Upcoming seminars for the academic year 2024/2025
: Joost Nuiten (Université Toulouse Ⅲ Paul Sabatier), The Van Est isomorphism for higher stacks
A classical theorem of Van Est relates the differentiable cohomology of a Lie group to the cohomology of its Lie algebra. I will discuss a version of this result for homotopy‐theoretic variants of Lie group(oid)s, such as the string group, relating their cohomology to the cohomology of their (higher) Lie algebroids. The main challenge is to provide a well‐behaved analogue of Lie differentiation for higher groupoids; I will discuss how one can use ideas from Lurie and Pridham to provide such a construction.: Eirik Eik Svanes (Universitetet i Stavanger), Heterotic distance conjectures and symplectic cohomology
Deformations of the heterotic superpotential give rise to a topological holomorphic theory with similarities to both Kodaira–Spencer gravity and holomorphic Chern–Simons theory. Although the action is cubic, it is only quadratic in the complex structure deformations (the Beltrami differential). Treated separately, for large fluxes, or alternatively at large distances in the background complex structure moduli space, these fields can be integrated out to obtain a new field theory in the remaining fields, which describe the complexified hermitian and gauge degrees of freedom. We investigate properties of this new holomorphic theory, and in particular connections to the swampland distance conjecture in the context of heterotic string theory. In particular, the quadratic action gives rise to an elliptic complex, similar to the symplectic cohomologies of Tseng and Yau, but where the Beltrami differential of the large complex structure modulus plays the role of the symplectic form.: Martin Wolf (University of Surrey), Homotopy Algebra Perspective on Quantum Field Theory
I will review recent progress on understanding perturbative quantum field theory within the realm of homotopy algebras. (slides): Kymani Tieral Keden Armstrong‐Williams (Queen Mary University of London), What can abelian gauge theories teach us about kinematic algebras?
The last twenty years or so have given rise to an interesting relationship (known as the double copy) at the level of scattering amplitudes between non-abelian gauge theories (such as Yang–Mills theory) and quantum gravity theories. After half a decade, this duality was seen to be present in classical physics for exact solutions in classical Yang–Mills theory and general relativity. Arising from the Double Copy, the phenomenon of BCJ duality implies that gauge theories are more complex than previously thought. They are now known to possess an abstract kinematic algebra, mirroring the non-abelian Lie algebra underlying the colour information. Although the nature of the kinematic algebra is known in certain cases, a full understanding is missing for arbitrary non‐abelian gauge theories, such that one typically works outwards from well‐known examples. In this talk, we explore using simpler abelian gauge theories to clarify our understanding of kinematic algebras. We first describe how classes of abelian gauge fields are associated with well‐defined subalgebras of the diffeomorphism algebra. By considering certain special subalgebras, we show that one may construct interacting theories, whose kinematic algebras are inherited from those already appearing in a related abelian theory. This talk will be based on work done in arXiv:2401.10750. (slides): Martina Bártová (Coláiste na Tríonóide, Baile Átha Cliath), Introduction to Quantum Integrability
This talk provides an accessible introduction to quantum integrability, exploring the mathematical structure of integrable quantum systems. We will discuss key concepts such as conserved quantities, exact solvability, and the role of symmetries.: Samuel Valach (Coláiste na Tríonóide, Baile Átha Cliath), Thermal Correlators, Holography and Black Hole Singularity
Black hole singularities in AdS are reflected in the so‐called bouncing geodesics. In the dual CFT this leads to the bouncing singularities in the thermal correlators. I will explain both sides of this problem and show how these singularities arise from the analytic properties of the stress tensor and its composites in the boundary CFT.: Sonja Klisch (Oilthigh Dhùn Èideann), A double copy from twistor space
In this talk I will present a new double copy relation at tree level between gauge theory and gravity amplitudes, graded by helicity.Amplitudes in gauge theory and gravity in can be partitioned into sectors according to their external helicity configurations. At tree level there are particularly beautiful expressions for these amplitudes coming from twistor strings known as the RSVW (Roiban–Spradlin–Volovich–Witten) formula for gauge theory, and Cachazo–Skinner formula for gravity. At the same time, the double copy gives an explicit relation between gauge theory and gravity amplitudes via . However, the explicit representation of the Cachazo-Skinner formula as a double copy of the RSVW formula has long been unknown.
The question answered in this talk is therefore: how do these helicity graded amplitudes manifest the double copy? In the journey there we will discover a new twistorial representation of biadjoint scalars, a way towards the double copy on non‐trivial backgrounds, and the importance of trees.
: Mehran Jalali Farahani مهران جلالی فراهاني (Oilthigh Heriot‐Watt, Maxwell Institute for Mathematical Sciences), Principal Lie groupoid bundles with connection (and a bit beyond)
In this talk, I present our recent work towards general theory of higher principal bundles with connection. In particular, we developed the notion of principal groupoid‐bundles with connection using a new framework that is suitable for generalization. In this setup, principal bundles with connection are principal bundles without connection in a different category, namely the category of NQ‐manifolds.: Catarina Carvalho (University of Hertfordshire), TBA
TBA: Mario García‐Fernández (Instituto de Ciencias Matemáticas, Universidad Autónoma de Madrid), TBA
TBA: Nivedita (University of Oxford), TBA
TBA: Yang‐Hui He 何杨辉 (University of Oxford; City, University of London), The AI mathematician: from physics, to geometry, to number theory
We summarize how AI can approach mathematics in three ways: theorem‐proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and string theory, we present a number of recent experiments on how various standard machine‐learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry to representation theory, to combinatorics, and to number theory. At the heart of the programme is the question: how does AI help with mathematical discovery?: Jingxiang Wu (University of Oxford), TBA
TBA: Pieter Bomans (University of Oxford), TBA
TBA: Niklas Garner (University of Oxford), TBA
TBA: Karapet Mkrtchyan Կարապետ Մկրտչյան (Imperial College London), TBA
TBA: Erica Bertolini (Institiúid Ard‐Léinn Bhaile Átha Cliath), TBA
TBA: Fedor Leonidovich Levkovich‐Maslyuk Фёдор Леонидович Левкович‐Маслюк (City, University of London), TBA
TBAPast years’ seminars
: Lukas Müller (Perimeter Institute for Theoretical Physics), Topological defects
Recently, the study of higher categories of topological defects in quantum field theory has gained significant attention due to their connection to categorical symmetries. These higher categories exhibit noteworthy additional structures, depending upon the specific theories and defects under consideration. For instance, in oriented 2‐dimensional field theories, they organize into a pivotal bicategory. Currently, we lack a comprehensive framework to systematically describe these intricate structures. In my talk I will argue that the theory of higher dagger categories provides such a framework. The talk is partially based on joint work in progress with Gio Ferrer, Brett Hungar, Theo Johnson‐Freyd, Cameron Krulewski, Nivedita, Dave Penneys, David Reutter, Claudia Scheimbauer, Luuk Stehouwer, and Chetan Vuppulury.: Pavol Ševera (Université de Genève), On non‐topological boundary conditions of (perturbative) topological field theories
In Witten’s CS/WZW correspondence the chiral WZW model appears from the chiral boundary condition imposed on the CS theory. I will describe how similar boundary conditions of the CS theory and of its analogs (given by the AKSZ constructions) produce many other interesting examples and how they explain, in particular, Poisson–Lie ‐duality. Unlike the CS/WZW correspondence, this generalization is only perturbative; on the other hand, the perturbative renormalization leads to the generalized Ricci tensor and hopefully still hides many interesting things. Based on joint works with J. Pulmann, F. Valach, and D. Youmans.: Daniel Waldram (Imperial College London), ‐maximisation and slope stability in generalised geometry
‐maximisation is a key property of 4d superconformal field theories that allows one to find the ‐symmetry and central charge. Understanding the generic gravitational dual of this process is a key question in holography. I will describe some work in progress with Anthony Ashmore, Stephanie Baines and Michela Petrini, where we argue that it can be understood in terms of a notion of slope stability for AdS backgrounds in generalised geometry, in a way that physically connects closely to renormalisation group flow. For simplicity we will focus on the special case of Sasaki–Einstein geometries, where, using generalised geometry, one reproduces (and extends) the volume minimisation procedure of Martelli–Sparks–Yau. Very broadly one can view the dual field theory and generalised geometry formulations as algebraic and differential descriptions of the same underlying geometry.: Bruno Giménez Umbert (University of Southampton, Perimeter Institute for Theoretical Physics), amplitudes from the positive tropical Grassmannian
The global Schwinger formula, introduced by Cachazo and Early as a single integral over the positive tropical Grassmannian, provides a way to uncover properties of scattering amplitudes which are hard to see in their standard Feynman diagram formulation. In this talk I will review how partial biadjoint scalar amplitudes are computed using the global Schwinger formula, and also how one can recover amplitudes from it by using a limiting procedure on the kinematic invariants. Moreover, I will show how the regions in the positive tropical Grassmannian contributing to amplitudes are obtained from non‐crossing ‐chord diagrams, which can be extended to objects whose triangulations are in bijection with Feynman diagrams. Finally, I will point out a connection between these extended diagrams and off‐diagonal partial amplitudes, implying that amplitudes can be expressed as a sum of products of cubic amplitudes.: Hyungrok Kim 김형록 (University of Hertfordshire), A witty title containing ‘homotopy algebras’, ‘scattering amplitudes’, ‘holography’ and ‘Schwinger–Keldysh’, such as this one
Abstract: ‘Give a child a hammer, and they will treat everything like a nail.’ As a child, the speaker was given homological perturbation theory (HPT) of ‐algebras, and he thus treats every perturbation theory as an instance of HPT. In particular, he will show how HPT naturally produces the Feynman diagrams of ‐matrices, the Witten diagrams of AdS/CFT, the cut diagrams of non‐time‐ordered Schwinger–Keldysh observables (as in the recent work by Caron‐Huot et al.) and more. He begs pardon for referring to himself in the third person. Based on ongoing joint work with Leron Borsten, David Simon Henrik Jonsson, Martin Wolf and (last but not least) Charles Alastair Stephen Young. (slides): Ján Pulmann (Oilthigh Dhùn Èideann), Lagrangian relations, half‐densities and quantum algebras
Lagrangian relations model maps and more general correspondences between physical systems. In Batalin–Vilkovisky formalism, it is natural to generalize Lagrangian relations to distributional half‐densities, as advocated by Ševera. We give a rigorous definition of linear distributional half‐densities and describe their composition, thus constructing a linear version of a quantum odd symplectic category. As an application, we describe the computation of the BV effective action as a composition in this category. Based on [arXiv:2401.06110], joint with B. Jurčo and M. Zika.: Carlos Shahin Shahbazi Alonso (Universidad Nacional de Educación a Distancia), The Heterotic‐Ricci flow and its three‐dimensional solitons
I will introduce a novel curvature flow, the Heterotic‐Ricci flow, as the two‐loop renormalization group flow of the Heterotic string common sector and study its three‐dimensional compact solitons. The Heterotic‐Ricci flow is a coupled curvature evolution flow, depending on a non‐negative real parameter , for a complete Riemannian metric and a three‐form on a manifold . Its most salient feature is that it involves a term quadratic in the curvature tensor of a metric connection with skew‐symmetric torsion . When the Heterotic‐Ricci flow reduces to the generalized Ricci flow and hence it can be understood as an extension of the latter via a higher order correction prescribed by string theory, whereas when and the Heterotic‐Ricci flow reduces to a constrained version of the RG‐2 flow and hence it can be understood as a generalization of the latter via the introduction of the three‐form . Solutions of Heterotic supergravity with trivial gauge bundle, which we call Heterotic solitons, define a particular class of self‐similar solutions of the Heterotic‐Ricci flow. I will present several structural results for three‐dimensional Heterotic solitons, providing the complete classification of three‐dimensional strong Heterotic solitons as quotients of the Heisenberg group equipped with a left‐invariant metric. Furthermore, I will prove that all Einstein three‐dimensional Heterotic solitons have constant dilaton, leaving as an open problem the construction of a Heterotic soliton with non‐constant dilaton. In this direction, I will prove that Heterotic solitons with constant dilaton are rigid and therefore they represent isolated points in moduli space that cannot be deformed into a soliton with non‐constant dilaton. Work in collaboration with Andrei Moroianu (Paris). (slides): Donal O’Connell (Oilthigh Dhùn Èideann), Amplitudes and waveforms
Waveforms are the most important precision observables in modern studies of general relativity. I will discuss how waveforms in gravitational scattering are closely very related to scattering amplitudes. The computation of next‐to‐leading order waveforms has revealed a number of interesting aspects of this connection, including generalisations of amplitudes, BMS dependence of the waveform, and the importance of a new class of functions involving iterated integrals of Bessel functions.: Silvia Nagy (Durham University), Asymptotic symmetries for subleading soft theorems
It is by now well understood how leading soft theorems follow as Ward identities of asymptotic symmetries defined at null infinity. For subleading infrared effects the connection is more subtle, but it turns out that this can be formalised, to all orders, by adapting the Stueckelberg procedure to construct an extended radiative phase space at null infinity. I will show this at all orders for self‐dual Yang–Mills, self‐dual gravity, and finally Yang–Mills.: Severin Bunk (University of Hertfordshire), Infinitesimal higher symmetries
Every bundle on a manifold has a universal symmetry group: for any Lie group , giving a ‐equivariant structure on the bundle is equivalent to giving a Lie group homomorphism from to this universal symmetry group. The existence of non‐trivial equivariant structures is usually obstructed. In this talk we modify this idea in two ways: we consider the infinitesimal version of universal symmetries and allow higher, or categorified bundles. These appear, for instance, in supergravity, higher gauge theory, more generally in geometry and topology. We will use a family‐version of the fundamental theorem of derived deformation theory to construct the higher, infinitesimal analogues of universal symmetry groups. We use this to provide a unified definition of connections on any higher bundle and an algebraic formulation of differential cohomology. This extends work by Baez, Schreiber, Waldorf, Kapranov and others. Here, the curvature of a higher connection appears as the infinitesimal version of the aforementioned obstruction to equivariance. We elaborate in particular on the case of higher ‐bundles, or ‐gerbes. This is joint work with Lukas Müller, Joost Nuiten and Richard Szabo. (slides): Nick Early (Max‐Planck‐Institut für Mathematik in den Naturwissenschaften), Scattering amplitudes and tilings of moduli spaces
In 2013, Cachazo, He and Yuan (CHY) discovered a remarkable framework for scattering amplitudes in Quantum Field Theory (QFT) which mixes the real, complex and tropical geometry associated to the moduli space of points on the projective line, . By duality, this moduli space has a twin moduli space of generic points in , leading to dual realization of scattering amplitudes, using a generalization of the CHY formalism introduced in 2019 by Cachazo, Early, Guevara and Mizera (CEGM). Any duality begs for an investigation: what structures lie between the twin moduli spaces? CEGM developed a framework to answer the question for moduli spaces of points in any , leading to the discovery of rich, recursive structures which seem to suggest a new frontier in the study of scattering amplitudes. In this talk, we discuss recent joint work with Cachazo and Zhang which uses ideas from oriented matroids to construct color‐dressed generalized biadjoint scalar amplitudes.: Ricardo Monteiro (Queen Mary University of London), On helicity violation in Yang–Mills and gravity
I will discuss topics related to helicity violation in Yang–Mills and gravity. Firstly, I will discuss the connection of helicity violation, well known from the context of scattering amplitudes, to the concept of electric–magnetic duality — and in particular to its failure. Secondly, I will focus on the self‐dual sectors of Yang–Mills and gravity. These sectors satisfy classically a version of the electric–magnetic duality, but this symmetry is anomalous, giving rise to the all‐plus one‐loop amplitudes. Finally, I will discuss some lessons and intriguing aspects.: Georgios Papathanasiou Γεώργιος Παπαθανασίου (City, University of London), Evaluating Feynman integrals with the help of the Landau equations
Feynman integrals are the core of theoretical physics predictions for collider experiments, and at the same time exhibit remarkably deep mathematical structure. In this talk, I will describe how the knowledge of their singularities, as captured by the Landau equations, may aid the formidable task of their evaluation much more than was previously expected. In particular, I will provide evidence that the latter equations capture the building blocks or ‘letters’ of the integrals, when recast as a polynomial of their variables known as the principal ‐determinant. (slides): Ömer C. Gürdoğan (University of Southampton), Antipodal dualities: reading form factors backwards
Our understanding of scattering amplitudes is in a process of perpetual refinement where new results lead to new insights, which in turn enable even more ambitious calculations. I will report on a recent observation that certain amplitudes and form factors are mapped to each other under the antipode of the Hopf algebra of multiple polylogarithms. This observation is based on an eight‐loop calculation of form factors, which turns out to be very amenable to the symbol bootstrap approach.: Benjamin Hoare (Durham University), Diamonds of integrable deformations
There are two well‐known origins of 2‐dimensional integrable QFTs from 4 dimensions — localisation of 4d Chern–Simons and symmetry reduction of 4d anti‐self‐dual Yang–Mills. It is a conjecture of Costello that these can be unified in a diamond of theories, starting from holomorphic Chern–Simons theory in 6 dimensions.It has been shown how this works for the simplest class of theories, including the principal chiral model with a Wess–Zumino term, by Bittleston and Skinner. In this talk I will discuss what happens if we try to deform and explore the new features that appear, including the role of novel boundary conditions in 4d Chern–Simons.
This talk is based on work with Lewis Cole, Ryan Cullinan, Joaquín Liniado and Dan Thompson.
: Javier José Murgas Ibarra (Universitetet i Stavanger), A heterotic Kodaira–Spencer theory
Taking extrema of the six‐dimensional heterotic superpotential gives the so‐called ‐term constraints of the Hull–Strominger system. Starting at a solution, one can consider ‘off‐shell’ fluctuations of the structure in the superpotential. This gives rise to a theory with similar features to Kodaira–Spencer gravity. A natural next step is then to quantise this theory. I will discuss the computation of the one‐loop partition function and find it can be expressed as a product of holomorphic Ray–Singer torsions. Finally, I will discuss its potential metric anomaly.: Leron Borsten (University of Hertfordshire), Higher symmetries and homotopy algebras: scattering amplitudes, colour–kinematics duality, ‐ and ‐algebras, the double copy and M2‐branes models
The string of (possibly cryptic) words in the title will be discussed. Neither prior knowledge of, nor interest in, all of the terms is assumed. We hope there is something for everyone! Essentially, two currently pervasive ideas that we find interesting will be brought together:- Scattering amplitudes are the most direct bridge between quantum field theory and particle collider experiments. They are also incredibly rich structures that provide deep physical/mathematical insights into the underlying theories. An example is provided by the colour–kinematics duality of gluon amplitudes. While in Yang–Mills theory the internal colour and spacetime Lorentz symmetries ostensibly live independent lives, it seems that they dance to the same tune in the scattering amplitudes. A consequence of this hidden property is that graviton scattering amplitudes are the “double copy” of amplitudes: !
- Homotopy algebras generalise familiar algebras (matrix, exterior, Lie…) by relaxing the defining identities up to homotopy. The homotopy maps form higher products in corresponding homotopy algebra. A key example is that of homotopy Lie algebras or ‐algebras. The violation of the familiar Lie bracket Jacobi identity is controlled by a unary and ternary bracket , which themselves satisfy nested Jacobi identities up to homotopies controlled by yet higher brackets and so on. They arise naturally and inevitably in a number of mathematical contexts, such as categorified symmetries. They also have deep connections to physics. Indeed, every perturbative Lagrangian quantum field theory corresponds to a homotopy Lie algebra, allowing one to move between the physics of scattering amplitudes and the mathematics of homotopy algebras.
We shall first review the remarkable correspondence between perturbative quantum field theory and homotopical algebras. We will then illustrate how the colour–kinematics duality of scattering amplitudes can be realised at the level of the Batalin–Vilkovisky action: assuming tree‐level colour–kinematics duality of the physical ‐matrix, there exist an action principle manifesting colour–kinematics duality as a (possible anomalous) conventional symmetry. In homotopy algebraic terms, the associated homotopy commutative algebra (aka the “colour‐stripped” homotopy algebra) carries a homotopy ‐algebra structure. This observation, in turn, allows for simple proofs of (tree‐level) colour–kinematics for a variety of theories, some old, some new, and progress in characterising what is and isn’t possible at the loop‐level. For example, we give a concise proof that the BLG and ABJM M2‐brane models have tree‐level colour–kinematics duality. (slides)
: Julian Mateo Kupka (University of Hertfordshire), Non‐relativistic geometry and why you might care
This talk explores non‐relativistic geometry, a mathematical framework allowing us to formulate Newtonian physics analogously to General Relativity, i.e. geometrizing it. We will build the geometry and theory from the ground up and explain how exactly it encodes Newtonian gravity in a diffeomorphism invariant fashion. We will then connect it to null reductions, which allows us to relate non‐relativistic to relativistic physics with a special null direction, highlighting why you might encounter it in your applications. (slides): Lewis Napper (University of Surrey), Monge–Ampère Geometry and the Navier–Stokes equations
Partial differential equations of Monge–Ampère type have been shown to correspond to specific choices of differential form on the associated phase space. Furthermore, Lagrangian submanifolds of said space may be viewed as a generalisation of solutions to a Monge–Ampère equation. Building from the observation that the Poisson equation for the pressure of an incompressible, two‐dimensional, Navier–Stokes flow can be presented as an equationof Monge–Ampère type, this talk introduces a framework for studying fluid dynamics using properties of the aforementioned submanifolds. In particular, it is noted that such a submanifold may be equipped with a metric whose signature acts as a diagnostic for the dominance of vorticity and strain. We discuss how this approach may be extended to fluid flows on an arbitrary Riemannian manifold and to higher dimensions, through the use of higher (categorified) symplectic geometry. We conclude with some comments on how this facilitates symmetry reductions and some open questions.: James Waldron (University of Newcastle), Skew Hecke Algebras
Given a finite group , a subgroup , and an action of on an algebra , we define an associated ‘skew Hecke algebra’. This construction provides a simultaneous generalisation of Hecke algebras of finite groups and of skew group algebras. I will explain some of the basic properties of these algebras, some results about their structure, and the relation to rings of invariants. I will also discuss the representation theory of these algebras and their homological properties such as global dimension. If there is time I will explain the relevance of these algebras to the study of quantum reference frames, and possible analogues of skew Hecke algebras for von Neumann algebras. This is based on work inprogress, joint with Leon Loveridge (University of South‐Eastern Norway) and Jan Głowacki (PAN Centre for Theoretical Physics, Warsaw).: Arthur Lipstein (Durham University), From amplitudes to cosmology
Whereas scattering amplitudes probe physics at the shortest distances, cosmology probes physics at the largest scales. Nevertheless, many concepts discovered in the study of scattering amplitudes generalise to cosmological observables. In this talk I will describe some aspects of this program which I have recently been investigating such as the double copy, scattering equations, and soft limits, and will discuss how these directions are interrelated.: Cheuk Yu Mak 麥焯如 (University of Southampton), Quantum cohomology and loop group action
Quantum cohomology is an invariant of symplectic manifolds. It is a commutative deformation of theusual cohomology as an algebra, where the deformation is controlled by counting certain holomorphic spheres. In this talk, we will first explain how it is related to the loop space, and to Floer cohomology. Then we will sketch that,in the presence of a Hamiltonian ‐action on a symplectic manifold with acompact Lie group , how we can obtain an action on the ‐‐equivariant quantum cohomology from the ‐equivariant Borel–Moore homology of the based loop space of . No knowledge on symplectic geometry is assumed. This is a joint work with Eduardo González and Dan Pomerleano.: Fridrich Valach (Imperial College London), ‐algebroids, embedding tensors, and Poisson–Lie duality
I will describe a class of structures, known as ‐algebroids, which arises as a natural generalisation of the ordinary, generalised, and exceptional tangent bundles from ordinary, generalised, and exceptional geometry, respectively. I will discuss aclassification result in the exceptional case and show how to obtain the possible fluxes/twists of the bracket. In the M‐theory and ⅡB setups, the twists organise themselves naturally into connections and covariantly constant differential forms, while in the ⅡA case one in particular recovers both the Romans mass and the deformation of Howe–Lambert–West. Finally, I will show how to use these algebroids to answer a question about the realisability of embedding tensors (providing a new perspective on the result of Inverso ’17) and to give a joint description of the Poisson–Lie ‐ and ‐duality. This is ajoint work with M. Bugden, O. Hulik, and D. Waldram.: Nils Martin Sten Cederwall (Chalmers tekniska högskola), Supersymmetry and Koszul duality
The pure spinor superfield formalism is a very general method to construct supermultiplets, and in certain cases, Batalin—Vilkovisky actions for them, using extra variables inthe form of constrained spinors. The main focus has typically been on thecircumvention of old “no‐go theorems” against manifestly supersymmetric off‐shell formulation of maximally supersymmetric models, such as super‐Yang—Mills theory and supergravity. In this talk, I will briefly review the formalism, and then describe how the constrained spinors are Koszuldual to certain superalgebras, generically algebras, constructed fromthe supermultiplets, and how these algebras introduce non‐linear structures on the supermultiplets. This is work in collaboration with S. Jonsson, J. Palmkvist and I. Saberi.: Emily Cliff (Université de Sherbrooke), Higher symmetries: smooth 2‐groups and their principal bundles
A 2‐group is acategorical generalization of a group: it’s a category with a multiplication operation which satisfies the usual group axioms only up to coherentisomorphisms. In this talk I will introduce the category of Lie groupoids and bibundles between them, in order to provide the definition of a smooth 2‐group. I will define principal bundles for such a smooth 2‐group and provide classification results that allow us to compare them to principal bundles for ordinary groups. I will conclude with some potential applications of these ideas to various areas, including string geometry, Chern–Simons theory, and equivariant elliptic cohomology. This talk is based on joint work with Dan Berwick‐Evans, Laura Murray, Apurva Nakade, and Emma Phillips. I will not assume any previous background on 2‐groups or Lie groupoids.: Christian Sämann (Oilthigh Heriot‐Watt), ‐duality with categorified principal bundles
The data appearing in the Buscher rules for ‐duality, the Kalb–Ramond field and the metric, can be unified to a connection on a categorified principal bundle governing all geometric T‐dualities. Interestingly, this picture allows for a direct extension to ‐dualities involving non‐geometric backgrounds. Before explaining all this, I will give a gentle introduction to categorified principal bundles. (slides): Katarzyna Anna Rejzner (University of York), Symmetries in mathematical perturbative quantum field theory
In this talk I will give an overview of (perturbative) algebraic quantum field theory and explain how it can be applied to describe interacting QFT models. I will then report on new progress in the description of renormalization group and symmetries, using nets of C*‐algebras.: Brian R. Williams (University of Edinburgh), A holomorphic approach to fivebranes
In almost all situations, the twist of a supersymmetric QFT has the structure of a holomorphic QFT. I’ll review general aspects of holomorphic QFT while drawing parallels to the familiarsituation of chiral CFT. I will then define a holomorphic model which wepropose describes the minimal twist of the six‐dimensional superconformal theory associated to the Lie algebra . A primary source of evidence forthis proposal is via an approach to holography in the twisted setting using Koszul duality.: Alexander Schenkel (University of Nottingham), BV and BFV formalism beyond perturbation theory
Modern approaches to quantum field theory, such as factorization algebras and homotopical AQFT, are based on cohomological methods in field theory that have been developed over the past decades in the context of the BV and BFV formalisms. While these techniques work pretty well for perturbative quantum field theories, they have intrinsic limitations to describe global features, such as the topology of the gauge group or moduli of bundles. Derived algebraic geometry is a powerful geometric framework in which one can attempt to globalize the BV and BFV formalisms. I will start this talk with a very basic introduction to derived algebraic geometry, focussing in particular on its more concrete and computational aspects. I will then illustrate the power and potential of this framework for new developments in mathematical physics by studying two applications:- The non‐perturbative BV formalism for a function on a quotient stack, and
- the quantization of a derived cotangent stack over a quotient stack, which is a global version of BFV quantization.
: Atul Sharma (University of Oxford), Twistors, gravity and celestial holography
I will describe a 2‐dimensional sigma model that is dual to self‐dual GR in 4‐dimensional flat space. Our sigma model governs holomorphic maps from the Riemann sphere to twistor space. Semiclassical correlators of the model take us beyond self‐dual GR and give rise to Hodges’ formula for tree‐level MHV graviton amplitudes. Whereas its 2d OPE algebra reproduces the infinite‐dimensional algebra recently discovered by Strominger to be the symmetry algebra of celestial holography.This talk is based on the works arXiv:2103.16984 and arXiv:2110.06066.: Pietro Benetti Genolini (King’s College London), Localization of the action in AdS/CFT
I will introduce a simple formula for the action of supersymmetric solutions to minimal gauged supergravity in the context of the AdS/CFT correspondence. Such solutions are equipped with a Killing vector, and I will show that the holographically renormalized action may be expressed entirely in terms of the weights of this vector field at its fixed points, together with certain topological data. In this sense, the classical gravitational partition function localizes in the bulk, and depends only on the topology of the fixed point set.: Congkao Wen 温从烤 (Queen Mary University of London), Integrated correlators in super Yang–Mills
In this talk, I will present an exact expression for an integrated correlator of four BPS operators in supersymmetric Yang–Mills (SYM) with any classical gauge group . I will show that this can be expressed as a simple two‐dimensional lattice sum, which is a function of and of the complex Yang–Mills coupling . This expression contains infinite orders of perturbative and non‐perturbative instanton terms, and manifests the Goddard–Nuyts–Olive duality of SYM. Furthermore, the integrated correlator satisfies a striking ‘Laplace difference’ equation, which determines the integrated correlator for all classical gauge groups in terms of the correlator. I will also comment on matching our result with the standard Feynman diagram computation, and the relation to four‐graviton amplitude in type ⅡB superstring in AdS space.: Severin Bunk (University of Oxford), Symmetries of gerbes, anomalies, and topology
Gerbes are geometric objects which describe the third integer cohomology of a manifold and the ‐field in string theory. Infinitesimal symmetries of gerbes on a manifold are associated with algebroids on . In this talk, we instead investigate finite symmetries of gerbes; we demonstrate how in the presence of a Lie group action on , a gerbe on induces an extension of the acting Lie group by a 2‐group. These extensions have various applications: they contain all information about equivariant structures on gerbes, explain non‐associative magnetic translations, describe anomalies in QFTs in odd dimensions, and give rise to new models for the string group.: Robert Gray (University of East Anglia), Undecidability of the word problem for one‐relator inverse monoids via right‐angled Artin subgroups of one‐relator groups
It is a classical result of Magnus proved in the 1930s that the word problem is decidable for one‐relator groups. This result inspired a series of investigations of the word problem in other one‐relator algebraic structures. For example, in the 1960s Shirshov proved the word problem is decidable in one‐relator Lie algebras. In contrast, it remains a longstanding open problem whether the word problem is decidable for one‐relator monoids. An important class of algebraic structures lying in between monoids and groups is that of inverse monoids. In this talk I will speak about a recent result which shows that there exist one‐relator inverse monoids of the form with undecidable word problem. This answers a problem originally posed by Margolis, Meakin and Stephen in . I will explain how this result relates to the word problem for one‐relator monoids, the submonoid membership problem for one‐relator groups, and to the question of which right‐angled Artin groups arise as subgroups of one‐relator groups.: Luigi Alfonsi (Queen Mary University of London), Towards an extended/higher correspondence – generalised geometry, bundle gerbes and global ‐duality
Extended Geometry is the geometric framework which makes ‐duality and ‐duality manifest symmetries. This is realised by lifting generalised geometry to a higher dimensional space which is, at least locally, the space underlying the fundamental representation of the duality group. However, despite its relevance in M‐theory, the global aspects of Extended Geometry are still unclear.I propose a global formulation of Extended Geometry via a generalisation of Kaluza–Klein principle which unifies a metric and a higher gauge field on a categorified principal bundle (e.g. a bundle gerbe). I will illustrate the relation of this formulation with usual generalised geometry and with the string ‐model. Finally, I will discuss how a global notion of generalised fluxes (geometric and non‐geometric) emerges from this picture and how this is related to a higher gauge theory with the string 2‐group.
: Guilherme Almeida (Scuola Internazionale Superiore di Studi Avanzati), An overview of Dubrovin Frobenius manifolds
Dubrovin Frobenius manifold is a geometric interpretation of a remarkable system of differential equations, called WDVV equations. Since the beginning of the nineties, there has been a continuous exchange of ideas from fields that are not trivially related to each other, such as: Topological quantum field theory, non‐linear waves, singularity theory, random matrices theory, integrable hierarchies of KdV type, and Painlevé equations. Dubrovin Frobenius manifolds theory is the bridge between them.In the first part of the talk, I will review the notion of Dubrovin Frobenius manifolds and its relationship with finite reflection groups and its extensions, isomonodromic deformation equations and Integrable hierarchy of KdV type. In the second part of the seminar, I will review the differential geometry of Orbit spaces of reflection groups and its extensions, which are one of the main examples of Dubrovin Frobenius manifolds. Thereafter, I will present a new extension of finite reflection group, which its orbit space has a Dubrovin Frobenius structure. This work is based on the papers arXiv:1907.01436 and arXiv:2004.01780.
: Tommaso Franzini (University of Hertfordshire), An introduction to Thermodynamic Bethe Ansatz with some applications
Since the Thermodynamic Bethe Ansatz was first introduced, it has always been a powerful tool to investigate several properties of integrable quantum field theories. In this talk I will give an introduction to the topic, explaining a general method to derive the TBA equations and focusing on some of their most relevant aspects. In this context, I will also present the work I have done for my master thesis about the structure of the TBA for a theory with quantum group symmetry, as part of a collaboration with Prof. Ravanini from Bologna University and Prof. Ahn from Ewha University.: Charles Strickland‐Constable (University of Hertfordshire), Generalised geodesics: the geometry of strings and branes
I will present recent work showing how the classical solutions of string and membrane sigma models can be understood as the analogue of integral curves of auto‐parallel vector fields in generalised geometry.I will try to make the first part of the talk accessible to a non‐string‐theory audience.