Abstract
We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general Kodaira deformation of the twistor complex structure. The Newton--Cartan connections can nevertheless be reconstructed from Merkulov's generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non--trivial on twistor lines. The resulting geometries agree with non--relativistic limits of anti-self-dual gravitational instantons.
Highlights
Over the last 6 years there has been large interest in the AdS/CFT correspondences providing the gravity duals of non–relativistic gauge theories relevant in solid–state physics [15]
There are several equivalent definitions of twistors for flat Minkowski space, but they lead to nonequivalent pictures once a non-relativistic limit is taken
The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non–trivial on twistor lines
Summary
Over the last 6 years there has been large interest in the AdS/CFT correspondences providing the gravity duals of non–relativistic gauge theories relevant in solid–state physics [15]. In the non-relativistic limit the normal bundle of the twistor curves corresponding to space-time points jumps from O(1) ⊕ O(1) to O ⊕ O(2), where O(n) is the holomorphic line bundle over CP1 with the Chern class n. This results in the conformal structure becoming degenerate. 6 we shall discuss the Kodaira deformation theory of the complex structure underlying the non-relativistic twistor space. In Appendix 2 we shall show how the spin connection in the Nonlinear Graviton construction arises as a Ward transform of a rank–two holomorphic vector bundle over the relativistic deformed twistor space
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