On Minimal Surface Extremals of Kahler Action

Abstract

The addition of the volume term to Kahler action has very nice interpretation as a generalization of equations of motion for a world-line extended to a 4-D space-time surface.  The field equations generalize in the same manner for 3-D light-like surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian, for 2-D string world sheets, and for  their 1-D boundaries defining world lines at the light-like 3-surfaces. For 3-D light-like surfaces the volume term is absent.  Either light-like 3-surface is freely choosable in which case one would have Kac-Moody symmetry as gauge symmetry or that the extremal property for Chern-Simons term fixes the gauge. The known non-vacuum extremals are minimal surface extremals of Kahler action and it might well be that the preferred extremal property realizing SH quite generally demands this. The addition of the volume term could however make Kahler coupling strength a manifest coupling parameter also classically when the phases of Λ and α_K are same. Therefore quantum criticality for Λ and α_K would have a precise local meaning also classically in the interior of space-time surface.  The equations of motion for a world line of U(1) charged particle would generalize to  field equations for a world line of 3-D extended particle. The conjecture is that α_K has zeros of zeta as its spectrum of critical values. If so all preferred extremals are minimal surface extremals of Kahler action. In the following the two options are compared. Also the implications of minimal surface property for conservation laws and for the possibility of solving field equations exactly using the analogy with 2-D minimal surfaces is considered.