Stability and gap phenomena for Yang-Mills fields

Abstract

It is shown that any weakly stable Yang-Mills field of type SU 2 or SU 3 on the four-sphere must be self-dual or anti-self-dual. Any Yang-Mills field on S n , n ≥ 5, is unstable. Examples of stable fields on S 4 and S n /Γ for n ≥ 5 and Γ ≠ { e } are given. It is also shown that, for any Yang-Mills field R on S 4 , 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 S n , n ≥ 3, that satisfies the pointwise condition ∥ R ∥ 2 < ½( 2 n ) is trivial. If n = 3 or 4, the condition ∥ R ∥ 2 ≤ ½( 2 n ) 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.