Renormalization and BRST symmetry 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.

arXiv comment32 pages + appendices, 2 figures; v2: minor changes, v3: corrected numerical value of 1-loop amplitude

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• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• Rademacher expansion of modular integrals
• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings