Topology
and Geometry Seminar
Speaker : Dr.
Zhe Sun, Tsinghua University
Time: 3:30-5:30pm Wednesday,
Nov 9, 2016
Place: X2415, Xipu campus SWJTU
Title: Quantization
of rank n swapping algebra for the PSL(n,R) Hitchin component (I)
Abstract:
Poisson algebra of the moduli space M of flat G-connections
on a Riemann surface S is used to formulate a quantization, addressing problems
in mapping class group representations, three dimensional topological
invariants, conformal field theory, topological quantum field theory,
Chern-Simon theory. In this talk, I will explain Atiyah-Bott-Goldman(ABG)
symplectic structure on M and its relation with lattice gauge theory,
Poisson-Lie group structure. The swapping algebra, introduced by F. Labourie
via pairs of points on a circle, gives a Poisson algebra on the Hitchin
component of M relevant to the ABG symplectic structure. I study its rank n
version by taking a quotient, which I call it rank n swapping algebra. Then I
Poisson embed the Fock-Goncharov coordinates(generalized Thurston shear
coordinates) for cluster X_{PSL(n,R),S} moduli space into the fraction algebra
of the rank n swapping algebra. Then I will give a quantization of the rank n
swapping algebra, which is the first step to generalize the Kashaev's 3D
invariant for the deformation parameter q at root of unit, Anderson-Kashaev's
TQFT for the deformation parameter not at root of unit.
