GreenSchwarz

GreenSchwarz refers to the Green-Schwarz mechanism, a cancellation mechanism for gauge and gravitational anomalies in certain ten-dimensional string theories. Developed by Michael Green and John Schwarz in 1984, it showed that otherwise inconsistent quantum theories could be rendered consistent by the dynamics of additional higher-form fields, specifically a two-form B-field, whose transformations compensate anomalous variations of the action.

The mechanism relies on the factorization of the ten-dimensional anomaly polynomial I12. When I12 can be written as a product I12 = X4 X8, with X4 depending on gauge and gravitational curvatures, the Green-Schwarz term can be arranged to cancel the anomaly. In practical terms, the theory introduces a modified field strength H3 = dB2 − α′(ω3YM − ω3L), where ω3YM and ω3L are Chern-Simons forms for the gauge and Lorentz connections, respectively. Under gauge or local Lorentz transformations, B2 shifts in just enough way to cancel the anomalous variation through a coupling of B2 to X8 in the action, effectively restoring gauge and gravitational consistency.

Historically, the mechanism is crucial for anomaly cancellation in the ten-dimensional heterotic string and Type I string with gauge groups such as SO(32) or E8 × E8. It has since influenced broader string theory frameworks, including generalized Green-Schwarz ideas in various dimensions and in the context of D-branes and flux compactifications. Today, the mechanism remains a foundational example of how higher-form fields can ensure quantum consistency in theories with chiral fermions.