The "Null-A" superintegrability for monomial matrix models

TL;DR

This paper demonstrates that superintegrability persists in a non-Gaussian monomial matrix model's exotic sector, revealing new algebraic structures and relations involving Schur functions and Virasoro algebra singular vectors.

Contribution

It uncovers the superintegrability property in the monomial matrix model's pure phase, connecting Schur averages to Virasoro singular vectors and extending the understanding of matrix model symmetries.

Findings

Superintegrability persists in the exotic sector of the monomial matrix model.

Schur averages are non-zero only for partitions with a specific r-core.

The superintegrability formula involves skew Schur functions at special points.

Abstract

We find that superintegrability (character expansion) property persists in the exotic sector of the monomial non-Gaussian matrix model, with potential $\Tr X^r$, in pure phase, where the naive partition function $\langle 1 \rangle$ vanishes. The role of the (anomaly-corrected) partition function is played by $\left\langle\chi_\rho\right\rangle$ -- the Schur average of the suitably chosen \textit{square} partiton $\rho$; such partitions are well-known to correspond to singular vectors of the Virasoro algebra. Further, non-zero are only Schur averages $\left\langle \chi_\mu\right\rangle$ for such $\mu$ that have $\rho$ as their $r$-core, and superintegrability formula features the value of the \textit{skew} Schur function $\chi_{\mu/\rho}$ at special point. The associated topological recursion and Harer-Zagier formula generalizations so far remain obscure.