Extension of the homotopy Maurer–Cartan framework to 3-gauge theories

Extend the homotopy Maurer–Cartan framework developed for balanced 2-term $L_\infty$-algebras to 3-gauge theories described by 2-crossed modules and related strict higher-gauge models, including the construction of appropriate higher invariant polynomials and invariant forms for the associated 3-term $L_\infty$-algebraic data.

Background

The paper constructs semistrict higher Chern–Simons theories in dimensions $2n+2$ from balanced 2-term $L_\infty$-algebras, using higher invariant multilinear forms, higher Pontryagin–Chern forms, transgression formulas, and the extended Cartan homotopy formula. The authors identify an unresolved extension of this framework to the next level of higher gauge theory, namely 3-gauge theories governed by 2-crossed modules and related strict models.

Solving this problem would require developing suitable higher invariant polynomials and corresponding invariant differential forms for 3-term $L_\infty$-algebraic structures, thereby extending the paper’s construction beyond semistrict Lie 2-algebras.