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

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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.

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

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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