Noncommutative differential operators, unparametrized paths and Hodge structures

Noncommutative differential operators, unparametrized paths and Hodge structures

-
Mikhail Kapranov, Yale University
417 Mathematics Hall, Columbia

A noncommutative differential operator (NCDO) on a manifold X is a compatible system of linear differential operators acting in all vector bundles with connections on X. The ring of such operators can be seen as a highly noncommutative version of the mildly noncommutative ring of usual differential operators: the partial derivatives are replaced by formal covariant derivatives which no longer commute and account for the curvature data.The talk will explain the relation of NCDO with the space of formal germs of unparametrized paths. In particular, we will make precise the statement that a connection is uniquely defined, up to a formal germ of isomorphism, by all the higher covariant derivatives of the curvature evaluated at one point. This relation allows us to give a "gauge-theoretic" description of the category of mixed Hodge structures.