Split-quaternionic Representation of Sdym Su(1, 1) Instantons in S

Using split-quaternions, we find explicit SDYM SU(1, 1) instanton solutions in S2 −×S 2 + which is the conformal compactification of the semi-Euclidean 4-spacetime R 2+2 of split-signature (−,−,+,+). The noncompact Lie group SU(1, 1) is naturally introduced as an appropriate gauge group for SDYM instantons in S2 −×S 2 +. It is shown that SDYM and ASDYM SU(1, 1) instanton solutions in S2 −×S 2 + lead to the absolute minima of Yang-Mills action and that Yang-Mills action satisfies the same quantization as in SU(2) non-abelian gauge theories in S4. It is also shown that SDYM and ASDYM field equations in S2 − ×S 2 + can be described as simple split-quaternionic 2-forms.