The Aarhus integral of rational homology 3-spheres III: The Relation with the Le-Murakami-Ohtsuki Invariant

TL;DR

This paper establishes a precise relationship between the Aarhus integral and the Le-Murakami-Ohtsuki invariant for rational homology 3-spheres, unifying their interpretations as integrated holonomies through formal Gaussian and negative-dimensional integrations.

Contribution

It proves the connection between the Aarhus integral and the LMO invariant, interpreting both as forms of integrated holonomies via different integration theories.

Findings

The Aarhus integral and LMO invariant are related through their interpretation as integrated holonomies.

The paper develops a new interpretation of the map j_m as formal negative-dimensional integration.

The relationship between the two invariants follows from the connection between their respective integration methods.

Abstract

Continuing the work started in Part I and II of this series (see q-alg/9706004 and math.QA/9801049), we prove the relationship between the Aarhus integral and the invariant $\Omega$ (henceforth called LMO) defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in q-alg/9512002. The basic reason for the relationship is that both constructions afford an interpretation as "integrated holonomies". In the case of the Aarhus integral, this interpretation was the basis for everything we did in Parts I and II. The main tool we used there was "formal Gaussian integration". For the case of the LMO invariant, we develop an interpretation of a key ingredient, the map $j_m$, as "formal negative-dimensional integration". The relation between the two constructions is then an immediate corollary of the relationship between the two integration theories.