(Fractional) Intersection Numbers, Tadpoles and Anomalies

TL;DR

This paper investigates tadpole charges and anomaly cancellation in string theory compactifications using the Witten index, revealing the importance of the gauge bundle's relation to the spin bundle and fractional intersection numbers.

Contribution

It demonstrates the connection between tadpole constraints, gauge bundles, and the spin bundle, providing solutions for noncompact orbifolds and clarifying anomaly cancellation mechanisms.

Findings

Standard embedding often fails tadpole conditions in fourfolds

Gauge bundle should be closely related to the spin bundle

Fractional intersection numbers explain anomaly coefficients

Abstract

We use the Witten index in the open string sector to determine tadpole charges of orientifold planes and D-branes. As specific examples we consider type I compactifications on Calabi Yau manifolds and noncompact orbifolds. The tadpole constraints suggest that the standard embedding is not a natural choice for the gauge bundle. Rather there should be a close connection of the gauge bundle and the spin bundle. In the case of a four fold, the standard embedding does not in general fulfill the tadpole conditions. We show that this agrees with the Green-Schwarz mechanism. In the case of noncompact orbifolds we are able to solve the tadpole constraints with a gauge bundle, which is related to the spin bundle. We compare these results to anomaly cancellation on the fixed plane of the orbifold. In the case of branes wrapping noncompact cycles, there are fractional intersection numbers and anomaly coefficients, which we explain in geometric terms.