Can D-Branes Wrap Nonrepresentable Cycles?

TL;DR

This paper investigates whether D-branes can wrap nonrepresentable cycles in string theory compactifications, providing examples where such branes carry K-theory charges despite the cycles' nonrepresentability.

Contribution

It presents the first explicit examples of Freed-Witten anomaly-free D6-branes wrapping nonrepresentable cycles in specific string theory backgrounds.

Findings

One example D6-brane carries a K-theory charge.

Another example D6-brane does not carry a K-theory charge.

Nonrepresentable cycles can be wrapped by D-branes with anomaly cancellation.

Abstract

Sometimes a homology cycle of a nonsingular compactification manifold cannot be represented by a nonsingular submanifold. We want to know whether such nonrepresentable cycles can be wrapped by D-branes. A brane wrapping a representable cycle carries a K-theory charge if and only if its Freed-Witten anomaly vanishes. However some K-theory charges are only carried by branes that wrap nonrepresentable cycles. We provide two examples of Freed-Witten anomaly-free D6-branes wrapping nonrepresentable cycles in the presence of a trivial NS 3-form flux. The first occurs in type IIA string theory compactified on the Sp(2) group manifold and the second in IIA on a product of lens spaces. We find that the first D6-brane carries a K-theory charge while the second does not.