1-loop graphs and configuration space integral for embedding spaces

TL;DR

This paper constructs differential forms on embedding spaces using configuration space integrals from 1-loop graphs, revealing new nontrivial cohomology classes in high-dimensional embedding spaces.

Contribution

It introduces a novel method to produce closed differential forms on embedding spaces via 1-loop graph integrals, leading to new cohomology classes in higher degrees.

Findings

Constructed closed forms on Emb(R^j,R^n) using 1-loop graph integrals.

Identified nontrivial cohomology classes in high-dimensional embedding spaces.

Extended results to homotopy fibers of embedding spaces.

Abstract

We will construct differential forms on the embedding spaces Emb(R^j,R^n) for n-j>=2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions. There are other dimensions in which we can show the closedness if we replace Emb(R^j,R^n) by fEmb(R^j,R^n), the homotopy fiber of the inclusion Emb(R^j,R^n) -> Imm(R^j,R^n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(R^j,R^n) and fEmb(R^j,R^n). In particular we obtain nontrivial cohomology classes (for example, in H^3(Emb(R^2,R^5))) of higher degrees than those of the first nonvanishing homotopy groups.