GDO

Abstract

This is a report on an ongoing project joint with T. Pantev, M. Vaquié and G. Vezzosi.

The purpose of this talk is to present the notions of n-shifted symplectic and n-shifted Poisson structures on derived algebraic stacks and to explain their relevance for the study of moduli spaces. I will start by the notion of n-shifted symplectic structures and present some existence results insuring that they exist in many examples. In a second part I will present the notion of n-shifted Poisson structures and explain its relation with the geometry of branes (maps from spheres to a fixed target). To finish I will explain what deformation quantization is in the n-shifted context.