# The Penrose transform, and volumes of polytopes

In twistor theory, the points of Minkowski space-time are picked out by the families of null geodesics through them. In relativity, each such family has the conformal structure of a sphere, interpreted in twistor theory as a Riemann sphere, or in other words a complex projective line. These complex projective lines lie inside the three complex dimensional manifold called twistor space.

The Penrose transform yields an isomorphism between analytic zero-rest-mass free fields on Minkowski space-time and elements of the first sheaf cohomology on twistor space. This transform arises from the double fibration shown in the figure.

The same double fibration appears in a quite different context: to an arrangement of hyperplanes (a polytope) one can associate both a matroid and an orbit in a Grassmannian. Then the double fibration can be used to calculate the Tutte polynomial of the matroid, and hence a volume of the polytope.

But volumes of polytopes also appear in recent work in twistor field theory, as certain scattering amplitudes. This project will examine some simple examples of these constructions, in order to investigate how they are related.

Supervisor: Prof Stephen Huggett
Second supervisor: Dr Daniel Robertz

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