Foams and KZ-equations in Rozansky-Witten theories

TL;DR

This paper offers a geometric perspective on foams and Rozansky-Witten theories, connecting algebraic structures like KZ equations with the geometry of target spaces, and explores applications in high-dimensional quantum field theories.

Contribution

It introduces a geometric framework for foams and Rozansky-Witten theories, linking algebraic and geometric aspects, and formulates the KZ equation within this context.

Findings

Derived braiding and associator morphisms from the KZ equation

Connected target space geometry to Coulomb branches of 6d SCFTs

Highlighted applications to compactified Little String Theory

Abstract

In this paper, we present a geometric description of foams, which are prevalent in topological quantum field theories (TQFTs) based on quantum algebra, and reciprocally explore the geometry of Rozansky-Witten (RW) theory from an algebraic perspective. This approach illuminates various aspects of decorated TQFTs via geometry of the target space $X$ of RW theory. Through the formulation of the Knizhnik-Zamolodchikov (KZ) equation within this geometric framework, we derive the corresponding braiding and associator morphisms. We discuss applications where the target space of RW theory emerges as the Coulomb branch of a compactified 6d SCFT or Little String Theory, with the latter being particularly intriguing as it results in a compact $X$.