From compact semi-toric systems to Hamiltonian $S^1$-spaces

Sonja Hohloch; Silvia Sabatini; Daniele Sepe

doi:10.3934/dcds.2015.35.247

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January 2015, 35(1): 247-281. doi: 10.3934/dcds.2015.35.247

From compact semi-toric systems to Hamiltonian $S^1$-spaces

Sonja Hohloch ^1, , Silvia Sabatini ^2, and Daniele Sepe ^3,

1.	Section de Mathématiques, EPFL, SB MATHGEOM CAG, Station 8, 1015 Lausanne, Switzerland
2.	CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal
3.	Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga, 24020-240 Niteroi, RJ, Brazil

Received December 2013 Revised March 2014 Published August 2014

Abstract
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We show how any labeled convex polygon associated to a compact semi-toric system, as defined by Vũ ngọc, determines Karshon's labeled directed graph which classifies the underlying Hamiltonian $S^1$-space up to isomorphism. Then we characterize adaptable compact semi-toric systems, i.e. those whose underlying Hamiltonian $S^1$-action can be extended to an effective Hamiltonian $\mathbb{T}^2$-action, as those which have at least one associated convex polygon which satisfies the Delzant condition.

Keywords: symplectic invariants., symplectic toric manifolds, integrable Hamiltonian systems, Hamiltonian torus actions, semi-toric systems.

Mathematics Subject Classification: Primary: 37J05, 37J35, 53D2.

Citation: Sonja Hohloch, Silvia Sabatini, Daniele Sepe. From compact semi-toric systems to Hamiltonian $S^1$-spaces. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 247-281. doi: 10.3934/dcds.2015.35.247

References:

[1]	M. F. Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi: 10.1112/blms/14.1.1. Google Scholar
[2]	O. Babelon and B. Doucot, Higher index focus-focus singularities in the Jaynes-Cummings-Gaudin model: symplectic invariants and monodromy,, preprint, (). doi: 10.1016/j.geomphys.2014.07.011. Google Scholar
[3]	A. V. Bolsinov and A. A. Oshemkov, Singularities of integrable Hamiltonian systems, in Topological Methods in the Theory of Integrable Systems, Camb. Sci. Publ., Cambridge, 2006, 1-67. Google Scholar
[4]	M. Chaperon, Quelques outils de la théorie des actions différentiables, in Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Astérisque, 107-108, Soc. Math. France, Paris, 1983, 259-275. Google Scholar
[5]	T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, (French) [Periodic Hamiltonians and convex images of the momentum mapping], Bull. Soc. Math. France, 116 (1988), 315-339. Google Scholar
[6]	J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602. Google Scholar
[7]	J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., 69 (1982), 259-269. doi: 10.1007/BF01399506. Google Scholar
[8]	L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, Ph.D thesis, University of Stockholm, 1984. Google Scholar
[9]	L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals - elliptic case, Comment. Math. Helv., 65 (1990), 4-35. doi: 10.1007/BF02566590. Google Scholar
[10]	A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Translated from the Russian by M. V. Tsaplina., Mathematics and its Applications (Soviet Series) 31, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-009-3069-8. Google Scholar
[11]	V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513. doi: 10.1007/BF01398933. Google Scholar
[12]	Y. Karshon, Periodic Hamiltonian flows on four dimensional manifolds, Mem. Amer. Math. Soc., 672 (1999). doi: 10.1090/memo/0672. Google Scholar
[13]	Y. Karshon, L. Kessler and M. Pinsonnault, A compact symplectic four-manifold admits only finitely many inequivalent toric actions, J. Symplectic Geom., 5 (2007), 139-166. doi: 10.4310/JSG.2007.v5.n2.a1. Google Scholar
[14]	Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc., 353 (2001), 4831-4861 (electronic). doi: 10.1090/S0002-9947-01-02799-4. Google Scholar
[15]	Y. Karshon and S. Tolman, Complete invariants for Hamiltonian torus actions with two dimensional quotients, J. Symplectic Geom., 2 (2003), 25-82. doi: 10.4310/JSG.2004.v2.n1.a2. Google Scholar
[16]	Y. Karshon and S. Tolman, Classification of Hamiltonian torus actions with two dimensional quotients,, , (). doi: 10.2140/gt.2014.18.669. Google Scholar
[17]	F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press, 1984. doi: 10.1007/BF01145470. Google Scholar
[18]	N. C. Leung and M. Symington, Almost toric symplectic four-manifolds, J. Symplectic Geom., 8 (2010), 143-187. doi: 10.4310/JSG.2010.v8.n2.a2. Google Scholar
[19]	E. Miranda and N. T. Zung, Equivariant normal form for non-degenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. Éc. Norm. Sup., 37 (2004), 819-839. doi: 10.1016/j.ansens.2004.10.001. Google Scholar
[20]	Á. Pelayo and S. Vũ Ngọc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 177 (2009), 571-597. doi: 10.1007/s00222-009-0190-x. Google Scholar
[21]	Á. Pelayo and S. Vũ Ngọc, Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125. doi: 10.1007/s11511-011-0060-4. Google Scholar
[22]	Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.), 48 (2011), 409-455. doi: 10.1090/S0273-0979-2011-01338-6. Google Scholar
[23]	D. A. Sadovskií and B. I. Zĥilinskií, Monodromy, diabolic points and angular momentum coupling, Phys. Lett. A., 256 (1999), 235-244. doi: 10.1016/S0375-9601(99)00229-7. Google Scholar
[24]	M. Symington, Four dimensions from two in symplectic topology, in Topology and Geometry of Manifolds (Athens, GA), Proc. Sympos. Pure Math., 71, Amer. Math. Soc., 2003, 153-208. doi: 10.1090/pspum/071/2024634. Google Scholar
[25]	S. Vũ Ngọc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X. Google Scholar
[26]	S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934. doi: 10.1016/j.aim.2006.04.004. Google Scholar
[27]	J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062. Google Scholar
[28]	N. T. Zung, Symplectic topology of integrable Hamiltonian systems I: Arnol'd-Liouville with singularities, Compositio Math., 101 (1996), 179-215. Google Scholar
[29]	N. T. Zung, A note on focus-focus singularities, Diff. Geom. Appl., 7 (1997), 123-130. doi: 10.1016/S0926-2245(96)00042-3. Google Scholar
[30]	N. T. Zung, Another note on focus-focus singularities, Lett. Math. Phys., 60 (2002), 87-99. doi: 10.1023/A:1015761729603. Google Scholar

show all references

References:

[1]	M. F. Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi: 10.1112/blms/14.1.1. Google Scholar
[2]	O. Babelon and B. Doucot, Higher index focus-focus singularities in the Jaynes-Cummings-Gaudin model: symplectic invariants and monodromy,, preprint, (). doi: 10.1016/j.geomphys.2014.07.011. Google Scholar
[3]	A. V. Bolsinov and A. A. Oshemkov, Singularities of integrable Hamiltonian systems, in Topological Methods in the Theory of Integrable Systems, Camb. Sci. Publ., Cambridge, 2006, 1-67. Google Scholar
[4]	M. Chaperon, Quelques outils de la théorie des actions différentiables, in Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Astérisque, 107-108, Soc. Math. France, Paris, 1983, 259-275. Google Scholar
[5]	T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, (French) [Periodic Hamiltonians and convex images of the momentum mapping], Bull. Soc. Math. France, 116 (1988), 315-339. Google Scholar
[6]	J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602. Google Scholar
[7]	J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., 69 (1982), 259-269. doi: 10.1007/BF01399506. Google Scholar
[8]	L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, Ph.D thesis, University of Stockholm, 1984. Google Scholar
[9]	L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals - elliptic case, Comment. Math. Helv., 65 (1990), 4-35. doi: 10.1007/BF02566590. Google Scholar
[10]	A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Translated from the Russian by M. V. Tsaplina., Mathematics and its Applications (Soviet Series) 31, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-009-3069-8. Google Scholar
[11]	V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513. doi: 10.1007/BF01398933. Google Scholar
[12]	Y. Karshon, Periodic Hamiltonian flows on four dimensional manifolds, Mem. Amer. Math. Soc., 672 (1999). doi: 10.1090/memo/0672. Google Scholar
[13]	Y. Karshon, L. Kessler and M. Pinsonnault, A compact symplectic four-manifold admits only finitely many inequivalent toric actions, J. Symplectic Geom., 5 (2007), 139-166. doi: 10.4310/JSG.2007.v5.n2.a1. Google Scholar
[14]	Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc., 353 (2001), 4831-4861 (electronic). doi: 10.1090/S0002-9947-01-02799-4. Google Scholar
[15]	Y. Karshon and S. Tolman, Complete invariants for Hamiltonian torus actions with two dimensional quotients, J. Symplectic Geom., 2 (2003), 25-82. doi: 10.4310/JSG.2004.v2.n1.a2. Google Scholar
[16]	Y. Karshon and S. Tolman, Classification of Hamiltonian torus actions with two dimensional quotients,, , (). doi: 10.2140/gt.2014.18.669. Google Scholar
[17]	F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press, 1984. doi: 10.1007/BF01145470. Google Scholar
[18]	N. C. Leung and M. Symington, Almost toric symplectic four-manifolds, J. Symplectic Geom., 8 (2010), 143-187. doi: 10.4310/JSG.2010.v8.n2.a2. Google Scholar
[19]	E. Miranda and N. T. Zung, Equivariant normal form for non-degenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. Éc. Norm. Sup., 37 (2004), 819-839. doi: 10.1016/j.ansens.2004.10.001. Google Scholar
[20]	Á. Pelayo and S. Vũ Ngọc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 177 (2009), 571-597. doi: 10.1007/s00222-009-0190-x. Google Scholar
[21]	Á. Pelayo and S. Vũ Ngọc, Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125. doi: 10.1007/s11511-011-0060-4. Google Scholar
[22]	Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.), 48 (2011), 409-455. doi: 10.1090/S0273-0979-2011-01338-6. Google Scholar
[23]	D. A. Sadovskií and B. I. Zĥilinskií, Monodromy, diabolic points and angular momentum coupling, Phys. Lett. A., 256 (1999), 235-244. doi: 10.1016/S0375-9601(99)00229-7. Google Scholar
[24]	M. Symington, Four dimensions from two in symplectic topology, in Topology and Geometry of Manifolds (Athens, GA), Proc. Sympos. Pure Math., 71, Amer. Math. Soc., 2003, 153-208. doi: 10.1090/pspum/071/2024634. Google Scholar
[25]	S. Vũ Ngọc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi: 10.1016/S0040-9383(01)00026-X. Google Scholar
[26]	S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934. doi: 10.1016/j.aim.2006.04.004. Google Scholar
[27]	J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062. Google Scholar
[28]	N. T. Zung, Symplectic topology of integrable Hamiltonian systems I: Arnol'd-Liouville with singularities, Compositio Math., 101 (1996), 179-215. Google Scholar
[29]	N. T. Zung, A note on focus-focus singularities, Diff. Geom. Appl., 7 (1997), 123-130. doi: 10.1016/S0926-2245(96)00042-3. Google Scholar
[30]	N. T. Zung, Another note on focus-focus singularities, Lett. Math. Phys., 60 (2002), 87-99. doi: 10.1023/A:1015761729603. Google Scholar

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From compact semi-toric systems to Hamiltonian $S^1$-spaces

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