Goldman algebra, opers and the swapping algebra

Labourie, François¹

¹ Laboratoire Jean-Alexandre Dieudonné, CNRS, Université Côte d’Azur, Nice, France

Geometry & topology, Tome 22 (2018) no. 3, pp. 1267-1348.

Résumé

We define a Poisson algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra, called the algebra of multifractions, as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of ${SL}_n\left(ℝ\right)$–opers with trivial holonomy. We relate this Poisson algebra to the Atiyah–Bott–Goldman symplectic structure and to the Drinfel’d–Sokolov reduction. We also prove an extension of the Wolpert formula.

Détail

DOI : 10.2140/gt.2018.22.1267

Classification : 32G15, 32J15, 17B63
Keywords: Poisson algebra, Teichmüller theory, gauge theory

Bibliographie

[1] M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523 | DOI

[2] M Bourdon, Sur le birapport au bord des CAT(−1)–espaces, Inst. Hautes Études Sci. Publ. Math. 83 (1996) 95

[3] M J Bridgeman, The Poisson bracket of length functions in the Hitchin component, preprint (2015)

[4] M Bridgeman, R Canary, F Labourie, A Sambarino, The pressure metric for Anosov representations, Geom. Funct. Anal. 25 (2015) 1089 | DOI

[5] L A Dickey, Lectures on classical W–algebras, Acta Appl. Math. 47 (1997) 243 | DOI

[6] V G Drinfel’D, V V Sokolov, Equations of Korteweg–de Vries type, and simple Lie algebras, Dokl. Akad. Nauk SSSR 258 (1981) 11

[7] V Fock, A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1 | DOI

[8] W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984) 200 | DOI

[9] W M Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986) 263 | DOI

[10] S Govindarajan, Higher-dimensional uniformisation and W–geometry, Nuclear Phys. B 457 (1995) 357 | DOI

[11] S Govindarajan, T Jayaraman, A proposal for the geometry of Wn–gravity, Phys. Lett. B 345 (1995) 211 | DOI

[12] P Guha, Euler–Poincaré flows on sln opers and integrability, Acta Appl. Math. 95 (2007) 1 | DOI

[13] O Guichard, Composantes de Hitchin et représentations hyperconvexes de groupes de surface, J. Differential Geom. 80 (2008) 391

[14] O Guichard, A Wienhard, Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012) 357 | DOI

[15] N J Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449 | DOI

[16] F Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006) 51 | DOI

[17] F Labourie, Cross ratios, surface groups, PSL(n, R) and diffeomorphisms of the circle, Publ. Math. Inst. Hautes Études Sci. 106 (2007) 139 | DOI

[18] F Labourie, Cross ratios, Anosov representations and the energy functional on Teichmüller space, Ann. Sci. Éc. Norm. Supér. 41 (2008) 437 | DOI

[19] F Labourie, An algebra of observables for cross ratios, C. R. Math. Acad. Sci. Paris 348 (2010) 503 | DOI

[20] F Labourie, Lectures on representations of surface groups, Eur. Math. Soc. (2013) | DOI

[21] F Ledrappier, Structure au bord des variétés à courbure négative, Sémin. Théor. Spectr. Géom. 13 (1995) 97 | DOI

[22] F Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978) 1156 | DOI

[23] P Van Moerbeke, Algèbres W et équations non-linéaires, from: "Séminaire Bourbaki, 1997/1998", Astérisque 252 (1998) 105

[24] G A Niblo, Separability properties of free groups and surface groups, J. Pure Appl. Algebra 78 (1992) 77 | DOI

[25] J P Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. 131 (1990) 151 | DOI

[26] J P Otal, Sur la géometrie symplectique de l’espace des géodésiques d’une variété à courbure négative, Rev. Mat. Iberoamericana 8 (1992) 441 | DOI

[27] A Sambarino, Hyperconvex representations and exponential growth, Ergodic Theory Dynam. Systems 34 (2014) 986 | DOI

[28] A Sambarino, Quantitative properties of convex representations, Comment. Math. Helv. 89 (2014) 443 | DOI

[29] G Segal, The geometry of the KdV equation, Internat. J. Modern Phys. A 6 (1991) 2859 | DOI

[30] E Witten, Surprises with topological field theories, from: "Strings ’90 : proceedings of the 4th International Superstring Workshop" (editors R L Arnowitt, R Bryan, M J Duff, D V Nanopoulos, C N Pope, E Sezgin), World Scientific (1991) 50 | DOI

[31] S Wolpert, The Fenchel–Nielsen deformation, Ann. of Math. 115 (1982) 501 | DOI

[32] S Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. 117 (1983) 207 | DOI