Exact correlators in the Gaussian Hermitian matrix model

TL;DR

This paper derives new $W_{1+ abla}$ and Virasoro constraints for the Gaussian Hermitian matrix model, leading to an effective formula for correlators and revealing underlying algebraic structures.

Contribution

It introduces $W_{1+ abla}$ constraints and a null 3-algebra structure, providing a novel method to compute correlators in the Gaussian Hermitian matrix model.

Findings

Derived $W_{1+ abla}$ constraints and $W_{1+ abla}$ $n$-algebra.

Established Virasoro constraints with null 3-algebra.

Presented a new effective formula for correlators.

Abstract

We present the $W_{1+\infty}$ constraints for the Gaussian Hermitian matrix model, where the constructed constraint operators yield the $W_{1+\infty}$ $n$-algebra. For the Virasoro constraints, we note that the constraint operators give the null 3-algebra. With the help of our Virasoro constraints, we derive a new effective formula for correlators in the Gaussian Hermitian matrix model.

Equations

$$Z_G$$

$$C_{i_1\cdots i_l}(N)=⟨tr{\varphi }^{i_1}\cdots tr{\varphi }^{i_l}⟩={\int }_{N\times N}d\varphi tr{\varphi }^{i_1}\cdots tr{\varphi }^{i_l}exp(-\frac{1}{2}tr{\varphi }^2).$$

$$Z_G=\sum _{s=0}^{\infty }{Z}_G^{(s)}=e^{{\hat{W}}_{-2}/2}{e}^{N{t}_0},$$

$$Z_G^{(s)}=e^{N{t}_0}\sum _{l=0}^{\infty }\sum _{\begin{array}{c}{i}_1+\cdots +i_l=s\\ i_1,\cdots ,i_l⩾1\end{array}}⟨tr{\varphi }^{i_1}\cdots tr{\varphi }^{i_l}⟩\frac{t_{i_1}\cdots t_{i_l}}{l!},$$

$${\hat{W}}_{-2}=\sum _{j_1,j_2=0}^{\infty }(j_1{j}_2{t}_{j_1}{t}_{j_2}\frac{\partial }{\partial t_{j_1+j_2-2}}+(j_1+j_2+2)t_{j_1+j_2+2}\frac{\partial }{\partial t_{j_1}}\frac{\partial }{\partial t_{j_2}}).$$

$${\hat{W}}_{-2}{Z}_G^{(s)}=(s+2)Z_G^{(s+2)}.$$

$$\hat{D}=\sum _{j=0}^{\infty }j{t}_j\frac{\partial }{\partial t_j},$$

$$\hat{D}{Z}_G^{(s)}=s{Z}_G^{(s)}.$$

$$[\hat{D},{\hat{W}}_{-2}]=2{\hat{W}}_{-2}.$$

$$\hat{D}{Z}_G={\hat{W}}_{-2}{Z}_G.$$

$$[\hat{D},\frac{\partial }{\partial t_2}]=-2\frac{\partial }{\partial t_2}.$$

$$\frac{\partial }{\partial t_2}{Z}_G=(\hat{D}+N^2)Z_G.$$

$$\frac{\partial }{\partial t_2}{Z}_G^{(s)}=(s-2+N^2)Z_G^{(s-2)}.$$

$$W_m^r=(-\frac{1}{2}{)}^{r-1}({\hat{W}}_{-2}{)}^m({\hat{W}}_{-2}-\hat{D}{)}^{r-1},m,r\in \mathbb{N},r⩾2,$$

$$W_m^r{Z}_G=0.$$

$$[W_{m_1}^{r_1},W_{m_2}^{r_2}]=(\sum _{k=0}^{r_1-1}{C}_{r_1-1}^k{m}_2^k-\sum _{k=0}^{r_2-1}{C}_{r_2-1}^k{m}_1^k)W_{m_1+m_2}^{r_1+r_2-1-k},$$

$$[W_{m_1}^{r_1},W_{m_2}^{r_2},\dots ,W_{m_n}^{r_n}]$$

$${ϵ}_{12\cdots (2n-1)}^{i_1{i}_2\cdots i_{2n-1}}[[A_{i_1},A_{i_2},\cdots ,A_{i_n}],A_{i_{n+1}},\cdots ,A_{i_{2n-1}}]=0.$$

$$[W_{m_1}^{n+1},W_{m_2}^{n+1},\dots ,W_{m_{2n}}^{n+1}]=\prod _{1⩽j<k⩽2n}(m_k-m_j)W_{m_1+\cdots +m_{2n}}^{n+1},$$

$$[W_{m_1}^{n+1},\dots ,W_{m_{2n+1}}^{n+1}]=0.$$

$${\tilde{W}}_m^r=(-\frac{1}{2}{)}^{r-1}(\frac{\partial }{\partial t_2}{)}^m(\hat{D}-\frac{\partial }{\partial t_2}+N^2{)}^{r-1},m,r\in \mathbb{N},r⩾2.$$

$${\tilde{W}}_m^r{Z}_G=0.$$

$$\frac{C_i}{(i-1)!!}=\text{coefficient of }{x}^i{\lambda }^N\text{ in }\frac{\lambda }{1-\lambda }\frac{1}{(1-\lambda )-(1+\lambda )x^2},$$

$$\frac{C_{(2k+1)(2m+1)}}{(2k+1)!!(2m+1)!!}=\text{coefficient of }{x}^{2k+1}{y}^{2m+1}{\lambda }^N\text{ in }\frac{\lambda }{(\lambda -1{)}^{3/2}}\frac{arctan(\frac{xy\sqrt{\lambda -1}}{\sqrt{\lambda -1+(\lambda +1)(x^2+y^2)}})}{\sqrt{\lambda -1+(\lambda +1)(x^2+y^2)}},$$

$$\frac{C_{(2k)(2m)}}{(2k-1)!!(2m-1)!!}=\text{coefficient of }{x}^{2k}{y}^{2m}{\lambda }^N\text{ in }$$

$$\frac{\lambda (\lambda +1)x^2{y}^2}{(1-\lambda )}(\lambda -1+(1+\lambda )(x^2+y^2){)}^{-1}(\lambda -1+(1+\lambda )(x^2+y^2)+(\lambda -1)x^2{y}^2{)}^{-1}$$

$$-\frac{\lambda (\lambda +1)xy}{(1-\lambda {)}^{3/2}}(\lambda -1+(\lambda +1)(x^2+y^2){)}^{-3/2}arctan(\frac{xy\sqrt{\lambda -1}}{\sqrt{\lambda -1+(\lambda +1)(x^2+y^2)}}).$$

$$C_{i_1{i}_2\cdots i_l}(N)\equiv {\mathcal{O}}_{\Lambda }=\sum _{R⊢|\Lambda |}\frac{1}{d_R}{\chi }_R\{t_n=\frac{1}{2}{\delta }_{n,2}\}\cdot D_R(N)\cdot {\psi }_R(\Lambda ),$$

$${\mathcal{O}}_{\sigma }=\sum _{R⊢m}{\phi }_R([2^m])\cdot D_R(N)\cdot {\psi }_R(\sigma ),$$

$${\tilde{W}}_l^2{Z}_G=0,l\in \mathbb{N}.$$

$$[{\tilde{W}}_{l_1}^2,{\tilde{W}}_{l_2}^2]=(l_2-l_1){\tilde{W}}_{l_1+l_2}^2,$$

$$[{\tilde{W}}_{l_1}^2,{\tilde{W}}_{l_2}^2,{\tilde{W}}_{l_3}^2]=0.$$

Full text

Keywords: Conformal and $W$ Symmetry, Matrix Models, $n$-algebra

1 Introduction

The various constraints for matrix models have attracted remarkable attention, such as Virasoro/$W$-constraints [1]-[5] and Ding-Iohara-Miki constraints [6, 7]. Due to the Bagger-Lambert-Gustavsson (BLG) theory of M2-branes [8, 9], $n$-algebra and its applications have aroused much interest [10]-[17]. In the context of matrix models, usually the Virasoro/W-constraint operators do not yield the closed $n$-algebra. Whether there exist such kind constraint operators leading to the closed $n$-algebra has recently been investigated for the (elliptic) Hermitian one-matrix models. By inserting the special multi-variable realizations of the $W_{1+\infty}$ algebra under the integral, it was found that the derived constraint operators for the Hermitian one-matrix model may yield the closed $W_{1+\infty}$ ($n$-)algebras [18]. For the case of the elliptic matrix model, one can obtain the constraint operators associating with the $q$-operators [19, 20]. The situation is different from that of the Hermitian one-matrix model, since the derived constraint operators do not yield the closed algebra. However, it was shown that the ($n$-)commutators of the constraint operators are compatible with the desired generalized $q$-$W_{\infty}$ ($n$-)algebras once we act on the partition function [20].

The partition functions of various matrix models can be obtained by acting on elementary functions with exponents of the given operators. For the Gaussian Hermitian matrix model, its partition function is generated by the operator $\hat{W}_{-2}$ [21]. This operator is also the constraint operator for the Hermitian one-matrix model which associates with the Lassalle operator and the potential of the $A_{N-1}$-Calogero model [18]. The correlators in the Gaussian Hermitian matrix model have been well investigated [22]-[28]. A compact formula for correlators has been given by finite sums over Young diagrams of a given size, which involve also the well known characters of symmetric group [27]. Moreover, the $2m$-fold Gaussian correlators of rank $r$ tensors have been given by $r$-linear combinations of dimensions with the Young diagrams of size $m$ [28]. In this letter, we reinvestigate the Gaussian Hermitian matrix model and present its Virasoro/$W$-constraints. We intend to further explore the properties of the constraints and derive a new formula for correlators in this matrix model.

2 $W_{1+\infty}$ constraints for the Gaussian Hermitian matrix model

Let us consider the Gaussian Hermitian matrix model

[TABLE]

where the coefficients $C_{i_{1}\cdots i_{l}}(N)$ are the so-called $l$-point correlators, which are given by the Gaussian integrals

[TABLE]

Due to the reflection symmetry of the action $tr\phi^{2}$, when $i_{1}+\cdots+i_{l}$ is odd, we have $C_{i_{1}\cdots i_{l}}(N)=0$.

The partition function of the Gaussian model (1) can also be expressed as [21]

[TABLE]

where

[TABLE]

and the operator $\hat{W}_{-2}$ is given by

[TABLE]

It indicates that the partition function (1) can indeed be generated by the operator $\hat{W}_{-2}$.

The action of the operator $\hat{W}_{-2}$ on $Z_{G}^{(s)}$ leads to increase the grading in the following sense:

[TABLE]

The operator preserving the grading is given by [21]

[TABLE]

which acting on $Z_{G}^{(s)}$ gives

[TABLE]

The commutation relation between $\hat{D}$ and $\hat{W}_{-2}$ is

[TABLE]

Note that the actions of $\hat{D}$ and $\hat{W}_{-2}$ on $Z_{G}$ give

[TABLE]

For the operators $\frac{\partial}{\partial t_{2}}$ and $\hat{D}$, there is the similar commutation relation as (9)

[TABLE]

The actions of $\frac{\partial}{\partial t_{2}}$ and $\hat{D}$ on $Z_{G}$ give

[TABLE]

By means of (11) and (12), it is easy to show that

[TABLE]

In contrast with the operator $\hat{W}_{-2}$, we see that the operator $\frac{\partial}{\partial t_{2}}$ decreases the grading in the sense (13).

Let us introduce the operators

[TABLE]

which obviously satisfy

[TABLE]

The remarkable property is that these constraint operators yield

[TABLE]

and $n$-algebra

[TABLE]

where $C_{r}^{k}=\frac{r(r-1)\cdots(r-k+1)}{k!}$, $\beta_{k}=\left\{\begin{aligned} &r_{i_{1}}-1,&k=1,\\ &\sum_{j=1}^{k}r_{i_{j}}-k-\sum_{i=1}^{k-1}\alpha_{i},&2\leqslant k\leqslant n-1,\\ \end{aligned}\right.$ and $\epsilon_{1\ 2\cdots n}^{i_{1}i_{2}\cdots i_{n}}$ is given by \epsilon_{j_{1}\cdots j_{p}}^{i_{1}\cdots i_{p}}=\det\left(\begin{array}[]{ccc}\delta_{j_{1}}^{i_{1}}&\cdots&\delta_{j_{p}}^{i_{1}}\\ \vdots&&\vdots\\ \delta_{j_{1}}^{i_{p}}&\cdots&\delta_{j_{p}}^{i_{p}}\end{array}\right).

It is noted that (16) and (17) completely match with the $W_{1+\infty}$ ($n$-)algebras presented in Ref.[18]. The $W_{1+\infty}$ $n$-algebra (17) with $n$ even is a generalized Lie algebra (or higher order Lie algebra), which satisfies the generalized Jacobi identity

[TABLE]

For the constraint operators $W_{m}^{r}$ (14) with fixed $r=n+1$, by taking the appropriate scaling transformations, it is not difficult to show that these operators constitute the subalgebras

[TABLE]

and

[TABLE]

For the $W_{1+\infty}$ constraints (15), it is noted that the constraint operators (14) contain the operators increasing and preserving the grading. Let us now introduce the following operators in terms of the operators decreasing and preserving the grading:

[TABLE]

Straightforward calculation shows that they also yield the $W_{1+\infty}$ algebra (16) and $n$-algebra (17). By carrying out the action of the operators (21) on the partition function of the Gaussian model, it gives another $W_{1+\infty}$ constraints

[TABLE]

3 Correlators in the Gaussian Hermitian matrix model

Let us first recall the correlators in the Gaussian Hermitian matrix model. Harer and Zagier presented a generating function for exact (all-genera) 1-point correlators in the Gaussian Hermitian matrix model [22, 23],

[TABLE]

where $i$ is even. By using Toda integrability of the model, Morozov and Shakirov derived the 2-point generalization of the Harer-Zagier 1-point function [26],

[TABLE]

However, it should be noted that it is difficult to give the higher correlators in this way. Recently Mironov and Morozov presented a compact formula for correlators by finite sums over Young diagrams of a given size [27],

[TABLE]

where $\Lambda=\{i_{1}\geqslant i_{2}\geqslant\cdots\geqslant i_{l}>0\}$ and $R$ are the Young diagrams of the given size $\sum_{k}i_{k}$, and $D_{R}(N)$, $\chi_{R}\{t\}$, $\psi_{R}(\Lambda)$ and $d_{R}$ are respectively the dimension of representation $R$ for the linear group $GL(N)$, the linear character (Schur polynomial), the symmetric group character and the dimension of representation $R$ of the symmetric group $S_{|R|}$ divided by $|R|!$. Furthermore, a representation of the correlators in terms of permutations is given by [28]

[TABLE]

where $\varphi_{R}([2^{m}])$ are the symmetric group characters.

Let us turn to consider the Virasoro constraints in (22)

[TABLE]

The constraint operators yield the Witt algebra

[TABLE]

and null $3$-algebra

[TABLE]

When $l\neq 0$, by using the expression (21) to calculate left-hand side of (27), we obtain

[TABLE]

From the fact that the constant term in the left-hand side of (3) should be zero, we have

[TABLE]

Taking the special constraint operator $\tilde{W}^{2}_{0}$ in (27), it is easy to obtain

[TABLE]

Thus from (31), we obtain

[TABLE]

By collecting the coefficients of $t_{i_{1}}t_{i_{2}}\cdots t_{i_{k}}$ in (3) and setting to zero, we have

[TABLE]

Let us take the constraint operator $W^{2}_{0}$ in (15), i.e.,

[TABLE]

After a straightforward calculation of the left-hand side of (35), we obtain

[TABLE]

By collecting the coefficients of $t_{1}^{2}$ and setting to zero, we obtain

[TABLE]

Similarly, for the case of the coefficients of $t_{1}^{l}$ with $l$ even, we have

[TABLE]

Substituting (37) into the recursive relation (38), we obtain

[TABLE]

Motivated by the exact $l$-point correlators $C_{\underbrace{1,\cdots,1}_{l}}(N)$ and $C_{\underbrace{2,\cdots,2}_{l}}(N)$, we now proceed to derive the general $l$-point correlators $C_{i_{1}\cdots i_{l}}(N)$. Let us consider the Virasoro constraints in (15)

[TABLE]

The constraint operators $W_{m}^{2}$ also yield the Witt algebra (28) and null $3$-algebra (29). By means of (9) and (10), we may rewrite (40) as

[TABLE]

Let us focus on the coefficients of $t_{i_{1}}t_{i_{2}}\cdots t_{i_{l}}$ with $\sum\limits_{j=1}^{l}i_{j}=2(m+1)$ on the both sides of (41). Note that the form of $(\hat{W}_{-2})^{m+1}$ appears to become more complicated very rapidly as one proceeds to higher power. We may formally express the $(m+1)$-th power of $\hat{W}_{-2}$ as

[TABLE]

where $\rho=\sum\limits_{n=1}^{k}j_{n}+2(m+1)$ and $P^{i_{1},i_{2},\cdots,i_{l}}_{j_{1},j_{2},\cdots,j_{k}}$ are polynomials in $i_{\alpha},\,\alpha=1,\cdots,l$ and $j_{\beta},\,\beta=1,\cdots,k$ .

When $j_{\beta}=0$ for $\beta=1,\cdots,k$ in (42), the corresponding terms acting on $Z_{G}$ give the coefficients of $t_{i_{1}}t_{i_{2}}\cdots t_{i_{l}}$ with $\sum\limits_{j=1}^{l}i_{j}=2(m+1)$ on the left-hand side of (41)

[TABLE]

where $\sigma$ denotes all distinct permutations of $(i_{1},i_{2},\cdots,i_{l})$. By means of (8), the right-hand side of (41) becomes

[TABLE]

From (3), we obtain that the coefficients of $t_{i_{1}}t_{i_{2}}\cdots t_{i_{l}}$ with $\sum\limits_{j=1}^{l}i_{j}=2(m+1)$ on the right-hand side of (41) are

[TABLE]

where we denote by $\lambda_{(i_{1}\cdots i_{l})}$ the number of distinct permutations of $(i_{1},i_{2},\cdots,i_{l})$.

By equating (43) and (45), we obtain the $l$-point correlators $C_{i_{1}\cdots i_{l}}(N)$

[TABLE]

where $\sum\limits_{j=1}^{l}i_{j}$ is even and $m=\frac{1}{2}\sum\limits_{j=1}^{l}i_{j}-1$.

When particularized to the 1-point correlators in (46), we have

[TABLE]

Comparing (46) with (25) and (26), we see that (46) is different from the other two expressions. Hence (46) is a new formula for correlators, where the operators $(\hat{W}_{-2})^{m+1}$ play an crucial role to determinate the polynomials in $i_{\alpha},\,\alpha=1,\cdots,l$ in the correlators.

For clarity of calculation, let us consider the $m=1$ case in (46), i.e., $\sum\limits_{j=1}^{l}i_{j}=4$. From the expression

[TABLE]

where $\tilde{t}_{j}=jt_{j}$, we have

[TABLE]

Substituting (3) into (46), we obtain

[TABLE]

where $\lambda_{(4)}=1$, $\lambda_{(1,3)}=2$, $\lambda_{(2,2)}=1$, $\lambda_{(1,1,2)}=3$ and $\lambda_{(1,1,1,1)}=1$.

4 Summary

It is known that the partition function of the Gaussian Hermitian matrix model can be obtained by acting on an elementary function with exponent of the operator $\hat{W}_{-2}$. This operator increases the grading in the sense (6). Based on the operators $\hat{W}_{-2}$ and $\hat{D}$ preserving the grading, we have constructed the $W_{1+\infty}$ constraints (15) for the Gaussian model, where the constraint operators yield not only the $W_{1+\infty}$ algebra, but also the closed $W_{1+\infty}$ $n$-algebra. In contrast with the operator $\hat{W}_{-2}$, we observed that the operator $\frac{\partial}{\partial t_{2}}$ decreases the grading in the sense (13). Another $W_{1+\infty}$ constraints (22) for the Gaussian model have been presented in terms of the operators $\frac{\partial}{\partial t_{2}}$ and $\hat{D}$, where the constraint operators also constitute the closed $W_{1+\infty}$ ($n$-)algebras. When particularized to the Virasoro constraints in (15) and (22), respectively, the corresponding constraint operators give the null 3-algebra.

Based on the Virasoro constraints (27), we have presented the exact correlators $C_{{2}\cdots{2}}(N)$ (33). However, it appears to be impossible to obtain arbitrary correlators from (27). With the help of another Virasoro constraints (40), we have derived a new formula (46) for correlators in the Gaussian Hermitian matrix model. Our results confirm that the constraint operators which lead to the higher algebraic structures provide new insight into the matrix models.

Acknowledgments

We would like to thank the referee for his/her helpful comments. This work is supported by the National Natural Science Foundation of China (Nos. 11875194, 11871350 and 11605096).