A dualization approach to the Ground State Subspace Classification of Abelian Higher Gauge Symmetry Models

TL;DR

This paper introduces a dualization method to classify the ground state subspace of abelian higher gauge symmetry models, revealing a new cohomological and homological framework for understanding their ground states.

Contribution

It develops a differential-geometric approach to classify ground states using cohomology and homology groups, providing a novel mathematical framework for abelian higher gauge models.

Findings

Ground state space classified by H^0(C,G) × H_0(C,G) groups

Ground state degeneracy linked to cohomology and homology

New classification scheme for higher gauge symmetry models

Abstract

In the literature, abelian higher gauge symmetry models are shown to be valid in all finite dimensions and exhibit the characteristic behavior of SPT phases models. While the ground state degeneracy and the entanglement entropy were thoroughly studied, the classification of the ground state space still remained obscure. Based on differentio-geometric approach and, anticipating the notation of the current paper, if $\left( C_{\bullet} , \partial^C_{\bullet} \right)$ is the chain complex associated to the geometrical content of these models, while $\left( G_{\bullet} , \partial^G_{\bullet} \right)$ is its symmetries counterpart, we show that the ground state space is classified by a $H^0 (C,G) \times H_0 (C,G)$ group, where $H^0(C,G)$ is the $0$-th cohomology and $H_0 (C,G)$ is the corresponding $0$-th homology group with coefficients in the chain complex.