Topology Seminar

Speaker: Leonid Chekhov (Steklov Math Inst, Moscow, and Niels Bohr Inst., Copenhagen)

Title: Quantum cluster algebras and geometry of string worldsheet

We describe the Teichmuller space $T_{g,s,n}$ of Riemann surfaces of genus g, s holes and n bordered cusps on boundaries of holes in the Poincare uniformization of constant negative curvature. The description uses structures on fat graphs, observables are geodesic functions and we derive Poisson and quantum structures on sets of observables relating them to quantum cluster algebras of Berenstein and Zelevinsky. We obtain MCG-invariant quantum Ptolemy transformation and quantum skein relations. We propose to associate coordinates of  $T_{g,s,n}$ (the Thurston shear coordinates and the Penner lambda-lengths) with the coordinates of a string worldsheet in which bordered cusps naturally correspond to open string asymptotic states. (joint with M.Mazzocco, Loughborough Univ., UK)