Quantum and classical fields interacting with geometry, Paris

18 mars 2024, 08:30 → 26 avr. 2024, 21:30 Europe/Paris

Institut Henri Poincaré

Thematic 6-weeks programme at Institut Henri Poincaré, Paris, March 18th to April 26th, 2024.

Program talks are in the new IHP building, bâtiment Perrin. Conference talks are in the old IHP building, bâtiment Borel.

Please also register for the workshop if you wish to attend. It takes place in the week April 8th to 12th, 2024. The link for the registration is here: Curved spacetimes, field theory and beyond.

The schedule can be found here.

_{Beware: It was reported to us that scammers are sending to participants fraudulent e-mails about accomodation/fees. Please be particularly cautious about e-mails not coming from the organizers nor from an @ihp.fr address.} 

^{Credit: ESO/M. Kornmesser}

Presentation of the programme:

In light of the progress in cosmological observations, in gravitational waves detection and in particle physics, there are exciting perspectives for discoveries at the interface between classical and quantum theories. To make the most of this variety of new data, it becomes increasingly important to understand and model how classical and quantum fields propagate and influence the spacetime geometry, and how quantum phenomena manifest themselves on the large scale. 

This imperative raises difficult mathematical questions which require a refined understanding of asymptotic structures, field propagation and spacetime dynamics, and of the relationships of quantum degrees of freedom with geometry. From the mathematical point of view it is now a particularly exciting time to address these interconnected problems because of the broad advances in partial differential equations and in field quantization: the former has resulted in a comprehensive array of methods to describe black hole physics and scattering phenomena on curved spacetimes, whereas the latter has provided crucial clarity into problematic concepts and formalisms, paving also new paths towards capturing quantum effects induced by gravity.

The main objective of the IHP programme will be to take these developments to the next level and to create an environment for an unprecedented exchange of ideas between expert mathematicians and physicists. The focus will be on topics among the following, both from the mathematical and physical perspective:

 

• Quantum Field Theory on curved spacetimes
• Semi-classical and effective theories of gravity
• Spectral action principles for gravity and beyond
• Asymptotic analysis and scattering on curved spacetimes 
• Geometry of null surfaces and the black hole entropy problem

 

Mini-lectures :

• Nguyen Viet Dang (Sorbonne Université) · Constructive Quantum Field Theory on manifolds
• Éric Gourgoulhon (Observatoire de Paris) · Geometry of Killing horizons and applications to black hole physics
• Peter Hintz (ETH Zürich) · Linear waves and spectral theory
• Stefan Hollands (Universität Leipzig) · Quantum Field Theory on black hole spacetimes
• Prahar Mitra (University of Amsterdam) · A primer on celestial holography
  
   

 The list of program speakers includes :

• Lars Andersson (BIMSA, Beijing)
• Spyros Alexakis (University of Toronto)
• Bernardo Araneda (Albert Einstein Institute, Potsdam)
• Abhay Ashtekar (Penn State)
• Rudrajit Banerjee (Okinawa Institute of Science and Technology) 
• Dean Baskin (Texas A&M)
• Lydia Bieri (University of Michigan)
• Vitor Cardoso (University of Copenhagen) *cancelled
• Marc Casals (Universität Leipzig)
• Sangmin Choi (University of Amsterdam)
• Piotr Chruściel (Universität Wien)
• Geoffrey Compère (Université Libre de Bruxelles)
• Claudio Dappiaggi (University of Pavia)
• Nguyen Viet Dang (Sorbonne Université) 
• Jan Dereziński (University of Warsaw) 
• Christopher Fewster (University of York)
• Christian Gérard (Université Paris-Saclay)
• Éric Gourgoulhon (Observatoire de Paris)
• Colin Guillarmou (Université Paris-Saclay)
• Peter Hintz (ETH Zürich)
• Stefan Hollands (Universität Leipzig)
• Cécile Huneau (Ecole Polytechnique)
• Vojkan Jaksic (McGill University)
• Igor Khavkine (Czech Academy of Sciences, Prague)
• Sergiu Klainerman (Princeton University)
• Christiane Klein (Université Grenoble Alpes)
• Eleni Kontou (King's College London)
• Albert Law (Stanford University)
• Hans Lindblad (Johns Hopkins University)
• Arthur Lipstein (Durham University)
• Roberto Longo (University of Rome Tor Vergata)
• Lionel Mason (University of Oxford)
• Pascal Millet (Ecole Polytechnique)
• Prahar Mitra (University of Amsterdam)
• Viatcheslav Mukhanov (LMU München)
• Nicola Pinamonti (University of Genoa)
• Harvey Reall (University of Cambridge)
• Kasia Rejzner (University of York)
• Romain Ruzziconi (University of Oxford)
• Pierre Schapira (Sorbonne Université)
• Michele Schiavina (University of Pavia)
• Jean-Marc Schlenker (Université du Luxembourg)
• Leonardo Senatore (ETH Zürich)
• Ali Seraj (Queen Mary University of London)
• Alexander Strohmaier (Universität Hannover)
• Grigalius Taujanskas (University of Cambridge)
• Lucas Tavares Cardoso (Federal University of Santa Maria)
• Rainer Verch (Universität Leipzig)
• Robert Wald (University of Chicago)
• Jared Wunsch (Northwestern University)

 

Scientific organizing committee:

• Dietrich Häfner (Université Grenoble Alpes)
• Frédéric Hélein (Université de Paris)
• Andrea Puhm (University of Amsterdam)
• András Vasy (Stanford University)
• Bernard Whiting (University of Florida)
• Elizabeth Winstanley (University of Sheffield)
• Michał Wrochna (Utrecht University)

lundi 18 mars

1 Mini-course: Geometry of Killing horizons and applications to black hole physics

I. Null hypersurfaces and non-expanding horizons

1. Basic geometry of null hypersurfaces
   a) null geodesic generators
   b) cross-sections
   c) expansion and shear
   d) null Raychaudhuri equation
2. Definition and main properties of non-expanding horizons
   a) marginally trapped cross-sections area invariance
   b) area invariance
   c) canonical affine connection
   d) total geodesicness

(see https://relativite.obspm.fr/blackholes/ihp24/ for details and slides)

Orateur: Eric Gourgoulhon (LUTH, Observatoire de Paris)

2 Mini-course: Geometry of Killing horizons and applications to black hole physics

II. Killing horizons

1. Isometry groups and Killing vectors
2. Definition and examples of Killing horizons
3. Killing horizons as non-expanding horizons
4. Vanishing of the shear
5. Surface gravity
6. Zeroth Law of black hole dynamics
7. Bifurcate Killing horizons

Orateur: Eric Gourgoulhon (LUTH, Observatoire de Paris)

3 Mini-course: Constructive Quantum Field Theory on manifolds

In this course, I will try to give an introduction to the nonperturbative construction of certain models of quantum field theories on manifolds. I will focus on the construction of the $P(\phi)_2$ theories on Riemann surfaces using both the Nelson bounds and the stochastic partial differential (SPDE) methods. Then I plan to discuss the Kontsevich-Segal axioms for this model following a recent construction of Jiasheng Lin. Finally, I will conclude the course by sketching the construction of the $\Phi^4_3$ measure on 3-manifolds based on SPDE methods which is joint work with Bailleul-Ferdinand-To.

(part 1)

Orateur: Nguyen Viet Dang (Sorbonne Université)

4 Mini-course: Constructive Quantum Field Theory on manifolds

(part 2)

Orateur: Nguyen Viet Dang (Sorbonne Université)

mardi 19 mars

5 Mini-course: Constructive Quantum Field Theory on manifolds

(part 3)

Orateur: Nguyen Viet Dang (Sorbonne Université)

6 Mini-course: Constructive Quantum Field Theory on manifolds

(part 4)

Orateur: Nguyen Viet Dang (Sorbonne Université)

7 Mini-course: Geometry of Killing horizons and applications to black hole physics

III. Stationary black holes

1. Stationary spacetimes
2. Black holes in stationary spacetimes
3. Topology theorems
4. Mass and angular momentum
5. Rigidity theorem: the event horizon as a Killing horizon
6. Smarr formula
7. No-hair theorems

Orateur: Eric Gourgoulhon (LUTH, Observatoire de Paris)

8 Mini-course: Geometry of Killing horizons and applications to black hole physics

IV. Degenerate Killing horizons and their near-horizon geometry

1. The extremal Reissner-Nordström example
2. The scri+ example
3. Near-horizon geometry of generic extremal black holes

Orateur: Eric Gourgoulhon (LUTH, Observatoire de Paris)

mercredi 20 mars

9 Mini-course: Quantum Field Theory on black hole spacetimes

I will outline the algebraic formulation of linear quantum field theories on globally hyperbolic spacetimes. A characterization of the physically allowed states (Hadamard states) is provided in terms of their short-distance behavior as well as in terms of wave
front sets. The latter concept requires certain notions from microlocal analysis/distribution theory, which are provided in a concise manner. The notion of KMS state is introduced and the relevance of this notion for the Unruh-effect in Rindler spacetime, and its counterparts in various other spacetimes with bifurcate Killing
horizons, is explained. I will discuss physically relevant states on black hole spacetimes such as the Unruh- and Boulware states, including their Hadamard property, and derive the presence of a steady, thermal flux of quantum-stress-energy in the Unruh state
(Hawking effect). Time permitting, I will outline the algebraic formulation of interacting quantum field theories on globally hyperbolic spacetimes, in the sense of formal power series.

(part 1)

Orateur: Stefan Hollands (Universität Leipzig)

10 Mini-course: Quantum Field Theory on black hole spacetimes

(part 2)

Orateur: Stefan Hollands (Universität Leipzig)

11 Mini-course: Linear waves and spectral theory

After an introduction to microlocal analysis, I will explain how to describe the long-time behavior of solutions of linear wave equations on spacetimes of interest in General Relativity using spectral theory and resonance expansions. Time permitting, I will also touch upon the asymptotically flat case (Schwarzschild, Kerr).

(part 1)

Orateur: Peter Hintz (ETH Zürich)

12 Mini-course: Linear waves and spectral theory

(part 2)

Orateur: Peter Hintz (ETH Zürich)

jeudi 21 mars

13 Mini-course: Constructive Quantum Field Theory on manifolds

(part 5)

Orateur: Nguyen Viet Dang (Sorbonne Université)

14 Mini-course: Geometry of Killing horizons and applications to black hole physics

V. Hands-on session: exploring the extremal Kerr near-horizon geometry with SageMath

1. Introduction to SageMath
2. The extremal Kerr throat
3. NHEK metric
4. Killing vectors and Lie algebra of the isometry group

Orateur: Eric Gourgoulhon (LUTH, Observatoire de Paris)

15 Mini-course: Quantum Field Theory on black hole spacetimes

(part 3)

Orateur: Stefan Hollands (Universität Leipzig)

16 Mini-course: Quantum Field Theory on black hole spacetimes

(part 4)

Orateur: Stefan Hollands (Universität Leipzig)

vendredi 22 mars

17 Mini-course: Quantum Field Theory on black hole spacetimes

(part 5)

Orateur: Stefan Hollands (Universität Leipzig)

18 Mini-course: Linear waves and spectral theory

(part 3)

Orateur: Peter Hintz (ETH Zürich)

19 Mini-course: Linear waves and spectral theory

(part 4)

Orateur: Peter Hintz (ETH Zürich)

20 Mini-course: Linear waves and spectral theory

(part 5)

Orateur: Peter Hintz (ETH Zürich)

lundi 25 mars

21 Estimates for rough wave maps on the Einstein cylinder via Peter—Weyl theory

Nonlinear wave equations of wave maps type are typically expected to be well-posed for initial data just above scaling critical regularity. For wave maps on Minkowski space, this is by now well-understood as a consequence of works of Klainerman—Machedon, Tataru, Tao, and others, and effectively relies on sharp null form estimates which exploit the special "null” structure of the nonlinearities. In Fourier space, these estimates capture cancellations between parallel propagating waves. I will introduce a new approach to obtain a wide range of analogous estimates on the Einstein cylinder, where traditional Fourier theory is unavailable, using instead the Lie group structure of SU(2), an emergent periodicity of the conformal wave equation on the Einstein cylinder, and Peter--Weyl theory. The estimates we obtain hold for a slightly different set of exponents than in flat space, including edge cases which are forbidden in flat space, with an arbitrarily small loss which we trace down to the non-commutativity of SU(2). Time permitting, I will outline how the estimates may be used to show almost optimal well-posedness of wave maps equations on the Einstein cylinder.

Orateur: Grigalius Taujanskas (University of Cambridge)

Peter_Weyl_null_forms.pdf

22 Semiclassical effects inside black holes

All black holes in the Universe are believed to be rotating. This poses interesting questions, since rotating black hole solutions of Einstein’s equations of General Relativity possess a so-called Cauchy horizon in their interior, beyond which Einstein’s equations cease to be predictable (i.e., the Cauchy value problem is no longer well-posed). However, these exact solutions may not model sufficiently accurately black holes in Nature, which have classical matter in their neighbourhood and, furthermore, are inevitably surrounded by a quantum vacuum (which, in the exterior, is responsible for Hawking radiation). It is generally found that effects on Cauchy horizons from quantum fields are in fact dominant over those from classical matter. In this talk, we will present recent results on effects due to a quantum field on the Cauchy horizon of rotating (Kerr and Kerr-de Sitter) black holes which are evaporating via the emission of Hawking radiation. In particular, we will show that the (renormalized) fluxes from a quantum scalar field generically diverge on the Cauchy horizon and cause latitudes of infinite twisting separating regions of infinite expansion and contraction for spheres approaching this horizon.

Orateur: Marc Casals (Leipzig University)

23 Mini-course: A primer on celestial holography

Holography in asymptotically flat spacetimes has received much attention recently and has been significantly developed along several different fronts. In one formulation known as celestial holography, quantum gravity in $d+2$-dimensional asymptotically flat spacetimes can be reformulated as a conformal field theory that lives on the $d$-dimensional celestial sphere. In this mini-lecture series, I will introduce the basics of celestial holography, starting with an identification of the relevant symmetries of the problem, Ward identities and their relationship to soft theorems, and ending with a discussion of celestial CFTs and their structure.

(part 1)

Orateur: Prahar Mitra (University of Amsterdam)

24 Mini-course: A primer on celestial holography

(part 2)

Orateur: Prahar Mitra (University of Amsterdam)

mardi 26 mars

25 Scattering on self-dual black holes

Penrose's twistor theory demonstrated that the self-dual sector of Einstein's general relativity is completely integrable. This talk will explain how this can be exploited to generate efficient scattering formulae by developing a perturbation theory about a fully nonlinear self-dual sector rather than around flat space. The talk will focus on the example of a self-dual black hole background. Rather than exploit the twistor theory directly, we find that the complete integrability can be used to solve for linear fields on the bacground using some novel geometric structures, charged helicity raising and lowering operators, whose existence reflects the underlying simplicity of the self-dual sector in this case. As a first step, these generate analogues of momentum eigenstates on the background and are simple enough to complete their nonperturbative two-point scattering off the background. I will briefly explain how the twistor theory can be used to to extend the construction to give formulae at all multiplicities.

Orateur: Lionel Mason (The Mathematical Institute, University of Oxford)

IHP-celestial-taub-nut.pdf

26 Mini-course: A primer on celestial holography

(part 3)

Orateur: Prahar Mitra (University of Amsterdam)

27 Mini-course: A primer on celestial holography

(part 4)

Orateur: Prahar Mitra (University of Amsterdam)

mercredi 27 mars

28 Sheaves for spacetime

We shall study the Cauchy problem on globally hyperbolic manifolds with the only tools of microlocal sheaf theory and the precise Cauchy-Kowalevski theorem.

A causal manifold is a manifold $M$ endowed with a closed convex proper cone $\lambda\subset T^*M$. On such a manifold, one defines the $\lambda$-topology and the associated notions of a causal pre-order and a causal path. One introduces the notion of a G-causal manifold, those for which there exists a time function. On a G-manifold, sheaves satisfying a suitable condition on their micro-support and defined on a neigborhood of a Cauchy hypersurface extend to the whole space. When the sheaf is the complex of hyperfunction solutions of a hyperbolic $\mathcal{D}$-module, this proves that the Cauchy problem is globally well-posed.

We will also describe a "shifted spacetime" associated with the quantization of an Hamiltonian isotopy.

This talk is partly based on papers in collaboration with Benoît Jubin, Stéphane Guillermou and Masaki Kashiwara.

Orateur: Pierre Schapira (Univ Paris 6)

SlideSpaceT.pdf

29 Asymptotic symmetries and log soft theorems in gauge theories and gravity

In the last few years, a remarkable link has been established between the soft theorems and asymptotic symmetries of quantum field theories: soft theorems are Ward identities of the asymptotic symmetry generators. In particular, the tree-level subleading soft theorems are the Ward identities of the subleading asymptotic symmetries of the theory, for instance divergent gauge transformation in QED and superrotation in gravity. However, it is known that the subleading soft theorems receive quantum corrections with logarithmic dependence on the soft photon/graviton energy. It is therefore natural to ask how the quantum effects affect the classical (tree-level) symmetry interpretation. In this talk, we explore this question in the context of scalar QED and perturbative quantum gravity, and show that the logarithmic soft theorems are the Ward identities of subleading asymptotic symmetries that arise from relaxed boundary conditions which take long-range interactions into account.

Orateur: Sangmin Choi (University of Amsterdam)

logsoft_IHP.pdf

30 Mini-course: A primer on celestial holography

(part 5)

Orateur: Prahar Mitra (University of Amsterdam)

jeudi 28 mars

31 Modular Theory: How and Why

In this talk I will review the Tomita-Takesaki modular theory from both mathematical and physical perspective and discuss some recent developments in non-equilibrium quantum statistical mechanics of open quantum systems where this theory has played a central role.

The talk is dedicated to the memory of Huzihiro Araki.
[See also Araki's lecture at McGill in the links.]

(part 1)

Orateur: Vojkan Jakšić (McGill University)

Araki lecture 1

Araki lecture 2

jaksic_ihp_2024.pdf

32 Modular Theory: How and Why

(part 2)

Orateur: Vojkan Jakšić (McGill University)

33 Carrollian amplitudes in flat space holography

Carrollian holography aims to express gravity in asymptotically flat space-time in terms of a dual Carrollian CFT living at null infinity. In this talk, I will review how Carrollian geometry arises at the conformal boundary of asymptotically flat spacetimes. I will then discuss some aspects of Carrollian holography and argue that this approach is naturally related to the AdS/CFT correspondence via a flat limit procedure. I will introduce the notion of Carrollian amplitude, which allows to encode massless scattering amplitudes into boundary correlators, and describe its connection to celestial amplitudes.

Orateur: Romain Ruzziconi (University of Oxford)

34 Classical superselection sectors, memory and soft symmetries from Hamiltonian reduction by stages in gauge theory

In recent years, increasing attention has been drawn to the behaviour of field theory with local (gauge) symmetries on manifolds with boundary and corners. On the physics side, this has led to the discovery of a relation between perturbative scattering results (e.g. Weinberg’s “soft photon” theorems) and certain soft/asymptotic/large gauge symmetries, with their associated conserved charges. However, on the mathematics side we have been lacking an exhaustive explanation of these phenomena.

In this talk I will attempt to provide one, through an application of Hamiltonian reduction by stages to gauge moduli problems in the presence of higher codimension strata, which is enabled by the existence of a particular normal gauge subgroup, and it is natural from a physical standpoint. The “first stage” implements the constraint reduction/moduli space of the gauge theory. Conversely, the residual momentum map governing further stages has generally no physically-preferred value, leading to the classical counterpart of the “superselection sectors” encountered in algebraic quantum field theory, here interpreted as symplectic leaves of the fully-reduced (Poisson) phase space.

As an example within this framework, I will describe the Hamiltonian assignment to a null manifold with boundary in Yang—Mills theory, and show how it provides a purely Hamiltonian explanation of the emergence of soft symmetries, and how the electromagnetic memory (or a non-Abelian counterpart thereof) is recovered as (a part of) the residual momentum map.
In passing, I will describe how the Ashtekar—Streubel phase space, usually treated as the true reduced phase space of the theory, is instead the result of a partial reduction.

This is based on joint work with A. Riello.

Orateur: Michele Schiavina (University of Pavia)

QFG24.pdf

vendredi 29 mars

35 The renormalized volume of 3-dimensional hyperbolic manifolds

Conformally compact Einstein manifolds have infinite volume, but a well-defined notion of renormalized volume, introduced by theoretical physicists. For 3-dimensional hyperbolic manifolds, this notion has interesting relations to other quantities studied by geometers, such as the volume of the convex core and the bending lamination on its boundary. We will describe how the relation between the physics and the mathematics points of view can bring new perspectives on both sides.

Orateur: Jean-Marc Schlenker (Université du Luxembourg)

36 Stability of spacetimes with supersymmetric compactifications

Spacetimes with compact directions, which have special holonomy such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss the global, non-linear stability for the vacuum Einstein equations on a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. I will start by giving a brief overview of related stability problems which have received a lot of attention recently, including the black hole stability problem. This is based on joint work with Pieter Blue, Zoe Wyatt and Shing-Tung Yau.

Orateur: Lars Andersson (BIMSA)

mardi 2 avril

37 Conformal Kähler geometry and non-vacuum gravitational instantons

Gravitational instantons are four-dimensional Riemannian geometries introduced by Hawking for the study of quantum fields on curved backgrounds and Euclidean Quantum Gravity. Their complete classification is a challenging open problem in geometry. After briefly reviewing recent developments in the Ricci-flat case, we will give a framework based on complex geometry that leads to major simplifications in the study of non-Ricci-flat instantons. Applications include generalisations of the Chen-Teo geometry (a recent counterexample to the Euclidean black hole uniqueness conjecture).

Orateur: Bernardo Araneda (MPI (Albert Einstein Institute))

38 Simplifying harmonic gauge perturbations around black holes

I will review a method of simplifying the separated equations of Maxwell and linear gravitational perturbations around a Schwarzschild black hole. The simplified form is a sparse upper triangular coupling of a system of Regge-Wheeler equations. The method takes advantage of a specific notion of equivalence between differential equations and of the special structure of ODEs with rational coefficients. I will also discuss the prospects of applying similar methods to the Kerr black hole.

Orateur: Igor Khavkine (Czech Academy of Sciences)

khavkine-ihp-2024.pdf

mercredi 3 avril

39 String-localized quantum field theory - some ideas and perspectives

The construction of string-localized free fields was rigorously accomplished almost twenty years ago. In this approach, the fields are operators in some Hilbert space, and therefore there are no unphysical degrees of freedom such as ghosts. In addition to allowing the construction of the fields entirely in a Hilbert space, the string-localized fields exhibit, in general, a good behavior in the ultraviolet regime and, among other features, the class (representation) of string-localized fields with m = 0 and s = ∞ are possible candidates to consistently describe dark matter. The interaction picture is obtained perturbatively via the Bogolyubov-Epstein-Glaser scheme together with the requirement that the S-matrix be string independent. This talk is intended to show some of the distinguishing features of sQFT along with future perspectives.

Orateur: Lucas Tavares Cardoso (Federal University of Santa Maria)

40 A spin correction to Coulomb interaction based on Kerr-Newmann solution

The complex shift introduced by Appell and used by Sommerfeld and Synge in introducing a nontrivial electromagnetic field with a ring singularity from the Coulomb field has an intimate relation to the Newman-Janis transformation in the context of exact solutions of GR. Such a constructed electromagnetic field, called the magic field by Lynden-Bell, coincides with the electromagnetic sector of the Kerr-Newman solution of Einstein-Maxwell theory. The Riemann surface of the analytic continuation of the magic field matches the analytic extension of massless Kerr or zero-G limit of Kerr-Newman. This field is also referred to as the square root of Kerr in more recent works on double-copy formalism. After briefly reviewing some of these points in this talk, I will report the results of calculating the interaction Lagrangian of two magic fields in a static situation, which will correct the Coulomb interaction with respect to the spin of the fields.

Orateur: Sajad Aghapour (Max Planck Institute for Gravitational Physics (AEI))

jeudi 4 avril

41 An Introduction to Quantum Fields and Local Measurements

An introduction to quantum field measurement in the local covariant framework of quantum field theory will be given. Concepts such as the local dynamical coupling between "system" and "probe" will be discussed, as well as selective and non-selective
state updates following probe measurements. Furthermore, Sorkin’s impossible measurement scenario will be sketched, and it will be argued that the impossible measurements of this scenario are also present in classical relativistic field theory. The material of this talk is based on joint works with C.J. Fewster (arXiv:1810.06512 and 2304.13356) and with A. Much (arXiv:2308.16673).

Orateur: Rainer Verch (Universität Leipzig)

42 Microlocal analysis near null infinity on asymptotically flat spacetimes

There are a number of reasons due to which it is advantageous to have a phase space based, or microlocal, approach available for analyzing wave propagation. In this talk I will explain a microlocal framework for wave propagation on asymptotically flat spacetimes of arbitrary dimension which in particular includes operator corresponding to Lorentzian metrics arising from solutions of Einstein’s equations in the 4 spacetime dimensional setting. On the compactification of Minkowski space that underlies this, which is a manifold with corners (with the usual null infinity, scri, being a boundary hypersurface), the operators lie in a combination of Melrose’s totally characteristic (also called b), and Mazzeo’s edge pseudodifferential operator algebras. I will give an introduction via a simpler setting (which includes Minkowski space and a different class of perturbations), and then explain the reasons for, and complications with, moving to the present setting. Along the way, I will also briefly describe the related Klein-Gordon work of Ethan Sussman. This is joint work with Peter Hintz.

Orateur: András Vasy (Stanford University)

vendredi 5 avril

43 Quantum fields on rotating black holes

Since astrophysical black holes are expected to be rotating, one would like to study quantum fields on such spacetimes. In this talk, we lay some of the mathematical foundations for this endeavor. We construct a physically motivated state, the Unruh state, for the free scalar quantum field on Kerr de Sitter and show that it is a Hadamard state. Moreover, we demonstrate that a similar construction for fermions on Kerr results in a Hadamard state for any subextremal black hole, extending previous results for small angular momentum. Finally, we discuss the divergence of the stress-energy tensor of the quantum scalar field towards the Cauchy horizon. We show that the leading divergence has a universal, state-independent behaviour, even in rotating black hole spacetimes, as long as there is a non-zero spectral gap for the corresponding equation of motion.

Orateur: Christiane Klein (Université Grenoble Alpes)

slides Paris 0324 v1.pdf

lundi 8 avril

44 Welcome / registration / coffee

45 The second law of black hole mechanics in effective field theory

The laws of black hole mechanics are an important part of the identification of black holes as thermodynamic objects. If this is correct then these laws should be robust against the inclusion of higher derivative corrections to the Einstein equation. For the first law, this was confirmed by the work of Wald et al in the early 1990s, which provides a definition of the entropy of a stationary black hole solution of any diffeomorphism invariant theory.
But it has remained an open problem to generalize this to obtain a definition of dynamical black hole entropy, that satisfies a second law of black hole mechanics. I shall describe a solution to this problem. The key ingredient is to treat higher derivative terms according to the rules of effective field theory.

Orateur: Harvey Reall (University of Cambridge)

20240408_Reall.pdf

10:10 Coffee break

46 The nonlinear stability of slowly rotating black holes

My talk will focus on the recent resolution of the Kerr stability conjecture for the particular case of slowly rotating Kerr black holes.

Orateur: Sergiu Klainerman (Princeton University)

IHP2024_Klainerman.pdf

47 Burnett's conjecture in General Relativity

In this talk, I will present a work in collaboration with Jonathan Luk, on the behaviour of weak limits of solutions to Einstein equations. In 1989, Burnett conjectured that a metric, obtained as a uniform limit of solutions to Einstein vacuum equation, whose derivatives converge only weakly, is a solution to Einstein- massless Vlasov equations. This conjecture has been back in the spotlight with the work of Green and Wald on perturbations in cosmology. In a recent work with Jonathan Luk, we prove this conjecture, without symmetry assumption, in generalized wave coordinates.

Orateur: Cécile Huneau (CNRS et Ecole Polytechnique)

12:30 Lunch break

48 Null Infinity Is a Weakly Isolated Horizon!

Null infinity, $\mathcal{I}^+$, arises as a boundary of the Penrose conformal completion $(\hat{M}, \hat{g}_{ab})$ of an asymptotically flat physical space-time $(M, g_{ab})$. I will begin by pointing out that $\mathcal{I}^+$ is a weakly isolated horizon (WIH) in $(\hat{M}, \hat{g}_{ab})$, and then show that all its familiar properties can be derived from the general WIH framework. This seems quite surprising because physics associated with black hole (and cosmological) WIHs $\Delta$ is very different from that extracted at $\mathcal{I}^+$. These differences can be directly traced back to the fact that while Einstein's equations hold at $\Delta$, conformal Einstein's equations hold at $\mathcal{I}^+$. In particular, the BMS group at $\mathcal{I}^+$ stems from the symmetry group of WIHs. There is also a unified procedure to arrive at fluxes and charges associated with the BMS symmetries at $\mathcal{I}^+$ and those associated with the symmetries at $\Delta$. This procedure differs from those commonly used in the literature and its novel elements seem interesting in their own right. This subtle interplay between geometry and physics is reminiscent of a musical fugue.

The fact that is there is a single mathematical framework underlying $\Delta$ and $\mathcal{I}^+$ paves the way to explore the relation between horizon dynamics in the strong field region and waveforms at infinity. It should also be useful in the analysis of black hole evaporation in quantum gravity. This work was carried out in collaboration with Simone Speziale and supported by Eberly and Atherton Funds of Penn State and the DVRC program of the Perimeter Institute.

Orateur: Abhay Ashtekar (Penn State)

20240408_Ashtekar.pdf

15:25 Coffee break

49 Black Holes Decohere Quantum Superpositions

We show that if a massive body is put in a quantum superposition of spatially separated states, the mere presence of a black hole in the vicinity of the body will eventually destroy the coherence of the superposition. This occurs because, in effect, the gravitational field of the body radiates soft gravitons into the black hole, allowing the black hole to harvest "which path'' information about the superposition. A similar effect occurs for quantum superpositions of electrically charged bodies. The effect is very closely related to the memory effect and infrared divergences at null infinity.

Orateur: Robert Wald (University of Chicago)

20240408_Wald.pdf

mardi 9 avril

50 The Euclidean vacuum state for linearized gravity on de Sitter spacetime

We will explain the construction of the euclidean vacuum state for linearized gravity on de Sitter spacetime by a rigorous version of Wick rotation. We will discuss issues related to gauge invariance, positivity and invariance under de Sitter isometries.

Orateur: Christian Gérard (Université Paris-Saclay)

slides-de-Sitter.pdf

10:10 Coffee break

51 On attempts and progresses towards proving the de Sitter no hair theorem using mean curvature flow

I will review how mean curvature flow is providing progress towards proving the so-called de Sitter no-hair theorem, concerning the onset of a de Sitter phase of the universe out of inhomogenous initial conditions. This is relevant for the initial condition of Inflation in
our universe.

Orateur: Leonardo Senatore (ETH Zürich)

seminar_senatore_geometry_Paris_v1.pdf

52 Gluing variation

I will review various ways of constructing spacetimes of interest using spacelike and gluing constructions methods

Orateur: Piotr Chruściel (University of Vienna)

Chrusciel-Gluing Variations Paris 2024.pdf

12:30 Lunch break

53 Asymptotics of Dynamical Spacetimes and Radiation

In this talk, I will discuss some results on the asymptotic behavior and radiation for a general class of spacetimes. This will include results on gravitational radiation and memory for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. We will also show that angular momentum at future null infinity is well defined for asymptotically-flat spacetimes with a term homogeneous of degree -1 in the initial data metric, that is it may include a non-isotropic mass term. Finally, we will also discuss peeling (that is, how peeling stops) for the Weyl curvature components at future null infinity for these spacetimes.

Orateur: Lydia Bieri (University of Michigan)

BieriParisIHPApril2024.pdf

15:25 Coffee break

54 On the Feynman propagator on curved spacetimes

The Feynman propagator is one of the main tools of Quantum Field Theory. I will review various definitions of this concept, including the operator-theoretic and out-in Feynman propagator. I will discuss various examples of spacetimes: stationary, asymptotically stationary, FLRW, de Sitter and anti-de Sitter. I will consider both the stable and tachyonic case. Based on joint work with Christian Gass.

Orateur: Jan Dereziński (University of Warsaw)

cft-slides.pdf

18:00 Cocktail

mercredi 10 avril

55 Secular growths and their relation to Equilibrium states in perturbative Quantum Field Theories

During this talk we discuss the emergence of secular growths in the correlation functions of interacting quantum field theories when treated with perturbation methods. It is known in the literature that these effects are present if the interaction Lagrangian density changes adiabatically in a finite interval of time. If this happens, the perturbative approach cannot furnish reliable results in the evaluation of scattering amplitudes or in the evaluation of various expectation values.
We show, during this talk, that these effects can be avoided for adiabatically switched-on interactions, if the spatial support of the interaction is compact and if the background state is suitably chosen. In particular, this is the case when the background state is chosen to be at equilibrium and when thermalisation occurs at late time. The same result holds also if the background state is only invariant under time translation or if the explicit time dependence is not too strong, in a precise sense which will be discussed in the talk.

(based on Galanda, Pinamonti, Sangaletti [arXiv: 2312.00556])

Orateur: Nicola Pinamonti (University of Genova)

20240410_Pinamonti.pdf

10:10 Coffee break

56 Bounds for the local entropy

Orateur: Roberto Longo (University of Rome Tor Vergata)

Longo-Paris24.pdf

57 Non-minimal coupling, negative null energy, and effective field theory

Even classical scalar fields, non-minimally coupled with the curvature, can violate energy conditions such as the null energy condition. In the context of quantum field theory, non-minimally coupled scalars can obey lower bounds, known as quantum energy inequalities, but these are always state dependent. In this talk I will discuss classical and quantum bounds on the null energy and consider possible violations. Further, I will examine the conformal transformation between Jordan and Einstein frames both classically and semiclassically. Finally, I will comment on extensions of this work and connections with self-interacting fields.

Orateur: Eleni Alexandra Kontou (King's College London)

12:30 Lunch break

58 Progress on the definition of asymptotically flat and de Sitter spacetimes

The definition of scattering in asymptotically flat spacetimes requires to consistently match its five asymptotic boundaries: past/future timelike infinity, past/future null infinity and spatial infinity. I will present a framework consistent with logarithmic corrections to soft theorems where a single BMS group acts on all boundaries and where individual ingoing/outgoing bodies are ascribed initial/final BMS charges. Using the post-Minkowskian expansion, I will demonstrate that non-radiative regions are entirely characterized by a set of conserved celestial charges that consist of the Geroch-Hansen multipole moments, the generalized BMS charges and additional non-stationary multipole moments. In the context of asymptotically de Sitter spacetimes, I will finally demonstrate that the presence of a quadrupole moment of any localized source in de Sitter leads to a fluctuating boundary metric that breaks the conformal asymptotic symmetry group. Dynamical Einstein gravity in de Sitter cannot therefore be modelled by a 3d CFT. Based on work performed with S. Gralla, H. Wei, R. Oliveri, A. Seraj, L. Blanchet, G. Faye, J. Hoque and E. Kutluk.

Orateur: Geoffrey Compère (Université Libre de Bruxelles)

Compère.pdf

15:25 Coffee break

59 Metric reconstruction

I will review recent developments in metric reconstruction theory, i.e., the task of parameterizing GR in terms of a single complex scalar field. I will focus on the GHZ (after Green, Hollands, Zimmerman) procedure of solving the sourced linearized (or $n$-th perturbative order-) Einstein equations in Kerr in terms of a Hertz potential. Time permitting, I will also outline other reconstruction methods such as ones based on the Lorenz gauge due to Dolan et al., or the AAB method (after Aksteiner, Anderson, Bäckdahl).

Orateur: Stefan Hollands (Universität Leipzig)

Folien.pdf

Discussion session: How to observe quantum effects on curved spacetimes?

Président de session: Bernard Whiting (University of Florida)

jeudi 11 avril

60 The timelike tube theorem and analytic vectors in QFT

The Bros-Iagolnitzer analysis of analytic singularities in QFT can be generalised to analytic curved spacetimes via the notion of the analytic wavefront set and the analytic microsupport. I will start with an elementary introduction into the topic and basic definitions. I will give some physics intuition that motivates the introduction of analytic states and will show that one can prove the timelike tube theorem in quantum field theory on curved spacetimes under plausible assumptions. I will also discuss some modified notions of analytic singularities that capture the structure of the n-point functions at infinity.
(Joint work with E. Witten)

Orateur: Alexander Strohmaier (Universität Hannover)

10:10 Coffee break

61 Imaginary Liouville conformal field theory

Conformal field theory in dim 2 allows to describe scaling limits of many statistical physics models, such as Ising model and its variants. CFT are parametrized by their central charge and conformal weights. We construct using probability (and in particular the compactified boson) a whole family of conformal field theories with central charge c<1 and discrete spectrum, which sets to be a good candidate to describe scaling limits of many statistical physics models, including non-unitary and logarithmic CFT, loop models. Joint work with Kupiainen and Rhodes.

Orateur: Colin Guillarmou (Univ. Paris Saclay & CNRS)

62 Wave propagators and traces on singular spaces

I will give a survey of recent progress in understanding wave and Schrödinger propagators in the presence of various kinds of geometric and analytic singularities; these singularities have the effect of diffracting energy of waves interacting with them, and thus may have a significant effect on scattering phenomena. In some cases we are able to understand the effects of diffractive propagation on existence of scattering resonances; one of the important tools here is a trace formula. I will, as time permits, discuss such results for conic singularities (joint with Melrose, Melrose-Vasy, Baskin, Ford, Hillairet); for the Dirac-Coulomb problem (joint with Baskin, Baskin-Wrochna); for rotating cosmic string spacetimes (joint with Morgan); and for semiclassical Schrödinger operators with singular potentials (joint with Gannot, Galkowski, Yang-Zou).

Orateur: Jared Wunsch (Northwestern University)

12:30 Lunch break

63 New proposal for Lorentzian renormalisation group flow equations on curved spacetimes

In this talk I will present recent results obtained in collaboration with d’Angelo, Drago and Pinamonti. We proposed a new formulation of renormalisation group flow equations that work on arbitrary globally hyperbolic spacetimes and for any chosen Hadamard state. The examples treated so far include the scalar field, Yang-Mills theories and gravity.

Orateur: Kasia Rejzner (University of York)

15:25 Coffee break

64 Scattering for wave equations with sources and slowly decaying data

We construct solutions with prescribed radiation fields for wave equations with polynomially decaying sources close to the lightcone. In this setting, which is motivated by semilinear wave equations satisfying the weak null condition, solutions to the forward problem have a logarithmic leading order term on the lightcone and non-trivial homogeneous asymptotics in the interior of the lightcone. The backward scattering solutions we construct from knowledge of the source and the radiation field at null infinity alone are given to second order by explicit asymptotic solutions which satisfy novel matching conditions close to the light cone. This requires a delicate analysis close to the light cone of the forward solution with sources on the light cone. We also relate the asymptotics of the radiation field towards space-like infinity to explicit homogeneous solutions in the exterior of the light cone for slowly polynomially decaying data corresponding to mass, charge and angular momentum in the applications. The somewhat surprising discovery is that these data can cause the same logarithmic radiation field as the source term. This requires a delicate analysis of the forward homogeneous solution close to the light cone using the invertibility of the Funk transform. This is joint work with Volker Schlue.

Orateur: Hans Lindblad (Johns Hopkins University)

Lindblad-Scattering24.pdf

vendredi 12 avril

65 Interplay between Stochastic Partial Differential Equations and Quantum Field Theory on Curved Backgrounds

We review a novel framework for the study of a large class of non-linear stochastic partial differential equations (SPDEs), which is inspired by microlocal analysis and by the algebraic approach to quantum field theory (AQFT). In particular, we show that AQFT and SPDEs share similar problems, most notably the existence of pathological divergences. These are dealt with adapting to this framework the Epstein-Glaser renormalization procedure which is used in the analysis of locally covariant quantum field theories. As a concrete example we shall discuss the stochastic $\Phi^3_d$ model and we shall comment on its applicability to other models such as the stochastic non-linear Schrödinger equation, the stochastic sine-Gordon equation as well as the stochastic Thirring equation.

Based on joint works with A. Bonicelli, B. Costeri, N. Drago, P. Rinaldi and L. Zambotti

Orateur: Claudio Dappiaggi (University of Pavia)

Dappiaggi - Paris - IHP 2024.pdf

10:10 Coffee break

66 Double copy and soft limits in (A)dS

The study of the double copy relating gluon to graviton amplitudes and their soft limits has been a major driving force in the study of scattering amplitudes in flat space. I will describe recent progress in generalising these ideas to (A)dS, which may have interesting implications for holography and cosmology.

Orateur: Arthur Lipstein (Durham University)

67 Local measurement theory for quantum fields

Many presentations of quantum mechanics include a postulate that the state of a system undergoes an instantaneous change following a measurement. This is clearly incompatible with special and general relativity and raises questions concerning the description of measurement in quantum field theory (QFT). Attempts to extend measurement postulates to QFT by hand have produced pathologies, such as the "impossible measurements" described long ago by Sorkin. I will present a recent operational approach to these questions, which models measurement of one quantum field (the system) by coupling it to another (the probe). This is all accomplished in a model-independent way within algebraic quantum field theory (AQFT). The resulting framework provides a description of measurement in QFT that is causal, covariant and consistent, and includes state update rules that are derived from the formalism, and works equally well in flat or curved spacetimes. As well as covering the basics of the formalism I will touch on some more recent developments, including asymptotic measurement schemes, and how one may describe Bell inequality violation in this framework.

The talk is mostly based on joint works with Rainer Verch, Henning Bostelmann, Maximilian Ruep and Ian Jubb.

Orateur: Christopher Fewster (University of York)

Fexster.pdf

12:30 Lunch break

68 The Klein—Gordon equation on asymptotically flat spacetimes

We consider the Klein—Gordon equation on Minkowski space with a potential that decays suitably fast in spatial directions and approaches a limiting potential as $t$ tends to infinity. The aim of this talk is to describe a global construction of the Feynman inverse for this problem. This is joint work with Moritz Doll and Jesse Gell-Redman (Melbourne).

Orateur: Dean Baskin (Texas A&M University)

15:25 Coffee break

69 The Calderón and nonlinear scattering problems for wave equations

We present results on reconstruction of linear and nonlinear wave operators from knowledge of their near field and far field effect on incoming waves. Connections with unique continuation and nonlinear Goursat problems will be made. Based on joint work with Ali Feizomohammadi, Lauri Oksanen, Hiroshi Isozaki, Matti Lassas, and Teemu Tyni.

Orateur: Spyros Alexakis (University of Toronto)

Alexakis.pdf

lundi 15 avril

70 Kaluza-Klein theories without a priori fibration hypotheses

I will present a theory based on a variational principle, the critical points of which lead to solutions of the Einstein-Yang-Mills equations, in the spirit of Kaluza-Klein theories. The novelty is that the a priori fibration hypothesis is not required: fields are defined on a
"space-time" $Y$ of dimension $4+r$ without a priori principal bundle structure, where $r$ is the dimension of the structure group. If this group is compact and simply connected solutions it allows to construct a 4-dimensional manifold $X$ which can be interpreted as our space-time, in such a way that $Y$ acquires a principal bundle structure over $X$ and produce solutions of the Einstein-Yang-Mills system. If the structure group is $U(1)$ (the case which corresponds to the Einstein-Maxwell system) the situation is slightly degenerated and supplementary hypotheses are necessary.

Orateur: Frédéric Hélein (Université Paris Cité, IMJ-PRG)

mardi 16 avril

71 Leading-order term expansion for the Teukolsky equation on subextremal Kerr black holes

The study of wave propagation on black hole spacetimes has been an intense field of research in the past decades. This interest has been driven by the stability problem for black holes and by questions related to scattering theory. On Kerr black holes, the analysis of Maxwell's equations and the equations of linearized gravity, can be simplified by introducing the Teukolsky equation, which offers the advantage of being scalar in nature. After explaining this reduction, I will present a result providing the large time leading-order term for initially localized and regular solutions of the Teukolsky equation, valid for the full subextremal range of black hole parameters and for all spins. I will explain how such a development follows naturally from the precise analysis of the resolvent operator on the real axis. Recent advances in microlocal analysis are used to establish the existence and mapping properties of the resolvent.

Orateur: Pascal Millet (Ecole Polytechnique)

72 Horizon-horizon scattering and black hole thermodynamics

The thermal canonical partition function for a free scalar field outside a black hole horizon is ill-defined due to the continuous nature of the normal mode spectrum. In this talk, I will explain how non-trivial spectral information can be extracted from the near-horizon asymptotics of the normal modes through a relativistic generalization of the Krein-Friedel-Lloyd formula. These considerations lead to a family of well-defined renormalized canonical partition functions parametrized by a choice of reference scattering problem. Remarkably, for a specific choice of reference, the renormalized canonical partition function is equal to the 1-loop Euclidean black hole determinant, leading to a precise statistical interpretation of the latter. Time permitting, I will comment on the generalizations to spinning fields and interacting theories.

The talk is mostly based on joint works with Dionysios Anninos, Frederik Denef, Manvir Grewal, Klaas Parmentier, and Zimo Sun.

Orateur: Albert Law (Stanford University)

mercredi 17 avril

Junior talks

73 An analogue of non-interacting quantum field theory in Riemannian signature

Recent advances using microlocal tools have led to constructions, for wave operators on various classes of spacetimes, of four distinguished Fredholm inverses which have the singular behavior required of retarded, advanced, Feynman, and anti-Feynman propagators in QFT. Vasy and Wrochna have used these to define a QFT on asymptotically Minkowski spacetimes, for which they construct Hadamard states described by asymptotic data at infinity. I will describe an analogue of this construction on Riemannian manifolds with two asymptotically conic ends, defining quantum fields satisfying $(\Delta-\lambda^2)\phi=0$ and using scattering data to construct states satisfying a wavefront mapping-property version of the Hadamard condition. The absence of a spacetime interpretation lends itself to a sharper focus on the theory's analytic structure, from whose perspective the Feynman propagators are no less natural than the advanced/retarded ones. I will also highlight some differences between Feynman propagators defined as distinguished inverses and as time-ordered expectations. Based on joint work with András Vasy.

Orateur: Mikhail Molodyk (Stanford University)

74 Functional Renormalization and the Nash-Moser theorem

The Renormalization Group (RG) Equation determines the flow of the effective action under changes in an artificial energy scale, which roughly corresponds to the scale of the system under consideration. I report on a rigorous construction of a non-perturbative RG flow for the effective action in Lorentzian manifolds. I give the main ideas of a proof of local existence of solutions for the RG equation, when a suitable Local Potential Approximation is considered. The proof is based on an application of the renown Nash-Moser theorem. Time permitting, I also discuss an application of the RG equation to the non-perturbative renormalizability of quantum gravity.

Orateur: Edoardo D'Angelo (Università di Genova)

lundi 22 avril

75 TBA

Orateur: Ali Seraj (Queen Mary University of London)

mardi 23 avril

76 Local covariance in Epstein-Glaser renormalisation

In a seminal work from 2000, Brunetti and Fredenhagen built on the causal perturbation theory of Epstein and Glaser to prove the renormalisability of scalar field theory on curved spacetimes. Their construction was criticized in subsequent publications by Hollands and
Wald for failing to satisfy local covariance. I will show that with appropriate modifications, local covariance can be achieved in the Brunetti-Fredenhagen construction, for distributions which resemble (classical) pseudodifferential operators. The crucial feature of such distributions is the existence of uniform asymptotic expansions.

Orateur: Arne Hofmann (Universität Hannover)

77 Anti-de Sitter space interacting with quantum fields

We compute the renormalized expectation value of the stress-energy tensor operator (RSET) for a quantum scalar field on three- and four-dimensional anti-de Sitter space-time (adS). Since adS is not a globally hyperbolic space-time, boundary conditions have to be applied to the field. We explore the effect of the boundary conditions on the RSET. The RSET then acts as a source term in the Einstein equations governing the quantum-corrected adS metric (QCadS). We describe some of the qualitative features of the QCadS space-time solutions.

Orateur: Elizabeth Winstanley (University of Sheffield)

mercredi 24 avril

78 Wick rotation in the lapse: admissible complex metrics and the Wick rotated heat kernel

A Wick rotation in the lapse (not in time) is introduced for foliated metric geometries. This interpolates between Lorentzian and Riemannian metrics on the same underlying smooth real manifold, passing through "admissible" complex metrics dampening the exponential of the action of a real scalar field. Moreover, strictly away from the Lorentzian metric, the associated Laplace-Beltrami operator generates a "Wick rotated heat semigroup", an analytic semigroup generalizing the usual heat semigroup on a Riemannian manifold. I will discuss the properties of the Wick rotated heat semigroup, including: (i) existence and uniqueness, (ii) the existence and uniqueness of an integral kernel, the "Wick rotated heat kernel", (iii) the kernel's diagonal admits a small semigroup time asymptotic expansion, and (iv) when the d'Alembertian of the Lorentzian metric is essentially self-adjoint, the Wick rotated heat semigroup converges to the unitary Schrödinger group generated by the d'Alembertian in the strict Lorentzian limit. This talk is based on joint work with Max Niedermaier.

Orateur: Rudrajit Banerjee (Okinawa Institute of Science and Technology)