1307.3991 (Yasha Savelyev)

Global Fukaya Category    [PDF]

Yasha Savelyev

We construct an analogue of the Fukaya category of a symplectic manifold, for smooth Hamiltonian fibrations $M \hookrightarrow P \to X$. This is the global Fukaya category of $P$. This is an invariant of $P$ with a full local to global (in $X$) reconstruction principle, which at the same time recovers (at least formally) other known invariants of Hamiltonian fibrations, like the quantum characteristic classes, or Seidel representation when $X= S ^{2}$, and possibly Hutchings' family Floer homology. By projectivizing the complexified tangent bundle we obtain a new Floer theoretic invariant of a smooth manifold, with full locality. Reconstruction of these "classical" invariants, relies on a certain convergence of formalism combining Toen's "derived Morita theory", with the theory of quasi-categories particularly after Lurie, combined with his dg-nerve construction further developed by Hiro Tanaka in his thesis in the $A_{\infty} $-context. As a consequence symplectic geometry will motivate a purely algebraic prediction in Toen's theory. One of the author's main personal motivations for developing this construction, is a certain rigidity conjecture in Hofer geometry, which we review here.

View original: http://arxiv.org/abs/1307.3991