Exact correlators in the Gaussian Hermitian matrix model

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.


Introduction
The various constraints for matrix models have attracted remarkable attention, such as Virasoro/Wconstraints [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 nalgebra has recently been investigated for the (elliptic) Hermitian one-matrix models. By inserting the special multi-variable realizations of the W 1+∞ algebra under the integral, it was found that the derived constraint operators for the Hermitian one-matrix model may yield the closed W 1+∞ (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 ∞ (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 operatorŴ −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.

W 1+∞ constraints for the Gaussian Hermitian matrix model
Let us consider the Gaussian Hermitian matrix model where the coefficients C i 1 ···i l (N ) are the so-called l-point correlators, which are given by the Gaussian integrals Due to the reflection symmetry of the action trφ 2 , when i 1 + · · · + i l is odd, we have C i 1 ···i l (N ) = 0.
The partition function of the Gaussian model (1) can also be expressed as [21] where Z (s) and the operatorŴ −2 is given bŷ It indicates that the partition function (1) can indeed be generated by the operatorŴ −2 .
The action of the operatorŴ −2 on Z (s) G leads to increase the grading in the following sense: The operator preserving the grading is given by [21] which acting on Z The commutation relation betweenD andŴ −2 is Note that the actions ofD andŴ −2 on Z G givê For the operators ∂ ∂t 2 andD, there is the similar commutation relation as (9) [D, The actions of ∂ ∂t 2 andD on Z G give By means of (11) and (12), it is easy to show that In contrast with the operatorŴ −2 , we see that the operator ∂ ∂t 2 decreases the grading in the sense (13).

Let us introduce the operators
which obviously satisfy The remarkable property is that these constraint operators yield and n-algebra and ǫ i 1 i 2 ···in 1 2···n is given by It is noted that (16) and (17) completely match with the W 1+∞ (n-)algebras presented in Ref. [18]. The W 1+∞ n-algebra (17) with n even is a generalized Lie algebra (or higher order Lie algebra), which satisfies the generalized Jacobi identity For the constraint operators W r m (14) with fixed r = n + 1, by taking the appropriate scaling transformations, it is not difficult to show that these operators constitute the subalgebras and For the W 1+∞ 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: Straightforward calculation shows that they also yield the W 1+∞ algebra (16) and n-algebra (17).
By carrying out the action of the operators (21) on the partition function of the Gaussian model, 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], 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], 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], where Λ = {i 1 i 2 · · · i l > 0} and R are the Young diagrams of the given size k i k , and D R (N ), χ R {t}, ψ R (Λ) 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] where ϕ R ([2 m ]) are the symmetric group characters.
Let us turn to consider the Virasoro constraints in (22) The constraint operators yield the Witt algebra and null 3-algebra When l = 0, by using the expression (21) to calculate left-hand side of (27), we obtain From the fact that the constant term in the left-hand side of (30) should be zero, we have Taking the special constraint operatorW 2 0 in (27), it is easy to obtain Thus from (31), we obtain By collecting the coefficients of t i 1 t i 2 · · · t i k in (30) and setting to zero, we have Let us take the constraint operator W 2 0 in (15), i.e., After a straightforward calculation of the left-hand side of (35), we obtain By collecting the coefficients of t 2 1 and setting to zero, we obtain Similarly, for the case of the coefficients of t l 1 with l even, we have Motivated by the exact l-point correlators C 1, · · · , 1 l (N ) and C 2, · · · , 2 l (N ), we now proceed to derive the general l-point correlators C i 1 ···i l (N ). Let us consider the Virasoro constraints in (15) The constraint operators W 2 m also yield the Witt algebra (28) and null 3-algebra (29). By means of (9) and (10), we may rewrite (40) as Let us focus on the coefficients of t i 1 t i 2 · · · t i l with l j=1 i j = 2(m + 1) on the both sides of (41).
When j β = 0 for β = 1, · · · , k in (42), the corresponding terms acting on Z G give the coefficients of t i 1 t i 2 · · · t i l with l j=1 i j = 2(m + 1) on the left-hand side of (41) where σ denotes all distinct permutations of (i 1 , i 2 , · · · , i l ). By means of (8), the right-hand side of (41) becomes From (44), we obtain that the coefficients of t i 1 t i 2 · · · t i l with l j=1 i j = 2(m + 1) on the right-hand side of (41) are where we denote by λ (i 1 ···i l ) the number of distinct permutations of (i 1 , i 2 , · · · , i l ).
By equating (43) and (45), we obtain the l-point correlators C i 1 ···i l (N ) where l j=1 i j is even and m = 1 2 l j=1 i j − 1.
When particularized to the 1-point correlators in (46), we have Comparing (46) with (25) and (26), we see that (46) is different from the other two expressions.

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 operatorŴ −2 . This operator increases the grading in the sense (6). Based on the operatorsŴ −2 andD preserving the grading, we have constructed the W 1+∞ constraints (15) for the Gaussian model, where the constraint operators yield not only the W 1+∞ algebra, but also the closed W 1+∞ n-algebra. In contrast with the operator W −2 , we observed that the operator ∂ ∂t 2 decreases the grading in the sense (13). Another W 1+∞ constraints (22) for the Gaussian model have been presented in terms of the operators ∂ ∂t 2 andD, where the constraint operators also constitute the closed W 1+∞ (n-)algebras. When particularized to the Virasoro constraints in (15) and (22), respectively, the corresponding constraint operators give the null 3-algebra.
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.