Geometric Structures Laboratory

In this talk (in two parts), I will introduce derived algebraic geometry and the theory of shifted symplectic and Poisson structures. Such structures appear naturally on a broad class of moduli stacks, in particular, shifted symplectic structures have a really important role in the enumerative geometry of Donaldson–Thomas invariants of Calabi–Yau 3-folds. It is believed that these invariants are sort of (-1)-shifted deformation quantizations of the stack of perfect sheaves on a Calabi–Yau 3-fold.

Shifted Poisson structures are a generalization of shifted symplectic structures. Together with formality, knowing existence of those structures give automatically higher version of deformation quantization on stacks. It was conjectured that most existence theorems for shifted symplectic structures also hold for shifted Poisson structures. I will present my recent work on the so called AKSZ theorem for shifted Poisson structures, asserting existence of shifted Poisson structures on a lot of interesting moduli stacks. If time permits, I will also talk about relationships between derived geometry and BV formalism.

This is the first in a series of two talks. The talk will be about 50 minutes. The remainder of the time is meant for questions/discussion.