Vacuum Structures Revisited

TL;DR

This work shows that higher-form symmetries, particularly a $\mathbb{Z}_{p}$ $d$-form symmetry, control vacuum structure in SUSY gauge theories by enlarging the vacuum moduli to include the field strength $F_{0\cdots d}$ and analyzing the semi-classical potential $U(\phi, F_{0\cdots d})$. It demonstrates a dynamical decomposition: at an intermediate scale, vacua split into $p$ distinct universes labeled by discrete phases, with infinite barriers that prevent inter-universe domain walls, and it provides a UV completion in 4d where instanton sectors are restricted to topological numbers multiple of $p$. The analysis spans 2d linear sigma models, 3d Chern-Simons matter theories via KK reduction, and a proposed 4d UV theory for generalized SYM, linking the decomposition to both physical dynamics and mathematical structures such as Gromov-Witten theory and Picard-Fuchs systems. The results offer a coherent picture of how higher-form symmetries shape vacuum sectors and how UV completions realize the intermediate-scale decomposition, with potential extensions to non-Lagrangian theories and higher dimensions.

Abstract

We consider the relationship between the higher symmetry and the dynamical decomposition in supersymmetric gauge theory in various dimensions by studying the semi-classical potential energy. We observe that besides the scalar moduli we shall also include the field strength $F_{0\cdots d}$ in the vacuum moduli in the 1+d dimensional theory along with a $\mathbb{Z}_{p}$ $d$-form symmetry. In gauge theory for charge-$p$ matters with this symmetry, we find that the vacua decompose into $p$ different universes at an intermediate scale, which means no dynamical domain wall can interpolate between them. In our setup, we re-derive the existing results on the decomposition in various dimensions. In four dimensions, we propose a UV gauge theory for the generalized super Yang-Mills theory, whose instanton sectors are restricted to the topological number with integer multiples of $p$.