Abstract
A geometric flow on (2, 2)-forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established, using Hamilton’s version of the Nash–Moser implicit function theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The usual balanced condition for a Hermitian metric \(\omega \) in dimension n is \(d\omega ^{n-1}=0\). For simplicity, we use the same terminology for the slight modification used in (2.5).
References
Andreas, B., Garcia-Fernandez, M.: Heterotic non-Kahler geometries via polystable bundles on Calabi–Yau threefolds. J. Geom. Phys. 62(2), 183–188 (2012)
Andreas, B., Garcia-Fernandez, M.: Solutions of the Strominger system via stable bundles on Calabi–Yau threefolds. Commun. Math. Phys. 315, 153–168 (2012)
Becker, K., Becker, M., Fu, J.X., Tseng, L.S., Yau, S.T.: Anomaly cancellation and smooth non-Kahler solutions in heterotic string theory. Nucl. Phys. B 751(1), 108–128 (2006)
Carlevaro, L., Israel, D.: Heterotic resolved conifolds with torsion, from supergravity to CFT. J. High Energy Phys. 1, 1–57 (2010)
Dasgupta, K., Rajesh, G., Sethi, S.S.: M-theory, orientifolds and G-flux. J. High Energy Phys. 1999(08), 023 (1999)
Donaldson, S.: Infinite determinants, stable bundles, and curvature. Duke Math. J. 54, 231–247 (1987)
Fei, T.: A construction of non-Kähler Calabi–Yau manifolds and new solutions to the Strominger system. Adv. Math. 302, 529–550 (2016)
Fei, T.: Some torsional local models of heterotic strings, preprint, arXiv:1508.05566
Fei, T., Yau, S.T.: Invariant solutions to the Strominger system on complex Lie groups and their quotients. Commun. Math. Phys. 338(3), 1–13 (2015)
Fernandez, M., Ivanov, S., Ugarte, L., Vassilev, D.: Non-Kahler heterotic string solutions with non-zero fluxes and non-constant dilaton. J. High Energy Phys. 6, 1–23 (2014)
Fernandez, M., Ivanov, S., Ugarte, L., Villacampa, R.: Non-Kahler heterotic string compactifications with non-zero fluxes and constant dilaton. Commun. Math. Phys. 288, 677–697 (2009)
Fu, J.X., Tseng, L.S., Yau, S.T.: Local heterotic torsional models. Commun. Math. Phys. 289, 1151–1169 (2009)
Fu, J.X., Wang, Z.Z., Wu, D.M.: Form-type equations on Kähler manifolds of nonnegative orthogonal bisectional curvature. Calc. Var. Partial Differ. Equ. 52(1–2), 327–344 (2015)
Fu, J.X., Yau, S.T.: The theory of superstring with flux on non-Kahler manifolds and the complex Monge–Ampere equation. J. Differ. Geom. 78(3), 369–428 (2008)
Fu, J.X., Yau, S.T.: A Monge–Ampère type equation motivated by string theory. Commun. Anal. Geom. 15(1), 29–76 (2007)
Goldstein, E., Prokushkin, S.: Geometric model for complex non-Kähler manifolds with SU(3) structure. Commun. Math. Phys. 251(1), 65–78 (2004)
Grantcharov, G.: Geometry of compact complex homogeneous spaces with vanishing first Chern class. Adv. Math. 226, 3136–3159 (2011)
Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)
Hull, C.: Superstring compactifications with rorsion and space-time supersymmetry. In: D’Auria, R., Fre, P. (eds.) Proceedings of the First Torino Meeting on Superunification and Extra Dimensions. World Scientific, Singapore (1986)
Hull, C.: Compactifications of the heterotic superstring. Phys. Lett. B 178(4), 357–365 (1986)
Li, J., Yau, S.T.: The existence of supersymmetric string theorey with torsion. J. Differ. Geom. 70(1), 143–181 (2005)
Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 149(3–4), 261–295 (1982)
Otal, A., Ugarte, L., Villacampa, R.: Invariant solutions to the Strominger system and the heterotic equations of motion on solvmanifolds, preprint arXiv:1604.02851
Phong, D.H., Picard, S., Zhang, X.W.: On estimates for the Fu–Yau generalization of a Strominger system. J. Reine Angew. Math, arXiv:1507.08193 (to appear)
Phong, D.H., Picard, S., Zhang, X.W.: The Fu–Yau equation with negative slope parameter. Invent. Math, arXiv:1602.08838 (to appear)
Phong, D.H., Picard, S., Zhang, X.W.: Anomaly flows, arXiv:1610.02739
Phong, D.H., Picard, S., Zhang, X.W.: The anomaly flow and the Fu–Yau equation, arXiv:1610.02740
Popovici, D.: Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds. Bulletin de la Société Mathématique de France 143(4), 763–800
Siu, Y.-T.: Lectures on Hermitian–Einstein metrics for stable bundles and Kähler- Einstein metrics, DMV Seminar, 8. Birkhäuser Verlag, Basel (1987)
Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274(2), 253–284 (1986)
Tosatti, V., Weinkove, B.: The Monge–Ampère equation for \((n-1)\)-plurisubharmonic functions on a compact Kähler manifold. J. Am. Math. Soc. 30(2), 311–346 (2017)
Ugarte, L., Villacampa, R.: Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry. Asian J. Math. 18(2), 229–246 (2014)
Uhlenbeck, K., Yau, S.T.: On the existence of Hermitian Yang–Mills connections on stable vector bundles. Commun. Pure Appl. Math. 39, 257–293 (1986)
Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
Work supported in part by the National Science Foundation under Grant DMS-12-66033 and DMS-1308136.
Rights and permissions
About this article
Cite this article
Phong, D.H., Picard, S. & Zhang, X. Geometric flows and Strominger systems. Math. Z. 288, 101–113 (2018). https://doi.org/10.1007/s00209-017-1879-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-017-1879-y