Chern-Simons couplings, modular duality, and anomaly cancellation in abelian F-theory

TL;DR

This paper demonstrates how three-dimensional Chern-Simons couplings derived from F-theory compactifications encode four-dimensional abelian anomalies and their cancellation, linking flux, duality, and spectrum analysis.

Contribution

It provides two independent methods to determine Chern-Simons couplings in F-theory, clarifying anomaly cancellation and duality compatibility.

Findings

Chern-Simons couplings encode 4D abelian anomalies.

Agreement between flux-based and spectrum-based calculations.

Compatibility with type IIB duality confirmed.

Abstract

F-theory compactifications with a nontrivial Mordell-Weil group realize abelian gauge symmetry through rational sections, but their consistency is ultimately a statement about the quantum effective action. We show that compactification on a circle makes this statement concrete: the quantized, parity-odd Chern-Simons couplings of the resulting three-dimensional theory provide a one-loop exact and scheme-independent encoding of all local four-dimensional abelian anomalies, including the mixed gauge-gravitational terms, together with their Green-Schwarz cancellation. We determine these Chern-Simons couplings in two logically independent ways, first from flux-induced terms in the M-theory dual description, and second from an explicit one-loop integration over the complete massive spectrum, including Kaluza-Klein towers and Coulomb-branch states. The agreement fixes all normalizations and clarifies how large gauge transformations reorganize the spectrum. We then show compatibility with type IIB modular duality once the known ten-dimensional duality counterterm is included, and we present a fully explicit rank-two example over projective three-space.