Pauli reductions of supergravities in six and five dimensions
Arash Azizi
1,*
and C. N. Pope
1,2,†
1
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,
Texas A&M University, College Station, Texas 77843 , USA
2
DAMTP, Center for Mathematical Sciences, Cambridge University, Wilberforce Road,
Cambridge CB3 OWA, United Kingdom
(Received 25 June 2018; published 13 August 2018)
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, e.g., the S
7
reduction of 11-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
¼ SUð2Þ reductions of a theory in one dimension higher. By this means, we establish the nonexistence
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.
DOI: 10.1103/PhysRevD.98.046010
I. INTRODUCTION
The idea of using a dimensional reduction on a curved
space such as a sphere in order to obtain a lower-
dimensional theory with non-Abelian gauge symmetries
has a long history, which seems to have originated with an
unpublished communication by Pauli in 1952 [1,2] .His
idea was that by dimensionally reducing six-dimensional
Einstein gravity on the 2 sphere, one might obtain a four-
dimensional theory describing gravity coupled to gauge
fields that we would now call SOð3Þ Yang-Mills. Pauli
realized, however, that there were difficulties with this idea,
which we can recognize as being inconsistencies in the
higher-dimensional components of the Einstein equations.
This is in fact a classic example of an inconsistent
truncation in Kaluza-Klein.
One can always, of course, make a consistent Kaluza-
Klein “reduction” in which one simply expands all the
higher-dimensional fields in terms of the complete set of
harmonics (spherical harmonics, in the case of S
2
) and
retains the entire infinite towers of lower-dimensional
massless and massive fields. The consistency issue
arises if one wants to set all except a subset of the
lower-dimensional fields to zero. For this to be a consistent
truncation, it is necessary that the full set of equations of
motion for the infinite towers of fields should be compat-
ible with setting to zero the fields one wishes to truncate.
The danger is that, because of the nonlinear nature of the
theory, the fields one is retaining might act as sources for
the fields one wishes to set to zero. This is exactly what
happens in the case Pauli considered; the Yang-Mills fields
one wishes to retain actually act as sources for a set of
massive spin-2 fields that one wishes to set to zero. From
the higher-dimensional point of view, this reflects itself as
an inconsistency between the truncation Ansatz and the
higher-dimensional components of the Einstein equations.
The inconsistency of the truncation to a finite number of
lower-dimensional fields that include gravity and the Yang-
Mills fields of the isometry group of the compactifying
manifold is not restricted to the case where one starts from
pure higher-dimensional gravity. It is in fact what happens
generically when compactifying any higher-dimensional
theory that includes gravity. There are, however, some
notable exceptions, of which the most celebrated is the
reduction of 11-dimensional supergravity on the 7 sphere.
In this case, the inconsistencies that would arise if 11-
dimensional pure gravity alone were reduced on S
7
are
removed, because of further contributions in the higher-
dimensional equations of motion coming from the 3-form
potential. The consistency of this remarkable reduction was
demonstrated in [3]. The exceptional situations where a
consistent reduction on a curved internal manifold M,
retaining a finite number of fields including gravity and the
Yang-Mills gauge fields of the isometry group of M,have
*
sazizi@physics.tamu.edu
†
pope@physics.tamu.edu
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D 98, 046010 (2018)
2470-0010=2018=98(4)=046010(10) 046010-1 Published by the American Physical Society
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