arXiv 2110.040902021-10-05hep-th

具有Lorentz不变性和规范协变星积的Seiberg-Witten映射

Seiberg-Witten Map with Lorentz-Invariance and Gauge-Covariant Star Product

M. Chaichian, M. N. Mnatsakanova, M. Oksanen

备注：: 32 pages
DOI：: 10.1016/j.nuclphysb.2022.115831

摘要：

我们使用规范协变星积和非对易张量$θ^{μν}(x)$来发展Seiberg-Witten映射。后者保证了理论的洛伦兹不变性。这个映射的通常形式和它的其他最近的推广都没有考虑这样的协变星积。我们构造了非对易参数$θ^{μν}(x)$中规范参数、规范场和强度张量的Seiberg-Witten映射。文中还给出了将该映射推广到高阶的方法。有趣的是，协变星积的结合性在一阶和二阶都需要相同的约束，即在$θ^{μν}(x)$和时空连接上。这一事实表明，相同的约束可能足以确保所有顺序中的结合性。由此产生的塞伯格-威滕映射既适用于内部规范理论，也适用于时空规范理论。与其他（非协变）星积的Seiberg-Witten映射进行了比较，并给出了一些特征性质。作为应用，我们考虑了引力的$GL(2, C)$非对易规范理论，其中证明了连接决定了具有辛结构的时空（如Zumino等人[AIP Conf. Proc. 1200（2010），204，arXiv：0910.0459]所提出的）。这个例子表明，规范协变星积的结合性所需的约束可以得到满足。本文还将所提出的$GL(2, C)$非对易引力规范理论与Chamseddine [Phys.Rev.D69（2004），024015，hep-th/0309166]给出的非协变星积引力规范理论进行了比较.

英文摘要：

We develop the Seiberg-Witten map using the gauge-covariant star product with the noncommutativity tensor $θ^{μν}(x)$. The latter guarantees the Lorentz invariance of the theory. The usual form of this map and its other recent generalizations do not consider such a covariant star product. We construct the Seiberg-Witten map for the gauge parameter, the gauge field and the strength tensor to the first order in the noncommutativity parameter $θ^{μν}(x)$. Prescription for the generalization of the map to higher orders is also given. Interestingly, the associativity of the covariant star product both in the first and second orders requires the same constraints, namely, on the $θ^{μν}(x)$ and on the space-time connection. This fact suggests that the same constraints could be enough to ensure the associativity in all orders. The resulting Seiberg-Witten map applies both to the internal and space-time gauge theories. Comparisons with the Seiberg-Witten map based on other (non-covariant) star products are given and some characteristic properties are also presented. As an application, we consider the $GL(2, C)$ noncommutative gauge theory of gravitation, in which it is shown that the connection determines a space-time with symplectic structure (as proposed by Zumino et al [AIP Conf. Proc. 1200 (2010), 204, arXiv:0910.0459]). This example shows that the constraints required for the associativity of the gauge-covariant star product can be satisfied. The presented $GL(2, C)$ noncommutative gauge theory of gravitation is also compared to the one (given by Chamseddine [Phys. Rev. D 69 (2004), 024015, hep-th/0309166]) with non-covariant star product.

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摘要：

英文摘要：

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