Two and three loop alpha' corrections to T-duality: Kasner and Schwarzschild

TL;DR

Two- and three-loop $\alpha'$ corrections to T-duality are computed for diagonal Kasner and Schwarzschild backgrounds in the critical bosonic string at tree level. The authors derive linear ($\alpha'^1$) and quadratic ($\alpha'^2$) corrections to the Kasner geometry and its T-dual, and formulate $\alpha'$-modified T-duality rules in covariant form; these rules are then applied to Schwarzschild in $D=4$ and its time-dual, revealing horizon-related divergences and a massless time-dual geometry. A key result is that consistency between the corrected backgrounds and the duality map fixes the quadratic dilaton transformation under T-duality, enabling a unique, covariant description of higher-order duality. The work emphasizes frame choices and field redefinitions necessary to preserve duality symmetries at higher orders and provides insights for string cosmology and the pre-big bang scenario.

Abstract

Two and three loop alpha' corrections are calculated for Kasner and Schwarzschild metrics, and their T-duals, in the bosonic string theory. These metrics are used to obtain the two and three loop alpha' corrections to T-duality. It is noted in particular that the inclusion of alpha' corrections and the requirement of consistency with the alpha'-corrected T-duality for the Kasner and Schwarzschild metrics enables one to fix uniquely the covariant form of the T-duality rules at three loops. As a generalization of the T-dual of the Schwarzschild geometry a class of massless geometries is presented.

Figures (1)

• Figure 1: T-duality and the $\alpha'$ corrections. By requiring that $\alpha'$ modified T-duality maps the $\alpha'$ corrected dual backgrounds to each other the $\alpha'$ modifications to the rules of the T-duality can be identified.