dc.description.abstract | In twistor theory the nonlinear graviton construction realises four-dimensional antiself-
dual Einstein manifolds as Kodaira moduli spaces of rational curves in threedimensional
complex manifolds. We establish a Newtonian analogue of this procedure,
in which four-dimensional Newton-Cartan manifolds arise as Kodaira moduli
spaces of rational curves with normal bundle O + O(2) in three-dimensional complex
manifolds. The isomorphism class of the normal bundle is unstable with respect
to general deformations of the complex structure, exhibiting a jump to the Gibbons-
Hawking class of twistor spaces. We show how Newton-Cartan connections can be
constructed on the moduli space by means of a splitting procedure augmented by an
additional vector bundle on the twistor space which emerges when considering the
Newtonian limit of Gibbons-Hawking manifolds. The Newtonian limit is thus established
as a jumping phenomenon.
Newtonian twistor theory is extended to dimensions three and five, where novel features
emerge. In both cases we are able to construct Kodaira deformations of the flat
models whose moduli spaces possess Galilean structures with torsion. In five dimensions
we find that the canonical affine connection induced on the moduli space can
possess anti-self-dual generalised Coriolis forces.
We give examples of anti-self-dual Ricci-flat manifolds whose twistor spaces contain
rational curves whose normal bundles suffer jumps to O(2 - k) + O(k) for arbitrarily
large integers k, and we construct maps which portray these big-jumping twistor
spaces as the resolutions of singular twistor spaces in canonical Gibbons-Hawking
form. For k > 3 the moduli space itself is singular, arising as a variety in an ambient
complex space. We explicitly construct Newtonian twistor spaces suffering similar jumps.
Finally we prove several theorems relating the first-order and higher-order symmetry
operators of the Schrödinger equation to tensors on Newton-Cartan backgrounds,
defining a Schrödinger-Killing tensor for this purpose. We also explore the role of conformal
symmetries in Newtonian twistor theory in three, four, and five dimensions. | |