Dimensional reduction, truncations, constraints and the issue of consistency

TL;DR

This paper reviews dimensional reduction methods for diffeomorphism invariant theories, emphasizing the distinction between physical compactification and mathematical truncation, and discusses conditions for consistency in various reduction scenarios.

Contribution

It clarifies the criteria for consistent truncations, especially on group and coset spaces, and highlights known examples of such truncations in theoretical physics.

Findings

Consistency in group manifold reductions with unimodularity condition

Reduction of gauge group in certain truncations

Examples of remarkable consistent truncations on coset spaces

Abstract

A brief overview of dimensional reductions for diffeomorphism invariant theories is given. The distinction between the physical idea of compactification and the mathematical problem of a consistent truncation is discussed, and the typical ingredients of the latter --reduction of spacetime dimensions and the introduction of constraints-- are examined. The consistency in the case of of group manifold reductions, when the structure constants satisfy the unimodularity condition, is shown in a clear way together with the associated reduction of the gauge group. The problem of consistent truncations on coset spaces is also discussed and we comment on examples of some remarkable consistent truncations that have been found in this context.