Symplectic field theory and quantum backgrounds

Abstract

We derive an L∞ structure associated to a polarized quantum background and characterize the obstructions to finding a versal solution to the quantum master equation (QME). We illustrate how symplectic field theory (SFT) is an example of a polarized quantum background and discuss the L∞ structure in the SFT context. The discussion may be summarized as follows: given a contact manifold M with contact homology H, one can define an L∞ algebra on A[[~]], where A = A(M) is the free symmetric algebra on the vector space of Reeb orbits of M. The obstructions to finding a versal solution to the quantum master equation in A[[~]] are organized into what we call the kappa invariant, which is a new differential κ : H[[~]] → H[[~]]. Also, a quantum background associated to an arbitrary manifold is defined which does not use any contact structure. It agrees with the one from SFT of the unit cotangent bundle of the manifold in some cases, but might, in general, be different.