A Moduli Space of the Quaternionic Hopf Surface Encodes Standard Model Physics

Abstract

The quaternionic Hopf surface, HL, is associated with a non-compact moduli space, ML, of stable holomorphic SL(2,C) bundles. ML is open in MLc, the corresponding compact moduli space of holomorphic SL(2,C) bundles, and naturally fibers over an open set of the quaternionic projective line HP^1. We pull back to ML natural locally conformal kaehler and hyperkaehler structures from MLc, and lift natural sub-pseudoriemannian and optical structures from HP^1. Unexpectedly, the holomorphic maps connecting these structures solve the the classical Dirac-Higgs equations of the unbroken Standard Model. These equations include: all observed fermionic and bosonic fields of all three generations with the correct color, weak isospin, and hypercharge values; a Higgs field coupling left and right fermion fields; and a pp-wave gravitational metric. We hypothesize that physics is essentially the geometry of ML, both algebraic (quantum) and differential (classical). We further show that the Yang-Mills equations with fermionic currents also naturally emerge, along with an induced action on the ML structure sheaf equivalent to the time-evolution operator of the associated quantum field theory.