I have done a considerable amount of speculative guesswork to identify what I have used to call preferred extremals of Kähler action. The problem is that the mathematical problem at hand is extremely non-linear and that there is no existing mathematical literature. One must proceed by trying to guess the general constraints on the preferred extremals which look physically and mathematically plausible. The hope is that this net of constraints could eventually chrystallize to Eureka! Certainly the recent speculative picture involves also wrong guesses. The need to find explicit ansatz for the deformations of known extremals based on some common principles has become pressing. The following considerations represent an attempt to combine the existing information to achieve this.
What might be the common features of the deformations of known extremals?
The dream is to discover the deformations of all known extremals by guessing what is common to all of them. One might hope that the following list summarizes at least some common features.
Effective three-dimensionality at the level of action
- Holography realized as effective 3-dimensionality also at the level of action requires that it reduces to 3-dimensional effective boundary terms. This is achieved if the contraction jαAα vanishes. This is true if jα vanishes or is light-like, or if it is proportional to instanton current in which case current conservation requires that CP2 projection of the space-time surface is 3-dimensional. The first two options for j have a realization for known extremals. The status of the third option - proportionality to instanton current - has remained unclear.
- As I started to work again with the problem, I realized that instanton current could be replaced with a more general current j=*B∧J or concretely: jα= εαβγδBβJγδ, where B is vector field and CP2 projection is 3-dimensional, which it must be in any case. The contractions of j appearing in field equations vanish automatically with this ansatz.
- Almost topological QFT property in turn requires the reduction of effective boundary terms to Chern-Simons terms: this is achieved by boundary conditions expressing weak form of electric magnetic duality. If one generalizes the weak form of electric magnetic duality to J=Φ *J one has B=dΦ and j has a vanishing divergence for 3-D CP2 projection. This is clearly a more general solution ansatz than the one based on proportionality of j with instanton current and would reduce the field equations in concise notation to Tr(THk)=0.
- Any of the alternative properties of the Kähler current implies that the field equations reduce to Tr(THk)=0, where T and Hk are shorthands for Maxwellian energy momentum tensor and second fundamental form and the product of tensors is obvious generalization of matrix product involving index contraction.
Could Einstein's equations emerge dynamically?
For jα satisfying one of the three conditions, the field equations have the same form as the equations for minimal surfaces except that the metric g is replaced with Maxwell energy momentum tensor T.
- This raises the question about dynamical generation of small cosmological constant Λ: T= Λ g would reduce equations to those for minimal surfaces. For T=Λ g modified gamma matrices would reduce to induced gamma matrices and the modified Dirac operator would be proportional to ordinary Dirac operator defined by the induced gamma matrices. One can also consider weak form for T=Λ g obtained by restricting the consideration to sub-space of tangent space so that space-time surface is only "partially" minimal surface but this option is not so elegant although necessary for other than CP2 type vacuum extremals.
- What is remarkable is that T= Λ g implies that the divergence of T which in the general case equals to jβJβα vanishes. This is guaranteed by one of the conditions for the Kähler current. Since also Einstein tensor has a vanishing divergence, one can ask whether the condition to T= κ G+Λ g could the general condition. This would give Einstein's equations with cosmological term besides the generalization of the minimal surface equations. GRT would emerge dynamically from the non-linear Maxwell's theory although in slightly different sense as conjectured (see this)! Note that the expression for G involves also second derivatives of the imbedding space coordinates so that actually a partial differential equation is in question. If field equations reduce to purely algebraic ones, as the basic conjecture states, it is possible to have Tr(GHk)=0 and Tr(gHk)=0 separately so that also minimal surface equations would hold true.
What is amusing that the first guess for the action of TGD was curvature scalar. It gave analogs of Einstein's equations as a definition of conserved four-momentum currents. The recent proposal would give the analog of ordinary Einstein equations as a dynamical constraint relating Maxwellian energy momentum tensor to Einstein tensor and metric.
- Minimal surface property is physically extremely nice since field equations can be interpreted as a non-linear generalization of massless wave equation: something very natural for non-linear variant of Maxwell action. The theory would be also very "stringy" although the fundamental action would not be space-time volume. This can however hold true only for Euclidian signature. Note that for CP2 type vacuum extremals Einstein tensor is proportional to metric so that for them the two options are equivalent. For their small deformations situation changes and it might happen that the presence of G is necessary. The GRT limit of TGD discussed in kenociteallb/tgdgrt kenocitebtart/egtgd indeed suggests that CP2 type solutions satisfy Einstein's equations with large cosmological constant and that the small observed value of the cosmological constant is due to averaging and small volume fraction of regions of Euclidian signature (lines of generalized Feynman diagrams).
- For massless extremals and their deformations T= Λ g cannot hold true. The reason is that for massless extremals energy momentum tensor has component Tvv which actually quite essential for field equations since one has Hkvv=0. Hence for massless extremals and their deformations T=Λ g cannot hold true if the induced metric has Hamilton-Jacobi structure meaning that guu and gvv vanish. A more general relationship of form T=κ G+Λ G can however be consistent with non-vanishing Tvv but require that deformation has at most 3-D CP2 projection (CP2 coordinates do not depend on v).
- The non-determinism of vacuum extremals suggest for their non-vacuum deformations a conflict with the conservation laws. In, also massless extremals are characterized by a non-determinism with respect to the light-like coordinate but like-likeness saves the situation. This suggests that the transformation of a properly chosen time coordinate of vacuum extremal to a light-like coordinate in the induced metric combined with Einstein's equations in the induced metric of the deformation could allow to handle the non-determinism.
Are complex structure of CP2 and Hamilton-Jacobi structure of M4 respected by the deformations?
The complex structure of CP2 and Hamilton-Jacobi structure of M4 could be central for the understanding of the preferred extremal property algebraically.
- There are reasons to believe that the Hermitian structure of the induced metric ((1,1) structure in complex coordinates) for the deformations of CP2 type vacuum extremals could be crucial property of the preferred extremals. Also the presence of light-like direction is also an essential elements and 3-dimensionality of M4 projection could be essential. Hence a good guess is that allowed deformations of CP2 type vacuum extremals are such that (2,0) and (0,2) components the induced metric and/or of the energy momentum tensor vanish. This gives rise to the conditions implying Virasoro conditions in string models in quantization:
gξiξj=0 , gξ*iξ*j=0 , i,j=1,2 .
Holomorphisms of CP2 preserve the complex structure and Virasoro conditions are expected to generalize to 4-dimensional conditions involving two complex coordinates. This means that the generators have two integer valued indices but otherwise obey an algebra very similar to the Virasoro algebra. Also the super-conformal variant of this algebra is expected to make sense.
These Virasoro conditions apply in the coordinate space for CP2 type vacuum extremals. One expects similar conditions hold true also in field space, that is for M4 coordinates.
- The integrable decomposition M4(m)=M2(m)+E2(m) of M4 tangent space to longitudinal and transversal parts (non-physical and physical polarizations) - Hamilton-Jacobi structure- could be a very general property of preferred extremals and very natural since non-linear Maxwellian electrodynamics is in question. This decomposition led rather early to the introduction of the analog of complex structure in terms of what I called Hamilton-Jacobi coordinates (u,v,w,w*) for M4. (u,v) defines a pair of light-like coordinates for the local longitudinal space M2(m) and (w,w*) complex coordinates for E2(m). The metric would not contain any cross terms between M2(m) and E2(m): guw=gvw= guw* =gvw*=0.
A good guess is that the deformations of massless extremals respect this structure. This condition gives rise to the analog of the constraints leading to Virasoro conditions stating the vanishing of the non-allowed components of the induced metric. guu= gvv= gww=gw*w* =guw=gvw= guw* =gvw*=0. Again the generators of the algebra would involve two integers and the structure is that of Virasoro algebra and also generalization to super algebra is expected to make sense. The moduli space of Hamilton-Jacobi structures would be part of the moduli space of the preferred extremals and analogous to the space of all possible choices of complex coordinates. The analogs of infinitesimal holomorphic transformations would preserve the modular parameters and give rise to a 4-dimensional Minkowskian analog of Virasoro algebra. The conformal algebra acting on CP2 coordinates acts in field degrees of freedom for Minkowskian signature.
Field equations as purely algebraic conditions
If the proposed picture is correct, field equations would reduce basically to purely algebraically conditions stating that the Maxwellian energy momentum tensor has no common index pairs with the second fundamental form. For the deformations of CP2 type vacuum extremals T is a complex tensor of type (1,1) and second fundamental form Hk a tensor of type (2,0) and (0,2) so that Tr(THk)= is true. This requires that second light-like coordinate of M4 is constant so that the M4 projection is 3-dimensional. For Minkowskian signature of the induced metric Hamilton-Jacobi structure replaces conformal structure. Here the dependence of CP2 coordinates on second light-like coordinate of M2(m) only plays a fundamental role. Note that now Tvv is non-vanishing (and light-like). This picture generalizes to the deformations of cosmic strings and even to the case of vacuum extremals.
For background see the chapter Basic Extremals of Kähler action. See also the article About deformations of known extremals of Kähler action.