# Effective gauge theories of superfluidity with topological order

## TL;DR

This paper develops a generalized topological $BF$ theory to describe superfluidity with topological order in 3+1 dimensions, incorporating higher-form symmetries, and applies it to dense QCD phases.

## Contribution

It introduces a non-square $K$ matrix in $BF$ theory to model superfluidity with topological order and analyzes its symmetry structure and anomalies.

## Key findings

- Generalized $BF$ theory describes massless Nambu-Goldstone bosons and anyonic statistics.
- Identifies topological order via higher-form symmetries and 't Hooft anomalies.
- Applies framework to dense QCD, revealing anyonic statistics without topological order.

## Abstract

We discuss the low-energy dynamics of superfluidity with topological order in $(3+1)$ spacetime dimensions. We generalize a topological $BF$ theory by introducing a non-square $K$ matrix, and this generalized $BF$ theory can describe massless Nambu-Goldstone bosons and anyonic statistics between vortices and quasiparticles. We discuss the general structure of discrete and continuous higher-form symmetries in this theory, which can be used to classify quantum phases. We describe how to identify the appearance of topological order in such systems and discuss its relation to a mixed 't Hooft anomaly between discrete higher-form symmetries. We apply this framework to the color-flavor locked phase of dense QCD, which shows anyonic particle-vortex statistics while no topological order appears. An explicit example of superfluidity with topological order is discussed.