Organizers: Michele Del Zotto, Constantin Teleman, Yifan Wang
PRELIMINARY SCHEDULE
MONDAY
14:00 – 14:30 ::: Vasily Krylov (Harvard)
14:30 – 15:00 ::: Andrea Grigoletto (Uppsala)
15:00 – 15:30 ::: Will Stewart (TMU)
coffee/tea and discussion
16:00 – 16:30 ::: Jake McNamara (SCGP)
16:30 – 17:00::: Lorenzo Riva (Harvard)
17:00 – 17:30 ::: Bowen Yang (Harvard)
17:30 – 18:00 ::: Matthew Yu (Oxford)
TUESDAY
9:00 – 10:00 ::: Shu-Heng Shao (MIT)
coffee break
10:30 – 11:30 ::: Zhenghan Wang (UCSB)
11:30 – 12:30 ::: Nikita Sopenko (IAS)
lunch break and discussions (coffee/tea 15:30)
16:00 – 17:00 ::: Noah Snyder (Indiana)
WEDNESDAY
9:00 – 10:00 ::: Nathan Tantivasadakarn (Caltech and Stony Brook)
coffee break
10:30 – 11:30 ::: Nicolai Reshetikhin (UC Berkeley and BIMSA)
12:00 – 13:00 ::: Zohar Komargodski (SCGP)
lunch break and discussions
14:00 – 15:00 ::: NYU Seminar: Clay Córdova
15:30 – 16:30 ::: Wine and cheese reception and discussions
ABSTRACTS
Clay Córdova Electron-Monopole Scattering from Conformal Field Theory — S-wave scattering of electrons off of heavy magnetic monopoles is an important problem both in formal theory and in particle phenomenology. It has long been understood that this scattering can be effectively studied in two-dimensional conformal field theory with a recently appreciated crucial role played by topological line defects and generalized global symmetry. We use this formalism to compute the S-matrix to leading order in electromagnetic coupling and find agreement with expectations from unitarity.
Andrea Grigoletto Higher representations and the strip algebra — Building on the correspondence between C-modules and C-symmetric TQFTs, the strip-algebra method provides a useful way to organize the representation theory underlying the spectra of massive two-dimensional theories. In this talk, I will describe how to extend this viewpoint to higher dimensions. The first step is to formulate an analogue of the correspondence between C-symmetric higher-dimensional TQFTs and projective modules. I will then illustrate how this works in a simple higher-dimensional setting, using scalar QCD as an interesting example.
Zohar Komargodski Spin-Flux duality: An application of symmetries, anomalies, and renormalization group flows on defects
Vasily Krylov Graded traces on quantized Coulomb branches via 3D mirror symmetry — In 2016, Braverman, Finkelberg, and Nakajima gave a mathematical definition of the Coulomb branch of a three-dimensional topological quantum field theory associated with a reductive group and its complex representation. The resulting Coulomb branch is an affine Poisson variety, and they also constructed its quantizations. The notion of a graded trace on a quantized Coulomb branch can be viewed as a natural generalization of a character. Graded traces appear in both mathematical physics and representation theory, and are closely related to partition functions of the corresponding field theories, the “short” star products of Beem–Peelaers–Rastelli, and the Frenkel-Reshetikhin q-characters. I will describe a conjectural framework for understanding graded traces (as well as a certain D-module on which these traces are functionals), using the geometry of Higgs branches. Time permitting, I will sketch a proof of the conjecture in the case of ADE quiver gauge theories. Based on joint work with Dinkins and Karpov.
Jake McNamara (T)QFT Reconstruction from Reflection Positivity — Using ideas from quantum gravity and Tannakian reconstruction, I will sketch a proof of the following theorem: any reflection positive partition function is the partition function of a (T)QFT. The main technical step is the construction of the symmetric tensor category of gravitational sectors, viewing the given partition function as the partition function of a holographically dual quantum theory of gravity. For physicists, this category should be viewed as a categorification of the baby universe Hilbert space of Coleman, Giddings, and Strominger. For mathematicians, it should be viewed as a generalization of the category GL(t) of Deligne and Milne.
Nicolai Reshetikhin Invariants of knots with flat connections in the complement — This talk is an overview of a construction of such invariants from representation theory of quantum groups at a root of unity,
and of some new results.
Lorenzo Riva How to add adjoints to a category — A fundamental concept in categorical algebra is the notion of an adjunction between two 1-morphisms in a 2-category, whose definition is a generalization of adjoint functors between categories. The concept extends down to objects in a monoidal category (after delooping, it recovers the familiar notion of the dual of an object) and up to k-morphisms in an n-category. Consider the collection of symmetric monoidal n-categories that are fully dualizable, i.e. with the property that all of their objects admit a dual and all of their k-morphisms, for k < n, admit a left and a right adjoint. The cobordism hypothesis tells us that the simplest (non-trivial) fully dualizable symmetric monoidal n-category, the one generated by a single object, is the n-category of iterated framed cobordisms. It is really surprising that the categorical/combinatorial procedure of repeatedly adding duals and adjoints ends up giving a very concrete, geometric gadget. In this talk we will try to see the emerging geometry in the simpler case of adding adjoints to a 1-category, which thankfully can be drawn in just two dimensions, and we will sketch what might happen in higher dimensions. This is based on joint work with Martina Rovelli.
Shu Heng Shao CPT and anomalies — CPT (or more precisely, CRT) is a universal symmetry in any relativistic, local QFT. I will discuss how CRT can be used as a crutch to detect ‘t Hooft anomalies of global symmetries in field theory and lattice models. Examples include a mod 8 anomaly relevant for type II superstring theory and the Kitaev chain.
Nikita Sopenko Reflection positivity and invertible phases of quantum many-body systems — Reflection positivity is a property that is usually taken as an assumption in the classification of topological phases of matter via continuous quantum field theories. For general quantum many-body systems, this property does not hold. This raises the question of whether it somehow emerges in the effective theory from the microscopic description, thereby justifying the field-theoretic approach. In this talk, I will discuss reflection positivity in the context of invertible phases of two-dimensional lattice systems. I will explain why every such phase admits a reflection-positive representative, and why inverse phases are represented by complex conjugate states. I will also introduce an index that distinguishes these phases and is conjecturally related to the chiral central charge.
Will Stuart Relative homotopy field theories — I will give a topological definition of left and right relative field theories. Using this topological perspective, I will define two classes of relative field theories in the setting of homotopy field theory. These theories relax the usual pi-finiteness condition. I will illustrate how this provides a generalization of the usual symmetry sandwich of Freed-Moore-Teleman.
Nat Tantivasadakarn Color code and non-invertible symmetries —
The SymTFT is a powerful framework to study gapped and gapless phases with a given symmetry in the continuum. However, on the lattice, a single SymTFT can admit multiple distinct microscopic realizations. I will present an example where an appropriate choice of microscopic model can help constrain the phase diagram by exploiting emanant symmetries without additional fine-tuning. Specifically, I will construct a simple qubit Hamiltonian with Rep(D₈) symmetry and discuss exact and numerical results of the phase diagram. The lattice SymTFT is closely related to an error correcting code called the color code.
Bowen Yang Extracting TQFT from Pauli Stabilizer codes — I’ll explain a natural procedure to extract topological field theory from a class of 2+1d toy models. The whole setup is equivalent to a problem in commutative algebra of independent interest. The proofs rely on key results about Cohen-Macaulay rings.
Matthew Yu Geometric Categories for Continuous Gauging — We present a unified categorical framework which encodes gauging of continuous and finite symmetries in arbitrary spacetime dimension. We show how our framework can identify electric and magnetic symmetries expected in $G$-gauge theory, and how to capture electric symmetry breaking resulting from the addition of charged matter. We work with geometric categories, i.e. categories internal to stacks. This allows us to extend (de)equivariantization of fusion categories to continuous groups, construct a functorial SymTFT and boundaries for this theory, and compute the relevant categories of endomorphisms and Drinfeld centers.
Zhenghan Wang Weak Hopf monad symmetries of anyon models — Finite group symmetries of anyon models and their gauging are well-understood, and we might ask what are the most general symmetries of anyon models beyond groups. I would argue they are weak Hopf monads generalizing categorical weak Hopf algebras. A theory generalizing the finite group case will be outlined based on joint work with Cui and Shokrian-Zini, and further on-going work with Cui, Galindo,…
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.
