Noncommutative deformation of spinor zero mode and Atiyah-Drinfeld-Hitchin-Manin construction

Abstract

A method to construct noncommutative instantons as deformations from commutative instantons was provided in [1]. Using this noncommutative deformed instanton, we investigate the spinor zero modes of the Dirac operator in a noncommutative instanton background on noncommutative R 4 , and we modify the index of the Dirac operator on the noncommutative space slightly and show that the number of the zero mode of the Dirac operator is preserved under the noncommutative deformation. We prove the existence of the Green's function associated with instantons on noncommutative R 4 , as a smooth deformation of the commutative case. The feature of the zero modes of the Dirac operator and the Green's function derives noncommutative ADHM equations which coincide with the ones introduced by Nekrasov and Schwarz. We show a one-to-one correspondence between the instantons on noncommutative R 4 and ADHM data. An example of a noncommutative instanton and a spinor zero mode are also given.

Exact construction of noncommutative instantons

“…We further prove one-to-one correspondence between the moduli space of the U(N) k-instantons and the moduli space of the ADHM data labeled by (N, k) [25]. In the proof, we apply Furuuchi's observation and several properties on noncommutative field theory (especially NC-deformed index theorem by Maeda and Sako [34]) to all the other ingredients of the ADHM construction and prove the existence of them in the operator sense. As a result, we obtain the following formula (a noncommutative version of the This directly shows an origin of the instanton number from the language of the ADHM data.…”

“…(A beautiful treatment can be also found in [11].) Noncommutative version of the ADHM duality is discussed in [25,34,41]. We start this section by giving a brief review on the Nahm transformation and then argue the ADHM/Nahm duality by taking certain limits.…”

Hermitian-Einstein metrics from noncommutative U(1) instantons

“…Another method to construct noncommutative instantons as smooth deformations of commutative instantons was provided in [48,49,50]. The correspondence between the smooth deformation and the ADHM construction are discussed in [51]. On the other hand, there exist instanton solutions which are not smoothly connected to commutative instantons.…”

Noncommutative Instantons Revisited

“…(A beautiful treatment can be also found in [9].) Noncommutative version of the ADHM duality is discussed in [17,27,32]. We start this section by giving a brief review on the Nahm transformation and then argue the ADHM/Nahm duality by taking certain limits.…”