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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 PSL2(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
Perroni, Fabio 1 ; Rahm, Alexander 2
@article{10_2140_agt_2019_19_2715,
author = {Perroni, Fabio and Rahm, Alexander},
title = {On {Ruan{\textquoteright}s} cohomological crepant resolution conjecture for the complexified {Bianchi} orbifolds},
journal = {Algebraic and Geometric Topology},
pages = {2715--2762},
publisher = {mathdoc},
volume = {19},
number = {6},
year = {2019},
doi = {10.2140/agt.2019.19.2715},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2715/}
}
TY - JOUR AU - Perroni, Fabio AU - Rahm, Alexander TI - On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds JO - Algebraic and Geometric Topology PY - 2019 SP - 2715 EP - 2762 VL - 19 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2715/ DO - 10.2140/agt.2019.19.2715 ID - 10_2140_agt_2019_19_2715 ER -
%0 Journal Article %A Perroni, Fabio %A Rahm, Alexander %T On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds %J Algebraic and Geometric Topology %D 2019 %P 2715-2762 %V 19 %N 6 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2715/ %R 10.2140/agt.2019.19.2715 %F 10_2140_agt_2019_19_2715
Perroni, Fabio; Rahm, Alexander. On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds. Algebraic and Geometric Topology, Tome 19 (2019) no. 6, pp. 2715-2762. doi: 10.2140/agt.2019.19.2715
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