The Ising Mannequin Coupled to 2D Gravity: Larger-order Painlevé Equations/The (3,4) String Equation
Authors: Nathan Hayford
Summary: In continuation of the work [1], we research a higher-order Painlevé-type equation, arising as a string equation of the third order discount of the KP hierarchy. This equation seems on the multi-critical level of the 2-matrix mannequin with quartic interactions, and describes the Ising part transition coupled to 2D gravity. We characterize this equation when it comes to the isomonodromic deformations of a specific rational connection on P1. We additionally establish the (nonautonomous) Hamiltonian construction related to this equation, and write an appropriate τ-differential for this technique. This τ-differential may be prolonged to the canonical coordinates of the related Hamiltonian system, permitting us to confirm Conjectures 1. and a pair of. of [2] in our case. We additionally current a reasonably common formulation for the τ-differential of a particular class of resonant connections, which is considerably easier than that of [3]. [1] M. Duits, N. Hayford, and S.-Y. Lee. “The Ising Mannequin Coupled to 2D Gravity: Genus Zero Partition Perform”. arXiv preprint, 2023. [2] A.R. Its and A. Prokhorov. “On some Hamiltonian properties of the isomonodromic tau capabilities”. Rev. Math. Phys. 30.7 (2018). [3] M. Bertola and M.Y. Mo. “Isomonodromic deformation of resonant rational connections”. Int. Math. Res. Pap. 11 (2005).
