Boundary theory of the X-cube model in the continuum

TL;DR

This paper investigates the boundary theory of the $ olinebreak\mathbb{Z}_N$ X-cube model in the continuum, revealing how boundary conditions influence ground state degeneracy and anomaly inflow, and showing the model's non-uniqueness in anomaly cancellation.

Contribution

It provides a continuum framework for the boundary theory of the X-cube model, analyzing boundary conditions, ground state degeneracies, and anomaly inflow, highlighting the model's non-uniqueness.

Findings

Boundary conditions affect ground state degeneracy on $T^2\times I$

Different boundary conditions lead to variations in extensive ground state degeneracy

The X-cube model is not unique in canceling boundary 't Hooft anomalies

Abstract

We study the boundary theory of the $\mathbb{Z}_N$ X-cube model using a continuum perspective, from which the exchange statistics of a subset of bulk excitations can be recovered. We discuss various gapped boundary conditions that either preserve or break the translation/rotation symmetries on the boundary, and further present the corresponding ground state degeneracies on $T^2\times I$. The low-energy physics is highly sensitive to the boundary conditions: even the extensive part of the ground state degeneracy can vary when different sets of boundary conditions are chosen on the two boundaries. We also examine the anomaly inflow of the boundary theory and find that the X-cube model is not the unique (3+1)d theory that cancels the 't Hooft anomaly of the boundary.