Equivalence of projections as gauge equivalence on noncommutative space. (English) Zbl 1012.81049
By using gauge theory with projection [cf. the author, Prog. Theor. Phys. 103, 1043-1068 (2000)], the noncommutative analog of the BPST instanton is constructed as a noncommutative resolution of the singular gauge transformation in ordinary \(\mathbb{R}^4\) [cf. N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198, 689-703 (1998; Zbl 0923.53062)].
For this purpose, gauge theory with projection is reviewed in Sect. 2, as follows: Let \({\mathcal H}\) be the Hilbert space on which \(z_1=x_2 +ix_1\), \(z_2= x_4+ix_3\) and \(\overline z_1\), \(\overline z_2\) are represented as creation and annihilation operators, \({\mathcal A}\) the algebra of smooth operators on \({\mathcal H}\), \({\mathcal A}^n =\mathbb{C}^n \otimes {\mathcal A}\) and \(\mathbb{M}_n ({\mathcal A})\) the algebra of \({\mathcal A}\)-valued \((n,n)\)-matrices. Then projections \(p\) and \(q\) in \(\mathbb{M}_n({\mathcal A})\) are said to be Murray-von Neumann (MvN) equivalent when there exists a unitary matrix \(U\) such that \(p=U^† U\), \(q=UU^†\). \(U\) maps \(p{\mathcal H}^n\) to \(q {\mathcal H}^n\). The transformation \(A_p=U^† A_qU+U^† (dU)p\), \(A_p=pAp\) is called the MvN gauge transformation. The MvN gauge transformation can be understood as a mixture of gauge transformation and coordinate transformation on noncommutative \(\mathbb{R}^4\).
In Sect. 3, first the ADHM construction is reviewed and interpreted as the projection to the zero- mode space of a Dirac-like operator. Then the BPST instanton \(\Psi_{\text{BPST}}\), having the factor \(1/\sqrt {r^2+\rho^2}\), and a singular zero-mode \(\Psi_{\text{sing}}\), having the factor \(1/r\sqrt {r^2+\rho^2}\), are constructed. These fields are related by the singular gauge transformation \(g(x)=x^\mu \sigma_\mu /r\), \(\sigma_\mu= (i\tau_1, i\tau_2,i \tau_3,1)\), where \(\tau_i\) are the Pauli matrices. Then the noncommutative analogues of these constructios are presented. The operator zero-mode corresponding to the noncommutative BPST instanton has the factor \(1/\sqrt {\widehat N}\), which is defined omitting the kernel \(\sqrt{ \widehat N}\), that is by \[ {1\over \sqrt{\widehat N}}=\sum_{(n_1,n_2) \neq(0,0)} {1\over \sqrt{n_1+n_2}} |n_1,n_2 \rangle\langle n_1,n_2|. \] By virtue of this definition of \(1/\sqrt{\widehat N}\), the MvN gauge transformation for the noncommutative BPST instanton is defined which describes the noncommutative resolution of the singular gauge transformation \(g(x)\). It is remarked the similar to the commutative BPST instanton, the noncommutative instanton is classified by \(\pi^3 (U(2))\). But the corresponding connection in the gauge theory with projection \(p\) is not classified by \(\pi^3(U(2))\), but classified by the dimension of the projection \((1-p)\).
For this purpose, gauge theory with projection is reviewed in Sect. 2, as follows: Let \({\mathcal H}\) be the Hilbert space on which \(z_1=x_2 +ix_1\), \(z_2= x_4+ix_3\) and \(\overline z_1\), \(\overline z_2\) are represented as creation and annihilation operators, \({\mathcal A}\) the algebra of smooth operators on \({\mathcal H}\), \({\mathcal A}^n =\mathbb{C}^n \otimes {\mathcal A}\) and \(\mathbb{M}_n ({\mathcal A})\) the algebra of \({\mathcal A}\)-valued \((n,n)\)-matrices. Then projections \(p\) and \(q\) in \(\mathbb{M}_n({\mathcal A})\) are said to be Murray-von Neumann (MvN) equivalent when there exists a unitary matrix \(U\) such that \(p=U^† U\), \(q=UU^†\). \(U\) maps \(p{\mathcal H}^n\) to \(q {\mathcal H}^n\). The transformation \(A_p=U^† A_qU+U^† (dU)p\), \(A_p=pAp\) is called the MvN gauge transformation. The MvN gauge transformation can be understood as a mixture of gauge transformation and coordinate transformation on noncommutative \(\mathbb{R}^4\).
In Sect. 3, first the ADHM construction is reviewed and interpreted as the projection to the zero- mode space of a Dirac-like operator. Then the BPST instanton \(\Psi_{\text{BPST}}\), having the factor \(1/\sqrt {r^2+\rho^2}\), and a singular zero-mode \(\Psi_{\text{sing}}\), having the factor \(1/r\sqrt {r^2+\rho^2}\), are constructed. These fields are related by the singular gauge transformation \(g(x)=x^\mu \sigma_\mu /r\), \(\sigma_\mu= (i\tau_1, i\tau_2,i \tau_3,1)\), where \(\tau_i\) are the Pauli matrices. Then the noncommutative analogues of these constructios are presented. The operator zero-mode corresponding to the noncommutative BPST instanton has the factor \(1/\sqrt {\widehat N}\), which is defined omitting the kernel \(\sqrt{ \widehat N}\), that is by \[ {1\over \sqrt{\widehat N}}=\sum_{(n_1,n_2) \neq(0,0)} {1\over \sqrt{n_1+n_2}} |n_1,n_2 \rangle\langle n_1,n_2|. \] By virtue of this definition of \(1/\sqrt{\widehat N}\), the MvN gauge transformation for the noncommutative BPST instanton is defined which describes the noncommutative resolution of the singular gauge transformation \(g(x)\). It is remarked the similar to the commutative BPST instanton, the noncommutative instanton is classified by \(\pi^3 (U(2))\). But the corresponding connection in the gauge theory with projection \(p\) is not classified by \(\pi^3(U(2))\), but classified by the dimension of the projection \((1-p)\).
Reviewer: Akira Asada (Takarazuka)
MSC:
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 46L87 | Noncommutative differential geometry |
| 47N50 | Applications of operator theory in the physical sciences |
| 58B34 | Noncommutative geometry (à la Connes) |
