Workshop talk 1: PAVEL PUTROV - Q/Z symmetry and non-invertible topological defects from mixed anomalies Room: Albano 3: 4204 - SU Conference Room (56 seats) Q/Z symmetry and non-invertible topological defects from mixed anomalies Abstract: in the beginning of the talk I will review how anomalies of different symmetries can be related to each other. I will then show how such relations can be used to determine the classification of anomalies of Q/Z (i.e. torsion subgroup of U(1)) symmetry with discrete topology. I will then review the construction of topological defects in 4d associated to mixed U(1) gauge -- U(1) global anomaly and describe its generalization to the case of mixed gravitational -- U(1) global anomaly. In both cases the topological defects act as Q/Z symmetry on local operators.

Workshop talk 2: IÑAKI GARCÍA ETXEBARRIA - Holography and categorical symmetries Room: Albano 3: 4204 - SU Conference Room (56 seats) Holography and categorical symmetries Abstract: I will review how some aspects of categorical symmetries arise from the dynamics of D-branes in string theory constructions, focusing on the case of holographic setups for N \geq 3 S-folds.

Discussion Session on Non-Invertible Symmetries Room: Albano 3: 6228 - Mega (22 seats)

Workshop talk 3: WENJIE JI Room: Albano 3: 4204 - SU Conference Room (56 seats)

Workshop seminar - LIANG KONG (online) - A morphism between two QFTs A morphism between two QFTs Abstract: A morphism between two mathematical objects of the same type (e.g. groups, algebras, representations, categories, etc.), which preserves the defining structures of the objects, is one of the most important notions in mathematics. However, how to define such a morphism between two QFT's (or quantum phases) had never been considered in physics until arXiv:1502.01690. In this talk, I will give a review of this notion and discuss its applications in the study of topological orders and more general quantum liquids. I will also clarify its relation with "topological symmetries'' or SymTFTs.

Workshop talk 4: JÜRGEN FUCHS Room: Albano 3: 4204 - SU Conference Room (56 seats) I will review various categorical structures that arise in the study of two-dimensional rational conformal field theory. The focus will be on structures needed for describing symmetries that are realized by topological defects, and thus on Frobenius algebras in tensor categories and their representations. At the end I will mention aspects of more general situations: beyond rationality, beyond two dimensions, and beyond rigidity.

Workshop talk 5: YUNQIN ZHENG - An application of SymTFT: group theoretical duality defects Room: Albano 3: 4204 - SU Conference Room (56 seats) An application of SymTFT: group theoretical duality defects Abstract: Group theoretical duality defects are a special type of duality defects that can be mapped to invertible defects by a sequence of twisted gaugings. After giving several motivations of studying such defects, we explain how to determine whether a duality defect is group theoretical. The Symmetry TFT (SymTFT) turns out to be a very useful tool to address this problem. We then first give a brief overview the SymTFT, and use it to provide a computable criteria for the group-theoretical-ness. Then we apply the criteria to two examples: the duality defects associated with gauging Z_N 0- and 1-form symmetries in 2d and 4d respectively. I will end by commenting on the relation between the group theoretical condition, and the condition for free of anomaly as well as the obstruction to duality preserving gapped phases.

Workshop talk 6: LAKSHYA BHARDWAJ - Generalized Charges, Symmetry TFT and Phases Protected by Non-Invertible Symmetries Room: Albano 3: 4204 - SU Conference Room (56 seats) Generalized Charges, Symmetry TFT and Phases Protected by Non-Invertible Symmetries Abstract: I will discuss how non-invertible symmetries act on (both local and extended) operators in a theory and present a generalized version of Landau-Ginzburg paradigm applicable to non-invertible symmetries. The central tool unifying all these considerations is that of Symmetry TFT. We will see that possible actions of a symmetry, referred to as generalized charges of that symmetry, are parametrized by topological defects of the Symmetry TFT. These generalized charges can confine or deconfine, and hence act as order parameters distinguishing gapped and gapless phases protected by non-invertible symmetries.

Jing-Yuan Chen - Instanton Operator in Lattice QCD from Higher Anafunctor -- or, How to Put Continuum QFT onto Lattice Room: Albano 3: 4204 - SU Conference Room (56 seats) Instanton Operator in Lattice QCD from Higher Anafunctor -- or, How to Put Continuum QFT onto Lattice Abstract: A long standing problem in lattice QCD is there is no natural lattice definition of instanton operator. This problem hampers our understanding of, e.g., QCD confinement. I will show how this traditional problem is -- and has to be -- solved by higher category theory, in particular by the use of higher anafunctors. Related to this, one can also define the Chern-Simons term in Yang-Mills theory, as well as the skyrmion operator and Wess-Zumino-Witten term in SU(N) non-linear sigma model. More broadly, using the language higher anafunctors, I will sketch a systematic program to put continuum QFT (note, not just TQFT in the IR, but QFT with general dynamics) onto the lattice, especially for non-linear sigma models and gauge theories with continuous-valued fields, while retaining all the topological operators. I will discuss two important future prospects: The first is how one might unify this general picture and the usual use of categories in the TQFT context with discrete degrees of freedom and/or symmetries. The second is why this picture has the hope to lead to non-trivial progress in the field of constructive QFT, which aims at defining QFT in the continuum, such as 4D Yang-Mills.

Francesco Benini - Aspects of self-duality symmetries Room: Albano 2: C2207 - Auditorium 4 (80 seats)

Yuya Tanizaki - Non invertible solitonic symmetry Room: Albano 2: C2207 - Auditorium 4 (80 seats) Abstract: Conventionally, the selection rule of solitons has been believed to be controlled by the homotopy group. In recent papers (https://arxiv.org/abs/2210.13780, https://arxiv.org/abs/2307.00939), we found that the solitonic symmetry also becomes noninvertible because of the interplay between different dimensional objects. After detailed explanations of the example of the 4d CP1 sigma model, I will explain our proposal to describe the general form of noninvertible solitonic symmetry.

Po-Shen Hsin - Anomalies of Non-Invertible Symmetry Anomalies of Non-Invertible Symmetry

Hayashi - Non-invertible symmetries of Cardy-Rabinovici model and mixed gravitational anomaly Non-invertible symmetries of Cardy-Rabinovici model and mixed gravitational anomaly Abstract: In this talk, we discuss noninvertible symmetries of the Cardy-Rabinovici model. The Cardy-Rabinovici model is the 4d U(1) gauge theory with electric and magnetic matters. We find that the $SL(2,\mathbb{Z})$ electromagnetic transformations can be understood as dualities between the Cardy-Rabinovici model and its appropriately $\mathbb{Z}_N$ 1-form gauged model. Based on this observation, we can construct noninvertible symmetry defects with non-group-like fusion rules at self-dual parameters. As an application, we show that the mixed gravitational anomaly of this symmetry rules out the trivially gapped vacuum for some parameters. We also reveal how the conjectured phase diagram of the Cardy-Rabinovici model is consistent with this anomaly.

Brennan - Anomalies of Discrete 1-Form Symmetries in QCD-like Theories Anomalies of Discrete 1-Form Symmetries in QCD-like Theories Abstract: In this talk we will discuss a new class of non-perturbative anomalies of discrete 1-form global symmetries in 4D QCD-like theories. This generalizes the techniques developed by Wang-Wen-Witten to more general theories that allow for discrete 1-form global symmetries including chiral gauge theories. We will demonstrate several new anomalies and comment on their implication on symmetric mass generation in 3+1D.

Reception and Poster Session @ Nordita (featuring posters by: Antinucci, Cao, Koizumi, Leung, Li, Pasquarella, Rizi) Room: Albano 3: 6228 - Mega (22 seats) Andrea Antinucci Duality defects in two and four dimensional theories, anomalies and gauging Abstract: Duality defects are ubiquitous in two and four dimensions, where they generate a 0-form symmetry, which is often non-invertible. However, sometimes it becomes invertible in specific global variants. In such cases, the duality is a non-intrinsic non-invertible symmetry. By employing the powerful tool of symmetry TFT, we study and classify obstructions to gauging the duality symmetries. We show that in the intrinsic non-invertible case, they are necessarily anomalous, hence implying a strong constraint on the IR of duality-preserving RG flows. In the non-intrinsic case, we found that the anomaly is not uniquely determined, depending on further data, namely the choice of an equivariantization of a Lagrangian algebra of the Drinfeld center. We propose and verify in several examples that the boundary counterpart of this ambiguity is a choice of symmetry fractionalization of the duality symmetry on the global variant where it becomes invertible. === Weiguang Cao Subsystem Kramars-Wannier duality and non-invertible symmetry Abstract: Recently, the notion of symmetry has been generalized by relaxing the dimensions, invertibility and topologicalness of the symmetry operators. In this poster, I will introduce a new generalization, subsystem non-invertible symmetry, by lifting both the invertibility and topologicalness of the ordinary global symmetry. I will first review the simplest non-invertible symmetry in (1+1)d from the ordinary Kramers-Wannier transformation. Then I will explore non-invertible symmetries in two-dimensional lattice models with subsystem Z_2 symmetry by introduce a subsystem Z_2-gauging procedure, called the subsystem Kramers-Wannier transformation. For both case, the corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. I will derive the fusion rules, check the mobility of the defects and comment on the anomaly. Finally, I will comment on generalizing the results to subsystem Z_n symmetry in (2+1)d and further to subsystems in arbitrary dimensions. I will also give examples in continuum field theory. === Saki Koizumi Anomaly Inflow of Rarita-Schwinger Field in 3 Dimensions Abstract: We study the anomaly inflow of the Rarita-Schwinger field with gauge symmetry in $3$ dimensions. We find that global anomalies of the Rarita-Schwinger field are obtained by the spectral flow, which is similar to Witten's $SU(2)$ global anomaly for a Weyl fermion. The Rarita-Schwinger operator is shown to be a self-adjoint Fredholm operator, and its spectral flow is determined by a path on the set of self-adjoint Fredholm operators with the gap topology. From the spectral equivalence of the spectral flow, we find that the spectral flow of the Rarita-Schwinger operator is equivalent to that of the spin-$3/2$ Dirac operator. From this fact, we confirm that the anomaly of the $3$-dimensional Rarita-Schwinger field is captured by the anomaly inflow. Finally, we find that there are no global anomalies of gauge-diffeomorphism transformations on spin manifolds with any gauge group. We also confirm that the anomalous phase of the partition function which corresponds to the generator of $\Omega_4^{{\rm Pin}^+}(pt)=\mathbb{Z}_{16}$ is $\exp(3i\pi /8)$ for the Rarita-Schwinger theory on unorientable ${\rm Pin}^+$ manifolds without gauge symmetry. === Enoch Leung Brane Fusion Frenzy: Non-Invertible Defect Fusion and Tachyon Condensation Abstract: It has been recently appreciated in the literature that non-invertible symmetry defects in QFTs can be realized holographically as certain D-brane configurations. The hallmark of non-invertible defects is two-fold: 1) the fusion “coefficients” are generally decoupled TQFTs, 2) the fusion of a defect with its dual gives rise to a “condensation defect” comprising localized lower-dimensional defects. We show that both of these field-theoretic features are fully characterized by brane kinematics/dynamics, namely, the former corresponds to the relative motion of two stacks of D-branes, and the latter corresponds to tachyon condensation on a brane-antibrane pair. === Linhao Li Non-Invertible Kennedy-Tasaki Transformation and Applications to Gapless-SPT Abstract: In this poster, I propose a way to define it on a closed chain, by sacrificing unitarity. The operator realizing such a non-unitary transformation satisfies non-invertible fusion rule, and implements a generalized gauging of the Z_2×Z_2 global symmetry. By choosing free boundary on the open chain, this generalization will reduce to the original KT transformation. Besides, we further apply the KT transformation to systematically construct gapless symmetry protected topological phases. This construction reproduces the known examples of (intrinsically) gapless SPT where the non-trivial topological features come from the gapped sectors by means of decorated defect constructions. We also construct new (intrinsically) purely gapless SPTs where there are no gapped sectors, hence are beyond the decorated defect construction. === Veronica Pasquarella Drinfeld Centers from Magnetic Quivers The present work shows that magnetic quivers encode the necessary information for determining the Drinfeld center in the symmetry topological field theory constructions (SymTFT) associated to a given absolute theory. The crucial argument resides in their common aim of generalising homological mirror symmetry. === Giovanni Rizi Symmetries and topological operators, on average. Abstract: We study Ward identities and selection rules for local correlators in disordered theories where a 0-form global symmetry of a QFT is explicitly broken by a random coupling hh but it re-emerges after quenched average. We consider hh space-dependent or constant. In both cases we construct the symmetry operator implementing the group action, topological after average. In the first case, relevant in statistical systems with random impurities, such symmetries can be coupled to external backgrounds and can be gauged, like ordinary symmetries in QFTs. We also determine exotic selection rules arising when symmetries emerge after average in the IR, explaining the origin of LogCFTs from symmetry considerations. In the second case, relevant in AdS/CFT to describe the dual boundary theory of certain bulk gravitational theories, the charge operator is not purely codimension-1, it can be defined only on homologically trivial cycles and on connected spaces. Selection rules for average correlators exist, yet such symmetries cannot be coupled to background gauge fields in ordinary ways and cannot be gauged. When the space is disconnected, in each connected component charge violation occurs, as expected from Euclidean wormholes in the bulk theory. Our findings show the obstruction to interpret symmetries emergent after average as gauged in the bulk.

Gukov - Categorical symmetries of T[M] theories

Jonathan Heckman - Symmetries via Branes: To Infinity and Beyond Symmetries via Branes: To Infinity and Beyond Abstract: We discuss some recent progress on the construction and study of defects and topological symmetry operators via top down (i.e., stringy) methods, including possibly non-trivial fusion rules (e.g., non-invertible symmetries). We explain how branes "wrapped at infinity" implement topological symmetry operators in a string-engineered quantum field theory (QFT), and then determine the consequences of coupling such QFTs to other QFTs as well as gravity. This leads to contributions "beyond infinity" which we systematically study via the associated extra-dimensional geometry. We also present an application of these methods to large N averaging in holography.

Hübner - Generalized Symmetries and Gravity Brane constructions of generalized symmetry operators are well understood in many cases. For example, given a QFT engineered by a local geometry in string theory such operators arise via brane wrappings on asymptotic cycles. When the local geometry is embedded into a global one the notion of asymptotic cycle is lost, gravity is turned on and symmetries are either gauged or broken following the no global symmetries folk theorem. We give a simple geometric characterization of these gauging and breaking phenomena.

Tizzano - Non-invertible symmetries along 4d RG flows I will present novel examples of RG flows that preserve a non-invertible duality symmetry, with a primary focus on N=1 quadratic superpotential deformations of N=4 SYM. A famous theory that can be obtained in this way is N = 1* SYM, where all adjoint chiral multiplets possess a finite mass term. This IR theory exhibits a a rich structure of vacua that I will describe. Through this analysis, I will elucidate the physics underlying spontaneous duality symmetry breaking which occurs in the degenerate gapped vacua. Finally, I will briefly comment on various generalizations of these ideas for RG flows resulting in gapless IR theories.

Roumpedakis - Non-Invertible Symmetries in Maxwell Theory (Part I) This is the first of a two-part talk on non-invertible symmetries in Maxwell theory in four dimensions. The second one will be given by Pierluigi Niro. I will argue that Maxwell theory has three infinite sets of non-invertible defects. The first set can be obtained by higher-gauging a discrete subgroup of the electric one-form symmetry along a codimension-1 defect. Similarly, we can higher-gauge a discrete subgroup of the magnetic one-form symmetry and construct a second infinite set of topological defects. Then, I will argue that we can combine elements of the SL(2,Z) duality group with the gauging of one-form symmetries in half-space to define a third infinite set of non-invertible symmetries.

Pierluigi Niro - Non-Invertible Symmetries in Maxwell Theory (Part II) This is the second of a two-part talk on non-invertible symmetries in Maxwell theory in four dimensions. The first will be given by Konstantinos Roumpedakis. I will show how to realize topological codimension-one defects in Maxwell theory with a constructive Lagrangian approach. While the action of such defects on local operators is invertible, the action on line operators is generically non-invertible.

Carqueville - Orbifold data as gaugeable non-invertible symmetries Orbifold data are defects in topological quantum field theories which can be gauged to obtain a new TQFT. Examples include gaugings of (higher) group actions, state sum models, and more generally gaugings of "non-invertible symmetries". The defining conditions of orbifold data encode invariance unter the choice of defect network used in the gauging process, which has been rigorously developed in arbitrary dimensions. The talk gives an introduction to the orbifold construction and illustrates it with examples in dimensions 2,3, and 4.

Teleman - Quantization commutes with reduction in 2 dimensions After historical review of classical results in 1D (quantum mechanics), I will discuss "Quantum GIT conjecture", recently proved in joint work with Dan Pomerleano, describing the quantum cohomology of (smooth) GIT quotients of a Fano manifold X in terms of equivariant cohomology of X and its twisted sectors.

Moudgalya - Symmetries as Commutant Algebras Symmetries as Commutant Algebras Abstract: The study of symmetry lies at the heart of various parts of physics. However, the symmetries conventionally studied in a lot of the literature are mostly restricted to either on-site unitary symmetries or lattice symmetries. While such symmetries are sufficient to explain several physical phenomena, the recent discoveries of weak ergodicity breaking phenomena such as Hilbert space fragmentation and quantum many-body scars have called for a reconsideration of the definition of symmetry in quantum many-body physics. In this talk, I will discuss a general mathematical framework to define symmetries based on so-called commutant algebras. This leads to a generalization of the conventional notion of symmetry and explains weak ergodicity breaking in terms of unconventional non-local symmetries. In addition, it reveals a novel interpretation of symmetries as ground states of local superoperators, leading to insights on the nature of symmetries realizable in systems with locality. Title: Symmetries as Commutant Algebras Abstract: The study of symmetry lies at the heart of various parts of physics. However, the symmetries conventionally studied in a lot of the literature are mostly restricted to either on-site unitary symmetries or lattice symmetries. While such symmetries are sufficient to explain several physical phenomena, the recent discoveries of weak ergodicity breaking phenomena such as Hilbert space fragmentation and quantum many-body scars have called for a reconsideration of the definition of symmetry in quantum many-body physics. In this talk, I will discuss a general mathematical framework to define symmetries based on so-called commutant algebras. This leads to a generalization of the conventional notion of symmetry and explains weak ergodicity breaking in terms of unconventional non-local symmetries. In addition, it reveals a novel interpretation of symmetries as ground states of local superoperators, leading to insights on the nature of symmetries realizable in systems with locality. References: https://arxiv.org/abs/2108.10324 https://arxiv.org/abs/2209.03370

Tiwari - Twisted Theta Symmetries I will describe a uniform construction to gauge a discrete invertible symmetry S in possibly non-topological (D+1)-dimensional quantum systems with higher-categorical and possibly non-invertible symmetries. The symmetry in the theory obtained after gauging contains a sub-category of universal topological defects, which we term as Theta defects. These defects are universal in the sense that they exist in any system that can be obtained by gauging the symmetry S of some other system. These are equivalent to the D-category of lower (<= D dimensional) dimensional quantum systems with S-symmetry. The non-universal symmetry defects within the gauged theory termed Twisted Theta defects correspond to an S-gauging of non-trivial defects in the pre-gauged theory stacked with lower (<= D dimensional) dimensional TQFTs. I will exemplify this construction by gauging invertible 0-form symmetries in quantum theories with 2-categorical symmetries in 2+1 dimensions. The 0-form groups I will discuss will be Z4, Z2 x Z2 and S3.

Wen (online) - Symmetry/Topological-Order (Symm/TO) correspondence -- A number theoretical approach to gapless liquid phases

Copetti - Comments on the higher structure of chiral symmetry

Chang - Non-Invertible Symmetries in 2d Fermionic CFTs I will discuss topological defect lines (TDLs) in 2d CFTs, that generate invertible and non-invertible symmetries. Besides inheriting all the properties of TDLs in bosonic CFTs, TDLs in fermionic CFTs could host fermionic defect operators at their endpoints and junctions. Furthermore, there is a new type of TDLs, called q-type TDLs, that have no analog in bosonic CFTs. Their distinguishing feature is an extra one-dimensional Majorana fermion living on the TDLs. The properties of TDLs in fermionic CFTs are captured in the mathematical language of the super fusion category. I will present a classification of the rank-2 super fusion categories generalizing the Z8 classification for the anomalies of Z2 symmetry. Finally, I will discuss an interesting relation between q-type symmetries and fermionic symmetries.

Apruzzi - Aspects of categorical symmetries from branes: Symmetry TFT

Bonetti - Aspects of Categorical Symmetries from Branes: Condensation Defects and Generalized Charges Branes in geometric engineering and holography have a striking connection with generalized global symmetries. In particular, non-trivial aspects of non-invertible symmetries are encoded in brane physics. In this talk, I will discuss the brane origin of condensation defects and generalized charges. Hanany-Witten brane configurations in string theory play a crucial role in the analysis. The general discussion is illustrated with examples from holography and geometric engineering. As an application, I will consider duality/triality defects in 4d and show how Hanany-Witten transitions provide a powerful diagnostic of their intrinsic/non-intrinsic character.

Tillim

Grigoletto - Tubes and representation theory for categorical symmetries A systematic study of representation theory for higher categorical symmetries is necessary in order to access all the potentiality that these symmetries have to offer. Based on 2305.17165 (see also 2305.17159 for related work), in this talk I will explain how to properly formalize such notion from a mathematical perspective, describing representations for both local and extended operators via tube categories and showing their connection with the SymTFT construction.

Meynet - Comments on topological field theory and anomalies Comments on topological field theory and anomalies Abstract: In the last years the paradigm for understanding the symmetries of quantum field theories has shifted towards the concept of the so called SymmetryTFT, a topological field theory encoding the data of all possible realizations of global and higher group structures. In this brief talk we will review some aspects of topological field theory and how standard techniques allow to reinterpret anomalies in term of topological invariants. As an example, we will discuss the case of 5d SCFT constructed via geometric engineering, commenting in particular the relation between wrapped branes, topological operators and generalized membranes link.

Aguilera-Damia - Duality symmetries and multicriticality in 2d CFT Duality symmetries and multicriticality in 2d CFT Abstract: Conformal field theories in two dimensions are the ideal playground to explore the reach of (non-invertible) duality symmetries. In this talk, we will focus on the interesting interplay between such symmetries and the phenomenon of multicriticality. Multicritical points are characterized by the appearance of additional marginal operators and are quite ubiquitous along two dimensional conformal manifolds. After an account of the well known case of the KT point at c=1, we will comment on some recent findings taking place at the c=2 conformal manifold.