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 Ševera. 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

References in corpus (20)

• Introduction to string field theory
• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• Perturbative quantum gauge theories on manifolds with boundary
• Classical BV theories on manifolds with boundary
• Remarks on Chern-Simons invariants
• On the origin of the BV operator on odd symplectic supermanifolds
• Modular operads and Batalin-Vilkovisky geometry
• Quantum $L_{\infty }$ Algebras and the Homological Perturbation Lemma
• Effective Batalin--Vilkovisky theories, equivariant configuration spaces and cyclic chains
• Symplectic Microgeometry I: Micromorphisms
• Feynman diagrams and minimal models for operadic algebras
• Shifted Coisotropic Correspondences
• "Nonlinear pullbacks" of functions and $L_{\infty }$-morphisms for homotopy Poisson structures
• Batalin-Vilkovisky Integrals in Finite Dimensions
• Abstract Hodge decomposition and minimal models for cyclic algebras
• A Graphical Calculus for Lagrangian Relations
• Minimal models of quantum homotopy Lie algebras via the BV-formalism
• Solving the noncommutative Batalin-Vilkovisky equation
• Topological defects as lagrangian correspondences
• Decomposition of (co)isotropic relations