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# Mathematics Physics Seminar : Fall 2009 – Winter 2010

### Stress-energy tensor, Schwarzian, and conformal loop ensembles

#### Benjamin Doyon, Department of Mathematical Sciences, Durham University

Conformal loop ensembles (CLE) provide a provable probability theory for the scaling or « continuum » limits of critical models, through random sets of non-intersecting loops. These scaling limits are also believed to be described by conformal field theory (CFT). I will overview my recent works on relating the two theories. It is based on the CLE construction of the stress-energy tensor, the most fundamental quantum field in CFT. I will briefly explain how the corresponding conformal Ward identities, and in general stress-energy tensor insertions, can be compactly written using the notion of conformal differentiability (a particular type of Hadamard differentiability that I developed); how such conformal derivatives are related to objects in CLE; and how from this relation and from a simple but nice lemma about conformal transformations, their transformation properties can be shown to involve the Schwarzian.

### Classification of solvable algebras with the given nilradical – can the knowledge of solvable extensions of its nilpotent subalgebra be useful?

#### Libor Snobl, Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engine

We construct all solvable Lie algebras with a specific n- dimensional nilradical n_{n,3} which contains the previously studied filiform nilpotent algebra n_{n-2,1} as a subalgebra but not as an ideal. Rather surprisingly it turns out that the classification of such solvable algebras can be reduced to the classification of solvable algebras with the nilradical n_{n-2,1} together with one additional case. Also the sets of invariants of coadjoint representation of n_{n,3} and its solvable extensions are deduced from this reduction. In several cases they have polynomial bases, i.e. the invariants of the respective solvable algebra can be chosen to be Casimir invariants in its enveloping algebra.

### Indecomposable modules for the Virasoro algebra

#### David Ridout, CRM

Recent progress in the study of statistical models (logarithmic conformal field theory and Schramm-Loewner evolution) has led to a need to understand representation theory beyond the highest weight category. Here, we report on the classification of the simplest class of such representations of the Virasoro algebra, illustrated with examples of physical significance.

### Making sense of non-Hermitian Hamiltonians

#### Carl Bender, Physics Department, Washington University in St. Louis

The average quantum physicist believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (symmetric under combined matrix transposition and complex conjugation) so that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian$tex: H=p^2+ix^3$, for example, which is obviously not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent quantum mechanics. Evidently, Dirac Hermiticity is too restrictive. While $tex: H=p^2+ix^3$ is not Dirac Hermitian, it is PT symmetric — symmetric under combined space reflection P and time reversal T. In general, if H is not Dirac Hermitian but has an unbroken PT symmetry, there is a procedure for determining the adjoint operation under which H is Hermitian. (One should not assume that the adjoint operation that interchanges bra and ket vectors in the Hilbert space of states is the Dirac adjoint. This would be like postulating a priori what the metric $tex:g^{\mu\nu}$ in curved space is before solving Einstein’s equations.) Non-Dirac-Hermitian PT-symmetric Hamiltonians have remarkable properties!
This talk will be presented at an elementary colloquium-style level and will be broadly accessible to both theoreticians and experimantalists.

### Group Field Theory

#### Razvan GURAU, Perimeter Institute

Group field theory is the higher-dimensional generalization of random matrix models. As it has built-in scales and automatically sums over metrics and discretizations, it provides a combinatoric origin for space time. Its graphs facilitate a new approach to algebraic topology. I exemplify this approach by introducing a graph’s cellular structure and associated homology.

### Homogeneous operators, jet construction and similarity

#### Subrata Shyam Roy, Indian Statistical Institute, Kolkata

In this talk we show, starting with the jet construction, how to construct all the irreducible homogeneous operators in the Cowen-Douglas class whose associated representations are multiplicity-free.

### Coupling constant metamorphosis and Nth order symmetries in classical and quantum mechanics

#### Sarah Post, CRM

In this talk, I will discuss coupling constant metamorphosis and the Stäckel transform, in particular their generalization to higher order symmetry operators. I will present specializations of these actions which preserve polynomial symmetry operators and the structure of the symmetry algebras. I will also give examples of superintegrable systems on spaces of non-constant curvature and their symmetry algebras.

### Toutes les solutions elliptiques d’une équation différentielle algébrique quelconque

#### Robert Conte, École normale supérieure de Cachan

Étant donné une équation différentielle ordinaire algébrique et autonome admettant au moins une série de Laurent, nous donnons un algorithme exhibant explicitement toutes ses solutions elliptiques ou dégénérées d’elliptiques (rationnelles en une exponentielle, rationnelles). Les seuls ingrédients en sont : la série de Laurent, deux théorèmes de Briot et Bouquet, un algorithme de Poincaré implanté en Maple. Les méthodes existantes n’étaient que suffisantes (obtention de quelques telles solutions), alors que celle-ci est nécessaire. De possibles généralisations seront évoquées :équations discrètes, fonctions de Painlevé.

### Brownian motions and integrable equations

#### Mattia Cafasso

The study of 1-dimensional non-intersecting brownian motions leads to the analysis of some Fredholm determinants with integrable kernels such as, for instance, the celebrated Airy and Pearcey kernels. In this talk I will explain how to obtain some differential equations for such determinants starting from Gelfan’d Dickey equations, i.e. some solitonic equations such as KdV and Boussinesq.

### Universality in the profile of the nonlinear Schrödinger equation at the first breaking curve

#### Marco Bertola, CRM et Concordia University

We consider the zero-dispersion limit of the focusing nonlinear one-dimensional Schrödinger equation with smooth, decaying initial data. The space-time plane subdivides into regions with qualitatively different behavior, with the boundary between them consisting typically of collection of (breaking curve(s)). For small time and/or large distance, the asymptotics is ruled by modulation equations (Whitham equations) whereby the amplitude is a smooth function and the phase is fastly rotating at the scale of the dispersion parameter; for any time greater than the time of gradient catastrophe, there is a compact subset of the x-axis where the asymptotic solution develops fast, quasiperiodic behavior, and the amplitude becomes fastly oscillating at scales of order epsilon. We study the asymptotic behavior of the left and right edges of the interface between these two regions at any time after the gradient catastrophe. The main finding is that the first oscillations in the amplitude are of nonzero asymptotic size even as epsilon tends to zero, and display two separate natural scales; of order O(epsilon) in the parallel direction to the breaking curve in the (x,t)-plane, and of order O(epsilon log epsilon) in a transversal direction.

### Propriétés homotopiques de plages de Fortuin-Kasteleyn sur un tore

#### Alexi Morin-Duchesne, Département. de physique et CRM

On dit qu’un cluster FK sur un tore est dans le groupe d’homotopie {a,b} s’il est possible de dessiner une courbe dans le cluster qui entoure le tore a fois dans une direction et b fois dans l’autre. Pour le modèle de FK avec $tex:\beta\in [0,2]$, nous étudions, au point critique, la probabilité, $tex:\pi({a,b})$, qu’il y ait un cluster du groupe {a,b} et identifions le comportement asymptotique lorsque le tore devient infiniment mince. Les exposants décrivant le comportement critique sont liés aux poids $tex:h_{r,s}$ de Kac pour r,s entiers, mais aussi demi-entiers.

### Matrix model topological expansion with a non-simple branchpoint

#### Aleix Prats-Ferrer, Concordia et CRM

The topological expansion of matrix models has been understood for some years now with only one constraint: all the branch-points of the underlying algebraic curve must be simple. In this work we present the Cauchy matrix model for which this condition is not satisfied and extend the usual topological expansion to allow for a branch-point of branching number 2. The method can be easily extended to any branching number.

### Lie algebras and automorphic forms from vertex algebras

#### Thomas Creutzig, The University of North Carolina, NC Chapel Hill

Vertex algebras of central charge 24 are a key ingredient in relating sporadic groups and automorphic forms. The most prominent example being the monster. I want to explain how to construct out of certain vertex algebras some generalized Kac-Moody algebras whose denominator identity is an automorphic product and which correspond to the Mathieu group M_23.

### Relations de récurrence invariantes associées aux modèles $tex: CP^{N-1}$

#### Michel Grundland, UQTR et CRM

Dans cet exposé nous présentons des relations de récurrence invariantes pour le modèle sigma euclidien $tex: CP^{N-1}$ complètement intégrable en deux dimensions défini sur la sphère de Riemann $tex: S^2$ lorsque sa fonctionnelle d’action est finie. Nous déterminons les liens entre les opérateurs de projection successifs, les fonctions d’ondes du problème linéaire spectral, et les fonctions de plongement des surfaces dans l’algèbre su(N). Notre formulation conserve l’invariance conforme de ces quantités. Certains aspects géométriques de ces relations seront présentés. Nous étudions également les singularités des solutions méromorphes du modèle $tex: CP^{N-1}$ et démontrons qu’elles n’ont aucun impact sur les quantités invariantes. Nous présentons des exemples de la méthode de construction, plus précisemment les modèles $tex: CP^2$ et $tex: CP^3$.

### Monopoles, Periods and Problems

#### Harry W. Braden, School of Mathematics, University of Edinburgh

The modern approach to integrability proceeds via a Riemann surface, the spectral curve. In many applications this curve is specified by transcendental constraints in terms of periods. I will highlight some of the problems this leads to in the context of monopoles, problems including integer solutions to systems of quadratic forms, questions of real algebraic geometry and conjectures for elliptic functions. Several new results will be presented including the uniqueness of the tetrahedrally symmetric monopole.

### États cohérents et comprimés pour des systèmes quantiques et potentiel de Morse

#### Véronique HUSSIN, DMS et CRM

Les états cohérents ont été introduits par R. Glauder dans les années 1950 comme des états quantiques de l’oscillateur harmonique qui minimisent la relation d’incertitude de Heisenberg. Ils sont aussi connus comme des états quasi-classiques. Par la suite, plusieurs généralisations de ceux-ci ont été obtenues tant du point de vue mathématique que physique. Ils trouvent à présent des applications pour de nombreux systèmes quantiques. Ces états cohérents pour l’oscillateur harmonique sont largement utilisés en optique quantique, par exemple. En plus de minimiser la relation d’incertitude de Heisenberg, ils ont des dispersions identiques pour les observables position et impulsion. Il est possible de généraliser ces états afin de réduire la dispersion d’une des observables (au prix d’augmenter celle sur l’autre) tout en maintenant la minimisation de la relation d’incertitude. Ces nouveaux états sont appelés comprimés. Dans les années 1980, des expériences ont permis de mettre en évidence l’existence de tels états pour la lumière. Le potentiel de Morse constitue une meilleure approximation que l’oscillateur harmonique pour décrire les interactions au sein de molécules diatomiques et il possède un spectre discret fini. Les états cohérents et comprimés peuvent également Ãªtre construits pour ce modèle. Dans cet exposé, je commencerai par rappeler différentes définitions des états cohérents et comprimés associés à l’oscillateur harmonique et j’expliquerai brièvement leur intérÃªt en optique quantique. Ensuite, ces définitions seront étendues au contexte du potentiel de Morse. J’insisterai sur la faÃ§on d’ajuster les paramètres introduits pour décrire ces états afin d’assurer une bonne localisation de ceux-ci en termes de l’opérateur position notamment. Je terminerai en donnant quelques avenues pour décrire ces états dans le contexte de modèles à plusieurs dimensions spatiales.

### The Global Geometry of Stochastic Loewner Evolutions

#### Roland Friedrich, Max Planck Inst. Bonn & Leipzig

In this talk we develop a concise description of the global geometry which is underlying the universal construction of all possible generalised Stochastic Loewner Evolutions. The main ingredient is the Universal Grassmannian of Sato-Segal-Wilson. We illustrate the situation in the case of univalent functions defined on the unit disc and the classical Schramm-Loewner stochastic differential equation. In particular we show how the Virasoro algebra acts on probability measures. This approach provides the natural connection with Conformal Field Theory and Integrable Systems.

Monday, February 1st 2010, 15h30

### Algebraic Structure of Univalent Functions and Integrable Systems

#### Dr Roland Friedrich, Max Planck Inst. Bonn & Leipzig

In this seminar we provide algebraic and categorical structures which intrinsically and deeply connect the classical theory of Univalent Functions with the theory of Integrable Systems, as arising in the KP hierarchy. A pivotal role is played by the Faà di Bruno polynomials & the Sato-Segal-Wilson Grassmannian. In our picture we see the Witt algebra & the Schwarzian emerge, and we unravel the very combinatorial nature of all this objects. Once these basics are established, the field is wide open towards other directions, which we shall also mention.

### Illusion of space-time and quantum mechanics

#### Pavel K. Smrz, The University of Newcastle, Australia

General Relativity and Quantum Mechanics do not go well with each other. One is a strictly local theory, while the other predicts non-local effects confirmed by experiments. In this talk it will be demonstrated that a generalized structure of space-time allowing non- locality may be a way to resolve the problem. After a brief introduction to basic concepts of differential geometry the most important results contained in recent publications of the speaker will be described without going into technical details.

### Recent Developments in Non-Equilibrium Quantum Statistical : An overview

#### Vojkan Jaksic, McGill et CRM

In this talk I shall discuss mathematical foundations of non- equilibrium quantum statistical mechanics focusing on a class of recent developments which fall roughly into two categories:
(A) Axiomatic results that concern mathematical structure of the theory;
(B) Study of concrete physically relevant models.
In the first part of the talk I shall focus on (A) and discuss the entropy production observable, entropy production balance equation, non-equilibrium steady states and linear response theory (Kubo formulas, Onsager relations) in the abstract framework of algebraic quantum statistical mechanics. In the second part of the talk I will discuss some concrete physically relevant models for which the axioms of (A) can be verified.

### Affine sl(2) Modules and Applications to Conformal Field Theory

#### David Ridout, CRM

We report on progress in studying fractional-level theories with affine sl(2) symmetry. These are non-unitary cousins of the (positive-integer-level) Wess-Zumino-Witten models which have formed the basis of an enormous amount of interaction between mathematics and physics. Mathematically, the most significant difference when going to fractional levels is that one is obliged to admit representations more general than highest weight. Our aim in this talk is to introduce these more general types of representations and explain why their structure theory is relevant to physics.

### Fondements de la mécanique quantique sur bases de théorie de l’information

#### Gilles Brassard, Dépt. d’informatique et de recherche opérationnelle, Université de Montréal

La théorie de l’information quantique pourrait-elle servir de base é de nouveaux fondements pour la mécanique quantique? Ceux-ci auraient l’avantage d’&ehat;tre plus homogènes et, de mon point de vue, plus fondamentaux que ceux qui ont été développés pendant la première moitié du vingtième siècle.

### Periodic orbits for an infinite family of classical superintegrable systems

#### Frédérick Tremblay, DMS, UdeM et CRM

We show that all bounded trajectories of an infinite family of classical integrable systems are closed for all integer and rational values of k. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.

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