9 Apr 2024

Speaker :  Anirban Chowdhury (IBM) : Quantum Counting and the Symmetric Group

I will talk about connections between the representation theory of the symmetric group and counting problems inspired by quantum computation and many-body physics. The main result that I will present concerns the Kronecker coefficients, which are the analogs of the Clebsch-Gordon coefficients for the symmetric group. It is an open question whether computing the Kronecker coefficients reduces to a classical counting problem, i.e., whether the Kroneckers can be expressed as the number of solutions to NP-complete problems. I will show that approximating the Kronecker coefficients to within a multiplicative error reduces to approximate versions of quantum counting problems. This is interesting also because the quantum counting problems are questions about the eigenvalues of operators, such as Hamiltonians, POVM elements, or density operators. And spectral properties, specifically of density operators, are closely related to the representation theory of the symmetric group. In this talk I will explain our result and also discuss some of these connections.

19 Mar 2024

Speaker :  Garreth Kemp (Johannesburg U) : A generalized dominance ordering for 1/2-BPS states

I discuss a generalized dominance ordering for irreducible representations of the symmetric group $S_{n}$ with the aim of distinguishing the corresponding states in the 1/2-BPS sector of $U(N)$ Super Yang-Mills theory when a certain finite number of Casimir operators are known. Having knowledge of a restricted set of Casimir operators was proposed as a mechanism for information loss in this sector and its dual gravity theory. It is well-known that the states in this sector are labeled by Young diagrams with n boxes. I will propose a generalization of the well-known dominance ordering of Young diagrams. Using this generalization, I posit a conjecture to determine an upper bound for the number of Casimir operators needed to distinguish between the 1/2-BPS states and thus also between their duals in the gravity theory. I offer numerical and analytic evidence for the conjecture. Lastly, I discuss implications of this conjecture when the energy $n$ of the states is asymptotically large.

12 Mar 2024

Speaker :  Jingxiang Wu (Oxford) : MTC[M3,G]: 3d Topological Order Labeled by Seifert Manifolds

We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group G. Topological order in 2+1d is known to be characterised in terms of modular tensor categories (MTCs), and we thus propose a relation between MTCs and Seifert three-manifolds. The correspondence defines for every Seifert manifold and choice of G a fusion category, which we conjecture to be modular whenever the Seifert manifold has trivial first homology group with coefficients in the centre of G. The construction determines the spins of anyons and their S-matrix, and provides a constructive way to determine the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realising all MTCs (unitary or non-unitary) with rank r<=5 in terms of Seifert manifolds and a choice of Lie group G.

20 Feb 2024

Speaker :  Anindya Banerjee (QMUL) : On the 4d/3d/2d view of the SCFT/VOA correspondence

Several classes of supersymmetric field theories in four, three and two spacetime dimensions have been found to contain protected subsectors described by vertex operator algebras (VOAs) in recent years. We will discuss the construction in hep-th/2312.17747 that seeks to tie together these theories and their VOAs, placing 4d N=2 theories and their associated VOAs at the center of this web of theories, and deriving the SQFT/VOA correspondences from these theories by a sequence of manipulations of the background data.

13 Feb 2024

Speaker :  Mahesh Balasubramanian (QMUL) : Non-Invertible Anyon Condensation and Level-Rank Dualities

I will discuss the paper [2312.16317] Non-Invertible Anyon Condensation and Level-Rank Dualities (arxiv.org). In the paper Taming the conformal Zoo, Moore and Seiberg conjectured that all RCFTs can be obtained from Chern-Simons theories by appropriate choice of gauge groups. However, RCFTs corresponding to Conformal embeddings, Maverick Cosets have extra selection rules and do not fit the paradigm proposed by Moore and Seiberg. The paper adopts a modern parlance to understand these issues and proposes a resolution using non-abelian anyon condensations with some examples, which we will go over.

6 Feb 2024

Speaker :  Zhihao Duan (QMUL) : A SymTFT for Continous Symmetries

I will discuss the recent paper 2401.06128 on constructing a symmetry topological field theory for abelian symmetries. After introducing the main idea, I will also talk about some applications.

12 Dec 2023

Speaker :  Ginestra Bianconi (Queen Mary University of London) : The mass of simple and higher-order networks

We propose a theoretical framework that explains how the mass of simple and higher-order networks emerges from their topology and geometry. We use the discrete topological Dirac operator to define an action for a massless self-interacting topological Dirac field inspired by the Nambu–Jona-Lasinio model. The mass of the network is strictly speaking the mass of this topological Dirac field defined on the network; it results from the chiral symmetry breaking of the model and satisfies a self-consistent gap equation. Interestingly, it is shown that the mass of a network depends on its spectral properties, topology, and geometry. Due to the breaking of the matter–antimatter symmetry observed for the harmonic modes of the discrete topological Dirac operator, two possible definitions of the network mass can be given. For both possible definitions, the mass of the network comes from a gap equation with the difference among the two definitions encoded in the value of the bare mass. Indeed, the bare mass can be determined either by the Betti number β0 or by the Betti number β1 of the network. We provide numerical results on the mass of different networks, including random graphs, scale-free, and real weighted collaboration networks. We also discuss the generalization of these results to higher-order networks, defining the mass of simplicial complexes. The observed dependence of the mass of the considered topological Dirac field with the topology and geometry of the network could lead to interesting physics in the scenario in which the considered Dirac field is coupled with a dynamical evolution of the underlying network structure.

8 Dec 2023

Speaker :  Bowen Shi (UCSD) : Entanglement bootstrap for gapped topological phases II

In the 2nd lecture, we dig into the underlying logic of entanglement bootstrap. Illustrative examples are aimed to be simple but nontrivial. The following will be included: (1) We explain a few basic uses of strong subadditivity and quantum Markov states; we explain why axiom A0 is crucial to protect coherence. We derive the information convex set of the sphere as an application. (2) Related to the information convex set of the annulus, we explain the definition of quantum dimensions, why the vacuum has the smallest entropy, and why a certain "merged state" has the maximum entropy. (3) We classify immersed annuli on a sphere and explain why some puzzles of figure-8 annulus are not solved in naive ways. (4) In the context the reference state has a 0-form symmetry, we sketch a way to create a symmetry defect line, which (in some models) permutes anyons. This lecture is given using an ipad and is, thus, flexible. We also discuss topics from questions (feedbacks) during the 1st (and 2nd) lecture.

5 Dec 2023

Speaker :  Brian Swingle (Maryland) : A Bosonic Model of Quantum Holography

We analyze a model of qubits which we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model is generic in that it includes all possible q-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. We will discuss implications for Hamiltonian complexity, the factorization problem in quantum gravity, and quantum simulations of holography. Based on 2311.01516 with Mike Winer.

4 Dec 2023

Speaker :  Bowen Shi (UCSD) : Entanglement bootstrap for gapped topological phases

Topological quantum field theory can emerge in gapped many-body quantum systems at low energies. In 2+1D systems, anyons can emerge, and in 3+1D, emergent excitations, including point-particles and loops-like excitations, possibly knotted or linked. In this lecture, we introduce an ongoing effort to understand (in fact, derive) laws of the emergent theory in 2+1D, 3+1D, (and higher D) gapped systems from a few axioms about the entanglement of a many-body ground state wave function. This research program, referred to as entanglement bootstrap, is an approach independent of quantum field theory, and it uses nontrivial quantum information and topology ideas. We explain the axioms and key concepts. We sketch the proof of several main theorems, including the definition of superselection sectors (anyons in 2+1D, point and loop excitations in 3+1D), the fusion spaces, and their constraints. We explain why immersion (i.e., local embedding) is valuable for, e.g., putting systems on closed space manifolds and what we hope to learn next.

28 Nov 2023

Speaker :  Andrea Ferrari (Edinburgh and DESY) : On a geometrisation of generalised symmetries

There has recently been a lot of activity in the field of generalised symmetries. In the context of supersymmetric gauge theories with interesting moduli spaces of vacua, such as 3d N=4 theories, global symmetries may enjoy a geometric interpretation: they act as isometries of the moduli space. In this talk I will informally put forward the idea that by enhancing the notion of moduli space one can in a similar fashion geometrise generalised symmetries. I will focus on simple 3d N=4 abelian example and talk about various things including what I'll call 0- and 1-form resolutions as well as automorphism 2-groups.

21 Nov 2023

Speaker :  Vishnu Jejjala (Witwatersrand) : Modular factorization of superconformal indices

The Bekenstein-Hawking entropy of 1/16-BPS AdS_5 black holes is captured by a superconformal index. Such indices exhibit SL(3,Z) modular properties, which are explicated in terms of ambiguities in the Heegaard splitting of an associated Hopf surface. We conjecture a "modular factorization" of superconformal indices of general N=1 gauge theories and provide evidence for this conjecture by studying the free chiral multiplet and SQED.

14 Nov 2023

Speaker :  Jackson Fliss (Cambridge) : Essential topological entanglement

Long range entanglement is a conceptually useful notion in the physics of quantum phases of matter. E.g. in (2+1) dimensions, ground states display area law entanglement with a potential constant correction: the "topological entanglement entropy" (TEE) which is a smoking gun of topological order. Through the lens of the IR effective field theory, described by topological quantum field theory (TQFT), we encounter the following puzzle: how does a field theory with a finite dimensional Hilbert space support a divergent area law? The simple resolution to this puzzle will also suggest an alternative perspective on topological entanglement. Utilizing the algebraic formulation of entanglement I will define a quantity I will call "essential topological entanglement." It is (i) strictly topological, (ii) positive, (iii) finite, and (iv) displays more long-range features than traditional TEE. Working with Abelian p-form BF theory as an example, I will explain general aspects of essential topological entanglement. I will elaborate on potential further applications of essential topological entanglement, as well as describe some follow-up work regarding the entanglement carried by edge-modes in BF theory.

7 Nov 2023

Speaker :  Matthew Yu (Oxford) : Tambara Yamagami fusion 2-categories and fiber 2-functors

Tambara-Yamagami (TY) 1-categories provide the mathematical framework to describe the algebra of extended operators of (1+1)-d theories that admit a duality defect. In this talk I will define what is the generalization of TY 1-categories for fusion 2-categories, and how to construct them from fusion 2-categories that are group-theoretical. I will also explain that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Using this characterization, I will show when a fusion 2-category admits a fiber 2-functor.

17 Oct 2023

Speaker :  Zhihao Duan (Queen Mary University of London) : ℤ_N duality and parafermions revisited

The Symmetry Topological field theory (SymTFT) has played an important role in recent studies of symmetries in physics. As one application, it provides a clear and unifying picture of bosonization and fermionization in two dimensions. On the other hand, for theories having a non-anomalous ℤ_N symmetry with N > 2, people have long speculated about the presence of parafermions. In this informal talk, we revisit the bosonization/parafermionization procedure from the SymTFT point of view, and explain some subtleties and peculiarities involved. It will be based on 2309.01913 and (hopefully) accessible to a broad audience.

26 May 2022

Speaker :  Amihay Hanany (Imperial College) : Phase structure of theories with 8 supercharges — Higgs branch

The moduli space of Supersymmetric gauge theories with 8 supercharges has a rich structure of symplectic singularities associated with degenerations in which additional massless states arise. These are neatly arranged into phase diagrams that encode the different sets of massless states, with information on the moduli that are needed to be tuned in order to move from one phase to another. We will present results from studies of families of quivers, including those on the affine grassmanian of finite dimensional Lie algebras.

19 May 2022

Speaker :  Francesco Benini (SISSA) : Factorization and global symmetries in holography

There exist low-dimensional models of holography in which the bulk gravitational theory is dual to an ensemble average of boundary quantum field theories (as opposed to a single theory). In the case of three-dimensional gravitational theories based on topological field theories, we draw a connection between the ensemble averaging (and the lack of factorization of the partition function) and the presence of global symmetries. Once the global symmetries are removed (by a suitable gauging procedure), the gravitational theory behaves as a unitary quantum system.

12 May 2022

Speaker :  Vladimir Dobrev (Bulgarian Academy of Sciences : Tenfold Way for Holography : AdS/CFT and Beyond

The main purpose of the present talk is to lay the foundations of generalizing the AdS/CFT (holography) idea beyond the conformal setting, where it is very natural. The main tool is to find suitable realizations of the bulk and boundary via group theory. We use all ten families of classical real semisimple Lie groups G and Lie algebras g. For this are used several group and algebra decompositions: the global Iwasawa decomposition and the local Bruhat and Sekiguchi decomposititions, which we introduce first on easy examples. The same analysis is applied to the exceptional real semisimple Lie algebras. We present the boundary-to-bulk operators first in the Euclidean conformal setting and then outline the various generalizations.

5 May 2022

Speaker :  Zohar Komargodski (Simons Center for Geometry and Physics) : Comments on Wilson Lines in Massless Gauge Theories.

28 Apr 2022

Speaker :  Anindya Banerjee (Rutgers) : Comments on Summing over Bordisms in TQFT

Recent works in quantum gravity, motivated by the factorization problem and baby universes, have considered sums over bordisms with fixed boundaries in topological quantum field theory. I will discuss this construction, its scope and its limitations, and describe the total amplitude in this class of theories in terms of a curious splitting formula.

21 Apr 2022

Speaker :  Gabriel Cuomo (Simons Center for Geometry and Physics) : Simulation of quantum field theories by quantum computers

I will discuss line defects in d-dimensional Conformal Field Theories (CFTs). I will first review the definitions and some properties of defect CFTs and defect RG flows, including a recent result on the monotonicity of the defect RG flow. I will then discuss in detail two examples relevant for three-dimensional critical systems: magnetic field defects, which arise from a localized external field in a lattice system, and spin defects, that describe doping impurities in magnets. I will show in particular that impurities with large spin are “effectively’’ equivalent to a magnetic field defect. I will close with a comment on Wilson lines in conformal gauge theories.

14 Apr 2022

Speaker :  Zhenghan Wang (Microsoft Station Q and UC Santa Barbara) : Simulation of quantum field theories by quantum computers

The quantum Church-Turing thesis would imply that there is a unique model of quantum computing. It follows that quantum computers could simulate quantum field theories efficiently.After a review on the simulation of topological quantum field theories, we will focus on a lattice approach to conformal field theories from anyonic chains.

31 Mar 2022

Speaker :  Ying-Hsuan Lin (Harvard) : Numerical evidence for a Haagerup conformal field theory

We numerically study an anyon chain based on the Haagerup fusion category, and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is ~2.

24 Mar 2022

Speaker :  David Vegh (QMUL) : Liouville on the lattice

The Liouville equation has many applications: it describes surfaces of constant negative curvature and plays an important role in non-critical string theory. In this talk we discuss how to put the Liouville equation on the lattice in a completely integrable way (based on Faddeev, Takhtajan: Liouville model on the lattice, 1988).

17 Mar 2022

Speaker :  Tudor Dimofte (UC Davis and U. Edinburgh) : A QFT for non-semisimple TQFT

Topological twists of 3d 𝒩=4 gauge theories naturally give rise to non-semisimple 3d TQFT's. In mathematics, prototypical examples of the latter were constructed in the 90's (by Lyubashenko and others) from representation categories of small quantum groups at roots of unity; they were recently generalized in work of Costantino-Geer-Patureau Mirand and collaborators. I will introduce a family of physical 3d quantum field theories that (conjecturally) reproduce these classic non-semisimple TQFT's. The physical theories combine Chern-Simons-like and 3d 𝒩=4-like sectors. They are also related to Feigin-Tipunin vertex algebras, much the same way that Chern-Simons theory is related to WZW vertex algebras. (Based on work with T. Creutzig, N. Garner, and N. Geer).

10 Mar 2022

Speaker :  Vincent Vargas (ENS, Paris) : Liouville conformal field theory: from probability theory to the conformal bootstrap

Liouville conformal field theory (LCFT) was introduced by Polyakov in 1981 as an essential ingredient in his path integral construction of string theory. Since then Liouville theory has appeared in a wide variety of contexts ranging from random conformal geometry to 4d Yang-Mills theory with supersymmetry. Recently, a probabilistic construction of LCFT on general Riemann surfaces was provided using the 2d Gaussian Free Field. This construction can be seen as a rigorous construction of the 2d path integral introduced in Polyakov's 1981 work. In contrast to this construction, modern conformal field theory is based on representation theory and the so-called bootstrap procedure (based on recursive techniques) introduced in 1984 by Belavin-Polyakov-Zamolodchikov. In particular, a bootstrap construction for LCFT has been proposed in the mid 90's by Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ) on the sphere. The aim of this talk is to review a recent series of work which shows the equivalence between the probabilistic construction and the bootstrap construction of LCFT on general Riemann surfaces. In particular, the equivalence is based on showing that LCFT satisfies a set of natural geometric axioms known as Segal's axioms.

3 Mar 2022

Speaker :  Alessandro Torrielli (U. Surrey) : Integrable scattering of massless particles and the AdS/CFT correspondence

After a brief introduction to some of the impact which integrable methods and the Bethe ansatz have had on the study of the AdS/CFT correspondence in string theory, we will focus on the axiomatic approach to S-matrix theory in 1+1 dimensions. We will highlight the issues that arise when the particles are massless, and how this is in fact connected to Zamolodchikov's way of describing two-dimensional conformal field theories by means of integrability techniques. We will then mention how the axiomatic approach extends to form-factors, which are the gate to access the n-point functions of the theory. If time permits, we will briefly depict how this finds a contemporary application in the area of the AdS_3/CFT_2 correspondence.

24 Feb 2022

Speaker :  André Henriques (Oxford) : Every conformal net has an associated VOA

We show that every conformal net has an associated vertex algebra, thus identifying the class of conformal nets with a sub-class of the class of unitary vertex algebras. We also characterise those unitary vertex algebras that arise from a conformal net. (We conjecture that every unitary vertex algebras arises in this way, and hence that there is a bijective correspondence between conformal nets and unitary vertex algebras.) To construct the correspondence between conformal nets and unitary vertex algebras, we introduce a new notion of "field localised in a segment embedded in a Riemann surface", which could be of independent interest. This is joint work with James Tener.

10 Feb 2022

Speaker :  Gernot Akemann (U. Bielefeld) : Random Matrix Theory and Quantum Chromodynamics – Recent Developments

The application of random matrix techniques in QCD and non-Abelian gauge theories in general has a long history e.g. in counting Feynman diagrams, going back to ‘t Hooft and others. In this talk I will focus on a different aspect that relates the two in the low energy spectrum of the QCD Dirac operator, as initiated by Shuryak and Verbaarschot. First, I will explain what is the approximation studied here where spectral statistics of random matrices apply, and where for example the technique of orthogonal polynomials can be useful in comparing to QCD lattice data. It is given by a particular finite volume low energy limit, the epsilon regime of chiral perturbation theory of Gasser and Leutwyler. I will mention how QCD parameters like quark masses, zero-modes, finite lattice spacing or chemical potential can be incorporated into the random matrix ensemble. In the last part I will discuss some recent work with my former student Tim Würfel on the inclusion of finite temperature, that leads out of the standard classes of random matrices, but still remains analytically tractable. This talk is mainly based on the review arXiv:1603.06011 and the paper with Tim arXiv:2110.03617

3 Feb 2022

Speaker :  George Barnes (QMUL) : Large N Factorization for Matrix Model Observables With Continuous and Discrete Symmetry

We will discuss the combinatoric and algebraic structure of matrix models with continuous and discrete symmetries. These models often simplify in the large N limit, with N the size of the matrices. The fact of large N factorisation is familiar for U(N) matrix models, we provide a combinatoric proof of this result and prove an analogous result for matrix models with discrete SN symmetry. In both the continuous and discrete cases Schur-Weyl duality provides a method of efficiently organising the observables of the theory. The dual algebra of SN is a diagram algebra called the partition algebra which we describe and employ to describe the space of SN invariant observables.

27 Jan 2022

Speaker :  Adrian Padellaro (QMUL) : Schur-Weyl Duality, Diagram Algebras and Matrix Observables

Matrix models can be usefully viewed as zero-dimensional quantum field theories and serve as computational tools which capture the combinatoric and algebraic structure underlying correlators of gauge invariant operators in higher-dimensional quantum field theories. I will review the correspondence between symmetric groups and U(N) invariant matrix polynomials and its application to AdS/CFT. The U(N) case is a single instance of a more general correspondence between Schur-Weyl dual algebras and invariant matrix polynomials. Along the way, I will introduce a diagrammatic description for gauge invariant operators. The diagrams are related to interesting ldquo;diagram algebras rdquo;, which contain useful information about correlators. A useful background reference is https://arxiv.org/abs/hep-th/0205221 and recent work exploiting the diagrammatic technique for permutation invariant matrix models is https://arxiv.org/abs/2112.00498

20 Jan 2022

Speaker :  Lakshya Bhardwaj (Oxford) : Relative Defects and Trapped Higher-Form Symmetries

Relative theories are non-topological theories living at the boundaries of TQFTs in one higher dimension. An interesting and well-studied class of relative theories are 6d 𝒩=(2,0) theories. I will introduce the notion of relative defects in relative theories, which are non-topological defects of the relative theory living at the boundary of a topological defect of the above-mentioned TQFT in one higher dimension. I will argue that codimension two defects of 6d 𝒩=(2,0) theories are relative defects. Relative defects carry “trapped” higher-form symmetries localized on their world-volume which are independent from the higher-form symmetries of the bulk theory. When the bulk theory is compactified with the insertion of relative defects, the trapped higher-form symmetries provide extra contributions to the higher-form symmetries of the lower-dimensional theory resulting from the compactification. For example, when 6d 𝒩=(2,0) theories are compactified on a Riemann surfaces with punctures (which are relative codimension-two defects) then the 1-form symmetry of the resulting 4d 𝒩=2 Class S theory obtains contributions from the 1-form symmetries trapped at the punctures, along with the well-known contribution coming from the 2-form symmetry of the 6d 𝒩=(2,0) theory.

13 Jan 2022

Speaker :  Hongliang Jiang (QMUL) : TQFTs from SCFTs in 3D

I will discuss the paper Non-unitary TQFTs from 3D 𝒩=4 rank 0 SCFTs , where 3D non-unitary TQFTs are obtained from 3D 𝒩=4 rank 0 SCFTs

6 Jan 2022

Speaker :  Rajath Radhakrishnan (QMUL) :

On “Non-Invertible Duality Defects in 3+1 Dimensions” and “Kramers-Wannier-like duality defects in (3+1)d gauge theories”