Holomorphic unified field theory of gravity and the standard model
Bibliographic record
Abstract
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 $$ y^\mu = 0 ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>μ</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> a real symmetric metric $$ g_{(\mu \nu )}(x) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> satisfying the vacuum Einstein’s equations, while its imaginary, antisymmetric part $$ g_{[\mu \nu ]}(x) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> reproduces both the homogeneous and inhomogeneous Maxwell identities with explicit coupling to external four-currents. A single holomorphic gauge connection for a simple group $$ G_{\text {GUT}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mtext>GUT</mml:mtext> </mml:msub> </mml:math> 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 $$\nabla _\mu F^{\mu \nu }_A = J^\nu _A.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∇</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>J</mml:mi> <mml:mi>A</mml:mi> <mml:mi>ν</mml:mi> </mml:msubsup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Chiral fermions are introduced via a holomorphic Dirac Lagrangian that, on $$ y = 0 ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> 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 $$G_{\text {GUT}} \rightarrow SU(3) \times SU(2) \times U(1),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mtext>GUT</mml:mtext> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> ensuring $$ g_3 = g_2 = g_1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> at the unification scale. A second holomorphic Higgs doublet then breaks $$SU(2)_L \times U(1)_Y \rightarrow U(1)_{\text {EM}},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>Y</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>EM</mml:mtext> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> generating $$ W^\pm ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi>
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".