Holography = Holomorphy Vision & a More Precise View of Partonic Orbits

Abstract

In this article a more precise view about the 3-D light-like trajectories of 2-dimensional parton surfaces is developed on the basis of holograpy = holomorphy hypothesis. Partonic orbits are identified as light-like 3-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian so that the metric determinant vanishes. The technical problems related to the finding of the roots of (f_1, f_2) appearing in the Euclidean space-time regions are discussed. In Minkowskian regions one has two kinds of solutions for which hypercomplex coordinate u or v appears in f_i. These should correspond to a single solution and the only way is to consider their union. The two regions in question have a natural identification as Minkowskian space-time sheets connected by a wormhole contact with an Euclidean signature of the induced metric. In Euclidean regions, the realization of holography= holomorphy principle using (f_1,f_2) = (0,0) ansatz assuming that either hypercomplex coordinate u or v is dynamical variable, leads to a problem. Either u or v is a complex analytic function f of CP_2 coordinates and its reality implies Im(f) =0 so that CP_2 projection is 3-dimensional. Wick rotation does not help. It is an open question how to generalize (f_1,f_2)=(0,0) ansatz to obtain CP_2 extremals with 4-D CP_2 projection and 1-D light-like curve or geodesic as M^4 projection. The light-like u or v coordinate lines can have edges at the partonic orbits. This has led to a proposal for how exotic smooth structures necessary for defining fermion pair creation vertices emerge via partonic orbits as defects of the standard smooth structure. Fermion pair as a fermion returning backwards in time would correspond to the edge of u (or v) coordinate line. These conditions generalize the Virasoro conditions for 1-dimensional light-like curves to the 3-dimensional light-like partonic orbits. The light-likeness condition for the partonic orbits generalizes the Virasoro conditions for 1-dimensional light-like curves to the 3-dimensional light-like partonic orbits. Also an explicit procedure for finding the partonic orbits is discussed.