Correlators in the supereigenvalue model in the Ramond sector

We investigate the supereigenvalue model in the Ramond sector. We prove that its partition function can be obtained by acting on elementary functions with exponents of the given operators. The Virasoro constraints for this supereigenvalue model are presented. The remarkable property of these bosonic constraint operators is that they obey the Witt algebra and null 3-algebra. The compact expression of correlators can be derived from these Virasoro constraints.


Introduction
Matrix models play important roles in physics and mathematics. Generally speaking they are quantum field theories where the field is an N ×N real or complex matrix. Supereigenvalue models can be regarded as supersymmetric generalizations of matrix models. They have attracted considerable attention [1]- [11]. The supereigenvalue model in the Ramond sector is given by [9] where d N zd N θ = N a=1 dz a dθ a , N is even, z a are positive real variables, θ a are Grassmann variables, ∆ R (z, θ) is the Vandermonde-like determinant, and ξ k are Grassmann coupling constants, V B (z) and V F (z) are the bosonic and fermionic potentials, respectively.
The various constraints for matrix models have been constructed, such as Virasoro constraints [12]- [15], W 1+∞ constraints [16,17] and Ding-Iohara-Miki constraints [18,19]. They are useful in analyzing the structures of matrix models. For the partition function (1), it is known that there are the super Virasoro constraints [9] L n Z = 1 16 δ n,0 Z, where The operators (5) and (6) obey the super Virasoro algebra Recently a formal supereigenvalue model in the Ramond sector is investigated [11] where and the bosonic variables z a are integrated from −∞ to +∞. To calculate the correlation functions of the model (8), the recursive formalism has been derived. It was found that the correlation functions obtained from the recursion formalism have no poles at the irregular ramification point due to a supersymmetric correction.
The partition functions of various matrix models can be obtained by acting on elementary functions with exponents of the given operators, such as Gaussian Hermitian and complex matrix models and the given W operators called W -representations [20]- [23]. For the case of supersymmetric generalizations, to our best knowledge, it has not been reported so far in the existing literature. In this letter, we investigate the supereigenvalue model in the Ramond sector and derive its W -representations. We also give the correlators in this matrix model.
2 Generation of the supereigenvalue model in the Ramond sector byŴ -operator Let us consider the supereigenvalue model in the Ramond sector which can be obtained by taking the shift t 1 → t 1 + 1 in the bosonic potential V B (z) of (1), the normalization factor Λ is given by We note that the partition function (10) is invariant under with an infinitesimal bosonic parameter ǫ. It leads to the bosonic loop equation where the expectation value is taken with respect to the partition function (10). The loop equation (13) can be derived by applying the following differential operators to the partition function (10) The partition function (10) is also invariant under which leads to another bosonic loop equation Similarly, (17) can be also obtained by applying the following differential operators to the partition function Combining (14) and (18), we have whereD =D 1 +D 2 ,Ŵ =Ŵ 1 +Ŵ 2 and their commutation relation is Since the partition function (10) only depends on even numbers of the fermionic variables, it can be formally expanded as wherē m is even and the coefficients C s 1 ,··· ,sm k 1 ,··· ,kn are the correlators defined by For the cases of m = 0 and n = 0 in (24), respectively, we denote and Due to the properties of the fermionic variables, we have and The operatorD acting onZ (s) giveŝ By means of (20), (21) and (29), we obtain The partition function (10) is graded by the total (t, ξ)-degree. From (29) and (30), we see that theD andŴ are indeed the operators preserving and increasing the grading, respectively.
In terms of the operatorŴ , (22) can be rewritten as It indicates that the supereigenvalue model in the Ramond sector can be obtained by acting on elementary functions with exponents of the given bosonic operatorsŴ .
When particularized to the m = 0 and n = 0 cases in (34), respectively, we have For examples, let us list some correlators.
(III) When l = 2 in (32), by direct calculations, it is easy to obtain the precise expression of the 3-th power ofŴ . Then we have the final results from (34) 3 Virasoro constraints for the supereigenvalue model in the Ra-

mond sector
It is known that the partition function (1) is invariant under two pairs of the changes of integration variables (z a → z a +ǫz n+1 a , θ a → θ a + 1 2 ǫnz n a θ a ) and (z a → z a +z n a √ z a θ a δ, θ a → θ a +z n a √ z a δ), where ǫ and δ are the infinitesimal bosonic and fermionic constants, respectively. These invariances, respectively, lead to the bosonic and fermionic loop equations which give the super Virasoro constraints (4). Taking the shift t 1 → t 1 + 1 in (4), we have the super Virasoro constraints for (10)L nZ = 1 16 δ n,0Z ,Ḡ nZ = 0, n ∈ N.
The super Virasoro algebra (7) still holds for the constraint operatorsL n andḠ n .
From the super Virasoro constraints (42), the recursive formulas for correlators can be obtained. In principle, we can calculate the correlators step by step from the recursive formulas.
However, the compact expression of correlators (34) can not be derived from them.
Let us introduce the bosonic operatorŝ These operators are different fromL n . They obey not only the Witt algebra (7a), but also the null Witt 3-algebra [24] [L l 1 ,L l 2 ,L l 3 ] : The action of the operators (43) on the partition function (10) leads to the Virasoro con-straintsL lZ = 0.
Recently similar Virasoro constraints without the Grassmann variables have been presented for the Gaussian Hermitian matrix model and they have been used to derive the correlators of the matrix model [25].
We have derived the special correlators from (20). It is known that the compact expression of correlators (34) can not be derived from the super Virasoro constraints (42). However, it should be pointed out that the special correlators (48), (52) and (55) can be still obtained from (42).
Let us consider the case of (45) with l = 0. By means of (20) and (21), (45) can be rewritten Substituting (32) into (58), by collecting the coefficients of t k 1 · · · t kn ξ s 1 · · · ξ sm with n µ=1 k µ + m ν=1 s ν = l + 1 and setting to zero, we may also derive the correlators (34). We have achieved the desired correlators from the Virasoro constraints (45). Unlike the operatorsL n in (42), the remarkable property of the constraint operators (43) is that these bosonic operators yield the higher algebraic structures. It should be noted that the closure of the super algebra does not hold for (43) and the fermionic operatorsḠ n in (42).

Summary
We have investigated the supereigenvalue model in the Ramond sector and proved that its partition function can be obtained by acting on elementary functions with exponents of thê W operators. In terms of the operatorsD andŴ preserving and increasing the grading, respectively, we have constructed the Virasoro constraints for this supereigenvalue model, where the constraint operators obey the Witt algebra and null 3-algebra. The compact expression of correlators (34) can be derived from these Virasoro constraints. It should be noted that this desired result can not be derived from the well known super Virasoro constraints (42). For the supereigenvalue model in the Neveu-Schwarz sector, whether its partition function can be expressed in terms of W -representation still deserves further study.
We have only constructed the Virasoro constraints for the supereigenvalue model (10). The remarkable property of these bosonic constraint operators is that they yield the higher algebraic structures. It is certainly worth to construct the super (Virasoro) constraints for supereigenvalue models, where the super higher algebraic structures hold for the bosonic and fermionic constraint operators. It would be interesting to study further properties of supereigenvalue models from these constraints.