Abstract
In this paper we begin revisiting the little string theories (LSTs) which govern the dynamics of the instantonic heterotic E8 × E8 five-branes probing ALE singularities, building on and extending previous results on the subject by Aspinwall and Morrison as well as Blum and Intriligator. Our focus are the cases corresponding to choices of non-trivial flat connections at infinity. The latter are in particular interesting for the exceptional ALE singularities, where a brane realization in Type I′ is lacking. Our approach to determine these models is based on 6d conformal matter: we determine these theories as generalized 6d quivers. All these LSTs have a higher-one form symmetry which forms a 2-group with the zero-form Poincaré symmetry, the R-symmetry and the other global symmetries: the matching of the R-symmetry two-group structure constant is a stringent constraint for T-dualities, which we use in combination with the matching of 5d Coulomb branches and flavor symmetries upon circle reduction, as a consistency check for the realization of the 6d LSTs we propose.
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 [arXiv:1312.5746] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
M. Del Zotto, J.J. Heckman, D.R. Morrison and D.S. Park, 6D SCFTs and Gravity, JHEP 06 (2015) 158 [arXiv:1412.6526] [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and 6d(1,0) → 4d(𝒩 =2), JHEP 11 (2015) 123 [arXiv:1504.08348] [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].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d 𝒩 = (1, 0) theories on T2 and class S theories: Part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d 𝒩 = (1, 0) theories on S1/T2 and class S theories: part II, JHEP 12 (2015) 131 [arXiv:1508.00915] [INSPIRE].
L. Bhardwaj, M. Del Zotto, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, F-theory and the Classification of Little Strings, Phys. Rev. D 93 (2016) 086002 [arXiv:1511.05565] [INSPIRE].
M. Del Zotto, J.J. Heckman and D.R. Morrison, 6D SCFTs and Phases of 5D Theories, JHEP 09 (2017) 147 [arXiv:1703.02981] [INSPIRE].
N. Mekareeya, K. Ohmori, Y. Tachikawa and G. Zafrir, E8 instantons on type-A ALE spaces and supersymmetric field theories, JHEP 09 (2017) 144 [arXiv:1707.04370] [INSPIRE].
K. Ohmori, Y. Tachikawa and G. Zafrir, Compactifications of 6d N = (1, 0) SCFTs with non-trivial Stiefel-Whitney classes, JHEP 04 (2019) 006 [arXiv:1812.04637] [INSPIRE].
M. Dierigl, P.-K. Oehlmann and F. Ruehle, Non-Simply-Connected Symmetries in 6D SCFTs, JHEP 10 (2020) 173 [arXiv:2005.12929] [INSPIRE].
M. Del Zotto, J.J. Heckman, D.S. Park and T. Rudelius, On the Defect Group of a 6D SCFT, Lett. Math. Phys. 106 (2016) 765 [arXiv:1503.04806] [INSPIRE].
C. Córdova, T.T. Dumitrescu and K. Intriligator, Exploring 2-Group Global Symmetries, JHEP 02 (2019) 184 [arXiv:1802.04790] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories, JHEP 04 (2021) 252 [arXiv:2009.00138] [INSPIRE].
M. Del Zotto and K. Ohmori, 2-Group Symmetries of 6D Little String Theories and T-Duality, Annales Henri Poincare 22 (2021) 2451 [arXiv:2009.03489] [INSPIRE].
L. Bhardwaj and S. Schäfer-Nameki, Higher-form symmetries of 6d and 5d theories, JHEP 02 (2021) 159 [arXiv:2008.09600] [INSPIRE].
F. Apruzzi, L. Bhardwaj, D.S.W. Gould and S. Schafer-Nameki, 2-Group symmetries and their classification in 6d, SciPost Phys. 12 (2022) 098 [arXiv:2110.14647] [INSPIRE].
P.S. Aspinwall, Point - like instantons and the spin (32) / Z(2) heterotic string, Nucl. Phys. B 496 (1997) 149 [hep-th/9612108] [INSPIRE].
P.S. Aspinwall and D.R. Morrison, Point - like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].
J.D. Blum and K.A. Intriligator, New phases of string theory and 6-D RG fixed points via branes at orbifold singularities, Nucl. Phys. B 506 (1997) 199 [hep-th/9705044] [INSPIRE].
K.A. Intriligator, New string theories in six-dimensions via branes at orbifold singularities, Adv. Theor. Math. Phys. 1 (1998) 271 [hep-th/9708117] [INSPIRE].
A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE].
I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, JHEP 03 (1998) 003 [hep-th/9712143] [INSPIRE].
A. Sagnotti, Open Strings and their Symmetry Groups, in NATO Advanced Summer Institute on Nonperturbative Quantum Field Theory (Cargese Summer Institute), (1987) [hep-th/0208020] [INSPIRE].
M. Bianchi and A. Sagnotti, On the systematics of open string theories, Phys. Lett. B 247 (1990) 517 [INSPIRE].
A. Font and C. Mayrhofer, Non-geometric vacua of the Spin(32)/ℤ2 heterotic string and little string theories, JHEP 11 (2017) 064 [arXiv:1708.05428] [INSPIRE].
A. Font, I.n. Garcia-Etxebarria, D. Lüst, S. Massai and C. Mayrhofer, Non-geometric heterotic backgrounds and 6D SCFTs/LSTs, PoS CORFU2016 (2017) 123 [arXiv:1712.07083] [INSPIRE].
L. Bhardwaj, Discovering T-Dualities of Little String Theories, arXiv:2209.10548 [INSPIRE].
M. Del Zotto, M. Liu and P.-K. Oehlmann, Back to Heterotic String on ALE spaces. Part III. F-theory Geometry and T-duality, to appear.
M. Del Zotto and G. Lockhart, Universal Features of BPS Strings in Six-dimensional SCFTs, JHEP 08 (2018) 173 [arXiv:1804.09694] [INSPIRE].
J.J. Heckman and T. Rudelius, Top Down Approach to 6D SCFTs, J. Phys. A 52 (2019) 093001 [arXiv:1805.06467] [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].
E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].
J.D. Blum and K.A. Intriligator, Consistency conditions for branes at orbifold singularities, Nucl. Phys. B 506 (1997) 223 [hep-th/9705030] [INSPIRE].
M. Berkooz, R.G. Leigh, J. Polchinski, J.H. Schwarz, N. Seiberg and E. Witten, Anomalies, dualities, and topology of D=6 N=1 superstring vacua, Nucl. Phys. B 475 (1996) 115 [hep-th/9605184] [INSPIRE].
L. Bhardwaj, Revisiting the classifications of 6d SCFTs and LSTs, JHEP 03 (2020) 171 [arXiv:1903.10503] [INSPIRE].
P. Horava and E. Witten, Heterotic and type I string dynamics from eleven-dimensions, Nucl. Phys. B 460 (1996) 506 [hep-th/9510209] [INSPIRE].
O.J. Ganor and A. Hanany, Small E(8) instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
J.J. Heckman, T. Rudelius and A. Tomasiello, Fission, Fusion, and 6D RG Flows, JHEP 02 (2019) 167 [arXiv:1807.10274] [INSPIRE].
D.D. Frey and T. Rudelius, 6D SCFTs and the classification of homomorphisms ΓADE → E8, Adv. Theor. Math. Phys. 24 (2020) 709 [arXiv:1811.04921] [INSPIRE].
M. Del Zotto, M. Liu and P.-K. Oehlmann, Back to Heterotic String on ALE spaces. Part II. Geometric Engineering Limit of the Heterotic String, to appear.
V.G. Kac, Infinite Dimensional Lie Algebras, Birkhäuser Boston (1983), https://doi.org/10.1007/978-1-4757-1382-4.
J. Polchinski and E. Witten, Evidence for heterotic - type I string duality, Nucl. Phys. B 460 (1996) 525 [hep-th/9510169] [INSPIRE].
A. Sen, Dynamics of multiple Kaluza-Klein monopoles in M and string theory, Adv. Theor. Math. Phys. 1 (1998) 115 [hep-th/9707042] [INSPIRE].
K.A. Intriligator, RG fixed points in six-dimensions via branes at orbifold singularities, Nucl. Phys. B 496 (1997) 177 [hep-th/9702038] [INSPIRE].
B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Triality in Little String Theories, Phys. Rev. D 97 (2018) 046004 [arXiv:1711.07921] [INSPIRE].
B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Beyond Triality: Dual Quiver Gauge Theories and Little String Theories, JHEP 11 (2018) 016 [arXiv:1807.00186] [INSPIRE].
A. Hanany and A. Zaffaroni, Issues on orientifolds: On the brane construction of gauge theories with SO(2n) global symmetry, JHEP 07 (1999) 009 [hep-th/9903242] [INSPIRE].
J.J. Heckman, T. Rudelius and A. Tomasiello, 6D RG Flows and Nilpotent Hierarchies, JHEP 07 (2016) 082 [arXiv:1601.04078] [INSPIRE].
M. Fazzi and S. Giri, Hierarchy of RG flows in 6d (1, 0) orbi-instantons, JHEP 12 (2022) 076 [arXiv:2208.11703] [INSPIRE].
H. Elvang, D.Z. Freedman, L.-Y. Hung, M. Kiermaier, R.C. Myers and S. Theisen, On renormalization group flows and the a-theorem in 6d, JHEP 10 (2012) 011 [arXiv:1205.3994] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Anomalies, renormalization group flows, and the a-theorem in six-dimensional (1, 0) theories, JHEP 10 (2016) 080 [arXiv:1506.03807] [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: 2209.10551
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
Del Zotto, M., Liu, M. & Oehlmann, PK. Back to heterotic strings on ALE spaces. Part I. Instantons, 2-groups and T-duality. J. High Energ. Phys. 2023, 176 (2023). https://doi.org/10.1007/JHEP01(2023)176
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2023)176