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.
Highlights
The idea of using a dimensional reduction on a curved space such as a sphere in order to obtain a lowerdimensional theory with non-Abelian gauge symmetries has a long history, which seems to have originated with an unpublished communication by Pauli in 1952 [1,2]
Our second example is provided by the six-dimensional gauged supergravity of Salam and Sezgin [11]. This has the intriguing feature that it admits a supersymmetric S2 × ðMinkowskiÞ4 vacuum. It was shown in [12] that there is a consistent S2 Pauli reduction of the Salam-Sezgin model, which yields a four-dimensional supergravity with SUð2Þ Yang-Mills fields originating from the isometry group of the 2 sphere, and whose Minkowski vacuum corresponds to the six-dimensional S2 × ðMinkowskiÞ4 vacuum found in [11]
We have employed a relation between consistent DeWitt reductions and Pauli reductions that was established in [4], applying it to two instances of S2 reductions in supergravity theories
Summary
The idea of using a dimensional reduction on a curved space such as a sphere in order to obtain a lowerdimensional theory with non-Abelian gauge symmetries has a long history, which seems to have originated with an unpublished communication by Pauli in 1952 [1,2]. This has the intriguing feature that it admits a supersymmetric S2 × ðMinkowskiÞ4 vacuum It was shown in [12] that there is a consistent S2 Pauli reduction of the Salam-Sezgin model, which yields a four-dimensional supergravity with SUð2Þ Yang-Mills fields originating from the isometry group of the 2 sphere, and whose Minkowski vacuum corresponds to the six-dimensional S2 × ðMinkowskiÞ4 vacuum found in [11]. We find that, at least at the bosonic level, there is a different way to embed the Salam-Sezgin theory into the seven-dimensional SOð2; 2Þ gauged supergravity, in which the Kaluza-Klein vector plays an active role In this new embedding, it supplies the necessary twist of the S1 fibers so that the lift of the S2 Pauli reduction becomes an SUð2Þ DeWitt reduction from seven dimensions. Cpoffinffi sistent 3Að1Þ so with the equations that the remaining gauge field has a canonical normalization, the Lagrangian (2.8) reduces to that for the bosonic sector of pure minimal supergravity, ÃFð2Þ ð2:10Þ
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