Geometric QCD II: The Confining Twistor String and Meson Spectrum

Abstract: We develop a continuum framework for confining planar QCD ($N_c\to\infty$) by quantizing the Fermi-string (1981) degrees of freedom on the rigid Hodge-dual minimal surface of Geometric QCD I. Worldsheet Majorana fermions provide the algebraic mechanism required by the Makeenko-Migdal loop equations: the Pauli principle cancels non-planar self-intersections, preserving planar factorization. The Liouville instability of random surfaces is avoided since the bulk geometry is rigidly fixed holographically by the boundary loop, without summing over fluctuating metrics. Formulating the theory in momentum loop space integrates out coordinate-space cusp singularities, yielding a finite local limit for quark-loop amplitudes. Changing variables in the phase-space path integral, after fixing reparametrization symmetry (Virasoro constraint), parametrizes the reduced measure by boundary twistors and an induced Jacobian. This yields a twistor-string integral derived directly from the planar QCD amplitude. In the local limit, the theory becomes an analytic twistor string: a boundary sigma model $S^1\to (S^3\times S^3)/S^1$ coupled to a holographically determined Liouville field $ρ(z,\bar z)=\log(|λ(z)||μ(z)|)$ obtained by analytic continuation of boundary twistor data. The meson spectrum is encoded in a 1D functional integral over the boundary twistor trajectory $λ(z),μ(z)$ on $|z|=1$. While evaluated numerically via a reweighted Complex Langevin approach (extracting linear Regge trajectories as a proof of concept), our ultimate finding is analytical. Analyzing complexified effective action monodromies reveals the discrete mass spectrum is exactly governed by Catastrophe Theory and classified by the topological number of twistor poles inside the unit circle. The 1-pole sector exactly yields $m^2 = πσ(n + 1/12)$.