Séminaire de physique mathématique SEMINAIRE/20162017

Année 2016-2017

~~30/06/2017~~ ANNULÉ : Ariane Carrance (ICJ)
Modèles aléatoires de trisps colorés.
<a>RésuméLes graphes \((D+1)\)-colorés sont des graphes \((D+1)\)-réguliers, munis d'une coloration propre de leurs arêtes. Ils encodent des espaces topologiques linéaires par morceaux, de dimension \(D\), appelés trisps colorés. Ces structures ont suscité l'intérêt de physiciens théoriciens au cours des dernières années, car elles sont au cœur d'une nouvelle approche à la gravité quantique, les modèles de tenseurs colorés. Ceux-ci généralisent les modèles de matrices aux dimensions supérieures. Je présenterai des modèles aléatoires de graphes colorés et des trisps associés, motivés par la recherche d'une limite continue (et aléatoire) pour l'espace-temps, décrit dans les modèles de tenseurs colorés par un trisp aléatoire.</a>.
16/06/2017 : Thierry Dauxois (CNRS & ENS de Lyon)
Instabilities of Internal Gravity Wave Beams.
<a>RésuméInternal gravity waves play a primary role in geophysical fluids: they contribute significantly to mixing in the ocean and they redistribute energy and momentum in the middle atmosphere. Until recently, most of the studies were focused on plane-wave solutions. However, these solutions are not a satisfactory description of most geophysical manifestations of internal gravity waves, and it is now recognized that internal wave beams with a locally confined profile are ubiquitous in the geophysical context. We will discuss the reason for their ubiquity in stratified fluids, since they are solutions of the nonlinear governing equations. Moreover, in the light of the recent experimental and analytical studies of those internal gravity beams, it is timely to discuss the two main mechanisms of instability for those beams: the triadic resonant instability and the streaming instability.</a>.
7-9/06/2017 : Journées de physique mathématique de Lyon.
19/05/2017 : Boris Tsygan (Northwestern University)
Microlocal category of a symplectic manifold.
<a>RésuméGiven a symplectic manifold \(M\), one can consider its deformation quantization, i.e. an associative multiplication law on functions on \(M\) that depends on a formal parameter \(h\). When \(M\) is the cotangent bundle of a manifold \(X\), one essentially recovers the algebra of differential operators on \(X\), or rather of \(h\)-differential operators \(P(x, hd/dx)\). Modules over differential operators are well known to have interesting applications to PDE and other topics, so it is natural to hope that modules over deformation quantization would be interesting as well. However, what one really encounters in PDE and quantum mechanics is some sort of modules where two formal parameters are involved: \(h\) and \(\exp(1/h)\). What we present in the talk is a construction of modules over a bigger algebra that contains not only expressions \(P(q,p,h)\) as in deformation quantization but also \(\exp(f(q,p)/h)\). Here \(q\) and \(p\) are local Darboux coordinates on \(M\). Note that from a module over differential operators on \(X\) one can pass to a sheaf on \(X\) (the module is a generalized bundle with a flat connection, and the sheaf is its De Rham complex). Since our construction (for a cotangent bundle) is the enlarged algebra of differential operators, one can ask whether its sheaf-theoretic counterpart exists. This is indeed the case. This sheaf-theoretical counterpart is given by Tamarkin's category of sheaves on \(X\times R\) (\(t\) on \(R\) roughly corresponds to \(\exp(t/h)\)). This construction was recently generalized by Tamarkin to a general symplectic manifold. The above constructions are conjecturally connected by some version of a Riemann-Hilbert correspondence (as \(D\)-modules and sheaves are). They have some of the features possessed by the Fukaya category of a symplectic manifold. We do not know of any functor going in either direction. The talk will give a broad overview of the topic, not assuming any prior knowledge and using the example of the plane for much of the construction.</a>.
05/05/2017 : Moulay Tahar Benameur (Institut Montpellierain Alexander Grothendieck)
Conjecture du label des gaps magnétique pour les quasi-cristaux.
<a>RésuméJe formulerai une conjecture pour le label des gaps des opérateurs de Schrödinger euclidiens avec champ magnétique constant et avec potentiel apériodique. Des résultats en petites dimensions ainsi que le cas périodique confirment la conjecture, et seront explicités. Il s'agit d'un travail en collaboration avec V. Mathai.</a>.
07/04/2017 : Nicolò Drago (Università degli Studi di Genova)
Perturbative methods in Algebraic QFT with applications to Thermal Field Theory.
<a>RésuméThe Principle of Perturbative Agreement (PPA), as introduced by Hollands & Wald, is a renormalisation condition in quantum field theory on curved spacetimes. This principle states that the perturbative and exact constructions of a field theoretic model given by the sum of a free and an exactly tractable interaction Lagrangian should agree. We develop a proof of the validity of the PPA in the case of scalar fields and quadratic interactions without derivatives which differs in strategy from the one given by Hollands & Wald for the case of quadratic interactions encoding a change of metric. Afterwards, we prove a generalization of the PPA and show that considering an arbitrary quadratic contribution of a general interaction either as part of the free theory or as part of the perturbation gives equivalent results. Finally, we will briefly describe how our findings can be used in order to provide a mathematically rigorous description of the "thermal mass" idea used in Physics.</a>. Diapositives.
24/03/2017
14h00 - 15h00 : Francesco Bei (ICJ), Scattering theory for the Hodge Laplacian under a conformal pertubation. <a>RésuméLet \(g\) and \(h\) be two conformally equivalent Riemannian metrics on a noncompact manifold \(M\). We will show that under some mild first order control on the conformal factor, the wave operators associated to the Hodge-Laplacians \(\Delta_g\) and \(\Delta_h\) acting on differential forms exist and are complete. Then we will show some applications and we will provide some explict calculations of the absolutely continuous spectrum in the setting of Riemannian manifolds with bounded geometry.</a>. Diapositives.
15h15 - 16h00 : Sara Azzali (IFM, Potsdam), The class of a flat vector bundle and a noncommutative generalisation. <a>RésuméLet \(G\) be the fundamental group of a closed manifold \(X\) and \(\alpha: G\to U(n)\) a finite dimensional unitary representation, i.e. a flat unitary vector bundle over \(X\). To these data, Atiyah, Patodi and Singer associated a class \([\alpha]\) in the K-theory group with \(\mathbb R/\mathbb Z\)-coefficients and investigated its relation to spectral \(\rho\) invariants. In this talk, we take an operator algebraic point of view on the class \([\alpha]\) and generalise it to the noncommutative setting of a discrete group \(G\) suitably acting on a \(C^*\)-algebra \(A\). The condition on the action is encoded by KK-theory with real coefficients, which will be introduced, and can be called K-theoretical free and properness. We exhibit natural classes of \(G\)-algebras satisfying this property. Based on joint work with Paolo Antonini and Georges Skandalis.</a>.
17/03/2017 : Takeo Kojima (Yamagata University, Japan)
A bosonization of \(U_q(\widehat{sl}(M|N))\).
<a>RésuméBosonization is a powerful method to study representation theory of infinite-dimensional algebras and its application to mathematical physics, such as calculation of correlation functions of exactly solvable models. For level \(k=1\), bosonization has been constructed for the quantum affine algebra \(U_q(g)\) in many cases. Bosonization of an arbitrary level \(k \in {\bf C}\) is completely different from those of level \(k=1\). For level \(k \in {\bf C}\), bosonization has been constructed only for \(U_q(\hat{sl}(N))\) and \(U_q(\hat{sl}(M|1))\). In this talk we give a bosonization of the quantum affine superalgebra \(U_q(\hat{sl}(M|N))\ (M,N=1,2,3,\cdots)\) for an arbitrary level \(k\). For the level \(k \neq -M+N\) we give screening operators that commute with \(U_q(\hat{sl}(M|N))\) and propose a realization of the vertex operator. This talk is based on arXiv:1701.03645.</a>.
10/03/2017 : Luca Lionni (LPT Orsay & LIPN)
Generalized \(p\)-angulations in higher dimensions.
<a>RésuméI will talk about generalized \(p\)-angulations in dimension higher than 2, which are gluings of elementary building blocks with \(p\) external faces. Such discretized spaces are generated by so called enhanced random tensor models. In two dimensions, \(p\)-angulations which maximize the number of vertices at fixed number of \(p\)-gons are the planar ones. They belong to the same universality class in the sense that their generating functions exhibit the same critical behavior. The situation in higher dimension appears to be richer. I'll present counting results and the critical behaviors of a number of such models.</a>. Diapositives.
17/02/2017 : Gianluca Panati (Università di Roma "La Sapienza")
Optimal decay of Wannier functions in Chern and Quantum Hall insulators.
<a>RésuméWe investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e.g. in Chern insulators and in Quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as \(|x| \to \infty\), of the composite Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy, in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is \(+\infty\). In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity. The talk is based on a joint work with Domenico Monaco, Adriano Pisante, and Stefan Teufel.</a>.
03/02/2017 : Carlos Ignacio Pérez (:html:) Sánchez(:htmlend:) (Universität Münster)
The full Ward Takahashi Identity for arbitrary coloured tensor models: Focus on 3D and 4D quartic interactions.
<a>RésuméHaving a common backbone with matrix models, coloured tensor models is a random geometry approach to quantum gravity in arbitrary dimensions. In particular, regularly edge-coloured bipartite graphs—precisely the Feynman graphs of these theories—generalise ribbon graphs and represent orientable PL-manifolds of dimension one less than the number of colours. Also, an integer called Gurau’s degree, which generalises the genus, controls the \(1/N\) expansion. We introduce a graph theoretical representation of the connected sum that is additive with respect to Gurau’s degree and compatible with the QFT-structure. Moreover, we find the general exact non-perturbative Ward Takahashi Identities and then discuss particular (with so-called melonic, or dominant, interaction vertices) quartic 3D and 4D coloured tensor theories. Recursions for the correlation functions of these theories are obtained.</a>. Diapositives.
06/01/2017 : Elba Garcia-Failde (MPIM, Bonn)
Nesting statistics in the \(O(n)\) loop model on random maps of any topology.
<a>RésuméIn this talk, I will call maps a certain class of graphs embedded on surfaces. We will consider random maps equipped with a statistical physics model: the \(O(n)\) loop model. We investigate the nesting properties of loops by associating to every map with a loop configuration a so-called nesting graph, which encodes all the interesting nesting information. We study the generating series of maps of genus \(g\) with \(k\) boundaries and \(k'\) marked points realizing a fixed nesting graph, which are amenable to explicit computations in the loop model with bending energy on triangulations. We describe their critical behavior in the dense and dilute phases, arriving to some interesting qualitative conclusions about the most probable kind of maps. This was first studied for maps with the topology of disks and cylinders, and we generalize it to arbitrary topologies making use of a procedure called topological recursion, which will be introduced and applied to our analysis.</a>. Diapositives.

16/12/2016 : Rémy Mosseri (LPTMC, UPMC)
Geometry of entanglement for few qubits systems: a view from Hopf fibrations and Majorana spheres.
<a>RésuméDans cet exposé, je voudrais montrer comment une propriété non triviale de la théorie quantique, l'intrication, s'illustre, dans le cas le plus simple rencontré (celui de deux systèmes à deux niveaux) à l'aide d'un objet géométrique, lui aussi non trivial, la fibration de Hopf (plus précisément ici, la fibration de de la sphère \(S^7\)). Cette mise en correspondance permet de feuilleter l'espace de Hilbert du système en fonction du taux d'intrication. Je décrirai au passage les difficultés du traitement de l'intrication trois qubits, lorsque l'on cherche à impliquer la fibration de la sphère \(S^{15}\). Enfin, dans le cas où l'on s'intéresse à des systèmes de quelques qubits dans la représentation symétrique, la représentation (sphère) de Majorana permet d'analyser certaines familles d'invariants que l'on peut alors exprimer de façon géométrique. De façon un peu inattendue, l'invariant modulaire de Klein apparait dans cette description.</a>.
09/12/2016 : Daniel Parra (ICJ)
Théorie spectrale et de la diffusion pour des graphes périodiques perturbés.
<a>RésuméOn considère des analogues discrets du Laplacien de Hodge et de l'opérateur de Gauss-Bonnet sur des graphes périodiques munis d'une mesure périodique \(m_0\). Grâce à une transformation de Floquet-Bloch, on montre que ces opérateurs ont un spectre purement absolument continu en dehors d'un ensemble discret. En utilisant la méthode à commutateurs on étudie des perturbations du graphe. Cette perturbation est encodée par une mesure non-periodique \(m\) qui converge à l'infini vers \(m_0\). On montre que si \(m\) converge suffisamment vite vers \(m_0\), la structure spectrale est préservée et les opérateurs d'onde locaux existent et sont complets.</a>. Diapositives.
02/12/2016 : Ajay Chandra (University of Warwick)
An analytic BPHZ theorem for regularity structures.
<a>RésuméI will give a light introduction to the theory of regularity structures and then discuss recent developments with regards to renormalization within the theory - in particular I will describe joint work with Martin Hairer where multiscale techniques from constructive field theory are adapted to provide a systematic method of obtaining needed stochastic estimates for the theory.</a>.
25/11/2016 : Andras Szenes (Université de Genève)
Higgs bundles and residues.
<a>RésuméI will present some recent and not so recent joint work with Tamas Hausel on the topology of Higgs bundles, Bethe Ansatz and hyperplane arrangements.</a>.
18/11/2016 : Etera Livine (ÉNS Lyon)
Dualité entre le modèle d'Ising 2d et la gravité 3d : vers une formule géométrique pour les zéros de Fisher.
<a>RésuméPour une topologie de bulk triviale, les amplitudes pour la gravité quantique 3d ne dépendent que de l'état choisi sur la frontière 2d. Elles s'expriment en tant qu'évaluations de spin networks vivant sur un graphe défini sur cette frontière 2d et se calculent en termes de symboles 3nj de recoupling de spin. On montre que la fonction génératrice pour ces symboles 3nj est égale à l'inverse du carré de la fonction de partition du modèle d'Ising sur ce graph frontière. Cela permet une approche géométrique à la détermination des zéros de la fonction de partition d'Ising (Fisher zeroes), mais également offre un prototype d'implémentation de la correspondance AdS/CFT dans un cadre discret en 3d/2d.</a>. Diapositives.
04/11/2016 : Sylvain Lavau (ICJ) - soutenance de thèse
Lie infini-algébroides et feuilletages singuliers.
<a>RésuméOn dit qu’une variété M est feuilletée lorsqu’il existe une partition de celle-ci en sous-variétés immergées connexes. La théorie des feuilletages a des applications très profondes dans divers champs des Mathématiques et de la Physique. Il semble d’autant plus intéressant de pouvoir analyser le feuilletage à partir de ce qui semble être une donnée plus fondamentale : sa distribution de champs de vecteurs associée. Certaines distributions sont engendrées par des sous modules du module des champs de vecteurs sur M, et nous choisirons de les étudier. C’est ainsi que nous avons observé que si ce sous module est résolu par un fibré gradué sur M, on peut relever le crochet de Lie des champs de vecteurs en une structure de Lie \(\infty\)-algébroide sur les sections de ce fibré. D’autre part, cette structure est universelle dans le sens où toute autre résolution du feuilletage sera isomorphe à celle-ci dans un sens L-\(\infty\), mais seulement à homotopie près. Lorsqu’on se limite à l’étude au dessus d’un point, on observe que la cohomologie associée à la résolution devient potentiellement non triviale. La structure de Lie \(\infty\)-algébroide universelle se réduit alors à une algèbre de Lie graduée sur cette cohomologie. Cette structure algébrique peut être transportée (non canoniquement) tout le long de la feuille, transformant la cohomologie au dessus d’une feuille en algébroide de Lie gradué. Cela nous permet de retrouver des résultats déjà connus par ailleurs et de déduire des applications prometteuses.</a>.
14/10/2016 : Simon Covez (Laboratoire Jean Leray, Nantes)
Sur l'intégration des algèbres de Leibniz.
<a>RésuméUne algèbre de Leibniz est un espace vectoriel muni d'un crochet vérifiant l'identité de Leibniz (identité exprimant que le crochetage avec un élément est une dérivation du crochet) mais qui n'est pas forcément antisymétrique. Cette notion, qui généralise celle d'algèbre de Lie, est apparue naturellement dans les travaux de Loday concernant les problèmes de périodicité en K-théorie algébrique. Le problème des coquecigrue posé par Loday est de savoir si la correspondance groupe-algèbre de Lie se généralise aux algèbres de Leibniz, et, dans le cas où cette généralisation existe, quelle est la théorie de (co)homogie naturellement associée aux objets intégrant ces algèbres. Le but de cet exposé est de présenter des résultats concernant ce problème. Nous verrons que toute algèbre de Leibniz s'intègre localement en un rack de Lie, et que (co)homologie de rack et (co)homologie de Leibniz possèdent des propriétés communes.</a>.
30/09/2016 : Vincent Vargas (DMA, ÉNS Paris)
Ward and Belavin-Polyakov-Zamolodchikov identities for Liouville quantum field theory on the Riemann sphere.
<a>RésuméThe foundations of modern conformal field theory (CFT) were introduced in a 1984 seminal paper by Belavin, Polyakov and Zamolodchikov (BPZ). Though the CFT formalism is widespread in the physics literature, it remains a challenge for mathematicians to make sense out of it. Liouville CFT (or quantum field theory) is an important class of CFTs which can be seen as a random version of the theory of Riemann surfaces. In a recent work, we constructed the correlation functions of Liouville CFT in the Feynman path formalism using probabilistic techniques. In this talk, I will present a rigorous derivation of the so-called Ward and BPZ identities for Liouville CFT. These identities are the basis to compute the correlations and to establish the correspondence between the Feynman path formalism of Liouville CFT and the algebraic formalism based on the Virasoro algebra. Based on joint works with F. David, A. Kupiainen and R. Rhodes.</a>.
16/09/2016 : Pierre Clavier (IFM, Potsdam)
Procédure de branchement et régularisation de zêtas arborifiés.
<a>RésuméEn utilisant la propriété universelle des arbres enracinés, nous définissons une procédure permettant de relever une fonction sur une algèbre A en une fonction sur l'algèbre des arbres décorés par A. En choisissant A comme étant une algèbre de fonctions méromorphes à valeurs dans les symboles et dotée d'un produit partiel, nous pouvons définir des sommes itérées respectant la structure de poset d'arbres. Ceci nous amène à une régularisation de fonctions zétas arborifiées, dont nous étudions certaines propriétés, en particulier leurs pôles.</a>.
09/09/2016 : Sara Azzali (IFM, Potsdam)
Large time limit of the heat operator for families of cocompact manifolds.
<a>RésuméThe heat kernel’s supertrace of a Dirac operator interpolates between the local geometry of a closed manifold and a global invariant, the index of the Dirac operator. This fundamental property, first applied by Atiyah, Bott and Patodi to give a proof of the Atiyah–Singer index theorem, is a crucial tool in local index theory. In the case of a fibre bundle and a family of fibrewise Dirac operators, heat kernel techniques are combined with the use of superconnections and were introduced by Quillen and Bismut: the heat operator’s supertrace is here a differential form on the parameter space (the base space of the fibration) and the computation of its large time limit provides a differential forms refinement of the cohomological Atiyah–Singer formula. The case when the fibres are noncompact is particularely tricky as in general the large time limit is not convergent: Heitsch, Lazarov and Benameur studied this problem on a foliated manifold and found regularity conditions on the spectrum which ensure the convergence at large time. In a joint work with Sebstian Goette and Thomas Schick, we have studied the particular case of a family of fibrewise signature operators for families of manifolds with a cocompact group action. We prove that there exists a way to carefully estimate all the terms appearing in the Volterra expansion of the heat operator’s supertace and compute explicitely its large time limit, without requiring extra regularity assumption on the spectrum. With some regularity conditions (involving the determinant class or the positivity of the Novikov-Shubin invariants) we obtain the \(L^2\) local index formula.</a>.

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