On matrix models and their q-deformations

Abstract

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical β-deformed Hermitean matrix model and the q-deformed matrix models associated to 3d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories on D2×qS1 and $$ {S}_b^3 $$ S b 3 by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the W -algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results.

W-representation of Rainbow tensor model

“…We observe that (5.21) is similar with the case of non-Gaussian matrix model [30]. It shows that the dual expression for the non-Gaussian red tensor model (5.8) through differentiation can also be formulated.…”

“…The correlators in the matrix models including supereigenvalue models have attracted considerable attention. Much interest has also been attributed to the non-Gaussian cases [26]- [30]. But so far, no investigation has been made for the non-Gaussian tensor models.…”

Virasoro Constraints Revisited

“…In particular, it is well known that in the case of a Gaussian potential the constraints can be solved combinatorially through repeated application of cut-and-join operators to some specific initial data [4]. In the case of higher polynomial potentials this technique can be generalized but the solution one finds is no longer unique, and one finds a larger space of possible initial data [5,6].…”

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

“…Similar to the representation for the non-Gaussian Hermitian matrix model presented in ref. [40], we see thatŴ in (3.13) is not a homogeneous operator. SinceŴ contains the noncommutative operators with degrees ranging from 1 2 to 2p + 2, it not only leads to the fact that the partition function (3.1) can not be obtained by acting on elementary functions with exponents of the operatorŴ, but also makes the handling of the correlators quite difficult from (3.13).…”

“…For the non-Gaussian Hermitian matrix model, the correlators can be evaluated by the recursive formulas derived from the Virasoro constraints and the additional constraints. However, it is hard to give the compact expressions of the correlators [36][37][38][39][40].…”