Abstract
We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.L. Shatashvili and C. Vafa, Superstrings and manifold of exceptional holonomy, Selecta Math.1 (1995) 347 [hep-th/9407025] [INSPIRE].
J.M. Figueroa-O’Farrill, A note on the extended superconformal algebras associated with manifolds of exceptional holonomy, Phys. Lett.B 392 (1997) 77 [hep-th/9609113] [INSPIRE].
D.D. Joyce, Compact Riemannian 7-manifolds with holonomy g2. I, J. Diff. Geom.43 (1996) 291.
D.D. Joyce, Compact Riemannian 7-manifolds with holonomy g2. II, J. Diff. Geom.43 (1996) 329.
D.D. Joyce, Compact 8-manifolds with holonomy Spin(7), Inv. Math.123 (1996) 507.
B.S. Acharya, On mirror symmetry for manifolds of exceptional holonomy, Nucl. Phys.B 524 (1998) 269 [hep-th/9707186] [INSPIRE].
B.S. Acharya, Dirichlet Joyce manifolds, discrete torsion and duality, Nucl. Phys.B 492 (1997) 591 [hep-th/9611036] [INSPIRE].
A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys.B 479 (1996) 243 [hep-th/9606040] [INSPIRE].
M.R. Gaberdiel and P. Kaste, Generalized discrete torsion and mirror symmetry for G2manifolds, JHEP08 (2004) 001 [hep-th/0401125] [INSPIRE].
W.-y. Chuang, A note on mirror symmetry for manifolds with Spin(7) holonomy, J. Phys.A 43 (2010) 235403 [hep-th/0406151] [INSPIRE].
G. Papadopoulos and P.K. Townsend, Compactification of D = 11 supergravity on spaces of exceptional holonomy, Phys. Lett.B 357 (1995) 300 [hep-th/9506150] [INSPIRE].
A. Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math.565 (2003) 125.
A. Corti, M. Haskins, J. Nordström and T. Pacini, Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds, Geom. Topol.17 (2013) 1955.
A. Corti, M. Haskins, J. Nordström and T. Pacini, G2-manifolds and associative submanifolds via semi-Fano 3-folds, Duke Math. J.164 (2015) 1971 [arXiv:1207.4470] [INSPIRE].
A.P. Braun and M. Del Zotto, Mirror symmetry for G2-manifolds: twisted connected sums and dual tops, JHEP05 (2017) 080 [arXiv:1701.05202] [INSPIRE].
A.P. Braun and M. Del Zotto, Towards generalized mirror symmetry for twisted connected sum G2manifolds, JHEP03 (2018) 082 [arXiv:1712.06571] [INSPIRE].
D. Joyce, Compact manifolds with special holonomy, Oxford mathematical monographs, Oxford University Press, Oxford U.K. (2000).
V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom.3 (1994) 493 [alg-geom/9310003] [INSPIRE].
A.P. Braun, Tops as building blocks for G2manifolds, JHEP10 (2017) 083 [arXiv:1602.03521] [INSPIRE].
P.S. Aspinwall and D.R. Morrison, String theory on K 3 surfaces, hep-th/9404151 [INSPIRE].
A.P. Braun and S. Schäfer-Nameki, Compact, singular G2-holonomy manifolds and M/Heterotic/F-theory duality, JHEP04 (2018) 126 [arXiv:1708.07215] [INSPIRE].
J. Halverson and D.R. Morrison, The landscape of M-theory compactifications on seven-manifolds with G2holonomy, JHEP04 (2015) 047 [arXiv:1412.4123] [INSPIRE].
J. Halverson and D.R. Morrison, On gauge enhancement and singular limits in G2compactifications of M-theory, JHEP04 (2016) 100 [arXiv:1507.05965] [INSPIRE].
T.C. da C. Guio, H. Jockers, A. Klemm and H.-Y. Yeh, Effective Action from M-theory on Twisted Connected Sum G2-Manifolds, Commun. Math. Phys.359 (2018) 535 [arXiv:1702.05435] [INSPIRE].
A.P. Braun et al., Infinitely many M 2-instanton corrections to M-theory on G2-manifolds, JHEP09 (2018) 077 [arXiv:1803.02343] [INSPIRE].
B.S. Acharya, A.P. Braun, E.E. Svanes and R. Valandro, Counting associatives in compact G2orbifolds, JHEP03 (2019) 138 [arXiv:1812.04008] [INSPIRE].
A.P. Braun and S. Schäfer-Nameki, Spin(7)-manifolds as generalized connected sums and 3d \( \mathcal{N} \) = 1 theories, JHEP06 (2018) 103 [arXiv:1803.10755] [INSPIRE].
M.-A. Fiset, Superconformal algebras for twisted connected sums and G2mirror symmetry, JHEP12 (2018) 011 [arXiv:1809.06376] [INSPIRE].
C. Vafa, Modular invariance and discrete torsion on orbifolds, Nucl. Phys.B 273 (1986) 592.
E. Witten, Supersymmetry and Morse theory, J. Diff. Geom.17 (1982) 661 [INSPIRE].
P.S. Aspinwall, D.R. Morrison and M. Gross, Stable singularities in string theory, Commun. Math. Phys.178 (1996) 115 [hep-th/9503208] [INSPIRE].
M. Gross, Special Lagrangian Fibrations II: geometry, math/9809072 (1998).
D. Crowley, S. Goette and J. Nordström, An analytic invariant of G2manifolds, arXiv:1505.02734 [INSPIRE].
C. Vafa and E. Witten, On orbifolds with discrete torsion, J. Geom. Phys.15 (1995) 189 [hep-th/9409188] [INSPIRE].
D. Joyce, A new construction of compact 8-manifolds with holonomy Spin(7), J. Diff. Geom.53 (1999) 89 [math/9910002] [INSPIRE].
D. Joyce and S. Karigiannis, A new construction of compact torsion-free G2-manifolds by gluing families of Eguchi-Hanson spaces, arXiv:1707.09325.
R. Gopakumar and S. Mukhi, Orbifold and orientifold compactifications of F-theory and M-theory to six-dimensions and four-dimensions, Nucl. Phys.B 479 (1996) 260 [hep-th/9607057] [INSPIRE].
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1905.01474
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Braun, A.P., Majumder, S. & Otto, A. On mirror maps for manifolds of exceptional holonomy. J. High Energ. Phys. 2019, 204 (2019). https://doi.org/10.1007/JHEP10(2019)204
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)204