Definition of algebraic geometry on quantum curves. (Définition de la géométrie algébrique sur les courbes quantiques)
Bertrand Eynard
IPhT
Lundi 28/09/2009, 11:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers

We shall introduce the definition of quantum algebraic geometry''.
We shall define the notions of genus, cycles, meromorphic forms of 1st kind, 2nd kind, 3rd kind, branchpoints, Bergman kernel, Riemann bilinear identities, Rauch formula, symplectic invariants $F_g$,...etc, on quantum curves.
A classical algebraic curve is given by a polynomial equation $0=P(x,y)=\sum_{i,j} P_{i,j} x^i y^j$.
A quantum curve (D-module) is given by a non-commutative polynomial $\sum_{i,j} P_{i,j} x^i y^j$, where $y$ and $x$ don't commute: $[y,x]=\hbar$.
The definition of algebro-geometric quantities on quantum curves, is suggested from non-hermitian matrix models (beta ensembles).
We shall show, that almost all relations existing in classical geometry, continue to hold for quantum curves.
A remarkable point, is that the Bethe ansatz plays a crucial role in the very definition of all those geometric quantities. This provides a natural geometric interpretation for the Bethe ansatz.

This new quantum algebraic geometry'' can be expected to have applications to integrable systems, string theory, enumerative geometry, matrix models, and maybe more. \\ \\ \\(The seminar will be in english if there are non-french speaking people.)

Contact : Vincent PASQUIER