The heterotic \(\operatorname{G}_2\) system on contact Calabi-Yau 7-manifolds. (English) Zbl 1530.53038
The heterotic \(G_2\) system is a system of PDEs over a \(G_2\)-manifold that includes instanton-like equations and a prescribed Chern-Simons defect. It may therefore be regarded as part of the theory of the Hull-Strominger (HS) system on 6-, 7- and 8-manifolds with special geometry, versions of which have been quite popular for a few years now. For example, A. Fino et al. [Commun. Math. Phys. 388, No. 2, 947–967 (2021; Zbl 1485.53042)] found solutions to the HS system using \(T^2\)-bundles over \(K3\) orbifolds, extending the work of J.-X. Fu and S.-T. Yau [J. Differ. Geom. 78, No. 3, 369–428 (2008; Zbl 1141.53036)]. The appearance of the HS system in the mathematics literature goes back to M. Fernández et al. [Adv. Theor. Math. Phys. 15, No. 2, 245–284 (2011; Zbl 1261.81091)] where partial explicit solutions were found on nilmanifolds.
The present paper’s approach to the heterotic \(G_2\) system follows the work of X. de la Ossa et al. [J. High Energy Phys. 2016, No. 11, Paper No. 16, 47 p. (2016; Zbl 1390.81475); Commun. Math. Phys. 360, No. 2, 727–775 (2018; Zbl 1395.58015); in: Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 2. Oxford: Oxford University Press. 503–517 (2018; Zbl 1420.81022), pp. 503–517].
The focus is on contact Calabi-Yau 7-manifolds [G. Habib and L. Vezzoni, Differ. Geom. Appl. 41, 12–32 (2015; Zbl 1348.53032)], which are Sasakian total spaces of circle bundles over CY 3-orbifolds and carry standard cocalibrated \(G_2\)-structures. The gauge theory of such was studied in [O. Calvo-Andrade et al., Rev. Mat. Iberoam. 36, No. 6, 1753–1778 (2020; Zbl 1462.53014)] and [L. Portilla and H. N. Sá Earp, Q. J. Math. 74, No. 3, 1027–1083 (2023)].
The authors produce and study in detail approximate solutions to the heterotic \(G_2\) system over compact 2-connected contact Calabi-Yau 7-manifolds. ‘Approximate’ means the difference to an exact solution can be made as small as one wants by shrinking the fibres’ size (the string constant \(\alpha\)). Each approximate structure carries with it the relevant fields (non-trivial scalar field, constant dilaton and \(H\)-flux with non-trivial Chern-Simons defect). Each one also yields a family of connections that obey the heterotic Bianchi identity and are \(G_2\) instantons up to quadratic corrections in \(\alpha\).
The presentation is extremely clear and will be of use to geometers and physicists alike.
The present paper’s approach to the heterotic \(G_2\) system follows the work of X. de la Ossa et al. [J. High Energy Phys. 2016, No. 11, Paper No. 16, 47 p. (2016; Zbl 1390.81475); Commun. Math. Phys. 360, No. 2, 727–775 (2018; Zbl 1395.58015); in: Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 2. Oxford: Oxford University Press. 503–517 (2018; Zbl 1420.81022), pp. 503–517].
The focus is on contact Calabi-Yau 7-manifolds [G. Habib and L. Vezzoni, Differ. Geom. Appl. 41, 12–32 (2015; Zbl 1348.53032)], which are Sasakian total spaces of circle bundles over CY 3-orbifolds and carry standard cocalibrated \(G_2\)-structures. The gauge theory of such was studied in [O. Calvo-Andrade et al., Rev. Mat. Iberoam. 36, No. 6, 1753–1778 (2020; Zbl 1462.53014)] and [L. Portilla and H. N. Sá Earp, Q. J. Math. 74, No. 3, 1027–1083 (2023)].
The authors produce and study in detail approximate solutions to the heterotic \(G_2\) system over compact 2-connected contact Calabi-Yau 7-manifolds. ‘Approximate’ means the difference to an exact solution can be made as small as one wants by shrinking the fibres’ size (the string constant \(\alpha\)). Each approximate structure carries with it the relevant fields (non-trivial scalar field, constant dilaton and \(H\)-flux with non-trivial Chern-Simons defect). Each one also yields a family of connections that obey the heterotic Bianchi identity and are \(G_2\) instantons up to quadratic corrections in \(\alpha\).
The presentation is extremely clear and will be of use to geometers and physicists alike.
Reviewer: Simon Chiossi (Niteroi)
MSC:
| 53C10 | \(G\)-structures |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 81T45 | Topological field theories in quantum mechanics |
Citations:
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