Beschreibung
BV algebras, they don't really exist
Given a differential graded Gerstenhaber algebra $(\mathcal{G},d,\cdot,\{,\})$ and a bigraded homotopy Cartan calculus for it, the choice of a Poincaré duality element enhances the Gerstenhaber algebra structure on $H(\mathcal{G})$ to a BV-algebra structure. Every BV algebra can be realized, essentially in a tautological way, as an instance of homotopy Cartan calculus with Poincaré duality. Possibly more interestingly, the dependence of the BV-laplacian on the choice of the Poincaré duality element provides a justification for the typically observed phenomenon that the BV-bracket is "more canonical" than the BV-laplacian. Classical examples of BV-algebras from homotopy Cartan calculus with Poincaré duality include the divergence of multivector fields on Riemannian manifolds and the Ginzburg-Menichi BV algebra structure on Hochschild cohomology. Possibly less known is the fact that Ševera's derivation of the BV-algebra structure on the Gerstenhaber algebra of functions on an odd symplectic manifold is an instance of homotopy Cartan calculus. No surprise, Poincaré duality elements in this setting are precisely compatible half-densities. Joint work with Eugenia Boffo.