Line and surface defects in Reshetikhin-Turaev TQFT

TL;DR

This paper extends Reshetikhin-Turaev TQFTs to include surface and line defects labeled by algebraic structures, enabling the distinction of certain non-isotopic embeddings of 2-tori.

Contribution

It introduces a new framework for TQFTs with embedded defects labeled by Frobenius algebras and multi-modules, expanding the scope of topological invariants.

Findings

Invariant distinguishes non-isotopic embeddings of 2-tori

Invariant cannot distinguish non-isotopic embeddings of 2-spheres

Framework incorporates surface and line defects with algebraic labels

Abstract

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $\Delta$-separable symmetric Frobenius algebras and the line defects by "multi-modules" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where it distinguishes non-isotopic embeddings of 2-tori.