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Journal of High Energy Physics

, 2012:20 | Cite as

Towards the continuous limit of cluster integrable systems

  • Sebastián FrancoEmail author
  • Daniele Galloni
  • Yang-Hui He
Open Access
Article

Abstract

We initiate the study of how to extend the correspondence between dimer models and (0 + 1)-dimensional cluster integrable systems to (1 + 1) and (2 + 1)-dimensional continuous integrable field theories, addressing various points that are necessary for achieving this goal. We first study how to glue and split two integrable systems, from the perspectives of the spectral curve, the resolution of the associated toric Calabi-Yau 3-folds and Higgsing in quiver theories on D3-brane probes. We identify a continuous parameter controlling the decoupling between the components and present two complementary methods for determining the dependence on this parameter of the dynamical variables of the integrable system. Interested in constructing systems with an infinite number of degrees of freedom, we study the combinatorics of integrable systems built up from a large number of elementary components, and introduce a toy model capturing important features expected to be present in a continuous reformulation of cluster integrable systems.

Keywords

Brane Dynamics in Gauge Theories Conformal Field Models in String Theory Integrable Equations in Physics 

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Copyright information

© SISSA 2012

Authors and Affiliations

  • Sebastián Franco
    • 1
    • 2
    Email author
  • Daniele Galloni
    • 2
  • Yang-Hui He
    • 3
    • 4
    • 5
  1. 1.Theory Group, SLAC National Accelerator LaboratoryMenlo ParkU.S.A.
  2. 2.Institute for Particle Physics Phenomenology, Department of PhysicsDurham UniversityDurhamUnited Kingdom
  3. 3.Department of MathematicsCity UniversityLondonUnited Kingdom
  4. 4.School of PhysicsNanKai UniversityTianjinP.R. China
  5. 5.Merton CollegeUniversity of OxfordOxfordUnited Kingdom

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