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

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Anton Alekseev, Florian Naef, Xiaomeng Xu, Chenchang Zhu

**28/2/17**Published in : arXiv:1702.08857

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=⟨F,F⟩ where F is the curvature 2-form and \langle \cdot, \cdot\rangle is an invariant scalar product on the corresponding Lie algebra 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 ω0 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.