A geometrical description of untwisted 3d Dijkgraaf-Witten TQFT with defects

TL;DR

This paper presents a geometric, explicit construction of 3d untwisted Dijkgraaf-Witten TQFT with defects, avoiding triangulation sums and diagrammatic calculus, using homotopy theory and categorical methods.

Contribution

It introduces a new geometric and homotopy-theoretic approach to constructing defect TQFTs, providing explicit computations and connections to quantum double models.

Findings

Constructs a symmetric monoidal functor from defect cobordisms to vector spaces.

Provides explicit descriptions of defect surfaces and cobordisms via groupoids and bundles.

Shows how the 2d part relates to Kitaev's quantum double model.

Abstract

We give a simple, geometric and explicit construction of 3d untwisted Dijkgraaf-Witten theory with defects of all codimensions. It is given as a symmetric monoidal functor from a defect cobordism category into the category of finite-dimensional complex vector spaces. The objects of this category are oriented stratified surfaces and its morphisms are equivalence classes of stratified cobordisms, both labelled with higher categorical data. This TQFT is constructed in terms of geometric quantities such as fundamental groupoids and bundles and requires neither state sums on triangulations nor diagrammatic calculi for higher categories. It is obtained from a functor that assigns to each defect surface a representation of a gauge groupoid and to each defect cobordism a fibrant span of groupoids and an intertwiner between the groupoid representations at its boundary. It is constructed by homotopy theoretic methods and allows for an explicit computation of examples. In particular, we show how the 2d part of this defect TQFT gives a simple description of defects of all codimensions in Kitaev's quantum double model.