University of Hertfordshire Mathematics and Theoretical Physics Seminars

I will present a geometric perspective on superspace that proves fruitful for studying twisted supergravity, especially in eleven dimensions. This allows for a uniform description of the eleven-dimensional theory and its twists in terms of homotopy Poisson-Chern-Simons theory on appropriate superspaces. If time permits, I will further discuss higher symmetries in twisted supergravity theories and how they hint towards interesting global structures.

We analyze cosmological (in-in) correlators and the wave function of the universe in dS at tree and loop levels. We demonstrate a hidden simplicity in the structure of in-in correlators arising by adding various building blocks, which allows us to obtain a new integral representation for the in-in correlators in terms of scattering amplitudes in flat space. Therefore, enabling us to study cosmological graphs by exploiting amplitude tools and derive new relations for in-in correlators. Since dS wave functions are equivalent to AdS correlators by analytic continuation, a part of the analysis is also valid for AdS correlators. (slides)

Given a special holonomy manifold, we can obtain an associated superconformal algebra of symmetries (SW-algebra) by considering a string theory compactification. This is a fascinating correspondence between geometry and conformal field theory that has spectacular consequences: for instance, mirror symmetry was originally discovered this way! Although the correspondence is well understood for torsion-free G-structures and compactifications without fluxes, very little is known for more general G-structures in the presence of NS flux. In this talk I will present the first steps to identify these conformal algebras. We will do so through a study of the classical sigma model, focusing on the case of G2 manifolds. Time permitting, I will discuss how to reformulate these results in the language of Vertex Operator Algebras and present some explicit examples.

This talk is based on 2412.13904, joint with Xenia de la Ossa and Enrico Marchetto, and on 2502.02769, joint with Andoni de Arriba de la Hera and Mario Garcia-Fernandez. It also connects with some ongoing work with Leron Borsten and Hyungrok Kim. (slides)

In AdS/CFT, extremal couplings induce a mixing between single- and double-trace operators. I will diagnose this mixing and compute it coefficient using the conformal block expansion of 4-point Witten diagrams involving extremal couplings. I will then study the simplest dual pair in which such couplings arise: 4d N=2 theories dual to AdS5 x S5 with some 7-branes. Summing a tower of extremal Witten diagrams provides the final piece of the gluon 4-point function to order 1/N^2. (slides)

Geometry underpins gravity, while the quantum world is governed by algebras of observables. In string theory, (super)gravity and its quantum corrections emerge from specific vibration modes of relativistic strings, described by vertex operator algebras in two-dimensional conformal field theories. However, the precise map between these two is highly non-trivial and is only known explicitly to the first orders in a perturbative expansion. I will reveal a new, infinite-dimensional symmetry algebra that at least reproduces the leading two orders of quantum corrections. Remarkably, it has the potential to evade a no-go theorem that rules out competing approaches to uncover hidden symmetries in supergravity beyond second-order corrections. Given that many of the algebraic structures encountered here have a natural generalization from strings to membranes, this suggests a similar approach to analyzing quantum corrections in 11-dimensional supergravity - the low-energy limit of M-theory. Beyond fundamental insights, these results have broad applications, including the construction of new supergravity solutions, the analysis of their spectrum, the computation of renormalization group flows in two-dimensional σ-models, and the study of integrable strings. A generalization of Cartan geometry is found on the mathematical side.

Sen’s superstring field theory successfully formulates perturbative superstring theory but has a number of strange features.

The action is not fully background independent and it does not have standard diffeomorphism symmetry - instead there is an exotic gauge symmetry that acts on some fields but not others.

These issues greatly restrict the spacetimes that this action can be applied to.

A new action is found that reduces to Sen’s in a certain limit but which has diffeomorphism symmetry together with further exotic symmetries and is background independent. As a result, it can be applied to any spacetime. (slides)

Sen has proposed a novel action for self-dual fields that has been generalised by Hull to include two metrics. The resulting theory has many novel features including two independent diffeomorphism-like symmetries. Furthermore the action is quadratic in the fields and as a result the path-integral can be can be computed. We will discuss various aspect of this theory in two-dimensions such as the partition function and modular invariance. (slides)

The emergence of a universal pre-hydrodynamic attractor behaviour during fluid thermalisation is an important feature in the study of ultra-relativistic nuclear collisions and cold atom experiments: highly non-hydrodynamic behaviour dictated by initial conditions quickly decays towards this attractor solution, with the resulting hydrodynamic system independent of the initial state.

There are many approaches to study the emergence of this attractor. We will briefly review these approaches, then focusing on kinetic theory. Starting with the relativist Boltzmann equation in the so-called relaxation time approximation, we introduce a series of moments and their generating function, which we show solved a single PDE. We solve this PDE perturbatively for all moments, both in the small and large time approximations, and we then recover the free streaming and attractor regimes at each end via methods of analytic continuation.

Global symmetries can be generalised to transformations generated by topological operators, including cases in which the topological operator does not have an inverse. A family of such topological operators are intimately related to dualities via the procedure of half-space gauging. In this talk I will discuss the construction of non-invertible defects based on T-duality in two dimensions, generalising the well-known case of the free compact boson to any Non-Linear Sigma Model with Wess-Zumino term which is T-dualisable. I will also present several applications of this symmetry, which include the protection of some coupling constants of the sigma model from quantum corrections as well as new Ward identities. Time permitting, I'll discuss the target space interpretation of these defects, and their fate in String Theory. (slides)

While strong uniqueness theorems hold for stationary, asymptotically flat black holes in four dimensions, the results break down in higher dimensions and when the asymptotically flat condition is relaxed. Nonetheless, there has been significant progress in classifying the near-horizon geometries of extremal black holes. Motivated by this, we start with near-horizon geometries, which are better understood, and deform them to obtain extremal black hole solutions. We investigate whether a given near-horizon geometry uniquely corresponds to a black hole solution or if there are multiple black holes with the same near-horizon geometry. Our approach employs the homotopy algebraic framework, a powerful and increasingly influential framework in both classical and quantum field theory. Using homological perturbation theory, we recursively solve the deformation problem order by order. To illustrate the method, we present a concrete example of the deformation of the extremal Kerr horizon. (slides)

I will describe the use of methods of arithmetic geometry to find gauge theories in 4 dimensions that are free of anomalies. In many cases, it is possible to find all such theories.

In recent years, a growing community of researchers have been applying ideas from Quantum Information / Computing theory to high energy physics. Entanglement is one such quantity, but there are by now many others. This talk will examine the property of “magic” (also known as “non-stabliserness”) which, roughly speaking, measures whether quantum systems have a genuine computational advantage over their classical counterparts. I will review the origin of this concept, before describing recent applications from both collider physics and scattering amplitudes research. No prior knowledge of quantum computing will be assumed. (slides)

A colouring of a graph with $k$ colours is an assignment of colours to vertices so that no edge is monochromatic. As it is well-known colouring with 2 colours is in P while colouring with $\mathrm{k\; >\; 2}$ colours is NP-complete. This dichotomy was extended to the graph homomorphism problem, also called $H$-colouring, by Hell and Nešetřil [J. Comb. Theory B, 48(1):92-110, 1990]. More precisely, they proved that deciding whether there is a graph homomorphism from a given graph to a fixed graph $H$ is in P if $H$ is bipartite (or contains a self-loop), and is NP-complete otherwise. This dichotomy served as an important test-case for the Feder–Vardi dichotomy conjecture, and Bulatov–Zhuk dichotomy of complexity of finite-template CSPs.

In the talk, I will present a new proof of this theorem using tools from topological combinatorics based on ideas of Lovász [J. Comb. Theory, Ser. A, 25(3):319-324, 1978] and Brower’s fixed-point theorem. This is joint work with Sebastian Meyer (TU Dresden).

Symmetry is a recurring concept in both Science and Art. Upon hearing the term, one initially tends to think of rotations and reflections found in Geometry. However, if we recall that, formally, the symmetries of a geometric shape $X$ are just the distance preserving bijections $f:X\to X$, then we see that this idea naturally generalises from Geometry to virtually all areas of Mathematics: The symmetries of any kind of mathematical object $X$ are the bijections $f:X\to X$ which preserve the structure of $X$.

For example, the symmetries of a graph, group, or vector space are its automorphisms and the symmetries of a topological space its homeomorphisms. In each case, the symmetries of $X$ form a subgroup of the symmetric group $Sym\left(X\right)$ of all permutations of $X$. By Cayley's Theorem, all groups are isomorphic to subgroups of symmetric groups, and so Group Theory is essentially the study of symmetries.

There are two widely-studied and natural generalisations of the concept of symmetry above which correspond to generalisations of groups to semigroups and inverse semigroups, respectively. First, one might drop the requirement that symmetries need to be invertible. For example, one might consider all continuous transformations of some topological space $X$. This takes us from subgroups of $Sym\left(X\right)$ to subsemigroups of the Full Transformation Monoid $X^<mi>X</mi>$ of all functions from $X$ to $X$.

Alternatively, one might be interested not just in the symmetries of the whole structure $X$, but also in partial symmetries, i.e. symmetries between substructures of $X$. This approach takes us to studying inverse subsemigroups of the Symmetric Inverse Monoid $I$_X consisting of all bijections between subsets of $X$.

In this talk, we will consider $Sym\left(X\right)$, $X^<mi>X</mi>$, and $I$_X on an infinite set $X$. I will present a variety of results which highlight the connections between semigroups (including groups and inverse semigroups) on the one hand and Set Theory, Model Theory, and Topology on the other. No previous knowledge of Semigroup Theory will be assumed. (slides)

Kontsevich-Quinn (Finite Total Homotopy) TQFT is a topological quantum field theory defined for any dimension $B$ of space, depending on the choice of a homotopy finite space $B$. For instance, $B$ can be the classifying space of a finite group or a finite 2-group.

In this talk, I will report on recent joint work with Tim Porter on once-extended versions of Kontsevich-Quinn TQFT, taking values in the symmetric monoidal bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, yielding (0,1,2)-, (1,2,3)-, and (2,3,4)-extended TQFTs, given a homotopy finite space $B$. I will show how to compute these once-extended TQFTs when $B$ is the classifying space of a homotopy 2-type, represented by a crossed module of groups. Time permitting, I will address the twice-extended TQFTs also arising, as well as the ensuing representations of motion groups.

References:
Faria Martins J, Porter T: "A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids." arXiv:2301.02491 [math.CT]

Planar scattering amplitudes in N=4 super Yang-Mills theory are famously dual to the expectation values of null polygonal Wilson loops, and there has been much recent interest in studying closely related objects, such as a Wilson loop with a Lagrangian operator insertion.

After presenting some motivation for studying such objects and reviewing amplitude-Wilson loop duality, I will present evidence that much of the same mathematical structure is also enjoyed by correlators of multiple loop operators. This includes very natural generalisations of the Q-bar equation and BCFW recursion relations, and expressions for one-loop leading singularities in terms of tree-level objects. (slides)

A celebrated result of Håstad established that, for any constant ε>0, it is NP-hard to find an assignment satisfying a (1/|G|+ε)-fraction of the constraints of a given 3-LIN instance over an Abelian group G even if one is promised that an assignment satisfying a (1−ε)-fraction of the constraints exists. Engebretsen, Holmerin, and Russell showed the same result for 3-LIN instances over any finite (not necessarily Abelian) group. In other words, for almost-satisfiable instances of 3-LIN the random assignment achieves an optimal approximation guarantee. In this talk, I will talk about a recent result showing that the random assignment algorithm is still best possible under a stronger promise that the 3-LIN instance is almost satisfiable over an arbitrarily more restrictive group.

Joint work with Alberto Larrauri and Stanislav Živný. (slides)

The KLT double copy relates open string amplitudes to closed string amplitudes, which provides the string-theoretic version of the more familiar relation Gravity=(Yang-Mills)^2. This relation is mediated by the KLT Kernel. In field theory, this kernel is well-studied, and it is the inverse of a matrix of amplitudes in bi-adjoint scalar theory (BAS). These amplitudes have been central in many recent advances in the field of scattering amplitudes. Crucially, it is known that these amplitudes have a completely geometric description in terms of the ABHY associahedron.

By contrast, much less is known about the string theory KLT kernel. Its inverse defines some α'-completion of BAS amplitudes, and exhibits intrinsically stringy properties. In this talk, I will show that these 'stringy BAS' amplitudes also admit a description from positive geometry. We will see how the stringy features emerge from this geometric description. Furthermore, I will argue that this geometry also contains all pion amplitudes in the Non-Linear Sigma Model, as well as mixed pion/BAS amplitudes. This gives us a first glimpse into how the positive geometry framework can be used to capture truly stringy features, and can be extended beyond the realm of rational functions.

Based on: Phys.Rev.Lett. 136 (2026) 1, 011601 and Arxiv: 2508.20161 (slides)

In recent years there has been a renewed interest in understanding string theory close to BPS bounds using the tools of non-Lorentzian physics. In these regimes the underlying geometry is deformed, becoming a generalisation of Newton-Cartan geometry based around a brane-like foliation of spacetime. By considering such limits on both sides of a Lorentzian holographic duality we can construct dualities between non-Lorentzian QFT and string theory on non-Lorentzian spacetimes, which in the IR reduces to solutions of exotic non-Lorentzian supergravity theories. In this talk I will show how these theories, along with their BPS brane solutions, can be constructed using the explicit example of the M5-brane limit of eleven-dimensional supergravity. We will apply these lessons to D-brane limits from the perspective of both the gravitational solution and the worldvolume QFT to construct proposals for non-Lorentzian holography. We will see that both limits exhibit the same symmetry structure as the gravitational solutions, with the QFT dynamics reducing to motion on the moduli space of solitons representing quarter-BPS brane bound states. Time-permitting, I will finish with a discussion of upcoming work on including quantum effects on the QFT side using the example of Galilean Yang-Mills, which is proposed to be dual to non-relativistic string theory on a near-horizon D2-brane background. (slides)

One of the dozens of pieces of data one can use to define a matroid is its base polytope, the convex hull of the zero-one vectors encoding the bases. A matroid subdivision is a way to subdivide a matroid polytope into other matroid polytopes. I've been studying matroid subdivisions since my PhD. I suppose I should mention my most recent work on them, with Andy Berget and with Chris Eur and Matt Larson, giving two proofs of a 2005 conjecture of Speyer on the maximal number of cells in a matroid subdivision.

Matroid subdivisions that are obtained by projection from a polytope one dimension larger are in bijection with linear spaces in tropical geometry. More special yet, if $K$ is a field with a non-Archimedean valuation and we have a linear subspace of $Kn,\; then\; by\; applying\; the\; valuation\; to\; all\; maximal\; minors\; of\; the\; associated\; matrix,\; we\; get\; a\; matroid\; subdivision.\; The\; parameter\; spaces\; for\; these\; two\; constructions\; are\; called\; the$Dressian and the tropical Grassmannian.

The Dressian is strictly bigger than the tropical Grassmannian, and both are hard to explicitly compute; even constructing matroid subdivisions in other ways than by applying a valuation is hard. What's remarkable is that both spaces have a "totally positive" part, and the totally positive parts exactly agree. I'm aware the totally positive tropical Grassmannian is of interest in physics due to connections to amplitudes, and I can say a few things I understand to be of interest there (from conversations with Nick Early), but I'm sure you can tell me more!

I will present two recent results about supersymmetric quantum mechanics with 2 supercharges (1d N=2). First, I will explain new Seiberg-like dualities between 1d SQCD-like theories, which are distinct 1d gauge theories with the same supersymmetric ground states. Then I will discuss the coupling of 1d N=2 quivers to 3d N=2 gauge theories in the context of the GLSM/Quantum K-theory correspondence, explaining how such line defects engineer interesting objects (the so-called Schubert classes) in the quantum K-theory of partial flag varieties. (slides)

In the case of finite group G, the symmetries of 2d G gauge theories can be obtained using fusion categories and "higher representation-theoretic" methods.  However, gauge theories that appear in physics make use of Lie group symmetry.  In this talk, I will give a dictionary between gauge theory and category theory, and show how the fusion-categorical approach to gauge theory can be extended to recover the symmetries of flat continuous gauge theories in arbitrary dimension.

The problem of scale separation - i.e. whether String Theory admits AdS vacua where the size of the extra dimensions can be made parametrically smaller than the AdS radius - is an outstanding problem in string phenomenology. Moreover, it is also a fundamental question in Holography and Conformal Field Theory. After a pedagogical introduction covering the connections between these aspects, I will present a new consistency condition for the compatibility of a gravitational effective field theory in AdS with a dual holographic description. The condition amounts to the requirement that certain (properly defined) cubic interactions in the bulk must vanish. Remarkably, the constraint is satisfied for the well-known, scale-separated DGKT vacua in type IIA, thanks to a series of non-trivial cancellations.  I will conclude with a possible bulk interpretation, as well as a future outlook. (slides)

Recently, Bozec–Calaque–Scherotzke have articulated a noncommutative version of the AKSZ construction, which associates to a smooth Calabi–Yau category, a fully extended TQFT valued in a category of iterated Calabi–Yau cospans. In this talk, I will study a class of examples of such theories which arise in the context of conjectures of Costello and Li, which describe Type II strings in certain Ramond–Ramond backgrounds as topological strings. These TQFTs capture structural features of the BPS physics of D-branes that are universal in Chan–Paton factors. Conjecturally commutative limits of the values of such theories on closed manifolds can sometimes be geometrically quantized to yield algebraic structures with Hall-type products. Examples of this paradigm include CoHAs associated to complex 3-folds, CoHAs attached to local systems on 3-manifolds, and the categorified Hall algebras of Porta–Sala. (slides)

It is well known to mathematicians that there is a deep relationship between the arithmetic of algebraic varieties and their geometry.

These areas of mathematics have a fascinating connection with physical theories and vice versa. Examples include Feynman graphs and black hole physics. There are very many relationships however I will focus on the structure of black hole solutions of superstring theories on Calabi-Yau manifolds.

The main quantities of interest in the arithmetic context are the numbers of points of the variety, considered as varieties over finite fields, and how these numbers vary with the parameters of the varieties. The generating function for these numbers is the zeta function, about which much is known in virtue of the Weil conjectures. The first surprise, for a physicist, is that the numbers of these points, and so the zeta function, are given by expressions that involve the periods of the manifold. These same periods determine also many aspects of the physical theory, including the properties of black hole solutions.

I will discuss a number of interesting topics related to the zeta function, the corresponding L-function, and the appearance of modularity for one parameter families of Calabi-Yau manifolds. I will focus on an example for which the quartic numerator of the zeta function of a one parameter family factorises into two quadrics at special values of the parameter. These special values, for which the underlying manifold is smooth, satisfy an algebraic equation with coefficients in Q, so independent of any particular prime. The significance of these factorisations is that they are due to the existence of black hole attractor points in the sense of type II supergravity which predict the splitting of the Hodge structure over Q at these special values of the parameter. Modular groups and modular forms arise in relation to these attractor points, in a way that is familiar to mathematicians as a consequence of the Langland’s Program, but which is a surprise to a physicist. To our knowledge, the rank two attractor points that were found together with Mohamed Elmi and Duco van Straten by the application of number theoretic techniques, provide the first explicit examples of such attractor points for Calabi-Yau manifolds. (slides)

Positivity bounds are theoretical constraints on the Wilson coefficients of an effective field theory. These bounds emerge from the requirement that a given effective field theory must be the low-energy limit of a relativistic quantum theory that satisfies the fundamental principles of unitarity, locality, and causality. In this talk we will discuss how the task of deriving (forward scattering) positivity bounds can be reformulated as the geometric problem of finding the extremal rays of a convex cone of four-tensors with certain symmetries. We will give the solution to this problem for scatterings involving up to three flavors. Finally, we will see how the positivity bounds obtained this way give rise to new constraints that are not already captured by elastic bounds and hence improve the known forward scattering positivity bounds from the literature. Based on arXiv:2508.18165. (slides)

The double copy, which relates gravity to the square of Yang–Mills theory, is usually studied in flat spacetime and is known to hold both for scattering amplitudes and for classical field configurations. In this talk, I will present a new approach to extending these results to maximally symmetric curved spacetimes. The new relations are formulated in twistor space, a complex projective space that encodes solutions to the equations of motion as holomorphic data. First, I will discuss the case of AdS₃, where a double copy structure for bulk to boundary correlators and black hole solutions naturally arises in minitwistor space. Then, I will show how in (A)dS₄ one can construct bulk correlation functions using only twistors, dual twistors, and the infinity twistor as building blocks. The relation to coordinate space arises via nested Penrose transform. The boundary limit of these correlators yields CFT correlators that satisfy the expected Ward identities and obey a simple double copy relation. (slides)

Let G be a graph, i.e., a relational structure with a single symmetric binary relation. The quantified constraint satisfaction problem, QCSP(G), asks whether some input sentence, whose quantifier-free part is positive conjunctive (i.e. a conjunction of positive atoms), is true on G. Each input sentence is associated with a graph, and we investigate the complexity of QCSP when certain substructures are forbidden in the input graph.

The knot-quiver correspondence provided a picture of BPS state counts in 3d N=2 theories associated to symmetric quivers. On the other hand, we have another, well established construction of BPS states in 4d N=2 theories which also correspond to quivers, but not symmetric ones. In this talk I will give an answer for an obvious and long-standing question about the relation between these two perspectives.

I will give an introduction to functorial 2-dimensional topological field theories (TFTs), starting with the classical result that closed 2D TFTs valued in vector spaces are commutative Frobenius algebras. Along the way, I will indicate higher categorical generalisations, such as how taking values in linear categories one arrives at a notion of a modular functor. In these higher categorical or derived settings, it is generally difficult to fully classify TFTs, but one can still extract interesting data such as representations of mapping class groups.

Open-closed 2D TFTs are a generalisation where one allows surfaces with corners, and they offer a way of constructing closed 2D TFTs. In recent work with Barkan and Zhang, we classify open 2D TFTs and show that every open 2D TFT admits a canonical extension to an open-closed one. I will explain this statement and some of the ingredients that go into its proof.

Correlators and Amplitudes in the South East 2026 is a postgraduate meeting funded by the graduate school of the South East Physics Network (GRADnet). The goal is to bring together early-career researchers interested in correlators, scattering amplitudes and related areas of mathematical and theoretical physics. Everyone is welcome to give their contribution with a talk (20 to 30 minutes long) or a poster. We would like to create a friendly and informal environment where you can exchange ideas, discuss ongoing work, and build connections within the community. The meeting should also provide a good opportunity for people working in related areas to know each other ahead of Amplitudes 2026, which will take place at the end of June in Queen Mary University of London.

The notion of a superconformal structure on a supermanifold goes back some forty years. I will discuss some recent work that shows how these structures and their deformations govern supersymmetric and superconformal field theories in geometric fashion. A superconformal structure equips a supermanifold with a sheaf of dg commutative algebras; the tangent sheaf of this dg ringed space reproduces the Weyl multiplet of conformal supergravity (equivalently, the superconformal stress tensor multiplet), in any dimension and with any amount of supersymmetry. This construction is uniform under twists, and thus provides a classification of relations between superconformal theories, chiral algebras, higher Virasoro algebras, and more exotic examples.

$1$‐form symmetries are new types of symmetries, long known to even undergraduate students, but only recently formalized. Their distinguishing property is that (Wilson) loop operators carry charge under this global symmetry. As ordinary ($0$‐form) symmetries, they can be spontaneously broken or restored. Importantly such symmetries allow a rigorous definition of the confining phase, as the phase where the symmetry is restored, and Wilson loops obey area law. The alternative is that the symmetry is spontaneously broken, and Wilson loops get an expectation value, proportional to their length (the perimeter law). In 4d, the symmetry is discrete (e.g. ${\mathbb{Z}}_{𝑁}$) the spontaneously broken phase is a TQFT, while if it is continuous (e.g. $\mathrm{U}\left(1\right)$) is a photon phase, where photons are viewed as Goldstone bosons of the spontaneously broken symmetry group. Surprisingly, however, even systems with only a discrete $1$‐form symmetry generically have a photon phase. Moreover this photon phase can be seen as proliferation of string‐like objects — popularily called center vortices in QCD literature — which are commonly claimed to cause confinement. Rather I will argue theoretically, and demonstrate numerically that such objects cause the emergence of a $\mathrm{U}\left(1\right)$ theory, while another type of particle like object — the monopole — may cause confinement. I will give explicit examples in ${\mathbb{Z}}_{𝑁}$ lattice gauge theory of such phenomena.

Quantum toroidal algebras are the ‘double affine’ objects within the quantum setting. In particular, they contain — and are generated by — horizontal and vertical quantum affine subalgebras.

These quantum toroidal algebras are not known to possess a coproduct, but do have a topological coproduct mapping to a completion of the tensor square. However, this still fails to produce a tensor product on the category of integrable modules due to the presence of infinite sums.

In this talk, after introducing the quantum toroidal algebras, I shall explain how to overcome these issues — namely, we’ll obtain well‐defined tensor products satisfying various nice properties. (For example, there are compatibilities with both Drinfeld polynomials and $q$‐characters, as well as toroidal $R$‐matrices that solve the Yang–Baxter equation.) (slides)

I will first give a brief historical overview of the $D\left(2\right)$‐problem before explaining the importance of syzygy modules in solving this problem in low‐dimensional topology. We will see how a group structure is easily obtained, and I will conclude by discussing the syzygy modules for dihedral groups, using these to provide an affirmative answer for the $D\left(2\right)$‐problem, in this case. (slides)

To be held 10:30–16:30. See the seminar page for details.

I will discuss my recent work on a symplectic‐geometrical realisation of non‐invertible symmetries (alias codimension $1$ topological defects) and its application to Chern–Simons theory. In particular my collaborators and I have constructed non‐invertible symmetries in non‐abelian Chern–Simons theory using a Yang–Baxter type equation from the theory of integrable systems, and we have found a novel and simple kind of non-invertibility even in abelian Chern–Simons theory, based on the mathematical structure of a semigroup.

In this talk I get to tell you about my latest paper with my PhD advisor Marius de Leeuw in which we complete the classification of $4\times 4$ solutions of the Yang–Baxter equation. Regular solutions were recently classified and in this paper we found the remaining non‐regular solutions. I will present some of these new solutions, then consider regular and non‐regular Lax operators and their relation to the quantum Yang–Baxter equation.  I will show that for regular solutions there is a correspondence, which is lost in the non‐regular case. In particular, we will see that the non‐regular Lax operators whose $R$‐matrix from the fundamental commutation relations is regular but does not satisfy the Yang–Baxter equation. These $R$‐matrices satisfy a modified Yang–Baxter equation instead.

Beginning with simple examples, my talk will introduce a new approach to understanding Atiyah–Patodi–Singer indices for manifolds with boundary that casts them within the framework of an associated topological quantum field theory. The main result interprets these indices in terms of the determinant line bundle of the induced differential operator over the boundary and proves a gluing property. Based on arXiv:2312.06818.

Localization is a powerful technique that simplifies drastically the evaluation of the path integrals defining certain protected observables in $𝒩=2$ supersymmetric theories in $𝑑=4$. To apply localization, the theory must be defined on a compact space like ${𝑆}_4$. For theories which are conformal also at the quantum level, the results on ${𝑆}_4$ can be mapped onto flat space. In recent works, we have been investigating the situation for theories with a non‐zero $𝛽$ function. Naively one would expect the conformal anomaly to prevent the extension of localization results on ${𝑆}_4$ to flat space. However, we show that this is actually the case within in a well specified perturbative regime. We perform explicit checks with flat space Feynman diagrams up to order ${𝑔}^6$ for observables like circular Wilson loops and correlators of chiral operators.

I shall demonstrate that all scale invariant two‐dimensional sigma models with compact target space are conformally invariant. The proof relies on techniques developed by Perelman for the proof of the Poincaré conjecture that I shall review.

The BV formalism was first introduced to deal with the perturbative quantization of gauge theories, generalizing the BRST procedure. It is well known that, in the case of supergravity, the supersymmetry transformations close only on‐shell, leading to a rank‐$2$ (i.e. quadratic in the anti‐fields) BV action.

After a short introduction to the BV formalism, I will present the fully covariant BV analysis of $𝒩=1$, $𝐷=4$ supergravity in the first order formalism, where the spin connection is treated as a dynamical field.

If time allows it, I will highlight the advantages of such description in the presence of boundaries, where one can expect to obtain a BFV structure describing the boundary theory. (slides)

In the dynamic setting of competitive multi‐agent systems, strategic decision‐making and resource allocation are key to achieving optimal outcomes. This research explores the optimisation of power and mobility in adversarial multi‐agent games, modelled as a Nash zero‐sum framework. By utilising advanced game‐theoretic methods, we analyse the interactions between the agents of two teams of autonomous vehicles, with decision variables encompassing communication, interference, and movement. The model incorporates power, movement, and minimum distance constraints, employing fixed‐point formulations and block‐descent algorithms to determine equilibrium strategies. Additionally, we introduce the concept of team preferences and discuss ongoing work on individual agent preference.

I will report on a work in progress, join with Kang Lu, Yang‐ning Peng, Lukas Tappeiner and Weiqiang Wang, which connects the two algebras in the title of this talk. Finite $W$‐algebras are an interesting family of algebras, built from a complex semisimple Lie algebra and a nilpotent element by a process of quantum Hamiltonian reduction. Twisted Yangians are a family of algebras of a totally different pedigree. They were introduced by Olshanskii in order to quantize the twisted current algebra coming from symmetric pairs, and they arise as coideal subalgebras of the Yangians studied extensively by Drinfeld. A famous theorem of Brundan and Kleshchev states that the finite $W$‐algebras of type A arise as quotients of the (shifted) Yangian associated to general linear algebras. For finite W-algebras associated to even nilpotent elements in types B, C, D it turns out that the (shifted) twisted Yangians are the correct replacement. Our work generalises results of Brown which dealt with the rectangular case. I will attempt to make the talk as accessible as possible.

Quantum Field Theory (QFT) is an exciting yet elusive domain within mathematics and physics. Despite the lack of rigorous foundations to support many advancements made by physicists, mathematicians have engaged in a fruitful endeavor to formalize QFTs. In current times, we find ourselves at a crossroads: while we still lack the comprehensive techniques and language to fully grasp QFT, numerous distinct axiomatic frameworks are attempting to capture its essence. A natural question arises: how are these approaches connected? This talk will focus on two such frameworks: Algebraic Quantum Field Theories (AQFTs) and Factorization Algebras (FAs). The significance of these frameworks is motivated, among other things, by rigorous programs led by Fredenhagen–Rejzner and Costello–Gwilliam to construct perturbative QFTs using AQFTs and FAs, respectively. Recent contributions from Gwilliam–Rejzner and Benini–Musante–Schenkel establish a relationship between these two programs, vaguely speaking. At a more structural level, Benini–Perin–Schenkel have established an equivalence of 1-categories between specific subcategories of AQFTs and (time‐orderable pre‐)FAs. The goal of this talk is to present our proof of an even broader equivalence between $\infty$‐categories of AQFTs and tpFAs.

This talk is based on arXiv:2412.07318 and ongoing joint work with M. Benini, A. Grant‐Stuart and A. Schenkel.

A common direction in algorithmic graph theory is to investigate the complexity of hard problems restricted to specific classes of graphs. This raises the question of whether large sets of such problems can be explained by common problem conditions. For a set of graphs $\mathscr{H}$, the class of $\mathscr{H}$‐subgraph‐free graphs is the class of all graphs not containing any graph from $\mathscr{H}$ as a subgraph. We present a framework to classify the complexity of certain problems on $\mathscr{H}$‐subgraph‐free graph classes. In particular, we give a meta‐theorem classifying the complexity of problems satisfying three conditions.

We extend the powerful property of Yangian invariance to a new large class of conformally invariant multi-loop Feynman integrals. This leads to new highly constraining differential equations for them, making integrability visible at the level of individual Feynman graphs. Our results apply to planar Feynman diagrams in any spacetime dimension dual to an arbitrary network of intersecting straight lines on a plane (Baxter lattice), with propagator powers determined by the geometry. The graphs we consider determine correlators in the recently proposed “loom” fishnet CFTs. The construction unifies and greatly extends the known special cases of Yangian invariance to likely the most general family of integrable scalar planar graphs. We also relate these equations in certain cases to famous GKZ (Gelfand–Kapranov–Zelevinsky) hypergeometric operators, opening the way to using new powerful solution methods. (slides)

In this talk, I will discuss different mathematical properties of exotic ($7$‐)spheres and their implications in physics. Namely, I will cover some known facts as well as recent results regarding:

• The geometry of Milnor’s bundles and a number of analytic results obtained in arXiv:2309.01703 (J. High Energ. Phys. 2023, 100) and arXiv:2410.01909; I will discuss the relevance of these results in the context of supergravity.
• The differentiable structure of exotic spheres together with its conjectured implications in general relativity and shock waves (work in progress).
• A numerical technique for finding and approximating Einstein metrics, to be applied eventually to the case of exotic spheres, which is based on state‐of‐the‐art machine learning tools (work in progress with Edward Hirst and Alex Stapleton). The possible uses of this numerical scheme in (super)gravity theories are also discussed.

(slides)

The concept of a fracton quasiparticle has recently emerged in many areas of physics, from mathematical physics to condensed matter and quantum information, unified by the general feature of restricted mobility.

In this seminar I will introduce the basic notion of fracton quasiparticles, focussing on the gauge field theory construction developed in Pretko's original papers [1604.05329,1606.08857], mentioning the physical applications in elasticity theory and connections with gravitational models. In the second part of the seminar I will then step into the building of a covariant model for fractons in which a strong relation with Linearized Gravity naturally emerges. The fracton phenomenology is reproduced from first principles of QFT through a symmetry‐based approach, opening the doors to a new class of covariant gauge field theories. This formulation can be generalized to any spacetime dimension, allowing for instance the study of lower‐dimentional fracton models and showing relations with some condensed matter systems. (slides)

We will summarize a recently developed approach to $𝑝$‐form dynamics, which allows for a democratic (treating electric and magnetic degrees of freedom on an equal footing) Lagrangian formulation for electrodynamics and generalization to (chiral) $𝑝$‐forms, including arbitrary abelian interactions. In particular, a Lagrangian for arbitrary duality‐symmetric non‐linear electrodynamics in $3+1$ dimensions is derived where both $\mathrm{SO}\left(2\right)$ duality and Lorentz covariance are manifest. Democratic formulation for (type Ⅱ) SUGRA in 10d and 11d is also achieved. We also present a simple derivation of this formulation from a topological theory in one higher dimension. Time permitting, more discussion on an ongoing work on generalizations of these results will be provided. (slides)

The Higgs and Coulomb branches of vacua are central objects in the mathematics of three-dimensional $𝒩=4$ supersymmetric quantum field theories and, in particular, three-dimensional mirror symmetry. In this talk, I will describe how these interesting ($2$‐shifted) Poisson schemes can be extracted by exploiting a natural action of an infinite‐dimensional Lie algebra that is a three‐dimensional analogue of the $𝒩=2$ superconformal algebra familiar from two‐dimensional superconformal field theory. This is based off of joint work with S. Raghavendran and B. Williams. (slides)

I will give an overview of higher operations in quantum field theories which control their deformations, (generalized) OPEs, and anomalies. Particular attention is paid to holomorphic-topological theories where we can systematically describe and regularize the Feynman diagrams which compute these higher operations in free and perturbative scenarios, including examples with defects.

In this talk I will introduce the holomorphic twist for generic 4d $𝒩=1$ SQFTs and discuss the infinite dimensional symmetry enhancement. Consequentially, holomorphically twisted theories come equipped with a variety of mathematical structures. After discussing the relevant structures, I will explain how to relate them to data in the physical theory. In particular. I will discuss how the central charges can be obtained from the ternary bracket. Time permitting, I will finish by demonstrating how the associated VOA can be obtained from the holomorphic twist in the presence of extended supersymmetry. (slides)

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? (slides)

We introduce $W*-Cat$, the bicategory of $W*$‐categories, functors and natural transformations and discuss its equivalence with the Morita category of von Neumann algebras ($vN2$). We highlight some analogies of $W*$‐categories with Hilbert spaces. We introduce categorified von Neumann algebras, building up to a definition of a Bicommutant Category and discuss a string calculus on a cylinder. We finally sketch some examples of bicommutant categories associated to chiral CFTs. (slides)

I will overview joint work with Raúl González Molina and Jeffrey Streets in arXiv:2408.11674 about a natural extension of pluriclosed flow, where the evolution of the hermitian metric on the complex manifold $X$ is further coupled to a hermitian metric on a holomorphic vector bundle $V$ over $X$. Flow lines are special solutions of the generalized Ricci flow in generalized geometry, and hence of the renormalization group flow in the heterotic sigma model. Using this, we interpret the flow as a version of Donaldson’s Hermitian–Yang–Mills flow in the realm of higher gauge theory. We will comment on the higher regularity structure of the flow and on global existence and convergence results on special backgrounds. If time allows, we will also discuss the relationship to the Hull–Strominger system and, conjecturally, to the geometrization of Reid’s fantasy. (slides)

We search for the class of digraphs whose obstruction sets, in the digraph homomorphism problem, consist of caterpillars. The conjecture being that this class of digraphs should be the same as the class of digraphs that are preserved by majority and set function polymorphisms. This is work in progress. (slides)

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. (slides)

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 $4d$ 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 $\text{GR}={\left(\text{YM}\right)}^2$. 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. (slides)

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.

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.

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)

I will review recent progress on understanding perturbative quantum field theory within the realm of homotopy algebras. (slides)

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.

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.

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.

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 $𝑇$‐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.

$𝑎$‐maximisation is a key property of 4d superconformal field theories that allows one to find the $𝑅$‐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 $\mathrm{SL}(𝑛+1)$ 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.

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 $(𝑝-2)$‐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.

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 ${𝐿}_{\infty }$‐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 $𝑆$‐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)

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.

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 $𝜅$, for a complete Riemannian metric and a three‐form $𝐻$ on a manifold $𝑀$. Its most salient feature is that it involves a term quadratic in the curvature tensor of a metric connection with skew‐symmetric torsion $𝐻$. When $𝜅=0$ 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 $𝐻=0$ and $𝜅>0$ 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 $𝐻$. 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)

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. (slides)

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.

Every bundle on a manifold has a universal symmetry group: for any Lie group $𝐺$, giving a $𝐺$‐equivariant structure on the bundle is equivalent to giving a Lie group homomorphism from $𝐺$ 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 $U\left(1\right)$‐bundles, or $𝑛$‐gerbes. This is joint work with Lukas Müller, Joost Nuiten and Richard Szabo. (slides)

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 $𝑛$ points on the projective line, ${ℳ}_{0,𝑛}$. By duality, this moduli space has a twin moduli space of $𝑛$ generic points in ${\mathbb{P}}^{𝑛-3}$, 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 $𝑛$ points in any ${\mathbb{P}}^{𝑘-1}$, 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.

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.

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 $𝐴$‐determinant. (slides)

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.

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.

Taking extrema of the six‐dimensional heterotic superpotential gives the so‐called $𝐹$‐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.

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:

1. 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: $\text{Einstein}=\text{Yang–Mills squared}$!
2. 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 ${𝐿}_{\infty }$‐algebras. The violation of the familiar Lie bracket $[\text{–},\text{–}]$ Jacobi identity is controlled by a unary $\left[\text{–}\right]$ and ternary bracket $[\text{–},\text{–},\text{–}]$, 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 $𝑆$‐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 ${\mathrm{BV}}^{◽️}$‐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)

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)

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.

Given a finite group $𝐺$, a subgroup $𝐻$, and an action of $𝐺$ on an algebra $𝐴$, 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).

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.

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 $𝐺$‐action on a symplectic manifold $𝑀$ with acompact Lie group $𝐺$, how we can obtain an action on the $𝐺$‐‐equivariant quantum cohomology from the $𝐺$‐equivariant Borel–Moore homology of the based loop space of $𝐺$. No knowledge on symplectic geometry is assumed. This is a joint work with Eduardo González and Dan Pomerleano.

I will describe a class of structures, known as $𝐺$‐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 $𝑇$‐ and $𝑈$‐duality. This is ajoint work with M. Bugden, O. Hulik, and D. Waldram.

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 $𝐷=10$ super‐Yang—Mills theory and $𝐷=11$ supergravity. In this talk, I will briefly review the formalism, and then describe how the constrained spinors are Koszuldual to certain superalgebras, generically ${𝐿}_{\infty }$ 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.

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.

The data appearing in the Buscher rules for $𝑇$‐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 $𝑇$‐dualities involving non‐geometric backgrounds. Before explaining all this, I will give a gentle introduction to categorified principal bundles. (slides)

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.

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 $\mathrm{𝔰𝔩}\left(2\right)$. A primary source of evidence forthis proposal is via an approach to holography in the twisted setting using Koszul duality.

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:

1. The non‐perturbative BV formalism for a function $𝑓:[𝑋∕𝐺]\to 𝑘$ on a quotient stack, and
2. the quantization of a derived cotangent stack ${\mathrm{T}}^{\ast }[𝑋∕𝐺]$ over a quotient stack, which is a global version of BFV quantization.

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 ${𝑤}_{1+\infty }$ 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.

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.

In this talk, I will present an exact expression for an integrated correlator of four BPS operators in $𝒩=4$ supersymmetric Yang–Mills (SYM) with any classical gauge group ${𝐺}_{𝑁}$. I will show that this can be expressed as a simple two‐dimensional lattice sum, which is a function of $𝑁$ and of the complex Yang–Mills coupling $𝜏=𝜃/2\pi +\mathrm{i}4\pi /{𝑔}_{\text{YM}}^2$. This expression contains infinite orders of perturbative and non‐perturbative instanton terms, and manifests the Goddard–Nuyts–Olive duality of $𝒩=4$ 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 $\mathrm{SU}\left(2\right)$ 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.

Gerbes are geometric objects which describe the third integer cohomology of a manifold and the $𝐵$‐field in string theory. Infinitesimal symmetries of gerbes on a manifold $𝑀$ are associated with algebroids on $𝑀$. In this talk, we instead investigate finite symmetries of gerbes; we demonstrate how in the presence of a Lie group action on $𝑀$, a gerbe on $𝑀$ 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.

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 $\mathrm{Inv}⟨𝐴|𝑤=1⟩$ with undecidable word problem. This answers a problem originally posed by Margolis, Meakin and Stephen in 1987. 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.

Extended Geometry is the geometric framework which makes $𝑇$‐duality and $𝑈$‐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 $𝜎$‐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.

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.

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.

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.

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