U-duality (sub-)groups and their topology

TL;DR

This paper explores how the topology of symmetry groups in compactified supergravity influences the interpretation of sigma models, constrains M-theory symmetry proposals, and relates to generalized holonomy and E-series conjectures.

Contribution

It provides criteria based on group topology to determine when sigma models correspond to higher-dimensional theories and analyzes implications for M-theory symmetry proposals.

Findings

Topology of symmetry groups constrains M-theory symmetry proposals.

Criteria established for interpreting 3D sigma models as higher-dimensional reductions.

Certain M-theory symmetry proposals are ruled out by topological considerations.

Abstract

We discuss some consequences of the fact that symmetry groups appearing in compactified (super-)gravity may be non-simply connected. The possibility to add fermions to a theory results in a simple criterion to decide whether a 3-dimensional coset sigma model can be interpreted as a dimensional reduction of a higher dimensional theory. Similar criteria exist for higher dimensional sigma models, though less decisive. Careful examination of the topology of symmetry groups rules out certain proposals for M-theory symmetries, which are not ruled out at the level of the algebra's. We conclude with an observation on the relation between the ``generalized holonomy'' proposal, and the actual symmetry groups resulting from E_10 and E_11 conjectures.