Higher Structure Conferences Brochier (changes)

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Quantum D-modules and topological field theories

Abstract: Given a braided tensor category C and a surface S, the formalism of factorization homology produces a category equipped with a canonical action of the braid group of S. The main goal of the talk is to explain how in the case of a punctured surface this category can be identified with the category of modules over a very explicit algebra. In the case C is the category of modules over a quantum group, this provides a uniform quantization of character varieties of punctured surfaces. Remarkably, if S is a punctured torus, one get a certain deformation of the category of D-modules over the underlying algebraic group. If S is a closed torus, one get a category of equivariant quantum D-module, closely related to the category of modules over the double affine Hecke algebra (DAHA). This construction should be thought of as the 2d part of a (at least partially defined) 4d topological field theory closely related to Witten-Reshetikhin-Turaev 3d TFT and topological Yang-Mills theory. As such, we expect this formalism to produce DAHA-modules valued knots invariant and to explain the relation between the Jones polynomial and the quantized A-polynomial. This is a joint project with David Ben-Zvi and David Jordan.