Symplectic and Poisson Geometry Seminar: Nikolay Shesko

Abstract: Symplectic reduction is a procedure of reducing the phase space of a physical system with symmetries by passing to the quotient of a level set of the moment map. Classically this requires the level set to be a smooth submanifold on which the group acts freely. I will describe a derived version of symplectic reduction that removes both assumptions, working at a singular value of the moment map with arbitrary stabilizers. The symplectic quotient is modeled by the action dg-groupoid over the derived zero locus of the moment map, following Carchedi and Behrend–Liao–Xu. I will construct the reduced symplectic form inside the triple Bott–Shulman–Stasheff complex of this dg-groupoid and show that it is 0-shifted symplectic. If time permits, I will discuss the relation to the classical BRST complex and to recent work of Cueca, Dorsch, Sjamaar and Zhu.