Cumulants, Koszul brackets and homological perturbation theory for commutative $BV_\infty$ and $IBL_\infty$ algebras

Abstract

We explore the relationship between the classical constructions of cumulants and Koszul brackets, showing that the former are an expontial version of the latter. Moreover, under some additional technical assumptions, we prove that both constructions are compatible with standard homological perturbation theory in an appropriate sense. As an application of these results, we provide new proofs for the homotopy transfer Theorem for L∞ and IBL∞ algebras based on the symmetrized tensor trick and the standard perturbation Lemma, as in the usual approach for A∞ algebras. Moreover, we prove a homotopy transfer Theorem for commutative BV∞ algebras in the sense of Kravchenko which appears to be new. Along the way, we introduce a new definition of morphism between commutative BV∞ algebras. Contents 37 3.1. IBL ∞ [1] algebras 37 3.2. Homotopy transfer for IBL ∞ [1] algebras 40 References 42

Homotopy transfer for QFT on non-compact manifold with boundary: a case study

“…Example 4.1.3 motivates us to discuss derived BV algebras. We base our discussion on [Ban20]. For the case we consider, Proposition 5.0.3 concludes that the renormalized observable complexes and the specific homotopies (renormalization) between them are objects and morphisms in the category of derived BV algebras, respectively.…”

“…In Section 5 we review the definition and homotopy transfer of derived BV algebras based on [Ban20], then apply them to our constructions in previous sections.…”

“…In order to describe it, we resort to a generalization of BV algebra called "derived BV algebra", introduced by Olga Kravchenko [Kra99] (the terminology there is "commutative BV ∞ algebra"). In [Ban20], Ruggero Bandiera showed that derived BV algebra structures can be transferred along certain kind of SDR's via homological perturbation theory. We will review this result (also simplify it a little) in this section, and conclude that constructions in previous sections fit into derived BV algebra structure (see Proposition 5.0.3 and Proposition 5.0.8).…”

“…In this subsection we collect basics to prepare for discussions of derived BV algebras. We refer to [Ban20] and references therein for details.…”

“…Inspired by [Ban20, Section 3], if ( Sym(V )[[ ]], D) is a derived BV algebra, we may call it a "dual IBL ∞ [1] algebra". We might modify Theorem 3.6 in [Ban20] and use it to prove Proposition 5.0.8 without refering to the condition (5.17). We leave this consideration for later work.…”