Symplectic Homology as Hochschild Homology

In the wake of Donaldson’s pioneering work [6], Picard-Lefschetz theory has been extended from its original context in algebraic geometry to (a very large class of) symplectic manifolds. Informally speaking, one can view the theory as analogous to Kirby calculus: one of its basic insights is that one can give a (non-unique) presentation of a symplectic manifold, in terms of a symplectic hypersurface and a collection of Lagrangian spheres (vanishing cycles) in it. This is particularly impressive in the four-dimensional case, since the resulting data are easy to encode combinatorially; but the formalism works just as well in higher dimensions. These kinds of presentations are instructive and useful in some respects, but hard to work with in others. For instance, it is not obvious how to recover the known symplectic invariants, such as Gromov-Witten invariants, from vanishing cycle data. In these notes, we ask a simpler version of this question, regarding one of the basic invariants of symplectic manifolds with boundary, namely symplectic homology as defined by Viterbo [17] (a closely related construction is due to Cieliebak-Floer-Hofer [4]).