Jin, Ren, Yang, and Yu take the number of colors in Yang-Mills theory — an integer in physics, 3 for QCD — and make it complex. The analytic continuation of N_c endows the dilatation operator with a non-Hermitian structure, and the spectrum of anomalous dimensions develops exceptional points: degeneracies where eigenvalues collide and eigenstates coalesce. These exceptional points are topological defects in the complex N_c plane, generating non-Abelian geometric phases and enforcing nontrivial monodromies among gauge-invariant operators. Encircling an exceptional point permutes the operators — the spectrum has branch-cut structure in a parameter that was never supposed to be continuously varied. Near these points, correlation functions develop logarithmic scaling, the signature of logarithmic conformal field theories. Standard unitary Yang-Mills at N_c = 3 sits within a larger complexified landscape whose topological features constrain even the physical theory.

The connection between non-Hermitian physics and gauge theory is the structural payload. Exceptional points are well-studied in optics and condensed matter, where gain-loss systems naturally produce non-Hermitian Hamiltonians. Here they emerge from a purely mathematical operation — analytic continuation of a discrete parameter — applied to the most physical of theories. The PT symmetry that governs the non-Hermitian dilatation operator maps directly to the spacetime PT symmetry of the gauge theory, linking a formal mathematical structure to a physical symmetry. The logarithmic CFT behavior near exceptional points means the operator product expansion acquires logarithmic corrections, which are observable consequences if one could probe the theory at non-integer N_c. The topology of the complexified parameter space is not a curiosity; it organizes the structure of the physical spectrum.

Analytically continuing a discrete parameter reveals structure that the discrete values alone cannot show. The integers are isolated points; the complex plane between them contains the branch cuts, monodromies, and topological defects that explain why the integers behave as they do. This is the method’s power: embed the physical case in a continuous family, and let the family’s topology constrain the physics. The pattern recurs in random matrix theory, in statistical mechanics at non-integer dimension, in the replica trick. The discrete system inherits constraints from a continuous structure it never asked to belong to.

(arXiv:2603.19006)