Non-abelian gauge theories on non-commutative spaces (Q5940413)

This paper sketches the construction of a non-Abelian gauge theory on non-commutative spaces and shows additional couplings to appear if its dynamics formulated with a Lagrangian. A non-commutative space \({\mathcal A}_x\) is the algebra generated by the coordinates \(\{x_1,\dots, x^N\}\) with the relation \({\mathcal R}\). If the relations are of the type \([\widehat x_i,\widehat x_j]= i\theta^{ij}\), \(\theta^{ij}\in \mathbb{C}\), \({\mathcal A}_x\) is said to be a canonical structure, and in this paper, non-Abelian gauge theory is constructed on such spaces. Non-commutative space of Lie structure \([\widehat x^i,\widehat x^j]= i\theta^{ij}_k\widehat x^k\), and of quantum space structure, \([\widehat x^i,\widehat x^j]= i\theta^{ij}_{kl}\widehat x^k\widehat x^l\), are also explained in the introduction. After reviewing gauge transformations of non-commutative space (including Lie structure) in Section 2, it is remarked that composition of infinitesimal transformations appear as commutators and anticommutators of the group. So not the Lie algebra, but the enveloping algebra of the Lie algebra is the proper setting for non-Abelian gauge theory on non-commutative spaces. In general, an enveloping algebra valued infinitesimal transformation depends on infinitely many parameters. But with the help of the Seiberg Witten map [\textit{N. Seiberg} and \textit{E. Witten}, String theory and non-commutative geometry, JHEP, 9909, 032 (1999)], enveloping algebra-valued infinitesimal transformations and gauge fields that depend on a finite number of paramters and components are defined [cf. \textit{B. Jurčo} and \textit{P. Schuppe}, Non-commutative Yang-Mills form equivariance of star product, Eur. Phys. J. C14, 367 (2000), \textit{B. Jurčo}, \textit{P. Schuppe} and \textit{J. Wess}, Non-commutative gauge theory for Poisson manifolds, Nucl. Phys. B584, 784 (2000)]. Field strength \(F_{\nu,\rho}\) of this theory is similarly defined on commutative spaces, but the product is replaced to the Moyal-Weyl \(*\)-product, which is defined in Section 2. The Lagrangian of a gauge theory is defined to be \(L= 1/4\text{ Tr }F_{\alpha\beta}* F^{\alpha\beta}\). After remarking invariance under gauge transformation of \(L\), it is remarked that the first-order in \(\theta\) of the field strength takes the form \[ F_{\kappa\lambda}= F^1_{\kappa\lambda, a} T^a+ \theta^{\mu\nu} \textstyle{{1\over 2}} (T^a T^b+ T^b T^a)\{F^1_{\kappa,\mu, a} F^1_{\lambda\nu,b}- \textstyle{{1\over 2}} a^1_{\mu,a} (2\partial_\nu F^1_{\kappa\lambda, b}+ a_{\nu,c} F^1_{\kappa\lambda, d}f^{cd}_b)\}. \] The author says this should serve as an example of how a non-commutative space-time structure manifests itself in the coupling of a gauge theory, formulated on ordinary space in terms of ordinary fields. These are given in Section 3, the last Section.