Non-Abelian W-representation for GKM

TL;DR

This paper extends the concept of W-representations to generalized Kontsevich models, introducing a non-Abelian structure with ordered P-exponentials for non-commuting operators, enabling exact partition function formulations.

Contribution

It introduces a non-Abelian W-representation framework for GKM, incorporating ordered P-exponentials for non-commuting operators, expanding the applicability of W-representations.

Findings

Established non-Abelian W-representation for GKM

Demonstrated use of ordered P-exponentials for non-commuting operators

Extended known W-representations to more complex matrix models

Abstract

$W$-representation is a miraculous possibility to define a non-perturbative (exact) partition function as an exponential action of somehow integrated Ward identities on unity. It is well known for numerous eigenvalue matrix models when the relevant operators are of a kind of $W$-operators: for the Hermitian matrix model with the Virasoro constraints, it is a $W_3$-like operator, and so on. We extend this statement to the monomial generalized Kontsevich models (GKM), where the new feature is the appearance of an ordered P-exponential for the set of non-commuting operators of different gradings.