Abstract

A TQFT in terms of general gauge fixing functions is discussed. In a covariant gauge it yields the Donaldson-Witten TQFT. The theory is formulated on a generalized phase space where a simplectic structure is introduced. The Hamiltonian is expressed as the anticommutator of off-shell nilpotent BRST and anti-BRST charges. Following the original ideas of Witten a time reversal operation and the corresponding inner product are defined. We present a non-covariant gauge fixing that gives rise to a manifestly time reversal invariant Lagrangian and a positive definite Hamiltonian, with the previously introduced inner product. As a consequence, the indefiniteness problem of some of the kinetic terms of the Witten's action is resolved. The construction allows then a consistent interpretation of Floer groups in terms of the cohomology of the BRST charge which is explicitly independent of the background metric. The relation between the BRST cohomology and the ground states of the Hamiltonian is then completely established. The topological theories arising from the covariant, Donaldson-Witten, and non-covariant gauge fixing are shown to be quantum equivalent by using the operatorial approach.