A note on the algebraic engineering of 4D $\mathcal{N}=2$ super Yang-Mills theories

TL;DR

This paper develops an algebraic framework for 4D $ ext{N}=2$ super Yang-Mills theories using a degenerate DIM algebra, connecting instanton partition functions, W-algebras, and integrable models.

Contribution

It introduces a degenerate DIM algebra approach to 4D $ ext{N}=2$ theories, constructing $ ext{T}$-operators and linking to W-algebras and integrable systems.

Findings

Constructed $ ext{T}$-operators reproducing instanton partition functions.

Identified degenerate Kimura-Pestun's quiver W-algebra as a limit of q-Virasoro.

Connected the algebraic structures to the sine-Gordon model's Faddeev algebra.

Abstract

Some BPS quantities of $\mathcal{N}=1$ 5D quiver gauge theories, like instanton partition functions or qq-characters, can be constructed as algebraic objects of the Ding-Iohara-Miki (DIM) algebra. This construction is applied here to $\mathcal{N}=2$ super Yang-Mills theories in four dimensions using a degenerate version of the DIM algebra. We build up the equivalent of horizontal and vertical representations, the first one being defined using vertex operators acting on a free boson's Fock space, while the second one is essentially equivalent to the action of Vasserot-Shiffmann's Spherical Hecke central algebra. Using intertwiners, the algebraic equivalent of the topological vertex, we construct a set of $\mathcal{T}$-operators acting on the tensor product of horizontal modules, and the vacuum expectation values of which reproduce the instanton partition functions of linear quivers. Analysing the action of the degenerate DIM algebra on the $\mathcal{T}$-operator in the case of a pure $U(m)$ gauge theory, we further identify the degenerate version of Kimura-Pestun's quiver W-algebra as a certain limit of q-Virasoro algebra. Remarkably, as previously noticed by Lukyanov, this particular limit reproduces the Zamolodchikov-Faddeev algebra of the sine-Gordon model.