Quantum differential operators, and the torus T2
David Jordan (University of Edinburgh)
Wednesday 2nd October, 2013 16:00-17:00 Maths 204
The algebra Dq(G) is a q-deformation of the algebra D(G) of differential operators on a semi-simple algebraic group. In this talk, I will explain an intimate relationship between Dq(G) and the torus T^2: namely, Dq(G) carries an action by algebra automorphisms of the torus mapping class group SL2(Z), and also yields representations of the torus braid group extending the well-known action of the planar braid group on tensor powers of quantum group representations. Finally, the so-called Hamiltonian reduction of Dq(G) quantizes the moduli space Loc_G(T^2) of G-local systems on T^2, or equivalently, homomorphisms π_1(T^2)→G,and this observation allows us to generalize the construction of D_q(G) to quantize Loc_G(Σ_g,r), for an arbitrary surface with genus g and r punctures.
Time permitting, I will outline work in progress with David Ben-Zvi and Adrien Brochier putting all of the above into the context of topological field theories.