Superintegrability in $β$-deformed Gaussian Hermitian matrix model from $W$-operators

Abstract

This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and W -representation. We apply it to prove the formula for the Jack functions averages -appropriate analogue of characters for the β-deformed Hermitian Gaussian matrix model. We also sketch the construction of W -operators from Calogero-Ruijsenaars Hamiltonians.

The "Null-A" superintegrability for monomial matrix models

“…Since Jack polynomials of square Young diagrams are also singular vectors of Virasoro algebra for β = 1, one can postulate the straightforward Jack generalization for formula (21), restore the corresponding Ward identities and, hopefully, recover the correct integral definition, with proper branch selections. This, then, would imply the correct form of the Ŵ -operator, and the proof of thus obtained superintegrability formulas would be possible along the lines of [34].…”

Bilinear character correlators in superintegrable theory

“…, one immediately obtains from this formula relation (25). Indeed, let us look, for instance, at the case of K [2] .…”

“…In the case of matrix models, even this language is still to be developed: our original definition of superintegrability in [1,2] (based on the phenomenon earlier observed in [3]- [12], see also some preliminary results in [26]- [30] and later progress in [13]- [25], [23,31]) implies the mapping between a big space X (functions of matrix eigenvalues or time-variables p k ) to a small one, Z (functions of the matrix size N ) with the Schur functions S R being a kind of eigenfunctions of this contraction map. Despite the setting looks very poor, the phenomenon clearly exists: a minor deformation of the Gaussian measure (which preserves integrability) is not compensated by a small deformation of the Schur functions so that the superintegrability property S R {p} ∼ S R {N } is preserved.…”

Superintegrability for ($β$-deformed) partition function hierarchies with $W$-representations

“…For the Gaussian tensor model [19] and (fermionic) rainbow tensor models [8,20], they can still be expressed as the W -representations. Recently it was shown that the superintegrability for (βdeformed) matrix models can be analyzed from their W -representations [15,16]. In this letter, we construct the partition function hierarchies with W -representations and analyze the superintegrability property.…”