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We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of generalised orbifolds for any $n$-dimensional defect TQFT: Given a defect TQFT $mathcal{Z}$, one obtains a new TQFT $mathcal{Z}_{mathcal{A}}$ by decorating the Poincare duals of triangulated bordisms with certain algebraic data $mathcal{A}$ and then evaluating with $mathcal{Z}$. The orbifold datum $mathcal{A}$ is constrained by demanding invariance under $n$-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.
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We initiate a systematic study of 3-dimensional `defect topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.
We specialise the construction of orbifold graph TQFTs introduced in Carqueville et al., arXiv:2101.02482 to Reshetikhin-Turaev defect TQFTs. We explain that the modular fusion category ${mathcal{C}}_{mathcal{A}}$ constructed in Muleviv{c}ius-Runkel, arXiv:2002.00663 from an orbifold datum $mathcal{A}$ in a given modular fusion category $mathcal{C}$ is a special case of the Wilson line ribbon categories introduced as part of the general theory of orbifold graph TQFTs. Using this, we prove that the Reshetikhin-Turaev TQFT obtained from ${mathcal{C}}_{mathcal{A}}$ is equivalent to the orbifold of the TQFT for $mathcal{C}$ with respect to the orbifold datum $mathcal{A}$.
A generalised orbifold of a defect TQFT $mathcal{Z}$ is another TQFT $mathcal{Z}_{mathcal{A}}$ obtained by performing a state sum construction internal to $mathcal{Z}$. As an input it needs a so-called orbifold datum $mathcal{A}$ which is used to label stratifications coming from duals of triangulations and is subject to conditions encoding the invariance under Pachner moves. In this paper we extend the construction of generalised orbifolds of $3$-dimensional TQFTs to include line defects. The result is a TQFT acting on 3-bordisms with embedded ribbon graphs labelled by a ribbon category $mathcal{W}_{mathcal{A}}$ that we canonically associate to $mathcal{Z}$ and $mathcal{A}$. We also show that for special orbifold data, the internal state sum construction can be performed on more general skeletons than those dual to triangulations. This makes computations with $mathcal{Z}_{mathcal{A}}$ easier to handle in specific examples.
We prove the $r$-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer $r$: The 2-groupoid of 2-dimensional fully extended $r$-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced $textrm{Spin}_2^r$-action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the $r$-th power of their Serre automorphisms. For $r=1$ we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to $r=2$. To construct examples, we explicitly describe $textrm{Spin}_2^r$-homotopy fixed points in the equivariant completion of any symmetric monoidal 2-category. We also show that every object in a 2-category of Landau--Ginzburg models gives rise to fully extended spin TQFTs, and that half of these do not factor through the oriented bordism 2-category.
These notes offer an introduction to the functorial and algebraic description of 2-dimensional topological quantum field theories `with defects, assuming only superficial familiarity with closed TQFTs in terms of commutative Frobenius algebras. The generalisation of this relation is a construction of pivotal 2-categories from defect TQFTs. We review this construction in detail, flanked by a range of examples. Furthermore we explain how open/closed TQFTs are equivalent to Calabi-Yau categories and the Cardy condition, and how to extract such data from pivotal 2-categories.
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