This entry describes classes of examples of A-∞ category-valued FQFTs defined on a version of the symplectic category.
Let be a compact symplectic manifold. At least in good cases to this is associated a Fukaya category of Lagrangian submanifolds and an enlarged version .
Write for the symplectiv manifold .
Now if for are two Lagrangian submanifolds and a Lagrangian correspondence then we get an A-∞ functor
For and Lagrangian submanifolds, assuming monotonicity and Maslov conditions we have an -homotopy
where on the right we have a natural notion of composition of Lagrangian submanifolds.
(Wehrheim-Woodward 07, section 3.2)
This is the symplectic version of Mukai functors?.
For and a compact Riemann surfaces and
their moduli spaces of fixed determinant rank -bundles, and for a cobordism (compact, oriented) from to then consider
If is elementary in that there exists a Morse function with critical points then is a Lagrangian correspondence.
defines a 2+1-dimensional FQFT for connected cobordisms with values in A-∞ categories.
This is supposed to be the 2+1-dimensional part of Donaldson theory.
Other theories that fit into this framework:
symplectic Khovanov theory? (Seidel-Smith and Rezazodegab)
elementary cobordisms vanishing cycle
Write for a symplectic manifold with its symplectic form reversed.
For two symplectic manifolds, a Lagrangian correspondence is a Lagrangian submanifold of , that is
where are the two projections out of the product.
The composition of two Lagrangian submanifolds is
which is a Lagrangian correspondence in if everything is suitably smoothly embedded by .
For a symplectomorphism we have
is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.
Let be a manifold, the unitary group, a -principal bundle and a -bundle with connection.
Then there is the moduli space of connections on with central curvature and given determinant.
For example if has genus then
Let be a cobordism from to with extension
is a Lagrangian correspondence if is sufficiently simple. Further assuming this we have for composition that