On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

Perroni, Fabio; Rahm, Alexander

doi:10.2140/agt.2019.19.2715

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On Ruan's cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

Fabio Perroni and Alexander D Rahm

Algebraic & Geometric Topology 19 (2019) 2715–2762

DOI: 10.2140/agt.2019.19.2715

Abstract

We give formulae for the Chen–Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic $3$ –space.

The Bianchi groups are the arithmetic groups ${PSL}_{2} (O)$ , where $O$ is the ring of integers in an imaginary quadratic number field. The underlying real orbifolds which help us in our study, given by the action of a Bianchi group on real hyperbolic $3$ –space (which is a model for its classifying space for proper actions), have applications in physics.

We then prove that, for any such orbifold, its Chen–Ruan orbifold cohomology ring is isomorphic to the usual cohomology ring of any crepant resolution of its coarse moduli space. By vanishing of the quantum corrections, we show that this result fits in with Ruan’s cohomological crepant resolution conjecture.

Keywords

Chen–Ruan orbifold cohomology, Bianchi orbifolds

Mathematical Subject Classification 2010

Primary: 55N32

References

Publication

Received: 5 December 2016

Revised: 14 November 2018

Accepted: 6 December 2018

Published: 20 October 2019

Authors

Fabio Perroni
Department of Mathematics and Geosciences University of Trieste Trieste Italy
http://perroni.dmg.units.it/
Alexander D Rahm
Mathematics Research Unit Université du Luxembourg Esch-sur-Alzette Luxembourg
http://math.uni.lu/~rahm/

Volume 19, issue 6 (2019)