Decomposition in diverse dimensions

TL;DR

This work analyzes the relationship between gauge theories with trivially-acting center subgroups and theories with restricted nonperturbative sectors across $2$-D and $4$-D. In $2$-D, the decomposition conjecture extends to nonabelian theories with center-invariant matter, yielding a disjoint union of sectors labeled by discrete theta angles, exemplified by $SU(2)=SO(3)_+ + SO(3)_-$. In contrast, in $4$-D these notions generally differ, as demonstrated by dyons and instanton-restriction analyses and illustrated by Vafa–Witten TFT partition functions. The paper further discusses coupling to Dijkgraaf–Witten theories, showing that such couplings can modify or obstruct decomposition, and provides concrete checks via exact partition functions and dyon spectra. Overall, the results unify known 2D decompositions, clarify 4D distinctions, and highlight nonperturbative-sector restrictions as a tool to distinguish gauge theories, with implications for related topics like gerbes and DW theories.

Abstract

This paper discusses the relationships between gauge theories defined by gauge groups with finite trivially-acting centers, and theories with restrictions on nonperturbative sectors, in two and four dimensions. In two dimensions, these notions seem to coincide. Generalizing old results on orbifolds and abelian gauge theories, we propose a decomposition of two-dimensional nonabelian gauge theories with center-invariant matter into disjoint sums of theories with rotating discrete theta angles; for example, schematically, SU(2) = SO(3)_+ + SO(3)_-. We verify that decomposition directly in pure nonsupersymmetric two-dimensional Yang-Mills as well as in supersymmetric theories. In four dimensions, by contrast, these notions do not coincide. To clarify the relationship, we discuss theories obtained by restricting nonperturbative sectors. These theories violate cluster decomposition, but we illustrate how they may at least in special cases be understood as disjoint sums of well-behaved quantum field theories, and how dyon spectra can be used to distinguish, for example, an SO(3) theory with a restriction on instantons from an SU(2) theory. We also briefly discuss how coupling various analogues of Dijkgraaf-Witten theory, as part of a description of instanton restriction via coupling TQFT's to QFT's, may modify these results.