Second order bosonic string effective action from $O(d,d)$

TL;DR

This paper derives a simplified second-order effective action for the bosonic string consistent with $O(d,d)$ symmetry, revealing structural patterns across multiple orders in $\alpha'$.

Contribution

It provides a more streamlined form of the second-order effective action and identifies recurring structures in the Lagrangian at multiple orders.

Findings

Effective action form is simplified compared to previous literature.

Compatibility with $O(d,d)$ symmetry constrains the effective action.

Structural patterns in the Lagrangian appear at least up to third order in $\alpha'$.

Abstract

The corrections to the tree-level effective action for the bosonic string up to second order in $\alpha'$ are fixed by requiring its dimensional reduction to $26-d$ dimensions to be compatible with $O(d,d)$ symmetry. The result is in agreement with the literature, but takes a simpler form than previously know expressions. We identify some structures in the Lagrangian which appear at least at the first three orders in $\alpha'$.