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.
The article examines 2-D minimal surfaces X2 in the 4-D space X4 assumed to have twistor space. From superminimality which looks somewhat peculiar assumption, it follows that in the twistor space of X4 (assuming that it exists) there is a Lagrangian surface, which is also a minimal surface. Superminimality means that the normal spaces of the 2-surface form a 1-D curve in the space of all normal spaces, which for the Euclidian signature is the 4-D Grassmannian SO(4)/SO(2)× SO(2)= S2× S2 (SO(1,3)/SO(1,1)× SO(2) for M4). Superminimal surface is therefore highly flattened. Of course, already the minimal surface property favours flatness.
Why the result is interesting from the TGD point of view?
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. Certainly, if M4Käher form vanishes, 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.
Superminimal surfaces look like the opposite of holomorphic minimal surfaces (this expectation turned tou to be wrong!). If M4Käher form vanishes, their counterparts give a huge vacuum degeneracy and non-determinism for the pure Kähler action, which turned out to be mathematically undesirable. The cosmological constant, which follows from twistoralization, was believed to correct the situation.
I had not noticed that the Kähler action, whose existence for T(H)=T(M4)× T(CP2) fixes the choice of H, gives a huge number of 6-D Lagrangian manifolds! Are they consistent with dimensional reduction, so that 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? Furthermore, could one extend the very promising looking holography=holomorphy picture by allowing also Lagrangian 6-surfaces T(H)?
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!? And what about the Lagrangian minimal surfaces possibly obtained when Λ is non-vanising?
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.
Lagrangian surfaces in H=M4× CP2 and its twistor space
So let's consider the 12-D twistor space T(H)=T(M4)× T(CP2) and its 6-D Lagrangian surfaces having a local decomposition X6=X4× S2. Assume a twistor lift with Kähler action on T(H). It exists only for H=M4× 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).
- Since X6=X4× S2 is Lagrangian, the induced symplectic form of the for it must vanish. This is also true in S2. Fibers S2 together with T(M4) and T(CP2) are identified by an 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 6-D Kähler action, its contribution to the 6-D Kähler action vanishes. The cosmological constant is zero because the S2 contribution to the 4-D action vanishes.
- 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 in its original form, involving only CP2 Kähler form, 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, and of course holography=holomorphy principle, is lost. This option is not physically acceptable.
- 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 much more general surface M4 projection, even 4-D, was Lagrangian. There is no loss of holography! Holography = holomorphy principle is however lost. Holography would be replaced with the Lagrangian property.
- The symplectic transformations of H produce new Lagrangian vacuum surfaces. If they are allowed, one might talk of symplectic phase. The second phase would be the holomorphic phase. The two major symmetry groups of physics would both be involved. For Λ= 0 these Lagrangian surfaces are classical vacua and also fermionic vacua because the modified gamma matrices appearing in the modified Dirac action vanish identically. Therefore Λ=0 sector does not contribute to physics at all. For non-vanishing Λ one has only minimal Lagrangian surfaces, which are string like entities and they contribute to physics.
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 this phase induced Kähler form and induced color gauge fields vanish and it would not involve monopole fluxes. This phase might be called Maxwell phase for nonvanishing Λ. Could it correspond to the Coulomb phase as the perturbative phase of the gauge theories, while the monopole flux tubes (large heff and dark matter) would correspond to the non-perturbative phase in which magnetic monopole fluxes are present. If so, there would be an analogy with the electric-magnetic duality of gauge theories although the two phases does not look like two equivalent descriptions of one and the same thing unless one restricts the consideration to fermions.
I have not yet considered the question whether the Lagrangian surfaces can be minimal surfaces as they should be for a non-vanishing Λ. In the theorem the minimal Lagrangian surfaces were superminimal surfaces.
- For super-minimal surfaces, a unit vector in the normal direction defines a 1-D very specific curve in normal space. It should be noted that for minimal surfaces, however, the second fundamental form disappears and cannot be used to define the normal vector. Lagrangian surfaces in twistor space also turned out to be minimal surfaces.
- The field equations for the Kähler action do not force the Lagrangian surfaces to be minimal surfaces. However, there exists a lot of minimal Lagrangian surfaces.
- In CP2, a homologically trivial geodesic sphere is a minimal surface. Note that the geodesic spheres obtained by isometries are regarded here as equivalent. Also g=1 minimal Lagrangian surface in CP2 is known. There are many other minimal Lagrangian surfaces and second order differential equations for these surfaces are known (see this).
- In M4, the plane M2 is an example of a minimal surface, which is a Lagrangian surface. Are there others? Could Hamilton-Jacobi structures (see this) that also involve the symplectic form and generalized Kähler structure (more precisely, their generalizations) define Lagrangian surfaces in M4? There is a general construction for Lagrangian minimal surfaces in M4 allowing to construct them from the solutions of a massless Dirac equation.
- In the case considered now, the Lagrangian surfaces in H would be products X2 × Y2. Interestingly, in the 2-D case the induced metric always defines a holomorphic structure. Now, however, this holomorphic structure would not be the same as the one related to the holomorphic ansatz for which it is induced from H.
These findings raise several questions related to the detailed understanding of TGD. Should one allow only non-vanishing values of Λ? This would allow minimal Langrangian surfaces X2× Y2 besides the holomorphic ansatz. The holomorphic structure due to the 2-dimensionality of X2 and Y2 means that holography=holomorphy principle generalizes.
If one allows Λ=0, all Lagrangian surfaces X2× Y2 are allowed but also would have a holomorphic structure due to the 2-dimensionality of X2 and Y2 so that holography=holomorphy principle would generalize also now! Minimal surface property is obtained as a special case. Classically the extremals correspond to a vacuum sector and also in the fermionic sector modified Dirac equation is trivial. Therefore there is no physics involved.
Minimal Lagrangian surfaces are favored by the physical interpretation in terms of a geometric analog of the field particle duality. The orbit of a particle as a geodesic line (minimal 1-surface) generalizes to a minimal 4-surface and the field equations inside this surface generalizes massless field equations.
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.