Three lectures on classical integrable systems and gauge field theories

In these lectures I consider the Hitchin integrable systems and their relations with the self-duality equations and the twisted super-symmetric Yang-Mills theory in four dimension follow Hitchin and Kapustin-Witten. I define the Symplectic Hecke correspondence between different integrable systems. As an example I consider Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL(N,C) and explain the Symplectic Hecke correspondence for these systems.Comment: 36 pages, Lectures given at Advanced Summer School on Integrable Systems and Quantum Symmetries (Prague, June, 2007

Branes: from free fields to general backgrounds

Motivated by recent developments in string theory, we study the structure of boundary conditions in arbitrary conformal field theories. A boundary condition is specified by two types of data: first, a consistent collection of reflection coefficients for bulk fields on the disk; and second, a choice of an automorphism $\omega$ of the fusion rules that preserves conformal weights. Non-trivial automorphisms $\omega$ correspond to D-brane configurations for arbitrary conformal field theories. The choice of the fusion rule automorphism $\omega$ amounts to fixing the dimension and certain global topological features of the D-brane world volume and the background gauge field on it. We present evidence that for fixed choice of $\omega$ the boundary conditions are classified as the irreducible representations of some commutative associative algebra, a generalization of the fusion rule algebra. Each of these irreducible representations corresponds to a choice of the moduli for the world volume of the D-brane and the moduli of the flat connection on it.Comment: 56 pages, LaTeX2e. Typos corrected; two references adde