Advertisement

M-theoretic derivations of 4d-2d dualities: from a geometric Langlands duality for surfaces, to the AGT correspondence, to integrable systems

  • Meng-Chwan Tan
Article

Abstract

In part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations — which relate some cohomology of the moduli space of certain (“ramified”) G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group — can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

In part II, to the setup in part I, we introduce Omega-deformation via fluxbranes and add half-BPS boundary defects via M9-branes, and show that the celebrated AGT correspondence in [2, 3], and its generalizations — which essentially relate, among other things, some equivariant cohomology of the moduli space of certain (“ramified”) G-instantons to the integrable representations of the Langlands dual of certain affine \( \mathcal{W} \)-algebras — can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

In part III, we consider various limits of our setup in part II, and connect our story to chiral fermions and integrable systems. Among other things, we derive the NekrasovOkounkov conjecture in [4] — which relates the topological string limit of the dual Nekrasov partition function for pure G to the integrable representations of the Langlands dual of an affine G-algebra — and also demonstrate that the Nekrasov-Shatashvili limit of the “fullyramified” Nekrasov instanton partition function for pure G is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine G-algebra. Via the case with matter, we also make contact with Hitchin systems and the “ramified” geometric Langlands correspondence for curves.

Keywords

Differential and Algebraic Geometry Conformal and W Symmetry M-Theory String Duality 

References

  1. [1]
    A. Braverman and M. Finkelberg, Pursuing the double affine Grassmannian I: transversal slices via instantons on A k-singularities, Duke Math. 152 (2010) 175 [arXiv:0711.2083].MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. [3]
    D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [INSPIRE].
  4. [4]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  5. [5]
    H. Nakajima, Instantons on ALE Spaces, Quiver Varieties, and Kac-Moody Algebras, Duke Math. 76 (1994) 365.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    C. Vafa, Instantons on D-branes, Nucl. Phys. B 463 (1996) 435 [hep-th/9512078] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    J.A. Harvey and G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489 [hep-th/9609017] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. [9]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].MathSciNetGoogle Scholar
  10. [10]
    R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, Supersymmetric gauge theories, intersecting branes and free fermions, JHEP 02 (2008) 106 [arXiv:0709.4446] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    I. Mirkovic and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, math.RT/0401222.
  12. [12]
    A. Beilinson and V. Drinfeld, Quantization of Hitchins integrable system and Hecke eigensheaves, preprint (ca. 1995).Google Scholar
  13. [13]
    E. Witten, Duality from Six-Dimensions I, II, III, lectures delivered at the IAS in Feb. 2008. Notes for the lectures taken by D. Ben-Zvi can be found at: http://www.math.utexas.edu/users/benzvi/GRASP/lectures/IASterm.html.
  14. [14]
    E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [INSPIRE].
  15. [15]
    M.-C. Tan, Five-Branes in M-theory and a Two-Dimensional Geometric Langlands Duality, Adv. Theor. Math. Phys. 14 (2010) 179 [arXiv:0807.1107] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  16. [16]
    D. Gaiotto, \( \mathcal{N} \) = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    N. Wyllard, A(N-1) conformal Toda field theory correlation functions from conformal N =2 SU(N) quiver gauge theories,JHEP 11 (2009) 002[arXiv:0907.2189] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of Instantons and W-algebras, JHEP 03 (2012) 045 [arXiv:1111.5624] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. [20]
    A. Braverman, B. Feigin, M. Finkelberg and L. Rybnikov, A Finite analog of the AGT relation I: Finite W -algebras and quasimapsspaces, Commun. Math. Phys. 308 (2011) 457 [arXiv:1008.3655] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. [21]
    O. Schiffmann and E. Vasserot, Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on \( {{\mathbb{A}}^2} \), arXiv:1202.2756.
  22. [22]
    D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE].
  23. [23]
    J. Yagi, On the Six-Dimensional Origin of the AGT Correspondence, JHEP 02 (2012) 020 [arXiv:1112.0260] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    J. Yagi, Compactification on the Ω-background and the AGT correspondence, JHEP 09 (2012) 101 [arXiv:1205.6820] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    N. Wyllard, Instanton partition functions in N = 2 SU(N ) gauge theories with a general surface operator and their W-algebra duals, JHEP 02 (2011) 114 [arXiv:1012.1355] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    A. Braverman, Instanton counting via affine Lie algebras. 1. Equivariant J functions of (affine) flag manifolds and Whittaker vectors, math/0401409 [INSPIRE].
  28. [28]
    O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [arXiv:1203.2930] [INSPIRE].MathSciNetADSGoogle Scholar
  29. [29]
    V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories, JHEP 07 (2011) 079 [arXiv:1105.5800] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    G. Bonelli, K. Maruyoshi and A. Tanzini, Instantons on ALE spaces and Super Liouville Conformal Field Theories, JHEP 08 (2011) 056 [arXiv:1106.2505] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    A. Belavin, V. Belavin and M. Bershtein, Instantons and 2d Superconformal field theory, JHEP 09 (2011) 117 [arXiv:1106.4001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    G. Bonelli, K. Maruyoshi and A. Tanzini, Gauge Theories on ALE Space and Super Liouville Correlation Functions, Lett. Math. Phys. 101 (2012) 103 [arXiv:1107.4609] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. [33]
    N. Wyllard, Coset conformal blocks and N = 2 gauge theories, arXiv:1109.4264 [INSPIRE].
  34. [34]
    Y. Ito, Ramond sector of super Liouville theory from instantons on an ALE space, Nucl. Phys. B 861 (2012) 387 [arXiv:1110.2176] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M.N. Alfimov and G.M. Tarnopolsky, Parafermionic Liouville field theory and instantons on ALE spaces, JHEP 02 (2012) 036 [arXiv:1110.5628] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    A. Belavin and B. Mukhametzhanov, N=1 superconformal blocks with Ramond fields from AGT correspondence, JHEP 01 (2013) 178 [arXiv:1210.7454] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    T. Nishioka and Y. Tachikawa, Central charges of para-Liouville and Toda theories from M-5-branes, Phys. Rev. D 84 (2011) 046009 [arXiv:1106.1172] [INSPIRE].ADSGoogle Scholar
  38. [38]
    N. Proudfoot, Research Statement, http://pages.uoregon.edu/njp/research.pdf.
  39. [39]
    H. Nakajima, Quiver Varieties and Branching, SIGMA 5 (2009) 3 [arXiv:0809.2605].MathSciNetGoogle Scholar
  40. [40]
    S. Reffert, General Omega Deformations from Closed String Backgrounds, JHEP 04 (2012) 059 [arXiv:1108.0644] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    S. Hellerman, D. Orlando and S. Reffert, The Omega Deformation From String and M-theory, JHEP 07 (2012) 061 [arXiv:1204.4192] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    E.A. Bergshoeff, G.W. Gibbons and P.K. Townsend, Open M5-branes, Phys. Rev. Lett. 97 (2006) 231601 [hep-th/0607193] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    C. Vafa, Geometric origin of Montonen-Olive duality, Adv. Theor. Math. Phys. 1 (1998) 158 [hep-th/9707131] [INSPIRE].MathSciNetGoogle Scholar
  44. [44]
    A. Hanany and B. Kol, On orientifolds, discrete torsion, branes and M-theory, JHEP 06 (2000) 013 [hep-th/0003025] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    A. Sen, A Note on enhanced gauge symmetries in M- and string-theory, JHEP 09 (1997) 001 [hep-th/9707123] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, Curr. Dev. Math. 2006 (2008) 35 [hep-th/0612073] [INSPIRE].MathSciNetGoogle Scholar
  47. [47]
    V.B. Mehta and C.S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980) 205.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    I. Biswas, Parabolic bundles as orbifold bundles, Duke Math. 88 (1997) 305.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, BPS quantization of the five-brane, Nucl. Phys. B 486 (1997) 89 [hep-th/9604055] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44 [hep-th/9512059] [INSPIRE].MathSciNetADSGoogle Scholar
  51. [51]
    R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, BPS spectrum of the five-brane and black hole entropy, Nucl. Phys. B 486 (1997) 77 [hep-th/9603126] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    J.H. Schwarz, Selfdual superstring in six-dimensions, hep-th/9604171 [INSPIRE].
  53. [53]
    R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, 5 − D black holes and matrix strings, Nucl. Phys. B 506 (1997) 121 [hep-th/9704018] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    O. Aharony, M. Berkooz, S. Kachru, N. Seiberg and E. Silverstein, Matrix description of interacting theories in six-dimensions, Adv. Theor. Math. Phys. 1 (1998) 148 [hep-th/9707079] [INSPIRE].MathSciNetGoogle Scholar
  55. [55]
    P.S. Howe, N. Lambert and P.C. West, The Selfdual string soliton, Nucl. Phys. B 515 (1998) 203 [hep-th/9709014] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    R. Dijkgraaf, The Mathematics of five-branes, Doc. Math. J. DMV (1998) [hep-th/9810157] [INSPIRE].Google Scholar
  57. [57]
    Y. Tachikawa, On S-duality of 5d super Yang-Mills on S 1, JHEP 11 (2011) 123 [arXiv:1110.0531] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    M.R. Douglas, Branes within branes, hep-th/9512077 [INSPIRE].
  59. [59]
    C. Johnson, D-branes, Cambridge Monographs on Mathematical Physics, Cambridge University Press, New York, U.S.A. (2003).Google Scholar
  60. [60]
    S. Wu, S-duality in Vafa-Witten theory for non-simply laced gauge groups, JHEP 05 (2008) 009 [arXiv:0802.2047] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  62. [62]
    N. Hitchin, L 2 cohomology of hyperKähler quotients, Commun. Math. Phys. 211 (2000) 153 [math/9909002] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  63. [63]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York, U.S.A. (1999).Google Scholar
  64. [64]
    M. Goresky, L 2 -cohomology is Intersection Cohomology, http://www.math.ias.edu/ goresky/pdf/zucker.pdf.
  65. [65]
    K. Hori et al., Mirror Symmetry, Clay Mathematics Monographs, Volume 1 (2003).Google Scholar
  66. [66]
    C.P. Bachas, M.B. Green and A. Schwimmer, (8, 0) quantum mechanics and symmetry enhancement in type-Isuperstrings, JHEP 01 (1998) 006 [hep-th/9712086] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    L.-Y. Hung, Comments on I1-branes, JHEP 05 (2007) 076 [hep-th/0612207] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    M.B. Green, J.A. Harvey and G.W. Moore, I-brane inflow and anomalous couplings on D-branes, Class. Quant. Grav. 14 (1997) 47 [hep-th/9605033] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  69. [69]
    E. Kiritsis, String Theory in a Nutshell, Princeton University Press (2007).Google Scholar
  70. [70]
    V.G. Kac, Infinite Dimensional Lie Algebras, Third Edition, Cambridge University Press (1994).Google Scholar
  71. [71]
    E.G. Gimon and J. Polchinski, Consistency conditions for orientifolds and d manifolds, Phys. Rev. D 54 (1996) 1667 [hep-th/9601038] [INSPIRE].MathSciNetADSGoogle Scholar
  72. [72]
    N. Itzhaki, D. Kutasov and N. Seiberg, I-brane dynamics, JHEP 01 (2006) 119 [hep-th/0508025] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  73. [73]
    K. Hasegawa, Spin Module Versions of Weyls Reciprocity Theorem for Classical Kac-Moody Lie Algebras - An Application to Branching Rule Duality, RIMS, Kyoto Univ. 25 (1989) 741.Google Scholar
  74. [74]
    E. Witten, On Holomorphic factorization of WZW and coset models, Commun. Math. Phys. 144 (1992) 189 [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  75. [75]
    S.V. Ketov, Conformal Field Theory, World Scientific Press, Singapore (1997).Google Scholar
  76. [76]
    J. McKay, Graphs, singularities, and finite groups, Proc. Symp. Pure Math. 37 (1980) 183.MathSciNetCrossRefGoogle Scholar
  77. [77]
    A. Braverman and M. Finkelberg, Pursuing the double affine Grassmannian II: Convolution, Adv. Math. 230 (2012) 414 [arXiv:0908.3390].MathSciNetzbMATHCrossRefGoogle Scholar
  78. [78]
    G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
  79. [79]
    G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].
  80. [80]
    D.H. Collingwood and W.M. McGovern, Nilpotent Orbits in Semisimple Lie Algebra: An Introduction, Van Nostrand Reinhold Press (1993).Google Scholar
  81. [81]
    A. Braverman, M. Finkelberg and D. Gaitsgory, Uhlenbeck spaces via affine Lie algebras, Prog. Math. 244 (2006) 17 [math.AG/0301176].MathSciNetCrossRefGoogle Scholar
  82. [82]
    N. Nekrasov, Lectures on nonperturbative aspects of supersymmetric gauge theories, Class. Quant. Grav. 22 (2005) S77 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  83. [83]
    M.-C. Tan, Equivariant Cohomology Of The Chiral de Rham Complex And The Half-Twisted Gauged σ-model, Adv. Theor. Math. Phys. 13 (2009) 897 [hep-th/0612164] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  84. [84]
    P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  85. [85]
    N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  86. [86]
    V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer (1999).Google Scholar
  87. [87]
    M.F. Atiyah and R. Bott, The Moment map and equivariant cohomology, Topology 23 (1984) 1 [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  88. [88]
    W. Lerche, Introduction to Seiberg-Witten theory and its stringy origin, Nucl. Phys. Proc. Suppl. 55B (1997) 83 [hep-th/9611190] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  89. [89]
    J. de Boer and T. Tjin, Quantization and representation theory of finite W algebras, Commun. Math. Phys. 158 (1993) 485 [hep-th/9211109] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  90. [90]
    J. de Boer and T. Tjin, The Relation between quantum \( \mathcal{W} \) -algebras and Lie algebras, Commun. Math. Phys. 160 (1994) 317 [hep-th/9302006] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  91. [91]
    L. Feher, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, On the general structure of Hamiltonian reductions of the WZNW theory, hep-th/9112068 [INSPIRE].
  92. [92]
    D. Nemeschansky and N. Warner, Topological matter, integrable models and fusion rings, Nucl. Phys. B 380 (1992) 241 [hep-th/9110055] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  93. [93]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  94. [94]
    J. Harnad, Tau functions, integrable systems, random matrices and random processes, BIRS Workshop on Quadrature Domains and Laplacian Growth in Modern Physics, Banff, July 15–20, 2007.Google Scholar
  95. [95]
    P. Etingof, Whittaker functions on quantum groups and q-deformed Toda operators, math.QA/9901053.
  96. [96]
    E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  97. [97]
    E. D’Hoker and D.H. Phong, Seiberg-Witten theory and Calogero-Moser systems, Prog. Theor. Phys. Suppl. 135 (1999) 75 [hep-th/9906027] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  98. [98]
    R.Y. Donagi, Seiberg-Witten integrable systems, alg-geom/9705010 [INSPIRE].
  99. [99]
    D. Nanopoulos and D. Xie, Hitchin Equation, Singularity and N = 2 Superconformal Field Theories, JHEP 03 (2010) 043 [arXiv:0911.1990] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  100. [100]
    N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  101. [101]
    B. Enriquez, V. Rubtsov, Hitchin systems, higher Gaudin operators and r-matrices, Math. Res. Lett. 3 (1996) 343 [alg-geom/9503010].MathSciNetzbMATHGoogle Scholar
  102. [102]
    K. Becker, M. Becker, J.H Schwarz. String Theory and M-theory: A Modern Introduction, Cambridge Monographs on Mathematical Physics, Cambridge University Press, New York, U.S.A. (2007).Google Scholar
  103. [103]
    D. Tong, NS5-branes, T duality and world sheet instantons, JHEP 07 (2002) 013 [hep-th/0204186] [INSPIRE].ADSCrossRefGoogle Scholar
  104. [104]
    M. Atiyah and N.J. Hitchin, Low-Energy Scattering of Nonabelian Monopoles, Phys. Lett. A 107 (1985) 21 [INSPIRE].MathSciNetADSGoogle Scholar
  105. [105]
    M. Atiyah and N.J. Hitchin, Low-energy scattering of nonAbelian magnetic monopoles, Phil. Trans. Roy. Soc. Lond. A 315 (1985) 459 [INSPIRE].MathSciNetADSGoogle Scholar
  106. [106]
    M. Atiyah and N.J. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton Univ. Press (1988).Google Scholar
  107. [107]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
  108. [108]
    N. Seiberg, IR dynamics on branes and space-time geometry, Phys. Lett. B 384 (1996) 81 [hep-th/9606017] [INSPIRE].MathSciNetADSGoogle Scholar
  109. [109]
    J. Polchinski, String Theory Vol 2: Superstring Theory and Beyond, Cambridge Monographs on Mathematical Physics, Cambridge University Press, New York, U.S.A. (2003).Google Scholar
  110. [110]
    E. Witten, On holomorphic factorization of WZW and coset models, Comm. Math. Phys. 144 (1992) 189.MathSciNetADSzbMATHCrossRefGoogle Scholar
  111. [111]
    B.L. Feigin and E. Frenkel, Quantization of the Drinfeld-Sokolov reduction, Phys. Lett. B 246 (1990) 75 [INSPIRE].MathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Department of PhysicsNational University of SingaporeSingaporeSingapore

Personalised recommendations