Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 003, 32 pages      arXiv:1711.03379

Note on Character Varieties and Cluster Algebras

Kazuhiro Hikami
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

Received July 25, 2018, in final form January 10, 2019; Published online January 20, 2019

We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structure of cluster algebra. It is also shown that cluster mutations give the automorphism of the character varieties. Motivated by a work of Chekhov-Mazzocco-Rubtsov, we revisit confluences of punctures on sphere from cluster algebraic viewpoint, and we obtain associated affine cubic surfaces constructed by van der Put-Saito based on the Riemann-Hilbert correspondence. Further studied are quantizations of character varieties by use of quantum cluster algebra.

Key words: cluster algebra; character variety; Painlevé equations; Goldman Poisson algebra.

pdf (681 kb)   tex (126 kb)


  1. Aigner M., Markov's theorem and 100 years of the uniqueness conjecture, Springer, Cham, 2013.
  2. Allegretti D.G.L., Kim H.K., A duality map for quantum cluster varieties from surfaces, Adv. Math. 306 (2017), 1164-1208, arXiv:1509.01567.
  3. Berenstein A., Zelevinsky A., Quantum cluster algebras, Adv. Math. 195 (2005), 405-455, math.QA/0404446.
  4. Berest Yu., Samuelson P., Affine cubic surfaces and character varieties of knots, J. Algebra 500 (2018), 644-690, arXiv:1610.08947.
  5. Bershtein M., Gavrylenko P., Marshakov A., Cluster integrable systems, $q$-Painlevé equations and their quantization, J. High Energy Phys. 2018 (2018), no. 2, 077, 34 pages, arXiv:1711.02063.
  6. Bonahon F., Wong H., Quantum traces for representations of surface groups in ${\rm SL}_2(\mathbb C)$, Geom. Topol. 15 (2011), 1569-1615, arXiv:1003.5250.
  7. Bonahon F., Wong H., Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations, Invent. Math. 204 (2016), 195-243, arXiv:1206.1638.
  8. Bullock D., Przytycki J.H., Multiplicative structure of Kauffman bracket skein module quantizations, Proc. Amer. Math. Soc. 128 (2000), 923-931, math.QA/9902117.
  9. Chekhov L., Mazzocco M., Shear coordinate description of the quantized versal unfolding of a $D_4$ singularity, J. Phys. A: Math. Theor. 43 (2010), 442002, 13 pages, arXiv:1007.3854.
  10. Chekhov L., Mazzocco M., Rubtsov V., Painlevé monodromy manifolds, decorated character varieties, and cluster algebras, Int. Math. Res. Not. 2017 (2017), 7639-7691, arXiv:1511.03851.
  11. Chekhov L., Mazzocco M., Rubtsov V., Algebras of quantum monodromy data and decorated character varieties, arXiv:1705.01447.
  12. Chekhov L., Shapiro M., Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables, Int. Math. Res. Not. 2014 (2014), 2746-2772, arXiv:1111.3963.
  13. Cherednik I., Double affine Hecke algebras, London Mathematical Society Lecture Note Series, Vol. 319, Cambridge University Press, Cambridge, 2005.
  14. Cherednik I., DAHA-Jones polynomials of torus knots, Selecta Math. (N.S.) 22 (2016), 1013-1053, arXiv:1406.3959.
  15. Coman I., Gabella M., Teschner J., Line operators in theories of class $\mathcal S$, quantized moduli space of flat connections, and Toda field theory, J. High Energy Phys. 2015 (2015), no. 10, 143, 89 pages, arXiv:1505.05898.
  16. Felikson A., Shapiro M., Tumarkin P., Cluster algebras and triangulated orbifolds, Adv. Math. 231 (2012), 2953-3002, arXiv:1111.3449.
  17. Felikson A., Shapiro M., Tumarkin P., Skew-symmetric cluster algebras of finite mutation type, J. Eur. Math. Soc. (JEMS) 14 (2012), 1135-1180, arXiv:0811.1703.
  18. Fock V.V., Chekhov L.O., A quantum Teichmüller spaces, Theoret. and Math. Phys. 120 (1999), 1245-1259, math.QA/9908165.
  19. Fock V.V., Goncharov A.B., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006), 1-211, math.AG/0311149.
  20. Fock V.V., Goncharov A.B., Dual Teichmüller and lamination spaces, in Handbook of Teichmüller Theory, Vol. I, IRMA Lect. Math. Theor. Phys., Vol. 11, Eur. Math. Soc., Zürich, 2007, 647-684, math.DG/0510312.
  21. Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930, math.AG/0311245.
  22. Fock V.V., Goncharov A.B., The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2009), 223-286, math.QA/0702397.
  23. Fomin S., Shapiro M., Thurston D., Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), 83-146, math.RA/0608367.
  24. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, math.RT/0104151.
  25. Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121, math.RA/0208229.
  26. Gabella M., Quantum holonomies from spectral networks and framed BPS states, Comm. Math. Phys. 351 (2017), 563-598, arXiv:1603.05258.
  27. Gaiotto D., Moore G.W., Neitzke A., Framed BPS states, Adv. Theor. Math. Phys. 17 (2013), 241-397, arXiv:1006.0146.
  28. Gekhtman M., Shapiro M., Vainshtein A., Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), 291-311, math.QA/0309138.
  29. Gekhtman M., Shapiro M., Vainshtein A., Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, Vol. 167, Amer. Math. Soc., Providence, RI, 2010.
  30. Goldman W.M., Ergodic theory on moduli spaces, Ann. of Math. 146 (1997), 475-507.
  31. Goldman W.M., The modular group action on real ${\rm SL}(2)$-characters of a one-holed torus, Geom. Topol. 7 (2003), 443-486, math.DG/0305096.
  32. Goldman W.M., Neumann W.D., Homological action of the modular group on some cubic moduli spaces, Math. Res. Lett. 12 (2005), 575-591, math.GT/0402039.
  33. Hikami K., DAHA and skein algebra on surface: double-torus knots, arXiv:1901.02743.
  34. Hikami K., Inoue R., Cluster algebra and complex volume of once-punctured torus bundles and 2-bridge links, J. Knot Theory Ramifications 23 (2014), 1450006, 33 pages, arXiv:1212.6042.
  35. Horowitz R.D., Induced automorphisms on Fricke characters of free groups, Trans. Amer. Math. Soc. 208 (1975), 41-50.
  36. Iwasaki K., A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), 131-135.
  37. Iwasaki K., An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation, Comm. Math. Phys. 242 (2003), 185-219.
  38. Jimbo M., Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), 1137-1161.
  39. Kashaev R.M., Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), 105-115, q-alg/9705021.
  40. Koornwinder T.H., Zhedanov's algebra $\rm AW(3)$ and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra, SIGMA 4 (2008), 052, 17 pages, arXiv:0711.2320.
  41. Lê T.T.Q., Triangular decomposition of skein algebras, Quantum Topol. 9 (2018), 591-632, arXiv:1609.04987.
  42. Nagao K., Terashima Y., Yamazaki M., Hyperbolic 3-manifolds and cluster algebras, Nagoya Math. J., to appear, arXiv:1112.3106.
  43. Nekrasov N., Rosly A., Shatashvili S., Darboux coordinates, Yang-Yang functional, and gauge theory, Nuclear Phys. B Proc. Suppl. 216 (2011), 69-93, arXiv:1103.3919.
  44. Oblomkov A., Double affine Hecke algebras of rank 1 and affine cubic surfaces, Int. Math. Res. Not. 2004 (2004), 877-912, math.RT/0306393.
  45. Ohyama Y., Okumura S., A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations, J. Phys. A: Math. Gen. 39 (2006), 12129-12151, math.CA/0601614.
  46. Penner R.C., Decorated Teichmüller theory, QGM Master Class Series, European Mathematical Society (EMS), Zürich, 2012.
  47. Przytycki J.H., Sikora A.S., On skein algebras and ${\rm Sl}_2({\bf C})$-character varieties, Topology 39 (2000), 115-148, q-alg/9705011.
  48. Terwilliger P., The universal Askey-Wilson algebra and DAHA of type $(C^\vee_1,C_1)$, SIGMA 9 (2013), 047, 40 pages, arXiv:1202.4673.
  49. Turaev V.G., Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4) 24 (1991), 635-704.
  50. van der Put M., Saito M.H., Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611-2667, arXiv:0902.1702.

Previous article  Next article   Contents of Volume 15 (2019)