Lorentz Center - The Interface of Integrability and Quantization from 12 Apr 2010 through 16 Apr 2010
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    The Interface of Integrability and Quantization
    from 12 Apr 2010 through 16 Apr 2010

 

Abstracts

 

Gleb Arutyunov
The Mirror TBA: Through the Looking-Glass, and, What Alice Found There

I will discuss the construction of the mirror Thermodynamic Bethe Ansatz
as the tool to determine the spectrum of strings in the
AdS_5 x S^5 space-time and, correspondingly, the spectrum of the dual
gauge theory.

 

 

O.C. Ceyhan

Connes-Marcolli's renormalization group in Epstein-Glaser scheme

 

In this talk we describe the deformations of QFTs in terms of the distributions supported on the diagonals of the configuration spaces of points on spacetime, and then give an action of the pro-algebraic group which appears in Connes–Marcolli’s setting, on the 

finite QFTs constructed by Epstein-Glaser renormalization scheme.

 

 

V.K. Dobrev

Intertwining Operator Realization of Non-Relativistic Holography

 

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrodinger algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov-Witten we give the conditions when both boundary fields are

physical. Finally, we recover in our formalism earlier non-relativistic results for scalar fields by Son and others.

 

 

Tamas Hausel

Topology of the Hitchin system and the arithmetic of the character variety

In this talk I will discuss the perverse filtration on the cohomology of the total space of the Hitchin integrable system; which is obtained from a study of the topology of the singular fibers. We show that this filtration agrees with the weight filtration on the cohomology of a diffeomorphic variety of representations of the fundamental group of the Riemann surface. This implies very strong properties of the topology of the Hitchin system. This is joint work with Mark de Cataldo and Luca Migliorini.

 


Sergey Igonin

Lie algebras associated with PDEs and Backlund transformations

We introduce a new geometric invariant of PDEs: with any analytic system of PDEs we associate naturally a certain system of Lie algebras. These Lie algebras are responsible for Backlund transformations (a tool to construct exact solutions for nonlinear PDEs) and zero-curvature representations (including 2-dimensional Lax pairs) in the theory of integrable systems. Using infinite jet spaces, we regard PDEs as infinite-dimensional manifolds with involutive distributions and study their special morphisms called Krasilshchik-Vinogradov coverings, which generalize the classical concept of coverings from topology and provide a geometric framework for Backlund transformations, Lax pairs, and some other constructions in soliton theory. Recall that topological coverings of a manifold M can be described in terms of the fundamental group of M. We show that a similar description exists for finite-rank Krasilshchik-Vinogradov coverings of PDEs.
However, the "fundamental group of a PDE" is not a group, but a certain system of Lie algebras, which we call fundamental algebras. We have computed these algebras for a number of well-known nonlinear PDEs. As a result, one obtains infinite-dimensional Lie algebras of Kac-Moody type and Lie algebras of matrix-valued functions on algebraic curves. Applications to construction and classification of Backlund transformations will be also presented.

 

 

Arthemy Kiselev (joint with J.W.van de Leur)
The Leibnitz II rule

We compare the well-known property of commutation closure for images of Hamiltonian differential operators in the Lie algebras of evolutionary vector fields with a similar condition upon the variational bi-vectors that act on Hamiltonian functionals. Regarding the Jacobi identity for the Schouten bracket as the Leibnitz rule, we analyse the parallel between the two involutivity conditions. We show that the solution to the still open problem -- whether all Poisson brackets on the spaces of functionals are the derived brackets with respect to variational Poisson bi-vectors -- will properly state the Darboux lemma in the jet bundle setup.

 

 

Stefan Kolb

Classification of coideal subalgebras for quantum groups

In the theory of quantum groups, Lie subalgebras of semisimple Lie algebras should be realised as coideal subalgebras of quantised enveloping algebras. While many classes of such coideal subalgebras of are known, there is so far no general classification. In this talk, a classification of coideal subalgebras of the positive Borel part of a quantised enveloping algebra is presented. The result is expressed in terms of characters of quantisations of nilpotent Lie subalgebras, which were introduced by de Concini, Kac, and Procesi for any element in the Weyl group. The study of such characters naturally leads to fun Weyl group combinatorics. The talk is based on joint work with I. Heckenberger.

 

 

M. Marvan

Classification of integrable PDE in the differential geometry of surfaces

 

The purpose of this talk is to present a recent project, undertaken with my collaborators
from the Silesian university in Opava, and the results obtained so far.

Recognizing integrability is among the important unsolved problems in soliton theory.
Numerous beautiful results have been obtained by indirect approaches like the singularity analysis and the symmetry analysis. Yet obtaining a reasonably complete classification of integrable PDE remains a challenge. In particular, the majority of classification problems in differential geometry seem to be beyond the scope of the methods mentioned.

To cope with the problem, we apply the method of characteristic elements dating back to 1992.
Classifying equations possessing a zero curvature representation with values in a prescribed unsolvable Lie algebra is equivalent to solving a quasilinear system of equations in total derivatives. However, unless in simplest settings, the calculations are prohibitively resource-demanding.
Moreover, one-parametric families of zero curvature representations, which are characteristic of integrability, have to be selected from the vast corpus of calculation results.

Nevertheless, if a PDE comes with a known non-parametric zero curvature representation, then the problem reduces to solving a linear system of equations in total derivatives, and is much easier.
This is the case, e.g., with the Gauss--Mainardi--Codazzi equations of immersed surfaces.
Classification results obtained so far include previously unknown integrable classes of
Weingarten surfaces. Subcases such as surfaces of constant curvature have been known since the nineteenth century. Others, such as surfaces of ``constant astigmatism,'' have fallen into oblivion.

 

 

A. Mironov

AGT relations, Nekrasov functions and conformal field theories

The recently proposed AGT conjecture which relates two-dimensional conformal field theories and supersummetric gauge theories will be discussed. More precisely, the conjecture identifies the conformal blocks and the Nekrasov functions. An important represenation for these is in terms of Dotsenko-Fateev (free field) type integral discriminants. These latters are naturally treated within the context of the Dijkgraaf-Vafa phases of matrix models, in particular, the internal dimensions of the conformal blocks (or vevs of the scalar field in the gauge theory) are associated with the choice of contours.

 

 

Pierre van Moerbeke

Non-intersecting Brownian motions and Integrable systems 

 

By introducing Ornstein-Uhlenbeck dynamics in Gaussian Hermitian random matrices, Dyson showed that their spectrum evolves according to non-intersecting Brownian particles on the real line. By letting the size of the matrices grow very large, or what is the same, by letting the number of particles go to infinity, a number of (universal) critical infinite-dimensional diffusions have appeared. Their transition probabilities are described, on the one hand by the Fredholm determinant of certain kernels, and on the other hand by the solution of certain non-linear partial dfferential equations. These transition probabilities are intimately related to integrable systems.

 

 

A.K. Pogrebkov (in collaboration with M.Boiti and F.Pempinelli)

Asymptotic properties of the Jost solutions of the heat equation

 

Properties of the tau-function, Jost and dual Jost solutions of the heat equation in the case of a pure solitonic potential are studied in detail. We describe their analytical properties with respect to the spectral parameter and their asymptotic behavior on the x-plane and show that the extended version of the heat operator with a pure solitonic potential has left and right annihilators, that are exponentially localized on the x-plane.

 

 

Thomas Quella

Conformal Superspace Sigma-models

Conformally invariant sigma-models on superspaces are two-dimensional supersymmetric quantum field theories which play a prominent role in a number of recent developments in mathematical physics. Apart from their applications in string theory and condensed matter physics (especially disordered systems) they also provide a geometric road towards logarithmic conformal field theories and the representation theory of (affine) Lie superalgebras. Last but not least, they arise as critical continuum limits of certain super spin chains.

In my talk I will review recent progress on this subject, with special emphasis on supergroups and supercosets as superspaces. Employing the examples of superspheres and projective superspaces it will be indicated how exact spectra of anomalous critical dimensions can be calculated as a function of some geometric modulus. These results are then used to argue for new and highly non-trivial dualities between geometric and non-geometric supersymmetric conformal field theories such as supersphere sigma-models and OSP Gross-Neveu models. Our results provide a non-abelian generalization of the duality between the compactified free boson and the massless Thirring model.

 

 

Volodya Roubtsov

Polynomial and Poisson elliptic algebras, Heisenberg group and Cremona transformations

We study different algebraic and geometric properties of Heisenberg ($H-$)invariant Poisson polynomial algebras. We show that these algebras are unimodular. The elliptic
Sklyanin-Odesskii-Feigin Poisson algebras $q_{n;k}(\mathcal E)$ are the main important example. We classify all quadratic $H-$invariant Poisson tensors on ${\mathbb C}^n$ with $n\leq 6$. For the Sklyanin algebras $q_{5,1}(\mathcal E)$ and $q_{5;2}(\mathcal E)$ we explicitly write down the Poisson morphisms on the
moduli space of vector bundles on the normal elliptic curve $\mathcal E$ in $\mathbb P4$, studied by Polishchuk and Odesskii-Feigin as the quadro-cubic Cremona transformation on $\mathbb P4$.

 

 

Gerry Schwarz

Characteristic invariants of reductive groups


Let G be a reductive complex group and V a finite dimensional G-module. Associated to G there are various invariant objects: orbits, fibers of the quotient mapping, invariant polynomial functions, etc. Following Raïs we say that an object is characteristic if the subgroup of GL(V) preserving it is G or at least has identity component contained in G. We discuss some examples of characteristic objects with special attention to orbits which are characteristic. For many V it turns out that all nonzero orbits are characteristic.

 

 

Jasper Stokman
Quantum spherical functions

Harmonic analysis on noncompact Riemannian symmetric spaces is closely related to the spectral analysis of a particular commuting family of differential operators on Euclidean space, the so-called hypergeometric differential operators associated
to root systems. These differential operators, for root systems of type A, are gauge-equivalent to quantum conserved integrals for a quantum integrable many-body system on the line, the so-called quantum hyperbolic Calogero-Moser system.

When deforming these structures, we formally get harmonic analysis on quantum noncompact Riemannian symmetric spaces and a link to the spectral problem of a commuting family of q-difference operators, known as the Cherednik-Macdonald-Ruijsenaars q-difference operators. These operators, for root system A, are gauge-equivalent to the quantum conserved integrals for Ruijsenaars' quantum relativistic hyperbolic Calogero-Moser system.

In this talk the spectral analysis of the Cherednik-Macdonald-Ruijsenaars operators
will be discussed. A fundamental role in the talk is played by the affine Hecke algebra.
I will in particular derive a q-analogue of Harish-Chandra's c-function expansion
of the spherical function.

 

 

Alessandro Torrielli

The Hopf superalgebra of AdS/CFT

Abstract: We will define the infinite-dimensional Hopf superalgebra
emerging in the context of integrability of the AdS/CFT correspondence,
and discuss its typical and atypical representations.   

 

 

A.Verbovetsky, P.Kersten, J.Krasil'shchik and R.Vitolo

On Hamiltonian geometry of PDEs

 

The talk, done by A.Verbovetsky, will survey geometrical ideas underlying the Hamiltonian  approach to integrable systems.  Tangent and cotangent coverings

 over the infinite jet spaces and differential equations,  variational forms, multivectors, de Rham differential, Schouten  bracket, etc. will be discussed after a short introduction to the geometry of differential equations.

 

 

Anton Zabrodin

Classical integrable structures in quantum integrable models

Classical integrable equations are known to exist in quantum integrable problems as exact relations even for non-zero Planck's constant. One face of this general phenomenon is appearance of discrete integrable classical dynamics governed by the Hirota equation in the space of commuting transfer matrices (integrals of motion) for integrable quantum spin chains. The spectrum of the quantum spin model can be found by solving this discrete dynamical system via a chain of Backlund transformations, which appears to be equivalent to the nested Bethe ansatz procedure. Bethe equations themselves acquire then the meaning of dynamical equations for Ruijsenaars-type many-body problems in discrete time. This nontrivial interplay between quantum and classical integrability becomes even more meaningful and natural for supersymmetric quantum spin chains. In the talk, we outline the construction of quantum transfer matrices for GL(K|M)-invariant spin chains and Baxter's Q-operators and show how their eigenvaluescan be found in terms of the discrete Hirota dynamics.



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