State sum models with defects based on spherical fusion categories. (English) Zbl 07716345
Summary: 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 labelled with bimodule categories over spherical fusion categories with bimodule traces. Line and point defects form directed graphs on these surfaces and labelled 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.
MSC:
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 18M20 | Fusion categories, modular tensor categories, modular functors |
| 57R56 | Topological quantum field theories (aspects of differential topology) |
| 81-XX | Quantum theory |
Keywords:
state sum models; defects; spherical categories; fusion categories; bimodule categories; diagrammatic calculusReferences:
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