Geometry & Topology

Quantum characteristic classes and the Hofer metric

Yasha Savelyev

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Abstract

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),ℚ), 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800127

Digital Object Identifier
doi:10.2140/gt.2008.12.2277

Mathematical Reviews number (MathSciNet)
MR2443967

Zentralblatt MATH identifier
1146.53070

Subjects
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]

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

Citation

Savelyev, Yasha. Quantum characteristic classes and the Hofer metric. Geom. Topol. 12 (2008), no. 4, 2277--2326. doi:10.2140/gt.2008.12.2277. https://projecteuclid.org/euclid.gt/1513800127


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Geometry & Topology

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