Multi-oriented props and homotopy algebras with branes

Abstract

We introduce a new category of differential graded multi-oriented props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of k linear subspaces in that space, k being the number of extra orientations (if $$k=0$$ this structure recovers an ordinary prop); symplectic vector spaces equipped with k Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context of this paper, Manin triples are precisely symplectic Lagrangian representations of the 2-oriented generalization of the classical operad of Lie algebras. In a sense, the theory of multi-oriented props provides us with a far reaching strong homotopy generalization of Manin triples type constructions. The homotopy theory of multi-oriented props can be quite non-trivial (and different from that of ordinary props). The famous Grothendieck–Teichmuller group acts faithfully as homotopy non-trivial automorphisms on infinitely many multi-oriented props, a fact which motivated much the present work as it gives us a hint to a non-trivial deformation quantization theory in every geometric dimension $$d\ge 4$$ generalizing to higher dimensions Drinfeld–Etingof–Kazhdan’s quantizations of Lie bialgebras (the case $$d=3$$) and Kontsevich’s quantizations of Poisson structures (the case $$d=2$$).