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## The Interface of Integrability and Quantization |

The Mirror TBA: Through the
Looking-Glass, and, What Alice Found ThereI will discuss the construction of the mirror Thermodynamic Bethe Ansatzas 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.
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.
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.
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.
Lie algebras associated with PDEs and Backlund
transformations
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.
The purpose of this talk is to present a recent project,
undertaken with my collaborators
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.
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.
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.
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.
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. [Back] |