Topological Modes in Dual Lattice Models

TL;DR

This paper explores topological modes in dual lattice models, revealing their connection to cohomology groups and demonstrating their significance through numerical analysis in various models.

Contribution

It uncovers the presence of topological modes in dual lattice gauge theories and relates them to cohomology, extending understanding of dualities in lattice models.

Findings

Topological modes correspond to cohomology groups in dual lattice models.

Dual theories exhibit twisted actions influenced by topological sectors.

Numerical evidence confirms the importance of topological sectors in the 2D Ising model.

Abstract

Lattice gauge theory with gauge group $Z_{P}$ is reconsidered in four dimensions on a simplicial complex $K$. One finds that the dual theory, formulated on the dual block complex $\hat{K}$, contains topological modes which are in correspondence with the cohomology group $H^{2}(\hat{K},Z_{P})$, in addition to the usual dynamical link variables. This is a general phenomenon in all models with single plaquette based actions; the action of the dual theory becomes twisted with a field representing the above cohomology class. A similar observation is made about the dual version of the three dimensional Ising model. The importance of distinct topological sectors is confirmed numerically in the two dimensional Ising model where they are parameterized by $H^{1}(\hat{K},Z_{2})$.