3-manifold invariants from cosets

TL;DR

This paper constructs unitary modular categories from coset conformal field theories and explores their associated 3-manifold invariants, revealing complex behaviors and finer invariants than simple product decompositions.

Contribution

It introduces a new framework for understanding 3-manifold invariants from coset theories using subfactor theory, including cases with fixed point resolutions.

Findings

Link invariants factorize under certain conditions

3-manifold invariants can be finer than product invariants

Framework provides new representation theory insights

Abstract

We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain index conditions the link invaraints colored by the representations of coset factorize into the products of the the link invaraints colored by the representations of the two groups in the coset. But the 3-manifold invariants do not behave so simply in general due to the nontrivial branching and selection rules of the coset. Examples in the parafermion cosets and diagonal cosets show that 3-manifold invariants of the coset may be finer than the products of the 3-manifold invariants associated with the two groups in the coset, and these two invariants do not seem to be simply related in some cases, for an example, in the cases when there are issues of ``fixed point resolutions''. In the later case our framework provides a representation theory understanding of the underlying unitary modular categories which has not been obtained by other methods.