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.
Related
- On Weakly Contractible Non-Contractible Finite Topological Spaces of Ten Points
- Intrinsic Non-Reductive Geometric Invariant Theory and Moduli Spaces of Principal Bundles with Non-Reductive Structure Groups
- From Gradient Descent to Potential Theory: A Geometric Dictionary for Binary Classification
- Maximal Center of Distances of Finite Ultrametric Spaces and Perfect Binary Trees
- Geometric structures, orbit closures and invariant measures in moduli spaces
BibTeX
@misc{vasii2026yangmillshiggs,
title = {{Yang-Mills-Higgs: A Geometric Theory of Binary Labels on Non-Contractible Spaces}},
author = {Catalin Vasii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.00999},
eprint = {2607.00999},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.00999},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Yang-Mills-Higgs: A Geometric Theory of Binary Labels on Non-Contractible Spaces
AU - Catalin Vasii
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2607.00999
UR - https://arxiv.org/abs/2607.00999
AB - 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.
ER -