Sheaf lines of Yang–Mills instanton sheaves

Abstract

We calculate a sheaf line in CP 3 which is the real line supporting sheaf points on CP 3 of SL(2, C) Yang-Mills instanton (or SU (2) complex Yang-Mills instanton) sheaves for some given ADHM data we obtained previously. We found that this sheaf line is indeed a special jumping line over S 4 spacetime. In addition, we calculate the singularity structure of the connection A and the field strength F at the corresponding singular point on S 4 of this sheaf line. We found that the order of singularity at the singular point on S 4 associated with the sheaf line in CP 3 is higher than those of other singular points associated with normal jumping lines. We conjecture that this is a general feature for sheaf lines among jumping lines. * Electronic address:

Extended complex Yang–Mills instanton sheaves

“…Similar discussion can be done for cases with β not surjective [1,6]. That is, for some ADHM data at some sheaf points on CP 3 , there exists a common eigenvector v in the stable…”

“…(4.163) is to be compared to the CFTW [10] real 2instanton solution [28] which is lengthy and quite complicated. Presumably, there are many simplified mechanisms [6] for the calculation of complex YM instanton sheaves which make the solvability available. Moreover, the two terms in Eq.…”

“…Moreover, one sheaf line which is the real line supporting sheaf points on CP 3 of SU(2) YM 2-instanton sheaf was identified and found to be a special jumping line over S 4 spacetime [6]. In addition, the order of the singularity structure of the connection A and the field strength F at the corresponding singular point on S 4 associated with this sheaf line in CP 3 was found to be higher than those of other singular points associated with the normal jumping lines [6]. These results suggest that the existence of YM instanton sheaves on CP 3 is closely related to the new singular structure of the corresponding A and F on S 4 spacetime.…”

“…Recent developments of complex Yang-Mills (YM) instantons [1][2][3][4][5][6] have revealed many new mathematical structures of YM instantons which were not realized in the previous real YM instantons [7][8][9][10][11][12][13][14][15][16][17]. In particular, for the case of SU(2) complex YM instantons (or non-compact SL(2, C) [18] real YM instantons [19,20]), it was shown that for some ADHM data at some points on CP 3 the vector bundle description of 2-instanton breaks down, and one is led to use a description in terms of torsion free sheaves for these "instanton sheaves" on CP 3 [4].…”

“…On the other hand, one unexpected result obtained in [6] was the great simplification or "solvability" of calculation of the field strength F associated with the sheaf ADHM data, and the explicit form of a class of SU(2) complex YM 2-instanton field strength without removable singularities can be exactly calculated ! This "solvability" was not available for the previous real YM k-instanton calculation with k ≥ 2, and was presumably related to the imposing of the sheaf condition [1,2] on YM instanton sheaves.…”