Skip to main content
SpringerLink
Log in

Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy \(\mathrm {SU}(3)\) and solutions of the Strominger system.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Abbati, M.C., Cirelli, R., Manià, A., Michor, P.W.: The Lie group of automorphisms of a principal bundle. J. Geom. Phys. 6, 215–235 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandrov, B., Ivanov, S.: Vanishing theorems on hermitian manifolds. Differ. Geom. Appl. 14, 251–265 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson, L.B., Gray, J., Sharpe, E.: Algebroids, heterotic moduli spaces and the Strominger system. JHEP 07, 37 (2014)

    Article  Google Scholar 

  4. Andreas, B., Garcia-Fernandez, M.: Solutions of the Strominger system via stable bundles on Calabi-Yau threefolds. Commun. Math. Phys. 315(1), 153–168 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baraglia, D.: Leibniz algebroids, twistings and exceptional generalized geometry. J. Geom. Phys. 62, 903–934 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baraglia, D., Hekmati, P.: Transitive courant algebroids, string structures and T-duality. Adv. Theor. Math. Phy. 19(3), 613–672 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Becker, K., Tseng, L.: Heterotic Flux Compactifications and their moduli. Nucl. Phys. B 741, 162–179 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Becker, K., Tseng, L., Yau, S.T.: Moduli space of torsional manifolds. Nucl. Phys. B 786, 119–134 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berline, N., Getzler, E., Vergne, M.: Heat kernels and dirac operators, Grundlehren text editions. Springer, New York (1992)

    Book  MATH  Google Scholar 

  10. Bursztyn, H., Cavalcanti, G., Gualtieri, M.: Reduction of Courant algebroids and generalized complex structures. Adv. Math. 211(2), 726–765 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Calabi, E.: The space of Kähler metrics. Proc. Int. Congr. Math. Amst. 2, 206–207 (1954)

    Google Scholar 

  12. Candelas, P., Horowitz, G., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nucl. Phys. B 258, 46–74 (1985)

    Article  MathSciNet  Google Scholar 

  13. Chen, Z., Stienon, M., Xu, P.: On regular Courant algebroids. J. Symp. Geom. 11, 1–24 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Coimbra, A., Strickland-Constable, C., Waldram, D.: Supergravity as generalised geometry I: type II theories. JHEP 11, 91 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Coimbra, A., Minasian, R., Triedl, H., Waldram, D.: Generalized geometry for string corrections. JHEP 11, 160 (2014)

    Article  MATH  Google Scholar 

  16. Cyrier, M., Lapan, J.M.: Towards the massless spectrum of non-Kähler heterotic compactifications. Adv. Theor. Math. Phys. 10, 853–877 (2007)

    Article  MATH  Google Scholar 

  17. De la Ossa, X., Svanes, E.: Holomorphic bundles and the moduli space of \(N=1\) supersymmetric heterotic compactifications. JHEP 10, 123 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Donaldson, S.K.: Anti-self-dual Yang-Mills connections on a complex algebraic surface and stable vector bundles. Proc. Lond. Math. Soc. 50, 1–26 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Douglis, A., Niremberg, L.: Interior estimates for elliptic systems of partial differential equations. Comm. Pure App. Math. 8(4), 503–538 (1955)

  20. Fei, T., Yau, S.-T.: Invariant solutions to the Strominger system on complex lie groups and their quotients. Commun. Math. Phys. 338(3), 1183–1195 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fernández, M., Ivanov, S., Ugarte, L., Vassilev, D.: Non-Kähler heterotic string solutions with non-zero fluxes and non-constant dilaton. JHEP 6, 73 (2014)

    Article  MATH  Google Scholar 

  22. Fernández, M., Ivanov, S., Ugarte, L., Villacampa, R.: Non-Kähler heterotic-string compactifications with non-zero fluxes and constant dilaton. Commun. Math. Phys. 288, 677–697 (2009)

    Article  MATH  Google Scholar 

  23. Fino, A., Tomassini, A.: On astheno-Kähler metrics. J. Lond. Math. Soc. 83, 290–308 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Freed, D.: Determinants, torsion and strings. Commun. Math. Phys. 107, 483–513 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fu, J.-X.: On non-Kähler Calabi-Yau Threefolds with Balanced Metrics. In: Proc. Int. Congress of Mathematicians, Hyderabad, India, vol. II, pp. 705–716. Hindustan Book Agency, New Delhi (2010)

  26. Fu, J.-X., Li, J., Yau, S.-T.: Balanced metrics on non-Kähler Calabi-Yau threefolds. J. Differ. Geom. 90, 81–129 (2012)

    Article  MATH  Google Scholar 

  27. Fu, J.-X., Yau, S.-T.: The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation. J. Differ. Geom. 78, 369–428 (2008)

    Article  MATH  Google Scholar 

  28. Fu, J.-X., Tseng, L.-S., Yau, S.-T.: Local heterotic torsional models. Commun. Math. Phys. 289, 1151–1169 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fu, J.-X., Wang, Z., Wu, D.: Form-type Calabi-Yau equations. Math. Res. Lett. 17, 887–903 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Garcia-Fernandez, M.: Generalized connections and heterotic supergravity. Commun. Math. Phys. 332, 89–115 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Garcia-Fernandez, M., Tipler, C.: Deformations of complex structures and the coupled Kähler-Yang-Mills equations. J. Lond. Math. Soc. 89(3), 779–796 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gauduchon, P.: Hermitian connections and Dirac operators. Boll. UMI 11-B(7), 257–288 (1997)

  33. Goto, R.: Moduli spaces of topological calibrations, Calabi-Yau, hyperKähler, G2 and Spin(7) structures. Int. J. Math. 15(3), 211–257 (2004)

    Article  MATH  Google Scholar 

  34. Green, M.B., Schwarz, J.H.: Anomaly cancellations in supersymmetric \(D = 10\) gauge theory and superstring theory. Phys. Lett. B 149, 117–122 (1984)

    Article  MathSciNet  Google Scholar 

  35. Gualtieri, M.: Generalized Complex Geometry, Ph.D. thesis, University of Oxford (2004). arXiv:math/0401221

  36. Gualtieri, M.: Branes on Poisson varieties, The many facets of geometry, pp. 368–394. Oxford University Press, Oxford (2010)

    Book  MATH  Google Scholar 

  37. Gualtieri, M.: Generalized Calabi-Yau manifolds. Q. J. Math 54, 281–308 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Huang, L.: On joint moduli spaces. Math. Ann. 302, 61–79 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hull, C.: Superstring compactifications with torsion and space-time supersymmetry. In: Turin 1985 Proceedings “Superunification and Extra Dimensions”, pp. 347–375 (1986)

  40. Ivanov, S.: Heterotic supersymmetry, anomaly cancellation and equations of motion. Phys. Lett. B 685(2–3), 190–196 (2010)

    Article  MathSciNet  Google Scholar 

  41. Ivanov, S., Papadopoulos, G.: Vanishing theorems and string backgrounds. Class. Quant. Grav. 18, 1089–1110 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ivanov, S., Papadopoulos, G.: A no-go theorem for string warped compactifications. Phys. Lett. B 497, 309–316 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kaledin, D., Verbitsky, M.: Non-hermitian Yang-Mills connections. Sel. Math. 4, 279–320 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kim, H.J.: Curvatures and holomorphic vector bundles, Ph.D. thesis, University of California, Berkeley (1985)

  45. Kobayashi, S.: Differential geometry of complex vector bundles. Princeton University Press, Princeton (1987)

  46. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. I. Interscience Publishers, New York (1963)

    MATH  Google Scholar 

  47. Lawson, H., Michelsohn, M.: Spin geometry, vol. 38 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ (1989)

    Google Scholar 

  48. Lee, H.: Strominger’s System on non-Kähler hermitian Manifolds, Ph.D. thesis, University of Oxford (2011)

  49. Li, J., Yau, S.-T.: Hermitian–Yang–Mills connections on non-Kähler manifolds, Mathematical aspects of string theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., vol. 1, pp. 560–573. World Scientific Publishing, Singapore (1987)

  50. Li, J., Yau, S.-T.: The existence of supersymmetric string theory with torsion. J. Differ. Geom. 70, 143–181 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  51. Lockhart, R.B., Mc Owen, R.C.: On elliptic systems in \({{\mathbb{R}}}^n\). Acta Math. 150(1), 125–135 (1983)

  52. Lockhart, R.B., Mc Owen, R.C.: Elliptic differential operators on non-compact manifolds. Ann. della Scuola Normale Superiore di Pisa 12(3), 409–447 (1985)

  53. Lübcke, M., Teleman, A.: The Kobayashi-Hitchin correspondence. World Scientific Publishing Co. Inc., Singapore (1995)

  54. Mackenzie, K.: General theory of Lie groupoids and Lie algebroids. London Mathematical Society Lecture Note Series, no. 213. Cambridge University Press, Cambridge (2005)

  55. Martelli, D., Sparks, J.: Non-Kähler heterotic rotations. Adv. Theor. Math. Phys. 15(1), 131–174 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  56. Matsuo, K., Takahashi, T.: On compact astheno-Kähler manifolds. Colloq. Math. 89(1), 213–221 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  57. Melnikov, I., Sharpe, E.: On marginal deformations of \((0,2)\) non-linear sigma models. Phys. Lett. B 705, 529–534 (2011)

    Article  MathSciNet  Google Scholar 

  58. Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 149(1), 261–295 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  59. Popovici, D.: Holomorphic deformations of balanced Calabi-Yau \({\partial }{\overline{\partial }}\)-Manifolds. arXiv:1304.0331 (preprint)

  60. Rubio, R.: \(B_n\)-generalized geometry and \(G_2^2\)-structures. J. Geom. Phys. 73, 150–156 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  61. Rubio, R.: Generalized geometry of type \(B_n\), Ph.D. thesis, University of Oxford (2014)

  62. Sati, H., Schreiber, U., Stasheff, J.: Twisted differential string and fivebrane structures. Commun. Math. Phys. 315, 169–213 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  63. Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274(2), 253–284 (1986)

    Article  MathSciNet  Google Scholar 

  64. Tseng, L.S., Yau, S.-T.: Non-Kähler Calabi-Yau Manifolds, String-Math 2011. Proc. Symp. Pure Math. 85, 241–254 (2012)

  65. Uhlenbeck, K., Yau, S.-T.: On the existence of Hermitian–Yang–Mills connections on stable bundles over compact Kähler manifolds. Commun. Pure Appl. Math. 39-S, 257–293 (1986)

  66. Wang, M.: Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7(1), 59–68 (1989)

  67. Witt, F.: Special metric structures and closed forms, Ph.D. thesis, University of Oxford (2005)

  68. Witt, F.: Calabi-Yau manifolds with B-fields. Rend. Sem. Mat. Univ. Politec. Torino 66, 1–21 (2008)

    MathSciNet  MATH  Google Scholar 

  69. Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA 74, 1798–1799 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  70. Yau, S.-T.: Complex geometry: Its brief history and its future. Sci. China Ser. A Math. 48, 47–60 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  71. Yau, S.-T.: Metrics on complex manifolds. Sci. China Math. 53(3), 565–572 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank Luis Álvarez-Cónsul, Bjorn Andreas, Vestislav Apostolov, Henrique Bursztyn, Ryushi Goto, Marco Gualtieri, Nigel Hitchin, Laurent Meersseman, Xenia de la Ossa, Dan Popovici, Brent Pym and Eirik Svanes for useful discussions. Part of this work was undertaken while CT was visiting IMPA, UFRJ, CRM, during visits of MGF and CT to CIRGET, and of RR to EPFL and ICMAT. We would like to thank these very welcoming institutions for providing a nice and stimulating working environment.

Author information

Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera 13–15, Cantoblanco, 28049, Madrid, Spain

    Mario Garcia-Fernandez

  2. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, 22460-320, Brazil

    Roberto Rubio

  3. Département de Mathématiques, LMBA, UMR CNRS 6205, Université de Bretagne Occidentale, 6, Avenue Victor Le Gorgeu, 29238, Brest Cedex 3, France

    Carl Tipler

Authors
  1. Mario Garcia-Fernandez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Rubio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carl Tipler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Garcia-Fernandez.

Additional information

Communicated by Ngaiming Mok.

This Project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 655162. This work is partially supported by an ESF—Short Visit Grant 5717 within the framework of the ITGP network. MGF is supported by a Marie Sklodowska-Curie Grant and was initially supported by ICMAT Severo Ochoa Project SEV-2011-0087 and by the École Polytechnique Fédéral de Lausanne. RR is supported by IMPA and was initially supported by QGM through its partnership with the Mathematical Institute of Oxford. CT is partially supported by Agence Nationale de la Recherche—ANR Project EMARKS.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garcia-Fernandez, M., Rubio, R. & Tipler, C. Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry. Math. Ann. 369, 539–595 (2017). https://doi.org/10.1007/s00208-016-1463-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-016-1463-5

Mathematics Subject Classification

Search

Navigation