Geodesic
Darboux coordinates, Yang–Yang functional, and gauge theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 181 (2014) no. 1, pp. 86-120 Cet article a éte moissonné depuis la source Math-Net.Ru

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The moduli space of flat $SL_2$ connections on a punctured Riemann surface $\Sigma$ with fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates in which the generating function of the variety of $SL_2$-opers is identified with the universal part of the effective twisted superpotential of the corresponding four-dimensional $\mathcal{N}=2$ supersymmetric theory subject to the two-dimensional $\Omega$-deformation. This allows defining the Yang–Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group and relating it to the instanton counting of the four-dimensional gauge theories in the rank-one case.
Keywords: gauge theory, supersymmetry, Hitchin integrable system, Darboux variable
Mots-clés : quantization. 
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N. A. Nekrasov; A. A. Roslyi; S. L. Shatashvili. Darboux coordinates, Yang–Yang functional, and gauge theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 181 (2014) no. 1, pp. 86-120. http://geodesic.mathdoc.fr/item/TMF_2014_181_1_a5/

[1] L. D. Faddeev, The Bethe Ansatz, Sonderforschungsbereich 288, Berlin, 1993

[2] C. N. Yang, C. P. Yang, J. Math. Phys., 10 (1969), 1115–1122 | DOI | MR | Zbl

[3] C. N. Yang, C. P. Yang, Phys. Rev., 150:1 (1966), 321–327 | DOI

[4] E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, TMF, 40:2 (1979), 194–220 | DOI | MR

[5] M. Jimbo, T. Miwa, F. Smirnov, J. Phys. A, 42:30 (2009), 304018, 31 pp., arXiv: 0811.0439 | DOI | MR | Zbl

[6] G. W. Moore, N. Nekrasov, S. Shatashvili, Commun. Math. Phys., 209:1 (2000), 97–121, arXiv: hep-th/9712241 | DOI | MR | Zbl

[7] A. A. Gerasimov, S. L. Shatashvili, Commun. Math. Phys., 277:2 (2008), 323–367, arXiv: hep-th/0609024 | DOI | MR | Zbl

[8] A. A. Gerasimov, S. L. Shatashvili, “Two-dimensional gauge theories and quantum integrable systems”, From Hodge Theory to Integrability and TQFT: tt*-geometry, Proceedings of Symposia in Pure Mathematics, 78, eds. R. Y. Donagi, R. Wendland, AMS, Providence, RI, 2008, 239–262, arXiv: 0711.1472 | DOI | MR | Zbl

[9] E. Witten, J. Geom. Phys., 9:4 (1992), 303–368, arXiv: hep-th/9204083 | DOI | MR | Zbl

[10] A. S. Gorskii, N. Nekrasov, TMF, 100:1 (1994), 97–103 | DOI | MR | Zbl

[11] A. Gorsky, N. Nekrasov, Nucl. Phys. B, 436:3 (1995), 582–608, arXiv: hep-th/9401017 | DOI | MR | Zbl

[12] A. Gorsky, N. Nekrasov, Elliptic Calogero–Moser system from two dimensional current algebra, arXiv: hep-th/9401021

[13] A. Gorsky, N. Nekrasov, Nucl. Phys. B, 414:1–2 (1994), 213–238, arXiv: hep-th/9304047 | DOI | MR | Zbl

[14] N. A. Nekrasov, S. L. Shatashvili, Nucl. Phys. B Proc. Suppl., 192–193 (2009), 91–112, arXiv: 0901.4744 | DOI | MR

[15] N. Nekrasov, S. Shatashvili, AIP Conf. Proc., 1134 (2009), 154–169 | DOI | Zbl

[16] N. A. Nekrasov, S. L. Shatashvili, Prog. Theor. Phys. Suppl., 177 (2009), 105–119, arXiv: 0901.4748 | DOI | Zbl

[17] N. A. Nekrasov, S. L. Shatashvili, “Quantization of integrable systems and four dimensional gauge theories”, XVIth International Congress on Mathematical Physics (Prague, Czech Republic, 3–8 August, 2009), ed. P. Exne, World Sci., Singapore, 2010, 265–289, arXiv: 0908.4052 | DOI | MR | Zbl

[18] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, A. Morozov, Phys. Lett. B, 355:3–4 (1995), 466–474, arXiv: hep-th/9505035 | DOI | MR | Zbl

[19] E. Martinec, N. Warner, Nucl. Phys. B, 459:1 (1996), 97–112, arXiv: hep-th/9509161 | DOI | MR | Zbl

[20] R. Donagi, E. Witten, Nucl. Phys. B, 460:2 (1996), 299–334, arXiv: hep-th/9510101 | DOI | MR | Zbl

[21] R. Y. Donagi, Seiberg–Witten integrable systems, arXiv: alg-geom/9705010 | MR

[22] N. A. Nekrasov, Adv. Theor. Math. Phys., 7:5 (2004), 831–864, arXiv: hep-th/0206161 | DOI | MR

[23] A. Losev, N. Nekrasov, S. L. Shatashvili, “Testing Seiberg–Witten solution”, Strings, Branes and Dualities (Cargése France, May 26 – June 14, 1997), eds. L. Baulieu, P. Di Francesco, M. Douglas, V. Kazakov, M. Picco, P. Windey, Kluwer Acad. Publ., Dordrecht, 1999, 359–372, arXiv: hep-th/9801061 | DOI | MR | Zbl

[24] A. Losev, N. Nekrasov, S. L. Shatashvili, Nucl. Phys. B, 534:3 (1998), 549–611, arXiv: hep-th/9711108 | DOI | MR | Zbl

[25] E. Witten, “Some comments on string dynamics”, Proceedings of Strings ' 95: Future Perspectives in String Theory (USC, Los Angeles, March 13–18, 1995), eds. I. Bars, P. Bouwknegt, J. Minahan, D. Nemeshansky, K. Pilch, H. Saleur, N. Warner, World Sci., Singapore, 1996, 501–523, arXiv: hep-th/9507121 | MR

[26] A. Strominger, Phys. Lett. B, 383:1 (1996), 44–47, arXiv: hep-th/9512059 | DOI | MR | Zbl

[27] E. Witten, Nucl. Phys. B, 500:1 (1997), 3–42, arXiv: hep-th/9703166 | DOI | MR | Zbl

[28] D. Gaiotto, $N=2$ dualities, arXiv: 0904.2715 | MR

[29] D. Gaiotto, G. W. Moore, A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, arXiv: 0907.3987 | MR

[30] N. Seiberg, E. Witten, Nucl. Phys. B, 431:1 (1994), 484–550, arXiv: hep-th/9408099 | DOI | MR | Zbl

[31] G. W. Moore, N. Nekrasov, S. Shatashvili, Commun. Math. Phys., 209:1 (2000), 77–95, arXiv: hep-th/9803265 | DOI | MR | Zbl

[32] E. Witten, Nucl. Phys. B, 443:1 (1995), 85–126, arXiv: hep-th/9503124 | DOI | MR | Zbl

[33] M. Atiyah, R. Bott, Phil. Trans. R. Soc. London Ser. A, 308:1505 (1982), 523–615 | DOI | MR

[34] E. Witten, Commun. Math. Phys., 121:3 (1989), 351–399 | DOI | MR | Zbl

[35] A. M. Polyakov, Modern Phys. Lett. A, 2:11 (1987), 893–898 | DOI | MR

[36] A. Alekseev, S. L. Shatashvili, Nucl. Phys. B, 323:3 (1989), 719–733 | DOI | MR

[37] E. Verlinde, H. Verlinde, Nucl. Phys. B, 348:3 (1991), 457–489 | DOI | MR

[38] H. Verlinde, Nucl. Phys. B, 337:3 (1990), 652–680 | DOI | MR

[39] E. Witten, Nucl. Phys. B, 311:1 (1988), 46–78 | DOI | MR | Zbl

[40] V. V. Fok, L. O. Chekhov, TMF, 120:3 (1999), 511–528 | DOI | DOI | MR | Zbl

[41] L. O. Chekhov, V. V. Fock, Czech. J. Phys., 50:11 (2000), 1201–1208 | DOI | MR | Zbl

[42] E. Witten, Three-dimensional gravity revisited, arXiv: 0706.3359

[43] N. Seiberg, E. Witten, “Gauge dynamics and compactification to three dimensions”, The Mathematical Beauty of Physics (Saclay, France, 5–7 June, 1996), Advanced Series in Mathematical Physics, 24, eds. J. M. Drouffe, J. B. Zuber, World Sci., Singapore, 1997, 333–366, arXiv: hep-th/9607163 | MR | Zbl

[44] N. J. Hitchin, Proc. London Math. Soc. (3), 55:1 (1987), 59–126 | DOI | MR | Zbl

[45] A. Kapustin, E. Witten, Commun. Number Theory Phys., 1:1 (2007), 1–236, arXiv: hep-th/0604151 | DOI | MR | Zbl

[46] N. Hitchin, Duke Math. J., 54:1 (1987), 91–114 | DOI | MR | Zbl

[47] R. Donagi, “Spectral covers”, Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), Mathematical Sciences Research Institute Publications, 28, eds. H. Clemens, J. Kollár, Cambridge Univ. Press, Cambridge, 1995, 65–86, arXiv: alg-geom/9505009 | MR | Zbl

[48] N. Nekrasov, E. Witten, JHEP, 09 (2010), 092, 82 pp., arXiv: 1002.0888 | DOI | MR

[49] D. Gaiotto, E. Witten, Adv. Theor. Math. Phys., 13:3, 721–896, arXiv: 0807.3720 | DOI | MR | Zbl

[50] D. Gaiotto, E. Witten, Supersymmetric boundary conditions in $N=4$ super Yang–Mills theory, arXiv: 0804.2902 | MR

[51] A. Beilinson, V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, not published

[52] S. Gukov, E. Witten, Branes and quantization, arXiv: 0809.0305 | MR

[53] L. F. Alday, D. Gaiotto, Y. Tachikawa, Lett. Math. Phys., 91:2 (2010), 167–197, arXiv: 0906.3219 | DOI | MR | Zbl

[54] J. Teschner, Adv. Theor. Math. Phys., 15:2 (2011), 471–564 | DOI | MR | Zbl

[55] P. G. Zograf, L. A. Takhtadzhyan, Matem. sb., 132(174):2 (1987), 147–166 | DOI | MR | Zbl

[56] V. Fock, A. Goncharov, Publ. Math. Inst. Hautes Études Sci. No., 103:1 (2006), 1–211 | DOI | MR | Zbl

[57] N. Nekrasov, Commun. Math. Phys., 180:3 (1996), 587–603, arXiv: hep-th/9503157 | DOI | MR | Zbl

[58] R. Donagi, E. Markman, “Spectral covers, algebraically completely integrable hamiltonian systems, and moduli of bundles”, Integrable Systems and Quantum Groups (Montecatini Terme, Italy, June 14–22, 1993), Lecture Notes in Mathematics, 1620, eds. M. Francaviglia, S. Greco, 1996, 1–119, arXiv: alg-geom/9507017 | DOI | MR | Zbl

[59] A. Kapustin, S. Sethi, Adv. Theor. Math. Phys., 2:3 (1998), 571–591, arXiv: hep-th/9804027 | DOI | MR | Zbl

[60] E. Markman, Compositio Math., 93:3 (1994), 255–290 | MR | Zbl

[61] A. Gorsky, N. Nekrasov, V. Rubtsov, Commun. Math. Phys., 222:2 (2001), 299–318, arXiv: hep-th/9901089 | DOI | MR | Zbl

[62] V. V. Fock, A. A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix, preprint ITEP-72-92, 1992 ; AMS Transl. Ser. 2, 191 (1999), 67–86, arXiv: math/9802054 | MR | Zbl

[63] V. V. Fock, A. A. Roslyi, TMF, 95:2 (1993), 228–238 | DOI | MR | Zbl

[64] V. V. Fock, A. A. Rosly, Internat. J. Modern Phys. B, 11:26–27 (1997), 3195–3206 | DOI | MR | Zbl

[65] W. M. Goldman, Adv. Math., 54:2 (1984), 200–225 | DOI | MR | Zbl

[66] W. M. Goldman, Invent. Math., 85 (1986), 263–302 | DOI | MR | Zbl

[67] V. G. Turaev, Ann. Sci. École Norm. Sup. (4), 24:6 (1991), 635–704 | DOI | MR | Zbl

[68] A. N. Tyurin, Quantization, Classical and Quantum FIeld Theory and Theta Functions, CRM Monograph Series, 21, AMS, Providence, RI, 2003 | DOI | MR | Zbl

[69] D. A. Derevnin, A. D. Mednykh, UMN, 60:2(362) (2005), 159–160 | DOI | DOI | MR | Zbl

[70] L. Schläfli, Theorie der vielfachen Kontinuität, Gesammelte Mathematishe Abhandlungen, 1, Birkhäuser, Basel, 1950 | DOI

[71] N. I. Lobatschefskij, “Imaginäre Geometrie und ihre Anwendung auf einige Integrale”, Deutsche Übersetzung von H. Liebmann, Teubner, Leipzig, 1904

[72] Yu. Cho, H. Kim, Discrete Comput. Geom., 22:3 (1999), 347–366 | DOI | MR | Zbl

[73] J. Milnor, “The Schläfli differential equality”, Collected Papers, v. I, Geometry, Publish or Perish, Houston, TX, 1994, 281–295 | MR | Zbl

[74] M. Kapovich, J. J. Millson, T. Treloar, The symplectic geometry of polygons in hyperbolic 3-space, arXiv: math/9907143 | MR

[75] A. A. Klyachko, “Spatial polygons and stable configurations of points in the projective line”, Algebraic Geometry and Its Applications (Yaroslavl, 1992), Aspects of Mathematics, 25, Vieweg, Braunschweig, 1994, 67–84 | DOI | MR | Zbl

[76] P. Foth, J. Geom. Phys., 58:7 (2007), 825–832, arXiv: math/0703525 | DOI | MR

[77] A. Beilinson, V. Drinfeld, Opers, arXiv: math.AG/0501398

[78] A. Bilal, I. Kogan, V. Fock, On the origin of $W$-algebras, Preprint CERN-TH.5965/90 | MR

[79] A. Gerasimov, A. Levin, A. Marshakov, Nucl. Phys. B, 360:2 (1991), 537–558 | DOI | MR

[80] N. Nekrasov, V. Pestun, Seiberg–Witten geometry of four dimensional $N=2$ quiver gauge theories, arXiv: 1211.2240

[81] E. K. Sklyanin, Zap. nauchn. sem. LOMI, 164, 1987, 151–169 | MR | Zbl

[82] E. Frenkel, “Affine algebras, langlands duality, and Bethe ansatz”, XIth International Congress on Mathematical Physics (Paris, France, 18–23 July, 1994), ed. D. Iagolnitzer, Internat. Press, Cambridge, MA, 1995, 606–642, arXiv: q-alg/9506003 | MR | Zbl

[83] A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov, S. L. Shatashvili, Internat. J. Modern Phys. A, 5:13 (1990), 2495–2589 | DOI | MR

[84] V. Schechtman, A. Varchenko, Integral Representations of $N$-point Conformal Correlators in the WZW Model, Preprint MPI/89-51, Max-Planck Institut, Bonn, 1989

[85] B. Feigin, E. Frenkel, N. Reshetikhin, Commun. Math. Phys., 166:1 (1994), 27–62, arXiv: hep-th/9402022 | DOI | MR | Zbl

[86] N. Reshetikhin, A. Varchenko, “Quasiclassical asymptotics of solutions to the $KZ$ equations”, Geometry, Topology, and Physics, Internat. Press, Cambridge, MA, 1995, 293–322, arXiv: hep-th/9402126 | MR

[87] G. Felder, A. Varchenko, Compositio Math., 107:2 (1997), 143–175, arXiv: hep-th/9511120 | DOI | MR | Zbl

[88] A. B. Zamolodchikov, Al. B. Zamolodchikov, Nucl. Phys. B, 477:2 (1996), 577–605, arXiv: hep-th/9506136 | DOI | MR | Zbl

[89] E. Aldrovandi, L. A. Takhtajan, Commun. Math. Phys., 227:2 (2002), 303–348, arXiv: math/0006147 | DOI | MR | Zbl

[90] L. F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa, H. Verlinde, JHEP, 01 (2010), 113, 50 pp., arXiv: 0909.0945 | DOI | MR

[91] D. Gaiotto, Asymptotically free $N=2$ theories and irregular conformal blocks, arXiv: 0908.0307

[92] E. Witten, Anal. Appl. (Singap.), 6:4 (2008), 429–501, arXiv: 0710.0631 | DOI | MR | Zbl

[93] B. Feigin, E. Frenkel, V. Toledano-Laredo, Adv. Math., 223:3 (2010), 873–948, arXiv: math/0612798 | DOI | MR | Zbl