. \end{aligned}$$
(4.1)

As shown in Sect. 3, the Hamiltonian of Calogero–Moser model with \(\displaystyle \beta =\frac{1}{2}\) is given as

$$\begin{aligned} H_{C_{\beta =\frac{1}{2}}}=&g \left( -\frac{1}{2}{{\mathcal {L}}}_{SD} \right) g^{-1}. \end{aligned}$$
(4.2)

Here \(g=e^{-\frac{1}{2}\sum _i y_i^2 }\prod _{j>k}(y_{j}-y_{k})^{\frac{1}{2}}\).

4.1 Review of the Virasoro algebra symmetry representation for the Calogero–Moser model

In this subsection, we review several results of [1]. As [1, 12], we define the creation, annihilation operators \( a_i^\dagger , a_i\), and the coordinate swapping operator \(K_{ij} \quad (i,j = 1,...,N)\) obeying the following relations:

$$\begin{aligned}{}[a_i, a_j ]= & {} [a^{\dagger }_i,a^{\dagger }_j ]=0, \quad [a_i,a^{\dagger }_j ] = A_{ij}:= \delta _{ij }\left( 1+\beta \sum _{l=1}^N K_{il}\right) -\beta K_{ij},\nonumber \\ \end{aligned}$$
(4.3)
$$\begin{aligned} K_{ij}K_{jl}= & {} K_{jl}K_{il}=K_{il}K_{ij}, \quad \text{ for } \text{ all } i \ne j, i \ne l, j \ne l, \end{aligned}$$
(4.4)
$$\begin{aligned} (K_{ij})^2= & {} I,\qquad K_{ij}=K_{ji}, \end{aligned}$$
(4.5)
$$\begin{aligned} K_{ij}K_{mn}= & {} K_{mn}K_{ij}, \quad \text{ if } \text{ all } \text{ indices } i,j,m,n \text{ are } \text{ different }, \end{aligned}$$
(4.6)
$$\begin{aligned} K_{ij}a^{(\dagger )}_j= & {} a^{(\dagger )}_i K_{ij}. \end{aligned}$$
(4.7)

Here, we chose \(\displaystyle \beta =\frac{1}{2}\) for our case, while \(K_{ij}\) are the elementary permutation operators of the symmetric group \(\mathfrak {S}_N\). \(K_{ij}\) means the replacement of coordinates as \(K_{ij}y_i = y_j\) in the following discussions. We use the standard convention that square brackets \([ *, * ]\) denote commutators and curly brackets \(\{ *, * \}\) anticommutators.

To make contact with the Calogero–Moser model, we chose these operators as

$$\begin{aligned} a_i = \frac{1}{\sqrt{2}} (y_i + D_i), \quad a^\dagger _i = \frac{1}{\sqrt{2}} (y_i - D_i), \end{aligned}$$
(4.8)

with Dunkl derivatives [5, 12]

$$\begin{aligned} D_i =\frac{\partial }{\partial y_i}+\beta \sum _{j=1, j\ne i}^N (y_i -y_j)^{-1} (1-K_{ij})\,. \end{aligned}$$
(4.9)

We can show it by direct calculations that the coordinates and the Dunkl derivatives satisfy the following commutation relations [3, 17]:

$$\begin{aligned}{}[y_i, y_j] = [D_i, D_j] = 0, \hspace{8.5359pt}[D_i, y_j] = A_{ij}, \end{aligned}$$
(4.10)

and then, we find that the relations (4.3) are also satisfied by (4.8) [1].

Let us introduce the following Hamiltonian like a harmonic oscillator system:

$$\begin{aligned} H = {1\over 2} \sum _{i=1}^N \{a_i ,a^\dagger _i\} . \end{aligned}$$
(4.11)

This Hamiltonian and \(H_{C_{\beta =\frac{1}{2}}}\) are related as

$$\begin{aligned} \textrm{Res}(H)=&\prod _{j>k}(y_{j}-y_{k})^{-\frac{1}{2}}\cdot H_{C_{\beta =\frac{1}{2}}}\cdot \prod _{j>k}(y_{j}-y_{k})^{\frac{1}{2}}\nonumber \\ =&\frac{1}{2}\sum _{j=1}^{N}\left( -\frac{\partial ^{2}}{\partial y_{j}^{2}}+y_{j}^{2}\right) -\frac{1}{4}\sum _{j\ne k}\frac{1}{y_{j}-y_{k}}\left( \frac{\partial }{\partial y_{j}}-\frac{\partial }{\partial y_{k}}\right) , \end{aligned}$$
(4.12)

where \(\displaystyle \textrm{Res}(H)\) means that operator H acts on symmetric function space. It is possible to represent any differential operator D including \(K_{ij}\)’s as placing the elements of \(S_n\) at the right end, i.e., \(D= \sum _{\omega \in S_{N}}D_{\omega }\omega \). Using this expression, \(\displaystyle \textrm{Res}\) is defined as \(\displaystyle \textrm{Res}\left( \sum _{\omega \in S_{N}}D_{\omega }\omega \right) =\sum _{\omega \in S_{N}}D_{\omega }\). The Hamiltonian satisfies

$$\begin{aligned} {[} H ~, ~ a^\dagger _i ] = a^\dagger _i, \qquad [ H~, ~ a_i ] = -a_i \end{aligned}$$
(4.13)

as in the harmonic oscillator case. Next we define the representation of the Virasoro generators using Dunkl operators:

$$\begin{aligned} L_{-n} = \sum _{i=1}^N \left( \alpha (a_i^\dagger )^{n+1} a_i + (1-\alpha ) a_i (a_i^\dagger )^{n+1} +\left( \lambda -{1\over 2}\right) (n+1) (a_i^\dagger )^{n} \right) , \nonumber \\ \end{aligned}$$
(4.14)

where \(\alpha , \lambda \) are arbitrary parameters. Or more generally, for any Laurent series \(\xi (a_i^\dagger ) \), we can define the Virasoro generators by

$$\begin{aligned} L_{\xi } = \sum _{i=1}^N \left( \alpha \xi (a_i^\dagger ) a_i + (1-\alpha ) a_i \xi (a_i^\dagger ) +\left( \lambda -{1\over 2}\right) \frac{\partial }{\partial a_i^\dagger } \xi (a_i^\dagger ) \right) . \end{aligned}$$
(4.15)

For simplicity, we chose \(\displaystyle \lambda =\frac{1}{2}\) in this paper; however, this choice is not essential in the following discussion. When \(\xi _1\) and \(\xi _2\) are arbitrary Laurent series, \([L_{\xi _1}, L_{\xi _2}]\) is as follows:

$$\begin{aligned} {[}L_{\xi _1}, L_{\xi _2}] = \sum _{i=1}^N\left( \alpha \xi _{1,2}(a_i^\dagger )a_i + (1-\alpha ) a_i \xi _{1,2}(a_i^\dagger )\right) , \end{aligned}$$
(4.16)

where \(\xi _{1,2}(a_i^\dagger )\) is defined by

$$\begin{aligned} \xi _{1,2}(a_i^\dagger ) = \xi _1(a_i^\dagger ) {\partial \over \partial a_i^\dagger } \xi _2(a_i^\dagger ) -\xi _2(a_i^\dagger ) {\partial \over \partial a_i^\dagger } \xi _1(a_i^\dagger ). \end{aligned}$$
(4.17)

Especially if \(L_{-n} = \sum _{i=1}^N \left( \alpha (a_i^\dagger )^{n+1} a_i + (1-\alpha ) a_i (a_i^\dagger )^{n+1}\right) \), their commutators are given by the ones of the Virasoro algebra with its central charge \(c=0\):

$$\begin{aligned} \displaystyle [L_{n},L_{m}]=(n-m)L_{n+m}. \end{aligned}$$
(4.18)

4.2 Virasoro algebra representation for real symmetric \(\Phi ^4\)-matrix model

We shall attempt to adapt the Virasoro algebra reviewed in the previous subsection to the matrix model we are considering.

From \(\displaystyle H=L_{0}-\biggl (\frac{1}{2}-\alpha \biggl )N+\frac{1}{2}\biggl (\alpha -\frac{1}{2}\biggl )\sum _{i\ne j}K_{ij}\), the commutator \([H ~, ~L_{-m}]\) is obtained as

$$\begin{aligned} {[}H,L_{-m}]=&mL_{-m}\!+\!\left[ \frac{1}{2}\!\Biggl (\alpha \!-\!\frac{1}{2}\Biggl )\sum _{i\!\ne \! j}K_{ij}~\!,\! ~ \sum _{i=1}^{N}\Biggl (\alpha (a_{i}^{\dagger })^{m+1}a_{i}\!+\!(1\!-\!\alpha )a_{i}(a_{i}^{\dagger })^{m\!+\!1}\Biggl )\right] . \end{aligned}$$
(4.19)

Let us calculate \([K_{pq}~, ~\sum _{i=1}^{N}(a_{i}^{\dagger })^{m}(a_{i})^{n} ]\). When \(p\ne q\),

$$\begin{aligned}&\left[ K_{pq}~, ~\sum _{i=1}^{N}(a_{i}^{\dagger })^{m}(a_{i})^{n}\right] \\&=\sum _{i\ne q, i\ne p}\left( K_{pq}(a_{i}^{\dagger })^{m}(a_{i})^{n}-(a_{i}^{\dagger })^{m}(a_{i})^{n}K_{pq}\right) \nonumber \\&+\Biggl (K_{pq}(a_{p}^{\dagger })^{m}(a_{p})^{n}-(a_{p}^{\dagger })^{m}(a_{p})^{n}K_{pq} +K_{pq}(a_{q}^{\dagger })^{m}(a_{q})^{n}-(a_{q}^{\dagger })^{m}(a_{q})^{n}K_{pq}\Biggl )\nonumber \\&=\Biggl ((a_{q}^{\dagger })^{m}(a_{q})^{n}K_{pq}-(a_{p}^{\dagger })^{m}(a_{p})^{n}K_{pq}+(a_{p}^{\dagger })^{m}(a_{p})^{n}K_{pq}-(a_{q}^{\dagger })^{m}(a_{q})^{n}K_{pq}\Biggl ) =0. \nonumber \end{aligned}$$
(4.20)

When \(p=q\), \([K_{pp} ~, ~\sum _{i=1}^{N}(a_{i}^{\dagger })^{m}(a_{i})^{n}]=0\) is trivial. For any p, q, \( \Bigg [K_{pq}, \sum _{i=1}^{N} (a_{i})^{m}(a_{i}^{\dagger })^{n} \Bigg ]=0 \) is calculated similarly. From these results, (4.19) is simplified as

$$\begin{aligned} {[}H~,~L_{-m}] =&m L_{-m}. \end{aligned}$$
(4.21)

From (4.2),

$$\begin{aligned} -\frac{1}{2} \mathcal {L}_{SD} = e^{\frac{1}{2}\sum _j y_j^2} \textrm{Res} ( H ) e^{-\frac{1}{2}\sum _j y_j^2} . \end{aligned}$$
(4.22)

Note that the functions \(e^{-\frac{1}{2}\sum _j y_j^2} \), \(e^{\frac{1}{2}\sum _j y_j^2} \), and the partition function \(Z(E, \eta )\) are invariants under \(\mathfrak {S}_N\) action, i.e. \(K_{ij} Z(E, \eta ) = Z(E, \eta )\), and so on, so that we can ignore \(\textrm{Res} \) in the following calculations. Let us introduce \(\displaystyle \widetilde{L}_{-m}:= e^{\frac{1}{2}\sum _j y_j^2} L_{-m} e^{-\frac{1}{2}\sum _j y_j^2}. \) The following is automatically satisfied:

$$\begin{aligned} {[}\widetilde{L}_{n} ~ ,~ \widetilde{L}_{m}]=(n-m) \widetilde{L}_{n+m}. \end{aligned}$$
(4.23)

More explicitly, using

$$\begin{aligned}&e^{\frac{1}{2}\sum _j y_j^2} D_i e^{-\frac{1}{2}\sum _j y_j^2} = D_i -y_i , \end{aligned}$$
(4.24)
$$\begin{aligned}&\widetilde{a}_{i} :=e^{\frac{1}{2}\sum _j y_j^2} a_i e^{-\frac{1}{2}\sum _j y_j^2} = \frac{1}{\sqrt{2}} D_i \end{aligned}$$
(4.25)
$$\begin{aligned}&\widetilde{a}_{i}^{\dagger } :=e^{\frac{1}{2}\sum _j y_j^2} a_i^{\dagger } e^{-\frac{1}{2}\sum _j y_j^2} = \frac{1}{\sqrt{2}} (2y_i - D_i) , \end{aligned}$$
(4.26)

\(\widetilde{L}_{-n} \) is expressed as

$$\begin{aligned} \widetilde{L}_{-n}&= \sum _{i=1}^N \left( \alpha (\widetilde{a}_i^\dagger )^{n+1} \widetilde{a}_i + (1-\alpha ) \widetilde{a}_i (\widetilde{a}_i^\dagger )^{n+1} \right) \nonumber \\&= \frac{1}{2^{(n+2)/2}}\sum _{i=1}^{N}\Biggl \{\alpha \left( -D_{i}+2y_{i}\right) ^{n+1}D_{i}+(1-\alpha ) D_{i}\left( -D_{i}+2y_{i}\right) ^{n+1}\Biggl \}. \end{aligned}$$
(4.27)

It is better to rewrite these operators using the original matrix model variables, \(E_i\) and \(\eta \). Let us introduce

$$\begin{aligned} D_i^E := \frac{\partial }{\partial E_i} +\frac{1}{2} \sum _{j=1, j\ne i}^N \frac{1}{(E_i - E_j)} (1-K_{ij}) = \sqrt{ \frac{2N}{\eta }} D_i . \end{aligned}$$

Of course, this operator \(D_i^E\) satisfies \([ D_i^E, E_j] = A_{ij}\) and \([ D_i^E, D_j^E]=0\). Using this \(D_i^E\), the operators \(\widetilde{a}_{i}, \widetilde{a}_{i}^{\dagger }\) and \(\widetilde{L}_{-n}\) are written as

$$\begin{aligned} \widetilde{a}_{i} =&\frac{1}{2} \sqrt{\frac{\eta }{N}} D_i^E , \qquad \widetilde{a}_{i}^{\dagger } = 2 \sqrt{\frac{N}{\eta }} E_i - \frac{1}{2}\sqrt{\frac{\eta }{N}}D_i^E , \end{aligned}$$
(4.28)
$$\begin{aligned} \widetilde{L}_{-n} =&\sum _{i=1}^N \left\{ \alpha \left( 2 \sqrt{\frac{N}{\eta }} E_i - \frac{1}{2}\sqrt{\frac{\eta }{N}}D_i^E\right) ^{n+1} \frac{1}{2} \sqrt{\frac{\eta }{N}} D_i^E \right. \nonumber \\&+\left. (1-\alpha ) \frac{1}{2} \sqrt{\frac{\eta }{N}} D_i^E \left( 2 \sqrt{\frac{N}{\eta }} E_i - \frac{1}{2}\sqrt{\frac{\eta }{N}}D_i^E \right) ^{n+1} \right\} . \end{aligned}$$
(4.29)

Recall \(\mathcal {L}_{SD} = -2 e^{\frac{1}{2}\sum _j y_j^2} \textrm{Res}(H) e^{-\frac{1}{2}\sum _j y_j^2} \) and (4.20), then

$$\begin{aligned} \left[ \mathcal {L}_{SD} ~, ~\widetilde{L}_{-m}\right] =&-2 e^{\frac{1}{2}\sum _j y_j^2} [ \textrm{Res}(H) ~ , ~ L_{-m} ] e^{-\frac{1}{2}\sum _j y_j^2} \nonumber \\ =&-2 e^{\frac{1}{2}\sum _j y_j^2} [ L_0 ~ , ~ L_{-m} ] e^{-\frac{1}{2}\sum _j y_j^2} =-2 m\widetilde{L}_{-m} . \end{aligned}$$
(4.30)

From Theorem 2.2 and (4.30), finally we get the following theorem.

Theorem 4.1

The partition function defined by (2.2) satisfies

$$\begin{aligned} \mathcal {L}_{SD}(\widetilde{L}_{-m}{Z}(E,\eta ))=&-2m (\widetilde{L}_{-m}{Z}(E,\eta )). \end{aligned}$$
(4.31)

This means that \(\widetilde{L}_{-m}{Z}(E,\eta )\) is an eigenfunction of \(\mathcal {L}_{SD}\) with the eigenvalue \(-2m\).

Footnote 2