##
**Noncommutative instantons: A new approach.**
*(English)*
Zbl 0989.46040

Modified ADHM construction for noncommutative instantons was presented by the author and N. Nekrasov [Commun. Math. Phys. 198, No. 3, 689-703 (1998; Zbl 0923.58062)]. In this paper, the author investigates noncommutative instantons and modified ADHM construction mainly from the viewpoints of Hilbert module (explained in the preliminaries) and \(C^*\)-deformations [M. Rieffel, “Deformation quantization for actions of \(\mathbb{R}^d\)”, Mem. Am. Math. Soc. 506 (1993; Zbl 0798.46053)]. The investigations do not use previous results on noncommutative instantons and are rich in ideas, but precise proofs are not given and some notational confusions exist.

The starting points of the paper is as follows: Consider fields on \(\mathbb{R}^4\) that are gauge equivalent to the trivial field at infinity in the commutative case, correspond to \({\mathcal S}(\mathbb{R}^4)\), the algebra of rapidly decreasing functions on \(\mathbb{R}^4\), while the working on \(S^4\) corresponds to add the unit to \({\mathcal S}(\mathbb{R}^4)\). Then their noncommutative version is obtained introducing the \(*\)-product on \({\mathcal S}(\mathbb{R}^d)\) by \[ (f* g)(x)= \iint f(x+\theta u) g(x+ v) e^{inv}du dv,\quad f,g\in{\mathcal S}(\mathbb{R}^d), \] where \(\theta\) is a purely imaginary matrix. The resulting algebra is denoted by \(\mathbb{R}^d_\theta\). \(\widetilde{\mathbb{R}}^d_\theta\) is the unital algebra obtained from \(\mathbb{R}^d_\theta\) by means of the addition of a unit element. Let \(\widehat x^k\) be Hermitian operators such that \([\widehat x^k,\widehat x^\ell]= i\theta^{k\ell}\). They are noncommutative coordinates of \(\mathbb{R}^d_\theta\), and if \(d= 2n\) and \(\theta\) is nondegeneate, these relations are equivalent to the canonical commutation relations. In this case, to define an operator \(\widehat\phi\) for \(\phi\in{\mathcal S}(\mathbb{R}^n)\) by \[ \widehat\phi= \int\psi(k) e^{ik\widehat x}dk,\quad \phi(x)= \int \psi(k) e^{ikx} dk, \] \({\mathcal S}(\mathbb{R}^n)\) becomes an \(\mathbb{R}^d_\theta\)-module (resp. \(\widehat{\mathbb{R}}^d_\theta\)-module). This (Hilbert) module is denoted by \({\mathcal F}\). It is a projective \(\widehat{\mathbb{R}}^d_\theta\)-module and every projective \(\widehat{\mathbb{R}}^d_\theta\) is isomorphic to \({\mathcal F}_{rs}\), \(r\)-copies of \({\mathcal F}\) and \(s\)-copies of \((\widehat{\mathbb{R}}^d_\theta)^1\), \(\widehat{\mathbb{R}}^d_\theta\) regarded as the right (Hilbert) \(\widehat{\mathbb{R}}^d_\theta\)-module. (This is proved in an Appendix by Connes.) The standard connection \(\nabla^{(0)}_k\) of \({\mathcal F}_{rs}\) acts as \(i(\theta^{-1})_{k\ell}\widehat x^\ell\) on \({\mathcal F}\) and as \(\partial_k\) on \((\widehat{\mathbb{R}}^d_\theta)^1\). Every connection \(\nabla_k\) takes the form \(\nabla^{(0)}_k+ \Phi\). Writing \(\Phi\) as a \((2,2)\)-matrix and devide its components as the sum operating on the copies of \({\mathcal F}\) and \((\widehat{\mathbb{R}}^d_\theta)^1\), the modified ADHM equations are derived as follows: \[ \alpha\cdot 1+[B_1,B_2]+ IJ= 0, \]

\[ \beta\cdot 1+ [B_1,B^+_1]+ [B_2, B^+_2]+ II^+- JJ^+= 0. \] Here \(B\to B^+\) is the antilinear involution (Section 1). In Section 2, instantons on \(\mathbb{R}^4_\theta\) are discussed. In this case, a gauge field \(\nabla_\mu\) is gauge trivial at infinity if \[ \nabla_\mu= T\circ\partial_\mu\circ T^++ (1- TT^+)\circ \partial_\mu\circ (1- TT^+)+ \sigma_\mu, \] where \(T^+T= 1\), \(\sigma_\mu\) is an endomorphism tending to zero at infinity faster than \(\|x\|^{-1}\) and \(T^+\) is a parametrix of \(T\), i.e. \(1-TT^+=\Pi\) is a matrix with entries from \({\mathcal S}(\mathbb{R}^4)\) (the existence condition of parametrix is discussed in Section 4). Then \(\text{Ker }T^+\oplus (\mathbb{R}^4_\theta)^n\cong (\mathbb{R}^4_\theta)^n\) as a Hilbert \(\mathbb{R}^4_\theta\)-module. Since \(\text{Ker }T^+\cong{\mathcal F}^k\), we have \({\mathcal F}^k\otimes (\mathbb{R}^4_\theta)^n\cong (\mathbb{R}^4_\theta)^n\) as a Hilbert \(\mathbb{R}^4_\theta\)-module. Hence, we can relate discussions in Section 1 to this case. The noncommutative ADHM condition is equivalent to the following diagonal form condition on the operator \(D^+D\), \[ D^+D= \begin{pmatrix}\Delta & 0\\ 0 &\Delta\end{pmatrix},\quad D^+= \begin{pmatrix} -B_2+ z_2 & B_1- z_1 & I\\ B^+_1- \overline z_1 & B^+_2-\overline z_2 & J^+\end{pmatrix}. \] To analyze \({\mathcal E}= \text{Ker }D^+\) as an \(\mathbb{R}^4_\theta\)-module \(\theta\) is assumed to be nondegenerate and having negative Pfaffian. Then elements of \(\mathbb{R}^4_\theta\) can be considered as pseudodifferential operators acting on functions defined on \(\mathbb{R}^2\), and it is shown \({\mathcal E}\cong{\mathcal F}^r\oplus(W\otimes \mathbb{R}^4_\theta)\) \((\cong{\mathcal F}^r\oplus(\mathbb{R}^4_\theta)^\ell)\) (Section 3).

In Section 4, some properties of star-product and pseudodifferential operators are listed and mathematical foundations of the discussions of preceding sections are given. Then generalizations of discussions in preceding sections to general noncommutative spaces such as \(\mathbb{R}^d_\theta\), for general \(d\), and noncommutative torus \(T^4_\theta\) are discussed. Effects to relax the selfdual condition to \(F\pm*F= \omega\cdot 1\), \(\omega\) is a scalar but may not be equal to \(0\), are also discussed including the commutative case mainly focussed on topological effects. Related to this effects, the existence of a noncommutative generalization of Donaldson’s Theorem is remarked [This generalization is given in A. Kapustin, A. Kuznetsov and D. Orlov, Commun. Math. Phys. 22, No. 2, 385-432 (2001; Zbl 0989.81127)].

The starting points of the paper is as follows: Consider fields on \(\mathbb{R}^4\) that are gauge equivalent to the trivial field at infinity in the commutative case, correspond to \({\mathcal S}(\mathbb{R}^4)\), the algebra of rapidly decreasing functions on \(\mathbb{R}^4\), while the working on \(S^4\) corresponds to add the unit to \({\mathcal S}(\mathbb{R}^4)\). Then their noncommutative version is obtained introducing the \(*\)-product on \({\mathcal S}(\mathbb{R}^d)\) by \[ (f* g)(x)= \iint f(x+\theta u) g(x+ v) e^{inv}du dv,\quad f,g\in{\mathcal S}(\mathbb{R}^d), \] where \(\theta\) is a purely imaginary matrix. The resulting algebra is denoted by \(\mathbb{R}^d_\theta\). \(\widetilde{\mathbb{R}}^d_\theta\) is the unital algebra obtained from \(\mathbb{R}^d_\theta\) by means of the addition of a unit element. Let \(\widehat x^k\) be Hermitian operators such that \([\widehat x^k,\widehat x^\ell]= i\theta^{k\ell}\). They are noncommutative coordinates of \(\mathbb{R}^d_\theta\), and if \(d= 2n\) and \(\theta\) is nondegeneate, these relations are equivalent to the canonical commutation relations. In this case, to define an operator \(\widehat\phi\) for \(\phi\in{\mathcal S}(\mathbb{R}^n)\) by \[ \widehat\phi= \int\psi(k) e^{ik\widehat x}dk,\quad \phi(x)= \int \psi(k) e^{ikx} dk, \] \({\mathcal S}(\mathbb{R}^n)\) becomes an \(\mathbb{R}^d_\theta\)-module (resp. \(\widehat{\mathbb{R}}^d_\theta\)-module). This (Hilbert) module is denoted by \({\mathcal F}\). It is a projective \(\widehat{\mathbb{R}}^d_\theta\)-module and every projective \(\widehat{\mathbb{R}}^d_\theta\) is isomorphic to \({\mathcal F}_{rs}\), \(r\)-copies of \({\mathcal F}\) and \(s\)-copies of \((\widehat{\mathbb{R}}^d_\theta)^1\), \(\widehat{\mathbb{R}}^d_\theta\) regarded as the right (Hilbert) \(\widehat{\mathbb{R}}^d_\theta\)-module. (This is proved in an Appendix by Connes.) The standard connection \(\nabla^{(0)}_k\) of \({\mathcal F}_{rs}\) acts as \(i(\theta^{-1})_{k\ell}\widehat x^\ell\) on \({\mathcal F}\) and as \(\partial_k\) on \((\widehat{\mathbb{R}}^d_\theta)^1\). Every connection \(\nabla_k\) takes the form \(\nabla^{(0)}_k+ \Phi\). Writing \(\Phi\) as a \((2,2)\)-matrix and devide its components as the sum operating on the copies of \({\mathcal F}\) and \((\widehat{\mathbb{R}}^d_\theta)^1\), the modified ADHM equations are derived as follows: \[ \alpha\cdot 1+[B_1,B_2]+ IJ= 0, \]

\[ \beta\cdot 1+ [B_1,B^+_1]+ [B_2, B^+_2]+ II^+- JJ^+= 0. \] Here \(B\to B^+\) is the antilinear involution (Section 1). In Section 2, instantons on \(\mathbb{R}^4_\theta\) are discussed. In this case, a gauge field \(\nabla_\mu\) is gauge trivial at infinity if \[ \nabla_\mu= T\circ\partial_\mu\circ T^++ (1- TT^+)\circ \partial_\mu\circ (1- TT^+)+ \sigma_\mu, \] where \(T^+T= 1\), \(\sigma_\mu\) is an endomorphism tending to zero at infinity faster than \(\|x\|^{-1}\) and \(T^+\) is a parametrix of \(T\), i.e. \(1-TT^+=\Pi\) is a matrix with entries from \({\mathcal S}(\mathbb{R}^4)\) (the existence condition of parametrix is discussed in Section 4). Then \(\text{Ker }T^+\oplus (\mathbb{R}^4_\theta)^n\cong (\mathbb{R}^4_\theta)^n\) as a Hilbert \(\mathbb{R}^4_\theta\)-module. Since \(\text{Ker }T^+\cong{\mathcal F}^k\), we have \({\mathcal F}^k\otimes (\mathbb{R}^4_\theta)^n\cong (\mathbb{R}^4_\theta)^n\) as a Hilbert \(\mathbb{R}^4_\theta\)-module. Hence, we can relate discussions in Section 1 to this case. The noncommutative ADHM condition is equivalent to the following diagonal form condition on the operator \(D^+D\), \[ D^+D= \begin{pmatrix}\Delta & 0\\ 0 &\Delta\end{pmatrix},\quad D^+= \begin{pmatrix} -B_2+ z_2 & B_1- z_1 & I\\ B^+_1- \overline z_1 & B^+_2-\overline z_2 & J^+\end{pmatrix}. \] To analyze \({\mathcal E}= \text{Ker }D^+\) as an \(\mathbb{R}^4_\theta\)-module \(\theta\) is assumed to be nondegenerate and having negative Pfaffian. Then elements of \(\mathbb{R}^4_\theta\) can be considered as pseudodifferential operators acting on functions defined on \(\mathbb{R}^2\), and it is shown \({\mathcal E}\cong{\mathcal F}^r\oplus(W\otimes \mathbb{R}^4_\theta)\) \((\cong{\mathcal F}^r\oplus(\mathbb{R}^4_\theta)^\ell)\) (Section 3).

In Section 4, some properties of star-product and pseudodifferential operators are listed and mathematical foundations of the discussions of preceding sections are given. Then generalizations of discussions in preceding sections to general noncommutative spaces such as \(\mathbb{R}^d_\theta\), for general \(d\), and noncommutative torus \(T^4_\theta\) are discussed. Effects to relax the selfdual condition to \(F\pm*F= \omega\cdot 1\), \(\omega\) is a scalar but may not be equal to \(0\), are also discussed including the commutative case mainly focussed on topological effects. Related to this effects, the existence of a noncommutative generalization of Donaldson’s Theorem is remarked [This generalization is given in A. Kapustin, A. Kuznetsov and D. Orlov, Commun. Math. Phys. 22, No. 2, 385-432 (2001; Zbl 0989.81127)].

Reviewer: Akira Asada (Takarazuka)

### MSC:

46L87 | Noncommutative differential geometry |

53C99 | Global differential geometry |

81R60 | Noncommutative geometry in quantum theory |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

58Z05 | Applications of global analysis to the sciences |

46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |