GEOMETRIC CONFINEMENT AND THE NAMBU–GOTO ACTION FROM THE SUPERALGEBRAIC GEOMETRY OF THE NONCOMMUTATIVE TORUS

We derive the linear confining potential of quantum chromodynamics and the Nambu–Goto string action from the geometric structure of the noncommutativesuper-torus T 2 θ . In our framework, quarks correspond to fermionic modes localised attorsion points zq on the elliptic curve Eτ, with colour charges arising from the matrix extension M3(Aθ).When a quark–antiquark pair is separated, the controlled Nash–Kuiper corrugation between them deforms into a narrow flux tube. The energy of this tube, computed from the Connes–Yang–Mills functional ∥∇∥^2 = Trω(R† ⋆ R), grows linearly with theseparation distance—a direct consequence of the minimal norm condition R = 0 that fixes the corrugation profile.We prove four principal results:(1) Linear confining potential: The static quark–antiquark potential is V (r) = σr+O(1/r), where the string tension σ is expressed in terms of the noncommutativity parameter θ and the modular parameter τ0.(2) Nambu–Goto action: In the thin-tube limit, the effective action for the flux tube reduces to the Nambu–Goto string action with tension σ. This provides a geometric derivation of string theory’s fundamental object from noncommutativegeometry.(3) Superconfinement: The same geometric mechanism that confines quarks also confines superpartners (gluinos, squarks) at the Planck scale. Superpartners are odd (Grassmann) modes of the superconnection Γ along the fermionic directionsdθa, and their separation is algebraically forbidden by the Berezinian of the Stepanov transformation.(4) Unified confinement: Quark confinement and superconfinement are governed by a single parameter—the tension of the Nash corrugation, fixed by the minimal norm principle. The QCD string tension and the Planck-scale superconfinement are two manifestations of the same geometric rigidity.The theory contains no free parameters. The string tension σ, the linear potential, and the Nambu–Goto action are determined by τ0 and θ = ℓ^2P, both fixed by independent measurements.