Superloop Equations and Minimal Surfaces I: Confining minimal surface in $4D, N=1$ SYM

Abstract

We formulate the loop equations of pure $4d, N=1$ super Yang--Mills (SYM) theory in a finite geometric form. The usual equal-point loop derivatives are singular because the order of gauge-field insertions is lost when contour points coincide. Our path-ordered operator calculus (POOC) keeps the insertions in distinct ordered slots, forms the graded commutators and Jacobi combinations, and only then takes the coincidence limit. This removes the spurious kinematical singularities without introducing a cutoff; the genuine ultraviolet contact remains a separate physical distribution. We apply POOC to the exact Itoyama--Takashino superloop hierarchy and construct a Lorentzian supersymmetric Hodge-dual (SHD) surface functional. Its Hodge-resolved area derivative is annihilated by the local chiral and anti-chiral loop operators. Exact additivity under surface-geodesic sewing shows that the exponential of this zero mode multiplies any solution of the complete finite-$N$ hierarchy without changing the equations. For planar contours the SHD functional is the geometric area. In particular, for a long rectangular Wilson loop, $W[C_{T,L}]\sim\exp[-iσ_kLT]$ and $E_k(L)=σ_kL>0$. Thus the construction gives an exact nonperturbative confining area-law factor in $ N=1$ SYM. After Euclidean continuation, the same SHD factor is an exact fixed point of the deterministic zero-noise SYM gradient flow. Its interpretation as a dynamically selected equilibrium still requires stability in the long-flow-time, large-volume, and zero-noise limits, but this does not affect the exact zero-mode and finite-$N$ dressing theorems. The complete planar Wilson-loop solution, including the undressed fluctuation factor and excitation spectrum, will be developed in the next papers of this series.