Abstract
M-theory compactified on G2-holonomy manifolds results in 4d \( \mathcal{N} \) = 1 supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle M3. We derive the BPS equations and find the massless spectrum for both abelian and non-abelian gauge groups in 4d. The mathematical tool that allows us to determine the spectrum is Morse theory, and more generally Morse-Bott theory. The latter generalization allows us to make contact with twisted connected sum (TCS) G2-manifolds, which form the largest class of examples of compact G2-manifolds. M-theory on TCS G2-manifolds is known to result in a non-chiral 4d spectrum. We determine the Higgs bundle for this class of G2-manifolds and provide a prescription for how to engineer singular transitions to models that have chiral matter in 4d.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.J. Heckman, D.R. Morrison and C. Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic Classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].
R. Donagi and M. Wijnholt, Model Building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].
A.P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles, Nucl. Phys. B 856 (2012) 129 [arXiv:1107.5337] [INSPIRE].
J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and Spectral Covers from Resolved Calabi-Yau’s, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].
S. Krause, C. Mayrhofer and T. Weigand, G 4 flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].
T.W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].
B.S. Acharya and S. Gukov, M theory and singularities of exceptional holonomy manifolds, Phys. Rept. 392 (2004) 121 [hep-th/0409191] [INSPIRE].
B.S. Acharya, N = 1 heterotic/M theory duality and Joyce manifolds, Nucl. Phys. B 475 (1996) 579 [hep-th/9603033] [INSPIRE].
B.S. Acharya, On realizing N = 1 superYang-Mills in M-theory, hep-th/0011089 [INSPIRE].
M. Atiyah, J.M. Maldacena and C. Vafa, An M-theory flop as a large N duality, J. Math. Phys. 42 (2001) 3209 [hep-th/0011256] [INSPIRE].
M. Atiyah and E. Witten, M theory dynamics on a manifold of G 2 holonomy, Adv. Theor. Math. Phys. 6 (2003) 1 [hep-th/0107177] [INSPIRE].
E. Witten, Anomaly cancellation on G 2 manifolds, hep-th/0108165 [INSPIRE].
B.S. Acharya and E. Witten, Chiral fermions from manifolds of G 2 holonomy, hep-th/0109152 [INSPIRE].
D.D. Joyce, Compact riemannian 7-manifolds with holonomy g 2 . I, J. Diff. Geom. 43 (1996) 291.
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, G2 -manifolds and associative submanifolds via semi-Fano 3-folds, Duke Math. J. 164 (2015) 1971 [arXiv:1207.4470] [INSPIRE].
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.
J. Halverson and D.R. Morrison, The landscape of M-theory compactifications on seven-manifolds with G 2 holonomy, JHEP 04 (2015) 047 [arXiv:1412.4123] [INSPIRE].
J. Halverson and D.R. Morrison, On gauge enhancement and singular limits in G 2 compactifications of M-theory, JHEP 04 (2016) 100 [arXiv:1507.05965] [INSPIRE].
A.P. Braun, Tops as building blocks for G 2 manifolds, JHEP 10 (2017) 083 [arXiv:1602.03521] [INSPIRE].
T.C. da C. Guio, H. Jockers, A. Klemm and H.-Y. Yeh, Effective Action from M-theory on Twisted Connected Sum G 2 -Manifolds, Commun. Math. Phys. 359 (2018) 535 [arXiv:1702.05435] [INSPIRE].
A.P. Braun and M. Del Zotto, Mirror Symmetry for G 2 -Manifolds: Twisted Connected Sums and Dual Tops, JHEP 05 (2017) 080 [arXiv:1701.05202] [INSPIRE].
A.P. Braun and S. Schäfer-Nameki, Compact, Singular G 2 -Holonomy Manifolds and M/Heterotic/F-Theory Duality, JHEP 04 (2018) 126 [arXiv:1708.07215] [INSPIRE].
A.P. Braun and M. Del Zotto, Towards Generalized Mirror Symmetry for Twisted Connected Sum G 2 Manifolds, JHEP 03 (2018) 082 [arXiv:1712.06571] [INSPIRE].
A.P. Braun, M. Del Zotto, J. Halverson, M. Larfors, D.R. Morrison and S. Schäfer-Nameki, Infinitely many M2-instanton corrections to M-theory on G 2 -manifolds, JHEP 09 (2018) 077 [arXiv:1803.02343] [INSPIRE].
M.-A. Fiset, Superconformal algebras for twisted connected sums and G 2 mirror symmetry, JHEP 12 (2018) 011 [arXiv:1809.06376] [INSPIRE].
B.S. Acharya, A.P. Braun, E.E. Svanes and R. Valandro, Counting Associatives in Compact G 2 Orbifolds, arXiv:1812.04008 [INSPIRE].
T. Pantev and M. Wijnholt, Hitchin’s Equations and M-theory Phenomenology, J. Geom. Phys. 61 (2011) 1223 [arXiv:0905.1968] [INSPIRE].
C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory — I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].
H. Hayashi, R. Tatar, Y. Toda, T. Watari and M. Yamazaki, New Aspects of Heterotic-F Theory Duality, Nucl. Phys. B 806 (2009) 224 [arXiv:0805.1057] [INSPIRE].
C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory — II: Experimental Predictions, JHEP 01 (2009) 059 [arXiv:0806.0102] [INSPIRE].
H. Hayashi, T. Kawano, R. Tatar and T. Watari, Codimension-3 Singularities and Yukawa Couplings in F-theory, Nucl. Phys. B 823 (2009) 47 [arXiv:0901.4941] [INSPIRE].
R. Donagi and M. Wijnholt, Higgs Bundles and UV Completion in F-theory, Commun. Math. Phys. 326 (2014) 287 [arXiv:0904.1218] [INSPIRE].
J. Marsano, N. Saulina and S. Schäfer-Nameki, Monodromies, Fluxes and Compact Three-Generation F-theory GUTs, JHEP 08 (2009) 046 [arXiv:0906.4672] [INSPIRE].
R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl. Phys. B 829 (2010) 325 [arXiv:0908.1784] [INSPIRE].
J. Marsano, N. Saulina and S. Schäfer-Nameki, Compact F-theory GUTs with U(1) (PQ), JHEP 04 (2010) 095 [arXiv:0912.0272] [INSPIRE].
H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, More on Dimension-4 Proton Decay Problem in F-theory — Spectral Surface, Discriminant Locus and Monodromy, Nucl. Phys. B 840 (2010) 304 [arXiv:1004.3870] [INSPIRE].
E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].
G. Chen, G 2 manifolds with nodal singularities along circles, arXiv:1809.02563.
J.J. Heckman, C. Lawrie, L. Lin and G. Zoccarato, F-theory and Dark Energy, arXiv:1811.01959 [INSPIRE].
P.B. Kronheimer, The construction of ALE spaces as hyper-Kählerquotients, J. Diff. Geom. 29 (1989) 665 [INSPIRE].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
E. Witten, Deconstruction, G 2 holonomy and doublet triplet splitting, in Supersymmetry and unification of fundamental interactions. Proceedings, 10th International Conference, SUSY’02, Hamburg, Germany, June 17–23, 2002, pp. 472–491, hep-ph/0201018 [INSPIRE].
K.C. Chang and J. Liu, A cohomology complex for manifolds with boundary, Topol. Methods Nonlinear Anal. 5 (1995) 325.
C. Godbillon, Éléments de topologie algébrique, Hermann, (1971).
K. Hori et al., Mirror symmetry, vol. 1 of Clay Mathematics Monographs, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, U.S.A., (2003).
J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells, Ann. Math. Stud., No. 51, Princeton University Press, Princeton, U.S.A., (1963).
D. Gaiotto, G.W. Moore and E. Witten, Algebra of the Infrared: String Field Theoretic Structures in Massive \( \mathcal{N} \) = (2, 2) Field Theory In Two Dimensions, arXiv:1506.04087 [INSPIRE].
M. Farber, Topology of closed one-forms, No. 108 in Mathematical surveys and monographs, American Mathematical Society, Providence, RI, U.S.A., (2004).
J.A. Harvey and G.W. Moore, Superpotentials and membrane instantons, hep-th/9907026 [INSPIRE].
C. Beasley and E. Witten, Residues and world sheet instantons, JHEP 10 (2003) 065 [hep-th/0304115] [INSPIRE].
D.M. Austin and P.J. Braam, Morse-Bott theory and equivariant cohomology, in The Floer memorial volume, Progr. Math. 133 (1995) 123.
K. Fukaya, Morse homotopy and its quantization, Geom. Topol. (1997).
M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
W. Lerche and N.P. Warner, Exceptional SW geometry from ALE fibrations, Phys. Lett. B 423 (1998) 79 [hep-th/9608183] [INSPIRE].
M. Billó et al., The rigid limit in special Kähler geometry: From K3 fibrations to special Riemann surfaces: A detailed case study, Class. Quant. Grav. 15 (1998) 2083 [hep-th/9803228] [INSPIRE].
P.B. Kronheimer, A Torelli type theorem for gravitational instantons, J. Diff. Geom. 29 (1989) 685 [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].
S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-Branes and Monodromy, JHEP 07 (2011) 030 [arXiv:1010.5780] [INSPIRE].
M. Cvetič and L. Lin, TASI Lectures on Abelian and Discrete Symmetries in F-theory, PoS(TASI2017)020 (2018) [arXiv:1809.00012] [INSPIRE].
R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].
A. Van Proeyen, Tools for supersymmetry, Ann. U. Craiova Phys. 9 (1999) 1 [hep-th/9910030] [INSPIRE].
G. Schwarz, Hodge decomposition — a method for solving boundary value problems, vol. 1607 of Lect. Notes Math., Springer-Verlag, Berlin, (1995), [https://doi.org/10.1007/BFb0095978].
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: 1812.06072
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., Cizel, S., Hübner, M. et al. Higgs bundles for M-theory on G2-manifolds. J. High Energ. Phys. 2019, 199 (2019). https://doi.org/10.1007/JHEP03(2019)199
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2019)199