The quantization problem in Scherk-Schwarz compactifications

TL;DR

This paper re-examines the quantization of structure constants in Scherk-Schwarz compactifications, clarifying when supersymmetry is broken or preserved, and classifying flat groups in various dimensions.

Contribution

It provides a detailed classification of flat groups and lattices in Scherk-Schwarz compactifications, and clarifies the effective theories associated with different vacua.

Findings

The low-energy effective theory is a gauged supergravity describing supersymmetry breaking.

When supersymmetry is preserved, the description is an artifact of truncation.

The Scherk-Schwarz algorithm for constructing flat groups is exhaustive up to six dimensions.

Abstract

We re-examine the quantization of structure constants, or equivalently the choice of lattice in the so-called flat group reductions, introduced originally by Scherk and Schwarz. Depending on this choice, the vacuum either breaks supersymmetry and lifts certain moduli, or preserves all supercharges and is identical to the one obtained from the torus reduction. Nonetheless the low-energy effective theory proposed originally by Scherk and Schwarz is a gauged supergravity that describes supersymmetry breaking and moduli lifting for all values of the structure constants. When the vacuum does not break supersymmetry, such a description turns out to be an artifact of the consistent truncation to left-invariant forms as illustrated for the example of ISO(2). We furthermore discuss the construction of flat groups in d dimensions and find that the Scherk--Schwarz algorithm is exhaustive. A classification of flat groups up to six dimensions and a discussion of all possible lattices is presented.