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Fun with F24

A preprint version of the article is available at arXiv.

Abstract

We study some special features of F24, the holomorphic c = 12 superconformal field theory (SCFT) given by 24 chiral free fermions. We construct eight different Lie superalgebras of “physical” states of a chiral superstring compactified on F24, and we prove that they all have the structure of Borcherds-Kac-Moody superalgebras. This produces a family of new examples of such superalgebras. The models depend on the choice of an \( \mathcal{N} \) = 1 supercurrent on F24, with the admissible choices labeled by the semisimple Lie algebras of dimension 24. We also discuss how F24, with any such choice of supercurrent, can be obtained via orbifolding from another distinguished c = 12 holomorphic SCFT, the \( \mathcal{N} \) = 1 supersymmetric version of the chiral CFT based on the E8 lattice.

References

  1. V. Anagiannis and M.C.N. Cheng, TASI Lectures on Moonshine, PoS TASI2017 (2018) 010 [arXiv:1807.00723] [INSPIRE].

  2. V. Anagiannis, M.C.N. Cheng, J.F.R. Duncan and R. Volpato, Vertex operator superalgebra/sigma model correspondences: The four-torus case, arXiv:2009.00186 [INSPIRE].

  3. J.-B. Bae, J.A. Harvey, K. Lee, S. Lee and B.C. Rayhaun, Conformal Field Theories with Sporadic Group Symmetry, arXiv:2002.02970 [INSPIRE].

  4. N. Berkovits and B. Zwiebach, On the picture dependence of Ramond-Ramond cohomology, Nucl. Phys. B 523 (1998) 311 [hep-th/9711087] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  5. R.E. Borcherds, The monster Lie algebra, Adv. Math. 83 (1990) 30.

    MathSciNet  MATH  Article  Google Scholar 

  6. R.E. Borcherds, The moduli space of Enriques surfaces and the fake Monster Lie superalgebra, Topology 35 (1996) 699.

    MathSciNet  MATH  Article  Google Scholar 

  7. R.E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992) 405.

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. R.E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998) 491 [alg-geom/9609022] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  9. R.E. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987) 133.

    MathSciNet  MATH  Article  Google Scholar 

  10. R.E. Borcherds, Lattices like the Leech lattice, J. Algebra 130 (1990) 219.

    MathSciNet  MATH  Article  Google Scholar 

  11. M.C.N. Cheng, X. Dong, J.F.R. Duncan, S. Harrison, S. Kachru and T. Wrase, Mock Modular Mathieu Moonshine Modules, Res. Math. Sci. 2 (2015) 13 [arXiv:1406.5502] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  12. M.C.N. Cheng, J.F.R. Duncan, S.M. Harrison and S. Kachru, Equivariant K3 Invariants, Commun. Num. Theor. Phys. 11 (2017) 41 [arXiv:1508.02047] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  13. M.C.N. Cheng and E.P. Verlinde, Wall Crossing, Discrete Attractor Flow, and Borcherds Algebra, SIGMA 4 (2008) 068 [arXiv:0806.2337] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  14. J.H. Conway and S.P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979) 308 [INSPIRE]

    MathSciNet  MATH  Article  Google Scholar 

  15. T. Creutzig, J.F.R. Duncan and W. Riedler, Self-Dual Vertex Operator Superalgebras and Superconformal Field Theory, J. Phys. A 51 (2018) 034001 [arXiv:1704.03678] [INSPIRE].

  16. B. Craps, M.R. Gaberdiel and J.A. Harvey, Monstrous branes, Commun. Math. Phys. 234 (2003) 229 [hep-th/0202074] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  17. R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Counting dyons in N = 4 string theory, Nucl. Phys. B 484 (1997) 543 [hep-th/9607026] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. J.F.R. Duncan, Super-moonshine for Conway’s largest sporadic group, Duke Math. J. 139 (2007) 255.

    MathSciNet  MATH  Article  Google Scholar 

  19. J.F.R. Duncan and S. Mack-Crane, The Moonshine Module for Conway’s Group, SIGMA 3 (2015) e10 [arXiv:1409.3829] [INSPIRE].

  20. J.F.R. Duncan and S. Mack-Crane, Derived Equivalences of K3 Surfaces and Twined Elliptic Genera, Res. Math. Sci. 3 (2016) 1 [arXiv:1506.06198] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  21. J.F.R. Duncan, M.J. Griffin and K. Ono, Moonshine, Res. Math. Sci. 2 (2015) 11 [arXiv:1411.6571] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  22. M. Eichler and D. Zagier, The Theory of Jacobi Forms, in Progress in Mathematics 55, Birkhäuser, Boston MA U.S.A. (1985).

  23. J.M. Figueroa-O’Farrill and T. Kimura, The BRST Cohomology of the NSR String: Vanishing and ‘No Ghost’ Theorems, Commun. Math. Phys. 124 (1989) 105 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. D. Gaiotto, T. Johnson-Freyd and E. Witten, A Note On Some Minimally Supersymmetric Models In Two Dimensions, arXiv:1902.10249 [INSPIRE].

  25. D. Gaiotto and T. Johnson-Freyd, Mock modularity and a secondary elliptic genus, arXiv:1904.05788 [INSPIRE].

  26. P. Goddard and D.I. Olive, Kac-Moody Algebras, Conformal Symmetry and Critical Exponents, Nucl. Phys. B 257 (1985) 226 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  27. V.A. Gritsenko, Fourier-Jacobi functions in n variables, Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. 168 (1988) 32.

  28. V. Gritsenko, 24 faces of the Borcherds modular form Φ12 , arXiv:1203.6503 [INSPIRE].

  29. S.M. Harrison, S. Kachru, N.M. Paquette, R. Volpato and M. Zimet, Heterotic sigma models on T8 and the Borcherds automorphic form Φ12 , JHEP 10 (2017) 121 [arXiv:1610.00707] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. S.M. Harrison, N.M. Paquette, D. Persson and R. Volpato, BPS-algebras in 2D string theory, in preparation.

  31. S.M. Harrison, N.M. Paquette and R. Volpato, A Borcherds-Kac-Moody Superalgebra with Conway Symmetry, Commun. Math. Phys. 370 (2019) 539 [arXiv:1803.10798] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  32. J.A. Harvey and G.W. Moore, Algebras, BPS states, and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  33. J.A. Harvey and G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489 [hep-th/9609017] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  34. J.A. Harvey and G.W. Moore, Conway Subgroup Symmetric Compactifications of Heterotic String, J. Phys. A 51 (2018) 354001 [arXiv:1712.07986] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  35. J.A. Harvey and G.W. Moore, Exact gravitational threshold correction in the FHSV model, Phys. Rev. D 57 (1998) 2329 [hep-th/9611176] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  36. A. Klemm and M. Mariño, Counting BPS states on the enriques Calabi-Yau, Commun. Math. Phys. 280 (2008) 27 [hep-th/0512227] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  37. A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic Symmetry Protected Topological Phases and Cobordisms, JHEP 12 (2015) 052 [arXiv:1406.7329] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  38. A. Kapustin and R. Thorngren, Fermionic SPT phases in higher dimensions and bosonization, JHEP 10 (2017) 080 [arXiv:1701.08264] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  39. A. Karch, D. Tong and C. Turner, A Web of 2d Dualities: Z2 Gauge Fields and Arf Invariants, SciPost Phys. 7 (2019) 007 [arXiv:1902.05550] [INSPIRE].

    ADS  Article  Google Scholar 

  40. B.H. Lian and G.J. Zuckerman, BRST Cohomology of the Supervirasoro Algebras, Commun. Math. Phys. 125 (1989) 301 [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  41. B.H. Lian and G.J. Zuckerman, New perspectives on the BRST algebraic structure of string theory, Commun. Math. Phys. 154 (1993) 613 [hep-th/9211072] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  42. N.M. Paquette, D. Persson and R. Volpato, Monstrous BPS-Algebras and the Superstring Origin of Moonshine, Commun. Num. Theor. Phys. 10 (2016) 433 [arXiv:1601.05412] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  43. N.M. Paquette, D. Persson and R. Volpato, BPS Algebras, Genus Zero, and the Heterotic Monster, J. Phys. A 50 (2017) 414001 [arXiv:1701.05169] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  44. D. Persson and R. Volpato, Fricke S-duality in CHL models, JHEP 12 (2015) 156 [arXiv:1504.07260] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  45. D. Persson and R. Volpato, Dualities in CHL-Models, J. Phys. A 51 (2018) 164002 [arXiv:1704.00501] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  46. J. Polchinski, String theory. Volume 1: An introduction to the Bosonic String, in Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1998) [INSPIRE].

  47. U. Ray, Automorphic Forms and Lie Superalgebras, in Algebra and Applications 5, Springer, Dordrecht The Netherlands (2007).

  48. N.R. Scheithauer, The Fake monster superalgebra, Adv. Math. 151 (2000) 226 [math.QA/9905113] [INSPIRE].

  49. T. Senthil, Symmetry Protected Topological phases of Quantum Matter, Ann. Rev. Condensed Matter Phys. 6 (2015) 299 [arXiv:1405.4015] [INSPIRE].

    ADS  Article  Google Scholar 

  50. A. Taormina and K. Wendland, The Conway Moonshine Module is a Reflected K3 Theory, arXiv:1704.03813 [INSPIRE].

  51. J.G. Thompson, Finite groups and modular functions, Bull. London Math. Soc. 11 (1979) 347.

    MathSciNet  MATH  Article  Google Scholar 

  52. J.G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979) 352.

    MathSciNet  MATH  Article  Google Scholar 

  53. D. Tong, Lectures on the Quantum Hall Effect, arXiv:1606.06687 [INSPIRE].

  54. D. Tong and C. Turner, Notes on 8 Majorana Fermions, SciPost Phys. Lect. Notes 14 (2020) 1 [arXiv:1906.07199] [INSPIRE].

    Google Scholar 

  55. J. Troost, Lie Algebra Fermions, J. Phys. A 53 (2020) 425401 [arXiv:2004.01055] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

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Correspondence to Roberto Volpato.

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ArXiv ePrint: 2009.14710

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Harrison, S.M., Paquette, N.M., Persson, D. et al. Fun with F24. J. High Energ. Phys. 2021, 39 (2021). https://doi.org/10.1007/JHEP02(2021)039

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Keywords

  • Conformal and W Symmetry
  • Conformal Field Models in String Theory
  • Conformal Field Theory
  • BRST Quantization