Abstract
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.
Highlights
Terms of these bi-partite graphs drawn on a torus has led to both the simplification of existing problems as well as development of new ideas in a plethora of directions ranging from physics to mathematics
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
We have taken the initial steps in extending the correspondence between dimer models and (0+1)-dimensional cluster integrable systems to continuous (1+1) and (2+1)-dimensional integrable theories
Summary
A remarkable correspondence linking dimer models to an infinite class of integrable systems, denoted cluster integrable systems, was recently introduced in [13]. The Poisson manifold of the integrable system is parametrized by oriented loops on the brane tiling. Ng, with Ng the number of gauge groups in the quiver) and the cycles z1 and z2 wrapping the two directions of the 2-torus provide one possible basis for loops. Ǫwi,wj is the antisymmetric oriented adjacency matrix that counts the number of arrows between gauge groups in the quiver dual to the brane tiling. The toric diagram of the Calabi-Yau 3-fold associated to the dimer model gives rise to a Riemann surface of genus equaling to the number of internal points. The latter is given by the zero locus of the Newton or characteristic polynomial. The full Poisson manifold of the integrable system is obtained by gluing different patches via cluster transformations, equivalently Seiberg duality in the associated quiver gauge theories
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
