Conservation laws for the minimal surface extremals of Kähler action
Consider first the basic conservation laws.
- Complex value of αK means that conserved quantities are complex: this brings strongly in mind twistor approach. The value of cosmological constant is assumed to be real. There are two separate local conservations laws associated with the volume term and Kähler action respectively in both Minkowskian and Euclidian regions. This need not mean separate global conservation laws in Minkowskian and Euclidian regions. If there is non canonical momentum current between Minkowskian (M) and Euclidian (E) space-time regions the real and imaginary parts of conserved quantum numbers correspond schematically to the sums
l Re(Q)= Re(1/αK)QK(E) + Im(1/αK)QK(M) +ρvac QV(M)
Im(Q)=Im(1/αK)QK(E) + Re(1/αK)QK(M) .
Here the subscripts V and K refer to the volume term and Kähler action respectively.
- If the canonical momentum current vanishes there both real and imaginary parts decompose to two separately conserved parts.
Re(Q1)= Re(1/αK)QK(E) ,
Re(Q2)= Im(1/αK)QK(M) +ρvac QV(M) ,
Im(Q1)= Im(1/αK)QK(E) ,
Im(Q2)= Re(1/αK)QK(M) .
This looks strange and the natural assumption is that canonical momentum currents can flow between the Euclidian and Minkowskian regions and boundary conditions equate the components of normal currents at both sides.
Are minimal surface extremals of Kähler action holomorphic surfaces in some sense?
I have considered several ansätze for the general solutions of the field equations for the preferred extremals. One proposal is that preferred extremals as 4-surfaces of imbedding space with octonionic tangent space structure have quaternionic tangent space or normal space (so called M8-H duality). Second proposal is that preferred extremals can be seen as quaternion analytic surfaces. Third proposal relies on a fusion of complex and hyper-complex structures to what I call Hamilton-Jacobi structure. In Euclidian regions this would correspond to complex structure. Twistor approach suggests that the condition that the twistor lift of the space-time surface to a 6-D surface in the product of twistor spaces of M4 and CP2 equals to the twistor space of CP2. This proposal is highly interesting since twistor lift works only for M4× CP2. The intuitive picture is that the field equations are integrable and all these views might be consistent.
Preferred extremals of Kähler action as minimal surfaces would be a further proposal. Can one make conclusions about general form of solutions assuming that one has minimal surface extremals of Kähler action?
In D=2 case minimal surfaces are holomorphic surfaces or they hyper-complex variants and the imbedding space coordinates can be expressed as complex-analytic functions of complex coordinate or a hypercomplex analog of this. Field equations stating the vanishing of the trace gαβHkαβ if the second fundamental form Hkαβ== Dα&partial;βhk are satisfied because the metric is tensor of type (1,1) and second fundamental form of type (2,0) ⊕ (2,0). Field equations reduce to an algebraic identity and functions involved are otherwise arbitrary functions. The constraint comes from the condition that metric is of form (1,1) as holomorphic tensor.
This raises the question whether this finding generalizes to the level of 4-D space-time surfaces and perhaps allows to solve the field equations exactly in coordinates generalizing the hypercomplex coordinates for string world sheet and complex coordinates for the partonic 2-surface.
The known non-vacuum extremals of Kähler action are actually minimal surfaces. The common feature suggested already earlier to be common for all preferred extremals is the existence of generalization of complex structure.
- For Minkowskian regions this structure would correspond to what I have called Hamilton-Jacobi structure. The tangent space of the space-time surface X4 decomposes to local direct sum T(X4)= T(X2)⊕ T(Y2), where the 2-D tangent places T(X2) and T(Y2) define an integrable distribution integrating to a decomposition X4=X2× Y2. The complex structure is generalized to a direct some of hyper-complex structure in X2 meaning that there is a local light-like direction defining light-like coordinate u and its dual v. Y2 has complex complex coordinate (w,wbar). Minkowski space M4 has similar structure. It is still an open question whether metric decomposes to a direct sum of orthogonal metrics assignable to X2 and Y2 or is the most general analog of complex metric in question. guv and gwbar are certainly non-vanishing components of the induced metric. Metric could allow as non-vanishing components also guw and gvbarw. This slicing by pairs of surfaces would correspond to decomposition to a product of string world sheet and partonic 2-surface everywhere.
In Euclidian regions ne would have 4-D complex structure with two complex coordinates (z,w) and their conjugates and completely analogous decompositions. In CP2 one has similar complex structure and actually Kähler structure extending to quaternionic structure. I have actually proposed that quaternion analyticity could provide the general solution of field equations.
- Assuming minimal surface property the field equations for Kähler action reduce to the vanishing of a sum of two terms. The first term comes from the variation with respect to the induced metric and is proportional to the contraction
A=Jαγ JγβHkαβ .
Second term comes from the variation with respect to induced Kähler form and is proportional to
B=jα PksJsl∂αhl .
Here Pkl is projector to the normal space of space-time surface and jα= Dβ Jαβ is the conserved Kähler current.
For the known extremals j vanishes or is light-like (for massless extremals) in which case A and B vanish separately.
- An attractive manner to satisfy field equations would be by assuming that the situation for 2-D minimal surface generalizes so that minimal surface equations are identically satisfied. Extremal property for Kähler action could be achieved by requiring that energy momentum tensor also for Kähler action is of type (1,1) so that one would have A=0. This implies jα ∂αsk=0. This can be true if j vanishes or is light-like as it is for the known extremals and if sk depend only on the light-like coordinate. In Euclidian regions one would have j=0.
- The proposed generalization is especially interesting in the case of cosmic string extremals of form X2× Y2, where X2⊂ M4 is minimal surface (string world sheet) and Y2 is complex homologically non-trivial sub-manifold of CP2 carrying Kähler magnetic charge. The generalization would be that the two transversal coordinates (w,wbar) in the plane orthogonal to the string world sheet defining polarization plane depend holomorphically on the complex coordinates of complex surface of CP2. This would transform cosmic string to flux tube.
- There are also solutions of form X2× Y2, where Y2 is Lagrangian sub-manifold of CP2 with vanishing Kähler magnetic charge and their deformations with (w,barw) depending on the complex coordinates of Y2 (see the slides of "On Lagrangian minimal surfaces on the complex projective plane" ). In this case Y2 is not complex sub-manifold of CP2 with arbitrary genus and induced Kähler form vanishes. The simplest choice for Y2 would be as homologically trivial geodesic sphere. Because of its 2-dimensionality Y2 has a complex structure defined by its induced metric so that solution ansatz makes sense also now.
For a summary of earlier postings see Latest progress in TGD.