##
**On noncommutative Nahm transform.**
*(English)*
Zbl 1024.81046

Authors’ introduction: It is shown in [A. Connes, M. R. Douglas and A. Schwarz, J. High Energy Phys. 1998, No. 2, Paper No. 3 (1998; Zbl 1018.81052), see also hep-th/9712117] that noncommutative geometry can be successfully applied to the analysis of M(atrix) theory. In particular, it is proven that one can compactify M(atrix) theory on a noncommutative torus; later, compactifications of this kind were studied in numerous papers. In the present paper, we analyze instantons in noncommutative toroidal compactifications. This question is important because instantons can be considered as BPS states of the compactified M(atrix) model. Instantons on a noncommutative \(\mathbb{R}^4\) were considered earlier in [N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198, No. 3, 689-703 (1998; Zbl 0923.58062)]. It was shown there that these instantons give some insight in the structure of the \((2,0)\) super-conformal six dimensional theory; the instantons on a noncommutative torus also should be useful in this relation. The main mathematical tool used in [Nekrasov-Schwarz, loc. cit.] is the noncommutative analogue of ADHM construction of instantons. The present paper is devoted to the noncommutative analogue of the Nahm transform (recall that the Nahm transform can be regarded as some kind of generalization of ADHM construction). We prove that some of the important pr operties of the Nahm transform remain correct in the noncommutative case.