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Chern-Simons, Wess-Zumino and other cocycles from Kashiwara-Vergne and associators | | Anton Alekseev
; Florian Naef
; Xiaomeng Xu
; Chenchang Zhu
; | | Date: |
28 Feb 2017 | | 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 $p= langle F, F
angle$ where $F$ is the
curvature 2-form and $langle cdot, cdot
angle$ 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. | | Source: | arXiv, 1702.8857 | | Services: | Forum | Review | PDF | Favorites | |
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