Algebraic Geometry Seminar: Theo Johnson-Freyd

AKSZ Theory is a topological version of the Sigma Model in quantum field theory, and includes many of the most important topological field theories.  I will present two generalizations of the usual AKSZ construction.  The first is closely related to the generalization from symplectic to Poisson geometry.  (AKSZ theory has already incorporated an analogous step from the geometry of cotangent bundles to the geometry of symplectic manifolds.)  The second generalization is to phrase the construction in an algebrotopological language (rather than the usual language of infinite-dimensional smooth manifolds), which allows in particular for lattice versions of the theory to be proposed.  From this new point of view, renormalization theory is easily recognized as the way one constructs strongly homotopy algebraic objects when their strict versions are unavailable.  Time permitting, I will end by discussing an application of lattice Poisson AKSZ theory to the deformation quantization problem for Poisson manifolds: a _one_-dimensional version of the theory leads to a universal star-product in which all coefficients are rational numbers.