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Geometric QCD II: The Confining Twistor String and Meson Spectrum

Alexander Migdal

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 $m^2 = πσ(n + 1/12)$.

Geometric QCD II: The Confining Twistor String and Meson Spectrum

Abstract

We develop a continuum framework for confining planar QCD () 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 coupled to a holographically determined Liouville field obtained by analytic continuation of boundary twistor data. The meson spectrum is encoded in a 1D functional integral over the boundary twistor trajectory on . 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 .
Paper Structure (111 sections, 3 theorems, 317 equations, 18 figures, 2 tables, 1 algorithm)

This paper contains 111 sections, 3 theorems, 317 equations, 18 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization of the paper and navigation points.
  3. Zero modes of Loop equation
  4. Self-dual area derivative and loop equation
  5. Compatibility with the MM loop equations.
  6. The Physical Vacuum and Multi-Instanton Resummation
  7. Hodge-dual minimal surface
  8. The external and internal geometry
  9. The Holomorphic Map
  10. The Virasoro Constraint and Uniformization
  11. The Elfin theory on a flat surface
  12. Conformal metric as a local gauge parameter
  13. The fermion determinants and the planar topology
  14. Mathematical justification of the signed sum over loops
  15. Random loops at the surface
  16. ...and 96 more sections

Key Result

Theorem 5.1

The functional integral is invariant under local variations of the conformal metric field $\rho$. That is, the theory depends only on the conformal class (moduli) of the surface, and the induced metric $e(\xi)$ but not on the local gauge parameter $\rho(\xi)$.

Figures (18)

  • Figure 1: The MM equation, with the loop diffusion operator on the left side, and the integral term on the right side. The double line on the right stands for the delta function $\delta^4(x-y)$ enforcing self-intersection.
  • Figure 2: The unit disk $\Sigma$ is mapped to $\mathbb{R}^3\otimes \mathbb{R}^4$ and then projected to $\mathbb{C}^4$ by a holomorphic map, minimizing the area functional.
  • Figure 3: Types of line collision: two planar (upper drawing) and one planar, another non-planar (lower drawing)
  • Figure 4: The hierarchical set of loops, some touching, but never intersecting each other nor self-intersecting
  • Figure 5: The first area derivative of Dirac determinant. There is one internal path $\Gamma$ which cut the surface $S$ into two pieces $S_{in} =S_1, S_{out} =S_2$
  • ...and 13 more figures

Theorems & Definitions (17)

  • Remark 1.1: Equation solving vs. model building
  • Remark 4.1: Holes, poles, and global signs
  • Theorem 5.1: Conformal Invariance of the Elf Determinant
  • proof : Proof
  • Corollary 5.1
  • Remark 5.1: Moduli and Topology
  • Remark 6.1
  • Lemma 7.1: Geodesic Inequality
  • proof
  • Remark 7.1
  • ...and 7 more