  On symplectic manifolds, we introduce a Morse-type complex with elements
generated by pairs of critical points of a Morse function. The differential of
the complex consists of gradient flows and an integration of the symplectic
structure over spaces of gradient flow lines. Using the Witten deformation
method, we prove that the cohomology of this complex is independent of both the
Riemannian metric and the Morse function used to define the complex and is in
fact isomorphic to the cohomology of differential forms of Tsai, Tseng and Yau
(TTY). We also obtain Morse-type inequalities that bound the dimensions of the
TTY cohomologies by the number of Morse critical points and the interaction of
symplectic structure with the critical points.
