The Aarhus integral of rational homology 3-spheres I: A highly non trivial flat connection on S^3

TL;DR

This paper introduces a rigorous construction of a universal finite-type invariant for rational homology 3-spheres, inspired by the conceptual framework of Chern-Simons theory and flat connections, linking it to known invariants like LMO, Rozansky, and Ohtsuki.

Contribution

It provides a new, mathematically rigorous approach to defining a universal invariant of rational homology spheres inspired by physical path integrals.

Findings

Invariant equals the LMO invariant

Recovers Rozansky and Ohtsuki invariants

Establishes a conceptual framework inspired by non-trivial flat connections

Abstract

Path integrals don't really exist, but it is very useful to dream that they do exist, and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly non-trivial mathematical theorems and theories. We argue that even though non-trivial flat connections on S^3 don't really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the non-existent Chern-Simons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finite-type invariant of rational homology spheres. We show that this invariant is equal to the LMO (Le-Murakami-Ohtsuki) invariant and that it recovers the Rozansky and Ohtsuki invariants. This is part I of a 4-part series, containing the introductions and answers to some frequently asked questions. Theorems are stated but not proved in this part, and it can be viewed as a "research announcement". Part II of this series is titled "Invariance and Universality" (see math/9801049), part III "The Relation with the Le-Murakami-Ohtsuki Invariant" (see math/9808013), and part IV "The Relation with the Rozansky and Ohtsuki Invariants".