Extended complex Yang-Mills instanton sheaves

In the search of Yang-Mills (YM) instanton sheaves with topological charge two, the rank of β matrix in the monad construction can be dropped from the bundle case with rank β = 2 to either rank β = 1 [S. H. Lai, J. C. Lee and I. H. Tsai, Yang-Mills instanton sheaves, Ann. Phys. 377 (2017) 446] or 0 on some points of CP3 of the sheaf cases. In this paper, we first show that the sheaf case with rank β = 0 does not exist for the previous construction of SU(2) complex YM instantons [S. H. Lai, J. C. Lee and I. H. Tsai, Biquaternions and ADHM construction of concompact SL(2,C) Yang-Mills instantons, Ann. Phys. 361 (2015) 14]. We then show that in the new "extended complex YM instantons" discovered in this paper, rank β can be either 2 on the whole CP3 (bundle) with some given ADHM data or 1, 0 on some points of CP3 with other ADHM data (sheaves). These extended SU(2) complex YM instantons have no real instanton counterparts. Finally, the potential applications to real physics systems are noted in the end of the paper.