On \(q\)-Painlevé VI and the geometry of Segre surfaces (Q6592183)

Very pleasant to read, this long article constitutes a rigorous, coherent description of the main features of the discrete \(q\)-\(\mathrm{P}_{\mathrm{VI}}\) of \textit{M. Jimbo} and \textit{H. Sakai} [Lett. Math. Phys. 38, No. 2, 145--154 (1996; Zbl 0859.39006)]: monodromy manifold, geometric description by an ``affine'' surface, asymptotic expansions near critical singularities, connection problem, etc.\N\NThe constant guideline is the continuum limit to \(\mathrm{P}_{\mathrm{VI}}\): the limit \(q \to 1\) of every property of \(q\)-\(\mathrm{P}_{\mathrm{VI}}\) should be a property of \(\mathrm{P}_{\mathrm{VI}}\). For instance, the nice discrete connection formula (2.41) is shown to admit the cubic of Fricke (2.44) as its continuum limit.\N\NAn important property -- of the highest interest in physics -- is however absent of this study. This is the discrete analogue of the ``quantum correspondence'' of \textit{B. I. Suleimanov} [Differ. Equations 30, No. 5, 726--732 (1994; Zbl 0844.34012); translation from Differ. Uravn. 30, No. 5, 791--796 (1994); Funct. Anal. Appl. 48, No. 3, 198--207 (2014; Zbl 1315.34099); translation from Funkts. Anal Prilozh. 48, No. 3, 52--62 (2014)] at the \(\mathrm{P}_{\mathrm{VI}}\) level (a linear parabolic partial differential equation not containing the \(q\)-\(\mathrm{P}_{\mathrm{VI}}\) fields). This requires considering a scalar, not matrix, Lax pair waiving the restriction \(q^{2 \theta_\infty}\not=1\) of page 7.\N\NA minor regret is the choice of the author for the monodromy exponents as twice their usually adopted values.\N\NTo summarize, this paper should remain a reference for several years.