Stability and gap phenomena for Yang-Mills fields


It is shown that any weakly stable Yang-Mills field of type SU2 or SU3 on the four-sphere must be self-dual or anti-self-dual. Any Yang-Mills field on Sn, n ≥ 5, is unstable. Examples of stable fields on S4 and Sn/Γ for n ≥ 5 and Γ ≠ {e} are given. It is also shown that, for any Yang-Mills field R on S4, the pointwise condition ∥R-∥ 2 < 3 (or ∥R+∥ 2 < 3) implies that R- = 0 (or respectively that R+ = 0). In general, any Yang-Mills field R on Sn, n ≥ 3, that satisfies the pointwise condition ∥R∥2 < ½(2n) is trivial. If n = 3 or 4, the condition ∥R∥2 ≤ ½(2n) implies that either R is the trivial field or it is the direct sum of a trivial field with a field of tangent spinors carrying the standard connection.

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