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Résumé |
We present the exact analytic solution of the Makeenko-Migdal loop equations,
solving planar QCD (Nc→∞) in the continuum limit. Quantizing internal Majorana
fermions (elves) on a rigid Hodge-dual minimal surface provides the algebraic
mechanism satisfying the unintegrated vector loop equations. The Pauli principle
exactly cancels non-planar intersections to reproduce planar factorization, while
iterating the equation generates the planar graphs of asymptotically free QCD.
Holographically fixing the bulk geometry by the boundary loop strictly avoids
Liouville instability.
Momentum loop space integrates out coordinate-space cusp singularities, yielding a
finite local limit. Gauge-fixing the Virasoro constraint parametrizes the reduced
phase-space measure by boundary twistors. The theory reduces to a confining
analytic twistor string: a boundary sigma model S1→(S3×S3)/S1 coupled to a
holographically determined Liouville field.
The meson spectrum becomes a 1D functional integral over boundary twistor
trajectories. Complexified action monodromies reveal a discrete mass spectrum
governed by Catastrophe Theory, classified by the topological number of twistor
poles inside the unit circle. This geometric localization is an infinite-
dimensional realization of Exact WKB analysis and Picard-Lefschetz resurgence,
where twistor poles act identically to Seiberg-Witten branch points.
The 1-pole sector yields the exact Regge spectrum m2=πσ2(n+1/24), matching
experimental π,K,ρ trajectories within 95% confidence. The empirically correct
open-string intercepts emerge directly from the conformal anomaly of the
microscopic elves rather than macroscopic string vibrations. Ultimately, this
explicitly realizes Witten's Master Field as a critical classical trajectory in
twistor space. |
