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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 $$(1-p)$$.

##### 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)
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