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

• Main Authors: Alekseev, Anton (Author), Naef, Florian (Author), Zhu, Chenchang (Author), Xu, Xiaomeng (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Springer Netherlands, 2018-07-27T13:50:21Z.
• Subjects: Article
• Online Access: Get fulltext

 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 ⟨⋅,⋅⟩ is an invariant scalar product on the corresponding Lie algebra g. The descent for p gives rise to an element ω=ω[subscript 3]+ω[subscript 2]+ω[subscript 1]+ω[subscript 0] of mixed degree. The 3-form part ω[subscript 3] is the Chern-Simons form. The 2-form part ω[subscript 2] is known as the Wess-Zumino action in physics. The 1-form component ω[subscript 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 ω[subscript 1] and ω[subscript 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 ω[subscript 1]=C(F) . Furthermore, the component ω[subscript 0] is related to the associator Φ corresponding to F. It is surprising that while F and Φ satisfy the highly nonlinear twist and pentagon equations, the elements ω[subscript 1] and ω[subscript 0] solve the linear descent equation. Keywords: Chern-Simons form, Kashiwara-Vergne theory, Associators, Kontsevich's non-commutative differential calculus