An SL(2,Z) anomaly in IIB supergravity and its F-theory interpretation

TL;DR

This paper investigates the SL(2,Z) duality anomaly in type IIB supergravity within F-theory, revealing topological restrictions on consistent backgrounds due to anomaly cancellation conditions.

Contribution

It identifies the origin of the SL(2,Z) anomaly in IIB supergravity and derives new topological constraints on F-theory backgrounds for anomaly cancellation.

Findings

Partial anomaly cancellation via a ten-form term involving the U(1) field and curvature.

Residual anomaly imposes topological restrictions on background manifolds.

F-theory on K3 x M8 requires M8's Euler character to be a positive multiple of 24.

Abstract

The SL(2,Z) duality transformations of type IIB supergravity are shown to be anomalous in generic F-theory backgrounds due to the anomalous transformation of the phase of the chiral fermion determinant. The anomaly is partially cancelled provided the ten-dimensional type IIB theory lagrangian contains a term that is a ten-form made out of the composite U(1) field strength and four powers of the curvature. A residual anomaly remains uncancelled, and this implies a certain topological restriction on consistent backgrounds of the euclidean theory. A similar, but slightly stronger, restriction is also derived from an explicit F-theory compactification on K3 x M8 (where M8 is an eight-manifold with a nowhere vanishing chiral spinor) where the cancellation of tadpoles for Ramond--Ramond fields is only possible if M8 has an Euler character that is a positive multiple of 24. The interpretation of this restriction in the dual heterotic theory on T2 x M8 is also given.