Green-Schwarz Mechanism for String Dualities

TL;DR

This paper derives the complete first-order spacetime action for bosonic string theory compactified on a torus, revealing a Green-Schwarz mechanism that ensures T-duality symmetry via an $ ext{O}(d,d, ext{R})$ invariant structure.

Contribution

It introduces a systematic method to express the action in minimal form and demonstrates the necessity of a Green-Schwarz type mechanism for $ ext{O}(d,d, ext{R})$ symmetry at first order in $ ext{alpha'}$.

Findings

Derived the first-order spacetime action for compactified bosonic string theory.

Established the requirement of a Green-Schwarz mechanism for T-duality invariance.

Modified the three-form curvature with a Chern-Simons term for gauge fields.

Abstract

We determine the complete spacetime action to first order in $\alpha'$ for the massless fields of bosonic string theory compactified on a $d$-dimensional torus. A fully systematic procedure is developed that brings the action into a minimal form in which all fields apart from the metric enter only with first-order derivatives. T-duality implies that this action must have a global $\mathrm{O}(d,d,\mathbb{R})$ symmetry, and we show that this requires a Green-Schwarz type mechanism for $\alpha'$-deformed $\mathrm{O}(d,d,\mathbb{R})$ transformations. In terms of a frame formalism with ${\rm GL}(d)\times {\rm GL}(d)$ gauge symmetry this amounts to a modification of the three-form curvature by a Chern-Simons term for composite gauge fields.