3d field theory, plane partitions and triple Macdonald polynomials

TL;DR

This paper explores the connection between 3d field theories, plane partitions, and triple Macdonald polynomials, proposing a new integrable model linked to the DIM algebra and its representations.

Contribution

It introduces a novel relationship between DIM algebra representations, plane partitions, and triple Macdonald polynomials, suggesting a new integrable 3d field theory framework.

Findings

Proposes a generalization of Bethe equations related to DIM algebra.

Identifies eigenstates as new triple Macdonald polynomials.

Hints at the existence of an integrable 3d field theory.

Abstract

We argue that MacMahon representation of Ding-Iohara-Miki (DIM) algebra spanned by plane partitions is closely related to the Hilbert space of a 3d field theory. Using affine matrix model we propose a generalization of Bethe equations associated to DIM algebra with solutions also labelled by plane partitions. In a certain limit we identify the eigenstates of the Bethe system as new triple Macdonald polynomials depending on an infinite number of families of time variables. We interpret these results as first hints of the existence of an integrable 3d field theory, in which DIM algebra plays the same role as affine algebras in 2d WZNW models.