Upper bound on the Atiyah-Singer index from tadpole cancellation

TL;DR

This paper establishes an upper limit on the Atiyah-Singer index in string theory compactifications, linking topological constraints to observable particle generations and Higgs fields, with implications for various string models.

Contribution

It introduces a novel upper bound on the Atiyah-Singer index derived from tadpole cancellation and supersymmetry, applicable to multiple string theory frameworks.

Findings

Upper bound on generation number in heterotic models.

Explicit bounds for Calabi-Yau threefolds in the Kreuzer-Skarke database.

Extension of bounds to Type IIB/F-theory compactifications.

Abstract

We propose an upper bound on the Atiyah-Singer index in the effective action of string theory. For $E_8 \times E_8^\prime$ and $SO(32)$ heterotic string theories on smooth Calabi-Yau threefolds with line bundles, we find that the tadpole cancellation and supersymmetry conditions lead to an upper bound on the generation number of quarks and leptons as well as Higgs doublets. By taking into account the observed value of four-dimensional gauge couplings and the supergravity approximation, we explicitly evaluate the bound on favorable complete intersection Calabi-Yau threefolds. The bound can be extended to Calabi-Yau threefolds in the Kreuzer-Skarke database. We also put the upper bound on the Atiyah-Singer index in Type IIB/F-theory compactifications.