Field theories with defects and the centre functor

TL;DR

Davydov, Kong, and Runkel present a coherent, higher-categorical framework for 2D quantum field theories with defects, framing field theories as functors from defect-enhanced bordism categories to vector spaces and showing how defects induce 2-categorical and bicategorical structures. They construct a concrete 2D lattice TFT with defects from Frobenius algebras with trace and bimodules, prove its independence from cell decompositions, and relate defect data to a functorial centre by passing to a bicategory of commutative algebras; this yields a functorial centre $Z:\mathcal{A}lg(k)\to \mathbf{CAlg}(k)$ (and a refined version) that encodes boundary–bulk relations in a defect setting. The work connects lattice TFTs with algebraic centers and Morita-type defect data, providing two versions of a centre-valued functor and a pathway to generalisations in braided monoidal categories and 2D conformal field theory. Overall, the paper clarifies how higher-categorical methods illuminate the algebraic content of field theories with defects and offers a concrete construction that links defect TFTs to functorial center constructions with potential applications in CFT and orbifolds.

Abstract

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.

Figures (17)

• Figure 1: Figures a)--c) show open subsets of a bordism in dimension $n=1$ and $n=2$. They give our orientation convention in the compatibility condition for the assignment of defect conditions in the case $n=1$ (figs. a, b) and $n=2$ (fig. c). The arrows represent positively oriented ordered bases.
• Figure 2: Illustration of the condition of cyclic composability of domain walls. Given the $n$-tuple $( (x_1,\varepsilon_1), \dots , (x_n,\varepsilon_n) )$, the $i$'th domain wall (counted anti-clockwise) is labelled by $x_i$ and is pointing towards the junction point if $\varepsilon_i=+$ and away from the junction point if $\varepsilon_i=-$. In the present example the 6-tuple is $((x_1,+),(x_2,-),(x_3,-),(x_4,+),(x_5,-),(x_6,+))$. The images under the maps $s$ and $t$ to $D_2$ have to agree as shown, e.g. $s(x_1)=s(x_2)$ and $t(x_2)=s(x_3)$. The junction point has orientation '$+$' and is labelled by $u \in D_0$.
• Figure 3: Illustration of a collar which forms part of the data for an object in the bordism category. In the notation of the text, the solid (blue) circle is a unit circle $U=S^1$, the shaded area is an open neighbourhood $A$, the solid (red) short lines form the oriented submanifold $A_1$ of $A$ which intersects $S^1$ in $U_0$. Our convention for the orientation of $A_1$ induced by that of $U_0$ (the signs '$\pm$') is as shown. The elements $a,b,c,d \in D_2$ label connected components of $U_1$ and their extension $A_2$; these labels have to agree with the source and target maps of the domain wall labels $w,x,y,z \in D_1$ as shown. E.g. $t(w)=a=s(x)$.
• Figure 4: Some domain walls and two junctions placed on $S^2$; we only display a fragment after projection to the plane. The two junctions are labelled by the same junction condition $u \in D_0$ but with opposite orientation '$\pm$'. Here, $j(u)$ is the cyclic permutation equivalence class of $((x_1,+),(x_2,+),(x_3,-),(x_4,+),(x_5,-))$. Thus, the junction labelled by $u$ with orientation '$+$' must have domain walls $(x_1,+),(x_2,+),(x_3,-),(x_4,+),(x_5,-)$ attached in anti-clockwise order, where for $(x_i,+)$ the domain wall is oriented towards the junction and for $(x_i,-)$ it is oriented away from the junction. The junction labelled by $u$ with orientation '$-$' must have domain walls $(x_5,+),(x_4,-),(x_3,+),(x_2,-),(x_1,-)$ attached in anti-clockwise order.
• Figure 5: Illustration of the condition for scale and translation invariant family of states: Let $\psi_{\underline{ x}}$ be such a family. The figure shows $Q$ applied to three annuli, understood as bordisms $O(\underline{ x},r_i) \to O(\underline{ x},R)$, for $i=1,2,3$, where $r_i$ denotes the radius of the inner disc of the $i$'th annulus shown above. All three annuli have the same outer radius $R$. Applying $Q$ to the bordism and evaluating the resulting linear map on $\psi_{\underline{ x},r_i} \in Q(O(\underline{ x},r_i))$, $i=1,2,3$, results always in the vector $\psi_{\underline{ x};R} \in Q(O(\underline{ x},R))$ of the same family.

Theorems & Definitions (55)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 3.4