The Twistor Space of H=M4 x CP2 Allows Lagrangian 6-Surfaces: What Does this Mean Physically?

Abstract

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 x CP₂ The 6-D Kahler action of the twistor space existing only for H=M⁴ x 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 Kahler 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 Kahler action. If the Kahler structure of M⁴ is non-trivial as strongly suggested by the notion of twistor space, these vacuum extremals are products X² x 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 $X^2$ and Y². Therefore holography=holomorphy principle generalizes. Λ can vanish and in this case the dimensionally reduced action equals Kahler 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.