TBA for non-perturbative moduli spaces

Open Access


Recently, an exact description of instanton corrections to the moduli spaces of 4d N = 2 supersymmetric gauge theories compactified on a circle and Calabi-Yau compactifications of Type II superstring theories was found. The equations determining the instanton contributions turn out to have the form of Thermodynamic Bethe Ansatz. We explore further this relation and, in particular, we identify the contact potential of quaternionic string moduli space with the free energy of the integrable system and the Kähler potential of the gauge theory moduli space with the Yang-Yang functional. We also show that the corresponding S-matrix satisfies all usual constraints of 2d integrable models, including crossing and bootstrap, and derive the associated Y-system. Surprisingly, in the simplest case the Y-system is described by the MacMahon function relevant for crystal melting and topological strings.


Supersymmetric gauge theory Superstrings and Heterotic Strings Bethe Ansatz 


  1. [1]
    J.A. Minahan and K. Zarembo, The Bethe-ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  2. [2]
    I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [SPIRES].MathSciNetADSGoogle Scholar
  3. [3]
    N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, The algebraic curve of classical superstrings on AdS 5 × S 5, Commun. Math. Phys. 263 (2006) 659 [hep-th/0502226] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  4. [4]
    N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [SPIRES].CrossRefMathSciNetGoogle Scholar
  5. [5]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [SPIRES].
  6. [6]
    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] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  7. [7]
    N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, arXiv:1002.0888 [SPIRES].
  8. [8]
    L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, arXiv:0911.4708 [SPIRES].
  9. [9]
    L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for Scattering Amplitudes, arXiv:1002.2459 [SPIRES].
  10. [10]
    Y. Hatsuda, K. Ito, K. Sakai and Y. Satoh, Thermodynamic Bethe Ansatz Equations for Minimal Surfaces in AdS 3, JHEP 04 (2010) 108 [arXiv:1002.2941] [SPIRES].CrossRefGoogle Scholar
  11. [11]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-Yang Models, Nucl. Phys. B 342 (1990) 695 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, arXiv:0807.4723 [SPIRES].
  14. [14]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [SPIRES].
  15. [15]
    S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, D-instantons and twistors, JHEP 03 (2009) 044 [arXiv:0812.4219] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  16. [16]
    S. Alexandrov, D-instantons and twistors: some exact results, J. Phys. A 42 (2009) 335402 [arXiv:0902.2761] [SPIRES].MathSciNetGoogle Scholar
  17. [17]
    C.-N. Yang and C.P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction, J. Math. Phys. 10 (1969) 1115 [SPIRES].MATHCrossRefADSGoogle Scholar
  18. [18]
    A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, hep-th/0309208 [SPIRES].
  19. [19]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three dimensions, hep-th/9607163 [SPIRES].
  20. [20]
    J. Bagger and E. Witten, Matter Couplings in \( \mathcal{N} = 2 \) Supergravity, Nucl. Phys. B 222 (1983) 1 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  21. [21]
    S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, Linear perturbations of quaternionic metrics - I. The HyperKähler case, Lett. Math. Phys. 87 (2009) 225 [arXiv:0806.4620] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  22. [22]
    S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, Linear perturbations of quaternionic metrics. II. The quaternionic-Kähler case, Commun. Math. Phys. 296 (2010) 353 [arXiv:0810.1675] [SPIRES].CrossRefADSGoogle Scholar
  23. [23]
    K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  24. [24]
    B. de Wit, M. Roček and S. Vandoren, Hypermultiplets, hyperKähler cones and quaternion-Kähler geometry, JHEP 02 (2001) 039 [hep-th/0101161] [SPIRES].CrossRefGoogle Scholar
  25. [25]
    I. Antoniadis, S. Ferrara, R. Minasian and K.S. Narain, R 4 couplings in M- and type-II theories on Calabi-Yau spaces, Nucl. Phys. B 507 (1997) 571 [hep-th/9707013] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    D. Robles-Llana, F. Saueressig and S. Vandoren, String loop corrected hypermultiplet moduli spaces, JHEP 03 (2006) 081 [hep-th/0602164] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  27. [27]
    M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
  28. [28]
    C. Korff, Lie algebraic structures in integrable models, affine Toda field theory, hep-th/0008200 [SPIRES].
  29. [29]
    O.A. Castro-Alvaredo and A. Fring, Integrable models with unstable particles, Prog. Math. 237 (2005) 59 [hep-th/0311148] [SPIRES].CrossRefMathSciNetGoogle Scholar
  30. [30]
    P. Dorey, Exact S matrices, hep-th/9810026 [SPIRES].
  31. [31]
    A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [SPIRES].MathSciNetADSGoogle Scholar
  32. [32]
    F. Ravanini, R. Tateo and A. Valleriani, Dynkin TBAs, Int. J. Mod. Phys. A 8 (1993) 1707 [hep-th/9207040] [SPIRES].MathSciNetADSGoogle Scholar
  33. [33]
    E. Frenkel and A. Szenes, Thermodynamics Bethe ansatz and dilogarithm identities. 1, hep-th/9506215 [SPIRES].
  34. [34]
    S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, hep-th/0111053 [SPIRES].
  35. [35]
    A. Kuniba, T. Nakanishi and J. Suzuki, T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras, SIGMA. 5 (2009) 108 [arXiv:0909.4618] [SPIRES].MathSciNetGoogle Scholar
  36. [36]
    M.J. Martins, Complex excitations in the thermodynamic Bethe ansatz approach, Phys. Rev. Lett. 67 (1991) 419 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  37. [37]
    P. Fendley, Excited-state thermodynamics, Nucl. Phys. B 374 (1992) 667 [hep-th/9109021] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  38. [38]
    B. Pioline and S. Vandoren, Large D-instanton effects in string theory, JHEP 07 (2009) 008 [arXiv:0904.2303] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  39. [39]
    D. Robles-Llana, M. Roček, F. Saueressig, U. Theis and S. Vandoren, Nonperturbative corrections to 4D string theory effective actions from SL(2, Z) duality and supersymmetry, Phys. Rev. Lett. 98 (2007) 211602 [hep-th/0612027] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    S. Alexandrov and F. Saueressig, Quantum mirror symmetry and twistors, JHEP 09 (2009) 108 [arXiv:0906.3743] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  41. [41]
    J. De Jaegher, B. de Wit, B. Kleijn and S. Vandoren, Special geometry in hypermultiplets, Nucl. Phys. B 514 (1998) 553 [hep-th/9707262] [SPIRES].CrossRefADSGoogle Scholar
  42. [42]
    B. de Wit, F. Vanderseypen and A. Van Proeyen, Symmetry structure of special geometries, Nucl. Phys. B 400 (1993) 463 [hep-th/9210068] [SPIRES].CrossRefADSGoogle Scholar
  43. [43]
    L. Bao, A. Kleinschmidt, B.E.W. Nilsson, D. Persson and B. Pioline, Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2, 1), arXiv:0909.4299 [SPIRES].
  44. [44]
    R. Gopakumar and C. Vafa, M-theory and topological strings. I, hep-th/9809187 [SPIRES].

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Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique & Astroparticules, CNRS UMR 5207Université Montpellier IIMontpellier Cedex 05France

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