# Gauge PDE and AKSZ-type Sigma Models

## TL;DR

This paper develops a geometric framework for gauge PDEs using Q-bundles, embedding them into super-jet bundles, and relates them to AKSZ-type sigma models, providing a globally consistent formulation.

## Contribution

It introduces a supergeometric language for gauge PDEs via Q-bundles and demonstrates their embedding into super-jet bundles, extending the parent formulation to AKSZ-type sigma models.

## Key findings

- Gauge PDEs can be embedded into super-jet bundles.
- The framework provides a globally well-defined formulation.
- Reparameterization-invariant systems relate to AKSZ sigma models.

## Abstract

A gauge PDE is a natural notion which arises by abstracting what physicists call a local gauge field theory defined in terms of BV-BRST differential (not necessarily Lagrangian). We study supergeometry of gauge PDEs paying particular attention to globally well-defined definitions and equivalences of such objects. We demonstrate that a natural geometrical language to work with gauge PDEs is that of $Q$-bundles. In particular, we demonstrate that any gauge PDE can be embedded into a super-jet bundle of the $Q$-bundle. This gives a globally well-defined version of the so-called parent formulation. In the case of reparameterization-invariant systems, the parent formulation takes the form of an AKSZ-type sigma model with an infinite-dimensional target space.