Light-Ray Radon Transform for Abelianin and Nonabelian Connection in 3 and 4 Dimensional Space with Minkowsky Metric

TL;DR

This paper introduces a light-ray Radon transform in 3 and 4 dimensional Minkowski space for reconstructing connections on trivial bundles, with applications to Yang-Mills equations, supersymmetry, and string theory.

Contribution

It develops a Radon transform based on light rays and 2-planes to reconstruct connections, linking it to Yang-Mills, supersymmetry, and representation theory.

Findings

Reconstruction of connections up to gauge transformations.

Application to self-dual Yang-Mills equations as zero curvature conditions.

Relation to supersymmetry and string theory via measures on planes.

Abstract

We consider a real manifold of dimension 3 or 4 with Minkovsky metric, and with a connection for a trivial GL(n,C) bundle over that manifold. To each light ray on the manifold we assign the data of paralel transport along that light ray. It turns out that these data are not enough to reconstruct the connection, but we can add more data, which depend now not from lines but from 2-planes, and which in some sence are the data of parallel transport in the complex light-like directions, then we can reconstruct the connection up to a gauge transformation. There are some interesting applications of the construction: 1) in 4 dimensions, the self-dual Yang Mills equations can be written as the zero curvature condition for a pair of certain first order differential operators; one of the operators in the pair is the covariant derivative in complex light-like direction we studied. 2) there is a relation of this Radon transform with the supersymmetry. 3)using our Radon transform, we can get a measure on the space of 2 dimensional planes in 4 dimensional real space. Any such measure give rise to a Crofton 2-density. The integrals of this 2-density over surfaces in R^4 give rise to the Lagrangian for maps of real surfaces into R^4, and therefore to some string theory. 4) there are relations with the representation theory. In particular, a closely related transform in 3 dimensions can be used to get the Plancerel formula for representations of SL(2,R).