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Unraveling T duality: Magnetic quivers in rank-zero little string theories

Lorenzo Mansi* and Marcus Sperling†

  • *Contact author: lorenzo.mansi@desy.de
  • †Contact author: marcus.sperling@univie.ac.at
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Phys. Rev. D 110, 126016 – Published 17 December, 2024

DOI: https://doi.org/10.1103/PhysRevD.110.126016

Abstract

An intriguing class of 6D supersymmetric theories are known as little strings theories, which exhibit a rich network of T-dualities. A robust feature of these theories are their Higgs branches. Focusing on the little string theories that are realized on a single curve of zero self-intersection, we utilize brane systems to derive the magnetic quivers. Using a variety of techniques (including branching rules, brane dynamics, F-theory geometry, quiver subtraction, and the decay and fission algorithm), we detail the Higgs branch Hasse diagram and determine the transverse slices for every elementary Higgs branch RG-flow. Building on these insights, we pursue two directions: first, we infer potential T-dual models by linking changes in 2-group structure constants along Higgs branch RG-flows to the RG-flow transition types encoded in the Hasse diagram. Second, we conjecture an algorithm for the non-Abelian flavor symmetry of the compactified little string theory by inspecting the magnetic quivers of all T-dual frames.

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References (98)

  1. K. Ohmori, H. Shimizu, Y. Tachikawa, and K. Yonekura, Anomaly polynomial of general 6D SCFTs, Prog. Theor. Exp. Phys. 2014, 103B07 (2014).
  2. K. Ohmori, H. Shimizu, and Y. Tachikawa, Anomaly polynomial of E-string theories, J. High Energy Phys. 08 (2014) 002.
  3. C. Cordova, T. T. Dumitrescu, and K. Intriligator, Anomalies, renormalization group flows, and the a-theorem in six-dimensional (1, 0) theories, J. High Energy Phys. 10 (2016) 080.
  4. H. Shimizu, Y. Tachikawa, and G. Zafrir, Anomaly matching on the Higgs branch, J. High Energy Phys. 12 (2017) 127.
  5. C.-M. Chang and Y.-H. Lin, Carving out the end of the world or (superconformal bootstrap in six dimensions), J. High Energy Phys. 08 (2017) 128.
  6. P. S. Aspinwall and M. Gross, The SO(32) heterotic string on a K3 surface, Phys. Lett. B 387, 735 (1996).
  7. P. S. Aspinwall, M theory versus F theory pictures of the heterotic string, Adv. Theor. Math. Phys. 1, 127 (1998).
  8. P. S. Aspinwall, Point—like instantons and the spin (32)Z2 heterotic string, Nucl. Phys. B496, 149 (1997).
  9. D. R. Morrison and C. Vafa, Compactifications of F theory on Calabi-Yau threefolds. 1, Nucl. Phys. B473, 74 (1996).
  10. D. R. Morrison and C. Vafa, Compactifications of F theory on Calabi-Yau threefolds. 2, Nucl. Phys. B476, 437 (1996).
  11. P. S. Aspinwall and D. R. Morrison, Point—like instantons on K3 orbifolds, Nucl. Phys. B503, 533 (1997).
  12. D. R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10, 1072 (2012).
  13. J. J. Heckman, D. R. Morrison, T. Rudelius, and C. Vafa, Atomic classification of 6D SCFTs, Fortschr. Phys. 63, 468 (2015).
  14. A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B529, 180 (1998).
  15. K. A. Intriligator, RG fixed points in six-dimensions via branes at orbifold singularities, Nucl. Phys. B496, 177 (1997).
  16. K. A. Intriligator, New string theories in six-dimensions via branes at orbifold singularities, Adv. Theor. Math. Phys. 1, 271 (1998).
  17. I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, J. High Energy Phys. 03 (1998) 003.
  18. E. Witten, Some comments on string dynamics, in STRINGS 95: Future Perspectives in String Theory (World Scientific, Singapore, 1995), pp. 501–523, arXiv:hep-th/9507121.
  19. N. Seiberg, New theories in six-dimensions and matrix description of M theory on T5 and  T5Z2, Phys. Lett. B 408, 98 (1997).
  20. 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, 086002 (2016).
  21. M. Del Zotto and K. Ohmori, 2-group symmetries of 6D little string theories and T-duality, Ann. Henri Poincare 22, 2451 (2021).
  22. L. Bhardwaj, Discovering T-dualities of little string theories, J. High Energy Phys. 02 (2024) 046.
  23. M. Del Zotto, M. Liu, and P.-K. Oehlmann, Back to heterotic strings on ALE spaces. Part I. Instantons, 2-groups and T-duality, J. High Energy Phys. 01 (2023) 176.
  24. M. Del Zotto, M. Liu, and P.-K. Oehlmann, Back to heterotic strings on ALE spaces. Part II. Geometry of T-dual little strings, J. High Energy Phys. 01 (2024) 109.
  25. M. Del Zotto, M. Liu, and P.-K. Oehlmann, 6D heterotic little string theories and F-theory geometry: An introduction, Proc. Symp. Pure Math. 107, 179 (2024).
  26. M. Del Zotto, M. Fazzi, and S. Giri, The Higgs branch of heterotic ALE instantons, J. High Energy Phys. 01 (2024) 167.
  27. H. Ahmed, P.-K. Oehlmann, and F. Ruehle, T-duality and flavor symmetries in little string theories, J. High Energy Phys. 08 (2024) 061.
  28. C. Lawrie and L. Mansi, Higgs branch of heterotic little string theories: Hasse diagrams and generalized symmetries, Phys. Rev. D 110, 026016 (2024).
  29. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, J. High Energy Phys. 02 (2015) 172.
  30. M. J. Kang and S. Kang, Central extensions of higher groups: Green-Schwarz mechanism and 2-connections, arXiv:2311.14666.
  31. C. Cordova, T. T. Dumitrescu, and K. Intriligator, 2-group global symmetries and anomalies in six-dimensional quantum field theories, J. High Energy Phys. 04 (2021) 252.
  32. S. Cabrera, A. Hanany, and M. Sperling, Magnetic quivers, Higgs branches, and 6d N=(1,0) theories, J. High Energy Phys. 06 (2019) 071.
  33. S. Cabrera, A. Hanany, and M. Sperling, Magnetic quivers, Higgs branches, and 6d N=(1,0) theories—orthogonal and symplectic gauge groups, J. High Energy Phys. 02 (2020) 184.
  34. M. Sperling and Z. Zhong, Balanced B and D-type orthosymplectic quivers—magnetic quivers for product theories, J. High Energy Phys. 04 (2022) 145.
  35. A. Hanany and M. Sperling, Magnetic quivers and negatively charged branes, J. High Energy Phys. 11 (2022) 010.
  36. M. Fazzi and S. Giri, Hierarchy of RG flows in 6d (1, 0) orbi-instantons, J. High Energy Phys. 12 (2022) 076.
  37. A. Bourget and J. F. Grimminger, Fibrations and Hasse diagrams for 6d SCFTs, J. High Energy Phys. 12 (2022) 159.
  38. M. Fazzi, S. Giacomelli, and S. Giri, Hierarchies of RG flows in 6d (1, 0) massive E-strings, J. High Energy Phys. 03 (2023) 089.
  39. A. Bourget, S. Cabrera, J. F. Grimminger, A. Hanany, M. Sperling, A. Zajac, and Z. Zhong, The Higgs mechanism—Hasse diagrams for symplectic singularities, J. High Energy Phys. 01 (2020) 157.
  40. S. Cabrera and A. Hanany, Quiver subtractions, J. High Energy Phys. 09 (2018) 008.
  41. A. Bourget, S. Giacomelli, J. F. Grimminger, A. Hanany, M. Sperling, and Z. Zhong, S-fold magnetic quivers, J. High Energy Phys. 02 (2021) 054.
  42. A. Bourget, J. F. Grimminger, A. Hanany, and Z. Zhong, The Hasse diagram of the moduli space of instantons, J. High Energy Phys. 08 (2022) 283.
  43. A. Bourget, M. Sperling, and Z. Zhong, Decay and fission of magnetic quivers, Phys. Rev. Lett. 132, 221603 (2024).
  44. A. Bourget, M. Sperling, and Z. Zhong, Higgs branch RG flows via decay and fission, Phys. Rev. D 109, 126013 (2024).
  45. J. J. Heckman, D. R. Morrison, T. Rudelius, and C. Vafa, Geometry of 6D RG flows, J. High Energy Phys. 09 (2015) 052.
  46. S. Katz, D. R. Morrison, S. Schafer-Nameki, and J. Sully, Tate’s algorithm and F-theory, J. High Energy Phys. 08 (2011) 094.
  47. M. Del Zotto, M. Fazzi, and S. Giri, A new vista on the heterotic moduli space from six and three dimensions, Phys. Rev. D 109, L021903 (2024).
  48. S. Cremonesi, G. Ferlito, A. Hanany, and N. Mekareeya, Coulomb branch and the moduli space of instantons, J. High Energy Phys. 12 (2014) 103.
  49. D. Gaiotto and E. Witten, S-duality of boundary conditions in N=4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13, 721 (2009).
  50. V. Borokhov, A. Kapustin, and X.-k. Wu, Topological disorder operators in three-dimensional conformal field theory, J. High Energy Phys. 11 (2002) 049.
  51. V. Borokhov, A. Kapustin, and X.-k. Wu, Monopole operators and mirror symmetry in three-dimensions, J. High Energy Phys. 12 (2002) 044.
  52. V. Borokhov, Monopole operators in three-dimensional N=4 SYM and mirror symmetry, J. High Energy Phys. 03 (2004) 008.
  53. A. Beauville, Symplectic singularities, Inventiones Math. 139, 541 (2000).
  54. D. Kaledin, Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. 2006, 135
  55. D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series (Van Nostrand Reinhold Co., New York, 1993).
  56. C. Vafa, Evidence for F theory, Nucl. Phys. B469, 403 (1996).
  57. J. J. Heckman, D. R. Morrison, and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, J. High Energy Phys. 05 (2014) 028.
  58. J. J. Heckman and T. Rudelius, Top down approach to 6D SCFTs, J. Phys. A 52, 093001 (2019).
  59. J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular Functions of One Variable IV, edited by B. J. Birch and W. Kuyk (Springer, Berlin, Heidelberg, 1975), pp. 33–52.
  60. L. Badescu, Algebraic Surfaces, 3rd ed. (Springer, New York, 2001).
  61. B. Conrad, Minimal models for elliptic curves (2005), https://math.stanford.edu/~conrad/papers/minimalmodel.pdf.
  62. K. Kodaira, On compact analytic surfaces. II, Ann. Math. 77, 563 (1963).
  63. K. Kodaira, On the structure of compact complex analytic surfaces. I, Am. J. Math. 86, 751 (1964).
  64. A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. l’IHÉS 21, 5 (1964).
  65. E. Witten, Physical interpretation of certain strong coupling singularities, Mod. Phys. Lett. A 11, 2649 (1996).
  66. J. A. Minahan, D. Nemeschansky, C. Vafa, and N. P. Warner, E strings and N=4 topological Yang-Mills theories, Nucl. Phys. B527, 581 (1998).
  67. O. J. Ganor and A. Hanany, Small E(8) instantons and tensionless noncritical strings, Nucl. Phys. B474, 122 (1996).
  68. J. A. Minahan, D. Nemeschansky, and N. P. Warner, Partition functions for BPS states of the noncritical E(8) string, Adv. Theor. Math. Phys. 1, 167 (1998).
  69. E. Witten, Phase transitions in M theory and F theory, Nucl. Phys. B471, 195 (1996).
  70. N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B471, 121 (1996).
  71. P. S. Aspinwall, S. H. Katz, and D. R. Morrison, Lie groups, Calabi-Yau threefolds, and F theory, Adv. Theor. Math. Phys. 4, 95 (2000).
  72. A. Hanany and A. Zaffaroni, Issues on orientifolds: On the brane construction of gauge theories with SO(2n) global symmetry, J. High Energy Phys. 07 (1999) 009.
  73. O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, J. High Energy Phys. 12 (2015) 163.
  74. E. Witten, Small instantons in string theory, Nucl. Phys. B460, 541 (1996).
  75. M. R. Douglas, Branes within branes, NATO Sci. Ser. C 520, 267 (1999).
  76. R. Feger and T. W. Kephart, lieart—A Mathematica application for Lie algebras and representation theory, Comput. Phys. Commun. 192, 166 (2015).
  77. R. Feger, T. W. Kephart, and R. J. Saskowski, lieart 2.0—A Mathematica application for Lie algebras and representation theory, Comput. Phys. Commun. 257, 107490 (2020).
  78. N. Yamatsu, Finite-dimensional Lie algebras and their representations for unified model building, arXiv:1511.08771.
  79. A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, G. Zafrir, and Z. Zhong, Magnetic quivers for rank 1 theories, J. High Energy Phys. 09 (2020) 189.
  80. A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, and Z. Zhong, Branes, quivers, and the affine Grassmannian, Adv. Stud. Pure Math. 88, 331 (2023).
  81. D. R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B483, 229 (1997).
  82. M. R. Douglas, S. H. Katz, and C. Vafa, Small instantons, Del Pezzo surfaces and type I-prime theory, Nucl. Phys. B497, 155 (1997).
  83. E. Gorbatov, V. S. Kaplunovsky, J. Sonnenschein, S. Theisen, and S. Yankielowicz, On heterotic orbifolds, M theory and type I-prime brane engineering, J. High Energy Phys. 05 (2002) 015.
  84. K. Ohmori and H. Shimizu, S1/T2 compactifications of 6d N=(1,0) theories and brane webs, J. High Energy Phys. 03 (2016) 024.
  85. G. Zafrir, Brane webs, 5d gauge theories and 6d N=(1,0) SCFT’s, J. High Energy Phys. 12 (2015) 157.
  86. H. Hayashi, S.-S. Kim, K. Lee, and F. Yagi, 6d SCFTs, 5d dualities and Tao web diagrams, J. High Energy Phys. 05 (2019) 203.
  87. R. Álvarez-García, S.-J. Lee, and T. Weigand, Non-minimal elliptic threefolds at infinite distance I: Log Calabi-Yau resolutions, J. High Energy Phys. 08 (2024) 240.
  88. R. Álvarez-García, S.-J. Lee, and T. Weigand, Non-minimal elliptic threefolds at infinite distance II: Asymptotic physics, arXiv:2312.11611.
  89. M. Bershadsky, K. A. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B481, 215 (1996).
  90. S. H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B497, 146 (1997).
  91. K. S. Narain, New heterotic string theories in uncompactified dimensions <10, Phys. Lett. 169B, 41 (1986).
  92. J. D. Blum and K. A. Intriligator, Consistency conditions for branes at orbifold singularities, Nucl. Phys. B506, 223 (1997).
  93. 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. B506, 199 (1997).
  94. B. McInnes, The semispin groups in string theory, J. Math. Phys. (N.Y.) 40, 4699 (1999).
  95. F. Apruzzi, M. Fazzi, J. J. Heckman, T. Rudelius, and H. Y. Zhang, General prescription for global U(1)’s in 6D SCFTs, Phys. Rev. D 101, 086023 (2020).
  96. M. B. Green and J. H. Schwarz, Anomaly cancellation in supersymmetric D=10 gauge theory and superstring theory, Phys. Lett. 149B, 117 (1984).
  97. A. Sagnotti, A note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294, 196 (1992).
  98. S. Cremonesi, A. Hanany, and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N=4 gauge theories, J. High Energy Phys. 01 (2014) 005.

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Phys. Rev. D 110, 126016– Published 17 December, 2024
  • Received 22 February 2024
  • Accepted 11 November 2024
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