I became again interested in finding preferred extremals of Kähler action, which would have 4-D CP2 and perhaps also M4 projections. This would correspond to Maxwell phase that I conjectured long time ago. Deformations of CP2 type vacuum extremals would correspond also to these extremals. The signature of the induced metric might be also Minkowskian. It however turns out that the
solution ansatz requires Euclidian signature and that M4 projection is 3-D so that original hope is not realized.
I proceed by the following arguments to the ansatz.
One could try to develop ansatz to a more detailed form. The most obvious guess is that the induced metric is apart from constant conformal factor the metric of CP2. This would guarantee self-duality apart from constant factor and jα=0. Metric would be in complex CP2 coordinates tensor of type (1,1) whereas CP2 Riemann connection would have only purely holomorphic or anti-holomorphic indices. Therefore CP2 contributions in Tr(THk) would vanish identically. M4 degrees of freedom however bring in difficulty. The M4 contribution to induced metric should be proportional to CP2 metric and this is impossible due to the different signatures.
The M4 contribution to the induced metric breaks its Kähler property.
- Effective 3-dimensionality for action (holography) requires that action decomposes to vanishing jαAα term + total divergence giving 3-D "boundary" terms. The first term certainly vanishes (giving effective 3-dimensionality and therefore holography) for
Empty space Maxwell equations, something extremely natural. Also for the proposed GRT limit these equations are true.
- How to obtain empty space Maxwell equations jα=0? Answer is simple: assume self duality or its slight modification:
holding for CP2 and CP2 type vacuum extremals or a more general condition
k some constant not far from unity. * is Hodge dual involving 4-D permutation symbol.k=constant requires that the determinant of the induced metric is apart from constant equal to that of CP2 metric. It does not require that the induced metric is proportional to the CP2 metric, which is not possible since M4 contribution to metric has Minkowskian signature and cannot be therefore proportional to CP2 metric.
- Field equations reduce with these assumptions to equations differing from minimal surfaces equations only in that metric g is replaced by Maxwellian energy momentum tensor T. Schematically:
where T is Maxwellian energy momentum tensor and Hk is the second fundamental form - asymmetric 2-tensor defined by covariant derivative of gradients of imbedding space coordinates.
- It would be nice to have minimal surface equations since they are the non-linear generalization of massless wave equations. This is achieved if one has
T= Λ g .
Maxwell energy momentum tensor would be proportional to the metric! One would have dynamically generated cosmological constant! This begins to look really interesting since it appeared also at the proposed GRT limit of TGD.
- Very skematically and forgetting indices and being sloppy with signs, the expression for T reads as
T= JJ -g/4 Tr(JJ) .
Note that the product of tensors is obtained by generalizing matrix product. This should be proportional to metric.
Self duality implies that Tr(JJ) is just the instanton density and does not depend on metric and is constant.
For CP2 type vacuum extremals one obtains
T= -g+g=0 .
Cosmological constant would vanish in this case.
- Could it happen that for deformations a small value of cosmological constant is generated? The condition would reduce to
JJ= (Λ-1)g .
Λ must relate to the value of parameter k appearing in the generalized self-duality condition. This would generalize the defining condition for Kähler form
JJ=-g (i2=-1 geometrically)
stating that the square of Kähler form is the negative of metric. The only modification would be that index raising is carried out by using the induced metric containing also M4 contribution rather than CP2 metric.
Jαμ Jμβ = (Λ-1)gαβ .
Cosmological constant would measure the breaking of Kähler structure.
A more realistic guess based on the attempt to construct deformations of CP2 type vacuum extremals is following.
What makes the ansatz attractive is that a special solutions of Euclidian Maxwell empty space equations are in question, equations reduces to non-linear generalizations of Euclidian massless wave equations Minkowskian coordinate variables, and cosmological constant pops up dynamically. What makes the ansatz attractive is that special solutions of Maxwell empty space equations are in question, equations reduces to non-linear generalizations of Euclidian massless wave equations, and cosmological constant pops up dynamically. These properties are true also for the GRT limit of TGD that I discussed in here.
- Physical intuition suggests that M4 coordinates can be chosen so that one has integrable decomposition to longitudinal degrees of freedom parametrized by two light-like coordinates u and v and to transversal polarization degrees of freedom parametrized by complex coordinate w and its conjugate. M4 metric would reduce in these coordinates to a direct sum of longitudinal and transverse parts. I have called these coordinates Hamilton Jacobi coordinates.
- w would be holomorphic function of CP2 coordinates and therefore satisfy massless wave equation. This would give hopes about rather general solution ansatz. u and v cannot be holomorphic functions of CP2 coordinates. Unless wither u or v is constant, the induced metric would have contributions of type (2,0) and (0,2) coming from u and v which would break Kähler structure and complex structure. These contributions would give no-vanishing contribution to all minimal surface equations. Therefore either u or v is constant: the coordinate line for non-constant coordinate -say u- would be analogous to the M4 projection of CP2 type vacuum extremal.
- With these assumptions the induced metric would remain (1,1) tensor and one might hope that Tr(THk) contractions vanishes for all variables except u because the there are no common index pairs (this if non-vanishing Christoffel symbols for H involve only holomorphic or anti-holomorphic indices in CP2 coordinates). For u one would obtain massless wave equation expressing the minimal surface property.
- The induced metric would contain only the contribution from the transversal degrees of freedom besides CP2 contribution. Minkowski contribution has however rank 2 as CP2 tensor and cannot be proportional to CP2 metric. It is however enough that its determinant is proportional to the determinant of CP2 metric with constant proportionality coefficient. This condition gives an additional non-linear condition to the solution. One would have wave equation for u (also w and its conjugate satisfy massless wave equation) and determinant condition as an additional condition.
The determinant condition reduces by the linearity of determinant with respect to its rows to sum of conditions involved 0,1,2 rows replaced by the transversal M4 contribution to metric given if M4 metric decomposes to direct sum of longitudinal and transversal parts. Derivatives with respect to derivative with respect to particular CP2 complex coordinate appear linearly in this expression they can depend on u via the dependence of transversal metric components on u. The challenge is to show that this equation has non-trivial solutions.
For background see the chapter Basic Extremals of Kähler action.