Cluster construction of the second motivic Chern class

Abstract

Let G be a split, simple, simply connected, algebraic group over Q. The degree 4, weight 2 motivic cohomology group of the classifying space BG of G is identified with Z. We construct cocycles representing the generator of this group, known as the second universal motivic Chern class. If G = SL(m), there is a canonical cocycle, defined by the first author (1993). For any group G, we define a collection of cocycles parametrised by cluster coordinate systems on the space of G-orbits on the cube of the principal affine space G/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 a canonical cocycle for the motivic generator of the degree 3 cohomology class of the complex manifold G(C). However to lift this component to the whole cocycle we need cluster coordinates: the construction of the first two components uses crucially the cluster structure of the moduli spaces A(G,S) related to the moduli space of G-local systems on S. In retrospect, it partially explains why the cluster coordinates on the space A(G,S) should exist. This construction has numerous applications, including an explicit construction of the universal extension of the group G by K_2, the line bundle on Bun(G) generating its Picard group, Kac-Moody groups, etc. Another application is an explicit combinatorial construction of the second motivic Chern class of a G-bundle. It is a motivic analog of the work of Gabrielov-Gelfand-Losik (1974), for any G. We show that the cluster construction of the measurable group 3-cocycle for the group G(C) gives rise to its quantum deformation.