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Integrable Systems in Quantum Theory
Speaker: Gleb Arutyunov
Title: Towards the solution of the AdS/CFT spectral problem
Abstract: I will discuss the recent progress towards finding the exact spectrum of the sigma model describing strings in the AdS_5 x S5 space-time.
Speaker: Harry Braden
Title: Cyclic Monopoles, Toda and Ramanujan
Abstract: Some years ago in the context of Seiberg-Witten theory Sutcliffe gave an ansatz for cyclic su(2) monopoles in terms of Toda theory. Here I will strengthen this connection and provide several new results arising from the relations between the spectral curves of these systems (and their covers) and the appropriate flows on their Jacobians. Some number theoretic problems are encountered in this work: a transcendental constraint on one family of curves is equivalent to a problem solved by Ramanujan.
Speaker: Ed Corrigan
Title: Some aspects of defects in integrable field theory
Abstract: The purpose of the talk is to discuss recent work concerning the possibility of allowing defects of 'shock' type in integrable field theory without destroying the feature of integrability. There are a number of examples though the talk will concentrate on and develop the classical and quantum features of just one of them (the sine-Gordon model).
Speaker: Andrey Marshakov
Title: First order string theory and the Kodaira-Spencer equations
Abstract: We consider the first-order bosonic string theory, perturbed by the primary operator, corresponding to the deformation of complex structure in target-space. We compute the effective action in this theory and find that its consistency with the world-sheet conformal invariance requires the Kodaira-Spenser equations to be satisfied by the target-space Beltrami differentials.
The same conclusion follows from studying directly the correlation functions in first-order conformal field theory. We discuss their and relation with the polyvertex structures in BRST approach, to be generally defined in terms of integrals over the moduli spaces of the world-sheet complex structures.
Speaker: Andrei Pogrebkov
Title: Commutator identities on associative algebras and integrability of nonlinear partial and difference equations
Abstract: It is shown that commutator identities on associative algebras generate
operator solutions of linearized versions of the integrable equations. A
dressing procedure in a special class of integral operators is introduced,
that enables to derive both nonlinear integrable equation itself and its
Lax pair from such commutator identity. As a specific example Hirota
bilinear difference equation and associated algebraic identity are
Speaker: Vladimir Rubtsov
Title: Manin matrices and Elliptic Gaudin revisited
Abstract: We construct a quadratic elliptic dynamical RLL algebra with Felder $R-$ elliptic matrix and show that $L$ is a Manin matrix. Then we obtain a quasi-commutative family from minors of $L$ and construct a quantum spectral curve for the quantum elliptic $gl_n$ Gaudin model. We revise the Separation of Variables in the case of the $gl_2$ model.
Speaker: Pol Vanhaecke
Title: Real and algebraic integrability
Abstract: After a short introduction to real integrable systems, I will discuss complex integrable systems, in particular the notion of algebraic integrability. As I will explain, the theory of Abelian varieties leads to a rigorous proof of the Kowalevski criterion.