Journal of High Energy Physics

, 2019:38 | Cite as

The Universal Geometry of heterotic vacua

  • Philip CandelasEmail author
  • Xenia de la Ossa
  • Jock McOrist
  • Roberto Sisca
Open Access
Regular Article - Theoretical Physics


We consider a family of perturbative heterotic string backgrounds. These are complex threefolds X with c1 = 0, each with a gauge field solving the Hermitian Yang-Mill’s equations and compatible B and H fields that satisfy the anomaly cancellation conditions. Our perspective is to consider a geometry in which these backgrounds are fibred over a parameter space. If the manifold X has coordinates x, and parameters are denoted by y, then it is natural to consider coordinate transformations \( x\to \tilde{x}\left(x,y\right)\kern0.5em \mathrm{and}\kern0.5em y\to \tilde{y}(y) \). Similarly, gauge transformations of the gauge field and B field also depend on both x and y. In the process of defining deformations of the background fields that are suitably covariant under these transformations, it turns out to be natural to extend the gauge field A to a gauge field \( \mathbb{A} \) on the extended (x, y)-space. Similarly, the B, H, and other fields are also extended. The total space of the fibration of the heterotic structures is the Universal Geometry of the title. The extension of gauge fields has been studied in relation to Donaldson theory and monopole moduli spaces. String vacua furnish a richer application of these ideas. One advantage of this point of view is that previously disparate results are unified into a simple tensor formulation. In a previous paper, by three of the present authors, the metric on the moduli space of heterotic theories was derived, correct through \( \mathcal{O} \)(α′), and it was shown how this was related to a simple Kähler potential. With the present formalism, we are able to rederive the results of this previously long and involved calculation, in less than a page.


Superstring Vacua Superstrings and Heterotic Strings 


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.


  1. [1]
    P. Candelas, X. de la Ossa and J. McOrist, A Metric for Heterotic Moduli, Commun. Math. Phys. 356 (2017) 567 [arXiv:1605.05256] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Atiyah and I. Singer, Dirac operators coupled to vector potentials, Proc. Natl. Acad. Sci. U.S.A. 81 (1984) 2597.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S.K. Donaldson, Polynomial invariants for smooth manifolds, Topology 29 (1990) 257 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S.K. Donaldson, The Orientation of Yang-Mills Moduli Spaces and 4-Manifold Topology, J. Diff. Geom. 26 (1987) 397.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J.A. Harvey and A. Strominger, String theory and the Donaldson polynomial, Commun. Math. Phys. 151 (1993) 221 [hep-th/9108020] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J.P. Gauntlett, Low-energy dynamics of N = 2 supersymmetric monopoles, Nucl. Phys. B 411 (1994) 443 [hep-th/9305068] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    N. Hitchin, What is a Gerbe?, Not. Am. Math. Soc. 50 (2003) 218.zbMATHGoogle Scholar
  9. [9]
    L. Anguelova, C. Quigley and S. Sethi, The Leading Quantum Corrections to Stringy Kähler Potentials, JHEP 10 (2010) 065 [arXiv:1007.4793] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, (1965).Google Scholar
  11. [11]
    P. Candelas, Yukawa Couplings Between (2,1) Forms, Nucl. Phys. B 298 (1988) 458 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    P. Candelas and X. de la Ossa, Moduli Space of Calabi-Yau Manifolds, Nucl. Phys. B 355 (1991) 455 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L.B. Anderson, J. Gray and E. Sharpe, Algebroids, Heterotic Moduli Spaces and the Strominger System, JHEP 07 (2014) 037 [arXiv:1402.1532] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    X. de la Ossa and E.E. Svanes, Holomorphic Bundles and the Moduli Space of N = 1 Supersymmetric Heterotic Compactifications, JHEP 10 (2014) 123 [arXiv:1402.1725] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    X. de la Ossa and E.E. Svanes, Connections, Field Redefinitions and Heterotic Supergravity, JHEP 12 (2014) 008 [arXiv:1409.3347] [INSPIRE].CrossRefGoogle Scholar
  16. [16]
    M. Garcia-Fernandez, R. Rubio and C. Tipler, Holomorphic string algebroids, arXiv:1807.10329 [INSPIRE].
  17. [17]
    A. Otal, L. Ugarte and R. Villacampa, Invariant solutions to the Strominger system and the heterotic equations of motion, Nucl. Phys. B 920 (2017) 442 [arXiv:1604.02851] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    P. Gauduchon, Hermitian connections and Dirac operators, Bollettino U.M.I 11-B 7 (1997) 257.Google Scholar
  19. [19]
    T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, Gauge Theories and Differential Geometry, Phys. Rept. 66 (1980) 213 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    K. Becker, M. Becker and J.H. Schwarz, String theory and M-theory: A modern introduction, Cambridge University Press, (2006).Google Scholar
  21. [21]
    A. Strominger, Superstrings with Torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    E.A. Bergshoeff and M. de Roo, The Quartic Effective Action of the Heterotic String and Supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    E. Bergshoeff and M. de Roo, Supersymmetric Chern-Simons Terms in Ten-dimensions, Phys. Lett. B 218 (1989) 210 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    J. McOrist, On the Effective Field Theory of Heterotic Vacua, Lett. Math. Phys. 108 (2018) 1031 [arXiv:1606.05221] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Philip Candelas
    • 1
    Email author
  • Xenia de la Ossa
    • 1
  • Jock McOrist
    • 2
  • Roberto Sisca
    • 2
  1. 1.Mathematical InstituteUniversity of OxfordOxfordU.K.
  2. 2.Department of MathematicsUniversity of SurreyGuildfordU.K.

Personalised recommendations