Saturday, January 06, 2024

The twistor space of H=M4× CP2 allows Lagrangian 6-surfaces: what does this mean physically?

This article was inspired by the article "A note on Lagrangian submanifolds of twistor spaces and their relation to superminimal surfaces" of Reinier Storm. For curiosity, I decided to look at Lagrangian surfaces in the twistor space of H=M4× CP2. The 6-D Kähler action of the twistor space existing only for H=M4× CP2 gives by a dimensional reduction rise to 6-D analog of twistor space assitable to a space-time surface. In the dimensional reduction the action reduces to 4-D Kähler action plus a volume term characterized by a dynamically determined cosmological constant Λ.

One can identify space-time surfaces, which are Lagrangian minimal surfaces and therefore have a vanishing Kähler action. If the Kähler structure of M4 is non-trivial as strongly suggested by the notion of twistor space, these vacuum extremals are products X2× Y2 of Lagrangian string world sheet X2 and 2-D Lagrangian surface Y2 of CP2, and are deterministic so that they allow holography. As minimal surfaces they allow a generalization of holography= holomorphy principle: now the holomorphy is not induced from that of H but by 2-D nature of X2 and Y2. Therefore holography=holomorphy principle generalizes.

Λ can vanish and in this case the dimensionally reduced action equals Kähler action. In this case, vacuum extremals are in question and symplectic transformations generate a huge number of these surfaces, which in general are not minimal surfaces. Holography= holomorphy principle is not however lost. Λ=0 sector contains however only classical vacua and also the modified gamma matrices appearing in the modified Dirac action vanish so that this sector contributes nothing to physics.

See the article The twistor space of H=M4× CP2 allows Lagrangian 6-surfaces: what does this mean physically? or the chapter Symmetries and Geometry of the ”World of Classical Worlds” .

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.

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