Daniel Mathews (Nantes)

Sutured Floer homology and contact-topological quantum field theory

We consider the topological quantum field theory properties of sutured
Floer homology, as introduced by Honda--Kazez--Matic. We present
several results in the ``dimensionally reduced" case of product
manifolds. The SFH of such manifolds reduces to that of solid tori,
and forms a ``categorification of Pascal's triangle". Contact
structures correspond to chord diagrams, and contact elements form
distinguished subsets of SFH of order given by the Catalan numbers. We
find natural ``creation and annihilation operators'' which allow us to
define a QFT-type basis of SFH, consisting of contact elements. In
fact sutured Floer homology in this case reduces to the combinatorics
of chord diagrams, and in a sense which can be made precise, is the
``quantum field theory of two non-commuting particles". The details of
this description have intrinsic contact-topological meaning, allowing
us for instance to compute certain contact categories, and to give a
``contact geometry free" proof that the contact element of a contact
structure with torsion is zero.