Dylan Butson, Pure Mathematics, University of Waterloo
"Factorization Algebras from Quantum Field Theory"
I will survey the construction of factorization algebras, a type of algebraic-topological data on a space, using methods from quantum field theory, following the work of Kevin Costello and Owen Gwilliam.
First, I will discuss how L-infinity algebras, a model for homotopy Lie algebras, encode the derived geometry of the formal neighbourhood of a point in a quotient space, and discuss symplectic structures on such formal derived spaces.
Next, I will explain how a classical field theory can be encoded perturbatively as a presheaf of formal derived spaces with 'local' symplectic structures, with corresponding precosheaf of homotopy Poisson algebras of functions on these spaces.
Finally, I will sketch an argument that deformation quantization of precosheaves of homotopy Poisson algebras into precosheaves of Beilinson-Drinfeld algebras, together with locality of the Poisson brackets, leads naturally to the structure of a factorization algebra on the cohomology precosheaf.
The main example throughout will be classical Chern-Simons theory with a fixed gauge group G, which corresponds mathematically to the presheaf of formal neighbourhoods of the trivial bundles in the moduli spaces of G bundles on the input open sets. The quantization of this in the way described above yields a factorization algebra which can be identified with the data of the quantum group U_q(g) as a quasi-triangular Hopf algebra, or equivalently the data of its category of representations as a braided monoidal category.
The amount of this material that I will attempt to cover will depend on the audience's background.