Functions on moduli spaces from cyclic homology

We discuss the 'moduli of objects' M_D in a dg category D and construct a map from cyclic homology of D to functions on the moduli space M_D. When D is a smooth, oriented dg category ('Calabi-Yau'), the cyclic homology HC(D) is endowed with a shifted Lie bracket ('algebraic string bracket') and the functions on M_D are endowed with a shifted Poisson bracket. We show that the map from cyclic homology to functions entwines the brackets. Examples include the Goldmann bracket of free loops on a surface, the string bracket of Chas-Sullivan, and the Hitchen system for Higgs bundles. This is joint work very much in progress with Nick Rozenblyum.