D-Branes and Spin^c Structures

TL;DR

The paper analyzes the Spin^c requirement for D-branes wrapping submanifolds in type II string theory and shows that, when the cosmological constant vanishes, the normal bundle to any supersymmetric (calibrated) cycle automatically supports a Spin^c structure. Using the relation $w_1(N)=0$ and $w_2(N)=w_2(TP)$ and the fact that calibrated submanifold tangents $TP$ are Spin^c in CY, G2, and Spin(7) settings, the authors conclude that $N$ is Spin^c. They provide case-by-case results: complex submanifolds of Calabi–Yau, special Lagrangian submanifolds in CY3 and CY4, associative and coassociative submanifolds in G2, and Cayley submanifolds in Spin(7); in many instances the normal bundle is even trivial. The work implies that the Spin^c constraint is automatically satisfied for these compactifications, while noting that in AdS backgrounds the constraint remains potentially nontrivial and thus more interesting.

Abstract

It was recently pointed out by E. Witten that for a D-brane to consistently wrap a submanifold of some manifold, the normal bundle must admit a Spin^c structure. We examine this constraint in the case of type II string compactifications with vanishing cosmological constant, and argue that in all such cases, the normal bundle to a supersymmetric cycle is automatically Spin^c.