YRS seminars

These were 30 minute seminars in two parallel sessions.

Parallel session 1
S. Cherkis Princeton, USA
R. Costa-Santos Utrecht, Netherlands
N. L. Harshman Washington, DC, USA
A. Hahn Bonn, Germany
T. Kuna Bielefeld, Germany
N. Reis Lyon, France
L. Rey-Bellet Amherst, MA, USA
Parallel session 2
J. Baptista Cambridge, UK
A. Hernández-Garduño Mexico City, Mexico
D. Krejcirik Lisbon, Portugal
G. Morsella Göttingen, Germany
G. Perelman Paris, France
O. Post Aachen, Germany
N. Romão Leipzig, Germany
João Baptista
Department of Applied Mathematics and Theoretical Physics
University of Cambridge

Some special Kahler metrics on SL(2,C) and their holomorphic quantization

The group SU(2)× SU(2) acts naturally on SL(2,C) by simultaneous right and left multiplication. I'll show how to study the Kähler metrics invariant under this action using global Kähler potentials. The volume growth and various curvature quantities are then explicitly computable. Examples include metrics of positive, negative and zero Ricci curvature, and the 1-lump metric of the CP1-model on a sphere.

I'll then look at the holomorphic quantization of these metrics, where some physically satisfactory results on the dimension of the Hilbert space can be obtained. These give rise to an interesting geometrical conjecture, regarding the dimension of this space for general Stein manifolds in the semi-classical limit.

Sergey Cherkis
Institute for Advanced Study

Asymptotically locally flat hyperkähler metrics of D-type

Self-Dual Gravitational Instantons are four-dimensional hyperkähler manifolds. These appear as moduli spaces of Seiberg-Witten gauge theories, compactifications of M and string theories that preserve supersymmetry, as well as moduli spaces of solutions of integrable systems. We present classification of these manifolds. Using string theory dualities and twistor geometry we compute metrics on all asymptotically locally flat gravitational instantons. Metrics of D-type that we present provide first examples of gravitational instantons without any isometries.

Ruben Costa-Santos
Spinoza Institute
Utrecht University

Discrete holomorphy, criticality and finite size corrections in free fermion models

A close connection between discrete complex analysis and statistical mechanics models has been recently unraveled; Mercat [1] has shown that the Ising model criticality condition on regular lattices can be understood in terms of isoradial embeddings and Kenyon [2] showed that the Dimer model becomes particularly simple in isoradial graphs allowing the evaluation of the partition function, the determinant of the discrete f-bar operator, in terms of the local lattice geometry.

In this talk I will discuss the Ising model on non-regular lattices, in particular the vanishing of determinants of Dirac operator and the relation between the criticality condition and isoradial embeddings. I will also discuss finite size corrections to the Ising model free energy on lattices with non-trivial topology and conical singularities [3, 4, 5, 6].

From this set of results it emerges that the purely combinatorial Pfaffian method [7] introduced by Kasteleyn for free fermions can be seen as a discrete formulation of some aspects of conformal field theory.

  1. C. Mercat, Commun. Math. Phys. 218 (2001), 177
  2. R. Kenyon, Invent. Math. 150 (2002), 409
  3. J.L. Cardy and I. Peschel, Nucl. Phys. B300 (1988), 377
  4. R. Costa-Santos and B.M. McCoy, Nucl. Phys. B623 (2002), 439
  5. R. Costa-Santos and B.M. McCoy, cond-mat/0210059, to be published in J. Stat. Phys. 112 (2003)
  6. R. Costa-Santos, cond-mat/0306396
  7. N.P. Dolbilin, A.S. Mishchenko, M.A. Shtan'ko, M.I. Shtogrin and Y.M. Zinoviev, Func. Anal. and Applic. 30 (1996), 163
Atle Hahn
Institut für Angewandte Mathematik der Universität Bonn
Wegelerstraße 6
53115 Bonn

A new approach for the path integral quantization of Chern-Simons models on S2 × S1

We study Abelian and Non-Abelian Chern-Simons models on manifolds M of the form M = Σ × S1 where Σ is a 2-dimensional oriented smooth manifold. For the special case Σ = S2 we develop a non-perturbative approach for the path integral quantization of the corresponding Chern-Simons model. This approach is based on the application of a gauge fixing procedure which we call "quasi-axial" as it is similar to axial gauge fixing for manifolds of the form Σ × R. We compute the Wilson loop observables explicitly in the case where the structure group G of the model is Abelian and find the same linking number expressions that have been obtained by other methods. Analogous computations for Non-Abelian groups like SU(N) or SO(N) might contribute to a better understanding of the well-known relations between Chern-Simons theory, knot polynomials, and quantum groups.

Nathan L. Harshman
American University
Washington, DC

Poincaré semigroup as an emergent property of unstable particles

Working with the particle concept of Wigner and the mathematical structure of rigged Hilbert spaces or Gel'fand triplets, the notion of microphysical causality is encoded in representations of the Poincaré semigroup. An open question is whether and how these representations emerge as limiting cases and/or extensions of the stable one-particle representations first classified by Wigner. The Clebsch-Gordon series for various representations of the (extended) Poincaré group will be discussed towards this goal.

Antonio Hernández-Garduño
Universidad Nacional Autónoma de México

Singular reduction and bifurcations in simple mechanical systems

In this talk we will summarize some results from the theory of singular reduction of hamiltonian systems with symmetry and bifurcatioins from equilibria and relative equilibria. We will discuss how the structure of cotangent bundles can be used to study bifurcations in the case of simple mechanical systems with symmetry.

David Krejcirik
Departamento de Matemática
Instituto Superior Técnico

Geometrically induced bound states in curved quantum layers

We consider a nonrelativistic quantum particle constrained to a curved hard-wall layer of constant width built over a non-compact surface embedded in R3. Under the assumption that the surface curvatures vanish at infinity, we find sufficient conditions which guarantee the existence of geometrically induced bound states. We focus on further spectral properties of the layers built over surfaces with a non-trivial genus or several ends.

Tobias Kuna

A relativized Dobrushin uniqueness condition and applications to Pirogov-Sinai models

Dobrushin's uniqueness approach is usually applied in high temperature regime to show uniqueness and decay of correlation for Gibbs measures. In this talk I would like to explain how this technique can be applied as well to study Gibbs measures below critical temperature. Obviously, one cannot expect to obtain contractivity on all configurations. We are forced to study a restricted class of configurations typical for the phase. One thus needs a relativized Dobrushin criterion, where the lack of validity of the original condition is compensated by an a priori estimates on the probability of bad configurations. In this talk I would like to demonstrate the applicability of this abstract result to two examples: Ising model and particles systems with Kac interactions. The latter application covers an essential step in the proof of existence of phase transition for this type of models by J. L. Lebowitz, A. Mazel, and E. Presutti.

The talk is based on a joint paper in preparation with I. Merola and E. Presutti.

Gerardo Morsella
Institut für Theoretische Physik

Scaling algebras for charge carrying quantum fields and superselection structure at short distances

I will present joint work with C. D'Antoni and R. Verch on the extension of the method of scaling algebras, which has been introduced earlier as a means for analyzing the short-distance behaviour of quantum field theories in the setting of the model-independent, operator algebraic approach, to the case of fields carrying superselection charges, of both DHR and BF type. This generalization will then be used to analyze the relation between the superselection structures of the underlying theory and the one of its scaling limit. Such a study is relevant in connection with the formulation of an intrinsic notion of charge confinement along the lines indicated by D. Buchholz. In particular, a physically motivated criterion for the preservance of superselection charges in the scaling limit will be proposed, leading to a natural identification between sectors of the underlying theory and a subset of sectors of the scaling limit one. Some consequences of this preservance of superselection charges will be discussed.

Galina Perelman
École Polytechnique

On the blow up phenomenon for the critical nonlinear Schrödinger equation

We consider the Cauchy problem for the one-dimensional critical nonlinear Schrödinger equation with initial data close to a soliton. We show that for a certain class of initial perturbations the solution develops a self-similar singularity in finite time T*, the profile being given by the ground state solitary wave and the limiting self-focusing law being of the form

λ(t) ∼ (ln |ln(T*-t)|)1/2 (T*-t)-1/2.

Olaf Post
Institut für Reine und Angewandte Mathematik

Spectral gaps on manifolds with non-commutative group actions

Consider the Laplacian ΔX on a non-compact manifold X with a discrete isometric group Γ acting on X such that the quotient X/Γ is a compact manifold. This situation is motivated by the analysis of periodic Schrödinger operators with periodic potentials. It was shown by the author that if Γ is abelian, then gaps necessarily appear in the essential spectrum of ΔX, where X is in a suitable family of manifolds satisfying the above conditions.

We generalize this result for certain classes of non-commutative groups Γ using a result of Sunada. We will give simple examples and comment on the rôle that the dual object Γ-hat plays in the non-abelian case. The essential ingredient in the abelian and non-abelian cases is the Min-Max-principle and Dirichlet-Neumann bracketing.

[Joint work with Fernando Lledó]

Nuno Reis
Laboratoire de Physique

WZW branes and gerbes

Bundle gerbes with connection are geometric objects inducing naturally line bundles on loop spaces and spaces of open paths ending on branes. In this talk, we explain how these geometric objects appear naturally in the Lagrangian approach to the WZW conformal field theories. In particular, they lead to a geometric determination of the spectrum of symmetric branes in the WZW models with groups covered by SU(N) as targets. We present a general relation, that seems at least partially new in the context of the WZW theory, between the open string amplitudes based on non-simply connected groups and on their covering groups.

Luc Rey-Bellet
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA 01003

Entropy production and its fluctuations in classical open systems

An open system is an Hamiltonian system with finitely many degrees of freedom coupled to one or several infinite reservoirs. One expects the system to relax (in the sense of ergodic theory) to a stationary state, a Gibbs equilibrium state in case of a single reservoir at positive temperature, a nonequilibrium stationary state if there are two or more reservoirs at different temperatures. In a nonequilibrium stationary state one expects, under general conditions, energy to flow through the system into the reservoirs (positivity of entropy production). We discuss universal properties of the fluctuations of entropy production. Around equilibrium the small fluctuations (of central limit theorem type) of the entropy production are connected to linear response (Kubo formula and Onsager relations). Far from equilibrium the large fluctuations (of large deviations type) of the entropy production exhibit a universal symmetry (the Gallavotti-Cohen fluctuation theorem) which generalizes Kubo formula and Onsager relations far from equilibrium. We give an example of a nonlinear Hamiltonian system coupled to linear Hamiltonian reservoirs for which all these properties can be proved rigorously.

Nuno M. Romão
Max-Planck-Institut für Mathematik in den Naturwissenschaften

Dynamics of CP1 lumps on a cylinder

There is a large class of field theories with second-order equations of motion for which the stable static solutions can be described by a simpler first-order PDE, usually called a Bogomol'nyi equation. One example of such "self-dual" theories is the CP1 sigma-model on a Riemann surface, for which the static solutions are precisely the harmonic maps, and their Dirichlet energy is minimised when they satisfy the Cauchy-Riemann equation or its conjugate. In my talk, I will address the adiabatic approximation for the dynamics of the topological solitons (usually called lumps) in this model, when space is taken to be an infinite cylinder. I will explain how this problem can be formulated in terms of the geodesic flow of certain Kaehler metrics on moduli spaces of rational maps. Some of these metrics can be computed explicitly in terms of elliptic integrals, and I will also present some preliminary results in this direction.