Generalized cohomological field theories in the higher order formalism

TL;DR

This paper extends the classical BV formalism to higher order operators, revealing new algebraic structures on homology and connecting them to a broader mathematical 'trinity' in the context of cohomological field theories.

Contribution

It introduces a generalized BV algebra framework with finite order operators and uncovers new algebraic structures on homology, expanding the understanding of cohomological field theories.

Findings

Homotopically trivial higher order BV operators induce new algebraic structures on homology.

The sequence of algebraic structures forms a 'trinity' aligned with Arnold's mathematical philosophy.

The work generalizes the classical BV formalism to higher order operators, revealing richer algebraic and geometric insights.

Abstract

In the classical Batalin--Vilkovisky formalism, the BV operator $\Delta$ 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 $\Delta$ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin--Vilkovisky algebras where the operator $\Delta$ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever $\Delta$ 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.