Graphs, strings, and actions

Abstract

In this paper, we revisit the formalism of graphs, trees, and surfaces which allows one to build cell models for operads of algebraic interest and represent them in terms of a dynamical picture of moving strings—hence relating string dynamics to algebra and geometry. In particular, we give a common framework for solving the original version of Deligne’s conjecture, its cyclic, A∞, and cyclic-A∞ versions. We furthermore study a question raised by Kontsevich and Soibelman about models of the little discs operad. On one hand, we give a new smooth model and on the other hand, a minimally small cell model for the A∞ case. Further geometric results these models provide are novel decompositions and realizations of cyclohedra as well as explicit simple cell representatives for Dyer-Lashof-Cohen operations. We also briefly discuss the generalizations to moduli space actions and applications to string topology as well as further directions.