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Symmetric IIB backgrounds

Monday, 24 September 2012 in hep-th, Maths&Physics by José | No comments

Noel Hustler and I have just arXived the paper Symmetric backgrounds of type IIB supergravity, containing a partial classification (up to local isometry) of homogeneous solutions of the field equations of type IIB supergravity whose underlying geometry is that of a ten-dimensional lorentzian symmetric space.

This is the continuation of the programme started in this paper, which was the subject of this blog post, to which I refer the reader for some of the background. In that paper I classified symmetric backgrounds of eleven-dimensional supergravity, whereas in the present paper we do the same for type IIB supergravity.

Despite the (modest) drop in dimension, which translates into substantially fewer candidate geometries, the profusion of fluxes in type IIB supergravity makes this problem rather more complicated and as a result in about one third of the geometries which required detailed analysis we have not been able to determine the full moduli space. We did in the end find at least some exact solutions for all geometries admitting such backgrounds, even though in order to get there we often had to resort to numerical optimization techniques.

In addition to the extra complexity of the analysis, there is one interesting feature of the IIB problem: namely how the duality of the theory behaves in a homogeneous background.

Recall that a supergravity background is said to be homogeneous if the underlying spacetime admits the transitive action of a Lie group preserving not just the metric but all the bosonic fields in the background. Since IIB supergravity is a gauge theory, we have to allow for invariance up to gauge transformation, but this is easily implemented by demanding invariance of the gauge invariant fields.

The bosonic fields of type IIB supergravity are the metric , the dilaton , the NSNS 3-form , the axion and the RR fieldstrengths: the 3-form and the self-dual 5-form . A homogeneous background is such that , , , , and are invariant. In particular, need not be invariant, only needs to be. This allows us to distinguish between two classes of homogeneous backgrounds: those for which is invariant, which we call strongly homogeneous in the paper, and those for which it is not. An equivalent characterisation of strongly homogeneous backgrounds are those homogeneous backgrounds for which, in addition, vanishes.

Since duality acts as fractional linear transformations on the axi-dilaton , a homogeneous background for which is not constant will transform in to a background which is not homogeneous, since the transformed dilaton is not constant. On the other hand, any dual background of a strongly homogeneous background is again strongly homogeneous.

Strongly homogeneous backgrounds are also singled out from supersymmetry. Indeed, it follows from the homogeneity theorem for IIB supergravity, proved in our recent paper and discussed in this recent blog post, and ealier work by EMPG alumna Emily Hackett-Jones and Douglas Smith, that any IIB background preserving more than one half of the total supersymmetry is strongly homogeneous. Therefore one may restrict to strongly homogeneous backgrounds in the classification effort for highly supersymmetric backgrounds. Nevertheless in this paper we do not restrict ourselves and indeed find many backgrounds with nonzero .

The backgrounds whose moduli space we have been able to determine fully have at most a 2-dimensional moduli space, hence we have not had the chance to draw beautiful pictures as in the eleven-dimensional case.

The backgrounds come in two flavours: those with underlying geometry a (possibly degenerate) Cahen-Wallach pp-wave and those of the form for . The latter list is given by the following geometries. First we have those for which we have determined the full moduli space:

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and then there are those which require further analysis:

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As in the eleven-dimensional case we are not discarding geometries which are not spin (such as those with , or factors).

Based on the available evidence, Patrick Meessen has conjectured (privately) that backgrounds preserving more than three quarters of the supersymmetry are symmetric. If this is true, then those backgrounds of eleven- and type II ten-dimensional supergravities preserving more than 24 supersymmetries are to be found among the ones we have already classified; although determining the precise loci in the moduli space where they preserve so much supersymmetry is not necessarily a trivial task.

We end with the word cloud de rigueur.

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