Chern-Simons, Wess-Zumino and other cocycles from Kashiwara-Vergne and associators

Abstract

Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g. in the Chern-Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as $p= \langle F, F\rangle$ where $F$ is the curvature 2-form and $\langle \cdot, \cdot\rangle$ is an invariant scalar product on the corresponding Lie algebra $\mathfrak{g}$. The descent for $p$ gives rise to an element $\omega=\omega_3 + \omega_2 + \omega_1 + \omega_0$ of mixed degree. The 3-form part $\omega_3$ is the Chern-Simons form. The 2-form part $\omega_2$ is known as the Wess-Zumino action in physics. The 1-form component $\omega_1$ is related to the canonical central extension of the loop group $LG$. In this paper, we give a new interpretation of the low degree components $\omega_1$ and $\omega_0$. Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara-Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class $p$. In more detail, we define a 1-cocycle $C$ which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara-Vergne equation $F$ is mapped to $\omega_1=C(F)$. Furthermore, the component $\omega_0$ is related to the associator corresponding to $F$. It is surprising that while $F$ and $\Phi$ satisfy the highly non-linear twist and pentagon equations, the elements $\omega_1$ and $\omega_0$ solve the linear descent equation.