TY - JOUR
T1 - Poisson brackets and symplectic invariants
AU - Buhovsky, Lev
AU - Entov, Michael
AU - Polterovich, Leonid
N1 - Funding Information:
Acknowledgments We thank Richard Hind for his help with symplectic field theory and in particular for providing us a short proof of Theorem 6.1. We are grateful to Paul Seidel and Ivan Smith for useful consultations on Donaldson–Fukaya category. In particular, Smith explained to us some interesting examples where operations µ2 and µ3 do not vanish. We thank Paul Biran for a number of stimulating discussions. We thank Strom Borman and an anonymous referee for comments and corrections—in particular, Borman suggested a considerable improvement of the numerical constants in Sect. 1.5. A part of this paper was written during our stay in MSRI, Berkeley. We thank MSRI for the hospitality. The third-named author thanks the Simons Foundation for sponsoring this stay. Michael Entov was partially supported by the Israel Science Foundation grant # 723/10 and by the Japan Technion Society Research Fund. Leonid Polterovich was partially supported by the National Science Foundation grant DMS-1006610.
PY - 2012/3
Y1 - 2012/3
N2 - We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the μ 3-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.
AB - We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the μ 3-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.
KW - Donaldson-Fukaya category
KW - Hamiltonian chord
KW - Poisson brackets
KW - Quasi-state
KW - symplectic manifold
UR - http://www.scopus.com/inward/record.url?scp=84857763456&partnerID=8YFLogxK
U2 - 10.1007/s00029-011-0068-9
DO - 10.1007/s00029-011-0068-9
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AN - SCOPUS:84857763456
SN - 1022-1824
VL - 18
SP - 89
EP - 157
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 1
ER -