2025/11/17 by Alexander Migdal, Migdal, Alexander
Physics and Astronomy · #Black Holes and Theoretical Physics #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Particle physics theoretical and experimental studies #Quantum Chromodynamics and Particle Interactions
paper · pdf · doi:10.48550/arxiv.2511.13688
openalex publication_date 2025/11/17 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/30
This is the first of two papers presenting a geometric framework for Planar QCD (Nc → ∞). In this part, we establish the kinematic foundation of the theory by constructing the unique stable vacuum of the loop equation. We demonstrate that the Makeenko-Migdal loop equation admits a solution of the form W[C] = Wpert[C] exp-κS[C], provided S[C] is a specific minimal surface possessing a self-dual area derivative. We prove that such a surface exists and corresponds to the Hodge-dual projection of a minimal surface in ℝ3 ⊗ ℝ4. Crucially, this confinement mechanism relies on the self-duality of the area derivative -- a property that exists exclusively in four dimensions. This geometric constraint ensures stability only in D=4, distinguishing the resulting theory from standard string models which require higher critical dimensions. We relate the string tension parameter κ to the gluon condensate via the Operator Product Expansion. The dynamical quantization of the Fermi string on this rigid surface and the resulting meson spectrum are derived in the companion paper \citeMigdal2026GeometricQCDII.