Beaudry: From Stabilizer Codes to Fracton Phases of Matter Fracton orders are quantum systems characterized by the presence of excitations with restricted mobility. Many examples are described by stabilizer codes, that is, lattice Hamiltonians constructed from commuting generalized Pauli matrices. In this lecture series, we will explore the algebraic framework for fracton phases and mutual statistics that emerges from the study of stabilizer codes. The course will be mostly based on work of Haah and Ruba-Yang, as well as joint work with Hermele, Shirley, Wickenden and Yang.

Wang: Seeing through Defects in Quantum Field Theory Defects provide powerful probes of quantum field theory. Supported on lower-dimensional subspaces, they host their own degrees of freedom, generalize ordinary local operator algebras, and exhibit novel phases connected by renormalization group flows. We will discuss modern developments in defect dynamics, focusing on their interplay with symmetries and anomalies, and explain how they encode structural information about the ambient QFT.

Wang: Seeing through Defects in Quantum Field Theory Defects provide powerful probes of quantum field theory. Supported on lower-dimensional subspaces, they host their own degrees of freedom, generalize ordinary local operator algebras, and exhibit novel phases connected by renormalization group flows. We will discuss modern developments in defect dynamics, focusing on their interplay with symmetries and anomalies, and explain how they encode structural information about the ambient QFT.

Beaudry: From Stabilizer Codes to Fracton Phases of Matter Fracton orders are quantum systems characterized by the presence of excitations with restricted mobility. Many examples are described by stabilizer codes, that is, lattice Hamiltonians constructed from commuting generalized Pauli matrices. In this lecture series, we will explore the algebraic framework for fracton phases and mutual statistics that emerges from the study of stabilizer codes. The course will be mostly based on work of Haah and Ruba-Yang, as well as joint work with Hermele, Shirley, Wickenden and Yang.

Wang: Seeing through Defects in Quantum Field Theory Defects provide powerful probes of quantum field theory. Supported on lower-dimensional subspaces, they host their own degrees of freedom, generalize ordinary local operator algebras, and exhibit novel phases connected by renormalization group flows. We will discuss modern developments in defect dynamics, focusing on their interplay with symmetries and anomalies, and explain how they encode structural information about the ambient QFT.

Tantivasadakarn: Symmetries and their gauging on the lattice Much progress of generalized symmetries has been formulated in the continuum. However, on the lattice, symmetries can behave quite differently, and its fundamental organizing principles are still in development. I will give a broad introduction to symmetries on the lattice and how to gauge them. Time permitting, I will discuss subtleties in matching symmetries on the lattice to those in the continuum.

Beaudry: From Stabilizer Codes to Fracton Phases of Matter Fracton orders are quantum systems characterized by the presence of excitations with restricted mobility. Many examples are described by stabilizer codes, that is, lattice Hamiltonians constructed from commuting generalized Pauli matrices. In this lecture series, we will explore the algebraic framework for fracton phases and mutual statistics that emerges from the study of stabilizer codes. The course will be mostly based on work of Haah and Ruba-Yang, as well as joint work with Hermele, Shirley, Wickenden and Yang.

Schlank: intro to stable Homotopy theory with a view towards quantum field theory In this series of talks I will give an introduction to the the subject of stable Homotopy theory, We shall discuss spectra, higher algebra, chromatic homotopy and other fundamental notion of this theory.

Bonetti: Aspects of SymTFT and Continuous Symmetries In the modern perspective, the generalized global symmetries of a quantum field theory (QFT) are encoded in the spectrum of its (extended) topological operators. Finite symmetries, in particular, are expected to be captured by higher fusion categories. The Symmetry Topological Field Theory (SymTFT) provides a powerful framework for analyzing many facets of these symmetry structures. While categorical methods and SymTFT techniques are by now well established for finite symmetries, a complete and systematic framework for continuous symmetries is still being developed. In these lectures, I will present some features of SymTFTs for continuous symmetries from a physicist’s perspective. As a complementary approach to better understanding the properties of symmetry operators associated with continuous symmetries, I will also discuss aspects of their realization in holography and string theory.

Tantivasadakarn: Symmetries and their gauging on the lattice Much progress of generalized symmetries has been formulated in the continuum. However, on the lattice, symmetries can behave quite differently, and its fundamental organizing principles are still in development. I will give a broad introduction to symmetries on the lattice and how to gauge them. Time permitting, I will discuss subtleties in matching symmetries on the lattice to those in the continuum.

Schlank: intro to stable Homotopy theory with a view towards quantum field theory In this series of talks I will give an introduction to the the subject of stable Homotopy theory, We shall discuss spectra, higher algebra, chromatic homotopy and other fundamental notion of this theory.

Bonetti: Aspects of SymTFT and Continuous Symmetries In the modern perspective, the generalized global symmetries of a quantum field theory (QFT) are encoded in the spectrum of its (extended) topological operators. Finite symmetries, in particular, are expected to be captured by higher fusion categories. The Symmetry Topological Field Theory (SymTFT) provides a powerful framework for analyzing many facets of these symmetry structures. While categorical methods and SymTFT techniques are by now well established for finite symmetries, a complete and systematic framework for continuous symmetries is still being developed. In these lectures, I will present some features of SymTFTs for continuous symmetries from a physicist’s perspective. As a complementary approach to better understanding the properties of symmetry operators associated with continuous symmetries, I will also discuss aspects of their realization in holography and string theory.

Tantivasadakarn: Symmetries and their gauging on the lattice Much progress of generalized symmetries has been formulated in the continuum. However, on the lattice, symmetries can behave quite differently, and its fundamental organizing principles are still in development. I will give a broad introduction to symmetries on the lattice and how to gauge them. Time permitting, I will discuss subtleties in matching symmetries on the lattice to those in the continuum.

Schlank: intro to stable Homotopy theory with a view towards quantum field theory In this series of talks I will give an introduction to the the subject of stable Homotopy theory, We shall discuss spectra, higher algebra, chromatic homotopy and other fundamental notion of this theory.

Bonetti: Aspects of SymTFT and Continuous Symmetries In the modern perspective, the generalized global symmetries of a quantum field theory (QFT) are encoded in the spectrum of its (extended) topological operators. Finite symmetries, in particular, are expected to be captured by higher fusion categories. The Symmetry Topological Field Theory (SymTFT) provides a powerful framework for analyzing many facets of these symmetry structures. While categorical methods and SymTFT techniques are by now well established for finite symmetries, a complete and systematic framework for continuous symmetries is still being developed. In these lectures, I will present some features of SymTFTs for continuous symmetries from a physicist’s perspective. As a complementary approach to better understanding the properties of symmetry operators associated with continuous symmetries, I will also discuss aspects of their realization in holography and string theory.

Opening Remarks

• Mariana Grana (Institut Henri Poincare)

McGreevy: Entanglement Bootstrap and Symmetric Conformal Field Theories Entanglement Bootstrap is a research program to understand and extract the universal data of quantum many body states from local entanglement information. The program proceeds by identifying conditions satisfied by the reduced density matrix of a ball for a given class of fixed-point states. I'll describe our progress at using this philosophy to learn about the space of conformal field theories in 1+1 dimensions and above, and how it can be refined by the imposition of symmetries. The talk is based on work with Xiang Li, Ting-Chun (David) Lin, and Rolando Ramirez Camasca.

Fidkowski: Chiral Lattice gauge theories from symmetry disentanglers We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers: constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for U (1) symmetries with mixed ’t Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional “3450” chiral gauge theory, and we argue that a related construction applies to the U (1) hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.

Córdova: Holomorphic CFT and Modular Tensor Categories We study the interplay between holomorphic conformal field theory and dualities of 3d topological quantum field theories generalizing the paradigm of level-rank duality. A holomorphic conformal field theory with a Kac-Moody subalgebra implies a topological interface between Chern-Simons gauge theories. Upon condensing a suitable set of anyons, such an interface yields a duality between topological field theories. We illustrate this idea using the c = 24 holomorphic theories classified by Schellekens, which leads to a list of novel sporadic dualities. Some of these dualities necessarily involve condensation of anyons with non-abelian statistics, i.e. gauging non-invertible one-form global symmetries. Several of the examples we discover generalize from c = 24 to an infinite series.

Wen: Chiral U(1) higher gauge theory and associated chiral (higher) U(1) anomaly on spacetime lattice I will discussion how to put chiral U(1) higher gauge theories (such as Chern-Simons theory) on spacetime lattice. This gives us lattice model with chiral (higher) U(1) anomaly that has no gapped phases, without explicitly breaking the U(1) symmetry.

Zhang: Spontaneous symmetry breaking in non-equilibrium dynamics We present simple toy models of mixed state phases and phase transitions beyond thermal transitions, observed from the steady state structure of strongly symmetric local Lindbladians. To diagnose these mixed state phases and study phase transitions, we introduce a local order parameter that can be computed efficiently to obtain critical exponents.

Leung: Basepoint anomalies and the vanishing of partition functions We discuss a sufficient (but not necessary) condition for partition functions of QFTs to vanish, meaning that generic correlators are zero unless suitable operators are inserted. Vanishing partition functions have been observed in multiple occasions in the past. Here we provide a systematic perspective from the moduli space to unify these seemingly independent results. In the context of D-branes, we recover the Freed-Witten anomaly cancellation condition, and our approach allows us to extend it to branes in non-perturbative string/M-theory backgrounds.

Krulewski: Invertible Field Theories and the Bott Spiral Reflection-positive fully extended invertible field theories on manifolds with twisted spin structures provide homotopical classifications for fermionic SPTs. Free fermion SPTs, a subset, are classified by K-theory. Generalizing work of Freed--Hopkins, we define and compute homotopical "free-to-interacting maps" between these classifications for the special case of the "Bott spiral" of SPTs studied originally by Queiroz--Khalaf--Stern. Time permitting, I will discuss some subtleties of our construction, such as compatibility with dimensional reduction, continuous versus discrete symmetry types, and the lack of interacting Morita invariance.

Kapustin: Higher symmetries and homotopy theory in quantum lattice models It is generally accepted that the interplay of symmetry and locality in Quantum Field Theory leads one to introduce higher or generalized symmetries. While ordinary (0-form) symmetries form a group, incorporating invertible higher symmetries requires one to replace groups with higher groups, that is, finite connected homotopy types. It is far from obvious how to attach such a gadget to a local QFT. In this talk I discuss this problem in the context of quantum lattice models. I will show how to attach a connected homotopy (d+1)-type to lattice models in d spatial dimensions by exploiting a construction which is a non-abelian analog of the Cech homology of a precosheaf. This homotopy type encodes all higher symmetries as well as all 't Hooft anomalies. A key ingredient in the construction is the equivalence between connected homotopy (d+1)-types and crossed d-cubes of groups due to Loday and Ellis-Steiner.

Copetti: When Symmetries Twist - Anomaly inflow on Monodromy Defects Symmetries can constrain exactly properties of dynamical defects. In this talk I will focus on Monodromy defects, which describe dynamical terminations of the symmetry operators themselves. I will review how they provide natural detectors of nontrivial disordered phases (SPTs), and employ anomaly inflow to then derive exact constraints on their physical properties in anomalous gapless theories.

Zhu: Towards Weinstein's category for shifted symplectic structures via groupoids Under the motto that “everything is Lagrangian”, Alan Weinstein proposed a category whose objects are symplectic manifolds and whose morphisms are Lagrangian correspondences. As Poisson geometry has developed over the past decades—motivated largely by classical mechanics—additional structures such as Poisson, Dirac, and Courant have emerged. Safronov, drawing on the shifted symplectic structures of Pantev–Toën–Vaquié–Vezzozi (PTVV), places these geometric structures into a unified framework by viewing them as symplectic structures of various shifts. The quantization of 1-shifted symplectic geometry, as developed through Meinrenken’s work and the Freed–Hopkins–Teleman theorem, naturally takes values in twisted K-theory and K-homology, and in particular in the Verlinde ring. In this sense, shifted symplectic geometry provides not only a conceptual framework for Poisson-type structures, but also a geometric counterpart to K-theoretic and representation-theoretic invariants. Calaque-Haugseng-Scheinbauer constructed a TFT with target a version of such Weinstein category for shifted symplectic structures using higher derived stacks in the sense of Toën–Vezzosi. While the language of stacks is intrinsic, it remains rather implicit for differential geometers and therefore difficult to use for concrete calculations. Pridham approaches higher derived stacks via presentations by groupoids, a setting far more familiar to researchers in differential geometry, dynamical systems, and noncommutative geometry. E.g. in nummeric Hamiltonian systems, explicit symplectic groupoids were in need to construct Poisson integrator. Inspired by Pridham’s approach, we work towards a Weinstein category for shifted symplectic structures using higher derived Lie groupoids. However, the topology we use for derived manifolds differs from that of Toën–Vezzosi and Pridham: we work with fibrations admitting local sections, which allow us to treat the odd line and cotangent groupoids—key examples in Poisson geometry. We prove that when intersections are transversal in a higher derived sense, the composition of Lagrangian correspondences remains Lagrangian. As always, there is technical difficulty when intersection is not transversal. Broadly, two methods are available: (1) perturbation, as in the Fukaya category, which yields explicit results but requires delicate analysis; or (2) (fibrant) replacement, as in PTVV, which is conceptual but requires a homotopical framework and typically leaves explicit fibrant replacements to be worked out. We take the second approach and build an iCFO (incomplete category of fibrant objects) for derived higher Lie groupoids and provide explicit fibrant replacements using a collection of tubular neighborhood theorems: Weinstein’s Lagrangian tubular neighborhood theorem, and the Hoyo–Fernandes version for Lie groupoids via the Crainic–Fernandes–Torres PMCT program. As applications, we use Calaque–Safronov’s trick, then singular symplectic reduction, quasi-symplectic reduction, and Lu–Weinstein reduction, all appear as shifted symplectic derived Lie groupoids. This is based on a joint work in progress with M. Cueca, F. Dorsch, and R. Sjamaar.

Debray: What are the possible anomalies of four-dimensional topological field theories? What are the possible anomalies of four-dimensional topological field theories? Abstract: Whether a given quantum field theory (QFT) can be approximated by a topological field theory (TFT) at long distances is an important qualitative aspect of the QFT. Since the anomaly of a QFT is invariant under such an approximation, we can attack this question by asking, "of all possible anomalies, which ones are anomalies of TFTs?", which is a rigorous mathematical question. In joint work with Weicheng Ye and Matthew Yu, we solve this question for four-dimensional Spin x_{\pm 1} G TFTs: the bottommost layer of the Atiyah--Hirzebruch spectral sequence is a complete obstruction to being the anomaly of a TFT. Moreover, when this obstruction vanishes, there is an algorithmic construction of a TFT realizing the anomaly. Our work builds on Décoppet--Yu's symmetry extension construction for fusion 2-categories as well as Córdova--Ohmori's answer to this question for cyclic G.

Decoppet: The Classification of 3+1d Symmetry Enriched Topological Order I will show that a 2-categorical version of (de-)equivariantization can be used to classify (3+1)d topological orders with a finite G-symmetry. As a consequence, in the genuinely fermionic case, I will explain how the categorical data necessary to define these theories agrees with that arising from a fermionic generalization of the Wang-Wen-Witten construction of bosonic topological theories with G-symmetry saturating an anomaly. This is joint work with Matthew Yu.

Calaque: A discrete 1d TFT with a point defect I will start by recalling a general result, obtained together with Victor Carmona (Leipzig), about "algebras over not too little disks". I will then illustrate this result in dimension 1, using a construction (of a discrete 1d TFT on the line with a point defect) from a work in progress that is joint with Maziar Farahzad (Toronto). I will finally explain how this construction in particular recovers the well-known quantization of linear coisotropic reduction.

Brochier: Swiss-Cheese, Poisson groups and topology As observed by many authors, the theory of finite-dimensional Poisson algebraic groups and of their quantizations has a natural interpretation in terms of 2d TFTs equipped with a pair of transverse boundary conditions. There are then various dualities in this theory which are reflected in symmetries of the associated TFT. In this talk we'll give an overview of this approach with an eye toward applications in low-dimensional topology. One of our main motivation was to understand the work of Alekseev--Enriquez--Torrossian and of Bar-Natan--Dancso which shows a surprising connection between the Kashiwara-Vergne conjecture and the Duflo isomorphism in Lie theory, and the braid groups actions on character varieties of punctured discs.

Bottini: Topological order enriched by non-invertible symmetry via anyon condensation In this talk, I will discuss a notion of topological order enriched by a non-invertible symmetry. For invertible symmetry enriched topological order, a well-established formalisation is available in terms of a G-crossed braided fusion category. By considering the condensation of an arbitrary algebra of charges in a quantum double model, a generalisation of this framework naturally emerges. In particular, I will show the topological order after condensation can be described as a hypergroup-graded extension of the category of deconfined excitations. This has a hypergroup symmetry which acts in a typically non-invertible manner on the confined and deconfined excitations in a way that is compatible with the grading. I will illustrate the general theory through a simple example.

Rayhaun: Hypergroup Symmetry in Relative Quantum Field Theories and Chiral Algebras There is by now a beautiful theory of generalized symmetries for 2D QFTs based in the physics of topological line defects and the mathematics of tensor categories. I will sketch a generalization of this theory to the setting of *relative* 2D QFTs (i.e. QFTs which live at the boundary of a bulk 3D topological order), emphasizing various new algebraic structures that arise, such as hypergroups. This theory is particularly rich when applied to the chiral half of a CFT, and I will spend most of the time unpacking this special case and discussing applications.

Antinucci: A Twist on Scattering from Defect Anomalies Scattering processes involving particles and very heavy probes, modeled by defects, have recently been shown to exhibit an unexpected feature: radiation can be emitted into twisted sectors rather than into the ordinary Hilbert space. In this talk, I will explain that the underlying mechanism is rooted in the subtle realization of bulk symmetries in the presence of defects. In particular, certain defect ’t Hooft anomalies allow charge to be trapped on the defect during scattering, thereby opening channels that would otherwise be forbidden by conservation laws. This viewpoint provides a systematic way to predict the phenomenon in new settings, including massive theories. I will present the general mechanism and discuss explicit examples in 1+1 dimensions.

Ji: Self-dual Higgs transitions: Toric code and beyond In (2+1)d, the transition from the Z_2 toric code phase to a trivial phase can be driven by the condensation of either the electric anyon e or magnetic anyon m. When the system has a global Z_2 “self-duality” symmetry that exchanges e and m anyons, the naive anyon-condensation picture no longer holds since e and m cannot simultaneously condense. Numerical studies have found a continuous transition from the toric code with the Z_2 self-duality symmetry to a trivial phase with the Z_2 spontaneously broken. Since then, a natural field-theoretic understanding of this transition has remained an open challenge. In this talk, I will propose an SO(4) Chern-Simons-Higgs (CSH) theory at level k=2 as a natural mean-field description of this self-dual transition. I will further show that varying the integer level k gives a broader series of analogous transitions involving non-Abelian topological orders, including the double Fibonacci order at k=3 and the S_3 quantum double at k=4. https://arxiv.org/abs/2601.20945

Zheng: Non-invertible Symmetries in Weyl Fermions, and Applications to Fermion-Boundary Scattering Problem We discuss a family of non-invertible topological defects in two-dimensional theories of n Weyl fermions. The construction relies on the existence of G-symmetric conformal boundary conditions for nDirac fermions. Upon unfolding, these boundary conditions become topological defects D of n Weyl fermions that intertwine the two G-representations, and they are generically non-invertible. We illustrate this construction when G= U(1)^n, where the topological defect D can be shown to be a duality defect associated with gauging certain finite abelian group Γ. By contrast, for certain non-Abelian symmetry including the G= SU(2) symmetry appearing in the 1-5-7-8-9 problem, we prove that D cannot be realized as a duality defect for gauging any finite Abelian group. We explain how the duality-defect perspective can be used to re-derive the fermion scattering from a conformal boundary.

Pace: Infinite-order lattice anomalies and CPT A key property of a global symmetry's anomaly is its order: the smallest integer n for which the diagonal symmetry of the n-copy system is anomaly-free. While many familiar lattice anomalies have finite order, perturbative anomalies in the continuum—those captured by Feynman diagrams—have infinite order. In this talk, we show that the Onsager symmetry, a lattice realization of the chiral symmetry of a 1+1d massless Dirac fermion, has an order-two anomaly. However, imposing lattice CPT symmetry enhances this anomaly from order two to infinite order, yielding a lattice chiral symmetry structure that more faithfully matches the continuum chiral anomaly. (This talk is based on arXiv:2606.12510 with Elijah Lew-Smith and Shu-Heng Shao.)

Nivedita: Constructing fully extended functorial Chiral CFTs Two-dimensional unitary chiral conformal field theories (CFTs) admit three distinct mathematical formulations: unitary vertex operator algebras (uVOAs), conformal nets, and Segal (functorial) chiral CFTs. With the aim of building a fully extended functorial chiral CFT from the data of a conformal net, we give its values on points and 1-dimensional cobordisms: to a point, we assign the category of solitonic representations of the net, and to a 1-dimensional cobordism, a bimodule category. We then prove that this assignment is functorial, with gluing of cobordisms corresponding to fusion of modules. The algebraic target of the construction is the 3-category BicomCat of bicommutant categories, the functional-analytic analogue of TensCat. We introduce the fusion of modules over bicommutant categories as 'categorified' Connes fusion.

Alison Warman: Intrinsic DQCP Transitions from Twin Phases We introduce the concept of twin phases for a symmetry S, defined as inequivalent phases, whose order parameters are part of the same generalized charge under S. Stable, direct transitions between such twin phases are never spontaneous-symmetry-breaking transitions, even after (partially) gauging the initial symmetry S: they are phase transitions without hidden symmetry breaking. We illustrate this with an (anomalous) finite group symmetry in 1+1d, which exhibits such intrinsically beyond Landau DQCP transitions. This is based on work with Alison Warman and Yuhan Gai.

Reutter: A higher category of topological quantum field theories It is commonly expected that n-dimensional topological quantum field theories and their interfaces assemble into an n-category. In this talk, I will suggest a list of reasonable assumptions satisfied by such an n-category of (discrete and semisimple) TQFTs, including Freed and Hopkins’ suggestion that invertible theories are completely determined by their partition functions on closed manifolds. I will then explain that there is a unique n-category satisfying these assumptions, and — if time permits — sketch how it looks in low dimensions. This is based on work in progress with Theo Johnson-Freyd.

Steffens: On some of the mathematics of continuous symmetries It is fair to say we have a rather complete mathematical framework for describing finite symmetries of field theories. I will discuss recent work on the mathematics of continuous symmetries. In particular, I will introduce a (in fact, several) higher category(ies) of `topological' (in the functional analytic sense) n-vector spaces together with, for each compact Lie group, a topological gauge theory (a BF theory) as a once-categorified fully extended theory with this target. Instead of invoking the cobordism hypothesis, I will write down a functor of n-categories explicitly; the construction may interpreted as a fully extended geometric quantization of the shifted cotangent of the group.

Müller: Fully local unitary topological quantum field theories The action of generalized symmetries on an n-dimensional quantum field theory can be conveniently encoded by realizing it as a boundary of a topological field theory in one dimension higher, called the symmetry TQFT, using a sandwich construction. The extended operators of the symmetry TQFT give a concrete implementation of the categorical symmetry of the boundary. In fully local topological quantum field theories, there exists an operator–state correspondence, which identifies the possible extended operators with the higher categories of states that the TQFT assigns to manifolds of lower codimension. Taking unitarity into account in this setting requires the symmetry TQFT to be unitary and fully extended, which in turn calls for a not-yet-fully-developed theory of unitary higher categories. In my talk, I will explain some recent progress towards the definition and classification of unitary fully local topological quantum field theories. This is based on several joint works, involving (in part) Giovanni Ferrer, Brett Hungar, Theo Johnson-Freyd, Cameron Krulewski, Nivedita, David Penneys, David Reutter, Claudia Scheimbauer, Luuk Stehouwer, and Chetan Vuppulury, as well as ongoing work with Theo Johnson-Freyd, Cameron Krulewski, and Luuk Stehouwer.

Parra-Martinez: Noether’s theorem, with a twist Noether’s theorem, which relates continuous global symmetries and conserved currents, is perhaps one of the most famous results in theoretical physics. While it is well established for theories with a path integral formulation, its status for abstract QFTs is not settled. In this talk I will present a proof of Noether’s theorem following from the usual axioms of two-dimensional conformal field theory. In higher dimensions, I will argue that it is equivalent to the existence of twist defects for continuous global symmetries.

Heckman: TQFTs are in the Swampland One of the recurring themes of recent years is the appearance of deep topological structures in the study of generalized global symmetries of quantum field theories. Perhaps surprisingly, then, there is also a general expectation that in quantum gravity, there are no global symmetries at all! In modern terminology, this is an example of a "Swampland Conjecture," namely that gravity imposes non-trivial constraints on the space of consistent low energy effective field theories. In this talk we give a brief overview of these expectations both from the top down and bottom up. We then present an argument showing that completely decoupled topological quantum field theory (TQFT) sectors are incompatible with basic features of holography. In particular, we argue that the fields of any TQFT necessarily couple to gravity both non-perturbatively and perturbatively in Newton's constant.

Yamashita: Towards categorical symmetry in TMF I present duality results in equivariant TMF in my joint works with Ying-Hsuan Lin and David Gepner, which can be thought of as shadows of categorical symmetries in TMF. I discuss the ideas towards actually realizing fusion-category-equivariant TMF, joint work with Tomer Schlank.

Utiralova: New examples of (non-semisimple) modular tensor categories The talk is based on joint work with Victor Ostrik. I will describe a construction of braided finite tensor categories that generalizes the usual semisimplification construction applied to tilting modules over quantum groups. The categories we obtain are no longer semisimple, but (after an appropriate de-equivariantization procedure) modular. (At least) one of these categories gives a rather nice and small example of a category not Witt equivalent to a semisimple category, existence of which was not known before.

Czenky: Cochain valued TQFTs from nonsemisimple modular tensor categories Consider a finite modular tensor category A. In [DGGPR] the authors exhibit a 3-dimensional topological field theory Z_A: Bord_A -> Vect, which, in the case where A is semisimple, recovers the usual Reshetikhin-Turaev TQFT. In the present work we show that this extends naturally to a TQFT which takes values in the symmetric tensor category of linear cochains. This cochain valued theory furthermore respects (certain classes of) homotopies. [DGGPR] M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, and I. Runkel. 3-dimensional TQFTs from non-semisimple modular categories. Sel. Math. New Ser., 28(2):42, 2022.

Hübner: On Frozen Singularities Among the various corners of the string landscape frozen singularities belong to the less well-explored backgrounds. The central example of frozen ADE singularities, in M-theory, exhibit challenges reminiscent to those of RR-backgrounds in type II string theory. For these reasons much analysis in the literature has focussed on mappings to alternative duality frames where these challenges are recast in more tractable form. In this talk we aim to describe freezing directly in the fluxed backgrounds, and characterize frozen theories as confined phases of unfrozen theories.

Sanford: A primer on fusion categories over R Fusion categories over R can be used to model time reversal symmetric (possibly composite) anyon systems. We will discuss several important aspects of the real setting that are different from what we have come to expect in the complex setting. Many examples will be given.

Open discussion about planning the week

Stewart: String nets for twisted pivotal categories

Organized Discussion Please add suggestions to https://docs.google.com/document/d/1C4teXxYc0RSsisdfJX9M1rgW8ulxnjtu1QueityBxJI/edit?usp=sharing

Freed: Fully Local Reshetikhin-Turaev theories

Kapustin: Higher symmetries and lattice models

Informal discussion Topics: SO-invariance and central charges in 3D TQFT / 2D CFT Classification of 4D TQFTs

Riva: An update on co-span constructions

Discussion: 4D TQFTs