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## Integrable Systems in Quantum Theory |

Speaker: Title: 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: Title: 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: Title: 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: Title: 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: Title: 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 considered. ------ Speaker: Title: 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: Title: 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. [Back] |