U-duality as General Coordinate Transformations, and Spacetime Geometry

TL;DR

This paper demonstrates that the global symmetry groups of maximal supergravities can be understood through internal general coordinate transformations of toroidal compactifications, linking different duality pictures and analyzing quantum effects on internal geometries.

Contribution

It provides a unified description of U-duality groups as closure of internal coordinate transformations and explores their quantum implications in supergravity compactifications.

Findings

U-duality groups arise from internal general coordinate transformations.

Quantum effects can alter internal torus size and shape.

Dualities relate large and small torus compactifications.

Abstract

We show that the full global symmetry groups of all the D-dimensional maximal supergravities can be described in terms of the closure of the internal general coordinate transformations of the toroidal compactifications of D=11 supergravity and of type IIB supergravity, with type IIA/IIB T-duality providing an intertwining between the two pictures. At the quantum level, the part of the U-duality group that corresponds to the surviving discretised internal general coordinate transformations in a given picture leaves the internal torus invariant, while the part that is not described by internal general coordinate transformations can have the effect of altering the size or shape of the internal torus. For example, M-theory compactified on a large torus T^n can be related by duality to a compactification on a small torus, if and only if n\ge 3. We also discuss related issues in the toroidal compactification of the self-dual string to D=4. An appendix includes the complete results for the toroidal reduction of the bosonic sector of type IIB supergravity to arbitrary dimensions D\ge3.