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Constructing integrable systems of semitoric type

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Acta Mathematica

Abstract

Let (M, ω) be a connected, symplectic 4-manifold. A semitoric integrable system on (M, ω) essentially consists of a pair of independent, real-valued, smooth functions J and H on M, for which J generates a Hamiltonian circle action under which H is invariant. In this paper we give a general method to construct, starting from a collection of five ingredients, a symplectic 4-manifold equipped a semitoric integrable system. Then we show that every semitoric integrable system on a symplectic 4-manifold is obtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors, this gives a classification of semitoric integrable systems on 4-manifolds, in terms of five invariants.

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, U.S.A.

    Álvaro Pelayo

  2. Washington University in St. Louis, Department of Mathematics, Campus box 1146, One Brookings Drive, St. Louis, MO, 63130, U.S.A.

    Álvaro Pelayo

  3. Institut de Recherches Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, FR-35042, Rennes cedex, France

    San Vũ Ngọc

Authors
  1. Álvaro Pelayo
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  2. San Vũ Ngọc
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Correspondence to Álvaro Pelayo.

Additional information

The first author was partially supported by an NSF post-doctoral fellowship. This work was done while the first author was at the Massachusetts Institute of Technology (2007–2008) and at the University of California, Berkeley (2008–2010).

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Pelayo, Á., Vũ Ngọc, S. Constructing integrable systems of semitoric type. Acta Math 206, 93–125 (2011). https://doi.org/10.1007/s11511-011-0060-4

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  • DOI: https://doi.org/10.1007/s11511-011-0060-4

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