The grain boundary morphology of a lamella-forming poly(styrene-b-isoprene) (SI) diblock copolymer was investigated by transmission electron microtomography. The twist grain boundary, at which two lamellar nanodomains orthogonally intersect, was successfully observed in three dimensions (3D). A twodimensional periodic minimal surface, the Scherk's first surface, was once hypothesized as a model of such a grain boundary morphology but never experimentally ascertained. The area-averaged curvatures of the interface between the PI and PS nanodomains as well as the interfacial area per unit volume suggested that the grain boundary morphology had characteristics of the saddlelike hyperbolic surface and was found to be quite similar to Scherk's first surface.
A 6-dimensional grand unified theory with the compact space having the topology of a real projective plane, i.e., a 2-sphere with opposite points identified, is considered. The space is locally flat except for two conical singularities where the curvature is concentrated. One supersymmetry is preserved in the effective 4d theory. The unified gauge symmetry, for example SU(5) , is broken only by the non-trivial global topology. In contrast to the Hosotani mechanism, no adjoint Wilson-line modulus associated with this breaking appears. Since, locally, SU(5) remains a good symmetry everywhere, no UV-sensitive threshold corrections arise and SU(5)-violating local operators are forbidden. Doublettriplet splitting can be addressed in the context of a 6d N = 2 super Yang-Mills theory with gauge group SU(6). If this symmetry is first broken to SU(5) at a fixed point and then further reduced to the standard model group in the above non-local way, the two light Higgs doublets of the MSSM are predicted by the group-theoretical and geometrical structure of the model.
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