Extended Cartan homotopy formula for higher Chern-Simons-Antoniadis-Savvidy theory

TL;DR

The paper extends the extended Cartan homotopy formula (ECHF) to higher gauge theory by formulating a higher ECHF for 2-connections $(A,B)$ with curvatures $(\mathcal{F}, \mathcal{G})$ in $(2n+2)$ dimensions. It introduces the higher invariant form $\mathcal{P}_{2n+3}=\langle \mathcal{F}^n, \mathcal{G} \rangle_{\mathcal{g}\mathcal{h}}$ and the $(2n+2)$-ChSAS form $\mathcal{C}^{2n+2}_{2ChSAS}$ via interpolations between 2-connections, then derives a higher Chern-Weil theorem and higher triangle equation through the higher ECHF. A Cartan homotopy approach shows that a 2-Antoniadis-Savvidy transgression form $\mathcal{Q}^{2n+2}$ can be written as the difference of two 2ChSAS forms minus an exact form, establishing a bulk-boundary separation in higher transgression theory. The framework unifies conventional and higher gauge theories and suggests routes toward higher transgression gauge theories and their quantization, with potential implications for boundary theories and anomalies in higher dimensions.

Abstract

We consider extended Cartan homotopy formula (ECHF) for higher gauge theory. Firstly, we construct an oriented simplex based on 2-connections and present differential and integral forms of the higher ECHF. Then, we study the higher Chern-Simons-Antoniadis-Savvidy (ChSAS) theory and prove that the higher ECHF can reproduce the higher Chern-Weil theorem and give higher triangle equation. We finally conclude from the higher ECHF that a higher transgression form can be written as the difference of two higher ChSAS forms minus an exact form.

Figures (1)

• Figure 1: Development path of ECHF framework and corresponding theories