M-Theory on S^1/Z_2 : New Facts from a Careful Analysis

TL;DR

This paper re-examines M-theory on S^1/Z_2, clarifying flux, anomaly cancellation, and boundary conditions, leading to new insights on flux quantization and anomaly inflow that correct previous misconceptions.

Contribution

It provides a careful analysis of flux quantization and anomaly cancellation in M-theory on S^1/Z_2, fixing previous misconceptions and revealing new contributions from the topological term.

Findings

Fixes the parameter b to 1 for global consistency.

Identifies an additional anomaly inflow contribution from the topological term.

Derives a quadratic relation between parameters for anomaly cancellation.

Abstract

We carefully re-examine the issues of solving the modified Bianchi identity, anomaly cancellations and flux quantization in the S^1/Z_2 orbifold of M-theory using the boundary-free "upstairs" formalism, avoiding several misconceptions present in earlier literature. While the solution for the four-form G to the modified Bianchi identity appears to depend on an arbitrary parameter b, we show that requiring G to be globally well-defined, i.e. invariant under small and large gauge and local Lorentz transformations, fixes b=1. This value also is necessary for a consistent reduction to the heterotic string in the small-radius limit. Insisting on properly defining all fields on the circle, we find that there is a previously unnoticed additional contribution to the anomaly inflow from the eleven-dimensional topological term. Anomaly cancellation then requires a quadratic relation between b and the combination lambda^6/kappa^4 of the gauge and gravitational coupling constants lambda and kappa. This contrasts with previous beliefs that anomaly cancellation would give a cubic equation for b. We observe that our solution for G automatically satisfies integer or half-integer flux quantization for the appropriate cycles. We explicitly write out the anomaly cancelling terms of the heterotic string as inherited from the M-theory approach. They differ from the usual ones by the addition of a well-defined local counterterm. We also show how five-branes enter our analysis.