Title:
Parenthesized chord diagrams, graph complexes, and Chern characters
Abstract:
I will give a broad overview of past joint work (arXiv:1211.4230) with V. Dolgushev and T. Willwacher which provides some interesting connections between geometric topology/quantum algebra and the deformation theory of algebraic varieties. The main player in this story is V. Drinfeld’s Grothendieck-Teichmueller group GRT_1. In our work, we describe how GRT}_1 acts as derived symmetries of the Gerstenhaber structure on the sheaf of polyvector fields T_{poly}X of any smooth complex algebraic variety X. GRT_1 itself can be understood as automorphisms of the topological operad of “parenthesized chord diagrams”, as demonstrated by D. Bar-Natan and D. Tamarkin. The connection to deformation theory is via Willwacher’s Theorem which identifies the corresponding Lie algebra grt_1 with the degree zero cohomology of M. Kontsevich’s graph complex GC_2. In particular, we show that the action on T_{poly}X by the graphs in GC_2 which represent the so-called “Deligne-Drinfeld elements” in grt_1 coincides with contracting polyvectors by the odd components of the Chern character of X, as conjectured by Kontsevich in 1999. After explaining these results in more detail, I hope to mention some new conjectural relationships between this work and the recent topological characterization of the Kashiwara-Vergne problem from Lie theory by Z. Dancso, I. Halacheva, and M. Robertson.