Algebra and Geometry Seminar

Consider the cosphere bundle of a manifold with the standard contact structure and Lagrangian cobordisms in its symplectization between Legendrian submanifolds. We construct a fully faithful functor between the categories of sheaves with singular support on the corresponding Legendrians, after enhancing the category at the negative end by local systems on the cobordism. In the special case where the contact manifold is the 1-jet bundle, we give a geometric model of the enhanced category as sheaves on the original manifold times the real line with microsupport on the cobordism. This gives a sheaf theoretic description of the maps between Legendrian contact homologies with coefficients enhanced by chains of the based loop spaces, defined using pseudoholomorphic curves.