Exact holomorphic solution of the Hodge-dual minimal surface: a confining quantum QCD string

Abstract: We present an exact analytic solution for the quaternionic Hodge-dual minimal surface in four dimensions, which provides a geometric fixed point of the Yang-Mills Wilson loop equation. The area of this surface is expressed through the eigenvalues of the Douglas-Gram matrix of the boundary loop, and is identical for the self-dual and anti-self-dual chiralities related by parity. We prove the Confinement Theorem by showing that the effective string tension sigma_effC is bounded as sigma_eff[C]>= sqrt(2) , remaining finite and positive for all regular loops. Consequently, the Wilson loop decays exponentially with the Euclidean area, providing a purely geometric proof of confinement. Quantization of the quadratic holomorphic action yields a quantum string, with a universal constant correction for each of the two chiral strings. The worldsheet theory is one-loop exact. This construction defines a solvable quantum Hodge-dual string theory reproducing the QCD area law with explicit tension dependence on the loop shape.