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Genus two modular bootstrap

  • Minjae Cho
  • Scott CollierEmail author
  • Xi Yin
Open Access
Regular Article - Theoretical Physics
  • 23 Downloads

Abstract

We study the Virasoro conformal block decomposition of the genus two partition function of a two-dimensional CFT by expanding around a ℤ3-invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two modular invariance and unitarity. In particular, we find “critical surfaces” that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.

Keywords

Conformal and W Symmetry Conformal Field 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]
    S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz 66 (1974) 23 [INSPIRE].Google Scholar
  3. [3]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
  4. [4]
    D. Friedan and S.H. Shenker, The Analytic Geometry of Two-Dimensional Conformal Field Theory, Nucl. Phys. B 281 (1987) 509 [INSPIRE].
  5. [5]
    G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, \( \mathcal{N}=4 \) superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].
  7. [7]
    Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, (2, 2) superconformal bootstrap in two dimensions, JHEP 05 (2017) 112 [arXiv:1610.05371] [INSPIRE].
  8. [8]
    S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    C.A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
  11. [11]
    S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, arXiv:1703.09805 [INSPIRE].
  13. [13]
    G. Mason and M.P. Tuite, On genus two Riemann surfaces formed from sewn tori, Commun. Math. Phys. 270 (2007) 587 [math/0603088] [INSPIRE].
  14. [14]
    O. Lunin and S.D. Mathur, Correlation functions for M N /S N orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
  15. [15]
    A. Maloney, Notes on the renyi surface, private communication.Google Scholar
  16. [16]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 0911 (2009) P11001 [arXiv:0905.2069] [INSPIRE].
  17. [17]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
  18. [18]
    J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP 10 (2017) 136 [arXiv:1705.05855] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C.A. Keller, G. Mathys and I.G. Zadeh, Bootstrapping Chiral CFTs at Genus Two, arXiv:1705.05862 [INSPIRE].
  20. [20]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A.B. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Sov. Phys. JETP 63 (1986) 1061.MathSciNetGoogle Scholar
  22. [22]
    C.-M. Chang and Y.-H. Lin, Bootstrap, universality and horizons, JHEP 10 (2016) 068 [arXiv:1604.01774] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
  26. [26]
    V.S. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
  27. [27]
    D. Simmons-Duffin, The Conformal Bootstrap, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., June 1-26, 2015, pp. 1-74 (2017) [DOI: https://doi.org/10.1142/9789813149441_0001] [arXiv:1602.07982] [INSPIRE].
  28. [28]
    D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    D. Whalen, An algorithm for evaluating Gram matrices in Verma modules of W-algebras, arXiv:1412.0759 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Jefferson Physical LaboratoryHarvard UniversityCambridgeU.S.A.

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