Towards construction of superintegrable basis in matrix models
Batukhan Azheev, Nikita Tselousov
TL;DR
This work develops a systematic framework for constructing superintegrable polynomials in matrix/eigenvalue models by exploiting the $W_{1+\infty}$ symmetry underlying the Gaussian Hermitian model. It introduces two core assumptions—the Pierri box-adding/removing rules and a finite-spin Hamiltonian for SIBs—and demonstrates how these constraints, together with Virasoro/W-constraints, fix the SIBs level by level. The method is first illustrated in detail for the Gaussian Hermitian model and then extended to the cubic Kontsevich model, where the SIBs correspond to $Q$-Schur polynomials and the Hamiltonian has integer eigenvalues. The authors further show that Virasoro operators act by adding/removing boxes and discuss potential generalizations to other matrix models and the limitations of the approach.
Abstract
We develop methods for systematic construction of superintegrable polynomials in matrix/eigenvalue models. Our consideration is based on a tight connection of superintegrable property of Gaussian Hermitian model and $W_{1 + \infty}$ algebra in Fock representation. Motivated by this example, we propose a set of assumptions that may allow one to recover superintegrable polynomials. The main two assumptions are box adding/removing rule (Pierri rule) and existence of Hamiltonian for superintegrable polynomials. We detail our method in case of the Gaussian Hermitian model, and then apply it to the cubic Kontsevich model.