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 $p=⟨F,F⟩$ where $F$ is the curvature 2-form and $⟨\cdot ,\cdot ⟩$ 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.

References in corpus (2)

• The cubical complex of a permutation group representation - or however you want to call it
• String principal bundles and Courant algebroids

Cited by in corpus (3)

• Towards holography in the BV-BFV setting
• String principal bundles and Courant algebroids
• Chern-Simons Type Characteristic Classes of Abelian Lattice Gauge Theory