Quantizing Derived Mapping Stacks

TL;DR

This paper reviews the process of quantizing low energy field substacks in sigma-models, using derived symplectic geometry and BV formalism, and explores implications for the AJ Conjecture in knot theory.

Contribution

It introduces a novel approach to quantizing specific substacks of mapping stacks, connecting topological invariants with derived geometric methods.

Findings

Quantization of low energy fields yields new topological invariants.

Framework connects gauge theory, derived geometry, and knot invariants.

Proposes a geometric perspective on the AJ Conjecture.

Abstract

In this review we discuss several topological and geometric invariants obtained by quantizing $\sigma$-models. More precisely, we don't quantize the entire mapping stack of fields, but rather only the substack of low energy fields. The theory restricted to this substack can be presented Lie theoretically and the problem is reduced to perturbative gauge theory. Throughout, we make extensive use of derived symplectic geometry and the BV formalism of Costello and Gwilliam. Finally, we frame the AJ Conjecture in knot theory as a question of quantizing character stacks.