Instantons from blow-up

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Abstract

We generalize Nakajima-Yoshioka blowup equations to arbitrary gauge group with hypermultiplets in arbitrary representations. Using our blowup equations, we compute the instanton partition functions for 4d \( \mathcal{N} \) = 2 and 5d \( \mathcal{N} \) = 1 gauge theories for arbitrary gauge theory with a large class of matter representations, without knowing explicit construction of the instanton moduli space. Our examples include exceptional gauge theories with fundamentals, SO(N ) gauge theories with spinors, and SU(6) gauge theories with rank-3 antisymmetric hypers. Remarkably, the instanton partition function is completely determined by the perturbative part.

A preprint version of the article is available at ArXiv.

Change history

  • 19 June 2020

    Corrected the authors��� affiliations. Affiliation a and c were swapped.

References

  1. [1]

    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys.B 426 (1994) 19 [Erratum ibid.B 430 (1994) 485] [hep-th/9407087] [INSPIRE].

  2. [2]

    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys.B 431 (1994) 484 [hep-th/9408099] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  3. [3]

    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  4. [4]

    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math.244 (2006) 525 [hep-th/0306238] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  5. [5]

    H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math.162 (2005) 313 [math.AG/0306198].

  6. [6]

    A. Braverman and P. Etingof, Instanton counting via affine Lie algebras. II. From Whittaker vectors to the Seiberg-Witten prepotential, math.AG/0409441.

  7. [7]

    M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Yu. I. Manin, Construction of instantons, Phys. Lett.A 65 (1978) 185 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  8. [8]

    G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys.209 (2000) 97 [hep-th/9712241] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  9. [9]

    U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP05 (2003) 054 [hep-th/0211108] [INSPIRE].

    ADS  Google Scholar 

  10. [10]

    N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys.252 (2004) 359 [hep-th/0404225] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  11. [11]

    M. Mariño and N. Wyllard, A note on instanton counting for N = 2 gauge theories with classical gauge groups, JHEP05 (2004) 021 [hep-th/0404125] [INSPIRE].

  12. [12]

    F. Fucito, J.F. Morales and R. Poghossian, Instantons on quivers and orientifolds, JHEP10 (2004) 037 [hep-th/0408090] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  13. [13]

    L. Hollands, C.A. Keller and J. Song, From SO/Sp instantons to W -algebra blocks, JHEP03 (2011) 053 [arXiv:1012.4468] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  14. [14]

    L. Hollands, C.A. Keller and J. Song, Towards a 4d/2d correspondence for Sicilian quivers, JHEP10 (2011) 100 [arXiv:1107.0973] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  15. [15]

    C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP07 (2015) 063 [arXiv:1406.6793] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  16. [16]

    C. Cordova and S.-H. Shao, An index formula for supersymmetric quantum mechanics, arXiv:1406.7853 [INSPIRE].

  17. [17]

    K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP01 (2015) 124 [arXiv:1407.2567] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  18. [18]

    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, Commun. Math. Phys.333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  19. [19]

    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys.104 (2014) 465 [arXiv:1305.0533] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  20. [20]

    H. Nakajima and K. Yoshioka, Lectures on instanton counting, in CRM Workshop on Algebraic Structures and Moduli Spaces, Montreal, Canada, 14–20 July 2003 [math.AG/0311058].

  21. [21]

    H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K -theoretic partition function, math.AG/0505553.

  22. [22]

    L. Gottsche, H. Nakajima and K. Yoshioka, K-theoretic Donaldson invariants via instanton counting, Pure Appl. Math. Quart.5 (2009) 1029 [math.AG/0611945] [INSPIRE].

  23. [23]

    H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blowup, III: blow-up formula from wall-crossing, Kyoto J. Math.51 (2011) 263 [arXiv:0911.1773] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  24. [24]

    L. Gottsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki’s formula and instanton counting, Publ. Res. Inst. Math. Sci. Kyoto47 (2011) 307 [arXiv:1001.5024] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  25. [25]

    R. Fintushel and R. J. Stern, The blowup formula for Donaldson invariants, Annals Math.143 (1996) 529.

    MathSciNet  MATH  Google Scholar 

  26. [26]

    G.W. Moore and E. Witten, Integration over the U plane in Donaldson theory, Adv. Theor. Math. Phys.1 (1997) 298 [hep-th/9709193] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  27. [27]

    M. Mariño and G.W. Moore, The Donaldson-Witten function for gauge groups of rank larger than one, Commun. Math. Phys.199 (1998) 25 [hep-th/9802185] [INSPIRE].

  28. [28]

    C.A. Keller and J. Song, Counting exceptional instantons, JHEP07 (2012) 085 [arXiv:1205.4722] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  29. [29]

    D. Gaiotto and S.S. Razamat, Exceptional indices, JHEP05 (2012) 145 [arXiv:1203.5517] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  30. [30]

    A. Grassi and J. Gu, BPS relations from spectral problems and blowup equations, Lett. Math. Phys.109 (2019) 1271 [arXiv:1609.05914] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  31. [31]

    J. Gu, M.-X. Huang, A.-K. Kashani-Poor and A. Klemm, Refined BPS invariants of 6d SCFTs from anomalies and modularity, JHEP05 (2017) 130 [arXiv:1701.00764] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  32. [32]

    M.-X. Huang, K. Sun and X. Wang, Blowup equations for refined topological strings, JHEP10 (2018) 196 [arXiv:1711.09884] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  33. [33]

    J. Gu, B. Haghighat, K. Sun and X. Wang, Blowup equations for 6d SCFTs. I, JHEP03 (2019) 002 [arXiv:1811.02577] [INSPIRE].

  34. [34]

    J. Gu, A. Klemm, K. Sun and X. Wang, Elliptic blowup equations for 6d SCFTs. II: exceptional cases, arXiv:1905.00864 [INSPIRE].

  35. [35]

    M. Honda, Borel summability of perturbative series in 4D N = 2 and 5D N = 1 supersymmetric theories, Phys. Rev. Lett.116 (2016) 211601 [arXiv:1603.06207] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  36. [36]

    J.D. Edelstein, M. Mariño and J. Mas, Whitham hierarchies, instanton corrections and soft supersymmetry breaking in N = 2 SU(N) super Yang-Mills theory, Nucl. Phys.B 541 (1999) 671 [hep-th/9805172] [INSPIRE].

  37. [37]

    J.D. Edelstein, M. Gómez-Reino, M. Mariño and J. Mas, N = 2 supersymmetric gauge theories with massive hypermultiplets and the Whitham hierarchy, Nucl. Phys.B 574 (2000) 587 [hep-th/9911115] [INSPIRE].

  38. [38]

    N.A. Nekrasov, Localizing gauge theories, in Mathematical Physics. Proceedings, 14thInternational Congress, ICMP 2003, Lisbon, Portugal, 28 July–2 August 2003, pg. 645 [INSPIRE].

  39. [39]

    L. Gottsche, H. Nakajima and K. Yoshioka, Instanton counting and Donaldson invariants, J. Diff. Geom.80 (2008) 343 [math.AG/0606180] [INSPIRE].

  40. [40]

    E. Gasparim and C.-C.M. Liu, The Nekrasov conjecture for toric surfaces, Commun. Math. Phys.293 (2010) 661 [arXiv:0808.0884] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  41. [41]

    G. Bonelli, K. Maruyoshi, A. Tanzini and F. Yagi, N = 2 gauge theories on toric singularities, blow-up formulae and W-algebrae, JHEP01 (2013) 014 [arXiv:1208.0790] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  42. [42]

    T. Sasaki, O(2) blow-up formula via instanton calculus on affine C2/Z2and Weil conjecture, hep-th/0603162 [INSPIRE].

  43. [43]

    Y. Ito, K. Maruyoshi and T. Okuda, Scheme dependence of instanton counting in ALE spaces, JHEP05 (2013) 045 [arXiv:1303.5765] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  44. [44]

    U. Bruzzo, M. Pedrini, F. Sala and R.J. Szabo, Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces, Adv. Math.288 (2016) 1175 [arXiv:1312.5554] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  45. [45]

    U. Bruzzo, F. Sala and R.J. Szabo, N = 2 quiver gauge theories on A-type ALE spaces, Lett. Math. Phys.105 (2015) 401 [arXiv:1410.2742] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  46. [46]

    E. Witten, Topological quantum field theory, Commun. Math. Phys.117 (1988) 353 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  47. [47]

    M. Bershtein, G. Bonelli, M. Ronzani and A. Tanzini, Exact results for N = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants, JHEP07 (2016) 023 [arXiv:1509.00267] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  48. [48]

    L. Baulieu, A. Losev and N. Nekrasov, Chern-Simons and twisted supersymmetry in various dimensions, Nucl. Phys.B 522 (1998) 82 [hep-th/9707174] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  49. [49]

    A. Losev, G.W. Moore, N. Nekrasov and S. Shatashvili, Four-dimensional avatars of two-dimensional RCFT, Nucl. Phys. Proc. Suppl.46 (1996) 130 [hep-th/9509151] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  50. [50]

    K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys.B 497 (1997) 56 [hep-th/9702198] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  51. [51]

    S. Shadchin, On certain aspects of string theory/gauge theory correspondence, Ph.D. thesis, LPT, Orsay, France (2005) [hep-th/0502180] [INSPIRE].

  52. [52]

    C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of instantons and W -algebras, JHEP03 (2012) 045 [arXiv:1111.5624] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  53. [53]

    L. Bhardwaj and Y. Tachikawa, Classification of 4d N = 2 gauge theories, JHEP12 (2013) 100 [arXiv:1309.5160] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  54. [54]

    P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards classification of 5d SCFTs: single gauge node, arXiv:1705.05836 [INSPIRE].

  55. [55]

    H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Rank-3 antisymmetric matter on 5-brane webs, JHEP05 (2019) 133 [arXiv:1902.04754] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  56. [56]

    M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys.254 (2005) 425 [hep-th/0305132] [INSPIRE].

  57. [57]

    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP10 (2009) 069 [hep-th/0701156] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  58. [58]

    N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett.B 388 (1996) 753 [hep-th/9608111] [INSPIRE].

    ADS  Google Scholar 

  59. [59]

    J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E6 global symmetry, Nucl. Phys.B 482 (1996) 142 [hep-th/9608047] [INSPIRE].

    ADS  MATH  Google Scholar 

  60. [60]

    J.A. Minahan and D. Nemeschansky, Superconformal fixed points with ENglobal symmetry, Nucl. Phys.B 489 (1997) 24 [hep-th/9610076] [INSPIRE].

    ADS  MATH  Google Scholar 

  61. [61]

    O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys.B 504 (1997) 239 [hep-th/9704170] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  62. [62]

    D.-E. Diaconescu and R. Entin, Calabi-Yau spaces and five-dimensional field theories with exceptional gauge symmetry, Nucl. Phys.B 538 (1999) 451 [hep-th/9807170] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  63. [63]

    P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On geometric classification of 5d SCFTs, JHEP04 (2018) 103 [arXiv:1801.04036] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  64. [64]

    L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: rank one, JHEP07 (2019) 178 [arXiv:1809.01650] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  65. [65]

    F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add flavor, part I: classification of 5d SCFTs, flavor symmetries and BPS states, arXiv:1907.05404 [INSPIRE].

  66. [66]

    H.-C. Kim, J. Kim, S. Kim, K.-H. Lee and J. Park, 6d strings and exceptional instantons, arXiv:1801.03579 [INSPIRE].

  67. [67]

    G. Zafrir, Brane webs and O5-planes, JHEP03 (2016) 109 [arXiv:1512.08114] [INSPIRE].

    ADS  MATH  Google Scholar 

  68. [68]

    H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 5-brane webs for 5d N = 1 G2gauge theories, JHEP03 (2018) 125 [arXiv:1801.03916] [INSPIRE].

    ADS  MATH  Google Scholar 

  69. [69]

    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP03 (2008) 069 [hep-th/0310272] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  70. [70]

    M. Del Zotto and G. Lockhart, On exceptional instanton strings, JHEP09 (2017) 081 [arXiv:1609.00310] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  71. [71]

    M. Del Zotto and G. Lockhart, Universal features of BPS strings in six-dimensional SCFTs, JHEP08 (2018) 173 [arXiv:1804.09694] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  72. [72]

    F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP09 (2009) 052 [arXiv:0906.0359] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  73. [73]

    P. Putrov, J. Song and W. Yan, (0, 4) dualities, JHEP03 (2016) 185 [arXiv:1505.07110] [INSPIRE].

  74. [74]

    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The superconformal index of the E6SCFT, JHEP08 (2010) 107 [arXiv:1003.4244] [INSPIRE].

    ADS  MATH  Google Scholar 

  75. [75]

    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys.319 (2013) 147 [arXiv:1110.3740] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  76. [76]

    A. Gadde, S.S. Razamat and B. Willett, “Lagrangian” for a non-Lagrangian field theory with N = 2 supersymmetry, Phys. Rev. Lett.115 (2015) 171604 [arXiv:1505.05834] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  77. [77]

    P. Agarwal, K. Maruyoshi and J. Song, A “Lagrangian” for the E7superconformal theory, JHEP05 (2018) 193 [arXiv:1802.05268] [INSPIRE].

    ADS  MATH  Google Scholar 

  78. [78]

    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett.B 387 (1996) 513 [hep-th/9607207] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  79. [79]

    S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories, JHEP01 (2014) 005 [arXiv:1309.2657] [INSPIRE].

    ADS  Google Scholar 

  80. [80]

    S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, Coulomb branch and the moduli space of instantons, JHEP12 (2014) 103 [arXiv:1408.6835] [INSPIRE].

    ADS  Google Scholar 

  81. [81]

    H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP12 (2011) 031 [arXiv:1110.2175] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  82. [82]

    Y. Hwang, J. Kim and S. Kim, M5-branes, orientifolds and S-duality, JHEP12 (2016) 148 [arXiv:1607.08557] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  83. [83]

    S.-J. Lee and P. Yi, D-particles on orientifolds and rational invariants, JHEP07 (2017) 046 [arXiv:1702.01749] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  84. [84]

    H.-C. Kim, S.-S. Kim and K. Lee, 5-dim superconformal index with enhanced ENglobal symmetry, JHEP10 (2012) 142 [arXiv:1206.6781] [INSPIRE].

    ADS  MATH  Google Scholar 

  85. [85]

    É. B. Vinberg and V.L. Popov, On a class of quasihomogeneous affine varieties, Math. USSR-Izv.6 (1972) 743.

    MATH  Google Scholar 

  86. [86]

    D. Garfinkle, A new construction of the Joseph ideal, chapter III, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, U.S.A. (1982).

  87. [87]

    S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert series of the one instanton moduli space, JHEP06 (2010) 100 [arXiv:1005.3026] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  88. [88]

    D. Gaiotto, N = 2 dualities, JHEP08 (2012) 034 [arXiv:0904.2715] [INSPIRE].

    ADS  Google Scholar 

  89. [89]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  90. [90]

    I. Coman, E. Pomoni and J. Teschner, Trinion conformal blocks from topological strings, arXiv:1906.06351 [INSPIRE].

  91. [91]

    R. Feger and T.W. Kephart, LieART — a mathematica application for Lie algebras and representation theory, Comput. Phys. Commun.192 (2015) 166 [arXiv:1206.6379] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

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Kim, J., Kim, S., Lee, K. et al. Instantons from blow-up. J. High Energ. Phys. 2019, 92 (2019). https://doi.org/10.1007/JHEP11(2019)092

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Keywords

  • Brane Dynamics in Gauge Theories
  • Differential and Algebraic Geometry
  • Field Theories in Higher Dimensions
  • Solitons Monopoles and Instantons