  We show that the BRST/anti-BRST invariant 3+1 dimensional 2-form gauge theory
has further nilpotent symmetries (dual BRST /anti-dual BRST) that leave the
gauge fixing term invariant. The generator for the dual BRST symmetry is
analogous to the co-exterior derivative of differential geometry. There exists
a bosonic symmetry which keeps the ghost terms invariant and it turns out to be
the analogue of the Laplacian operator. The Hodge duality operation is shown to
correspond to a discrete symmetry in the theory. The generators of all these
continuous symmetries are shown to obey the algebra of the de Rham cohomology
operators of differential geometry. We derive the extended BRST algebra
constituted by six conserved charges and discuss the Hodge decomposition
theorem in the quantum Hilbert space of states.
