Academic Year 2019/2020

Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser N-body system with a harmonic term and its trigonometric version. In the quantum case, the correspondence amounts to a specific operator on the algebra of symmetric functions in N variables that intertwines between the quantum integrals of these two systems. For special coupling parameter values, I will explain how to extend the Lassalle-Nekrasov correspondence from the symmetric to the much wider quasi-invariant setting and, time permitting, present a conceptual explanation of the correspondence using the rational Cherednik algebra. The talk is based on joint work with Misha Feigin and Alexander Veselov.

Recent work has reinterpreted and unified relations among tree-level string theory amplitudes known since the 80's in the language of twisted homology in the moduli space of punctured Riemann surfaces. This new geometric understanding allows not only for more efficient computations of explicit forms for these relations but also opens the way towards their generalization beyond tree-level, that is beyond genus 0. In this talk I'll review this interpretation of tree-level string amplitudes, in particular the monodromy relations as twisted homology relations and the Kawai-Lewellen-Tye (KLT) as twisted intersection numbers, and report on recent work by S. Mizera, P. Tourkine and I on generalizing these concepts to higher genus. I will also discuss their field theory limit and their relevance to field theory amplitude relations, with particular interest in the color-kinematics duality and the double-copy procedure.

If we alter the microscopic dynamics of a macroscopic system, the thermodynamics will change in a corresponding manner. I will discuss a simple set of exact and approximate relations which govern such thermodynamic corrections. The results are completely general, but are particularly interesting when applied to black holes as they reveal new implications of the Weak Gravity Conjecture, a string-theory-inspired constraint on low-energy physics. Based on https://arxiv.org/abs/1909.05254.

The q-deformed Haldane-Shastry model is a partially isotropic (XXZ-like) long-range spin chain that enjoys quantum-affine (really: quantum-loop) symmetries at finite size. I will start by introducing the model and its special properties. Next I will explain how its hierarchy of commuting Hamiltonians are obtained by 'freezing' the spin-version of Ruijsenaars--Macdonald operators. I will also present an explicit expression for the model's exact eigenvectors, which are of ('pseudo' or 'l-') highest weight in the sense that, in terms of the operators from the monodromy matrix, they are eigenvectors of A and D and annihilated by C. Each has a simple component featuring the 'symmetric square' of the q-Vandermonde times a Macdonald polynomial -- or more precisely its quantum spherical zonal special case. If time permits I will outline how we obtain these eigenvectors using the affine Hecke algebra. This is based on joint work with Vincent Pasquier and Didina Serban (IPhT CEA/Saclay), arXiv:2004.13210 and ongoing.

The last years have seen remarkable progress in understanding the scattering amplitudes of massless particles. At tree-level, they possess a simplicity obscured by the traditional Feynman diagram approach, as well as a structure clearly indicating a worldsheet origin. While many of these results can be extended to loop integrands – due to the availability of underlying worldsheet models known as ambitwistor strings – the integrand emerges in a non-standard representation. In this talk, I will discuss one strategy to overcome this difficulty, and formulate compact worldsheet formulas with standard Feynman propagators.

I will argue that a careful examination of the canonical formalism of gravity, together with reasonable assumptions about the UV theory, leads to insights into the origin of holography in anti-de Sitter space. The same techniques also lead to a clean conjecture about how quantum information is localized in asymptotically flat spacetimes. The conjecture is that, in quantum gravity, all information about massless excitations can be obtained from an infinitesimal neighbourhood of the past boundary of future null infinity and does not require observations over all of future null infinity. In the context of the information paradox, this suggests that the fine-grained von Neumann entropy of the state defined on a segment (-\infty,u) of future null infinity is independent of u. This conjecture also implies that the oft-discussed "Page curve" is a misleading target to aim for in analyses of black hole evaporation.

The talk is based on work with A. Sridhar and D. Keating. We show that differential equations defining the limit shape for the 6-vertex model on a cylinder have infinitely many conserved quantities (in the Euclidean time running along the cylinder).

Relations expressing gravity as a "double copy" of gauge theory appeared first in string theory, and have been used to compute scattering amplitudes in theories of gravity, with applications to both theory and phenomenology. I will discuss how the double copy extends to solutions to the equations of motion, including our best known black hole spacetimes, and how this story connects to the original story for scattering amplitudes.

Quantum cluster algebras are quantizations of cluster algebras, which are a class of algebras interpolating between integrable systems and combinatorics. These algebras were originally introduced to study positivity phenomena arising in the study of quantum groups, and so one of the key questions regarding them (and their quantum analogues) is whether they admit a basis for which the structure constants are positive. The classical version of this question was settled in the affirmative by Gross, Hacking, Keel and Kontsevich. I will present a proof of the quantum version of this positivity, due to joint work with Travis Mandel, based on results in categorified Donaldson-Thomas theory obtained in joint work with Sven Meinhardt.

In this talk I will examine the kernel of the Dirac operator on certain ALF gravitational instantons, namely Taub NUT, Euclidean Schwarzschild and Taub bolt. These spaces present many similarities, in particular they are all spherically symmetric and admit an additional U(1) isometric action. However the structure of their fixed point set under this U(1) isometry is different, something which affects the properties of harmonic spinors and forms. As the Dirac operator itself is either undefined or with trivial kernel, we will first need to twist it by some geometrically preferred Abelian connection, thus we will start by looking at the harmonic cohomology of these spaces.

A fundamental problem in the context of Einstein’s equations of general relativity is to understand the dynamical evolution of small initial data perturbations of known spacetimes. In this talk I will discuss some of the key results, open problems and mathematical techniques used in the study of the initial value problem in general relativity. I will also mention certain problems I have studied during my PhD that arise from particle physics, cosmology and string theory models and discuss how my results fit with the current literature.

In this talk, I will review the structure of non-relativistic supergravity in three dimensions. I will show how on-shell three-dimensional non-relativistic supergravity with zero bosonic torsion can be obtained by gauging suitable non-relativistic superalgebras. I will comment on how supergravity theories with bosonic torsion can be obtained starting from non-relativistic superconformal algebras. In the second part of the talk, I will focus on how rigid non-relativistic supersymmetry can be defined on curved manifolds, by analyzing suitable non-relativistic Killing spinor equations. I will give an example of such Killing spinor equations and discuss how they lead to a class of non-relativistic manifolds on which supersymmetry can be defined.

I will discuss the effective field theories of multiple interacting massive spin-2 fields with the highest possible cutoff scale. I will highlight two distinct classes of theories. In the first class, the mass eigenstates only interact through non-derivative operators. In the second class, also kinetic mixing between the mass eigenstates is included non-linearly. We perform decoupling limit and ADM analysis to show that both formulations can be used to consistently describe an EFT of interacting massive spin-2 fields up to some strong coupling scale Λ. We further check the consistency for the case of two interacting spin-2 fields by applying forward limit positivity bounds on the scattering amplitudes to exclude the region of parameter space devoid of a standard UV completion.

It is often suggested that integrable 2d sigma models should be renormalizable, however this relationship has only previously been checked in the 1-loop approximation. The aim of this work is to understand what happens beyond 1-loop. Based on the examples of the lambda- and eta-models, we confirm that classically integrable models appear to be 2-loop renormalizable if supplemented with particular finite local counterterms, i.e. quantum corrections to the target space geometry. The lambda-model is further studied as a sigma model on a "tripled" target space where extra symmetries become manifest, enforcing renormalizability without the need for extra counterterms. Its 2-loop beta-function is computed, matching the known results for groups and symmetric spaces in the limit when the lambda-model becomes the corresponding non-abelian dual model. This leads to the statement that non-abelian duality commutes with the RG flow beyond 1-loop order.

In this talk I will discuss symmetries of 2-dimensional Yang-Mills theory corresponding to outer automorphisms of the structure group G and the corresponding defects. I will argue that in the topological limit the partition function with defects computes the symplectic volume of the moduli space of twisted G-bundles. Using the defect approach to orbifolds the corresponding orbifold theory will be constructed. The results will be presented using lattice renormalization and the functorial approach to area-dependent QFTs via regularised Frobenius algebras introduced by Runkel and Szegedy. The talk is based on joint work in progress with R. Szabo and L. Szegedy.

I will discuss the Batalin-Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on the higher algebraic structures. In particular, I will explain how a field theory gives rise to an homotopy algebra and how quasi-isomorphisms between homotopy algebras correspond to physically equivalent field theories. I will also explain how recursion relations for scattering amplitudes, both at tree-level and loop-level, can be obtained by transitioning to the minimal model of that homotopy algebra.

Magnetic Skyrmions are nontrivial vortex-like whirls in the magnetisation field of ferromagnetic materials. They are often described as topological solitons, but the topological characterisation emerges only in the long-distance description and breaks down at short distances. This means that Skyrmions aren't quite as stable as one might expect: they can be created and destroyed by 'emergent magnetic (anti)monopoles', also known as Bloch points, which are particle-like objects with finite energy. These 'monopoles' have been (indirectly) observed experimentally but their short-distance behaviour has not yet been well-understood - they are instead often thought of as singular objects. In this talk, I'll give a model of ferromagnetic materials supporting '(anti)monopoles' of finite - but small - size. Extending recent work of Barton-Singer–Ross–Schroers, I'll describe a critically coupled model, which admits moduli spaces of energy-minimising configurations describing static 'monopoles' sourcing tubes of Skyrmion density.

I will review some recent progress on quantum symmetric pair coideal subalgebras of quantum groups, in particular the construction of the universal K matrix, which bears significant resemblance to the construction of the universal R-matrix for the quantum group.

The combinatorial R-matrix is a set-theoretic solution of the Yang-Baxter equation which arises in the context of Kashiwara’s crystal base theory, a combinatorial tool in the representation theory of Lie algebras. After some gentle introduction of crystal graphs, the R-matrix will be defined as a crystal graph isomorphism. Using the latter we define a special subset of crystal graph vertices which are shown to be in bijection with so-called cylindric tableaux (an affine version of ordinary Young tableaux). Counting cylindric tableaux yields quantum Kostka numbers, these are integers which are obtained as coefficients when multiplying a Schubert class with a Chern class in the small quantum cohomology ring of the Grassmannian.

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One of the biggest successes in the theory of (classical) integrable systems is that models that possess a Lax pair and are amenable to the inverse scattering method turn out to also possess a Hamiltonian description (several in fact). The connection between these two aspects is at the core of the theory and is well documented. Each aspect allows one to study different problems: geometric structure (Hamiltonian) and solution content (Lax pair). The question of "nontrivial" boundary conditions and the construction of solutions in such models was first considered as early as 1975 by Ablowitz and Segur but only from Lax pair point of view. In 1987, Sklyanin's seminal work laid the foundations to define and study integrable boundary conditions from both the Hamiltonian and Lax pair point of views. Both aspects seem to have developed independently though and the following points were never addressed properly: 1) Can one derive the Lax pair point of view from the Hamiltonian one in a way similar to what is known in the case without boundary conditions? 2) How can one understand the apparent gap between the two approaches that predict different integrable boundary conditions? I will present results obtained with J. Avan and N. Crampé which solve these 30 year-old problems. I will also discuss the implementation of the entire framework on the example of the Ablowitz-Ladik model which led us to new, time-dependent, integrable boundary conditions and the explicit construction of soliton solutions on the half-line with these conditions.

In this talk I will present some work in progress on the Yangian symmetry of Maldacena-Wilson loops in \mathcal{N}=4 super Yang-Mills theoryin the planar limit as an aid to study its integrability properties. First, I will motivate the usefulness of (quantum) integrable field theory models, the relation with Yangian symmetries and how we may do exact calculations in these theories. I then reviewYangian algebras, the realisation in field theory and subtleties associated with the construction. Following this I turn to Maldacena-Wilson loops, an interesting class of observables, and discuss some novel insights we have made. I will comment on the Yangianversions of Slavnov-Taylor identities, implementations and anomalies.

Temperature manifests itself within quantum field theories (QFTs) and conformal field theories (CFTs) via an identification of points in the Euclidean-time direction, which differ by an integer multiple of 1/T. Today, I will talk about finite-temperature path integrals for general QFTs and for two-dimensional CFTs (2d CFTs) on the compact two-torus. By definition, the latter path integrals are modular invariant. I will discuss why, propose an extension of the modular group from SL_2(\Z) to GL_2(\Z), introduce the notion of modular forms with poles, and discuss general properties of modular forms with and without poles that are defined on the extended group GL_2(\Z). Finally, I will discuss how this extension to GL_2(\Z) may introduce a new source of anomalies/consistency conditions in 2d CFTs (and beyond).

Topological semimetals are novel topological phases of matter that are under intensive theoretical and experimental studies. In three dimensions, they are characterised by band structures that support momentum-space topological defects, such as Dirac monopoles. In this talk, I will discuss their main physical features from the point of view of gauge theory, differential geometry and topology, by introducing the fundamental concepts of Berry connections and quantum metric. These are theoretical tools defined in the momentum space of quantum phases and allow us to derive the topological invariants, such as Chern and Stiefel-Whitney numbers, associated to topological systems. Moreover, I will introduce a generalisation of Berry connections, named tensor Berry connection, which behaves like a momentum-space Kalb-Ramond field and gives rise to the Dixmier-Douady invariant related to new topological phases. Finally, I will discuss novel kinds of topological semimetals, which are characterised by extended topological defects, such as nodal lines and nodal surfaces, which behave as extended magnetic monopoles.

Micromagnetic model of two-dimensional chiral magnet is studied. It is found that in the presence of a tilted external field, the model permits the existence of skyrmion and antiskyrmion in a wide range of parameters. The coexistence of these solutions and their dynamics are discussed in detail. In particular, I will present a numerical analysis of the interparticle interactions, their dynamics and a new phenomenon of skyrmion fusion. Presented results are obtained by means of direct energy minimization and simulations based on the Landau-Lifshitz equation.

The past years has seen a revival of interest in non-relativistic gravity theories in different communities (condensed matter physics, high-energy physics, gravity, mathematics) for a variety of reasons. Among these theories, Newton-Cartan gravity, which is a frame-independent reformulation of Newtonian gravity, is the most popular one. In this talk we will first give a review for non-experts of different aspects on Newton-Cartan gravity and its underlying Newton-Cartan geometry. We will next argue that non-relativistic strings sense a different kind of geometry which we will call string Newton-Cartan geometry. We will discuss a few new insights into the properties of non-relativistic string theory, in particular its relation to non-relativistic branes.

I will discuss applications of trinionic SCFTs in the context of geometric engineering and G(2) compactifications of M-theory to four dimensions.

Cardiac tissue and the Belousov-Zhabotinsky (BZ) chemical reaction are two examples of excitable media that host spiral wave vortex strings. The evolution of knotted and linked vortex strings will be described in the FitzHugh-Nagumo model of electrical activity in cardiac tissue, including applications to knot untangling. Experimental results on linked vortex strings in the BZ reaction will be presented and a novel swimming motion discussed.

We identify the boundary data on the null infinity of asymptotically flat spacetimes which describes the boundary geometry and the renormalized stress-energy tensor of the putative holographically dual field theory. The geometry on the boundary is a version of Carrollian geometry. The construction is carried out in the first order formalism which is well adapted to the non-Riemannian geometry on the null infinity.

The notion of Hamiltonian monodromy was introduced by Duistermaat, who defined this invariant as an obstruction to the existence of global action-angle coordinates in integrable Hamiltonian systems. Since then, non-trivial monodromy was observed in various specific examples of integrable systems, such as the spherical pendulum, the Lagrange top, and the hydrogen atom in crossed fields. In this talk, we shall recall the classical notion of Hamiltonian monodromy and discuss some of its generalisations. In particular, we shall discuss the so-called scattering monodromy, which generalises Hamiltonian monodromy to the case of non-compact systems; specifically, to systems that are both integrable and scattering. We present a new general description of this scattering invariant and compute it for the spatial Euler two-centre problem. This talk is based on joint works with H.W. Broer, H.R. Dullin, K. Efstathiou and H. Waalkens.

Originally developed to provide a geometric foundation for Newtonian gravity, Newton-Cartan geometry and its torsionful generalization have recently experienced a revival of interest, particularly in the contexts of non-AdS holography and various condensed matter problems -- notably the quantum Hall effect. In this talk, I will describe a general theory of Newton-Cartan submanifolds. A covariant description of non-relativistic fluids on surfaces is an important open problem with a wide range of applications in for example biophysics. Recasting `elastic' models, such as the Canham-Helfrich bending energy, in a Newton-Cartan setting allows for a covariant notion of non-relativistic time and provides the ideal starting point for a treatment of Galilean fluids on extremal submanifolds using the technology of hydrostatic partition functions.