Simons Collaboration on Global Categorical Symmetries
Simons Collaboration on Global Categorical Symmetries
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Simons Lectures on Categorical Symmetries

by |Published

Global Categorical Symmetries are a powerful new tool for analyzing quantum
field theories. The first volume compiles lecture notes from the 2022 and 2023
summer schools on Global Categorical Symmetries, held at the Perimeter
Institute for Theoretical Physics and at the Swiss Map Research Station in Les
Diableret.

Specifically, this volume collects the lectures:
* An introduction to symmetries in quantum field theory, Kantaro Ohmori
* Introduction to anomalies in quantum field theory, Clay C\’ordova
* Symmetry Categories 101, Michele Del Zotto
* Applied Cobordism Hypothesis, David Jordan
* Finite symmetry in QFT, Daniel S. Freed

These volumes are devoted to interested newcomers: we only assume (basic)
knowledge of quantum field theory (QFT) and some relevant maths. We try to give
appropriate references for non-standard materials that are not covered. Our aim
in this first volume is to illustrate some of the main questions and ideas
together with some of the methods and the techniques necessary to begin
exploring global categorical symmetries of QFTs.

Events

NYU Satellite Workshop 24

by |Published

November 11-13, 2024 @ NYU

726 Broadway, New York City 

Room 940

Organizers: Michele Del Zotto, Constantin Teleman, Yifan Wang

SCHEDULE

MONDAY

14:00 – 14:30 ::: Lukas Müller  
14:30 – 15:00 ::: Hector Peña Pollastri  
15:00 – 15:30 ::: Thibault Décoppet  
coffee/tea and discussion  
16:00 – 16:30 ::: Luuk Stehouwer  
16:30 – 17:00::: Shani Nadir Meynet  
17:00 – 17:30 ::: Pranay Gorantla  


TUESDAY

9:00 – 10:00 ::: Sakura Schäfer Nameki  
coffee break  
10:30 – 11:30 ::: Mayuko Yamashita  
11:30 – 12:30 ::: Anton Kapustin  
lunch break and discussions (coffee/tea 15:30)  
16:00 – 17:00 ::: Theo Johson Freyd  


WEDNESDAY

9:00 – 10:00 ::: Dan Freed  
coffee break  
10:30 – 11:30 ::: Pavel Etingof  
12:00 – 13:00 ::: Xiao-Gang Wen  
lunch break and discussions  
15:30 – 16:30 ::: Wine and cheese reception and discussions  

ABSTRACTS

Müller (PI)

A Higher Spin Statistics Theorem for Invertible Quantum Field Theories

The spin-statistics theorem asserts that in a unitary quantum field theory, the spin of a particle—characterized by its transformation under the central element of the spin group, which corresponds to a 360-degree rotation—determines whether it obeys bosonic or fermionic statistics. This relationship can be formalized mathematically as equivariance for a geometric and algebraic action of the 2-group $B\mathbf{Z}_2$. In my talk, I will present a refinement of these actions, extending from $B\mathbf{Z}_2$ to appropriate actions of the stable orthogonal group $O$, and demonstrate that every unitary invertible quantum field theory intertwines these $O$-actions.

Peña Pollastri (PI)

Exact Factorizations of fusion categories and the bicrossed product construction

In 2017 Gelaki introduced the notion of exact factorization of fusion categories, generalizing the notion of exact factorization of groups. We show some general results about the structure of such factorizations, their fusion rules and universal grading groups. Also we introduce a new construction, the bicrossed product, to produce new examples, including some exact factorizations involving Tambara-Yamagami categories and pointed categories.

Decoppet (Harvard U.)

Higher Verlinde Categories: The Mixed Case

In characteristic zero, the semisimple Verlinde categories are braided fusion categories constructed from quantum $\mathfrak{sl}_2$ at a root of unity. These categories have found many applications to low-dimensional topology, for instance, via the Reshetikhin-Turaev construction. I will explain how to construct the positive characteristic analogues of these categories, which are generally not semisimple, and discuss their fundamental properties.

Stehouwer (Dalhusie U.)

2-Hilbert spaces and extended unitarity

Hilbert spaces have two levels of equivalence; linear isomorphism and unitary isomorphism. The fact that 2-Hilbert spaces have three levels gives 2-Hilb the structure of a higher dagger category. This provides extra structure to the category of topological line operators in unitary theories, generalizing the fact that point operators form a *-algebra.

Meynet (Uppsala U.)

Emergent Non-Invertible Symmetries — The Adjoint QCD Example

After reviewing some general properties of the renormalization group (RG) flow for quantum field theories (QFT), I will describe how the recently descovered non-invertible symmetries can be used to study theories at strong coupling. I will illustrate these facts using (3+1)-dimensional Adjoint QCD with two flavors as an example. This theory can be obtained by mass deforming a pure N=2 super Yang-Mills theory. Relaying on supersymmetric results, dynamical abelianization and monopole condensation, one obtains an infrared (IR) phase characterized by an CP1 sigma model. In this scenario, the IR phase has an emergent non-invertible symmetry, which is matched with the non-invertible symmetry of the IR CP1 phase. This result illustrate how an emergent non-invertible symmetry can be used to provide a bridge connecting gauge theories at strong coupling and their IR via dynamical abelianization.

Gorantla (Chicago U.)

Duality-preserving deformation of 3+1d lattice Z2 gauge theory with exact gapped ground states

I will talk about a deformation of the 3+1d lattice Z2 that preserves the Wegner duality of the self-dual point. Surprisingly, there is a frustration-free point along the deformation where there are nine exact ground states (on a periodic cubic lattice): one of them is a trivial product state, whereas the rest are the topologically ordered ground states of the 3+1d toric code! Moreover, the gap at this point can be proved analytically. I will also briefly discuss generalization to lattice ZN gauge theory and what the phase diagram should look like for any N. Based on 2409.10612 with Tzu-Chen Huang.

Schafer-Nameki (Oxford U.)

Categorical Landau in 1+1d and 2+1d

Using the Symmetry TFT we systematically study the phases (gapped and gapless) of 1+1d and 2+1d systems with fusion 1-/2-categories.

Yamashita (PI)

Genuine refinements of elliptic genera

There is a classical construction of elliptic genera for SU-manifolds, which assigns Jacobi forms to SU-manifolds, and physically related to non-chiral 2d N=(1, 1) sigma models. In this talk, I explain my ongoing work with Ying-Hsuan Lin to give homotopy-theoretical refinements to it. The domain becomes “Topological Jacobi Forms (TJF)”, which is a spectrum (in homotopy theory) being developed by an ongoing work of T. Bauer and L. Meier. TJF refines the ring of Jacobi forms, just as TMF does for modular forms. In view of the Segal-Stolz-Teichner program, it is very natural to expect such a refinement of elliptic genera. The same strategy produces many variants of twisted equivariant TMF-valued genera including those from Sp-manifolds and Spin-manifolds. Having such refinements, we can detect torsion elements in the bordism groups, as well as deduce nontrivial divisibility results of characteristic numbers. In this talk I explain the motivation and construction, as well as why such refinements are more interesting beyond the classical elliptic genera.

Johnson Freyd (Dalhusie U. and PI)

The unitary cobordism hypothesis

A *dagger category* is a category equipped with extra “unitarity” data: among its isomorphisms, some are marked as “unitary”; to each 1-morphism, there is an “adjoint” 1-morphism, and this assignment sends unitary (but not general!) isomorphisms to their inverses. If a monoidal higher category has duals, it is interesting to ask for a choice of duality functor for which the units and counits are adjoints; whereas duality functors are unique up to contractible choice, so-defined “unitary duality” functors are not. I will explain a higher-categorical generalization of these ideas, and explain how, for any stable tangential structure H, a construction of Freed and Hopkins makes the extended bordism category Bord$_n^{H(n)}$ into a dagger symmetric monoidal n-category with unitary duality. This category satisfies a *unitary cobordism hypothesis*: whereas nonunitary functors Bord$_n^{H(n)} \to \mathcal{C}$ are classified by $H(n)$-fixed points in $\mathcal{C}$, *unitary* functors are classified by fixed points for the stablized group $H(\infty)$. This talk is based on joint work in preparation with Cameron Krulewski, Lukas Mueller, and Luuk Stehouwer.

Freed (Harvard U.)

Anomalies and quiches

Together with Greg Moore and Constantin Teleman we introduced an abstract notion of symmetry in QFT–quiches–analogous to abstract groups and algebras in other contexts. This leads to a natural question: How is the notion of an ‘t Hooft anomaly captured in the quiche framework? In joint work with Colleen Delaney, Julia Plavnik, and Constantin we address this question and also give applications to zesting of fusion categories. The first part of the talk will be a general discussion of anomalies in QFT.

Etingof (MIT)

Lie theory in tensor categories with applications to modular representation theory

Let $G$ be a group and $k$ an algebraically closed field of characteristic $p > 0$. Let $V$ be a finite dimensional representation of $G$ over $k$. Then by the classical Krull-Schmidt theorem, the tensor power $V^{\otimes n}$ can be uniquely decomposed into a direct sum of indecomposable represen- tations. But we know very little about this decomposition, even for very small groups, such as $G = (\mathbb Z/2)^3$ for $p = 2$ or $G = (\mathbb Z/3)^2$ for $p = 3$. For example, what can we say about the number $d_n(V)$ of such summands of dimension coprime to $p$? It is easy to show that there exists a finite limit $d(V) := \lim_{n\to \infty} d_n(V)^{1/n}$, but what kind of number is it? For example, is it algebraic or transcendental? Until recently, there was no techniques to solve such questions (and in particular the same question about the sum of dimensions of these summands is still wide open). Remarkably, a new subject which may be called “Lie theory in tensor categories” gives methods to show that $d(V)$ is indeed an algebraic number, which moreover has the form

$$ d(V) = \sum_{1 \leq j \leq p/2} n_j(V)[j]_q,$$

where $n_j(V) \in \mathbb N$, $q := \exp(i \pi /p)$, and $[j]_q := (q^j − q^{-j})/ (q−q^{−1})$. Moreover,

$$ d(V\oplus W ) = d(V ) + d(W ), \qquad d(V \otimes W ) = d(V )d(W ),$$

i.e., $d$ is a character of the Green ring of $G$ over $k$. Furthermore, $d_n(V ) \geq C_V d(V )^n$ for some $0 < C_V \leq 1$ and we can give lower bounds for $C_V$. In the talk I will explain what Lie theory in tensor categories is and how it can be applied to such problems. This is joint work with K. Coulembier and V. Ostrik.

Wen (MIT)

From sub-algebra of local operators, to generalized symmetry, and to their topological phases on lattice

TBA.

Kapustin (Caltech)

Topological invariants of gapped states as obstructions to gauging

Recently, an approach to constructing topological invariants of gapped ground-states of lattice systems has been developed in our joint work with N. Sopenko. In this talk I will explain an interpretation of these invariants as obstructions to gauging, i.e. to promoting a symmetry to a local symmetry. To formalize the notion of a local symmetry on the infinitesimal, I define a “local Lie algebra” as a pre-cosheaf of Lie algebras with additional properties. In the case of lattice systems, this pre-cosheaf lives on a Grothendieck site which encodes the large-scale geometry of the lattice.

PARTICIPANTS

Ibrahima Bah

Maissam Barkeshli

Francesco Benini 

Federico Bonetti

Lakshya Bhardwaj 

T. Daniel Brennan 

Christian Copetti  

Augustina Czenky 

Michele Del Zotto

Tudor Dimofte

Thibault Decoppet  

Pavel Etingof

Dan Freed

Pranay Gorantla 

Jonathan Heckman

Ken Intriligator

Theo Johnson-Freyd

Anton Kapustin

Liang Kong 

Zohar Komargodski

Vassily Krylov 

Tom Mainiero 

Shani Meynet 

Ruben Minasian

Greg Moore

Lukas Muller 

Kantaro Ohmori 

Hector Pena-Pollastri 

Konstantinos Roumpedakis 

Ingo Runkel 

Sakura Schafer-Nameki

Claudia Scheimbauer

Shu-Heng Shao 

Noah Snyder 

Luuk Stehouwer 

Pelle Steffens 

Will Stewart

Constantin Teleman

Jackson van Dyke

Kevin Walker 

Yifan Wang

Xiao-Gang Wen

Mayuko Yamashita

Bowen Yang

 

Events

GCS2024 School and Conference (APPLY NOW!)

by |Published

June 10-21, 2024 ICMS Edinburgh (UK)

More details can be found on the event webpage: https://www.icms.org.uk/Categoricalsymmetries

Apply here: https://www.smartsurvey.co.uk/s/ApplicationCSQFT/

Schedule

Summer school: Monday 10th – Friday 14th June. This will introduce students to the major themes of the workshop.

Workshop: Monday 17th – Friday 2th June.

Participation:

Public application is now open for both summer school and workshop. Please click the “Apply here” button on the top of this page. You can choose to attend in-person or online when you apply. 

Speakers

Workshop:

  • Francesco Benini, SISSA, Trieste
  • Jennifer Brown, University of Edinburgh
  • T. Daniel Brennan, UC San Diego 
  • Federico Bonetti, University of Durham
  • Ilka Brunner, University of Munich
  • Colleen Delaney, UC Berkeley
  • Andrea Grigoletto, University of Durham
  • Wenjie Ji, California Institute of Technology 
  • Theo Johnson-Freyd, Dalhousie University 
  • Max Metlitski,  Massachussetts Institute of Technology
  • Cris Negron, University of Southern California
  • Yuya Tanizaki, Kyoto University
  • Dave Penneys, Ohio State Univers
  • Tomer Schlank, Hebrew University of Jerusalem
  • Yunqin Zheng, Stony Brook

Summer School:

  • Claudia Scheimbauer, TU Munich
  • Ryan Thorngren, UCLA
  • Graeme Segal, University of Oxford
  • Nitu Kitchloo, Johns Hopkins
  • Ibrahama Bah, Johns Hopkins 
Events

NYU Satellite Workshop 23

by |Published

November 14-15, 2023 @ NYU

726 Broadway, New York City 

Room 940

Organizers: Ibrahima Bah, Michele Del Zotto, Constantin Teleman, Yifan Wang

TENTATIVE SCHEDULE

TUESDAY

9:30 – 11:00 ::: coffee/tea and informal discussion/short Q&A
11:00 – 11:50 ::: Shu-Heng Shao
12:00 – 12:50 ::: Michele Del Zotto
lunch break
2:30 – 3:20 ::: Victor Ostrik
coffee/tea and discussion
4:00 – 5:00 ::: Discussion session


WEDNESDAY

9:30 – 10:20 ::: Jonathan Heckman
coffee break
11:00 – 11:50 ::: Tomer Schlank
lunch break and discussion
2:30 – 3:20 ::: Yifan Wang
coffee/tea and discussion

ABSTRACTS

Michele Del Zotto (Uppsala University)

Higher Structure of Chiral Symmetry

In this talk we will discuss the higher structure of the chiral symmetry of massless QED. There are two main messages from our study: the first one is that the higher-associators have detectable consequences on correlators, and therefore are interesting properties of the symmetry. The second is that one can exploit partition functions on curved spaces with non-trivial topology to probe the higher structure of symmetry. In particular, we find an interesting strategy to derive Ward identities based on decorating morphisms in surgery theory with morphisms in the symmetry category. Along the way, we comment on a relation between (higher) associators and anomalies. Based on joint work with Christian Copetti, Kantaro Ohmori and Yifan Wang.

Jonathan Heckman (UPenn)

SymTrees and Multi-Sector QFTs

The global symmetries of a D-dimensional QFT can, in many cases, be captured in terms of a (D+1)-dimensional symmetry topological field theory (SymTFT). In this work we construct a (D+1)-dimensional theory which governs the symmetries of QFTs with multiple sectors which have connected correlators that admit a decoupling limit. The associated symmetry field theory decomposes into a SymTree, namely a treelike structure of SymTFTs fused along possibly non-topological junctions. In string-realized multi-sector QFTs, these junctions are smoothed out in the extra-dimensional geometry. Time permitting, we explain how we can use this perspective to study the fate of higher-form symmetries in the context of holographic large M averaging where the topological sectors of different large M replicas become dressed by additional extended operators associated with the SymTree. Based on joint work with F. Baume, M. Hubner, E. Torres, A.P. Turner, and X. Yu.

Victor Ostrik (U. Oregon)

Growth in tensor powers

This talk is based on joint work with K.~Coulembier, P.~Etingof, D.~Tubbenhauer. Let $\Gamma$ be any group and let $V$ be a finite dimensional representation of  $\Gamma$ over arbitrary field. We consider tensor powers $V^{\otimes n}$ of V and their decompositions into indecomposable summands. Let $b_n(V)$ be the total number of indecomposable summands in $V^{\otimes n}$. We prove that

$$\lim_{n\to \infty} \sqrt[n]{b_n(V)}=\dim(V).$$

Similarly let $d_n(V)$ be the number of indecomposable summands of $V^{\otimes n}$ with dimension not divisible by the characteristic of the field. Then we define

$$\delta(V):=\lim_{n\to \infty} \sqrt[n]{d_n(V)}.$$

The real number $\delta(V)$ is an interesting invariant of the representation $V$. Using theory of tensor categories we show that this invariant is additive (under direct sums), multiplicative and takes values in algebraic integers.

Tomer Schlank (Hebrew University, Jerusalem)

Notions of algebraic closeness in homotopy theory and quantum field theories.

A key paradigm in understanding fields (in the algebra sense) is to go to an algebraic closure and than consider the action of the Galois group. We shall discuss how this paradigm appears under more categorical gize in the worlds of homotopy theory and higher category theory  and how algebraic closeness seems to be a desired property for targets of quantum fields theories. This talk is based on several  joint works with T. Barthel, , B. Ben-Moshe, R. Burklund, C.Carmeli, S. Keidar,  A. yuan and L.  Yanovski,

Shu-Heng Shao (YITP, Stony Brook)

Non-invertible symmetries in 1+1d lattice models

I will discuss non-invertible symmetries in 1+1d lattice models with a tensor-product Hilbert space, including the transverse-field Ising model and its deformation. The Kramers-Wannier duality operator on the lattice mixes with the lattice translation, and obeys a different non-invertible algebra compared to the continuum one.

Yifan Wang (NYU)

On Gauging Non-Invertible Symmetries

I will discuss the physics of gauging non-invertible symmetries using concrete examples in d=2 quantum field theory and comment on connections to results in category theory and subfactor theory. 

LIST OF PARTICIPANTS

Bobby Acharya (ICTP)

Andrea Antinucci (SISSA)

Fabio Apruzzi (Padova) 

Francesco Benini (SISSA) 

Federico Bonetti (Durham U)

Daniel Brennan (UCSD)

Jennifer Brown (Edinburgh U)

Christian Copetti (Oxford U) 

Thibault Decoppet (Harvard U)

Colleen Delaney (Berkeley U)

Michele Del Zotto (Uppsala U)

Tudor Dimofte (Univ Edinburgh) 

Dan Freed (UT Austin, Harvard U)

Cesar Galindo  (Univ de los Andes)

Jonathan Heckman (UPenn) 

Ken Intriligator (UCSD)

Theo Johnson-Freyd (Dalhousie U, PI)

Zohar Komargodski (SCGP Stony Brook)

Lukas Mueller (PI)

Cris Negron (USC) 

Victor Ostrik (U Oregon)

Sunghyuk Park (Harvard U)

Hector Peña-Pollastri (IU Bloomington)

Julia Plavnik (IU Bloomington)

David Reutter (Hamburg)

Giovanni Rizi (SISSA)

Iordanis Romaidis (Edinburgh U)

Eric Rowell (TAMU) 

Konstantinos Roumpedakis (Edinburgh U)

Sean Sanford (Ohio State)?

Sakura Schafer-Nameki (Oxford U) 

Shu-Heng Shao (Stony Brook U)

Claudia Scheimbauer (TU Munich )

Tomer Schlank (HUJI)

Christopher Schommer-Pries (Notre Dame U) 

Nathan Seiberg (Princeton IAS) 

Noah Snyder (IU Bloomington) 

Constantin Teleman (Berkeley U)

Kevin Walker (Station Q) 

Yifan Wang (NYU)

Zhenghan Wang (UCSB)

Yunqin Zheng (YITP)

Lecture series

TASI 2023: Aspects of Symmetries

by |Published

The TASI school is a traditional school dedicated to US graduate students in theoretical physics. In 2023 the school has been co-organized by some of our collaboration members: Ibrahima Bah and Ken Intriligator, together with Shu-Heng Shao (Stony Brook) and Oliver DeWolfe (local organizer).

The TASI school features several beautiful lectures about some of the topics of interest for our collaboration (and beyond!) by many internationally renewed experts (including also some of our PIs).

YouTube Video Recordings are available. More details about the school can be found on the event’s website.

Events

GCS2023 conference and school

by |Published

SwissMAP research station, Les Diableret, Switzerland, Aug 28 – Sept 8, 2023

This will be a hybrid mode event: most speakers will be in-person, with some virtual speakers; about half of the participants will be in-person, and talks will also be live-streamed on Zoom. Videos will be archived on a youtube channel.

Further details will appear on the conference webpage and on the SwissMAP website.

Immediately preceeding GCS2023 there are two distinct events, a Nordita Program (August 14-25) and a conference in Regensburg (August 14-18). Although these events are logically separate, with different organizing teams and registrations, we purposefully scheduled these events close to each other to avoid too many transatlantic flights.

Program

28 August – 1 September 2023: Research Conference

Speakers:

Anton Kapustin

Catherine Meusburger

Chelsea Walton

Christoph Schweigert

David Ayala

David Tong*

Kenneth Intriligator

Mathew Bullimore

Mayuko Yamashita

Owen Gwilliam

Peter Teichner

Ryan Thorngren

Sakura Schafer-Nameki

Zohar Komargodski*

(*to be confirmed)

3-8 September 2023: School

Lectures:

Constantin Teleman – The Mathematics of TQFTs and Defects

David Jordan – Applied Cobordism Hypothesis

Michele Del Zotto – Symmetry Categories 101

Shu-Heng Shao – Non-invertible Symmetries

Events

NORDITA program on Categorical Aspects of Symmetries

by |Published

August 14 – August 25, 2023, NORDITA, Stockholm (Sweden)

Organizers: Michele Del Zotto, Ken Intriligator, Muyang Liu, Nicolai Reshetikhin

The program consists of a workshop (August 14-21) and a conference (August 22-25).

More informations can be found on the official webpage for the event.

Participants include:

Apoorv Tiwari

Chi Ming Chang

Christian Copetti

Constantin Teleman

Federico Bonetti

Francesco Benini

Iñaki Garcia Etxebarria

Jonathan Heckman

Jürgen Fuchs

Lakshya Bhardwaj

Nils Carqueville

Pavel Putrov

Po-Shen Hsin

Ruben Minasian

Sakura Schäfer Nameki (*)

Sergei Gukov

Si Li

Theodore Daniel Brennan

Tudor Dimofte

Xiao-Gang Wen (*)

Yunqin Zheng

Yuya Tanizaki

Wenjie ji

(*) online only

Events

GCS2022 School

by |Published

The GCS2022 School is the second week of the program Global Categorical Symmetries 2022, held at the Perimeter Institute for Theoretical Physics June 6-17. Videos are available at https://pirsa.org/C22008.

Minicourses

  • Clay Córdova: Introduction to anomalies in quantum field theory
  • Dan Freed: Finite symmetry in QFT
  • Mike Hopkins: Lattice systems and topological field theories
  • Kantaro Ohmori: Introduction to symmetries in quantum field theory

Problem sessions will be TAed by Arun Debray and Ho Tat Lam.

Suggested readings

Lecture notes

Problem sets

Detailed Schedule

Monday, June 13

9:30-10:30: Ohmori, Introduction to symmetries in quantum field theory

11:30-12:30: Freed, Finite symmetries in QFT

14:00-15:00: Cordova, Introduction to anomalies in quantum field theory

16:00-17:00: Hopkins, Lattice systems and topological field theories

Tuesday, June 14

9:00-10:00: Freed, Finite symmetries in QFT

11:00-12:00: Ohmori, Introduction to symmetries in quantum field theory

13:30-14:30: Cordova, Introduction to anomalies in quantum field theory

15:00-17:00: Problem session / open discussion

17:00-19:00: BBQ

Wednesday, June 15

9:00-10:00: Hopkins, Lattice systems and topological field theories

11:00-12:00: Freed, Finite symmetries in QFT

13:30-14:30: Ohmori, Introduction to symmetries in quantum field theory

15:00-17:00: Problem session / open discussion

Thursday, June 16

9:00-10:00: Cordova, Introduction to anomalies in quantum field theory

11:00-12:00: Ohmori, Introduction to symmetries in quantum field theory

13:30-14:30: Hopkins, Lattice systems and topological field theories

15:00-17:00: Problem session / open discussion

Friday, June 17

9:00-10:00: Freed, Finite symmetries in QFT

11:00-12:00: Hopkins, Lattice systems and topological field theories

13:30-14:30: Cordova, Introduction to anomalies in quantum field theory

15:00-17:00: Problem session / open discussion / goodbye

Events

GCS2022 conference schedule and abstracts

by |Published

This is the schedule for the first week of the GCS2022 conference and school. Videos are available at https://pirsa.org/C22008.

Schedule

(All times are in Eastern Daylight Time}

Monday, June 6

9:30-9:45: Welcome and opening remarks

9:45-10:45: Justin Kaidi — Non-Invertible Symmetries in d>2

10:45-11:45: Coffee break

11:45-12:45: Andrei Okounkov — Monodromy and derived equivalences

12:45-14:00: Lunch

14:00-15:00: Emily Nardoni — Lessons from SU(N) Seiberg-Witten Geometry

15:00-15:45: Coffee break

15:45-16:45: Kevin Costello — Vertex algebras and self-dual Yang-Mills theory

Tuesday, June 7

9:00-10:00: Shu-Heng Shao — Non-invertible Global Symmetries in the Standard Model

10:00-10:45: Coffee break

10:45-11:45: Pavel Etingof — Analytic Langlands correspondence over C and R

11:45-12:00: Short break

12:00-13:00: Colleen Delaney — Hopf algebras play an analogous role in some topological and non-topological QFTs

13:00-14:45: Lunch

14:45-17:00: Gong show

17:00-18:00: Reception 

Wednesday, June 8

9:00-10:00: Cris Negron — A (kind of) monoidal localization theorem for the small quantum group

10:00-11:00: Coffee break

11:00-12:00: Iñaki García-Etxebarria — Symmetries from string theory

12:00-14:00: Lunch

14:00-15:00: Nathan Seiberg (Colloquium) — Quantum Field Theory, Separation of Scales, and Beyond

15:00-16:00: Coffee break

Thursday, June 9

 9:00-10:00: André Henriques — All unitary 2D QFTs share the same state space

10:00-10:45: Coffee break

10:45-11:45: Mina Aganagic — Knot categorification from homological mirror symmetry

11:45-12:00: Short break

12:00-13:00 David Reutter — Higher S-matrices and higher modular categories

13:00-14:45: Lunch 

14:45-15:45: Federico Bonetti — ’t Hooft anomalies of QFTs realized in string theory

15:45-16:30: Coffee break

Friday, June 10

9:00-10:00: Sakura Schäfer-Nameki — Non-Invertible Higher-Categorical Symmetries

10:00-10:45: Coffee break

10:45-11:45: Thomas Dumitrescu — Line Defect Quantum Numbers and Anomalies

11:45-12:00: Short break

12:00-13:00: Tudor Dimofte — TQFT’s and flat connections

13-14:45: Lunch

14:45-16:00: Posters and goodbyes

Abstracts

Aganagic: Knot categorification from homological mirror symmetry

Khovanov showed in ‘99 that the Jones polynomial arises as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups and to explain their meaning: what are they homologies of?

Homological mirror symmetry, formulated by Kontsevich in ’94, naturally produces hosts of homological invariants. Sometimes, it can be made manifest, and then its striking mathematical power comes to fore. Typically though, it leads to invariants which have no particular interest outside of the problem at hand.

I will explain that there is a vast new family of mirror pairs of manifolds for which homological mirror symmetry does lead to interesting invariants, and solves the knot categorification problem.

Federico Bonetti: ’t Hooft anomalies of QFTs realized in string theory

String theory constructions allow one to realize vast classes of non-trivial quantum field theories (QFTs), including many strongly coupled models that elude a conventional Lagrangian description. ’t Hooft anomalies for global symmetries are robust observables that are particularly well-suited to explore QFTs realized in string theory. In this talk, I will discuss systematic methods to compute anomalies of theories engineered with branes, using as input the geometry and flux configuration transverse to the non-compact directions of the branes worldvolume. Examples from M-theory and Type IIB string theory illustrate the versatility of this approach, which can capture both ordinary and generalized symmetries, continuous or discrete.

Costello: Vertex algebras and self-dual Yang-Mills theory

I’ll discuss a vertex algebra whose correlators are scattering amplitudes (and form factors) of self-dual Yang-Mills theory, for certain gauge groups and matter. The vertex algebra is a kind of vertex quantum group, and is a cousin of the affine Yangian. This is joint work with Natalie Paquette.

Delaney: Hopf algebras play an analogous role in some topological and non-topological QFTs

A frequent theme in mathematical approaches to quantum field theory is being able to draw intuitive but rigorous pictures of particle interactions. For example, in theories like QED and QCD, Feynman diagrams can be organized into a Hopf algebra structure in such a way that models renormalization of Feynman integrals. One can draw an analogy with the role of Hopf algebras in (2+1)D topological quantum field theory, where in this second setting one can think of certain string diagrams as Feynman diagrams whose Witten-Reshetikhin-Turaev amplitudes are encoded by the representation theory of some (weak) Hopf algebra. This talk is intended to (1) be accessible to both mathematicians and physicians, (2) invite a dialogue about the meaningfulness of this analogy and (3) serve as an aperitif to the later talks.

Dimofte: TQFT’s and flat connections

I will discuss some of the (higher) structure of TQFT’s that can be deformed by flat connections for continuous global symmetries, focusing on examples coming from twists of 3d supersymmetric theories, and the manifestation of this structure in boundary VOA’s.

Dumitrescu: Line Defect Quantum Numbers and Anomalies

I will consider four-dimensional gauge theories whose global symmetries admit certain discrete ’t Hooft anomalies that are intimately related to the (fractionalized) global-symmetry quantum numbers of Wilson-’t Hooft line defects in the theory. Determining these quantum numbers is typically straightforward for Wilson lines, but requires a careful analysis of fermion zero modes for ’t Hooft lines, which I will describe for several classes of examples. This in turn leads to a calculation of the anomaly. Along the way I will comment on how this understanding relates to some classic and recent examples in the literature. 

Etingof: Analytic Langlands correspondence over C and R

I will review the analytic component of the geometric Langlands correspondence, developed recently in my joint work with E. Frenkel and D. Kazhdan (based on previous works by other authors, including A. Braverman, R. Langlands, J. Teschner, M. Kontsevich), with a special focus on archimedian local fields, especially R. This is based on our work with E. Frenkel and D. Kazhdan and insights shared by D. Gaiotto and E. Witten.

García-Etxebaria: Symmetries from string theory

It is possible to construct interesting field theories by placing string theory on suitable singular geometries, and adding branes. In the fairly special cases where Lagrangians are known for the resulting theories, field theory arguments often show that these theories have generalised symmetry structures. In this talk I will review recent work developing a dictionary, valid even in the absence of a known Lagrangian description, between properties of the string theory geometry and generalised symmetries of the associated field theories.

Henriques: All unitary 2D QFTs share the same state space

A unitary 1d QFT consists of a Hilbert space and a Hamiltonian. A group acting on a 1d QFT is a group acting on the Hilbert space, commuting with the Hamiltonian. Note that the *data* of an action only involves the Hilbert space. The Hamiltonian is only there to provide a constraint. Moreover, all 1d QFT have isomorphic Hilbert spaces (except in special cases, e.g. in the case of a 1d TQFT, when the Hilbert space is finite dimensional).

A unitary 2d QFT consists of the 0-dimensional and 1-dimensional part of the QFT, along with the data of the Stress-energy tensor. An action of a fusion category on a 2d QFT is again something where the *data* only involves the 0-dimensional and 1-dimensional part of the QFT, while the Stress-energy tensor is only there to provide a constraint. The upshot is that it makes sense to act on the 0-dimensional and 1-dimensional part of the QFT. Moreover, I conjecture that all 2d QFTs have isomorphic 0-dimensional + 1-dimensional parts (except in special cases, e.g. in the case of a chiral CFT).

Kaidi: Non-Invertible Symmetries in d>2

In this talk I will review some recent progress in the study of non-invertible symmetries in dimensions d>2. After introducing known constructions and describing how they lead to constraints on RG flows, I will discuss how non-invertible symmetries can also be used to obtain new RG flows. This involves the notion of “non-invertible twisted compactification,” which can be used to construct e.g. novel 3d N=6 theories from 4d N=4 SYM. Finally, I will describe upcoming work in which we give a partial criterion for determining whether a given non-invertible symmetry is “intrinsically non-invertible”, or whether it can be recast (in an appropriate sense) as a standard invertible symmetry with an anomaly.

Okounkov: Monodromy and derived equivalences

This will be an introductory discussion of our joint work with Roman Bezrukavnikov. Given a symplectic resolution X, one may study its Gromov-Witten theory and the monodromy group of the curve-counting functions in the Kähler variables. There is also a large group of derived autoequivalences of X coming from its quantization in large prime characteristic, as studied by Bezrukavnikov and collaborators. Conjecturally, the action of the latter group on K(X) is identified with the former group, and we prove this for many X.

Nardoni: Lessons from SU(N) Seiberg-Witten Geometry

Motivated by applications to soft supersymmetry breaking, we revisit the Seiberg-Witten solution for N=2 super Yang-Mills theory in four dimensions with gauge group SU(N). We present a simple exact Taylor series expansion for the periods obtained at the origin of moduli space, thereby generalizing earlier results for SU(2)  and SU(3). With the help of these analytic results and others, we analyze the global structure of the Kahler potential, presenting evidence for a conjecture that the unique global minimum is the curve at the origin of moduli space.

Two applications of these results are considered. Firstly, we analyze candidate walls of marginal stability of BPS states on special slices for which the expansions of the periods simplify. Secondly, we consider soft supersymmetry breaking of the N=2 theory to non-supersymmetric four-dimensional SU(N) gauge theory with two massless adjoint Weyl fermions (“adjoint QCD”). The Seiberg-Witten Kahler potential and strong coupling spectrum play a crucial role in this analysis, which ultimately leads to an exploration of the adjoint QCD phase diagram.

Negron: A (kind of) monoidal localization theorem for the small quantum group

I will talk about a monoidal localization theorem for the small quantum group u_q(G), where G is a reductive algebraic group and q is a root of unity.  In joint work with Julia Pevtsova, we show that the category of representations for u_q(G) admits a fully faithful tensor embedding into the category of coherent sheaves over a “quantum” flag variety.  This quantum flag variety is, essentially, some finitely fibered space over the classical flag variety G/B.  I will explain how this embedding theorem codifies certain relationships between the small quantum group and its quantum Borels.

Reutter: Higher S-matrices and higher modular categories

Every braided fusion category has a `framed S-matrix pairing’ which records the braiding between simple objects. Non-degeneracy/Morita invertibility of the category (aka `modularity’ in the oriented case) is equivalent to non-degeneracy of this pairing. I will define higher-dimensional versions of S-matrices which pair morphisms of complementary dimension in higher semisimple categories and sketch a proof that these pairings are non-degenerate if and only if the higher category is. Along the way, I will introduce higher semisimple categories and higher fusion categories and interpret these results in terms of the associated anomalous topological quantum field theories. This is based on joint work in progress with Theo Johnson-Freyd.

Schafer-Nameki: Non-Invertible Higher-Categorical Symmetries

I will discuss a proposal for generating non-invertible symmetries in QFTs in d>2, by gauging outer automorphisms. First this will be illustrated in 3d, where the framework is relatively well established, and then extended to higher dimensions. For 4d gauge theories, a comparison to other approaches to non-invertible symmetries is provided, in particular the map to gauging theories with mixed anomalies. This talk is based on work that appeared in 2204.06564 and in progress, with Lakshya Bharwaj (Oxford), Lea Bottini (Oxford) and Apoov Tiwari (Stockholm).

Seiberg: Quantum Field Theory, Separation of Scales, and Beyond

We will review the role of Quantum Field Theory (QFT) in modern physics.  We will highlight how QFT uses a reductionist perspective as a powerful quantitative tool relating phenomena at different length and energy scales.  We will then discuss various examples motivated by string theory and lattice models that challenge this separation of scales and seem outside the standard framework of QFT.  These lattice models include theories of fractons and other exotic systems.

Shao: Non-invertible Global Symmetries in the Standard Model

We identify infinitely many non-invertible generalized global symmetries in QED and QCD for the real world in the massless  limit.  In QED, while there is no conserved Noether current for the  axial symmetry because of the ABJ anomaly, for every rational angle, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively,  it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic U(1) gauge field.  These conserved symmetry operators do not obey a group multiplication law, but a non-invertible fusion algebra over  TQFT coefficients. These non-invertible symmetries lead to selection rules, which are consistent with the scattering amplitudes in QED. We further generalize our construction to QCD, and show that the neutral pion decay  can be understood from a matching condition of the non-invertible global symmetry.

Publications

Non-Invertible Chiral Symmetry and Exponential Hierarchies

by |Published

Córdova, Ohmori, https://arxiv.org/abs/2205.06243

Publications

Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions

by |Published

Choi, Córdova, Hsin, Lam, Shao, https://arxiv.org/abs/2204.09025

Events

GCS2022 Conference and School

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Perimeter Institute for Theoretical Physics, 6-17 June 2022

The conference and school will be “hybrid” style: most speakers will be in-person, with some virtual speakers; about half of the participants will be in-person, and talks will also be live-streamed on Zoom. Videos will be archived on pirsa.org. Further details can be found at Perimeter’s conference website. Registration has now closed.

Immediately following GCS2022 will be the school and conference QFT for Mathematicians. Although the two events are logically separate, with different organizing teams and registrations, we purposefully scheduled the two schools to be back-to-back so that students can attend both.

Program

6-10 June 2022: Research Conference

Speakers: Mina Aganagic, Federico Bonetti, Kevin Costello, Colleen Delaney, Tudor Dimofte, Thomas Dumitrescu, Pavel Etingof, Iñaki García Etxebarria, André Henriques, Justin Kaidi, Emily Nardoni, Cris Negron, Andrei Okounkov, David Reutter, Sakura Schäfer-Nameki, Nathan Seiberg, Shu-Heng Shao

Conference schedule, titles, and abstracts

13-17 June 2022: School

Lecturers: Clay Córdova, Dan Freed, Mike Hopkins, Kantaro Ohmori

TAs: Arun Debray, Ho Tat Lam

School schedule, problem sets, and lecture notes

Lecture series

Thorngren Scandinavian Lectures

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Uppsala University May 27-June 3, 2022 (POSTPONED)

The precise program will be announced on this page closer to the date. It is possible to follow these lectures from zoom. Please, contact the organizer to be added to the mailing list.

Lecture 1 – May 27, 2022

Lecture 2 – May 30, 2022

Lecture 3 – June 1st, 2022

Lecture 4 – June 3rd, 2022

Organizer: Michele Del Zotto

Events

San Diego Focus Meeting

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January 10-14, 2022

Meeting for members of the collaboration only.

PARTICIPANTS

Ibrahima Bah (JHU)

Alberto Cattaneo (University of Zurich)

Clay Córdova (University of Chicago)

Michele Del Zotto (Uppsala University)

Thomas Dumitrescu (UCLA)

Dan Freed (University of Texas, Austin)

Iñaki García Etxebarria (Durham University)

Michael Hopkins (Harvard)

Ken Intriligator (UCSD)

Theo Johnson-Freyd (Dalhousie University and PI)

David Jordan (University of Edinburgh)

Nytia Kitchloo (JHU)

Julia Plavnik (Indiana University)

Ryan Thorngren (KITP)

SCHEDULE

Mornings 9-12 and Afternoons 14-16: structured talks/discussions. All the other times: free discussions/collaboration. The precise agenda for each day is decided by the participants on the day prior.

Local organizer: Ken Intriligator

Lecture series

Claudia Scheimbauer on “Dualizabitility, higher categories, and topological field theories”

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Claudia has given a series of seven lectures at the Copenhagen Center for Geometry and Topology.

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.

Lecture series

Theo Johnson Freyd on “Operators and higher categories in quantum field theory”

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Theo has given a lecture series at the Korean Institute for Advance Studies, as part of a long-running series organized by Minhyong Kim. The abstract for the talks read:

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.

Events

GCS: Kick-Off Meeting

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October 11-13, 2021

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

Schedule

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

MONDAY

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

TUESDAY

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

WEDNESDAY

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

Program

Discussion sessions

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

Guest Lectures

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).

Lectures from our collaboration

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

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