State sum models with defects based on spherical fusion categories

TL;DR

This paper introduces a new state sum model for 3-manifolds with defects, utilizing spherical fusion categories, and demonstrates its invariance and sensitivity to defect surface topology and embedding.

Contribution

It defines a comprehensive state sum model incorporating surface, line, and point defects based on spherical fusion categories, with proofs of triangulation independence.

Findings

Model detects the genus of defect surfaces.

State sum is sensitive to defect surface embedding.

Defect lines define ribbon invariants for the center of the fusion category.

Abstract

We define a Turaev-Viro-Barrett-Westbury state sum model of triangulated 3-manifolds with surface, line and point defects. Surface defects are oriented embedded 2d PL submanifolds and are labeled with bimodule categories over spherical fusion categories with bimodule traces. Line and point defects form directed graphs on these surfaces and labeled with bimodule functors and bimodule natural transformations. The state sum is based on generalised 6j symbols that encode the coherence isomorphisms of the defect data. We prove the triangulation independence of the state sum and show that it can be computed in terms of polygon diagrams that satisfy the cutting and gluing identities for polygon presentations of oriented surfaces. By computing state sums with defect surfaces, we show that they detect the genus of a defect surface and are sensitive to its embedding. We show that defect lines on defect surfaces with trivial defect data define ribbon invariants for the centre of the underlying spherical fusion category.