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Investigate the spectral gap of the gauge-covariant Hodge Laplacian in Yang–Mills theory on the Poincaré homology sphere.

April 8, 2026Open Access

Linearized Yang–Mills Spectral Gap on the Poincaré Homology Sphere

Key Points

Investigate the spectral gap of the gauge-covariant Hodge Laplacian in Yang–Mills theory on the Poincaré homology sphere.
Computation of the spectral gap using the gauge-covariant Hodge Laplacian.
Analysis of the flat-connection moduli space with three isolated vacua.
Utilization of the McKay correspondence for spectral computation in the E₈ diagram.
Coexact 1-form spectral gap determined at 4/R² for trivial and standard vacua.
Galois conjugate vacuum reveals a larger gap of 36/R².
Icosaheral symmetry enhances the spectral gap by ninefold in the Galois sector.

Abstract

We compute the spectral gap of the gauge-covariant Hodge Laplacian for linearized SU(2) Yang–Mills theory on the Poincaré homology sphere S³/2I. The flat-connection moduli space consists of exactly three isolated vacua. Explicit spectral computation via the McKay correspondence for the extended E₈ diagram yields the coexact 1-form spectral gap at each vacuum: 4/R² for the trivial and standard vacua, and 36/R² for the Galois conjugate vacuum. The icosahedral symmetry filters the first four coexact levels from the Galois sector, producing a ninefold enhancement over the baseline gap.

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