ON THE EQUIVALENCE PROBLEM FOR TORIC CONTACT STRUCTURES ON S~3-BUNDLES OVER S~2

摘要

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',q') are inequivalent as contact structures if and only if p ≠ p'.