Supergeometry in locally covariant quantum field theory

TL;DR

This paper develops a categorical framework for supergeometric quantum field theories, incorporating supersymmetry transformations via enriched categories, and applies it to models like the superparticle and Wess-Zumino.

Contribution

It introduces enriched categories to include supersymmetry transformations in supergeometric quantum field theories, extending the categorical approach to supersymmetric models.

Findings

Constructed super-quantum field theories from geometric data.

Resolved the issue of small morphism sets by using enriched categories.

Analyzed specific models like the superparticle and Wess-Zumino model.

Abstract

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc --> S*Alg to the category of super-*-algebras which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc --> eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess-Zumino model in 3|2-dimensions.