Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4

TL;DR

This paper discusses the application of formal derived geometry to classical field theories, especially supersymmetric gauge theories and sigma-models, providing a natural mathematical framework for their description.

Contribution

It introduces a mathematically natural construction of Kapustin-Witten's P^1 of twisted supersymmetric gauge theories using derived geometry.

Findings

Provides a clear geometric description of supersymmetric theories

Connects classical field theories with derived geometry frameworks

Offers a natural construction of the P^1 family of theories

Abstract

These notes explore some aspects of formal derived geometry related to classical field theory. One goal is to explain how many important classical field theories in physics -- such as supersymmetric gauge theories and supersymmetric sigma-models -- can be described very cleanly using derived geometry. In particular, I describe a mathematically natural construction of Kapustin-Witten's P^1 of twisted supersymmetric gauge theories.