AI 中文总结
本文针对平衡2项$L_\tau$-代数，构造$2n+2$维半严格高阶陈-西蒙斯规范理论，建立高阶陈-韦伊定理，证明其为高阶迁越规范理论特例，明确扩展Cartan同伦公式为相关定理与方程的共同起源。
AI 中文摘要
我们构造了与平衡2项$L_\tau$-代数相关的$2n+2$维半严格高阶陈-西蒙斯（HCS）规范理论。从同伦Maurer-Cartan理论出发，我们首先引入2项$L_\tau$-代数规范理论，并证明存在4维HCS构造。随后，我们将不变双线性配对扩展为适当次数的不变多线性形式，定义了$(2n+3)$维高阶庞特里亚金-陈形式，该形式是闭的且在无穷小规范变换下不变。其迁越产生显式的$(2n+2)$维HCS形式。我们进一步建立生成高阶迁越形式的高阶陈-韦伊定理，并证明HCS理论是高阶迁越规范理论的一个特例。最后，我们在该半严格框架下应用扩展的Cartan同伦公式，证明其是高阶陈-韦伊定理及相关三角方程的共同起源。
英文摘要
We construct a semistrict higher Chern--Simons (HCS) gauge theory in $2n+2$ dimensions associated with balanced 2-term $L_\infty$-algebras. Starting from the homotopy Maurer--Cartan theory, we first introduce 2-term $L_\infty$-algebra gauge theory, and show that there is a four-dimensional HCS construction. Then we extend invariant bilinear pairings to invariant multilinear forms of the appropriate degree, and define a $(2n+3)$-dimensional higher Pontryagin--Chern form, which is closed and invariant under infinitesimal gauge transformations. Its transgression yields an explicit $(2n+2)$-dimensional HCS form. We further establish a higher Chern--Weil theorem that generates higher transgression forms, and prove that the HCS theory is a distinguished instance of the higher transgression gauge theory. Finally, we apply the extended Cartan homotopy formula in this semistrict setting, and show that it is a common origin of both the higher Chern--Weil theorem and the associated triangle equation.
