Duke Mathematical Journal

Moduli spaces of d-connections and difference Painlevé equations

D. Arinkin and A. Borodin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We show that difference Painlevé equations can be interpreted as isomorphisms of moduli spaces of difference connections (d-connections) on P1 with given singularity structure. In particular, we derive a difference equation that lifts to an isomorphism between A2(1)*-surfaces in Sakai's classification (see [29]); it degenerates to both difference Painlevé V and classical (differential) Painlevé VI equations. This difference equation has been known before under the name of asymmetric discrete Painlevé IV equation

Article information

Duke Math. J., Volume 134, Number 3 (2006), 515-556.

First available in Project Euclid: 28 August 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 39A10: Difference equations, additive
Secondary: 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05]


Arinkin, D.; Borodin, A. Moduli spaces of d-connections and difference Painlevé equations. Duke Math. J. 134 (2006), no. 3, 515--556. doi:10.1215/S0012-7094-06-13433-6. https://projecteuclid.org/euclid.dmj/1156771902

Export citation


  • M. Adler and P. Van Moerbeke, Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice, Comm. Math. Phys. 237 (2003), 397--440.
  • D. Arinkin and S. Lysenko, Isomorphisms between moduli spaces of, SL$(2)$-bundles with connections on ${\mathbb P}^1\setminus\{x_1,\ldots, x_4\}$, Math. Res. Lett. 4 (1997), 181--190.
  • —, On the moduli of,${\rm SL}(2)$-bundles with connections on ${\mathbb P}^1\setminus\{x_1,\ldots,x_4\}$, Internat. Math. Res. Notices 1997, no. 19, 983--999.
  • J. Baik, ``Riemann-Hilbert problems for last passage percolation'' in Recent Developments in Integrable Systems and Riemann-Hilbert Problems (Birmingham, Ala., 2000), Contemp. Math. 326, Amer. Math. Soc., Providence, 2003, 1--21.
  • A. Borodin, Discrete gap probabilities and discrete Painlevé equations, Duke Math. J. 117 (2003), 489--542.
  • —, Isomonodromy transformations of linear systems of difference equations, Ann. of Math. (2) 160 (2004), 1141--1182.
  • A. Borodin and D. Boyarchenko, Distribution of the first particle in discrete orthogonal polynomial ensembles, Comm. Math. Phys. 234 (2003), 287--338.
  • P. J. Forrester and N. S. Witte, Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE, Comm. Math. Phys. 219 (2001), 357--398.
  • —, Application of the $\tau$-function theory of Painlevé equations to random matrices: $\rm P_V$, $\rm P_ {III}$, the LUE, JUE, and CUE, Comm. Pure Appl. Math. 55 (2002), 679--727.
  • —, Discrete Painlevé equations and random matrix averages, Nonlinearity 16 (2003), 1919--1944.
  • —, Application of the $\tau$-function theory of Painlevé equations to random matrices: $\rm P\sb {\rm VI}$, the JUE, CyUE, cJUE and scaled limits, Nagoya Math. J. 174 (2004), 29--114.
  • —, Discrete Painlevé equations, orthogonal polynomials on the unit circle, and $N$-recurrences for averages over $U(N)$ --.-${\rm P}\sb {{\rm III}'}$ and ${\rm P}\sb {\rm V}\ \tau$-functions, Int. Math. Res. Not. 2004, no. 4, 160--183.
  • B. Grammaticos, F. W. Nijhoff, and A. Ramani, ``Discrete Painlevé equations'' in The Painlevé Property, CRM Ser. Math. Phys., Springer, New York, 1999, 413--516.
  • B. Grammaticos, Y. Ohta, A. Ramani, and H. Sakai, Degeneration through coalescence of the $q$-Painlevé VI equation, J. Phys. A 31 (1998), 3545--3558.
  • B. Grammaticos, A. Ramani, and Y. Ohta, A unified description of the asymmetric $q\text{-P}\sb {\rm V}$ and $d\text{-P}\sb {\rm IV}$ equations and their Schlesinger transformations, J. Nonlinear Math. Phys. 10 (2003), 215--228.
  • M.-A. Inaba, K. Iwasaki, and M.-H. Saito, Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence, Int. Math. Res. Not. 2004, no. 1, 1--30.
  • —, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, part I, preprint.
  • M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, II, Phys. D 2 (1981), 407--448.
  • M. Jimbo and H. Sakai, A $q$-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145--154.
  • I. M. Krichever, Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem, Uspekhi Mat. Nauk 59 (2004), no. 6, 111--150.; English translation in Russian Math. Surveys 59 (2004), 1117--1154.
  • G. Laumon, Transformation de Fourier généralisée, preprint.
  • M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), 281--295.
  • Y. Ohta, A. Ramani, B. Grammaticos, and K. M. Tamizhmani, From discrete to continuous Painlevé equations: A bilinear approach, Phys. Lett. A 216 (1996), 255--261.
  • K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. (N.S.) 5 (1979), 1--79.
  • —, Isomonodromic deformation and Painlevé equations, and the Garnier system, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986), 575--618.
  • A. Ramani, B. Grammaticos, and Y. Ohta, A geometrical description of the discrete Painlevé VI and V equations, Comm. Math. Phys. 217 (2001), 315--329.
  • M.-H. Saito and T. Takebe, Classification of Okamoto-Painlevé pairs, Kobe J. Math. 19 (2002), 21--50.
  • M.-H. Saito, T. Takebe, and H. Terajima, Deformation of Okamoto-Painlevé pairs and Painlevé equations, J. Algebraic Geom. 11 (2002), 311--362.
  • H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Comm. Math. Phys. 220 (2001), 165--229.