D-branes and topological field theories

In the presence of a D-brane a string theory develops a new subsector. We show that for curved D-branes the corresponding sector is a (partially twisted) topological field theory. We use this result to compute the degeneracy of 2-branes wrapped around K3 cycles as well as 3-branes wrapped around CY three-fold vanishing 3-cycles. in both cases we find the degeneracy is in accord with expectation. The counting of BPS states of a gas of 0-branes in the presence of a 4-brane in K3 is considered and it is noted that the effective 0-brane charge is shifted by 1, due to a quantum correction. This is in accord with string duality and the fact that the left-moving ground-state energy of the heterotic string starts at -1. We also show that all the three different topological twistings of four-dimensional N = 4 Yang-Mills theory do arise from curved 3-branes embedded in different spaces (Calabi-Yau manifolds and manifolds with exceptional holonomy groups).