Secondary string topology coproduct: geometric and algebraic constructions

I will describe a geometric chain level construction of a secondary coproduct operation on a suitable chain model for the free loop space of a manifold using the theory of De Rham chains. Such coproduct was described at the level of homology by Goresky and Higston using different methods. It is secondary in the sense that arises from a "1-parameter family of chain level intersections". The chain level theory around this secondary operation is useful for describing certain phenomena in symplectic topology and symplectic field theory. There are analogues of these geometric operations in the algebraic theory of Hochschild complexes of Frobenius algebras described in work by Wahl, Abbaspour, Tradler, Zeinalian, and others. I will discuss a new way of nterpreting both the algebraic loop product and the algebraic secondary coproduct in a single package. This is work in progress with Dingyu Yang (IMJ-PRG) and Zhengfeng Wang (IMJ-PRG).