Yang-Mills-Higgs: A Geometric Theory of Binary Labels on Non-Contractible Spaces

Abstract

We reformulate binary classification on a manifold M as a Yang-Mills-Higgs variational problem. Labelled data is encoded as a functor from the fundamental groupoid of M to the one-object groupoid B(Z_2), whose monodromy class in H^1(M, Z_2) is a topological obstruction to realising the classifier by a sign function. The classifier-section and the connection jointly minimise a Yang-Mills-Higgs energy subject to hard data conditions: the matter sector carries the classification content, while the Yang-Mills sector is bounded below in each topological class by the Bogomolny inequality and selects the gauge background. This recovers the companion paper's harmonic interpolation as the contractible-base, flat-connection reduction. Two structural payoffs follow. First, the curvature 2-form of the selected connection supplies a dictionary with transformer attention: by the plaquette formula it is a pairwise, antisymmetric, algebra-valued form on tangent directions, which we match - as a stipulated dictionary, not a derived identity - to the antisymmetric component of the attention bilinear; the abelian/non-abelian split of curvature corresponds to the single-head/multi-head split of attention. Second, XOR on the torus is realised by the covariantly harmonic section of the double-Mobius bundle, which the variational selector picks out: solving the regularised capacitance system on all four flat Z_2-bundles gives a strict energy ordering favouring the double-Mobius class, whereas an MLP trained on the same data finds a structurally different boundary that ignores the toroidal identifications. Worked examples run an example ladder (circle, torus, S^2 monopole, S^4 instanton). The connection is throughout either flat or pinned to the Bogomolny moduli, so curvature is data-decoupled by construction; whether freeing it makes curvature respond to label proximity is posed as an open question.