D-Branes and Topological Field Theories

TL;DR

This paper demonstrates that the sector of string theory with curved D-branes forms a topological field theory, enabling the computation of BPS state degeneracies and revealing connections between topological twists and brane embeddings.

Contribution

It establishes that curved D-branes induce a topological field theory sector and links topological twistings of 4D N=4 Yang-Mills to brane embeddings in special manifolds.

Findings

Degeneracy of 2-branes on K3 cycles matches expectations.

Degeneracy of 3-branes on CY threefolds matches expectations.

Effective 0-brane charge is shifted by 1 due to quantum correction.

Abstract

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 threefold 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 left-moving ground state energy of 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 three-branes embedded in different spaces (Calabi-Yau manifolds and manifolds with exceptional holonomy groups).