  We propose a new partially topological theory in three dimensions which
couples Chern-Simons theory to matter. The 3-manifolds needed for this
construction admit transverse holomorphic foliation (THF). The theory depends
only on the choice of such a structure, but not on a choice of metric and in
this sense, it is topological. We argue that this theory arises in topological
A-model string theory on Lagrangian 3-branes in the presence of additional
parallel coisotropic 5-branes. The theory obtained in this way is equivalent to
an N=2 supersymmetric Chern-Simons matter theory on the same 3-manifold, which
also only depends on the THF structure. The theory is a realization of a
topological theory of class H, which allows splitting of a temporal direction
from spatial directions. We briefly discuss potential condensed matter
applications.
