6d $\mathcal N=(1,0)$ anomalies on $S^1$ and F-theory implications

TL;DR

This paper demonstrates how 6d $ ext{(1,0)}$ gauge anomalies manifest as field-dependent Chern-Simons terms in 5d theories after circle compactification, linking anomaly cancellation to the structure of F-theory compactifications.

Contribution

It establishes a precise connection between 6d anomalies and 5d Chern-Simons terms, highlighting the importance of proper KK mode integration and confirming F-theory's automatic anomaly cancellation.

Findings

Chern-Simons terms vanish if and only if anomalies are canceled.

F-theory compactifications are anomaly-free when M/F-duality applies.

Quantum corrections to gauge couplings are derived from the 5d prepotential.

Abstract

We show that the pure gauge anomalies of 6d $\mathcal N=(1,0)$ theories compactified on a circle are captured by field-dependent Chern-Simons terms appearing at one-loop in the 5d effective theories. These terms vanish if and only if anomalies are canceled. In order to obtain this result, it is crucial to integrate out the massive Kaluza-Klein modes in a way that preserves 6d Lorentz invariance; the often-used zeta-function regularization is not sufficient. Since such field-dependent Chern-Simons terms do not arise in the reduction of M-theory on a threefold, six-dimensional F-theory compactifications are automatically anomaly free, whenever the M/F-duality can be used. A perfect match is then found between the 5d $\mathcal N=1$ prepotentials of the classical M-theory reduction and one-loop circle compactification of an anomaly free theory. Finally, from this potential, we read off the quantum corrections to the gauge coupling functions.