Correlators in the Gaussian and chiral supereigenvalue models in the Neveu-Schwarz sector

We analyze the Gaussian and chiral supereigenvalue models in the Neveu-Schwarz sector. We show that their partition functions can be expressed as the infinite sums of the homogeneous operators acting on the elementary functions. In spite of the fact that the usual W-representations of these matrix models can not be provided here, we can still derive the compact expressions of the correlators in these two supereigenvalue models. Furthermore, the non-Gaussian (chiral) cases are also discussed.


Introduction
The supereigenvalue models have attracted considerable attention.They can be regarded as supersymmetric generalizations of matrix models [1]- [13].Many of the important features of matrix model, such as the Virasoro constraints, the loop equations, the genus expansion and the moment description have their supersymmetric counterparts in the supereigenvalue model.The W -representations can be used to realize partition functions of various matrix models, such as the Gaussian Hermitian and complex matrix models [14]- [17], the Kontsevich matrix model [18] and the generalized Brezin-Gross-Witten model [19].Namely, by acting on elementary functions with exponents of the given W -operators, we can give the corresponding partition functions of the matrix models.For the supereigenvalue model in the Ramond sector, its super Virasoro constraints and topological recursion have been well investigated [10,12].Recently it was proved that this supereigenvalue model can be obtained in terms of the W -representation [20] Z = ( 1 Corresponding author: zhaowz@cnu.edu.cn where N is even, z i are bosonic variables and θ i are Grassmann variables, t k and ξ k are bosonic and fermionic coupling constants, respectively, ∆ R (z, θ) is the Vandermonde-like determinant, and the W -operator is Due to the W -representation, the compact expression of correlators in the supereigenvalue model (1) can be derived.The final result shows that the correlators are determined by the certain coefficients in the power of W .
The supereigenvalue model in the Neveu-Schwarz sector is [8] Z = ( where N is even, and ξ k+ 1 2 are fermionic coupling constants. There are the super Virasoro constraints for the partition function (4) where The generators of these constraints obey the super Witt algebra , When the bosonic variables z i in ( 4) are integrated from 0 to +∞, it gives the chiral supereigenvalue model which obeys the super Virasoro constraints (7) with n ≥ 0 [5].
The compact expression of correlators in the supereigenvalue model in the Ramond sector have been presented.However, for the cases of the supereigenvalue model in the Neveu-Schwarz sector, it still remains to be seen whether there are the similar results.In this paper, we analyze the (non)-Gaussian supereigenvalue model in the Neveu-Schwarz sector and give the correlators in these matrix models.Moreover, the cases of the chiral supereigenvalue model will be also investigated.
This paper is organized as follows.In section 2, we focus on the Gaussian supereigenvalue model in the Neveu-Schwarz sector and show that it can not be obtained by acting on elementary functions with exponents of the W -operator.Moreover, we derive the compact expression of the correlators.In section 3, we consider the non-Gaussian supereigenvalue model in the Neveu-Schwarz sector.In section 4, we consider the chiral supereigenvalue model and present the compact expression of the correlators.The non-Gaussian chiral case is analyzed in section 5.
We end this paper with the conclusions in section 6.

Gaussian supereigenvalue model in the Neveu-Schwarz sector
The Gaussian Hermitian matrix model is one of the most studied and best understood matrix models.Its partition function can be generated by the W -representation [14].The correlators in this matrix model have been well discussed [21]- [26] and the compact expressions of the correlators have been presented [25,26].Unlike the Gaussian case, the non-Gaussian Hermitian matrix model can not generated by the W -representation.Although the correlators in the non-Gaussian case can be evaluated by the recursive formulas derived from the Virasoro constraints and the additional constraints, it is hard to give the compact expressions of correlators [27]- [31].
In this section, we focus on the Gaussian supereigenvalue model which is obtained by taking the shift t 2 → t 2 + 1 2 in the potential ( 6) where the normalization factor Λ G is given by Let us now take the change of variables given by where ǫ is an infinitesimal bosonic parameter.By requiring that the partition function is invariant under the infinitesimal transformations (12), it leads to the constraint where Similarly, by requiring the invariance of the partition function (10) under it gives another constraint where Combining ( 13) and ( 16), we have where D = D1 + D2 , W = W1 + W2 .
The partition function (10) can be formally expanded as where and the coefficients Cs are the correlators defined by Let V be the infinite dimensional vector space on C[[t 0 ]] generated by the basis Thus the partition function Z G can be seen as a vector in V , the operators D and W are differential operators on V .Under the natural gradation deg( In addition, the kernel of D denoted as Ker( D) = {v ∈ V | Dv = 0}, is one dimensional.It is not difficult to see that Ker( D + W ) is still one dimensional.Therefore, the partition function Z G is uniquely determined by the constraint (18).As a consequence, the correlators (21) can be totally derived from (18).
Let us collect the coefficients of t l 1 and t l 2 in (18) and set to zero, respectively, we obtain where Ñ = βN + 1 − β, and the recursive relations Then it is easy to obtain and By collecting the coefficients of t l 1 t 2 and t l 1 ξ 1 2 ξ 3 2 in (18) and setting to zero, respectively, it gives the recursive relations Substituting ( 25) into (27), we obtain Similarly, for the cases of the coefficients of t l 2 t 2 1 and t l 2 ξ 1 2 ξ 3 2 in ( 18), we may also obtain the corresponding recursive relations and derive the correlators from ( 29) We have derived some special correlators from the constraint (18).In order to achieve more results, let us further analyze the constraint (18).Since De − √ β N t 0 = 0, the constraint ( 18) can be rewritten as From (31), we have Thus the partition function ( 10) can be expressed as We denote an operator O on V the degree operator if Of = deg(f )f for any homogeneous function f ∈ V .Note that although W is a homogeneous operator with degree 2 in (33), D is not the degree operator.It causes the W -representation of the matrix model (33) to fail.In other words, the partition function (10) can not be obtained by acting on elementary functions with exponents of the operator W .In spite of this negative result, by evaluating the action of the homogeneous operator (− D−1 W ) k on the function e − √ β N t 0 , we can still derive the compact expressions of the correlators from the representation (33).Let us continue to discuss the correlators.
Since D−1 W is an operator with degree 2, we can see from (33) that Z (s) In order to give the general expressions of the correlators for the s even case, we need to write out the action The operator D−1 can be expressed as where D is the degree operator and the relations [D, are used to give (34).
Thus the operators in (33) now take the following form: where Using the fact that for the homogeneous function The operator W l 2r−1 Wl 2r−2 • • • Wl 2 W l 1 with degree 2k can be formally expanded as Substituting (39) into (38) gives the final expression Similarly, we have where and Combining the actions (40) and (41), we obtain that the coefficients of 2 )) •P where σ 1 denotes all the distinct permutations of (k 2 ) and its inverse number is denoted as τ (σ On the other hand, the coefficients of where From the equivalence of ( 42) and (44), we obtain the final expression for the correlators where 2 )) P Note that (46) is complicated.For the special cases of (46), we may give their explicit forms.
For clarity of calculation of the correlators, let us now give an example.When k = 2 in (43), we have Taking k = 2 in (36), we may give the values of all the polynomial terms of the right-hand side of (49).Then substituting (49) into (46) gives Substituting (50) into (45), we finally obtain the correlators where N is even, p ≥ 1, a k and ε l are nonzero bosonic and fermionic coupling constants, respectively.
There are the super Virasoro constraints for the partition function ( 52) where By applying the changes of integration variables ( ) for the partition function (52), we may derive the constraint where and The partition function ( 52) is now viewed as a vector in the space Ṽ generated by the basis where and the correlators C (a, ε) are defined by It should be noted that the correlators C (a, ε) with the fermionic coupling constants are written on the right side of in (60) for the convenience of the following discussions.
Let us further analyze the constraint (56).We observe that in (56 has degree 0, Ŵ2p+2 , Ŵk and Ŵl+ 1 2 are operators with degrees 2p + 2, k and l + 1 2 , respectively.Since Ker( D + Ŵ ) is no longer one dimensional on Ṽ , the partition function (52) can not be uniquely determined by the constraint (56).On the other hand, there are the additional constraints for the partition function (52) Substituting (64) into (56), we obtain where Ŵ = Ŵ2p+2 + Ŵ2p+ , and Since the operators D and D + Ŵ are invertible on − Ŵe − √ β N t 0 Z N G (a, ε), by the constraint (65), we have Hence the partition function (52) can be expressed as Similar to the representation for the non-Gaussian Hermitian matrix model presented in Ref.
[31], we see that Ŵ in (68) is not a homogeneous operator.Since Ŵ contains the noncommutative operators with degrees ranging from 1 2 to 2p + 2, it not only leads to the fact that the partition function (52) can not be obtained by acting on elementary functions with exponents of the operator Ŵ, but also makes the handling of the correlators quite difficult from (68).We can in principle derive the correlators step by step from (68).
For examples, we list the correlators When s = 3 2 , the correlators are the correlators When s = 2, the correlators are the correlators When s = 5 2 , the correlators are the correlators When s = 3, the correlators are the correlators 4 Chiral supereigenvalue model in the Neveu-Schwarz sector Let us consider the chiral supereigenvalue model in the Neveu-Schwarz sector where N is even, the normalization factor Λ C is given by The correlators in the chiral supereigenvalue model are defined by Note that (76) is not convergent when the bosonic variables z i are integrated from −∞ to +∞.It is the reason that we consider the chiral case (74), but not the supereigenvalue model ( 4) in the Neveu-Schwarz sector.
By the invariances of the partition function (74) under two pairs of the changes of integration variables ( where Some specific types of the correlators can be recursively derived from the constraint (77).
Taking the coefficients of t l+1 1 , t l 1 ξ 1 2 ξ 3 2 , t l 1 t 2 in (77) and setting to zero, respectively, we obtain and the recursive relations From ( 79) and (80), we can further derive the correlators We observe that the operator D in (77) is the same as in (18), and the operator W in (77) is a homogeneous operator with degree 1.Following the similar considerations in the Gaussian supereigenvalue model, we reach the final expressions for the partition function ( 74) and the correlators where in which the functions where α − j = M in{α − j, l}, j = 0, • • • , 3.
5 Non-Gaussian chiral supereigenvalue model in the Neveu-

Schwarz sector
The non-Gaussian chiral supereigenvalue model in the Neveu-Schwarz sector is where N is even, p ≥ 1, a k and ε l are nonzero bosonic and fermionic coupling constants, respectively.The There are the super Virasoro constraints for the partition function (87 where Applying the changes of integration variables (z ) for the partition function (87), we derive the constraint where In similarity with the case of non-Gaussian supereigenvalue model (52), the partition function (87) can not be uniquely determined by the constraint (92).We have to consider the additional constraints.It is noted that the partition function (87) also satisfies the relations (98)

Conclusions
The W -representations of matrix models may realize the partition functions by acting on elementary functions with exponents of the given W operators.They play an important role in the calculation of the correlators.For the supereigenvalue model in the Ramond sector, one has found its W -representation and derived the compact expression of correlators.In this paper, we have analyzed the Gaussian and chiral supereigenvalue models in the Neveu-Schwarz sector.
We have also considered the non-Gaussian (chiral) supereigenvalue models in the Neveu-Schwarz sector.Similar to the case of the non-Gaussian Hermitian matrix model, they can not be obtained in terms of the W -representations. Unlike the cases of previous Gaussian and chiral supereigenvalue models, it was noted that the operators Ŵ and W in the (68) and (96) are not the homogeneous operators, which are constituted by the noncommutative operators with degrees ranging from 1 2 to 2p + 2 and 2p + 1, respectively.This makes it quite difficult to derive the compact expressions of the correlators from (68) and (96).
It is noted that the function − W e − √ β N t 0 on the right hand side of (31) has degree 2. Hence the operators D and D + W are invertible on − W e − √ β N t 0 .

) 3
Non-Gaussian supereigenvalue model in the Neveu-Schwarz sector Let us consider the non-Gaussian supereigenvalue model in the Neveu-Schwarz sector with coefficients on C[[t 0 , a, ε]].Let us define the degrees of the coupling constants a k and ε l are 0. Since m ∈ N in the basis (59), the partition function (52) is graded from 0, 1 2 , 1, 3 2 , • • • , ∞.Thus we have the expansion
and (74) can be expressed as the infinite sums of the homogeneous operators (− D−1 W ) k and (− D−1 W ) k acting on the function e − √ β N t 0 , i.e., (33) and (82), respectively.We noted that D is not the degree operator.It is clear that the W -representations of these two matrix models fail.In spite of this negative result, the notable feature has emerged from the studies of the correlators.Since the actions of the homogeneous operators (− D−1 W ) k and (− D−1 W ) k on the function e − √ β N t 0 can be evaluated explicitly, we have derived the compact expressions of the correlators (45) and (83) from (