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## No nontrivial circle-bundles on S4 and implication for mass gap

Since $H^2(S^4)=0$ and the first Chern class classifies line bundles, a circle bundle that can appear as the unit circle of a line bundle must be trivial.  In particular one cannot use the Atiyah-Singer index theorem to force nontrivial zero modes by line bundles.  But more importantly, the Dirac equation is

$(D - eA -m)\psi = 0$

Using an orthonormal basis $\phi_i$ satisfying $D\phi_i = \lambda_i \phi$ and letting $\beta_{ij}$ representing the coefficients of $eA$, we have

$\lambda_l - m - \sum_i e \beta_{il} = 0$

This implies that the Dirac equation is solvable only when $m - \sum_i e\beta_{il}$ belongs to the spectrum of $D$.