The "Null-A" superintegrability for monomial matrix models

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 1 vanishes. The role of the (anomaly-corrected) partition function is played by χρ -the Schur average of the suitably chosen square partiton ρ; such partitions are well-known to correspond to singular vectors of the Virasoro algebra. Further, non-zero are only Schur averages χµ for such µ that have ρ as their r-core, and superintegrability formula features the value of the skew Schur function χ µ/ρ at special point. The associated topological recursion and Harer-Zagier formula generalizations so far remain obscure.

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

“…Recently there has been increasing interest in the superintegrability for matrix models [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The superintegrability means that for the character expansions of the matrix models, the average of a properly chosen symmetric function is proportional to ratios of symmetric functions on a proper locus, i.e., < character >∼ character.…”