Skip to main content
SpringerLink
Log in

G2 Holonomy Manifolds are Superconformal

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the chiral de Rham complex (CDR) over a manifold M with holonomy G2. We prove that the vertex algebra of global sections of the CDR associated to M contains two commuting copies of the Shatashvili-Vafa G2 superconformal algebra. Our proof is a tour de force, based on explicit computations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ben-Zvi D., Heluani R., Szczesny M.: Supersymmetry of the chiral de Rham complex. Compos. Math. 144, 503–521 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Blumenhagen R.: Covariant construction of N = 1 super W-algebras. Nucl. Phys. B 381(3), 641–669 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  3. Borcherds R.E.: Vertex algebras, Kac–Moody algebras, and the Monster. Proc. Nat. Acad. Sci. USA 83(10), 3068–3071 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  4. Corti A., Haskins M., Nordström J., Pacini T.: G 2-manifolds and associative submanifolds via semi-Fano 3-folds. Duke Math. J. 164(10), 1971–2092 (2015)

    Article  MathSciNet  Google Scholar 

  5. D’Andrea A., Kac V.G.: Structure theory of finite conformal algebras. Sel. Math. 4(3), 377–418 (1998)

    Article  MathSciNet  Google Scholar 

  6. de Boer J., Naqvi A., Shomer A.: The topological G 2 string. Adv. Theor. Math. Phys. 12(2), 243–318 (2008)

    Article  MathSciNet  Google Scholar 

  7. De Sole A., Kac V.G.: Freely generated vertex algebras and non-linear Lie conformal algebras. Commun. Math. Phys. 254(3), 659–694 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  8. De Sole A., Kac V.G.: Finite vs affine W-algebras. Jpn. J. Math. 1(1), 137–261 (2006)

    Article  MathSciNet  Google Scholar 

  9. Ekstrand J., Heluani R., Källén J., Zabzine M.: Chiral de Rham complex on Riemannian manifolds and special holonomy. Commun. Math. Phys. 318(3), 575–613 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. Figueroa-O’Farrill J.M.: A note on the extended superconformal algebras associated with manifolds of exceptional holonomy. Phys. Lett. B 392, 77–84 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  11. Figueroa-O’Farrill J.M., Schrans S.: Extended superconformal algebras. Phys. Lett. B 257, 69–73 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  12. Figueroa-O’Farrill J.M., Schrans S.: The Conformal bootstrap and super W algebras. Int. J. Mod. Phys. A 7, 591–618 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  13. Heluani R.: Supersymmetry of the chiral de Rham complex 2: commuting sectors. Int. Math. Res. Not. 2009, 953–987 (2009)

    Article  MathSciNet  Google Scholar 

  14. Heluani R., Rodríguez Díaz LO: The Shatashvili-Vafa G 2 superconformal algebra as a quantum Hamiltonian reduction of \({D(2,1;\alpha)}\). Bull. Braz. Math. Soc. (N.S.) 46(3), 331–351 (2015)

    Article  MathSciNet  Google Scholar 

  15. Howe P.S., Papadopoulos G.: Holonomy groups and W-symmetries. Commun. Math. Phys. 151(3), 467–479 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  16. Howe P.S., Papadopoulos G.: A note on holonomy groups and sigma models. Phys. Lett. B 263(2), 230–232 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  17. Joyce, Dominic D.: Riemannian holonomy groups and calibrated geometry. Oxford Graduate Texts in Mathematics. Oxford Univ. Press, UK (2007)

  18. Kac, Victor: Vertex operator algebras for beginners. University Lecture Series, 10 (1998)

  19. Kac V.G., Wakimoto M.: Quantum reduction and representation theory of superconformal algebras. Adv. Math. 185(2), 400–458 (2004)

    Article  MathSciNet  Google Scholar 

  20. Karigiannis S.: Flows of G 2 Structures, I. Q. J. Math. 60(4), 487–522 (2009)

    Article  MathSciNet  Google Scholar 

  21. Malikov F., Schechtman V., Vaintrob A.: Chiral de Rham Complex. Commun. Math. Phys. 204(2), 439–473 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  22. Odake S.: Extension of N = 2 superconformal algebra and Calabi–Yau compactification. Mod. Phys. Lett. A 4, 557 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  23. Peeters K.: Cadabra: a field-theory motivated symbolic computer algebra system. Comput. Phys. Commun. 176(8), 550–558 (2007)

    Article  ADS  Google Scholar 

  24. Salamon, Simon: Riemannian geometry and holonomy groups. Pitman research notes in mathematics series. Longman Scientific & Technical New York (1989)

  25. Shatashvili S.L., Vafa C.: Superstrings and manifolds of exceptional holonomy. Sel. Math. 1(2), 347–381 (1995)

    Article  MathSciNet  Google Scholar 

  26. Thielemans K.: A Mathematica package for computing operator product expansions. Int. J. Mod. Phys. C 2, 787–798 (1991)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgement

The author would like to thank Reimundo Heluani for generously sharing his insights and ideas and for the encouragement despite the long computations; the present paper is influenced by his views about the subject.

Author information

Authors and Affiliations

  1. Institute of Pure and Applied Mathematics, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, CEP 22460-320, Brazil

    Lázaro O. Rodríguez Díaz

Authors
  1. Lázaro O. Rodríguez Díaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lázaro O. Rodríguez Díaz.

Additional information

Communicated by Y. Kawahigashi

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Díaz, L.O.R. G2 Holonomy Manifolds are Superconformal. Commun. Math. Phys. 364, 385–440 (2018). https://doi.org/10.1007/s00220-018-3264-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3264-z

Search

Navigation