BIMSA > BIMSA Lecture BIMSA Lecture Special solutions of a discrete Painlevé equation for quantum minimal surfaces
Special solutions of a discrete Painlevé equation for quantum minimal surfaces
Organizer
Anton Dzhamay
Speaker
Andy Hone
Time
Friday, September 18, 2026 3:00 PM - 4:00 PM
Venue
A3-2a-302
Online
Zoom 787 662 9899 (BIMSA)
Abstract
We consider solutions of a discrete Painlevé equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe and Kontsevich, and in earlier work of Cornalba and Taylor on static membranes. While the discrete equation admits a continuum limit to the continuous Painlevé I equation, we find that it has the same space of initial values as the Painlevé V equation with certain specific parameter values. We further explicitly show how each iteration of this discrete Painlevé I equation corresponds to a certain composition of Bäcklund transformations for Painlevé V, as was first remarked in work by Tokihiro, Grammaticos and Ramani. In addition, we show that some explicit special function solutions of Painlevé V, written in terms of modified Bessel functions, yield the unique positive solution of the initial value problem required for quantum minimal surfaces. This is joint work with Peter Clarkson, Anton Dzhamay and Ben Mitchell.