On the equivalence problem for toric contact structures on ${\bf S^3}$-bundles over ${\bf S^2}$

Charles P. Boyer and Justin Pati

Vol. 267 (2014), No. 2, 277–324

DOI: 10.2140/pjm.2014.267.277

Abstract

We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To show that two toric contact structures with the same first Chern class are contact inequivalent, we use Morse–Bott contact homology. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding $3$-tori belong to distinct conjugacy classes in the contactomorphism group. We treat a subclass of contact structures which includes the Sasaki–Einstein contact structures $Y^{p,q}$ studied by physicists with the anti-de Sitter/conformal field theory conjecture. In this case we give a complete solution to the contact equivalence problem by showing that $Y^{p,q}$ and $Y^{p^{\prime },q^{\prime }}$ are inequivalent as contact structures if and only if $p\ne p^{\prime }$.

Keywords

toric contact geometry, equivalent contact structures, orbifold Hirzebruch surface, contact homology, extremal Sasakian structures

Mathematical Subject Classification 2010

Primary: 53D10, 53D20, 53D42

Milestones

Received: 23 April 2012

Revised: 4 July 2013

Accepted: 9 July 2013

Published: 11 May 2014