Cluster construction of the second motivic Chern class

Abstract

Let $\textrm{G}$ be a split, simple, simply connected, algebraic group over ${\mathbb {Q}}$. The degree 4, weight 2 motivic cohomology group of the classifying space $\textrm{BG}$ of $\textrm{G}$ is identified with ${\mathbb {Z}}$. We construct cocycles representing the generator, known as the second universal motivic Chern class. If $\textrm{G}= \mathrm{SL(m)}$, there is a canonical cocycle, defined by Goncharov (Explicit construction of characteristic classes. Advances in Soviet mathematics, 16, vol 1. Special volume dedicated to I.M.Gelfand’s 80th birthday, pp 169–210, 1993). For any group G, we define a collection of cocycles parametrised by cluster coordinate systems on the space of $\textrm{G}$-orbits on the cube of the principal affine space $\textrm{G}/\textrm{U}$. Cocycles for different clusters are related by explicit coboundaries, constructed using cluster transformations relating the clusters. The cocycle has three components. The construction of the last one is canonical and elementary; it does not use clusters, and provides the motivic generator of $\textrm{H}^3(\textrm{G}({\mathbb {C}}), {\mathbb {Z}}(2))$. However to lift it to the whole cocycle we need cluster coordinates: construction of the first two components uses crucially the cluster structure of the moduli spaces $\mathcal{A}(\textrm{G},{{\mathbb {S}}})$ related to the moduli space of $\textrm{G}$-local systems on ${{\mathbb {S}}}$. In retrospect, it partially explains why cluster coordinates on the space $\mathcal{A}(\textrm{G},{{\mathbb {S}}})$ should exist. The construction has numerous applications, including explicit constructions of the universal extension of the group $\textrm{G}$ by $K_2$, the line bundle on $\textrm{Bun}(\textrm{G})$ generating its Picard group, Kac–Moody groups, etc. Another application is an explicit combinatorial construction of the second motivic Chern class of a $\textrm{G}$-bundle. It is a motivic analog of the work of Gabrielov et al. (1974), for any $\textrm{G}$. We show that the cluster construction of the measurable group 3-cocycle for the group $\textrm{G}({\mathbb {C}})$, provided by our motivic cocycle, gives rise to the quantum deformation of its exponent.