Chord diagrams, contact-topological quantum field theory, and contact categories

We consider contact elements in the sutured Floer homology of solid
tori with longitudinal sutures, as part of the (1+1)-dimensional
topological quantum field theory defined by Honda--Kazez--Matic. The
$SFH$ of these solid tori forms a ``categorification of Pascal's
triangle'', and contact structures correspond bijectively to chord
diagrams, or sets of disjoint properly embedded arcs in the disc;
contact elements form distinguished subsets of $SFH$ of order given by
the Narayana numbers. We find natural ``creation and annihilation
operators'' which allow us to define a QFT-type basis of each $SFH$
vector space, consisting of contact elements. Sutured Floer homology
in this case reduces to the combinatorics of chord diagrams. We prove
that contact elements are in bijective correspondence with comparable
pairs of basis elements with respect to a certain partial order, and
in a natural and explicit way. The algebraic and combinatorial
structures in this description have intrinsic contact-topological
meaning, related to the contact category of a disc. This leads us to
extend Honda's notion of contact category to a ``bounded'' contact
category. We compute this bounded contact category in certain cases.
We also use the algebraic structures arising among contact elements to
extend the notion of contact category to a 2-category.