Abstract
M-theory on local G2-manifolds engineers 4d minimally supersymmetric gauge theories. We consider ALE-fibered G2-manifolds and study the 4d physics from the view point of a partially twisted 7d supersymmetric Yang-Mills theory and its Higgs bundle. Euclidean M2-brane instantons descend to non-perturbative effects of the 7d supersymmetric Yang-Mills theory, which are found to be in one to one correspondence with the instantons of a colored supersymmetric quantum mechanics. We compute the contributions of M2-brane instantons to the 4d superpotential in the effective 7d description via localization in the colored quantum mechanics. Further we consider non-split Higgs bundles and analyze their 4d spectrum.
Article PDF
Similar content being viewed by others
References
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, M theory, Joyce orbifolds and superYang-Mills, Adv. Theor. Math. Phys. 3 (1999) 227 [hep-th/9812205] [INSPIRE].
B.S. Acharya, On Realizing N = 1 superYang-Mills in M-theory, hep-th/0011089 [INSPIRE].
E. Witten, Anomaly cancellation on G2 manifolds, hep-th/0108165 [INSPIRE].
M. Atiyah and E. Witten, M theory dynamics on a manifold of G2 holonomy, Adv. Theor. Math. Phys. 6 (2003) 1 [hep-th/0107177] [INSPIRE].
B.S. Acharya and E. Witten, Chiral fermions from manifolds of G2 holonomy, hep-th/0109152 [INSPIRE].
B.S. Acharya and S. Gukov, M theory and singularities of exceptional holonomy manifolds, Phys. Rept. 392 (2004) 121 [hep-th/0409191] [INSPIRE].
A. Kennon, G2-Manifolds and M-theory Compactifications, arXiv:1810.12659 [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.
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].
D. Joyce and S. Karigiannis, A new construction of compact torsion-free g2-manifolds by gluing families of eguchi-hanson spaces, J. Diff. Geom. 117 (2021) 255 [arXiv:1707.09325].
H. Sa Earp and T. Walpuski, G2-instantons on twisted connected sums, arXiv:1310.7933 [INSPIRE].
J. Halverson and D.R. Morrison, The landscape of M-theory compactifications on seven-manifolds with G2 holonomy, JHEP 04 (2015) 047 [arXiv:1412.4123] [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 and M. Del Zotto, Mirror Symmetry for G2-Manifolds: Twisted Connected Sums and Dual Tops, JHEP 05 (2017) 080 [arXiv:1701.05202] [INSPIRE].
A.P. Braun and S. Schäfer-Nameki, Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality, JHEP 04 (2018) 126 [arXiv:1708.07215] [INSPIRE].
M.-A. Fiset, Superconformal algebras for twisted connected sums and G2 mirror symmetry, JHEP 12 (2018) 011 [arXiv:1809.06376] [INSPIRE].
A.P. Braun, S. Cizel, M. Hübner and S. Schäfer-Nameki, Higgs bundles for M-theory on G2-manifolds, JHEP 03 (2019) 199 [arXiv:1812.06072] [INSPIRE].
F. Xu, On TCS G2 manifolds and 4D emergent strings, JHEP 10 (2020) 045 [arXiv:2006.02350] [INSPIRE].
M. Cvetič, J.J. Heckman, T.B. Rochais, E. Torres and G. Zoccarato, Geometric unification of Higgs bundle vacua, Phys. Rev. D 102 (2020) 106012 [arXiv:2003.13682] [INSPIRE].
T. Pantev and M. Wijnholt, Hitchin’s Equations and M-theory Phenomenology, J. Geom. Phys. 61 (2011) 1223 [arXiv:0905.1968] [INSPIRE].
R. Barbosa, M. Cvetič, J.J. Heckman, C. Lawrie, E. Torres and G. Zoccarato, T-branes and G2 backgrounds, Phys. Rev. D 101 (2020) 026015 [arXiv:1906.02212] [INSPIRE].
R.Y. Donagi, Spectral covers, alg-geom/9505009.
R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].
R. Donagi and M. Wijnholt, Model Building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [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].
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, Monodromies, Fluxes, and Compact Three-Generation F-theory GUTs, JHEP 08 (2009) 046 [arXiv:0906.4672] [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].
R. Donagi and M. Wijnholt, Higgs Bundles and UV Completion in F-theory, Commun. Math. Phys. 326 (2014) 287 [arXiv:0904.1218] [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].
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].
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].
P. Berglund and A. Brandhuber, Matter from G2 manifolds, Nucl. Phys. B 641 (2002) 351 [hep-th/0205184] [INSPIRE].
M. Aganagic and C. Vafa, G2 manifolds, mirror symmetry and geometric engineering, hep-th/0110171 [INSPIRE].
F. Cachazo, K.A. Intriligator and C. Vafa, A Large N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3 [hep-th/0103067] [INSPIRE].
G. Curio, Superpotentials for M-theory on a G2 holonomy manifold and triality symmetry, JHEP 03 (2003) 024 [hep-th/0212211] [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].
B.S. Acharya and C. Vafa, On domain walls of N = 1 supersymmetric Yang-Mills in four-dimensions, hep-th/0103011 [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].
J.A. Harvey and G.W. Moore, Superpotentials and membrane instantons, hep-th/9907026 [INSPIRE].
E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].
N.J. Hitchin, The self-duality equations on a riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].
D. Joyce, Compact Manifolds with Special Holonomy, Oxford mathematical monographs, Oxford University Press (2000).
D.D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford graduate texts in mathematics, 12, Oxford University Press, Oxford (2007).
H.-J. Chung, T. Dimofte, S. Gukov and P. Sułkowski, 3d-3d Correspondence Revisited, JHEP 04 (2016) 140 [arXiv:1405.3663] [INSPIRE].
R.P. Thomas and S.-T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Commun. Anal. Geom. 10 (2002) 1075 [math/0104197] [INSPIRE].
J. Erdmenger, Z. Guralnik, R. Helling and I. Kirsch, A World volume perspective on the recombination of intersecting branes, JHEP 04 (2004) 064 [hep-th/0309043] [INSPIRE].
H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M5 branes, Phys. Rev. D 94 (2016) 065012 [arXiv:1509.00847] [INSPIRE].
Y. Wang and D. Xie, Codimension-two defects and Argyres-Douglas theories from outer-automorphism twist in 6d (2, 0) theories, Phys. Rev. D 100 (2019) 025001 [arXiv:1805.08839] [INSPIRE].
S. Cecotti, M. Del Zotto and S. Giacomelli, More on the N = 2 superconformal systems of type Dp(G), JHEP 04 (2013) 153 [arXiv:1303.3149] [INSPIRE].
W. Lickorish, An Introduction to Knot Theory, Graduate texts in mathematics, Springer (1997) [DOI].
A.P. Braun, Tops as building blocks for G2 manifolds, JHEP 10 (2017) 083 [arXiv:1602.03521] [INSPIRE].
A.P. Braun, M-Theory and Orientifolds, JHEP 09 (2020) 065 [arXiv:1912.06072] [INSPIRE].
A.P. Braun and M. Del Zotto, Towards Generalized Mirror Symmetry for Twisted Connected Sum G2 Manifolds, JHEP 03 (2018) 082 [arXiv:1712.06571] [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].
C. Livingston and A.H. Moore, KnotInfo: table of Knot Invariants, http://www.indiana.edu/~knotinfo.
The Knot Atlas, http://katlas.org/.
S. Cecotti, C. Cordova and C. Vafa, Braids, Walls, and Mirrors, arXiv:1110.2115 [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational Anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
R.H. Rietdijk, Applications of supersymmetric quantum mechanics, Ph.D. Thesis, Amsterdam U. (1992).
J.J. Heckman, C. Lawrie, L. Lin and G. Zoccarato, F-theory and Dark Energy, Fortsch. Phys. 67 (2019) 1900057 [arXiv:1811.01959] [INSPIRE].
K. Hori et al., Mirror symmetry, vol. 1 of Clay Mathematics Monographs, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2003).
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].
F. Guedira and A. Lichnerowicz, Géométrie des algèbres de Lie locales de Kirilov, J. Math. Pures et Appl. 63 (1984) 407.
S.P. Novikov, Multivalued functions and functionals. an analogue of the morse theory, Dokl. Akad. Nauk SSSR 260 (1981) 31.
M. Farber, Topology of closed one-forms, Mathematical surveys and monographs, No. 108, American Mathematical Society, Providence, RI, January (2004).
K. Fukaya, Morse homotopy and its quantization, in Geometric topology, pp. 409–440, Amer. Math. Soc., Providence, RI (1997).
R. Barbosa, Deformations of g2-structures, string dualities and flat higgs bundles, Ph.D. Thesis, University of Pennsylvania (2019) [https://repository.upenn.edu/edissertations/3279].
R. Barbosa, Harmonic Higgs Bundles and Coassociative ALE Fibrations, arXiv:1910.10742 [INSPIRE].
X.-G. Wen and E. Witten, Electric and Magnetic Charges in Superstring Models, Nucl. Phys. B 261 (1985) 651 [INSPIRE].
D.S. Freed, Determinants, Torsion, and Strings, Commun. Math. Phys. 107 (1986) 483 [INSPIRE].
F. Marchesano, D6-branes and torsion, JHEP 05 (2006) 019 [hep-th/0603210] [INSPIRE].
P.G. Cámara, L.E. Ibáñez and F. Marchesano, RR photons, JHEP 09 (2011) 110 [arXiv:1106.0060] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].
T. Dimofte, 3d Superconformal Theories from Three-Manifolds, in New Dualities of Supersymmetric Gauge Theories, J. Teschner, ed. (2016), DOI [arXiv:1412.7129] [INSPIRE].
E. Falbel and A. Guilloux, Dimension of character varieties for 3-manifolds, arXiv:1510.0056.
M. Abouzaid and C. Manolescu, A sheaf-theoretic model for SL(2, ℂ) Floer homology, arXiv:1708.00289 [INSPIRE].
M. Farber, Topology of closed one-forms, vol. 108 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI (2004) [DOI].
S. Haller and T. Rybicki, On the group of diffeomorphisms preserving a locally conformal symplectic structure, Ann. Glob. Anal. Geom. 17 (1999) 475.
H. Seifert, Verschlingungsinvarianten, Sitzungsber. Preuß. Akad. Wiss. Phys.-Math. Kl. 1933 (1933) 811.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2009.07136
Rights and permissions
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.
About this article
Cite this article
Hübner, M. Local G2-manifolds, Higgs bundles and a colored quantum mechanics. J. High Energ. Phys. 2021, 2 (2021). https://doi.org/10.1007/JHEP05(2021)002
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)002