The addition of the volume term makes Kähler coupling strength a genuine coupling parameter also classically when the variation of Kähler action is non-vanishing. Therefore quantum criticality for Λ and αK gets precise meaning also classically. The equations of motion for a worldline of U(1) charged particle generalize to field equations for a “world line” of 3-D extended particle.
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.
For background see the chapter From principles to diagrams of “Towards M-matrix” or the article How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD?
For a summary of earlier postings see Latest progress in TGD.