Loop Models, Modular Invariance, and Three Dimensional Bosonization

TL;DR

This paper investigates 2+1D quantum loop models with gauge interactions, revealing how fractional spin breaks their statistical periodicity and connects to dualities in 2+1D field theories, clarifying their critical behavior.

Contribution

It demonstrates that including fractional spin in loop models breaks their statistical periodicity and aligns their dualities with recent 2+1D field theory dualities.

Findings

Fractional spin breaks statistical periodicity in loop models.

Loop model dualities mirror recent 2+1D field theory dualities.

Resolution of disagreement with conformal field theories at criticality.

Abstract

We consider a family of quantum loop models in 2+1 spacetime dimensions with marginally long-ranged and statistical interactions mediated by a U$(1)$ gauge field, both purely in 2+1 dimensions and on a surface in a 3+1 dimensional bulk system. In the absence of fractional spin, these theories have been shown to be self-dual under particle-vortex duality and shifts of the statistical angle of the loops by $2\pi$, which form a subgroup of the modular group, PSL$(2,\mathbb{Z})$. We show that careful consideration of fractional spin in these theories completely breaks their statistical periodicity and describe how this occurs, resolving a disagreement with the conformal field theories they appear to approach at criticality. We show explicitly that incorporation of fractional spin leads to loop model dualities which parallel the recent web of 2+1 dimensional field theory dualities, providing a nontrivial check on its validity.