Advertisement

Journal of High Energy Physics

, 2019:11 | Cite as

Quantum K-theory of Calabi-Yau manifolds

  • Hans Jockers
  • Peter MayrEmail author
Open Access
Regular Article - Theoretical Physics
  • 22 Downloads

Abstract

The disk partition function of certain 3d N = 2 supersymmetric gauge theories computes a quantum K-theoretic ring for Kähler manifolds X. We study the 3d gauge theory/quantum K-theory correspondence for global and local Calabi-Yau manifolds with several K¨ahler moduli. We propose a multi-cover formula that relates the 3d BPS world- volume degeneracies computed by quantum K-theory to Gopakumar-Vafa invariants.

Keywords

Supersymmetric Gauge Theory Chern-Simons Theories Sigma Models 

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]
    N. Nekrasov, Four Dimensional Holomorphic Theories, Ph.D. Thesis, (1996) [http://scgp.stonybrook.edu/people/faculty/bios/nikita-nekrasov].
  2. [2]
    N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl.192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, 3–8 August 2009, pp. 265–289 (2009) [ https://doi.org/10.1142/9789814304634_0015] [arXiv:0908.4052] [INSPIRE].
  4. [4]
    A. Givental, Permutation-equivariant quantum K-theory I. Definitions. Elementary K-theory of \( {\overline{\mathcal{M}}}_{0,n}/{S}_n \), arXiv:1508.02690.
  5. [5]
    A. Givental, Permutation-equivariant quantum K-theory II. Fixed point localization, arXiv:1508.04374.
  6. [6]
    A. Givental, Permutation-equivariant quantum K-theory III. Lefschetz’ formula on \( {\overline{\mathcal{M}}}_{0,n}/{S}_n \)and adelic characterization, arXiv:1508.06697.
  7. [7]
    A. Givental, Permutation-equivariant quantum K-theory IV. Dq -modules arXiv:1509.00830.
  8. [8]
    A. Givental, Permutation-equivariant quantum K-theory V. Toric q-hypergeometric functions, arXiv:1509.03903.
  9. [9]
    A. Givental, Permutation-equivariant quantum K-theory VI. Mirrors, arXiv:1509.07852.
  10. [10]
    A. Givental, Permutation-equivariant quantum K-theory VII. General theory, arXiv:1510.03076.
  11. [11]
    A. Givental, Permutation-equivariant quantum K-theory VIII. Explicit reconstruction, arXiv:1510.06116.
  12. [12]
    A. Givental, Permutation-equivariant quantum K-theory IX. Quantum Hirzebruch-Riemann-Roch in all genera, arXiv:1709.03180.
  13. [13]
    A. Givental, Permutation-equivariant quantum K-theory X. Quantum Hirzebruch-Riemann-Roch in genus 0, arXiv:1710.02376.
  14. [14]
    A. Givental, Permutation-equivariant quantum K-theory XI. Quantum Adams-Riemann-Roch, arXiv:1711.04201. [INSPIRE].
  15. [15]
    H. Jockers and P. Mayr, A 3d Gauge Theory/Quantum k-theory Correspondence, arXiv:1808.02040 [INSPIRE].
  16. [16]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys.B 403 (1993) 159 [hep-th/9301042] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    D.R. Morrison and M.R. Plesser, Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys.B 440 (1995) 279 [hep-th/9412236] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys.17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  19. [19]
    A. Kapustin and B. Willett, Wilson loops in supersymmetric Chern-Simons-matter theories and duality, arXiv:1302.2164 [INSPIRE].
  20. [20]
    C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, JHEP12 (2014) 177 [arXiv:1211.1986] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    S. Cecotti, D. Gaiotto and C. Vafa, tt geometry in 3 and 4 dimensions, JHEP05 (2014) 055 [arXiv:1312.1008] [INSPIRE].
  22. [22]
    D. Gaiotto and P. Koroteev, On Three Dimensional Quiver Gauge Theories and Integrability, JHEP05 (2013) 126 [arXiv:1304.0779] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, Mirror Symmetry and Symplectic Duality in 3d \( \mathcal{N} \) = 4 Gauge Theory, JHEP10 (2016) 108 [arXiv:1603.08382] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M. Aganagic and A. Okounkov, Elliptic stable envelopes, arXiv:1604.00423 [INSPIRE].
  25. [25]
    M. Aganagic and A. Okounkov, Quasimap counts and Bethe eigenfunctions, Moscow Math. J.17 (2017) 565 [arXiv:1704.08746] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    P. Koroteev, P.P. Pushkar, A. Smirnov and A.M. Zeitlin, Quantum k-theory of Quiver Varieties and Many-Body Systems, arXiv:1705.10419 [INSPIRE].
  27. [27]
    M. Bullimore, A. Ferrari and H. Kim, Twisted Indices of 3d \( \mathcal{N} \) = 4 Gauge Theories and Enumerative Geometry of Quasi-Maps, JHEP07 (2019) 014 [arXiv:1812.05567] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    B.H. Lian and S.-T. Yau, Mirror maps, modular relations and hypergeometric series 1, hep-th/9507151 [INSPIRE].
  29. [29]
    R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].
  30. [30]
    P. Candelas, X. De La Ossa, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 1., Nucl. Phys.B 416 (1994) 481 [hep-th/9308083] [INSPIRE].
  31. [31]
    S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys.167 (1995) 301 [hep-th/9308122] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    A. Gadde, S. Gukov and P. Putrov, Walls, Lines and Spectral Dualities in 3d Gauge Theories, JHEP05 (2014) 047 [arXiv:1302.0015] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    Y. Yoshida and K. Sugiyama, Localization of 3d \( \mathcal{N} \) = 2 Supersymmetric Theories on S 1× D 2 , arXiv:1409.6713 [INSPIRE].
  34. [34]
    D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI (1999).Google Scholar
  35. [35]
    Y. Ruan and M. Zhang, The level structure in quantum K-theory and mock theta functions, arXiv:1804.06552.
  36. [36]
    S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys.B 433 (1995) 501 [hep-th/9406055] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    A. Givental, Explicit reconstruction in quantum cohomology and K -theory, Ann. Fac. Sci. Toulouse Math.25 (2016) 419 [arXiv:1506.06431].MathSciNetCrossRefGoogle Scholar
  38. [38]
    A. Givental and V. Tonita, The Hirzebruch-Riemann-Roch theorem in true genus-0 quantum k-theory, arXiv:1106.3136 [INSPIRE].
  39. [39]
    Y.-P. Lee, Quantum K -theory. I. Foundations, Duke Math. J.121 (2004) 389 [math/0105014].
  40. [40]
    P.S. Aspinwall and D.R. Morrison, Topological field theory and rational curves, Commun. Math. Phys.151 (1993) 245 [hep-th/9110048] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    A. Givental, On the WDVV equation in quantum K -theory, Michigan Math. J.48 (2000) 295 [math/0003158].
  42. [42]
    H. Iritani, T. Milanov and V. Tonita, Reconstruction and Convergence in Quantum K -Theory via Difference Equations, Int. Math. Res. Not. (2015) 2887 [arXiv:1309.3750] [INSPIRE].
  43. [43]
    B.R. Greene, D.R. Morrison and M.R. Plesser, Mirror manifolds in higher dimension, Commun. Math. Phys.173 (1995) 559 [hep-th/9402119] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    P. Mayr, Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four folds, Nucl. Phys.B 494 (1997) 489 [hep-th/9610162] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    A. Klemm, B. Lian, S.S. Roan and S.-T. Yau, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys.B 518 (1998) 515 [hep-th/9701023] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    P. Mayr, N = 1 mirror symmetry and open/closed string duality, Adv. Theor. Math. Phys.5 (2002) 213 [hep-th/0108229] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  47. [47]
    H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys.B 577 (2000) 419 [hep-th/9912123] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [INSPIRE].
  49. [49]
    M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch.A 57 (2002) 1 [hep-th/0105045] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  50. [50]
    S.H. Katz and C.-C.M. Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys.5 (2001) 1 [math/0103074] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Bethe Center for Theoretical Physics, Physikalisches InstitutUniversität BonnBonnGermany
  2. 2.Arnold Sommerfeld Center for Theoretical PhysicsLudwig-Maximilians-UniversitätMunichGermany

Personalised recommendations