Field theories with defects and the centre functor

This note is intended as an introduction to the functorial formulation of quantum field theories with defects. After some remarks about models in general dimension, we restrict ourselves to two dimensions - the lowest dimension in which interesting field theories with defects exist. We study in some detail the simplest example of such a model, namely a topological field theory with defects which we describe via lattice TFT. Finally, we give an application in algebra, where the defect TFT provides us with a functorial definition of the centre of an algebra. This involves changing the target category of commutative algebras into a bicategory. Throughout this paper, we emphasise the role of higher categories - in our case bicategories - in the description of field theories with defects.

💡 Research Summary

This paper presents a systematic, functorial treatment of quantum field theories (QFTs) that incorporate defects, focusing on the simplest non‑trivial setting: two‑dimensional topological field theories (TFTs). The authors begin by recalling the standard Atiyah‑Segal framework, where a QFT is a symmetric monoidal functor from a bordism category (objects are (n‑1)‑manifolds with collars, morphisms are n‑dimensional manifolds with parametrised boundaries) to the category of vector spaces. They argue that many interesting physical phenomena—such as domain walls, phase boundaries, and junctions—cannot be captured within this bare framework and require an enrichment of the source category.

Defect‑enriched bordism category.
For a fixed dimension n, the authors introduce a hierarchy of defect‑condition sets (D_k) for (k=0,\dots,n). Elements of (D_n) label bulk phases, (D_{n-1}) label domain walls, and lower‑dimensional (D_k) label higher‑codimension junctions. An object of the defect bordism category is an ((n-1))-manifold (U) together with a disjoint decomposition into submanifolds (U_k) of dimension k, each component of which carries a label from (D_{k+1}). A morphism (M:U\to V) is an n‑manifold equipped with a similar decomposition (M_k) and a boundary identification (\partial M\cong -U\sqcup V) respecting the defect labels. The orientation of the ambient manifolds induces a left/right convention for domain walls, encoded by source and target maps (s,t:D_{n-1}\to D_n). Junctions are governed by a cyclic compatibility condition (equation (2.2) in the paper) that specifies which ordered collections of walls may meet at a given point. This data naturally assembles into a 2‑category (bicategory): objects are bulk phases, 1‑morphisms are domain walls (with prescribed source/target phases), and 2‑morphisms are junctions (equivalence classes of cyclically ordered wall configurations).

A lattice model of 2‑dimensional defect TFT.
To make the abstract framework concrete, the authors construct a lattice TFT based on a triangulation (or more generally a cell decomposition) of a compact oriented surface. The algebraic input consists of:

• a finite‑dimensional semisimple algebra (A) assigned to each 2‑cell,
• an (A)‑(A) bimodule (M) assigned to each 1‑cell,
• a compatible family of linear maps attached to each 0‑cell (essentially the evaluation of the bimodule tensors). Defects are introduced by allowing different bimodules on selected edges or by inserting special 0‑cells labelled by elements of (D_0). The authors define a functor from the category of bordisms with cell decompositions to vector spaces by evaluating a state‑sum that multiplies structure constants of (A) and the actions of the bimodules. A crucial technical result (Theorem 3.8) shows that the resulting linear map is independent of the chosen cell decomposition, i.e. the construction truly defines a topological field theory with defects.

From defect TFT to a functorial centre.
The most innovative contribution is the reinterpretation of the algebraic centre (Z(A)) in categorical terms. Traditionally, (Z(A)={z\in A\mid za=az\ \forall a\in A}) is a set with a commutative algebra structure but lacks a natural functorial behaviour with respect to algebra homomorphisms. By viewing the defect TFT as a bicategory, the authors embed commutative algebras as objects, bimodules as 1‑morphisms, and intertwiners as 2‑morphisms. Within this bicategory, the centre becomes a universal object characterized by a universal property involving defect lines that can be freely moved across bulk regions. Two constructions are presented:

1. Version 1 (Sections 4.1–4.4).
   The bicategory (\mathcal{C}_1) has objects = commutative algebras, 1‑morphisms = algebra homomorphisms regarded as bimodules, and 2‑morphisms = bimodule maps. The defect TFT yields a lax functor (Z_1:\mathcal{C}_1\to\mathcal{C}_1) sending an algebra (A) to its centre (Z(A)) and a homomorphism (f:A\to B) to the induced map (Z(A)\to Z(B)) obtained by “pull‑through” of defect lines. Theorem 4.12 proves that (Z_1) satisfies the expected functorial axioms and coincides with the ordinary centre on objects.

2. Version 2 (Sections 4.5–4.6).
   Motivated by 2‑dimensional conformal field theory, the authors enlarge the bicategory to (\mathcal{C}_2) where 1‑morphisms are not just homomorphisms but arbitrary (A)‑(B) bimodules, and 2‑morphisms are bimodule intertwiners. In this richer setting, the centre becomes a full centre (also known as the Drinfeld centre) of the monoidal category of (A)-modules. The associated lax functor (Z_2:\mathcal{C}_2\to\mathcal{C}_2) captures not only the commutative subalgebra but also the braiding data arising from defect junctions. This construction aligns with known results in modular tensor categories and provides a bridge between algebraic topology, higher category theory, and quantum field theory.

Outlook.
The paper concludes with several directions for future work: extending the defect framework to three dimensions (where defects become surfaces and lines, leading to 3‑categories), exploring connections with extended TQFTs à la Baez‑Dolan–Lurie, and applying the functorial centre to problems in representation theory, such as categorified quantum groups or boundary conformal field theories. The authors emphasize that higher categorical language is not merely a formal embellishment but a necessary tool for organizing the intricate web of defect data and for achieving functoriality of constructions that were previously only defined pointwise.

In summary, the authors provide a clear, concrete example of a 2‑dimensional topological field theory with defects, demonstrate how the defect data naturally forms a bicategory, and exploit this structure to give a fully functorial definition of the centre of an algebra. Their work showcases the power of higher categories in both mathematical physics and pure algebra, opening pathways for further interdisciplinary research.