Geometric QCD II: The Confining Twistor String and Meson Spectrum

Abstract

We present a local, asymptotically free solution of the planar Makeenko--Migdal loop equations in the continuum limit with full Lorentz invariance. The solution is constructed by quantizing internal Majorana fermions (referred to here as ``elves'') on a rigid Hodge-dual minimal surface. These worldsheet degrees of freedom provide the algebraic mechanism required to satisfy the unintegrated vector loop equations, with the Pauli principle enforcing planar factorization. In the local limit, the theory reduces to a confining analytic twistor-string representation. By analyzing the monodromy structure of the complexified effective action, we show that the discrete mass spectrum is organized by topological data associated with twistor singularities. The simplest sector with one branch point yields parametric Regge trajectories expressed in terms of trigonometric functions. These trajectories are non-linear but approximately linear over a broad range and are in agreement with experimental data for light mesons. The asymptotic behavior of the trajectory $J = \alpha(M^2)$ and its subleading corrections arise from the interaction between the Liouville term and the twistor monodromy, without introducing additional assumptions about string excitations. In our solution, the QCD mass spectrum follows from a generalized eigenvalue problem in complexified phase space, effectively reducing the problem to classical geometry. Within this framework, the large-$N_c$ Master Field is realized as a classical trajectory in twistor space.