Line and surface defects in Reshetikhin-Turaev TQFT

TL;DR

This work extends Reshetikhin-Turaev 3D TQFTs to incorporate surface and line defects labeled by $\Delta$-separable symmetric Frobenius algebras and cyclically symmetric multi-modules within a fixed modular tensor category $\mathcal{C}$. It constructs a defect TQFT $\mathcal{Z}^{\mathcal{C}}$ on a decorated 3D bordism category, built by encoding surface defects as algebra ribbons and line defects via module actions, then evaluating with the RT TQFT and taking an inverse limit to remove triangulation dependence. The authors clarify Morita-invariance subtleties and show how sphere embeddings remain invisible while certain genus-1 embeddings (tori) can be distinguished, providing explicit calculations (e.g., for $\widehat{\mathfrak{sl}}(2)_k$ at $k=16$) that connect to centers and Dynkin-type Morita classes. This framework paves the way for orbifold constructions in TQFT and potential applications in topological quantum computation, by enabling a robust algebraic encoding of defects in RT theory.

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 $Δ$-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.

Figures (7)

• Figure 1.1: Surface defects labelled by algebras $A_1,A_2,A_3 \in D_2^{\operatorname{\mathcal{C}}}$ meeting at a line defect labelled $M$. The arrangement on the left may be isotoped to the arrangement on the right by flipping the $A_1$-labelled surface clockwise around $M$ from the back to the front. Line defects only know the cyclic ordering of the surface defects adjacent to them; there is no total order.
• Figure 3.1: Local models for decorated stratified 3-manifolds are cylinders over the types of decorated stratified 2-manifolds shown. Here, $m \in \mathds{Z}_{\geqslant 0}$, $a,b,a_1,\dots, a_m \in D_3$, $f,f_1,\dots,f_m \in D_2$, and $x \in D_1$. For model (ii) we must have $a = s(f,+)=t(f,-)$ and $b=t(f,+)=s(f,-)$. For model (iii) the number of 2-strata can vary, as well as their orientations. The label of each 2-stratum has to have as source and target the labels on the two neighbouring 3-strata, e. g. $s(f_1,-)=a_m$. Finally, the junction map has to match the neighbouring 2-strata via $j(x,+) = ((f_1,\varepsilon_1),\dots,(f_m,\varepsilon_m))$. For the relation of the orientations in (iii) and of the strata in the corresponding cylinder, see CRS1.
• Figure 4.1: A ribbon diagram with $\phi \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X^\vee,V \otimes U^\vee \otimes W)$, embedded into a 3-ball with four marked points on the boundary.
• Figure 5.1: Rough outline of the construction of $\mathcal{Z}^{\mathcal{C}}(N)$. Only a patch of the defect bordism $N$ with one 1-stratum and one 2-stratum is shown.
• Figure 5.2: Two examples of orienting ribbons relative to $*$-decoration; the right picture involves the half-twist of \ref{['eq:halftwistAi']}.

Theorems & Definitions (36)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Definition 2.3
• Example 2.4
• Remark 2.5
• Definition 2.6
• Lemma 2.7
• proof