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Let $X$ be a Riemannian manifold. If $X$ is simply connected, irreducible, and not a symmetric space then we know that the possible holonomy groups of the metric on $X$ are:

1) $O(n)$ General Riemannian manifolds

2) $SO(n)$ Orientable manifolds

3) $U(n)$ Kahler manifolds

4) $Sp(n)Sp(1)$ Quaternionic Kahler manifolds

5) $SU(n)$ Calabi-Yau manifolds

6) $Sp(n)$ Hyperkahler manifolds

7) $G_{2}$ (if $X$ has dimension 7) $G_{2}$ manifolds

8) $Spin(7)$ (if X has dimension 8) $Spin(7)$ manifolds

Of these, cases 5-8 are important in string theory. The reduced holonomy implies that these manifolds have vanishing Ricci curvature and hence are automatically solutions to Einstein's equations of general relativity.

However, physics does not take place on a Riemannian manifold, but rather on a Lorentzian one. Thus my question is: what is known about special holonomy manifolds for metrics of general signature? (I am most interested in the (1,n) case!) In particular I would like to know if there is a classification of allowed holonomy groups and further if there are interesting examples that I should be aware of.

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2 Answers 2

up vote 14 down vote accepted

I think it would be more accurate to say that the real reason why Calabi-Yau, hyperkähler, $G_2$ and $\mathrm{Spin}(7)$ manifolds are of interest in string theory is not their Ricci-flatness, but the fact that they admit parallel spinor fields. Of course, in positive-definite signature, existence of parallel spinor fields implies Ricci-flatness, but the converse is still open for compact riemannian manifolds, as discussed in this question, and known to fail for noncompact manifolds as pointed out in an answer to that question.

The similar question for lorentzian manifolds has a bit of history. First of all, the holonomy principle states that a spin manifold admits parallel spinor fields if and only if (the spin lift of) its holonomy group is contained in the stabilizer subgroup of a nonzero spinor. Some low-dimensional (i.e., $\leq 11$, the cases relevant to string and M-theories) investigations (by Robert Bryant and myself, independently) suggested that these subgroups are either of two types: subgroups $G < \mathrm{Spin}(n) < \mathrm{Spin}(1,n)$, whence $G$ is the ones corresponding to the cases 5-8 in the question, or else $G = H \ltimes \mathbb{R}^{n-1}$, where $H < \mathrm{Spin}(n-1)$ is one of the groups in cases 5-8 in the question. Thomas Leistner showed that this persisted in the general case and, as Igor pointed out in his answer, arrived at a classification of possible lorentzian holonomy groups. Anton Galaev then constructed metrics with all the possible holonomy groups, showing that they all arise. Their work is reviewed in their paper (MR2436228).

The basic difficulty in the indefinite-signature case is that the de Rham decomposition theorem is modified. Recall that the de Rham decomposition theorem states that if $(M,g)$ is a complete, connected and simply connected positive-definite riemannian manifold and if the holonomy group acts reducibly, then the manifold is a riemannian product, whence it is enough to restrict to irreducible holonomy representations. This is by no means a trivial problem, but is tractable.

In contrast, in the indefinite signature situation, there is a modification of this theorem due to Wu, which says that it is not enough for the holonomy representation to be reducible, it has to be nondegenerately reducible. This means that it is fully reducible and the direct sums in the decomposition are orthogonal with respect to the metric. This means that it is therefore not enough to restrict oneself to irreducible holonomy representations. For example, Bérard-Bergery and Ikemakhen proved that the only lorentzian holonomy group acting irreducibly is $\mathrm{SO}_0(1,n)$ itself: namely, the generic holonomy group.

It should be pointed out that in indefinite signature, the integrability condition for the existence of parallel spinor fields is not Ricci-flatness. Instead, it's that the image of the Ricci operator $S: TM \to TM$, defined by $g(S(X),Y) = r(X,Y)$, with $r$ the Ricci curvature, be isotropic. Hence if one is interested in supersymmetric solutions of supergravity theories (without fluxes) one is interested in Ricci-flat lorentzian manifolds (of the relevant dimension) admitting parallel spinor fields. It is now not enough to reduce the holonomy to the isotropy of a spinor, but the Ricci-flatness equation must be imposed additionally.

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I am no expert in Lorentzian geometry, but according to MathSciNet review, Lorentzian holonomy groups has been completely classified by Thomas Leistner in [On the classification of Lorentzian holonomy groups. J. Diff. Geom. 76 (2007), no. 3, 423--484] who builds on classical work of Berger.

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