Lagrangian Relations and Quantum $L_{\infty }$ Algebras

Abstract

Quantum $L_{\infty }$ algebras are higher loop generalizations of cyclic $L_{\infty }$ algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of $(-1)$-shifted symplectic vector spaces and distributional half-densities, originally proposed by Å evera. Morphisms in this category can be given both by formal half-densities and Lagrangian relations; we prove that the composition of such morphisms recovers the construction of homotopy transfer of quantum $L_{\infty }$ algebras. Finally, using this category, we propose a new notion of a relation between quantum $L_{\infty }$ algebras.

43 pages; v4 - version accepted for publication; added some missing relevant references, changed the convention for degree shift, added minor clarifications including Remarks 4.8, 4.19