Academic Year 2020/2021

Maximally symmetric lorentzian spacetimes, like Minkowski and (Anti-)de Sitter space, are omnipresent in physics. We will review a classification of spacetimes that generalizes the classification of maximally symmetric lorentzian geometries and includes further physically interesting ones. Based on these spacetimes we will discuss limits and extensions of three-dimensional gravity and two-dimensional Jackiw-Teitelboim gravity.

Can the 32C-dimensional algebra R(x)C(x)H(x)O offer anything new for particle physics? Indeed it can. Here we identify a sequence of complex structures within R(x)C(x)H(x)O which sets in motion a cascade of breaking symmetries: Spin(10) -> Pati-Salam -> Left-Right symmetric -> Standard model + B-L (both pre- and post-Higgs-mechanism). These complex structures derive from the octonions, then from the quaternions, then from the complex numbers. Finally, we describe a left-right symmetric Higgs system which exhibits, we believe for the first time, an explicit demonstration of quaternionic triality.

At its critical point the Q-states Potts Model on random graphs is a discretised representation of a conformal field theory (otherwise known as 'matter') interacting with Liouville gravity. I will describe recent results obtained using matrix model techniques that enable computation of disk amplitudes with a large class of boundary conditions and discuss some implications.

In this talk I will present certain extensions, over Grassmann algebras, of some known maps which satisfy the set-theoretic Yang-Baxter or Entwining Yang-Baxter equations. Such maps can be constructed from refactorisation problems associated to products of Darboux matrices of certain integrable PDEs. I will discuss some integrability properties of the extended maps and open problems in relation to this work in progress.

The classification of equilibrium black holes is of fundamental interest in general relativity. The 4d vacuum case is well understood - all stationary, axisymmetric and asymptotically flat solutions are given by the Kerr metric. In 5d the picture is much less clear; whilst there is a uniqueness result for stationary, biaxisymmetric and asymptotically flat solutions, the corresponding existence problem is still open. In this talk I will discuss some recent results which give new tools for answering this problem. Using the integrability of this sector of gravity one can translate the problem into that of determining whether a particular system of polynomial equations has any solutions. These equations can be studied in the case of the simplest black lens - a black hole with lens space topology. A combination of analytic and numerical results show that there are no solutions to these equations, demonstrating that these spacetimes must be singular. This represents an important step towards the full classification in 5d.

I will give a basic introduction to the geometric aspects of Lie algebroids and show how they give the correct framework to discuss gauge theories. The analysis is solely based on geometry, and thus applies to every gauge theory, independently of their specific features and dynamics. By thoroughly re-formulating the physical content of gauge theories on Atiyah Lie algebroids, we will show that the BRST construction is part of the formalism, indicating a fascinating interplay between classical geometry and quantum physics. Time permitting, possible outlook to various domains in mathematical physics will be discussed.

Black hole perturbation theory leads to ordinary differential equations. Typically the resulting equations belong to Heun's differential equation and its confluent cousins. The same equations turn out to be found in a "quantization" of Seiberg-Witten theory. I will explain what we can predict from this relation to black hole physics.

High-energy states in large-N gauge theories obey interesting large-N scaling laws and are dual to black holes in holographic field theories. We derive the thermodynamic Euler equation for these high-energy states from the dependence of the extensive quantities on the number of colors N. This Euler equation relates the energy of the state to the temperature, entropy, number of degrees of freedom and its chemical potential, but not to the volume or pressure. In the context of the gauge/gravity duality we show that the Euler equation is dual to the generalized Smarr formula for black holes in the presence of a negative cosmological constant. We also match the fundamental variational equation of thermodynamics to the first law of black hole mechanics, when extended to include variations of the cosmological constant and Newton's constant.

Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations are system of partial differential equations which is a core structure of a Frobenius manifold. I will introduce these notions and review a special class of rational solutions. They correspond to special configurations of vectors forming a (rational) \check-system introduced by Veselov in 1999. For any Frobenius manifold there is an almost dual Frobenius manifold introduced by Dubrovin in 2004, in his work rational solutions for WDVV equations corresponding to root systems appear as almost dual prepotentials for the Coxeter orbit spaces polynomial Frobenius manifolds. Then I am going to discuss a class of trigonometric solutions which are determined by collections of vectors with multiplicities. They are closely related to trigonometric \check-systems introduced by Feigin in 2009. We show that similarly to the rational case such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. We also show that subsystems of trigonometric \check-systems are trigonometric \check-systems. This talk is based on joint work with M. Feigin.

I will talk about the program of extracting holographic geometries from path-integrals in conformal field theories. I will review basic examples and discuss the notion of path integral complexity. In the second part, I will present more recent works on path integral optimization from Hartle-Hawking wave functions in AdS/CFT.

We study the restoration of supersymmetry within systems where it is intrinsically non-linearly realized and we discuss physical applications of such a procedure related to de Sitter vacua.

I will present a holographic framework for reconstructing the experience of bulk observers in AdS/CFT. In particular, I will show how to recover the proper time and energy distribution measured along bulk worldlines, directly in the CFT via a universal, background-independent prescription. For an observer falling into an eternal AdS black hole, the proposal resolves a conceptual puzzle raised by Marolf and Wall. Notably, the prescription does not rely on an external dynamical Hamiltonian or the AdS boundary conditions and is, therefore, outlining a general framework for the emergence of time.

The moduli space of toroidal compactifications of the bosonic and heterotic string contains special points where the gauge symmetry is enhanced. We review this phenomenon in detail for the bosonic, heterotic and the so-called CHL string, and discuss its implications in the context of the swampland programme. For compactifications of the heterotic and CHL string on a circle, we show how to obtain all the the maximal gauge groups, as well as the point in moduli space where they arise, from a Generalized Dynkin diagram. We discuss its extension to compactifications on a 2-torus, which are dual to F-theory compactifications on K3.

Some theories have, in addition to momentum and energy, higher conserved charges. These theories are said to be integrable and can be exactly solved. Examples are the Kepler problem, Hubbard model and various versions of the AdS/CFT correspondence. The main ingredient of an integrable theory is an object called R-matrix which satisfies the so-called Yang-Baxter equation. In this talk, after a brief review of integrable models, we will introduce a new method to find solutions of the Yang-Baxter equation, using the so-called boost operator, and use it to classify R-matrices in various set-ups. Using this method we were able to find new interesting integrable models. We also constructed an integrable deformation of a Hubbard-like model and deformations of R-matrices from the AdS/CFT correspondence.

The microscopic motion of atoms and molecules can be well described using dynamical laws which are symmetric under time reversal; but the laws describing the actual behaviour of material objects made from these atoms (such as the thermodynamic law of entropy increase) do not inherit this symmetry. The apparent inconsistency is known as the irreversibility paradox. People have generally responded to the paradox by proposing explanations of why the behaviour should be asymmetric under time reversal, but my approach here will be instead to look for a better characterization of the asymmetry itself. The ideas will be illustrated using a simple model where the microscopic motion can be followed in detail.

In this talk I will discuss the main results of the papers arXiv:2001.10007 and arXiv:1907.11735. In these papers we studied both analytically and numerically the time dependence of the Rényi and von Neumann entropies in the integrable Ising spin chain, following different kinds of global quenches. We were interested in studying the continuum limit of these theories, namely the associated quantum field theories (QFTs) and we used QFT techniques, particularly, branch point twist fields, to perform our analytical computations. Using these techniques we have gained access not only to the precise leading linear large time dependence of the entropies that is observed in many integrable models but also to oscillatory behaviour that, depending on the quench, can become the leading feature of entanglement, at least for small quenches. Although there is still much to understand in this area of research, one of our conclusions is that the integrability of the quenched model is not the sole feature to determine its entanglement dynamics, in particular, whether or not the entropies will grow linearly with time or exhibit persistent undamped oscillations.

In this talk I will show how the BV-BFV formalism (used to quantize gauge theories with boundary) can be adapted to perturbative algebraic QFT and generalized to treat theories with non-trivial asymptotic limit "at infinity". As an example, I will discuss soft modes and conserved charges in QED on Minkowski spacetime.

There is a deep relationship between the scattering amplitudes of Yang–Mills theory and those of Einstein gravity: invoking the Bern–Carrasco–Johansson colour–kinematic (CK) duality, graviton amplitudes are the “square” or “double copy” of gluon amplitudes. This perspective is both conceptually suggestive and computationally powerful. Although the double copy is known to hold for tree-level amplitudes, the loop case has been a longstanding conjecture. In this series of two talks, we argue that the double-copy relation holds to all orders, tree and loop. To do so, we take an off-shell Lagrangian double copy point of view and capitalise on the rich connections between BV-quantisable field theories and homotopy algebras. In Part 2, using the factorisation properties of homotopy algebras, we split the CK-dual Yang–Mills BRST Lagrangian and its BRST charge into colour, kinematic and scalar factors. The colour factor can then be replaced by a second copy of the kinematic factor, producing a BRST Lagrangian and BRST charge that we argue are physically perturbatively equivalent to Einstein gravity coupled to a Kalb–Ramond 2-form and a dilaton. Based on 2007.13803, joint work by L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Sämann, M. Wolf.

There is a deep relationship between the scattering amplitudes of Yang–Mills theory and those of Einstein gravity: invoking the Bern–Carrasco–Johansson colour–kinematic (CK) duality, graviton amplitudes are the “square” or “double copy” of gluon amplitudes. This perspective is both conceptually suggestive and computationally powerful. Although the double copy is known to hold for tree-level amplitudes, the loop case has been a longstanding conjecture. In this series of two talks, we argue that the double-copy relation holds to all orders, tree and loop. To do so, we take an off-shell Lagrangian double copy point of view and capitalise on the rich connections between BV-quantisable field theories and homotopy algebras. In Part 1, we recast the Yang-Mills BRST Lagrangian into a form that manifests CK duality not only for the tree-level amplitudes of the physical gluons, but also for the unphysical longitudinal and ghost modes of the extended BRST Fock space. Using the structure theorems of homotopy algebras, this Lagrangian can be made to involve only cubic interactions. This final result is the CK-dual Yang–Mills BRST Lagrangian, ready to be double-copied. Based on 2007.13803, joint work by L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Sämann, M. Wolf.

We will give a pedagogical review on recent results, and work in progress, regarding the statistical origin of the Bekenstein-Hawking entropy of BPS black holes in AdS and string theory. The context will be that of the AdS_5/CFT_4 duality. Emphasis will be given to the field-theory side of the duality i.e. to the statistical ensemble of U(N) N=4 SYM that the asymptotic form of the geometry of the Gutowski-Reall black hole suggests exploring. The corresponding partition function, which coincides with the grand canonical superconformal index, can be represented as a unitary matrix integral. The large-N limit of that integral is signed by the competition among many complex eigenvalue/saddle point configurations. The Picard-Lefschetz method determines which, among the latter, define the large-N exponential growth in the — (-1)^F graded — number of BPS operators as a function of their charge. The recently noted oscillations around the Bekenstein-Hawking exponential curve, are shown to be a consequence of the presence of the abovementioned saddles. This effect suggests an interesting picture for the string theory/gravitational side of the duality.

The moduli space M_H of Higgs bundles on a Riemann surface is a (usually singular) hyperkähler manifold that plays a prominent role in many areas of mathematics and mathematical physics. The non-abelian Hodge correspondence relates Higgs bundles and harmonic metrics via Hitchin's self-duality equations. The hyperkähler manifold M_H has an associated twistor space Z, a complex manifold that fibers over the projective line, and we may view a solution of Hitchin's equations as a holomorphic section of this fibration satisfying a certain reality condition. In this talk we will first explain the above circle of ideas. Then we will explore the space holomorphic sections of Z and relate its geometry with the hyperkähler geometry of M_H.

The double copy is a web of relations between the scattering amplitudes of various theories. How symmetries emerge from the double copy construction is often subtle and interesting. In particular, Born-Infeld theory of nonlinear electromagnetism is recognized as the double copy of gluons and pions. Born-Infeld has electromagnetic duality symmetry and the goal of this talk will be to understand why this D-brane effective action has EM duality and to further study its interplay with the double copy at tree- and loop-level. Finally, we will also comment on the emergence of soft behaviour and locality from the double copy.

Four-dimensional Chern-Simons (4d CS) theory in the presence of surface defects was proposed by Costello and Yamazaki as a unifying framework for describing two-dimensional integrable field theories (2d IFTs). I will describe a new way of understanding the passage from 4d CS to 2d IFTs using techniques from homotopy theory. In particular, I will show that (a suitably regularised version of) the 4d CS action becomes gauge invariant if certain boundary conditions are imposed on the surface defects. Alternatively, the action can be made gauge invariant by coupling it to a new 2d field, the edge mode, localised on the surface defects. I will then explain how the edge mode perspective on 4d CS provides a direct link to 2d IFTs. The talk will be based on the joint work [arXiv:2008.01829] with M. Benini and A. Schenkel.

Simulations of uniformly random domino tilings of large Aztec diamonds give striking pictures due to the emergence of two macroscopic regions. These regions are often referred to as frozen and rough phases. A limiting curve separates these regions and interesting probabilistic features occur around this curve, which are related to random matrix theory. The two-periodic Aztec diamond features a third phase, often called the smooth phase. In this talk, we introduce the model and discuss some of the asymptotic behaviour at the rough-smooth boundary. This is based on joint works with Vincent Beffara (Grenoble), Kurt Johansson (Stockholm) and Benjamin Young (Oregon).

In this talk, I will review recent advances in the understanding of intriguing infinite-dimensional symmetries that emerge at the boundary of spacetimes. These symmetries play an important role in the description of many phenomena such as gravitational memory effects, scattering amplitudes in flat spacetime and black hole physics. By connecting seemingly unrelated topics in theoretical physics, infinite-dimensional symmetries have shed a new light on the fascinating universal features of gauge theories and gravity.

The late-time long-distance behaviour of a many-body system is usually described by the framework of hydrodynamics or similar condensed matter models based on the (possibly spontaneously broken) symmetries that the system enjoys. These models are characterised by a set of transport coefficients, such as viscosities and conductivity, that capture the transport properties of conserved charges over macroscopic scales. In this talk, we will investigate a new kind of transport coefficients that are not present in the usual formulation of hydrodynamics, but nonetheless affect the late time dynamics of conserved charges. Physically, these new coefficients arise due to the interaction of the hydrodynamic degrees of freedom with the background bath of thermal noise present out of equilibrium. For the purposes of this investigation, I will spend some time to review the newly developed Schwinger-Keldysh effective action formulation of non-equilibrium field theories, which allows one to systematically keep track of thermal noise and their correlations. The talk will be based on the recent preprint https://arxiv.org/abs/2009.01356