Generalized Cohomological Field Theories in the Higher Order Formalism

In the classical Batalin–Vilkovisky formalism, the BV operator Δ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by Δ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin–Vilkovisky algebras where the operator Δ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever Δ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a “trinity” of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s.