Loop spaces, loop groups and loop algebras 2011
May 30 - June 3, 2011
Preliminary list of speakers :
V. Toledano Laredo
We know from the work of Drinfeld-Sokolov (and before them, Zakharov-Shabat) in the 80s that
loop and affine Lie algebras play an important rôle in the study of solitons and integrable systems.
One of the main ingredients of the theory, namely the factorization property in the corresponding loop groups, has a deep interpretation in terms
of the Riemann-Hilbert problem and some appropriate moduli spaces (G-bundles, connections, local systems). Moreover, classical and quantum loop algebras,
as their multiple generalizations (current and toroidal algebras) play a growing rôle in representation theory, and naturally appear in the geometric Langlands program.
The representation theory of these infinite dimensional Lie algebras often involves vertex operators. A mathematical framework here is the one of vertex (Borcherds) and
chirals (Beilinson-Drinfeld) algebras. A typical example is given by the chiral de Rham complex of a complex algebraic variety, introduced par Malikov-Schechtman-Vaintrob,
which has a natural interpretation in terms of formal loops à la Kapranov-Vasserot. This chiral de Rham complex seems to be a serious candidate for a mathematical formulation
of some mirror symmetry phenomenons (Kapustin), as well as its conjectured relations with the geometric Langlands program (Frenkel et al.).
Finally, the topology of loop spaces recently encountered a renewal of interest thanks to the work of Chas-Sullivan in 2000, that emphazises the rich algebraic structures
one can get on the homology of the loop space of an oriented smooth manifold (as well as on its quotient by the action of the circle). More precisely, the homology of a loop space
naturally inherits a BV algebra structure, that one also encounters on the cohomology of a vertex algebra. Actually, it is the structure of a homological conformal field theory that one obtains.
There are important relations between this ''string topology'' and various field theories, notably symplectic field theory and Floer homology. E.g.~it is known that the homology of the loop space
of manifold is isomorphic as a BV algebra to the Floer homology of its cotangent bundle. To make these relations precise one often needs to work with strong homotopy algebraic structures on the chain level,
similarly to what happens with Deligne's conjecture for Hochschild cohomology. Here, homotopy and derived algebraic geometry (Töen, Lurie et al.) provides interesting perspectives:
homotopically, Hochschild homology of the structure sheaf of a variety is the structure sheaf of its loop space.
The aim of this conference is to bring together various people studying loops form different perspectives and aspects (algebra, geometry, topology). We will try to shed some light on
the richness of these ''extendeed objects'' (loops, strings, but also branes) that seem to be of growing importance in geometry (new invariants, geometric Langlands program), algebra
(higher structures, representation theory), or mathematical physics (field theories, mirror symmetry).