Algebra/Topology seminar Ezra Getzler

Title: The Gauss-Manin connection in noncommutative geometry and the Fedosov-Maurer-Cartan equation

 

Abstract: The Hochschild-Kostant-Rosenberg theorem motivates the introduction of a new A-infinity structure on differential forms. This leads to a deformation of the Maurer-Cartan equation to an equation that we call the Fedosov-Maurer-Cartan equation: if V is a graded vector space and A is a differential form with values in End(V) of total degree 1, the FMC equation is

u ( e^{dA/u} – 1 ) + A^2 = dA + (dA)^2/2u + … + A^2 = 0

This equation is the correct setting for the Gauss-Manin connection for periodic cyclic homology. Inspired by Tsygan’s 2007 paper on this subject, we present a surprisingly simple solution of this equation.