Holomorphic Unified Field Theory of Gravity and the Standard Model

Abstract

We present a holomorphic framework in which gravity, gauge interactions, and their couplings to charges and currents emerge from a single geometric action on a four-complex-dimensional manifold. The Hermitian metric yields on the real slice $y^{\mu }=0$, a real symmetric metric $g_{(\mu \nu )}$ giving the vacuum Einstein equations, and an antisymmetric part $g_{[\mu \nu ]}$ that reproduces Maxwell's equations with sources. A single holomorphic gauge connection for $G_{\text{GUT}}$, such as $SU(5)$ or $SO(10)$, encodes all gauge sectors; its Bianchi identities give homogeneous Yang--Mills equations, and variation imposes ${\nabla }_{\mu }{F}_A^{\mu \nu }=J_A^{\nu }$. Chiral fermions arise from a holomorphic Dirac Lagrangian and couple minimally to all gauge fields, reproducing the Standard Model spectrum. Anomaly cancellation follows from holomorphic gauge invariance. A holomorphic adjoint Higgs breaks $G_{\text{GUT}}\to SU(3)\times SU(2)\times U(1)$ with unified coupling, and a second Higgs breaks electroweak symmetry, generating $W^{\pm }$, $Z$, and fermion masses. Below the unification scale, couplings run by standard renormalization-group flow. This construction unifies Einstein gravity, Yang--Mills theory, electromagnetism, and chiral fermions into a single classical geometric framework, and admits quantization via a holomorphic path integral that reproduces standard Feynman rules.

Comments: 13 pages