Saturday, August 17, 2019

Could metaplectic group have some role in TGD framework?

Metaplectic group appears as a covering group of linear symplectic group Sp(2n,F) for any number field and its representations can be regarded as analogs of spinor representations of the rotation group. Since infinite-D symplectic group of δ M4+× CP2, where δ M4+ is light-cone boundary, appears as an excellent candidate for the isometries of the "world of classical worlds" in zero energy ontology (ZEO), one can ask whether and how the notion of metaplectic group could generalize to TGD framework.

The condition for the existence of metaplectic structure is same as of the spinor structure and not met in the case of CP2. One however expects that also the modified metaplectic structure exists if one couples spinors to an odd integer multiple of Kähler gauge potential. For triality 1 representation assignable to quarks one has n=1. The fact that the center of SU(3) is Z3 suggests that metaplectic group for CP2 is 3- or 6-fold covering of symplectic group instead of 2-fold covering.

Besides the ordinary representations of SL(2,C) also the possibly existing analogs of metaplectic representations of SL(2,C) = Sp(2,C) acting on wave functions in hyperbolic space H3 represented as a2=t2-r2 hyperbololoid of M4+ are cosmologically interesting since the many-sheeted space-time in number theoretic vision allows quantum coherence in even cosmological scales and there are indications for periodic redshift suggests tesselations of H3 analogous to lattices in E3 and defined by discrete subgroup of Sl(2,C). In this case one could require that only the subgroup SU(2) is represented projectively so that one would have an analogy with modular functions for discrete subgroup of SL(2,Z) would be represented in this manner.

See the chapter Recent View about K\"ahler Geometry and Spin Structure of WCW.

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