Abstract
We study a real symmetric \(\Phi ^4\)-matrix model whose kinetic term is given by \(\textrm{Tr}( E \Phi ^2)\), where E is a positive diagonal matrix without degenerate eigenvalues. We show that the partition function of this matrix model corresponds to a zero-energy solution of a Schrödinger type equation with Calogero–Moser Hamiltonian. A family of differential equations satisfied by the partition function is also obtained from the Virasoro algebra.
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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
The main theorem of this paper is:
Theorem 1.1
Let \(Z(E, \eta )\) be the partition function defined by
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
is a zero-energy solution of the Schrödinger type equation
where \({{\mathcal {H}}}_{CM}\) is the Hamiltonian for the Calogero–Moser model:
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
The partition function is defined by
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
which is expressed as
Similarly, for \(p\ne s\), from
the following is obtained:
From (2.3) and (2.5), after taking sum over the indices t, p, s, we get the following:
By using
a partial differential equation is obtained:
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:
Here, \({{\mathcal {L}}}_{SD}^H \) is a second-order differential operator defined by
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:
where \(B(x)=x~ Id_N - H\). Using this P(x), the formula
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,
Due to the fact that
and
we finally get
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
where \(\prod _{1\le k<l\le N}dO_{lk}\) is the Haar measure on O(n). Then, the Laplacian is rewritten as
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
where
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
Here, we denote the Hamiltonian of the Calogero–Moser model by \({{\mathcal {H}}}_{CM}\):
Proof
By direct calculations, we obtain
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:
Then, we obtain
Using
and
finally we obtain
\(\square \)
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]:
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 \(\frac{1}{2}\):
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
As shown in Sect. 3, the Hamiltonian of Calogero–Moser model with \(\displaystyle \beta =\frac{1}{2}\) is given as
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:
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
with Dunkl derivatives [5, 12]
We can show it by direct calculations that the coordinates and the Dunkl derivatives satisfy the following commutation relations [3, 17]:
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:
This Hamiltonian and \(H_{C_{\beta =\frac{1}{2}}}\) are related as
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
as in the harmonic oscillator case. Next we define the representation of the Virasoro generators using Dunkl operators:
where \(\alpha , \lambda \) are arbitrary parameters. Or more generally, for any Laurent series \(\xi (a_i^\dagger ) \), we can define the Virasoro generators by
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:
where \(\xi _{1,2}(a_i^\dagger )\) is defined by
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 \(c=0\):
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
Let us calculate \([K_{pq}~, ~\sum _{i=1}^{N}(a_{i}^{\dagger })^{m}(a_{i})^{n} ]\). When \(p\ne q\),
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
From (4.2),
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:
More explicitly, using
\(\widetilde{L}_{-n} \) is expressed as
It is better to rewrite these operators using the original matrix model variables, \(E_i\) and \(\eta \). Let us introduce
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
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
From Theorem 2.2 and (4.30), finally we get the following theorem.
Theorem 4.1
The partition function defined by (2.2) satisfies
This means that \(\widetilde{L}_{-m}{Z}(E,\eta )\) is an eigenfunction of \(\mathcal {L}_{SD}\) with the eigenvalue \(-2m\).
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Notes
“Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics – Geometry – Structure.”
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Acknowledgements
A.S. was supported by JSPS KAKENHI Grant Number 21K03258. R.W. was supported by the Cluster of Excellence Mathematics Münster and the CRC 1442 Geometry: Deformations and Rigidity. This study was supported by Erwin Schrödinger International Institute for Mathematics and Physics (ESI) through the project Research in Teams Project “Integrability.”
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A Appendix
A Appendix
We give the proof for (2.10) in Appendix A. (The first half of this proof consists of well-known facts. For example, (A.5) can be seen in [16]. However, for the reader’s convenience, the derivation of equation (A.5) has not been omitted.)
Proof
For a real symmetric matrix \(X={}^T\!X=(x_{ij})\), \(\displaystyle \frac{\partial X}{\partial x_{ij}}=E_{ij}+E_{ji}-E_{ij}E_{ji}\delta _{ij}=E_{ij}+E_{ji}-E_{ij}E_{ij}\), where \(E_{ij}\) is standard matrix basis with 1 on ij position, i.e., \(E_{ij}=(\delta _{ki}\delta _{lj})\). Or, equivalently it is written as \(\frac{\partial x_{kl}}{\partial x_{ij}}=\delta _{ki}\delta _{jl}+\delta _{kj}\delta _{il}-\delta _{ki}\delta _{jl}\delta _{ij}\). Then,
since X is symmetric. Next we calculate
By partial differentiation of \(\textrm{Tr}\left( X^{-1}\left( \exp \left( \log X\right) \right) \right) =\textrm{Tr} (\textrm{Id}) \) with respect to \(x_{ij}\), we obtain
Substituting (A.1) into (A.4), we get
We define \(\widetilde{B}\) as the cofactor matrix of B. Applying (A.5) for \(P(x)=\det (B)\),
On the other hand from \(P(x)=\prod _{i=1}^{N}(x-E_{i})\),
is obtained. Setting \(x= E_t\),
From (A.9), finally we get the result we want:
\(\square \)
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Grosse, H., Kanomata, N., Sako, A. et al. Real symmetric \( \Phi ^4\)-matrix model as Calogero–Moser model. Lett Math Phys 114, 25 (2024). https://doi.org/10.1007/s11005-024-01772-5
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DOI: https://doi.org/10.1007/s11005-024-01772-5