自旋-局部高自旋顶点$Υ^{ηη}_{ωCCC}$的表现形式
Manifest Form of the Spin-Local Higher-Spin Vertex $Υ^{ηη}_{ωCCC}$
摘要：
高自旋规范场方程的Vasiliev生成系统包含一个自由的复参数$η$，它允许重构高自旋规范场方程的非线性顶点。逐阶求解生成系统，得到与$η$和$\barη$的各个幂成比例的物理顶点。最近在2009.02811中以$Z$ -支配引理的$Z$ -支配形式给出了零形扇区中的$η^2$和$\barη^2$顶点，这暗示了它们的自旋局域性。然而，2009.02811的顶点具有依赖于辅助旋量变量$Z$的自旋局域项之和的形式，在理论中模所谓的$Z$ -支配项，提供了一种存在定理，而不是顶点的显式形式。本文的目的是详细说明一种方法，允许系统地考虑效果的$Z$主导的条款上的最终$Z$独立形式的顶点所需的任何实际分析。也就是说，本文得到了顶点$Υ^{ηη}_{ωCCC}$的$ωCCC$序部分的与$Z$无关的自旋局部形式，其中$ω$和$C$表示任意结合代数中的规范1形式和场强0形式的高自旋场，在这种情况下，顶点中乘积因子的序很重要.所发展的形式主义是基于本文中导出的广义三角恒等式，并且适用于顶点中的场的所有其他排序。
英文摘要：
Vasiliev generating system of higher-spin equations allowing to reconstruct nonlinear vertices of field equations for higher-spin gauge fields contains a free complex parameter $η$. Solving the generating system order by order one obtains physical vertices proportional to various powers of $η$ and $\barη$. Recently $η^2$ and $\barη^2$ vertices in the zero-form sector were presented in 2009.02811 in the $Z$-dominated form implying their spin-locality by virtue of $Z$-dominance Lemma of 1805.11941. However the vertex of 2009.02811 had the form of a sum of spin-local terms dependent on the auxiliary spinor variable $Z$ in the theory modulo so-called $Z$-dominated terms, providing a sort of existence theorem rather than explicit form of the vertex. The aim of this paper is to elaborate an approach allowing to systematically account for the effect of $Z$-dominated terms on the final $Z$-independent form of the vertex needed for any practical analysis. Namely, in this paper we obtain explicit $Z$-independent spin-local form for the vertex $Υ^{ηη}_{ωCCC}$ for its $ωCCC$-ordered part where $ω$ and $C$ denote gauge one-form and field strength zero-form higher-spin fields valued in an arbitrary associative algebra in which case the order of product factors in the vertex matters. The developed formalism is based on the Generalized Triangle identity derived in the paper and is applicable to all other orderings of the fields in the vertex.
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