Canonical Paths and Single Valued Iterated Integrals

Let X be a smooth connected complex variety and let a_0 and a_1 be two points in X. Since X can fail to be simply connected there may be many homotopy classes of paths from a_0 to a_1 in X, so that homotopy invariant iterated integrals of 1-forms on X along such paths need not be single-valued. In joint work with Daniel Litt, we define a "canonical path" p(a_0,a_1) in C[[π_1(X;a_0,a_1)]] the Mal'cev completion of the homotopy path torsor π_1(X;a_0,a_1), that is a certain formal C-linear combination of homotopy classes of paths from a_0 to a_1 in X, so that homotopy invariant iterated integrals of 1-forms on X along the "canonical path" are single-valued. We show that many examples of single-valued functions, for instance the single-valued polylogarithms of Bloch-Wigner-Ramakrishnan and their elliptic generalizations, arise from our formalism.