# On Pauli Reductions of Supergravities in Six and Five Dimensions

## TL;DR

This paper investigates the possibility of consistent Pauli reductions on S^2 in supergravity theories, demonstrating non-existence in one case and elucidating a known reduction in another through higher-dimensional group-manifold techniques.

## Contribution

It establishes the non-existence of a consistent S^2 Pauli reduction in five-dimensional supergravity and clarifies a known reduction in six dimensions via a higher-dimensional perspective.

## Key findings

- No consistent S^2 Pauli reduction for 5D minimal supergravity.
- A known 6D Salam-Sezgin reduction can be derived from 7D group-manifold reduction.
- Uses group-theoretic methods to analyze reduction consistency.

## Abstract

The dimensional reduction of a generic theory on a curved internal space such as a sphere does not admit a consistent truncation to a finite set of fields that includes the Yang-Mills gauge bosons of the isometry group. In rare cases, for example the $S^7$ reduction of eleven-dimensional supergravity, such a consistent "Pauli reduction" does exist. In this paper we study this existence question in two examples of $S^2$ reductions of supergravities. We do this by making use of a relation between certain $S^2$ reductions and group manifold $S^3=SU(2)$ reductions of a theory in one dimension higher. By this means we establish the non-existence of a consistent $S^2$ Pauli reduction of five-dimensional minimal supergravity. We also show that a previously-discovered consistent Pauli reduction of six-dimensional Salam-Sezgin supergravity can be elegantly understood via a group-manifold reduction from seven dimensions.