Abstract
We consider the non-perturbative superpotential for a class of four-dimensional \( \mathcal{N}=1 \) vacua obtained from M-theory on seven-manifolds with holonomy G2. The class of G2-holonomy manifolds we consider are so-called twisted connected sum (TCS) constructions, which have the topology of a K3-fibration over S3. We show that the non-perturbative superpotential of M-theory on a class of TCS geometries receives infinitely many inequivalent M2-instanton contributions from infinitely many three-spheres, which we conjecture are supersymmetric (and thus associative) cycles. The rationale for our construction is provided by the duality chain of [1], which relates M-theory on TCS G2-manifolds to E8 × E8 heterotic backgrounds on the Schoen Calabi-Yau threefold, as well as to F-theory on a K3-fibered Calabi-Yau fourfold. The latter are known to have an infinite number of instanton corrections to the superpotential and it is these contributions that we trace through the duality chain back to the G2-compactification.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
D.R. Morrison, What is F-theory?, to appear.
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.
R. Donagi, A. Grassi and E. Witten, A nonperturbative superpotential with E 8 symmetry, Mod. Phys. Lett. A 11 (1996) 2199 [hep-th/9607091] [INSPIRE].
S. Sethi, C. Vafa and E. Witten, Constraints on low dimensional string compactifications, Nucl. Phys. B 480 (1996) 213 [hep-th/9606122] [INSPIRE].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477] [hep-th/9906070] [INSPIRE].
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].
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 M. Del Zotto, Towards generalized mirror symmetry for twisted connected sum G 2 manifolds, JHEP 03 (2018) 082 [arXiv:1712.06571] [INSPIRE].
M. Dine, N. Seiberg, X.G. Wen and E. Witten, Nonperturbative effects on the string world sheet, Nucl. Phys. B 278 (1986) 769 [INSPIRE].
M. Dine, N. Seiberg, X.G. Wen and E. Witten, Nonperturbative effects on the string world sheet. 2., Nucl. Phys. B 289 (1987) 319 [INSPIRE].
J. Distler, Resurrecting (2, 0) compactifications, Phys. Lett. B 188 (1987) 431 [INSPIRE].
J. Distler and B.R. Greene, Aspects of (2, 0) string compactifications, Nucl. Phys. B 304 (1988) 1 [INSPIRE].
E. Silverstein and E. Witten, Criteria for conformal invariance of (0, 2) models, Nucl. Phys. B 444 (1995) 161 [hep-th/9503212] [INSPIRE].
C. Beasley and E. Witten, Residues and world sheet instantons, JHEP 10 (2003) 065 [hep-th/0304115] [INSPIRE].
E.I. Buchbinder and B.A. Ovrut, Non-vanishing superpotentials in heterotic string theory and discrete torsion, JHEP 01 (2017) 038 [arXiv:1611.01922] [INSPIRE].
E. Buchbinder, A. Lukas, B. Ovrut and F. Ruehle, Heterotic instanton superpotentials from complete intersection Calabi-Yau manifolds, JHEP 10 (2017) 032 [arXiv:1707.07214] [INSPIRE].
G. Curio and D. Lüst, A class of N = 1 dual string pairs and its modular superpotential, Int. J. Mod. Phys. A 12 (1997) 5847 [hep-th/9703007] [INSPIRE].
J.A. Harvey and G.W. Moore, Superpotentials and membrane instantons, hep-th/9907026 [INSPIRE].
R.C. Mclean, Deformations of calibrated submanifolds, Commun. Analy. Geom. 6 (1996) 705.
O.J. Ganor, A note on zeros of superpotentials in F-theory, Nucl. Phys. B 499 (1997) 55 [hep-th/9612077] [INSPIRE].
A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].
S. Gukov, S.-T. Yau and E. Zaslow, Duality and fibrations on G 2 manifolds, hep-th/0203217 [INSPIRE].
D.R. Morrison, Half K3 surfaces, talk given at Strings 2002, July 15-20, Cambridge, U.K. (2002).
S. Donaldson, Adiabatic limits of co-associative Kovalev Lefschetz fibrations, in Algebra, geometry, and physics in the 21st century, D. Auroux et al. eds., Progress in Mathematics volume 324, Birkhauser, Germany (2017).
C. Schoen, On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988) 177.
P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].
L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY threefolds, JHEP 10 (2017) 077 [arXiv:1708.07907] [INSPIRE].
E. Witten, Nonperturbative superpotentials in string theory, Nucl. Phys. B 474 (1996) 343 [hep-th/9604030] [INSPIRE].
L. Martucci, Warping the Kähler potential of F-theory/IIB flux compactifications, JHEP 03 (2015) 067 [arXiv:1411.2623] [INSPIRE].
L. Martucci, Warped Kähler potentials and fluxes, JHEP 01 (2017) 056 [arXiv:1610.02403] [INSPIRE].
L.B. Anderson et al., Instanton superpotentials, Calabi-Yau geometry and fibrations, Phys. Rev. D 93 (2016) 086001 [arXiv:1511.05188] [INSPIRE].
E. Witten, World sheet corrections via D instantons, JHEP 02 (2000) 030 [hep-th/9907041] [INSPIRE].
M.-H. Saito, Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry, in The arithmetic and geometry of algebraic cycles, B.B. Gordon et al. eds., NATO Sci. Ser. C Math. Phys. Sci. volume 548, Kluwer Academic Publishers, Dordrecht, The Netherland 2000
M. Cvetič, R. Donagi, J. Halverson and J. Marsano, On seven-brane dependent instanton prefactors in F-theory, JHEP 11 (2012) 004 [arXiv:1209.4906] [INSPIRE].
B. Andreas and G. Curio, Three-branes and five-branes in N = 1 dual string pairs, Phys. Lett. B 417 (1998) 41 [hep-th/9706093] [INSPIRE].
M.R. Gaberdiel and B. Zwiebach, Exceptional groups from open strings, Nucl. Phys. B 518 (1998) 151 [hep-th/9709013] [INSPIRE].
O. DeWolfe and B. Zwiebach, String junctions for arbitrary Lie algebra representations, Nucl. Phys. B 541 (1999) 509 [hep-th/9804210] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Matter from geometry without resolution, JHEP 10 (2013) 205 [arXiv:1306.1832] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Non-abelian gauge symmetry and the Higgs mechanism in F-theory, Commun. Math. Phys. 336 (2015) 1231 [arXiv:1402.5962] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Geometry and topology of string junctions, arXiv:1410.6817 [INSPIRE].
O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach, Uncovering infinite symmetries on [p, q] 7-branes: Kac-Moody algebras and beyond, Adv. Theor. Math. Phys. 3 (1999) 1835 [hep-th/9812209] [INSPIRE].
G. Lopes Cardoso, G. Curio, D. Lüst and T. Mohaupt, On the duality between the heterotic string and F-theory in eight-dimensions, Phys. Lett. B 389 (1996) 479 [hep-th/9609111] [INSPIRE].
J. McOrist, D.R. Morrison and S. Sethi, Geometries, Non-Geometries and Fluxes, Adv. Theor. Math. Phys. 14 (2010) 1515 [arXiv:1004.5447] [INSPIRE].
A.P. Braun, F. Fucito and J.F. Morales, U-folds as K3 fibrations, JHEP 10 (2013) 154 [arXiv:1308.0553] [INSPIRE].
A. Malmendier and D.R. Morrison, K3 surfaces, modular forms and non-geometric heterotic compactifications, Lett. Math. Phys. 105 (2015) 1085 [arXiv:1406.4873] [INSPIRE].
I. García-Etxebarria, D. Lüst, S. Massai and C. Mayrhofer, Ubiquity of non-geometry in heterotic compactifications, JHEP 03 (2017) 046 [arXiv:1611.10291] [INSPIRE].
P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, in Mirror Symmetry II, B. Greene and S.-T. Yau eds., International Press (1997), hep-th/9404151 [INSPIRE].
P.S. Aspinwall, K3 surfaces and string duality, in the proceedings of Fields, strings and duality. Theoretical Advanced Study Institute in Elementary Particle Physics (TASI’96), June 2-28, Boulder, U.S.A. (1996), hep-th/9611137 [INSPIRE].
N. Elkies and A. Kumar, K 3 surfaces and equations for Hilbert modular surfaces, Alg. Number Theor. 8 (2014) 2297.
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
D. Joyce, Conjectures on counting associative 3-folds in G 2 -manifolds, arXiv:1610.09836 [INSPIRE].
M. Bianchi, A. Collinucci and L. Martucci, Magnetized E3-brane instantons in F-theory, JHEP 12 (2011) 045 [arXiv:1107.3732] [INSPIRE].
J.J. Heckman et al., Instantons and SUSY breaking in F-theory, arXiv:0808.1286 [INSPIRE].
T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, On fluxed instantons and moduli stabilisation in IIB orientifolds and F-theory, Phys. Rev. D 84 (2011) 066001 [arXiv:1105.3193] [INSPIRE].
J. Marsano, N. Saulina and S. Schäfer-Nameki, G-flux, M 5 instantons and U(1) symmetries in F-theory, Phys. Rev. D 87 (2013) 066007 [arXiv:1107.1718] [INSPIRE].
M. Kerstan and T. Weigand, Fluxed M 5-instantons in F-theory, Nucl. Phys. B 864 (2012) 597 [arXiv:1205.4720] [INSPIRE].
L. Martucci and T. Weigand, Hidden selection rules, M 5-instantons and fluxes in F-theory, JHEP 10 (2015) 131 [arXiv:1507.06999] [INSPIRE].
E.I. Buchbinder, R. Donagi and B.A. Ovrut, Vector bundle moduli superpotentials in heterotic superstrings and M-theory, JHEP 07 (2002) 066 [hep-th/0206203] [INSPIRE].
E.I. Buchbinder, R. Donagi and B.A. Ovrut, Superpotentials for vector bundle moduli, Nucl. Phys. B 653 (2003) 400 [hep-th/0205190] [INSPIRE].
M. Cvetič, I. García Etxebarria and J. Halverson, Three looks at instantons in F-theory — New insights from anomaly inflow, string junctions and heterotic duality, JHEP 11 (2011) 101 [arXiv:1107.2388] [INSPIRE].
M. Berg, M. Haack and B. Körs, Loop corrections to volume moduli and inflation in string theory, Phys. Rev. D 71 (2005) 026005 [hep-th/0404087] [INSPIRE].
D. Baumann et al., On D3-brane potentials in compactifications with fluxes and wrapped D-branes, JHEP 11 (2006) 031 [hep-th/0607050] [INSPIRE].
R. Blumenhagen, M. Cvetič and T. Weigand, Spacetime instanton corrections in 4D string vacua: the seesaw mechanism for D-brane models, Nucl. Phys. B 771 (2007) 113 [hep-th/0609191] [INSPIRE].
L.E. Ibáñez and A.M. Uranga, Neutrino Majorana masses from string theory instanton effects, JHEP 03 (2007) 052 [hep-th/0609213] [INSPIRE].
B. Florea, S. Kachru, J. McGreevy and N. Saulina, Stringy instantons and quiver gauge theories, JHEP 05 (2007) 024 [hep-th/0610003] [INSPIRE].
A.P. Braun et al., The Hodge numbers of divisors of Calabi-Yau threefold hypersurfaces, arXiv:1712.04946 [INSPIRE].
B.S. Acharya, M theory, Joyce orbifolds and super-Yang-Mills, Adv. Theor. Math. Phys. 3 (1999) 227 [hep-th/9812205] [INSPIRE].
B.S. Acharya, On realizing N = 1 super-Yang-Mills in M-theory, hep-th/0011089 [INSPIRE].
J. Eckhard, S. Schäfer-Nameki and J.-M. Wong, An \( \mathcal{N}=1 \) 3d-3d correspondence, JHEP 07 (2018) 052 [arXiv:1804.02368] [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: 1803.02343
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., Del Zotto, M., Halverson, J. et al. Infinitely many M2-instanton corrections to M-theory on G2-manifolds. J. High Energ. Phys. 2018, 77 (2018). https://doi.org/10.1007/JHEP09(2018)077
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2018)077