A Large k Asymptotics of Witten's Invariant of Seifert Manifolds

TL;DR

This paper derives the large k asymptotic expansion of Witten's SU(2) invariant for Seifert manifolds, identifying contributions from all flat connections and connecting 2-loop corrections to Casson's invariant.

Contribution

It provides the first detailed large k asymptotic expansion of Witten's invariant for Seifert manifolds, including all flat connection contributions and 2-loop corrections.

Findings

Agreement with the 1-loop formula confirmed

Irreducible connection contributions are finite

2-loop correction linked to Casson's invariant

Abstract

We calculate a large $k$ asymptotic expansion of the exact surgery formula for Witten's $SU(2)$ invariant of Seifert manifolds. The contributions of all flat connections are identified. An agreement with the 1-loop formula is checked. A contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series. A 2-loop correction to the contribution of the trivial connection is found to be proportional to Casson's invariant.