Anomaly cancellation in M-theory: a critical review

TL;DR

Bilal and Metzger critically reanalyze anomaly cancellation in M-theory, focusing on 5-brane and S^1/{\bf Z}_2 anomalies and refining the signs and coefficients of topological terms by carefully treating orientation, chirality, and Euclidean continuation. They demonstrate that the Green-Schwarz and (modified) Chern-Simons terms must share the same sign and that a previously neglected factor forces a slight modification of the Chern-Simons term, enabling local anomaly cancellation on both branes and ten-planes and matching the heterotic limit at small radius via the $X_8$ structure. The analysis fixes the relation $\epsilon^3=\beta$ and shows anomaly cancellation persists in both the downstairs and upstairs formalisms, with local inflow providing a consistent ten-plane cancellation. Together these results yield a coherent, sign-consistent formulation of 11D supergravity with GS and CS terms and clarify the M-theory–heterotic duality via anomaly inflow.

Abstract

We carefully review the basic examples of anomaly cancellation in M-theory: the 5-brane anomalies and the anomalies on S^1/Z_2. This involves cancellation between quantum anomalies and classical inflow from topological terms. To correctly fix all coefficients and signs, proper attention is paid to issues of orientation, chirality and the Euclidean continuation. Independent of the conventions chosen, the Chern-Simons and Green-Schwarz terms must always have the same sign. The reanalysis of the reduction to the heterotic string on S^1/Z_2 yields a surprise: a previously neglected factor forces us to slightly modify the Chern-Simons term, similar to what is needed for cancelling the normal bundle anomaly of the 5-brane. This modification leads to a local cancellation of the anomaly, while maintaining the periodicity on S^1.