W -representations for multi-character partition functions and their β -deformations

In this letter we continue the development of W -representations. We propose several generalizations of the known models, such as the hypergeometric Hurwitz τ -functions. We construct W -representations for multi-character expansions, which involve a generic number of sets of time variables. We propose integral representations for such kind of partition functions which are given by tensor models and multi-matrix models with multi-trace couplings. We further propose the β -deformation of the discussed W -representation for the Hurwitz case for two sets of times as well as for the multi-character case


Introduction
Recently a lot of progress has been made in understanding the connection between W -representations for τ -functions and superintegrability of matrix models.In particular two main progressions have been made.
Firstly, a rather comprehensive construction of explicit W -operators, that generate a very large class of τ -functions has been obtained.The main idea of this construction is the use of iterated commutator in the W ∞ algebra to construct families of commutative operators W (m) n which are related to p n and ∂ ∂pn by an algebra automorphism.Second is the matrix model realization of the discussed τ -functions.The two matrix model and it's multimatrix generalization realise the character expansions generated by the W operators.In particular, the coefficients of the character expansions are given by the matrix averages of characters and, hence, the property of superintegraibility of these matrix models is crucial.
Here we would like to propose a few generalizations of the relation between W -operators, character expansions and matrix models.First is the generalization to multicharacter expansions.Such expansions would involve an arbitrary number of sets of time variables in contrast to the two sets of Toda τ -functions.It turns out that the generalisation is straightforward on the level of W -operators.We propose the question of finding integral representations for such partition functions and outline the first steps in this direction.Such multicharacter expansions naturally appear in tensor models.The simplest example is provided by the gaussian tensor model of [1].Hence, in our discussion we will provide a W -representation, for this tensor model, which was given only implicitly in [1].Another option, that we will explore are the multimatrix models with multitrace interaction vertices.Using ideas from [2] we provide an explicit multi-matrix realisation for a rather general multi-character expansion.
The other generalization is the so-called β-deformation.It has already been noticed in [3][4][5][6], that W -operators can be β-deformed.The main ingredient is the deformed operator W 0 , which is nothing but the Calogero-Sutherland hamiltonian: As we will demonstrate, the deformation on the side of W -operators is once-again straightforward.Combined with the results of [2] this hint at the existence of the corresponding deformation of the multimatrix models and their multicharacter generalization.
The paper is structured as follows.In section 2 we discuss the multicharacter expansions and their integral representations.In section 3 we disucss the β-deformation.

Multi-character expansion
Let us briefly remind the construction of W -operators from [2].As a starting point take: and Then, for each n one can construct a family of commuting operators: These operators are related to the time variables via an algebra automorphism, given by the operator Ô(u) defined by it's action on Schur functions [7] Ô then The W -representations in [2] where constructed as a counterpart to the Cauchy identity: where: Now, one can make use of the generalized Cauchy formula, which involves multiple sets of time variables p with the coefficient is given by where Clearly, these coefficients are nonzero only if the sizes |R i | of all the partitions are equal.Now, due to commutativity of the W -operators (4) we can freely substitute those operators instead some of the sets of time variables in the generalized Cauchy formula.Suppose we have r = r 1 + r 2 sets of times.Then, let us denote Finally, we construct the following multicharacter partition function, with the use of the Wrepresentation: S Rs {p (s) }, (11) where u = { u (1) Similarly we can construct the analog of the skew τ -functions of [2], which correspond to the positive branch of W -operators. Once again, take r 1 + r 2 sets of times p (i) k and r 3 sets of times g where g = {g (1) , • • • , g (r 3 ) }.
We would like to look for integral representations for these two types of partition functions.

Tensor models
A well known representative of the family (11) with r 1 = r, r 2 = 0 and n = 1, u , is the partition function of the Gaussian tensor model [1], given by: where a i =1 dM a 1 ,...,a r d M a 1 ,...,a r .It has been represented in a W -operator like form using the automorphism Ô: Here we used the expression for the avarages of generalized characters: with . . .
These generalized characters enjoy an analog of superintegrability, which was used in ( 14): This representation has its own drawbacks.First of all, in the tensor model itself, the generalized characters form an overcomplete basis of operators.It's limit towards the complex model requires a reduction of the set of time variables as explained in [1].Furthermore, it allows the introduction of only "half" of time variables, which correspond to r 2 = 0 in (11).
For completeness we provide two more examples of tensor model partition functions.These examples correspond to choose the locus in such way that only a single set of independent time variables is left.Our results on W -operators provide an explicit W -representation for these tensor models.
1 in (11), it reduces to the red tensor model [9,10] (3) where expansion: Note the different entrance of the couplings p, which is necessary to correctly reproduce the structure of the generalized Clebsh-Gordan coefficients.

β-deformation
In this section we would like to demonstrate that the β-deformation of all of the W -representations in section above are straightforward.This suggests to look for their integral representations as well [13].Here we focus on the operators and leave the subject of β-deformed integrals for the future.
Operators In order to proceed, we define the β-deformed diagonal operator: where u is an arbitrary parameter.Just as in the β = 1 case, it can be used to construct commutative families of operators: where The operators act on the corresponding deformation of characters -the Jack polynomials: where we introduced the β-deformed norm: with R ′ the conjugate partition of R.

Partition functions In terms of Ŵ(n)
−k ( u), one immediately construct the β-deformed partition functions of any type of those discussed in section 2. Beginning with the deformation of the Hurwitz τ -functions: Particular reduction of these expansions involves several cases well known in the literature such as: 27), it gives the supposed β-deformed of the complex (square) matrix model with a logarithmic insertion of [6,14]: It is obvious that when ν = 0, (28) is the β-deformed complex matrix model [15].
) in ( 27), it gives the β-deformed eigenvalue whose potential is the Gaussian potential with a linear term [6,14]: However these two examples are reductions are a very specific locus.Instead it would be interesting to construct a β-deformed analog of the multimatrix model [2,12], which was done in [13].Finishing with the negative branch partition functions, the multi-character expansions clearly also have a counterpart: