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A geometrical description of untwisted Dijkgraaf-Witten TQFT with defects

João Faría Martins, Catherine Meusburger

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.

A geometrical description of untwisted Dijkgraaf-Witten TQFT with defects

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.

Paper Structure

This paper contains 32 sections, 48 theorems, 222 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Spans of groupoids and groupoid representations
  3. Groupoids and groupoid representations
  4. Fibrant spans of groupoids and spans of groupoid representations
  5. The quantum Quinn construction
  6. Stratifications and graded graphs
  7. Graded graphs
  8. Stratifications
  9. Homogeneous stratifications
  10. Boundary stratifications
  11. Oriented, framed and fine stratifications
  12. Bundles of local strata
  13. Maps between sets of local strata
  14. Graded graphs from stratifications
  15. Defect cobordism category for untwisted Dijkgraaf-Witten TQFT
  16. ...and 17 more sections

Key Result

Lemma 2.1

Let $p:\mathcal{G}\to\mathcal{B}$ and $p':\mathcal{G}'\to\mathcal{B}$ be fibrations in $\mathrm{Grpd}$. Then any functor $f:p\to_{\mathcal{B}} p'$ of groupoids over $\mathcal{B}$ that is an equivalence of groupoids is an equivalence in $\mathrm{Grpd}_{\mathcal{B}}$.

Figures (5)

  • Figure 1: Neighbourhoods of $k$-strata in a stratified $n$-manifold for $k<n$. The maps $\iota_k: S_k^{\partial X}\to S_{k+1}^X$ from \ref{['eq:embedbound']} are induced by the inclusion $\iota: \mathbb R^2\to \mathbb R^3$, $(x_1,x_2)\mapsto (x_1,x_2,0)$.
  • Figure 2: Thickening $M_{th}$ of a stratified 3-manifold $M$ with vertices $u,v,w$, edges $d,e,f$ and a plane $p$.
  • Figure 3: Labelling the edges of the thickening $X_{th}$ with elements of the groups $G_{L(p)}$, $G_{R(p)}$ and the $G_{L(p)}\times G_{R(p)}^{op}$-set $M_p$. The labels are related by the conditions: $m_v=g_d\rhd m_w\lhd h_d^{-1}$, $m_w=g_f\rhd m_u\lhd h_f^{-1}$, $m_v=g_e\rhd m_u\lhd h_f^{-1}$, $g_e=g_d\cdot g_f$, $h_e=h_d\cdot h_f$.
  • Figure 4: Action of a gauge transformation for green planes in the thickening of an edge $e$.
  • Figure 5: Separating curve $e$ on a surface $\Sigma$ with $\Sigma\setminus e=\Omega_1\amalg \Omega_2$ and a vertex $v$ on $e$.

Theorems & Definitions (127)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 117 more