Séminaire Lotharingien de Combinatoire, 93B.67 (2025), 12 pp.
Houcine Ben Dali, Valentin Bonzom and Maciej Dołęga
>q,t)-Tau Functions and Path Operators
Abstract.
Motivated by weighted Hurwitz theory and its connection to integrability, we introduce a (q,t)-tau function that deforms the classical case of hypergeometric tau functions using Macdonald polynomials, while simultaneously generalizing several important series that have already appeared in enumerative geometry, gauge theory, and integrability. We prove that this function is uniquely characterized by a family of differential equations and demonstrate a positive combinatorial expansion of these PDEs in terms of a new family of operators encoded by alternating paths. As a byproduct of our techniques, we establish a connection between path operators and the Delta conjecture.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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