While there are several formalisms for quantum field theory, a rigorous mathematical description is still not available. In some areas, as for example in conformal field theory, one is better equipped and proper definitions like vertex algebras can be made.

By introducing the concept of a chiral algebra on an algebraic curve, Beilinson and Drinfeld have provided a reformulation of vertex algebras that centers around the similarity between the operator product expansion and the Jacobi identity for Lie algebras. Concretely, a chiral algebra is a Lie algebra in the category of D-modules. While for vertex algebras a generalisation to algebraic varieties of higher dimension seems very difficult, higher chiral algebras can be defined within the theory of quasi-categories. Via a polysimplicial model of the derived sections on configuration spaces, in joint work with Zhengping Gui and Charles Young, we have shown that the fundamental example reassembles a strong homotopy Lie algebra.

Lurie introduced a construction in the topological setting that links En-algebras to factorisation algebras, the Koszul dual of chiral algebras. Together with Keyou Zeng, our aim is to unify both pictures with the purpose of describing mixed topological/holomorphic theories in analogy to the notions of raviolo vertex algebras and configuration spaces as described in Alfonsi, Kim, Young.