# Renormalization and BRST symmetry in Donaldson-Witten theory

## TL;DR

This paper proposes a new renormalization method for Donaldson-Witten theory that confirms BRST exact observables decouple, resolving previous issues caused by boundary contributions in the Coulomb branch.

## Contribution

It introduces a novel renormalization prescription linking Coulomb branch integrals to modular domain integrals, ensuring BRST exact observables decouple.

## Key findings

- BRST symmetry leads to vanishing correlation functions of BRST exact observables
- A new renormalization procedure addresses boundary contributions in Coulomb branch
- Decoupling of BRST exact observables is established in Donaldson-Witten theory

## Abstract

The presence of a BRST symmetry in topologically twisted gauge theories makes a precise analysis of these theories feasible. While the global BRST symmetry suggests that correlation functions of BRST exact observables vanish, this decoupling might be obstructed due to a contribution from the boundary of field space. Motivated by divergent BRST exact observables on the Coulomb branch of Donaldson-Witten theory, we put forward a new prescription for the renormalization of correlation functions on the Coulomb branch. This renormalization is based on the relation between Coulomb branch integrals and integrals over a modular fundamental domain, and establishes that BRST exact observables indeed decouple in Donaldson-Witten theory.