Yang-Mills on Fractal Manifolds with boundary (PART A)

Fractal geometry provides a novel paradigm for describing complex systemsin modern physics. This paper, for the first time, constructs the mathematicalframework of Yang-Mills theory on fractal manifolds with boundary, overcoming theconstraint of traditional theories being limited to integer-dimensional manifolds. Byintroducing fractal differential structures, fractal chain complexes, and the fractalLaplace operator, we rigorously define the fractal Yang-Mills equation and energyfunctional. Monte Carlo numerical simulations further validate its effectivenessin fractal superconductors. This research lays the mathematical foundation for theexploration of fractal gauge field theories in quantum gravity and condensed matterphysics.Keywords: Sierpi´nski-type fractal manifold, Ahlfors regularity, Yang-Mills the?ory, string compactification, numerical validation, measure uniformity