Quivers and Difference Painlevé Equations
QUIVERS AND DIFFERENCE PAINLEVE´ EQUATIONS PHILIP BOALCH Dedicated to John McKay Abstract. We will describe natural Lax pairs for the difference Painlev´eequa- tions with affine Weyl symmetry groups of types E6, E7 and E8, showing that they do indeed arise as symmetries of certain Fuchsian systems of differential equations. 1. Introduction The theory of the Painlev´edifferential equations is now well established and their solutions, the Painlev´etranscendents, appear in numerous nonlinear problems of mathematics and physics, from Einstein metrics to Frobenius manifolds. However in certain applications these differential equations arise from a limiting process, starting with a discrete nonlinear equation. For example (see [41]) the difference equation an + b fn+1 + fn + fn−1 = + c fn arose in work of Br´ezin and Kazakov [12] and Gross and Migdal [23] in the calculation of a certain partition function of a 2D quantum gravity model, and it becomes a differential Painlev´eequation in a certain limit. The classification and study of such “discrete Painlev´eequations” has been an active area in recent years (see for example the reviews [22] or [41]). On one hand many such equations have been found by studying a discrete analogue of the Painlev´eproperty, such as singularity confinement. On the other hand Sakai [40] has discovered a uniform framework to classify all second order discrete Painlev´eequations in terms of rational surfaces obtained by blowing up P2 in nine, possibly infinitely near, points. In this framework, to each equation is associated an affine root system R embedded in the affine E8 root system, and thus an affine Dynkin diagram.
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