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Quantum Topology

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Volume 12, Issue 1, 2021, pp. 111–127
DOI: 10.4171/QT/146

Published online: 2021-03-15

A note on the $\Theta$-invariant of 3-manifolds

Alberto S. Cattaneo[1] and Tatsuro Shimizu[2]

(1) Universität Zürich, Switzerland
(2) Kyoto University, Japan

In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author in [4]. The $\Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern–Simons perturbation theory. The $\Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the $\Theta$-invariant that we can define even if the cohomology group is not vanishing.

Keywords: Invariants of 3-manifolds, Chern–Simons perturbation theory, configuration space integral

Cattaneo Alberto, Shimizu Tatsuro: A note on the $\Theta$-invariant of 3-manifolds. Quantum Topol. 12 (2021), 111-127. doi: 10.4171/QT/146