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Journal of High Energy Physics

, 2019:192 | Cite as

M5 branes and theta functions

  • Babak HaghighatEmail author
  • Rui Sun
Open Access
Regular Article - Theoretical Physics
  • 64 Downloads

Abstract

We propose quantum states for Little String Theories (LSTs) arising from M5 branes probing A- and D-type singularities. This extends Witten’s picture of M5 brane partition functions as theta functions to this more general setup. Compactifying the world-volume of the five-branes on a two-torus, we find that the corresponding theta functions are sections of line bundles over complex 4-tori. This formalism allows us to derive Seiberg-Witten curves for the resulting four-dimensional theories. Along the way, we prove a duality for LSTs observed by Iqbal, Hohenegger and Rey.

Keywords

Differential and Algebraic Geometry Field Theories in Higher Dimensions M-Theory String Duality 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

References

  1. [1]
    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].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys.325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Cordova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the squashed three-sphere, JHEP11 (2017) 119 [arXiv:1305.2891] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    C. Cordova and D.L. Jafferis, Toda theory from six dimensions, JHEP12 (2017) 106 [arXiv:1605.03997] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    E. Witten, Five-brane effective action in M-theory, J. Geom. Phys.22 (1997) 103 [hep-th/9610234] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    D. Belov and G.W. Moore, Holographic action for the self-dual field, hep-th/0605038 [INSPIRE].
  7. [7]
    S. Monnier, The anomaly field theories of six-dimensional (2, 0) superconformal theories, Adv. Theor. Math. Phys.22 (2018) 2035 [arXiv:1706.01903] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  8. [8]
    J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP05 (2014) 028 [Erratum ibid.06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
  9. [9]
    M. Del Zotto, J.J. Heckman, D.S. Park and T. Rudelius, On the defect group of a 6D SCFT, Lett. Math. Phys.106 (2016) 765 [arXiv:1503.04806] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    N. Nekrasov and E. Witten, The Ω deformation, branes, integrability and Liouville theory, JHEP09 (2010) 092 [arXiv:1002.0888] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings from F-theory and flop transitions, JHEP07 (2017) 112 [arXiv:1610.07916] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings and their partition functions, Phys. Rev.D 97 (2018) 106004 [arXiv:1710.02455] [INSPIRE].ADSMathSciNetGoogle Scholar
  13. [13]
    H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math.162 (2005) 313 [math.AG/0306198] [INSPIRE].
  14. [14]
    E. Witten, AdS/CFT correspondence and topological field theory, JHEP12 (1998) 012 [hep-th/9812012] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    E. Witten, Geometric Langlands from six dimensions, arXiv:0905.2720 [INSPIRE].
  16. [16]
    N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
  17. [17]
    C. Birkenhake and H. Lange, Complex Abelian varieties, Springer-Verlag, Berlin, Heidelberg, Germany (1980).zbMATHGoogle Scholar
  18. [18]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, JHEP02 (2015) 054 [arXiv:1407.6359] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  19. [19]
    K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6D N = (1, 0) theories on S 1/T 2and class S theories: part II, JHEP12 (2015) 131 [arXiv:1508.00915] [INSPIRE].
  20. [20]
    L. Bhardwaj, M. Del Zotto, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, F-theory and the classification of little strings, Phys. Rev.D 93 (2016) 086002 [Erratum ibid.D 100 (2019) 029901] [arXiv:1511.05565] [INSPIRE].
  21. [21]
    B. Haghighat, W. Yan and S.-T. Yau, ADE string chains and mirror symmetry, JHEP01 (2018) 043 [arXiv:1705.05199] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    H.W. Braden and T.J. Hollowood, The curve of compactified 6D gauge theories and integrable systems, JHEP12 (2003) 023 [hep-th/0311024] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    B. Runge, Theta functions and Siegel-Jacobi forms, Acta Math.175 (1995) 165.MathSciNetCrossRefGoogle Scholar
  24. [24]
    J. Manschot, On the space of elliptic genera, Commun. Num. Theor. Phys.2 (2008) 803 [arXiv:0805.4333] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  25. [25]
    E. Zaslow, Seidel’s mirror map for the torus, Adv. Theor. Math. Phys.9 (2005) 999 [math.SG/0506359] [INSPIRE].
  26. [26]
    K. Gunji, Defining equations of the universal Abelian surfaces with level three structure, Manuscripta Math.119 (2005) 61.MathSciNetCrossRefGoogle Scholar
  27. [27]
    A. Kanazawa and S.-C. Lau, Local Calabi-Yau manifolds of type à via SYZ mirror symmetry, J. Geom. Phys.139 (2019) 103 [arXiv:1605.00342] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc.66 (1949) 464.Google Scholar
  29. [29]
    J. de Boer, K. Hori, H. Ooguri and Y. Oz, Branes and dynamical supersymmetry breaking, Nucl. Phys.B 522 (1998) 20 [hep-th/9801060] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    K. Hori, Consistency condition for five-brane in M-theory on R 5/Z2 orbifold, Nucl. Phys.B 539 (1999) 35 [hep-th/9805141] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    Y. Tachikawa, Six-dimensional DN theory and four-dimensional SO-USp quivers, JHEP07 (2009) 067 [arXiv:0905.4074] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    B. Haghighat, J. Kim, W. Yan and S.-T. Yau, D-type fiber-base duality, JHEP09 (2018) 060 [arXiv:1806.10335] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    B. Bastian and S. Hohenegger, Dihedral symmetries of gauge theories from dual Calabi-Yau threefolds, Phys. Rev.D 99 (2019) 066013 [arXiv:1811.03387] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    W. Lerche, C. Vafa and N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys.B 324 (1989) 427 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    M. Gross and B. Siebert, Theta functions and mirror symmetry, arXiv:1204.1991 [INSPIRE].
  36. [36]
    M. Gross, P. Hacking, S. Keel and M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc.31 (2018) 497 [arXiv:1411.1394] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  37. [37]
    B. Haghighat, A. Iqbal, C. Koz¸caz, G. Lockhart and C. Vafa, M-strings, Commun. Math. Phys.334 (2015) 779 [arXiv:1305.6322] [INSPIRE].
  38. [38]
    J. Kim, S. Kim, K. Lee, J. Park and C. Vafa, Elliptic genus of E-strings, JHEP09 (2017) 098 [arXiv:1411.2324] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of minimal 6d SCFTs, Fortsch. Phys.63 (2015) 294 [arXiv:1412.3152] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    A. Gadde, B. Haghighat, J. Kim, S. Kim, G. Lockhart and C. Vafa, 6D string chains, JHEP02 (2018) 143 [arXiv:1504.04614] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    J. Kim and K. Lee, Little strings on Dn orbifolds, JHEP10 (2017) 045 [arXiv:1702.03116] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    J. Gu, B. Haghighat, K. Sun and X. Wang, Blowup equations for 6D SCFTs. I, JHEP03 (2019) 002 [arXiv:1811.02577] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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