Homotopy Transfer and Effective Field Theory I: Tree‐level

Abstract

We use the dictionary between general field theories and strongly homotopy algebras to provide an algebraic formulation of the procedure of integrating out of degrees of freedom in terms of homotopy transfer. This includes more general effective theories in which some massive modes are kept while other modes of a comparable mass scale are integrated out, as first explored by Sen in the context of closed string field theory. We treat L∞$L_\infty$‐algebras both in terms of a nilpotent coderivation and, on the dual space, in terms of a nilpotent derivation (corresponding to the BRST charge of the field theory) and provide explicit formulas for homotopy transfer. These are then shown to govern the integrating out of degrees of freedom at tree level, while the generalization to loop level will be explored in a sequel to this paper.

Homological quantum mechanics

“…[16]. (See also [17][18][19][20] for the L ∞ formulation of effective field theory in terms of homotopy transfer.) Given that ∂ 2 = 0 we can consider the cohomology of V : the space of ∂-closed vectors modulo ∂-exact vectors.…”

New Objects in Scattering Theory with Symmetries

“…We can now write down the central object of homotopy transfer -the induced differential (see, for example, [3] and [1])…”

“…Remark. This means that in what follows R will not be an arbitrary positive number, but an arbitrary multiple of 1 κ . Any level with positive n is double-degenerate, so ker H 0,E has dimension four.…”

Field theory equivalences as spans of L ∞ -algebras

“…If the latter space is embedded in the former, then a homotopy transfer amounts to integrating out fields, a well-known fact in BV quantisation 3 . For a recent discussion and application of this fact, see [29,30].…”