Koszul duality in quantum field theory

TL;DR

This paper introduces Koszul duality to a physics audience, illustrating its role in coupling quantum field theories to topological line defects through examples from supersymmetric theories.

Contribution

It provides an elementary introduction and new examples of Koszul duality in supersymmetric gauge theories with line defects, not previously documented in literature.

Findings

Koszul duality appears in supersymmetric theories with line defects

Examples from twists of simple supersymmetric theories are provided

The article suggests potential for generalizations to higher-dimensional defects

Abstract

In this article, we introduce basic aspects of the algebraic notion of Koszul duality for a physics audience. We then review its appearance in the physical problem of coupling QFTs to topological line defects, and illustrate the concept with some examples drawn from twists of various simple supersymmetric theories. Though much of the content of this article is well-known to experts, the presentation and examples have not, to our knowledge, appeared in the literature before. Our aim is to provide an elementary introduction for those interested in the appearance of Koszul duality in supersymmetric gauge theories with line defects and, ultimately, its generalizations to higher-dimensional defects and twisted holography.