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**Gauge theories on noncommutative Euclidean spaces.**
*(English)*
Zbl 1026.81062

Olshanetsky, M. et al., Multiple facets of quantization and supersymmetry. Michael Marinov memorial volume. Singapore: World Scientific. 794-803 (2002).

Applying the argument in the author’s paper [Commun. Math. Phys. 221, 433-450 (2001; Zbl 0989.46040)] the structure of the gauge group on noncommutative Euclidean spaces is studied. Let \(A\) be a noncommutative space, usually a (unital) Banach algebra over \(\mathbb{C}\), \(E\) a Hilbert \(A\)-module. Then the gauge group of the gauge \(E\) over \(A\) is \(G=U (\text{End}_AE)\), the group of unitary elements of \(\text{End}_A.\) Let \({\mathcal C}\) be the space of unitary connections of \(E\). To quantize the gauge theory, integration over \({\mathcal C}/G\) is used. This quotient space can be replaced by \({ \mathcal C}/G'\) if \(G'\subset G\) and \(G/G'\) is compact. The author says if \(G'\) acts freely on \({\mathcal C}\), e.g., in ordinary \(U(n)\)-gauge theory on a compact manifold \(X\), taking \(G'\) to be the subgroup fixing a point \(x_0\) of \(X\), is more convenient. Then taking \(A={\mathcal S}(\mathbb{R}^d_\theta)\), the non-unital algebra of Schwarz functions on \(\mathbb{R}^d\) equipped with a star-product
\[
(f*g) (x)= \iint f(x+\theta u) g(x+v)e^{iuv} du dv.
\]
By definition, \({\mathcal S}(\mathbb{R}^p)\), \(2p =d\) becomes an \(A\)-module, which is denoted by \({\mathcal F}\). As the finiteness condition, the gauge field is assumed to have the form
\[
\nabla_\mu= T\circ \partial_\mu \circ T^†+ \Pi\circ \partial_\mu\circ \Pi+ \rho_\mu,
\]
where \(T\in \Gamma_1^{0,0}\), \(\Pi=1- TT^†\in A\), \(\Pi=1- T^† T\in A\), \(\rho_\mu =\rho^†_\mu \in\Gamma\). Here \(T\in H\Gamma_\rho^{m,m_0}\) means \(T \in \Gamma^m_\rho\) and \(T^{-1}\in \Gamma_\rho^{-m_0}\), where \(a\in \Gamma^m_\rho \) if \(\|a\|\leq\text{const} \|x\|^m\) at infinity, and \(\Gamma= \bigcup_{m<-1} \Gamma_1^m\) [cf. M. Shubin, Pseudodifferential operators and spectral theory, Berlin (1987; Zbl 0451.47064), 2nd ed. (2001; Zbl 0980.35180)]. Gauge triviality at infinity implies \(T^† T=1\). Then there is an \(A\)-module isomorphism
\[
A^n\ni y\to(\Pi y,T^† y)\in {\mathcal F}\otimes A^n,
\]
whose inverses give a \(A\)-module isomorphism of \(\text{Ker} T^† \otimes A^n\). By using this isomorphism, the gauge group is approximated by unitary endomorphisms on \(A^n\), and how to recover the results in [J. A. Harvey, Topology of the Gauge Group in Noncommutative Gauge Theory, Proc. Strings 2001, Mumbai (2001)], is sketched. The gauge triviality at infinity is important. Because the author conjectures that calculating correlation function we can get functional integrals over fields that are gauge trivial at infinity, and “almost all” fields having finite action are gauge trivial at infinity.

In conclusion it is expected that almost all gauge fields having finite Euclidean action are gauge equivalent to the fields of the form \(\nabla_\mu= \partial_\mu+ \rho_\mu\), \(\rho_\mu\in \Gamma^m\), \(m=1-d/2\), when \(d\leq 4\).

For the entire collection see [Zbl 1002.00011].

In conclusion it is expected that almost all gauge fields having finite Euclidean action are gauge equivalent to the fields of the form \(\nabla_\mu= \partial_\mu+ \rho_\mu\), \(\rho_\mu\in \Gamma^m\), \(m=1-d/2\), when \(d\leq 4\).

For the entire collection see [Zbl 1002.00011].

Reviewer: Akira Asada (Matumoto)

### MSC:

81T75 | Noncommutative geometry methods in quantum field theory |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

81R15 | Operator algebra methods applied to problems in quantum theory |

46L87 | Noncommutative differential geometry |