Advertisement

Elliptic non-Abelian Donaldson-Thomas invariants of 3

  • Francesco BeniniEmail author
  • Giulio Bonelli
  • Matteo Poggi
  • Alessandro Tanzini
Open Access
Regular Article - Theoretical Physics
  • 32 Downloads

Abstract

We compute the elliptic genus of the D1/D7 brane system in flat space, finding a non-trivial dependence on the number of D7 branes, and provide an F-theory interpretation of the result. We show that the JK-residues contributing to the elliptic genus are in one-to-one correspondence with coloured plane partitions and that the elliptic genus can be written as a chiral correlator of vertex operators on the torus. We also study the quantum mechanical system describing D0/D6 bound states on a circle, which leads to a plethystic exponential formula that can be connected to the M-theory graviton index on a multi-Taub-NUT background. The formula is a conjectural expression for higher-rank equivariant K-theoretic Donaldson-Thomas invariants on 3.

Keywords

Brane Dynamics in Gauge Theories Differential and Algebraic Geometry Field Theories in Lower Dimensions Supersymmetric Gauge Theory 

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]
    F. Benini and S. Cremonesi, Partition functions of \( \mathcal{N} \) = (2, 2) gauge theories on S 2 and vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  2. [2]
    N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact results in D = 2 supersymmetric gauge theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    H. Jockers et al., Two-sphere partition functions and Gromov-Witten invariants, Commun. Math. Phys. 325 (2014) 1139 [arXiv:1208.6244] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    F. Benini, D.S. Park and P. Zhao, Cluster algebras from dualities of 2d \( \mathcal{N} \) = (2, 2) quiver gauge theories, Commun. Math. Phys. 340 (2015) 47 [arXiv:1406.2699] [INSPIRE].
  5. [5]
    F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1 [arXiv:1501.01310] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS 4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS 4, Phys. Lett. B 771 (2017) 462 [arXiv:1608.07294] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Gopakumar and C. Vafa, M theory and topological strings. 2, hep-th/9812127 [INSPIRE].
  11. [11]
    G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, The stringy instanton partition function, JHEP 01 (2014) 038 [arXiv:1306.0432] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants, Commun. Math. Phys. 333 (2015) 717 [arXiv:1307.5997] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) quantum intermediate long wave hydrodynamics, JHEP 07 (2014) 141 [arXiv:1403.6454] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories, J. Geom. Phys. 109 (2016) 3 [arXiv:1505.07116] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Poggi, Elliptic genus derivation of 4d holomorphic blocks, JHEP 03 (2018) 035 [arXiv:1711.07499] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    G. Bonelli, A. Tanzini and J. Zhao, Vertices, vortices and interacting surface operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    G. Bonelli, A. Tanzini and J. Zhao, The Liouville side of the Vortex, JHEP 09 (2011) 096 [arXiv:1107.2787] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    H. Ooguri, A. Strominger and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004) 106007 [hep-th/0405146] [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, in the proceedings of the Symposium on Geometric Issues in the Foundations of Science, June 25-29, Oxford, U.K. (1996).Google Scholar
  20. [20]
    D. Maulik, N. Nekrasov, A. Okounkov and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, I, Composit. Math. 142 (2006) 1263 [math/0312059].
  21. [21]
    D. Maulik, N. Nekrasov, A. Okounkov and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, II, Composit. Math. 142 (2006) 1286 [math/0406092v2].
  22. [22]
    Y. Toda, On a computation of rank two Donaldson-Thomas invariants, Commun. Number Theor. Phys. 4 (2010) 49 [arXiv:0912.2507].MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J. Stoppa, D0-D6 states counting and GW invariants, Lett. Math. Phys. 102 (2012) 149 [arXiv:0912.2923].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    J. Manschot, B. Pioline and A. Sen, Wall crossing from Boltzmann black hole halos, JHEP 07 (2011) 059 [arXiv:1011.1258] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    H. Awata and H. Kanno, Quiver matrix model and topological partition function in six dimensions, JHEP 07 (2009) 076 [arXiv:0905.0184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    N. Nekrasov, Instanton partition functions and M-theory, Japan J. Math. 4 (2009) 63.Google Scholar
  27. [27]
    N. Nekrasov and A. Okounkov, Membranes and sheaves, arXiv:1404.2323 [INSPIRE].
  28. [28]
    P.K. Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184 [hep-th/9501068] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d \( \mathcal{N} \) = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
  32. [32]
    N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    E. Witten, BPS bound states of D0-D6 and D0-D8 systems in a B field, JHEP 04 (2002) 012 [hep-th/0012054] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, arXiv:1512.07363 [INSPIRE].
  35. [35]
    F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  37. [37]
    A.N. Schellekens and N.P. Warner, Anomalies and modular invariance in string theory, Phys. Lett. B 177 (1986) 317 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    A.N. Schellekens and N.P. Warner, Anomaly cancellation and selfdual lattices, Phys. Lett. B 181 (1986) 339 [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    E. Witten, Elliptic genera and quantum field theory, Commun. Math. Phys. 109 (1987) 525 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  41. [41]
    F. Benini and B. Le Floch, Supersymmetric localization in two dimensions, J. Phys. A 50 (2017) 443003 [arXiv:1608.02955] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  42. [42]
    A. Gadde and S. Gukov, 2d index and surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291 [alg-geom/9307001].MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    R.J. Szabo, N = 2 gauge theories, instanton moduli spaces and geometric representation theory, J. Geom. Phys. 109 (2016) 83 [arXiv:1507.00685] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [arXiv:1406.6793] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    C. Cordova and S.-H. Shao, An index formula for supersymmetric quantum mechanics, arXiv:1406.7853 [INSPIRE].
  47. [47]
    K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  49. [49]
    B.R. Greene, A.D. Shapere, C. Vafa and S.-T. Yau, Stringy cosmic strings and noncompact Calabi-Yau manifolds, Nucl. Phys. B 337 (1990) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    A. Karch and E. Katz, Adding flavor to AdS/CFT, JHEP 06 (2002) 043 [hep-th/0205236] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    P. Ouyang, Holomorphic D7 branes and flavored N = 1 gauge theories, Nucl. Phys. B 699 (2004) 207 [hep-th/0311084] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    F. Benini et al., Unquenched flavors in the Klebanov-Witten model, JHEP 02 (2007) 090 [hep-th/0612118] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    F. Benini et al., Backreacting flavors in the Klebanov-Strassler background, JHEP 09 (2007) 109 [arXiv:0706.1238] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    F. Benini, A chiral cascade via backreacting D7-branes with flux, JHEP 10 (2008) 051 [arXiv:0710.0374] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    A. Sen, Dynamics of multiple Kaluza-Klein monopoles in M and string theory, Adv. Theor. Math. Phys. 1 (1998) 115 [hep-th/9707042] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    C.M. Hull, Gravitational duality, branes and charges, Nucl. Phys. B 509 (1998) 216 [hep-th/9705162] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    V.A. Kazakov, I.K. Kostov and N.A. Nekrasov, D particles, matrix integrals and KP hierarchy, Nucl. Phys. B 557 (1999) 413 [hep-th/9810035] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    O. Babelon, D. Bernard and M. Talon, Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2003).Google Scholar
  59. [59]
    M. Cirafici, A. Sinkovics and R.J. Szabo, Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory, Nucl. Phys. B 809 (2009) 452 [arXiv:0803.4188] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    Y. Zenkevich, 3d field theory, plane partitions and triple Macdonald polynomials, JHEP 06 (2019) 012 [arXiv:1712.10300] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    M. Cirafici and R.J. Szabo, Curve counting, instantons and McKay correspondences, J. Geom. Phys. 72 (2013) 54 [arXiv:1209.1486] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    M. Aganagic, A stringy origin of M 2 brane Chern-Simons theories, Nucl. Phys. B 835 (2010) 1 [arXiv:0905.3415] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M 2-branes at toric CY 4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    F. Benini, C. Closset and S. Cremonesi, Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS 4 /CFT 3, JHEP 09 (2011) 005 [arXiv:1105.2299] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  65. [65]
    A. Bawane, G. Bonelli, M. Ronzani and A. Tanzini, \( \mathcal{N} \) = 2 supersymmetric gauge theories on S 2 × S 2 and Liouville gravity, JHEP 07 (2015) 054 [arXiv:1411.2762] [INSPIRE].
  66. [66]
    M. Bershtein, G. Bonelli, M. Ronzani and A. Tanzini, Exact results for \( \mathcal{N} \) = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants, JHEP 07 (2016) 023 [arXiv:1509.00267] [INSPIRE].
  67. [67]
    M. Bershtein, G. Bonelli, M. Ronzani and A. Tanzini, Gauge theories on compact toric surfaces, conformal field theories and equivariant Donaldson invariants, J. Geom. Phys. 118 (2017) 40 [arXiv:1606.07148] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    A. Gholampour and M. Kool, Stable reflexive sheaves and localization, J. Pure Appl. Algebra 221 (2017) 1934 [arXiv:1308.3688] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  70. [70]
    L. Rastelli and S.S. Razamat, The supersymmetric index in four dimensions, J. Phys. A 50 (2017) 443013 [arXiv:1608.02965] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.International School of Advanced Studies (SISSA/ISAS) and INFN — Sezione di Trieste via Bonomea 265TriesteItaly

Personalised recommendations