Geometry & Topology

Quantum characteristic classes and the Hofer metric

Yasha Savelyev

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Given a closed monotone symplectic manifold M, we define certain characteristic cohomology classes of the free loop space LHam(M,ω) with values in QH(M), and their S1 equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring H(ΩHam(M,ω),), with its Pontryagin product to QH2n+(M) with its quantum product. As an application we prove an extension to higher dimensional geometry of the loop space LHam(M,ω) of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action.

Article information

Geom. Topol., Volume 12, Number 4 (2008), 2277-2326.

Received: 9 February 2008
Revised: 18 July 2008
Accepted: 5 June 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05]

quantum homology Hamiltonian group energy flow loop group Hamiltonian symplectomorphism Hofer metric


Savelyev, Yasha. Quantum characteristic classes and the Hofer metric. Geom. Topol. 12 (2008), no. 4, 2277--2326. doi:10.2140/gt.2008.12.2277.

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