T-duality and U-duality in toroidally-compactified strings

TL;DR

This paper investigates how T-duality and U-duality symmetries manifest in toroidally-compactified type IIA string theory, highlighting the non-local nature of U-duality on fundamental fields and proposing a perspective that preserves T-duality as a perturbative symmetry.

Contribution

It clarifies the action of U-duality in lower dimensions by considering un-dualized NS-NS fields, showing that T-duality remains intact while U-duality groups are modified in D≤6.

Findings

T-duality remains a perturbative symmetry in compactified strings.

U-duality groups are modified in dimensions D≤6 when NS-NS fields are not dualized.

U-duality acts non-locally on fundamental string fields in lower dimensions.

Abstract

We address the issue of T-duality and U-duality symmetries in the toroidally-compactified type IIA string. It is customary to take as a starting point the dimensionally-reduced maximal supergravity theories, with certain field strengths dualised such that the classical theory exhibits a global $E_{n(n)}$ symmetry, where n=11-D in D dimensions. A discrete subgroup then becomes the conjectured U-duality group. In dimensions D\le 6, these necessary dualisations include NS-NS fields, whose potentials, rather than merely their field strengths, appear explicitly in the couplings to the string worldsheet. Thus the usually-stated U-duality symmetries act non-locally on the fundamental fields of perturbative string theory. At least at the perturbative level, it seems to be more appropriate to consider the symmetries of the versions of the lower-dimensional supergravities in which no dualisations of NS-NS fields are required, although dualisations of the R-R fields are permissible since these couple to the string through their field strengths. Taking this viewpoint, the usual T-duality groups survive unscathed, as one would hope since T-duality is a perturbative symmetry, but the U-duality groups are modified in D\le 6.