Coaction and double-copy properties of configuration-space integrals at genus zero

We investigate configuration-space integrals over punctured Riemann spheres from the viewpoint of the motivic Galois coaction and double-copy structures generalizing the Kawai-Lewellen-Tye (KLT) relations in string theory. For this purpose, explicit bases of twisted cycles and cocycles are worked out whose orthonormality simplifies the coaction. We present methods to efficiently perform and organize the expansions of configuration-space integrals in the inverse string tension α′ or the dimensional-regularization parameter ϵ of Feynman integrals. Generating-function techniques open up a new perspective on the coaction of multiple polylogarithms in any number of variables and analytic continuations in the unintegrated punctures. We present a compact recursion for a generalized KLT kernel and discuss its origin from intersection numbers of Stasheff polytopes and its implications for correlation functions of two-dimensional conformal field theories. We find a non-trivial example of correlation functions in ($$ \mathfrak{p} $$ p , 2) minimal models, which can be normalized to become uniformly transcendental in the $$ \mathfrak{p} $$ p → ∞ limit.

Diagram Complexes, Formality, and Configuration Space Integrals for Spaces of Braids

We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen’s iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a commutative differential graded algebra suitable for integration. We show that this integration is compatible with Bott–Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.

On volume-preserving vector fields and finite-type invariants of knots

We consider the general non-vanishing, divergence-free vector fields defined on a domain in$3$-space and tangent to its boundary. Based on the theory of finite-type invariants, we define a family of invariants for such fields, in the style of Arnold’s asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.

CONFIGURATION SPACE INTEGRALS AND THE COHOMOLOGY OF THE SPACE OF HOMOTOPY STRING LINKS

Configuration space integrals have been used in recent years for studying the cohomology of spaces of (string) knots and links in ℝn for n > 3 since they provide a map from a certain differential graded algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links — the space of smooth maps of some number of copies of ℝ in ℝn with fixed behavior outside a compact set and such that the images of the copies of ℝ are disjoint — even for n = 3. We further study the case n = 3 in degree zero and show that our integrals represent a universal finite type invariant of the space of classical homotopy string links. As a consequence, we deduce that Milnor invariants of string links can be written in terms of configuration space integrals.

CONFIGURATION SPACE INTEGRALS FOR EMBEDDING SPACES AND THE HAEFLIGER INVARIANT

Let [Formula: see text] be the space of long j-knots in ℝn. In this paper, we introduce a graph complex [Formula: see text] and a linear map [Formula: see text] via configuration space integral, and prove that (1) when both n > j ≥ 3 are odd, I is a cochain map if restricted to graphs with at most one loop component, (2) when n - j ≥ 2 is even, I is a cochain map if restricted to tree graphs, and (3) when n - j ≥ 3 is odd, I added a correction term produces a (2n - 3j - 3)-cocycle of [Formula: see text] which gives a new formulation of the Haefliger invariant when n = 6k, j = 4k - 1 for some k.

MASSEY–MILNOR LINKING = CHERN–SIMONS–WITTEN GRAPHS

We express the first non-vanishing Massey–Milnor linkings in terms of Chern–Simons–Witten configuration space integrals in perturbative quantum field theory.

A SURVEY OF BOTT–TAUBES INTEGRATION

It is well-known that certain combinations of configuration space integrals defined by Bott and Taubes [11] produce cohomology classes of spaces of knots. The literature surrounding this important fact, however, is somewhat incomplete and lacking in detail. The aim of this paper is to fill in the gaps as well as summarize the importance of these integrals.