Coexact Spectral Gaps from McKay Distance for Flat Bundles on Homogeneous Spherical Space Forms

Abstract

Let S³/Γ be a homogeneous spherical space form, the quotient of the round three-sphere of radius R by a finite subgroup Γ ⊂ SU(2) acting by left translation, and let E_τ be the flat bundle determined by a finite-dimensional representation τ of Γ. We determine the lowest nonzero eigenvalue of the Hodge Laplacian on coexact E_τ-valued 1-forms, the twisted coexact spectral gap. For the adjoint bundle of an irreducible flat SU(2) connection, the coefficient system of the gauge-theoretic deformation problem, that gap is 4/R² across the nonabelian cases of the ADE classification of finite subgroups of SU(2), with a single exception over all pairs of group and connection: it is 36/R² for the connection given by the Galois-conjugate two-dimensional representation Q′ of the binary icosahedral group, on the Poincaré homology sphere. The mechanism is a first-occurrence rule. The gap is 1/R² times the square of the first level at which a constituent of τ enters the coexact spectrum of the round sphere, and whenever the nearest constituent lies at graph distance at least two from the trivial node of the affine ADE diagram of Γ, that level is the McKay distance itself.

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