Abstract
We discuss the modular symmetry and zeros of zero-mode wave functions on two-dimensional torus T 2 and toroidal orbifolds T 2/ℤN (N = 2, 3, 4, 6) with a background homogeneous magnetic field. As is well-known, magnetic flux contributes to the index in the Atiyah-Singer index theorem. The zeros in magnetic compactifications therefore play an important role, as investigated in a series of recent papers. Focusing on the zeros and their positions, we study what type of boundary conditions must be satisfied by the zero modes after the modular transformation. The consideration in this paper justifies that the boundary conditions are common before and after the modular transformation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].
L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds. 2, Nucl. Phys. B 274 (1986) 285 [INSPIRE].
L. E. Ibáñez, H. P. Nilles and F. Quevedo, Orbifolds and Wilson Lines, Phys. Lett. B 187 (1987) 25 [INSPIRE].
L. E. Ibáñez, J. E. Kim, H. P. Nilles and F. Quevedo, Orbifold Compactifications with Three Families of SU(3) × SU(2) × U(1)n, Phys. Lett. B 191 (1987) 282 [INSPIRE].
M. Cvetič, G. Shiu and A. M. Uranga, Three family supersymmetric standard - like models from intersecting brane worlds, Phys. Rev. Lett. 87 (2001) 201801 [hep-th/0107143] [INSPIRE].
T. Kobayashi, S. Raby and R.-J. Zhang, Searching for realistic 4d string models with a Pati-Salam symmetry: Orbifold grand unified theories from heterotic string compactification on a Z6 orbifold, Nucl. Phys. B 704 (2005) 3 [hep-ph/0409098] [INSPIRE].
W. Buchmüller, K. Hamaguchi, O. Lebedev and M. Ratz, Supersymmetric standard model from the heterotic string, Phys. Rev. Lett. 96 (2006) 121602 [hep-ph/0511035] [INSPIRE].
O. Lebedev et al., A Mini-landscape of exact MSSM spectra in heterotic orbifolds, Phys. Lett. B 645 (2007) 88 [hep-th/0611095] [INSPIRE].
L. E. Ibáñez, Hierarchy of Quark - Lepton Masses in Orbifold Superstring Compactification, Phys. Lett. B 181 (1986) 269 [INSPIRE].
T. Kobayashi, H. P. Nilles, F. Ploger, S. Raby and M. Ratz, Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys. B 768 (2007) 135 [hep-ph/0611020] [INSPIRE].
A. Font, L. E. Ibáñez, H. P. Nilles and F. Quevedo, Yukawa Couplings in Degenerate Orbifolds: Towards a Realistic SU(3) × SU(2) × U(1) Superstring, Phys. Lett. B 210 (1988) 101 [Erratum ibid. 213 (1988) 564] [INSPIRE].
K.-S. Choi and J. E. Kim, Quarks and leptons from orbifolded superstring, Lect. Notes Phys. 696 (2006) 1 [INSPIRE].
A. Abouelsaood, C. G. Callan Jr., C. R. Nappi and S. A. Yost, Open Strings in Background Gauge Fields, Nucl. Phys. B 280 (1987) 599 [INSPIRE].
C. Bachas, A Way to break supersymmetry, hep-th/9503030 [INSPIRE].
R. Blumenhagen, L. Görlich, B. Körs and D. Lüst, Noncommutative compactifications of type-I strings on tori with magnetic background flux, JHEP 10 (2000) 006 [hep-th/0007024] [INSPIRE].
C. Angelantonj, I. Antoniadis, E. Dudas and A. Sagnotti, Type I strings on magnetized orbifolds and brane transmutation, Phys. Lett. B 489 (2000) 223 [hep-th/0007090] [INSPIRE].
D. Cremades, L. E. Ibáñez and F. Marchesano, Computing Yukawa couplings from magnetized extra dimensions, JHEP 05 (2004) 079 [hep-th/0404229] [INSPIRE].
T.-H. Abe, Y. Fujimoto, T. Kobayashi, T. Miura, K. Nishiwaki and M. Sakamoto, ZN twisted orbifold models with magnetic flux, JHEP 01 (2014) 065 [arXiv:1309.4925] [INSPIRE].
C. Angelantonj and A. Sagnotti, Open strings, Phys. Rept. 371 (2002) 1 [Erratum ibid. 376 (2003) 407] [hep-th/0204089] [INSPIRE].
R. Blumenhagen, M. Cvetič, P. Langacker and G. Shiu, Toward realistic intersecting D-brane models, Ann. Rev. Nucl. Part. Sci. 55 (2005) 71 [hep-th/0502005] [INSPIRE].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional String Compactifications with D-branes, Orientifolds and Fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].
L. E. Ibáñez and A. M. Uranga, String theory and particle physics: An introduction to string phenomenology, Cambridge University Press (2012).
H. Abe, K.-S. Choi, T. Kobayashi and H. Ohki, Three generation magnetized orbifold models, Nucl. Phys. B 814 (2009) 265 [arXiv:0812.3534] [INSPIRE].
T.-h. Abe et al., Classification of three-generation models on magnetized orbifolds, Nucl. Phys. B 894 (2015) 374 [arXiv:1501.02787] [INSPIRE].
H. Abe, T. Kobayashi, K. Sumita and Y. Tatsuta, Gaussian Froggatt-Nielsen mechanism on magnetized orbifolds, Phys. Rev. D 90 (2014) 105006 [arXiv:1405.5012] [INSPIRE].
Y. Fujimoto, T. Kobayashi, K. Nishiwaki, M. Sakamoto and Y. Tatsuta, Comprehensive analysis of Yukawa hierarchies on T 2/ZN with magnetic fluxes, Phys. Rev. D 94 (2016) 035031 [arXiv:1605.00140] [INSPIRE].
T. Kobayashi, K. Nishiwaki and Y. Tatsuta, CP-violating phase on magnetized toroidal orbifolds, JHEP 04 (2017) 080 [arXiv:1609.08608] [INSPIRE].
W. Buchmüller and J. Schweizer, Flavor mixings in flux compactifications, Phys. Rev. D 95 (2017) 075024 [arXiv:1701.06935] [INSPIRE].
W. Buchmüller and K. M. Patel, Flavor physics without flavor symmetries, Phys. Rev. D 97 (2018) 075019 [arXiv:1712.06862] [INSPIRE].
H. Abe, K.-S. Choi, T. Kobayashi and H. Ohki, Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane Models, Nucl. Phys. B 820 (2009) 317 [arXiv:0904.2631] [INSPIRE].
M. Berasaluce-Gonzalez, L. E. Ibáñez, P. Soler and A. M. Uranga, Discrete gauge symmetries in D-brane models, JHEP 12 (2011) 113 [arXiv:1106.4169] [INSPIRE].
F. Marchesano, D. Regalado and L. Vazquez-Mercado, Discrete flavor symmetries in D-brane models, JHEP 09 (2013) 028 [arXiv:1306.1284] [INSPIRE].
H. Abe, T. Kobayashi, H. Ohki, K. Sumita and Y. Tatsuta, Non-Abelian discrete flavor symmetries of 10D SYM theory with magnetized extra dimensions, JHEP 06 (2014) 017 [arXiv:1404.0137] [INSPIRE].
W. Buchmüller, M. Dierigl, E. Dudas and J. Schweizer, Effective field theory for magnetic compactifications, JHEP 04 (2017) 052 [arXiv:1611.03798] [INSPIRE].
D. M. Ghilencea and H. M. Lee, Wilson lines and UV sensitivity in magnetic compactifications, JHEP 06 (2017) 039 [arXiv:1703.10418] [INSPIRE].
W. Buchmüller, M. Dierigl and E. Dudas, Flux compactifications and naturalness, JHEP 08 (2018) 151 [arXiv:1804.07497] [INSPIRE].
C. S. Lim, The implication of gauge-Higgs unification for the hierarchical fermion masses, PTEP 2018 (2018) 093B02 [arXiv:1801.01639] [INSPIRE].
T. Hirose and N. Maru, Cancellation of One-loop Corrections to Scalar Masses in Yang-Mills Theory with Flux Compactification, JHEP 08 (2019) 054 [arXiv:1904.06028] [INSPIRE].
W. Buchmüller and K. M. Patel, Proton decay in flux compactifications, JHEP 05 (2019) 196 [arXiv:1904.08810] [INSPIRE].
M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Am. Math. Soc. 69 (1969) 422 [INSPIRE].
M. B. Green, J. H. Schwarz and E. Witten, Superstring theory. vol. 2: loop amplitudes, anomalies and phenomenology (1988) [INSPIRE].
E. Witten, Some Properties of O(32) Superstrings, Phys. Lett. B 149 (1984) 351 [INSPIRE].
U. Venugopalkrishna, Fredholm operators associated with strongly pseudoconvex domains in cn, J. Funct. Anal. 9 (1972) 349 .
E. J. Weinberg, Index Calculations for the Fermion-Vortex System, Phys. Rev. D 24 (1981) 2669 [INSPIRE].
W. Buchmüller, M. Dierigl, F. Ruehle and J. Schweizer, Chiral fermions and anomaly cancellation on orbifolds with Wilson lines and flux, Phys. Rev. D 92 (2015) 105031 [arXiv:1506.05771] [INSPIRE].
W. Buchmüller, M. Dierigl and Y. Tatsuta, Magnetized orbifolds and localized flux, Annals Phys. 401 (2019) 91 [arXiv:1810.06362] [INSPIRE].
M. Sakamoto, M. Takeuchi and Y. Tatsuta, Zero-mode counting formula and zeros in orbifold compactifications, Phys. Rev. D 102 (2020) 025008 [arXiv:2004.05570] [INSPIRE].
M. Sakamoto, M. Takeuchi and Y. Tatsuta, Index theorem on T 2/ℤN orbifolds, Phys. Rev. D 103 (2021) 025009 [arXiv:2010.14214] [INSPIRE].
S. Kikuchi, T. Kobayashi, S. Takada, T. H. Tatsuishi and H. Uchida, Revisiting modular symmetry in magnetized torus and orbifold compactifications, Phys. Rev. D 102 (2020) 105010 [arXiv:2005.12642] [INSPIRE].
T. Kobayashi, S. Nagamoto, S. Takada, S. Tamba and T. H. Tatsuishi, Modular symmetry and non-Abelian discrete flavor symmetries in string compactification, Phys. Rev. D 97 (2018) 116002 [arXiv:1804.06644] [INSPIRE].
H. Ohki, S. Uemura and R. Watanabe, Modular flavor symmetry on a magnetized torus, Phys. Rev. D 102 (2020) 085008 [arXiv:2003.04174] [INSPIRE].
K. Hoshiya, S. Kikuchi, T. Kobayashi, K. Nasu, H. Uchida and S. Uemura, Majorana neutrino masses by D-brane instanton effects in magnetized orbifold models, arXiv:2103.07147 [INSPIRE].
F. Feruglio, Are neutrino masses modular forms?, in From My Vast Repertoire...: Guido Altarelli’s Legacy, A. Levy, S. Forte and G. Ridolfi eds. (2019), pp. 227–266 DOI [arXiv:1706.08749] [INSPIRE].
T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, Modular A4 invariance and neutrino mixing, JHEP 11 (2018) 196 [arXiv:1808.03012] [INSPIRE].
J. T. Penedo and S. T. Petcov, Lepton Masses and Mixing from Modular S4 Symmetry, Nucl. Phys. B 939 (2019) 292 [arXiv:1806.11040] [INSPIRE].
P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, Modular A5 symmetry for flavour model building, JHEP 04 (2019) 174 [arXiv:1812.02158] [INSPIRE].
G.-J. Ding, S. F. King and X.-G. Liu, Neutrino mass and mixing with A5 modular symmetry, Phys. Rev. D 100 (2019) 115005 [arXiv:1903.12588] [INSPIRE].
X.-G. Liu and G.-J. Ding, Neutrino Masses and Mixing from Double Covering of Finite Modular Groups, JHEP 08 (2019) 134 [arXiv:1907.01488] [INSPIRE].
M.-C. Chen, S. Ramos-Sánchez and M. Ratz, A note on the predictions of models with modular flavor symmetries, Phys. Lett. B 801 (2020) 135153 [arXiv:1909.06910] [INSPIRE].
Y. Almumin, M.-C. Chen, V. Knapp-Pérez, S. Ramos-Sánchez, M. Ratz and S. Shukla, Metaplectic Flavor Symmetries from Magnetized Tori, JHEP 05 (2021) 078 [arXiv:2102.11286] [INSPIRE].
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: 2104.03855
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
Tatsuta, Y. Modular symmetry and zeros in magnetic compactifications. J. High Energ. Phys. 2021, 54 (2021). https://doi.org/10.1007/JHEP10(2021)054
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2021)054