Abstract
We construct Seiberg-Witten curves for 5d \( \mathcal{N} \) = 1 gauge theories whose Type IIB 5-brane configuration involves an O7-plane and discuss an intriguing relation between theories with an O7+-plane and those with an O7−-plane and 8 D7-branes. We claim that 5-brane configurations with an O7+-plane can be effectively understood as 5-brane configurations with a set of an O7−-plane and eight D7-branes with some special tuning of their masses such that the D7-branes are frozen at the O7−-plane. We check this equivalence between SU(N) gauge theory with a symmetric hypermultiplet and SU(N) gauge theory with an antisymmetric with 8 fundamentals, and also between SO(2N) gauge theory and Sp(N) gauge theory with eight fundamentals. We also compute the Seiberg-Witten curves for non-Lagrangian theories with a symmetric hypermultiplet, which includes the local ℙ2 theory with an adjoint.
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O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, Del Pezzo surfaces and type I-prime theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
M. Aganagic, A. Klemm, M. Marino and C. Vafa, The Topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys. A 24 (2009) 2253 [arXiv:0805.0191] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. I, Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition function, Transform. Groups 10 (2005) 489 [math/0505553] [INSPIRE].
L. Gottsche, H. Nakajima and K. Yoshioka, K-theoretic Donaldson invariants via instanton counting, Pure Appl. Math. Quart. 5 (2009) 1029 [math/0611945] [INSPIRE].
C.A. Keller and J. Song, Counting Exceptional Instantons, JHEP 07 (2012) 085 [arXiv:1205.4722] [INSPIRE].
H.-C. Kim, S.-S. Kim and K. Lee, 5-dim Superconformal Index with Enhanced En Global Symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [INSPIRE].
M.-X. Huang, K. Sun and X. Wang, Blowup Equations for Refined Topological Strings, JHEP 10 (2018) 196 [arXiv:1711.09884] [INSPIRE].
J. Kim et al., Instantons from Blow-up, JHEP 11 (2019) 092 [Erratum ibid. 06 (2020) 124] [arXiv:1908.11276] [INSPIRE].
H.-C. Kim, M. Kim, S.-S. Kim and K.-H. Lee, Bootstrapping BPS spectra of 5d/6d field theories, JHEP 04 (2021) 161 [arXiv:2101.00023] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
K. Sakai, En Jacobi forms and Seiberg-Witten curves, Commun. Num. Theor. Phys. 13 (2019) 53 [arXiv:1706.04619] [INSPIRE].
C. Closset, S. Giacomelli, S. Schafer-Nameki and Y.-N. Wang, 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities, JHEP 05 (2021) 274 [arXiv:2012.12827] [INSPIRE].
N. Haouzi and J. Oh, On the Quantization of Seiberg-Witten Geometry, JHEP 01 (2021) 184 [arXiv:2004.00654] [INSPIRE].
C. Closset and H. Magureanu, The U-plane of rank-one 4d \( \mathcal{N} \) = 2 KK theories, SciPost Phys. 12 (2022) 065 [arXiv:2107.03509] [INSPIRE].
Q. Jia and P. Yi, Aspects of 5d Seiberg-Witten theories on 𝕊1, JHEP 02 (2022) 125 [arXiv:2111.09448] [INSPIRE].
A. Brini and K. Osuga, Five-dimensional gauge theories and the local B-model, Lett. Math. Phys. 112 (2022) 44 [arXiv:2110.11638] [INSPIRE].
F. Del Monte and P. Longhi, The threefold way to quantum periods: WKB, TBA equations and q-Painlevé, arXiv:2207.07135 [INSPIRE].
K. Sakai, E-strings, F4, and D4 triality, JHEP 07 (2023) 192 [arXiv:2304.04878] [INSPIRE].
B. Haghighat and W. Yan, M-strings in thermodynamic limit: Seiberg-Witten geometry, arXiv:1607.07873 [INSPIRE].
B. Haghighat, J. Kim, W. Yan and S.-T. Yau, D-type fiber-base duality, JHEP 09 (2018) 060 [arXiv:1806.10335] [INSPIRE].
X. Li and F. Yagi, Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane, JHEP 06 (2021) 004 [arXiv:2102.09482] [INSPIRE].
J. Chen, B. Haghighat, H.-C. Kim and M. Sperling, Elliptic quantum curves of class 𝒮k, JHEP 03 (2021) 028 [arXiv:2008.05155] [INSPIRE].
J. Chen et al., E-string quantum curve, Nucl. Phys. B 973 (2021) 115602 [arXiv:2103.16996] [INSPIRE].
J. Chen et al., Elliptic quantum curves of 6d SO(N) theories, JHEP 03 (2022) 154 [arXiv:2110.13487] [INSPIRE].
J. Chen, Y. Lü and X. Wang, D-type minimal conformal matter: quantum curves, elliptic Garnier systems, and the 5d descendants, JHEP 10 (2023) 045 [arXiv:2304.04383] [INSPIRE].
L. Bao et al., Non-Lagrangian Theories from Brane Junctions, JHEP 01 (2014) 175 [arXiv:1310.3841] [INSPIRE].
S.-S. Kim and F. Yagi, 5d En Seiberg-Witten curve via toric-like diagram, JHEP 06 (2015) 082 [arXiv:1411.7903] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Discrete theta angle from an O5-plane, JHEP 11 (2017) 041 [arXiv:1707.07181] [INSPIRE].
H. Hayashi et al., Seiberg-Witten curves for 6d SCFTs and Little string theories from 5-brane webs, in progress (2023).
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
A. Hanany and B. Kol, On orientifolds, discrete torsion, branes and M theory, JHEP 06 (2000) 013 [hep-th/0003025] [INSPIRE].
G. Zafrir, Brane webs and O5-planes, JHEP 03 (2016) 109 [arXiv:1512.08114] [INSPIRE].
A. Sen, F theory and orientifolds, Nucl. Phys. B 475 (1996) 562 [hep-th/9605150] [INSPIRE].
O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].
H. Hayashi et al., A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 5-brane webs for 5d \( \mathcal{N} \) = 1 G2 gauge theories, JHEP 03 (2018) 125 [arXiv:1801.03916] [INSPIRE].
A. Sen, Stable nonBPS states in string theory, JHEP 06 (1998) 007 [hep-th/9803194] [INSPIRE].
A. Sen, Stable nonBPS bound states of BPS D-branes, JHEP 08 (1998) 010 [hep-th/9805019] [INSPIRE].
A. Kapustin, D(n) quivers from branes, JHEP 12 (1998) 015 [hep-th/9806238] [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].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On Geometric Classification of 5d SCFTs, JHEP 04 (2018) 103 [arXiv:1801.04036] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Dualities and 5-brane webs for 5d rank 2 SCFTs, JHEP 12 (2018) 016 [arXiv:1806.10569] [INSPIRE].
P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards classification of 5d SCFTs: Single gauge node, SciPost Phys. 14 (2023) 122 [arXiv:1705.05836] [INSPIRE].
M. Akhond and F. Carta, Magnetic quivers from brane webs with O7+-planes, JHEP 10 (2021) 014 [arXiv:2107.09077] [INSPIRE].
A. Fayyazuddin and M. Spalinski, The Seiberg-Witten differential from M theory, Nucl. Phys. B 508 (1997) 219 [hep-th/9706087] [INSPIRE].
M. Henningson and P. Yi, Four-dimensional BPS spectra via M theory, Phys. Rev. D 57 (1998) 1291 [hep-th/9707251] [INSPIRE].
A. Mikhailov, BPS states and minimal surfaces, Nucl. Phys. B 533 (1998) 243 [hep-th/9708068] [INSPIRE].
K. Landsteiner and E. Lopez, New curves from branes, Nucl. Phys. B 516 (1998) 273 [hep-th/9708118] [INSPIRE].
H. Hayashi et al., More on 5d descriptions of 6d SCFTs, JHEP 10 (2016) 126 [arXiv:1512.08239] [INSPIRE].
L. Bao, E. Pomoni, M. Taki and F. Yagi, M5-Branes, Toric Diagrams and Gauge Theory Duality, JHEP 04 (2012) 105 [arXiv:1112.5228] [INSPIRE].
L. Bhardwaj, On the classification of 5d SCFTs, JHEP 09 (2020) 007 [arXiv:1909.09635] [INSPIRE].
M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M- and [p, q]-strings, JHEP 11 (2013) 112 [arXiv:1308.0619] [INSPIRE].
S.-S. Kim, X. Li, S. Nawata and F. Yagi, Higgsing and tuning D-branes with O-plane and BPS jumping, in progress.
H.-C. Kim, M. Kim, S.-S. Kim and G. Zafrir, Superconformal indices for non-Lagrangian theories in five dimensions, arXiv:2307.03231 [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
K. Landsteiner, E. Lopez and D.A. Lowe, N = 2 supersymmetric gauge theories, branes and orientifolds, Nucl. Phys. B 507 (1997) 197 [hep-th/9705199] [INSPIRE].
A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, M theory and Seiberg-Witten curves: Orthogonal and symplectic groups, Nucl. Phys. B 504 (1997) 175 [hep-th/9705232] [INSPIRE].
Acknowledgments
We thank Songling He, Hee-Cheol Kim, Minsung Kim, Xiaobin Li, Yongchao Lü, Satoshi Nawata, Xin Wang, and Gabi Zafrir for useful discussions and comments. SK thanks the hospitality of Postech where part of this work was done and KIAS for hosting “KIAS Autumn Symposium on String Theory 2022” where this work was presented. The work of HH is supported in part by JSPS KAKENHI Grant Number JP18K13543 and JP23K03396. SK is supported by the NSFC grant No. 12250610188. KL is supported by KIAS Individual Grant PG006904 and by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (No. 2017R1D1A1B06034369). FY is supported by the NSFC grant No. 11950410490 and by Start-up research grant YH1199911312101.
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Hayashi, H., Kim, SS., Lee, K. et al. Seiberg-Witten curves with O7±-planes. J. High Energ. Phys. 2023, 178 (2023). https://doi.org/10.1007/JHEP11(2023)178
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DOI: https://doi.org/10.1007/JHEP11(2023)178