Holomorphic unified field theory of gravity and the standard model

Abstract We present a single holomorphic framework in which gravity, all Standard Model interactions, and their couplings to charges and currents emerge from one geometric action on a four-complex dimensional manifold. The Hermitian metric yields, upon restriction to the real slice $$ y^\mu = 0 ,$$ y μ = 0 , a real symmetric metric $$ g_{(\mu \nu )}(x) $$ g ( μ ν ) ( x ) satisfying the vacuum Einstein’s equations, while its imaginary, antisymmetric part $$ g_{[\mu \nu ]}(x) $$ g [ μ ν ] ( x ) reproduces both the homogeneous and inhomogeneous Maxwell identities with explicit coupling to external four-currents. A single holomorphic gauge connection for a simple group $$ G_{\text {GUT}} $$ G GUT such as SU (5) or SO (10) encodes all non-Abelian and Abelian sectors, its Bianchi identities impose the homogeneous Yang–Mills equations, and variation of the same holomorphic action enforces $$\nabla _\mu F^{\mu \nu }_A = J^\nu _A.$$ ∇ μ F A μ ν = J A ν . Chiral fermions are introduced via a holomorphic Dirac Lagrangian that, on $$ y = 0 ,$$ y = 0 , yields exactly the curved-space Dirac equations with minimal coupling to all gauge fields, realizing inclusion of fermions with correct Standard Model charges. Holomorphic gauge invariance automatically imposes the standard anomaly-cancellation conditions. To achieve gauge-coupling unification, we add a holomorphic adjoint Higgs breaking $$G_{\text {GUT}} \rightarrow SU(3) \times SU(2) \times U(1),$$ G GUT → S U ( 3 ) × S U ( 2 ) × U ( 1 ) , ensuring $$ g_3 = g_2 = g_1 $$ g 3 = g 2 = g 1 at the unification scale. A second holomorphic Higgs doublet then breaks $$SU(2)_L

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