Symplectic/Contact Geometry VII at Les Diablerets, Day 1

(Partly joint work with Ivan Smith) See arXiv:1106.3975 and arXiv:1201.5880.

Faced with the monumental task of introducing (wrapped and unwrapped)
Floer homology, symplectic homology and Fukaya categories as well as
telling us about his theorem (all in the final hour before dinner),
Alex rose to the challenge with a beautiful set of highly illustrated
beamer slides.

The spaces Alex was talking about are the total spaces of line bundles
over symplectic manifolds (mostly \(\mathbf{CP}^n\)) such that the first
Chern class is a negative multiple of the cohomology class of the
symplectic form. For instance, \(\mathcal{O}(-1)\) over
\(\mathbf{CP}^1\). These are noncompact symplectic manifolds with
symplectic fibres and symplectic base and they are convex in the sense
that a sequence of holomorphic curves cannot escape to infinity. The
first theorem he proved was that the symplectic homology of such a
space is a quotient of the quantum homology (symplectic homology is a
Floer theory counting periodic orbits of a Hamiltonian which gets very
big quite quickly in the noncompact end of the manifold, quantum
homology just counts compact holomorphic spheres!). In particular you
quotient by the kernel of a certain map: the quantum cup product with
a high power of the first Chern class. When there are no spheres with
positive symplectic area in the base (and hence in the total space)
the quantum and classical cup products agree and hence a sufficiently
large power of the first Chern class vanishes, to the whole quantum
cohomology is in its kernel, which recovers an older result of Oancea
(that assuming there are no spheres with positive symplectic area the
symplectic homology vanishes). The idea of the proof was the
following: symplectic homology is defined as a limit of Floer
homologies for a sequence of Hamitonians. For a suitable choice of
these Hamiltonians you can ensure that each of these Floer homologies
is isomorphic to quantum cohomology (roughly speaking you rotate the
fibre in such a way as to ensure that all closed orbits lie in the
zero section) and the maps in the sequence are precisely
multiplications by the first Chern class.

Not satisfied with this, Alex raised the stakes algebraically and
introduced the “open-closed string map” (one of the more complicated,
though increasingly central, aspects of the Fukaya/Floer story). This
is a map from the Hochschild cohomology of the Fukaya category to the
quantum homology. I think (hope) I'm right in saying the
following. For a single Lagrangian \(L\) it takes a collection of cycles
(“inputs” – the Hochschild cohomology having as its $n$th chain group
the tensor product of \(n\) copies of the Floer chain group of \(L\)) to
an ambient cycle. The ambient cycle is traced out by a marked point on
a holomorphic disc with as many marked points as there are inputs
where each point point is required to be mapped to the corresponding
cycle. Mad. And then he raised the stakes yet more by introducing the
analogous map from the wrapped Fukaya category to the symplectic
homology.

Why? Well Abouzaid recently proved a criterion for when a Lagrangian
(or collection of Lagrangians) split-generates the Fukaya category (or
some part of it) by looking at the image of this open-closed string
map. Ritter and Smith have adapted this to the monotone setting they
need for these negative line bundles. Using this criterion (namely
that the image should contain some invertible element) they prove that
you only need a single Lagrangian to split-generate the wrapped Fukaya
category of a negative line bundle over \(\mathbf{CP}^n\) (for suitably
low Chern class of the line bundle). The Lagrangian in question is the
circle bundle living over the Clifford torus (which, when taken with
various flat connections, generates all the various parts of the
Fukaya category). In particular, the wrapped category is generated by
a compact Lagrangian, so this proves that all the potential
infinite-dimensionality introduced by wrapped Floer cohomology is
actually only finite-dimensional (in the same way that the symplectic
cohomology reduced to a quotient of quantum cohomology).

At this point, Alex brought the discussion back down to earth by
discussing the equator in the sphere (one of the most instructive
Lagrangian submanifolds, well worth your contemplation). Then we went
for tea.