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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 manifoldmanifolds

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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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 manifold

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 4-7 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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Let $X$ be a compact 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 manifold

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 4-7 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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