Generalized $\beta$ and $(q,t)$-deformed partition functions with $W$-representations and Nekrasov partition functions

TL;DR

This paper develops generalized $eta$ and $(q,t)$-deformed partition functions using $W$-representations, revealing deep links with Nekrasov partition functions and exploring higher Hamiltonians for these polynomials.

Contribution

It introduces a novel framework connecting deformed partition functions with $W$-operators and Nekrasov functions, expanding understanding of their algebraic and geometric structures.

Findings

Established relations between deformed partition functions and Nekrasov functions.

Revealed connections between $W$-operators and vertex operators.

Analyzed higher Hamiltonians for generalized Jack and Macdonald polynomials.

Abstract

We construct the generalized $\beta$ and $(q,t)$-deformed partition functions through $W$ representations, where the expansions are respectively with respect to the generalized Jack and Macdonald polynomials labeled by $N$-tuple of Young diagrams. We find that there are the profound interrelations between our deformed partition functions and the $4d$ and $5d$ Nekrasov partition functions. Since the corresponding Nekrasov partition functions can be given by vertex operators, the remarkable connection between our $\beta$ and $(q,t)$-deformed $W$-operators and vertex operators is revealed in this paper. In addition, we investigate the higher Hamiltonians for the generalized Jack and Macdonald polynomials.