About Lagrangian surfaces in the twistor space of M4×CP2

I received from Tuomas Sorakivi a link to the article “A note on Lagrangian submanifolds of twistor spaces and their relation to superminimal surfaces” (see this). The author of the article is Reinier Storm from Belgium.

The abstract of the article tells roughly what it is about.

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian 4-manifold and particular Lagrangian submanifolds of the twistor space over the 4-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration restricted to the constructed Lagrangian gives a circle bundle over the superminimal surface. Conversely, if a submanifold of the twistor space is Lagrangian for all the natural almost Hermitian structures, then the Lagrangian projects to a superminimal surface and is contained in the Lagrangian constructed from this surface. In particular this produces many Lagrangian submanifolds of the twistor spaces and with respect to both the Kähler structure as well as the nearly Kähler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds.

It is interesting to examine the generalization of the result to TGD because the interpretation for Lagrangian surfaces, which are vacuum extremals for the Kähler action with a vanishing induced symplectic form, has remained open. In any case, they do not fulfill the holomorphy=holography assumption, i.e. they are not surfaces for which the generalized complex structure in H induces a corresponding structure at 4-surface.

The article examines 2-D minimal surfaces in the 4-D space X⁴ assumed to have twistor space. String world sheets could be an example in TGD. From superminimalism, which looks like a peculiar assumption, it follows that in the twistor space of X⁴ there is a Lagrangian surface, which is also a minimal surface.

These surfaces are in a sense the opposite of holomorphic minimal surfaces. In TGD, they give a huge vacuum degeneracy for the pure Kähler action, which turned out to be mathematically undesirable. The cosmological constant, which follows from twistoralization, corrects the situation.

I had not noticed that the Kähler action, whose existence for T(H)=M⁴× T(CP₂) fixes the choice of H, gives a huge number of Lagrangian manifolds! They would be 6-D. Are they consistent with dimensional reduction, so they could be interpreted as induced twistor structures? Can a complex structure be attached to them? Certainly not as an induced complex structure. Does the Lagrangian problem of Kähler action make a comeback?

Do they have a physical interpretation, most naturally as vacuums? The volume term of the 4-D action characterized by the cosmological constant Λ does not allow vacuum extremals unless Λ vanishes. But Λ is dynamic for the twistor lift and can vanish! Do Lagrangian surfaces in twistor space correspond to 4-D minimal surfaces in H, which are vacuums and have a vanishing cosmological constant. Could even the original formulation of TGD using only Kähler action be an exact part of the theory and not just a long-length-scale limit? And does one really avoid the original problem due to huge non-determinism of vacuum extremals!?

The question is whether the result presented in the article could generalize to the TGD framework even though the super-minimality assumption does not seem physically natural at first.

So let’s consider the 12-D twistor space T(H)=T(M⁴)times T(CP₂) and its 6-D Lagrangian surfaces X⁶=X⁴× S² . Assume a twistor lift with Kähler action on T(H). It exists only for H=M²× CP₂.

Let us for a moment forget the requirement that these Lagrangian surfaces correspond to minimal surfaces in H. Let us first consider the situation in which there is no generalized Kähler and symplectic structure for M⁴.

One can actually identify Lagrangian surfaces in 12-D twistor space T(H).

1. Since X⁶=X⁴× S² is Lagrangian, the induced symplectic form of the for it war must vanish. This is also true in S². Strands S² together with T(M⁴) and T(CP₂) are identified by orientation-changing isometry. The induced Kähler form S² in the subset X⁶=X⁴× S² is zero as the sum of these two contributions of different signs. If this sum appears in the Kähler actiont, the contribution it determines to the 6-D Kähler action is zero. The cosmological constant is zero because the S² contribution to the 4-D action vanishes.
2. Therefore the 6-D Kähler action reduces in X⁴ to the 4-D Kähler action, which was the original guess for the 4-D action. The problem is that it involves a huge vacuum degeneracy. The CP₂ projection is a Lagrangian surface or its subset but the dynamics of M⁴ projection is essentially arbitrary, in particular with respect to time. One obtains a huge number of different solutions. Since the time evolution is non-deterministic, the holography is lost. Because of this, this option is not physically acceptable. Therefore these surfaces do not implement the holomorphy=holography principle.

How the situation changes when also M⁴ has a generalized Kähler form that the twistor space picture strongly suggests, and actually requires.

1. Now the Lagrangian surfaces would be products X²× Y², where X² and Y² are the Lagrangian surfaces of M⁴ and CP₂. The M⁴ projections of these objects look like string world sheets and in their ground state are vacuums.

Furthermore, the situation is deterministic! The point is that X² is Lagrangian and fixed as such. In the previous case it was a surface in M⁴ which was Lagrangian. There is no loss of holography! Holography = holomorphy principle is however lost. Holography would be replaced with the Lagrangian property. So this option cannot be ruled out.

It should be however made clear that the symplectic transformations are not isometries so that minimal surface property is not preserved. Minimal surface property would reduce the vacuum symmetries to isometries.

In fact, it is known that the so-called real K₃ surfaces are 2-D minimal Lagrangian surfaces in R⁴ (see this). A naive expectation is that the real projection of a complex K₃ surface defines a real K₃ surface. One can hope that real K₃ surfaces generalize from R⁴ to M⁴. According to the article of Yng-Ing Lee (see this) K₃ surfaces are algebraic surfaces and have the same properties as holomorphic curves for which complex structure is induced from embedding space: now it would be due to the 2-D nature of the real K₃ surface (see this). Real K₃ surfaces are also 2-D Calabi-Yau manifolds! As found, minimal surface property requires additional assumptions that could correspond to the somewhat strange-looking assumption of the theorem. Could super-minimalism be another way to state these assumptions?