# Defects and Symmetry

King's College London
23 - 24 June 2022

### Venue

The talks were in the Anatomy Lecture Theatre with Registration & coffee are in the Anatomy Museum, directly behind the Lecture Theatre.

### Programme

The programme was:

 Thursday, 23.6 12:30 - 13:00 Registration 13:00 - 14:00 Matt Buican Defects, Symmetries, and Theory Factorization [abstract] 14:00 - 14:30 Thibault Décoppet Fusion 2-Categories and Fully Extended 4d TQFTs [abstract] [slides] 14:30 - 14:50 coffee 14:50 - 15:50 Sakura Schafer-Nameki Generalized Symmetries in QFT and Strings [abstract] [slides] 15:50 - 16:20 Saghar Hosseinisamnani BF theory in TFTs from M/string theory [abstract] [slides] 16:20 - 16:30 Mini Gong Show 16:30 - 17:10 coffee & posters 17:10 - 17:40 Lakshya Bhardwaj Non-Invertible Higher-Categorical Symmetries From Outer-Automorphisms [abstract] 17:40 - 18:10 Rajath Radhakrishnan From Topological Defects to Quantum Error-Correcting Codes [abstract] [slides] Friday, 24.6: 9:00 - 10:00 David Jordan Langlands duality on 3-manifolds [abstract] 10:00 - 10:30 Ödül Tetik Geometric factorisation algebras and Morita theory [abstract] [slides] 10:30 - 11:00 coffee 11:00 - 12:00 Ilka Brunner Truncated affine Rozansky-Witten models as extended TQFTs [abstract] [slides] 12:00 - 12:30 Ziwen Kong Broken global symmetry and defect conformal manifolds [abstract] [slides] 12:30 - 14:00 lunch 14:00 - 15:00 Paul Fendley Defects, Duality, and Categorical Symmetries [abstract] 15:00 - 15:30 Guillermo Arias Non-invertible symmetries from discrete gauging and completeness of the spectrum [abstract] [slides] 15:30 - 16:00 Vincentas Mulevičius Internal Levin-Wen models [abstract] [slides]

### Abstracts

The abstracts for the talks are as follows:

 Arias Non-invertible symmetries from discrete gauging and completeness of the spectrum We study global $$1-$$ and $$(d-2)$$-form symmetries for gauge theories based on disconnected gauge groups which include charge conjugation. For pure gauge theories, the 1-form symmetries are shown to be non-invertible. In addition, being the gauge groups disconnected, the theories automatically have a $$Z_2$$ global $$(d-2)$$-form symmetry. We propose String Theory embeddings for gauge theories based on these groups. Remarkably, they all automatically come with twist vortices which break the $$(d-2)$$-form global symmetry. Bhardwaj Non-Invertible Higher-Categorical Symmetries From Outer-Automorphisms I will describe construction of non-invertible symmetries by gauging outer-automorphisms acting on invertible higher-form symmetries. I will also discuss how one can piece together the data of a higher-category describing finer properties of such non-invertible symmetries. Brunner Truncated affine Rozansky-Witten models as extended TQFTs Mathematicians formulate fully extended d-dimensional TQFTs in terms of functors between a higher category of bordisms and suitable target categories. Furthermore, the cobordism hypothesis identifies basic building blocks of such TQFTs. In this talk, I will discuss Rozansky Witten models with affine targets, also known as 3-dimensional topologically twisted N=4 theories of free hypermultiplets. I will show how in this simple example the cobordism hypothesis can be systematically applied to explicitly construct the (infinite dimensional) state spaces of this theory. Furthermore, a commutative Frobenius algebra will be identified that describes the extended TQFT restricted to circles and bordisms between them. (Based on work with Nils Carqueville and Daniel Roggenkamp) Buican Defects, Symmetries, and Theory Factorization In this talk we will explore what defects and symmetries tell us about how and when quantum field theories (QFTs) factorize (exactly and approximately) into components. We will consider both topological and non-topological QFTs in two, three, and four dimensions. Décoppet Fusion 2-Categories and Fully Extended 4d TQFTs I will recall the definition of a fusion 2-category, and give many examples. Then, I will define separable fusion 2-categories, and explain how they can be used to construct fully extended framed 4d TQFTs. Finally, I will examine what these TQFTs assign to certain low dimensional manifolds. Fendley Defects, Duality, and Categorical Symmetries Generalised/higher/topological/categorical symmetries are typically implemented by non-local and non-unitary operators. Kramers-Wannier duality is a canonical example of such a symmetry, in which case the corresponding operator is not even invertible. I will explain how these symmetries are naturally implemented by topological defects, and show how many such symmetries/defects can be constructed in 1+1 and 2+0 dimensional lattice models by utilising fusion categories. An advantage of this set-up is that various properties such as critical exponents and boundary g-factors can be computed exactly on the lattice, without utilising or even requiring integrability. Hosseinisemnani BF theory in TFTs from M/string theory (d+1)-dimensional topological field theories (TFTs) encode the higher symmetries, 't Hooft anomalies, higher structures, and the BF theory of d-dimensional field theories. These theories may be geometrically engineered from M/string theory. I will discuss how the BF theory, realising all the possible choices of higher symmetries, can be found by constructing a Chern-Simons action for the effective supergravity kinetic action. Time permitting, I will show how the kinetic action also encodes the anomalies and higher structures such as 2/3-group symmetries. I will do this by considering a specific example. Jordan Langlands duality on 3-manifolds I will first explain an elementary conjecture an equality between the generic dimension of the skein theories on a closed oriented 3-manifold attached to a group G and to its Langlands dual GL. Skein theories capture the 2- and 3-dimensional parts of the Markus/Kapustin--Witten topological twist of 4D N=4 gauge theories, and our conjecture extends the celebrated Langlands dualities from dimension 2 to dimension 3. I will however spend most of the time discussing an approach to the conjecture via defects attached to 1-form symmetries which allows us to directly verify it in a number of special cases, including $$SL_N/PGL_N$$ duality for $$T^3$$, and $$SL_2/PGL_2$$ duality for $$\Sigma_g \times S^1$$. This involves various joint works with Ben-Zvi, Gunningham, Safronov, Vazirani, and Yang. Kong Broken global symmetry and defect conformal manifolds I will present the recent work arXiv:2203.17157 with Nadav Drukker and Georgios Sakkas. Defects that break the global symmetry group provides some exactly marginal defect operators, which allow to deform a CFT along the defect conformal manifold, which is the symmetry breaking coset. Its Zamolodchikov metric is expressed as the 2-pt function of the exactly marginal operator and the Riemann tensor can be expressed as an integrated 4-pt functions. We examine in detail the case of the 1/2 BPS Maldacena-Wilson loop in N = 4 SYM, the 1/2 BPS surface operator of the 6d N=(2,0) theory, and the 1/2 BPS Fermionic Wilson in ABJM. Mulevičius Internal Levin-Wen models Given a spherical fusion category one can define an explicit (2+1)-dimensional lattice model, the Levin-Wen model, whose degenerate ground state space is the topological order given by the Drinfeld centre of the input category. In this talk we will discuss an analogous lattice system, built out of excitations in an arbitrary topological order, whose ground space state can be any other order, which is Witt-equivalent to the initial one. The construction relies on Reshetikhin-Turaev TQFTs with point, line, and surface defects, and yields the Levin-Wen models as a special case, namely when the initial order is vacuum. Based on joint work with Runkel and Voß. Radhakrishnan From Topological Defects to Quantum Error-Correcting Codes Topological line operators describe the 0-form symmetries of 1+1D quantum field theories. Recent results show that Rational Conformal Field Theories (RCFTs) and quantum error-correcting codes are closely related. It is then natural to wonder about the role of topological line operators in this context. In my talk, I will describe an explicit map relating RCFTs and quantum stabilizer codes. I will discuss the role of certain mixed anomalies in obtaining a consistent map. Using this map, the local operators of the RCFT will be related to the stabilizer group, and the non-genuine local operators living at the end of topological line operators will be related to the generalized Pauli group. Schafer-Nameki Generalized Symmetries in QFT and Strings I will provide an overview of recent developments in QFT and String theory realizing generalized symmetries -- higher-form, higher-group, higher-categorical. The constructions include 4d non-supersymmetric gauge theories, as well as superconformal field theories in 5d and 6d. Tetik Geometric factorisation algebras and Morita theory We propose two natural but structurally different definitions of a 'geometric' prefactorisation algebra on a stratified space X with a given stratified tangential structure B, sensitive to geometric information along inclusions of opens. In order to formulate a cosheaf condition, we define 'geometric' simplex categories that depend on X and B, and corresponding 'geometric' Čech complexes. With minor caveats, certain truncations of these objects recover the familiar ones, but otherwise they behave differently. In the locally-constant case, this leads to a construction of higher 'B-Morita' categories, which are conjecturally the correct target categories for extended functorial field theories constructed using factorisation homology, beyond the framed case. We end with a discussion of various open problems.

### Posters

The posters presented were as follows:

 Julien Barrat Bootstrapping Holographic defect correlator [poster] Gabriel Bliard The 1/2 BPS Wilson line defect in ABJM [poster] Davide Bonomi Conformal line defects and holography [poster] Julius Bootstrability for 1d defect CFT [poster] Arpit Das Higher-form symmetries, anomalous magnetohydrodynamics, and holography [poster]

### Registration

There was a registration fee of £25 for the meeting to cover coffee/tea/biscuits, reduced to £12.50 for members of the Institute of Physics.

### Photos

Here are a photo of the poster session on Thursday afternoon: and the meeting group photo on Friday.

### Support

There were limited funds available to support UK-based Research students and those with caring responsibilities. The Institute of Physics provides support to PhD students through its own internal funding mechanism.