Higher Chern--Simons Theory in $2n+2$ Dimensions for Balanced 2-term $L_{\infty }$-Algebras

Abstract

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.