**Dualizabitility, higher categories, and topological field theories**

In the past decade, using higher categories has proven to be an essential ingredient in the study of topological field theories (TFTs) from a mathematical perspective. The most prominent and seminal result is the Cobordism Hypothesis, which gives a beautiful classification of “fully extended” topological field theories. Here, fully extended means that our TFT can be evaluated at manifolds and bordisms of all dimensions below a given one; conversely, the mathematical language needed to describe the structure is that of higher categories and dualizability therein. In physics, we can interpret these values at all dimensions as (possibly higher) categories of boundary conditions, as point insertions/observables, line operators or higher dimensional operators, etc., depending on the stiuation.

The main goal of this master class will be to explain how to use the cobordism hypothesis to construct TFTs and variations thereof. One example we will look at in detail arises from factorization homology for E_n-algebras, which will also be a key tool in the parallel master class. We will discuss (∞,n)-categories and dualizability in detail, and, time permitting, some extensions and variations.

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I. A complete mathematical definition of quantum field theory does not yet exist. Following the example of quantum mechanics, I will indicate what a good definition in terms could look like. In this good definition, QFTs are defined in terms of their operator content (including extended operators), and the collection of all operators is required to satisfy some natural properties.

II. After reviewing some classic examples, I will describe the construction of Noether currents and the corresponding extended symmetry operators.

III. One way to build topological extended operators is by “condensing” lower-dimensional operators. The existence of this condensation procedure makes the collection of all topological operators into a semisimple higher category.

IV. Topological operators provide “noninvertible higher-form symmetries”. These symmetries assign charges to operators of complementary dimension. This assignment is a version of what fusion category theorists call an “S-matrix”.

V. The Tannakian formalism suggests a way to recognize higher gauge theories. It also suggests the existence of interesting higher versions of super vector spaces with more exotic tangential structures.

]]>**Simons Center for Geometry and Physics** (Stony Brook, NY), **International Centre for Mathematical Sciences** (Edinburgh, UK), and **Cyberspace**

Videos are available on the SCGP video repository.

If you are interested in attending this event from cyberspace you are welcome! Please register here

Indicated times are New York / Edinburgh. The program for the event with title and abstracts can be found below.

9 / 14 Welcome and Introduction (Teleman, Del Zotto)

9:30 / 14:30 * Discussion: What is QFT?* (Freed, Seiberg)

10:30 / 15:30 Coffee

11 / 16 Scheimbauer (Zoom)

12 / 17 Kitaev (live SCGP)

1 Lunch / 18 Dinner

2:30 / 19:30 Hopkins (live SCGP)

3:30 Tea / 20:30 Scottish Tea

4 / 21 Kapustin (Zoom)

5 / 22 free/ad hoc/informal discussions

9 / 14 Check-in with ICMS, free discussion

9:30 / 14:30 García Etxebarria (live ICMS)

10:30 / 15:30 Coffee

11 / 16 Plavnik (live SCGP)

12 / 17 Xiao-Gan Wen (Zoom)

1 Lunch / 18 Dinner

2:30 / 19:30 ** Discussion: Non-invertible symmetries **(Komargodski, Ohmori, Shao,Thorngren)

3:30 Tea / 20:30 Scottish Tea

4 / 21 Gaiotto (Zoom)

5 / 22 free/informal/ad hoc discussion

9 / 14 Check-in with ICMS

9:30 / 14:30 ** Discussion: Perspectives on gauge theories** (Moore, Nekrasov)

10:30 / 15:30 Coffee

11 / 16 Schäfer-Nameki (live ICMS)

12 / 17 Cordova/Dumitrescu (live SCGP)

1 / 18 Closing remarks, Lunch/Dinner, and Goodbyes

**What is quantum field theory?** – *Dan Freed and Nathan Seiberg*

**Non-invertible symmetries** – *Zohar Komargodski, Kantaro Ohmori, Shu-Heng Shao, and Ryan Thorngren*

**Perspectives on gauge theories** – *Greg Moore and Nikita Nekrasov*

**Davide Gaiotto:** **Condensation of topological phases**

I will review an higher-dimensional generalization of the notion of projector introduced in my work with Theo Johnson-Freyd.

**Anton Kapustin:** **A local Noether theorem for quantum lattice systems and topological invariants of gapped states**

In field theory, conserved currents have well-known ambiguities. Thanks to the Poincare lemma, these ambiguities are physically harmless. Similar issues arise for lattice systems, but have not been explored previously. I will explain some general results which both ensure the existence of local currents on a lattice and describe the corresponding ambiguities. A starring role in this problem is played by a certain 1-shifted differential graded Lie algebra attached to a quantum lattice system. A similar 1-shifted DG Lie algebra can also be attached to any gapped state of a quantum lattice system. I explain how to use this algebraic structure to extract a topological invariant out of a gaped state of a 2d lattice system invariant under a U(1) symmetry. This invariant is the zero-temperature Hall conductance. This is joint work with Nikita Sopenko.

**Alexei Kitaev: ****Short-range entangled quantum states.**

This ongoing work aims to understand topological properties of ground states of gapped lattice Hamiltonians. In the special case of free-fermion systems, the homotopy type of the space of states is given by a shifted K or KO spectrum (depending on imposed symmetry). I will consider more general short-range entangled states, focusing on Bose systems with no symmetry. By definition, short-range entangled states are “r-constructible” and “(2r,epsilon)-pure”, where r is some length characterizing the extent of entanglement and epsilon is an error parameter. Some important properties include invertibility (i.e. the existence of an “anti-state”), gluing, and error reduction. Using this toolkit, one can, hopefully, show that the space of short-range entangled states is an Omega-spectrum.

*Sakura Schäfer-Nameki: *Generalized Symmetries in 5d and 6d

I will give an overview of recent developments in 5d and 6d superconformal field theories, describing their higher-form symmetries, as we all as higher-group structures.

Most of the analysis will be based on the geometric realization of these field theories in M-theory or F-theory on canonical three-fold singularities, and we will discuss how the generalized symmetries are imprinted in the underlying geometry.

**Xiao-Gan Wen:** **A categorical view of symmetry and a holographic view of symmetry**

Symmetry are usually described by groups or higher groups. Here I will described a more general view of symmetry in terms of fusion higher category. Symmetries described by different groups and/or different fusion higher categories can be equivalent. There is an even better way to describe symmetries in terms of braided fusion higher category, ie in terms of topological order in one higher dimension. The equivalent symmetries will be described by the same braided fusion higher category (ie the same topological order in one higher dimension).

*Clay Córdova/Thomas Dumitrescu: *Higher Symmetry in Gauge Theory

We will review a few of the many ways in which higher form and higher group symmetries arise in quantum field theory and how they can be used to analyze and organize renormalization group flows. For concreteness, we focus on non-supersymmetric gauge theory examples in four dimensions. In addition to recent progress, we will also highlight some open problems of possible mathematical and physical interest.

**Iñaki García Etxebarria:** **Categorical Symmetries and String Theory**

String theory provides a way to associate field theories to

singular geometries. The resulting class of theories is very rich but

the individual theories are often poorly understood, and in particular

we generically don’t know how to define them starting from a

Lagrangian. In this talk I will review part of what we know about

deriving the symmetry structure of this class of theories from the

geometry of their string theory construction.

*Mike Hopkins: *Topology and quantum field theory

I will survey some of the interactions between homotopy theory and quantum field theory, and some mathematical questions it inspires.

*Julia Plavnik: *Gauging, condensation and zesting as quantum symmetries

In this talk, we will present the zesting construction for modular categories. Zesting was first introduced in 2012 and further developed for applications to fermionic theories in 2016. We will give some examples and properties of this construction and compare it with other constructions of modular categories like gauging and condensation.

*Claudia Scheimbauer: *Higher categorical tools for defects and boundary theories

Nowadays we have many mathematical tools, ie higher categorical, tools to construct and study TQFTs, boundary and defect theories (coupling). In this talk we will give an overview of the state of the art and discuss applications to various examples, such as Reshetikin-Turaev/Chern-Simons theory and lattice field theory and gauge theory.

** Curators: **Michele Del Zotto, Thomas Dumitrescu, David Jordan, Constantin Teleman