Abstract
Compactifications of the heterotic string on special Td/ℤ2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d + 8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II(d), which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d ≤ 2, and give a list of maximally enhanced points where the U(1)d+8 enhances to a rank d + 8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E10. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings of lattices into the dual of II(2). Our results easily generalize to d > 2.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Adams, O. DeWolfe and W. Taylor, String universality in ten dimensions, Phys. Rev. Lett. 105 (2010) 071601 [arXiv:1006.1352] [INSPIRE].
H.-C. Kim, G. Shiu and C. Vafa, Branes and the Swampland, Phys. Rev. D 100 (2019) 066006 [arXiv:1905.08261] [INSPIRE].
B. Fraiman, M. Graña and C. A. Núñez, A new twist on heterotic string compactifications, JHEP 09 (2018) 078 [arXiv:1805.11128] [INSPIRE].
A. Font, B. Fraiman, M. Graña, C. A. Núñez and H. P. De Freitas, Exploring the landscape of heterotic strings on Td, JHEP 10 (2020) 194 [arXiv:2007.10358] [INSPIRE].
K. S. Narain, New Heterotic String Theories in Uncompactified Dimensions < 10, Phys. Lett. B 169 (1986) 41 [INSPIRE].
P. Goddard and D. Olive, Algebras, lattices and strings, in Vertex Operators in Mathematics and Physics, J. Lepowsky, S. Mandelstam and I.M. Singer eds., New York, NY, pp. 51–96, Springer US (1985) [DOI].
F. A. Cachazo and C. Vafa, Type I’ and real algebraic geometry, hep-th/0001029 [INSPIRE].
I. Shimada and D. Q. Zhang, Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces, Nagoya Math. J. 161 (2001) 23 [math/0007171].
S. Chaudhuri, G. Hockney and J. D. Lykken, Maximally supersymmetric string theories in D < 10, Phys. Rev. Lett. 75 (1995) 2264 [hep-th/9505054] [INSPIRE].
S. Chaudhuri and J. Polchinski, Moduli space of CHL strings, Phys. Rev. D 52 (1995) 7168 [hep-th/9506048] [INSPIRE].
A. Mikhailov, Momentum lattice for CHL string, Nucl. Phys. B 534 (1998) 612 [hep-th/9806030] [INSPIRE].
H.-C. Kim, H.-C. Tarazi and C. Vafa, Four-dimensional \( \mathcal{N} \) = 4 SYM theory and the swampland, Phys. Rev. D 102 (2020) 026003 [arXiv:1912.06144] [INSPIRE].
M. Cvetič, M. Dierigl, L. Lin and H. Y. Zhang, String Universality and Non-Simply-Connected Gauge Groups in 8d, Phys. Rev. Lett. 125 (2020) 211602 [arXiv:2008.10605] [INSPIRE].
M. Cvetič, M. Dierigl, L. Lin and H. Y. Zhang, On the Gauge Group Topology of 8d CHL Vacua, arXiv:2107.04031 [INSPIRE].
E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].
W. Lerche, C. Schweigert, R. Minasian and S. Theisen, A Note on the geometry of CHL heterotic strings, Phys. Lett. B 424 (1998) 53 [hep-th/9711104] [INSPIRE].
J. de Boer et al., Triples, fluxes, and strings, Adv. Theor. Math. Phys. 4 (2002) 995 [hep-th/0103170] [INSPIRE].
L. Bhardwaj, D. R. Morrison, Y. Tachikawa and A. Tomasiello, The frozen phase of F-theory, JHEP 08 (2018) 138 [arXiv:1805.09070] [INSPIRE].
A. Dabholkar and J. Park, Strings on orientifolds, Nucl. Phys. B 477 (1996) 701 [hep-th/9604178] [INSPIRE].
O. Aharony, Z. Komargodski and A. Patir, The Moduli space and M(atrix) theory of 9d N = 1 backgrounds of M/string theory, JHEP 05 (2007) 073 [hep-th/0702195] [INSPIRE].
S. Elitzur and A. Giveon, Connection Between Spectra of Nonsupersymmetric Heterotic String Models, Phys. Lett. B 189 (1987) 52 [INSPIRE].
V. G. Kac, Automorphisms of finite order of semisimple Lie algebras, Funkcional. Anal. i Priložen. 3 (1969) 94.
C. Córdova, D. S. Freed, H. T. Lam and N. Seiberg, Anomalies in the Space of Coupling Constants and Their Dynamical Applications II, SciPost Phys. 8 (2020) 002 [arXiv:1905.13361] [INSPIRE].
I. Shimada, On elliptic k3 surfaces, Michigan Math. J. 47 (2000) 423 [math/0505140].
L. Chabrol, F-theory and Heterotic Duality, Weierstrass Models from Wilson lines, Eur. Phys. J. C 80 (2020) 944 [arXiv:1910.12844] [INSPIRE].
Y. Hamada and C. Vafa, 8d supergravity, reconstruction of internal geometry and the Swampland, JHEP 06 (2021) 178 [arXiv:2104.05724] [INSPIRE].
M. Bianchi, G. Pradisi and A. Sagnotti, Toroidal compactification and symmetry breaking in open string theories, Nucl. Phys. B 376 (1992) 365 [INSPIRE].
M. Montero and C. Vafa, Cobordism Conjecture, Anomalies, and the String Lamppost Principle, JHEP 01 (2021) 063 [arXiv:2008.11729] [INSPIRE].
M. Bianchi, A Note on toroidal compactifications of the type-I superstring and other superstring vacuum configurations with sixteen supercharges, Nucl. Phys. B 528 (1998) 73 [hep-th/9711201] [INSPIRE].
R. Blumenhagen, D. Lüst and S. Theisen, Basic concepts of string theory, Springer (2013) [DOI].
K. S. Narain, M. H. Sarmadi and C. Vafa, Asymmetric Orbifolds, Nucl. Phys. B 288 (1987) 551 [INSPIRE].
L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds. 2, Nucl. Phys. B 274 (1986) 285 [INSPIRE].
L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].
C. Vafa, Modular Invariance and Discrete Torsion on Orbifolds, Nucl. Phys. B 273 (1986) 592 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997) [DOI] [INSPIRE].
P. H. Ginsparg, Applied conformal field theory, in Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena, (1988) [hep-th/9108028] [INSPIRE].
I. B. Frenkel and V. G. Kac, Basic Representations of Affine Lie Algebras and Dual Resonance Models, Invent. Math. 62 (1980) 23.
G. Segal, Unitarity Representations of Some Infinite Dimensional Groups, Commun. Math. Phys. 80 (1981) 301 [INSPIRE].
H. Kawai, D. C. Lewellen and S. H. H. Tye, Classification of Closed Fermionic String Models, Phys. Rev. D 34 (1986) 3794 [INSPIRE].
P. Forgacs, Z. Horvath, L. Palla and P. Vecsernyes, Higher Level Kac-Moody Representations and Rank Reduction in String Models, Nucl. Phys. B 308 (1988) 477 [INSPIRE].
S. Hamidi and C. Vafa, Interactions on Orbifolds, Nucl. Phys. B 279 (1987) 465 [INSPIRE].
L. J. Dixon, D. Friedan, E. J. Martinec and S. H. Shenker, The Conformal Field Theory of Orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].
P. Goddard and D. I. Olive, Kac-Moody and Virasoro Algebras in Relation to Quantum Physics, Int. J. Mod. Phys. A 1 (1986) 303 [INSPIRE].
P. Goddard, W. Nahm, D. I. Olive and A. Schwimmer, Vertex Operators for Nonsimply Laced Algebras, Commun. Math. Phys. 107 (1986) 179 [INSPIRE].
D. Bernard and J. Thierry-Mieg, Level One Representations of the Simple Affine Kac-Moody Algebras in Their Homogeneous Gradations, Commun. Math. Phys. 111 (1987) 181 [INSPIRE].
M. Kuwahara, N. Ohta and H. Suzuki, Conformal field theories realized by free fields, Nucl. Phys. B 340 (1990) 448 [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: 2104.07131
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
Font, A., Fraiman, B., Graña, M. et al. Exploring the landscape of CHL strings on Td. J. High Energ. Phys. 2021, 95 (2021). https://doi.org/10.1007/JHEP08(2021)095
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2021)095