Abstract
We propose an explicit description of “duality walls” which encode at low energy the global symmetry enhancement expected in the UV completion of certain five-dimensional gauge theories. The proposal is supported by explicit localization computations and implies that the instanton partition function of these theories satisfies novel and unexpected integral equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [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].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [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].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, 5-Brane Webs, Symmetry Enhancement and Duality in 5d Supersymmetric Gauge Theory, JHEP 03 (2014) 112 [arXiv:1311.4199] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
D. Gaiotto and S.S. Razamat, \( \mathcal{N}=1 \) theories of class S k , JHEP 07 (2015) 073 [arXiv:1503.05159] [INSPIRE].
V.P. Spiridonov and S.O. Warnaar, Inversions of integral operators and elliptic beta integrals on root systems, math.CA/0411044.
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
K. Yonekura, Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories, JHEP 07 (2015) 167 [arXiv:1505.04743] [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
A. Hashimoto, P. Ouyang and M. Yamazaki, Boundaries and defects of \( \mathcal{N}=4 \) SYM with 4 supercharges. Part I: Boundary/junction conditions, JHEP 10 (2014) 107 [arXiv:1404.5527] [INSPIRE].
T. Dimofte and D. Gaiotto, An E 7 Surprise, JHEP 10 (2012) 129 [arXiv:1209.1404] [INSPIRE].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
S. Terashima, Supersymmetric gauge theories on S 4 × S 1, Phys. Rev. D 89 (2014) 125001 [arXiv:1207.2163] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
H. Nakajima, Homology of moduli spaces of instantons on ALE spaces. I, J. Diff. Geom. 40 (1994) 105 [INSPIRE].
N. Nekrasov and A.S. Schwarz, Instantons on noncommutative R 4 and (2, 0) superconformal six-dimensional theory, Commun. Math. Phys. 198 (1998) 689 [hep-th/9802068] [INSPIRE].
H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d T N partition functions, JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE].
L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi, Non-Lagrangian Theories from Brane Junctions, JHEP 01 (2014) 175 [arXiv:1310.3841] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, alg-geom/9307001.
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic Genera of 2d \( \mathcal{N}=2 \) Gauge Theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [Addendum ibid. 04 (2016) 094] [arXiv:1406.6793] [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
C. Cordova and S.-H. Shao, An Index Formula for Supersymmetric Quantum Mechanics, arXiv:1406.7853 [INSPIRE].
D. Tong and K. Wong, Instantons, Wilson lines and D-branes, Phys. Rev. D 91 (2015) 026007 [arXiv:1410.8523] [INSPIRE].
S. Shadchin, Saddle point equations in Seiberg-Witten theory, JHEP 10 (2004) 033 [hep-th/0408066] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, JHEP 10 (2016) 012 [arXiv:1412.2781] [INSPIRE].
N. Drukker, D. Gaiotto and J. Gomis, The Virtue of Defects in 4D Gauge Theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE].
M. Bullimore, H.-C. Kim and P. Koroteev, Defects and Quantum Seiberg-Witten Geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
Y. Tachikawa, Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories, Prog. Theor. Exp. Phys. 2015 (2015) 043B06 [arXiv:1501.01031] [INSPIRE].
G. Zafrir, Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories, JHEP 07 (2015) 087 [arXiv:1503.08136] [INSPIRE].
O. Bergman and G. Zafrir, Lifting 4d dualities to 5d, JHEP 04 (2015) 141 [arXiv:1410.2806] [INSPIRE].
S.-S. Kim, M. Taki and F. Yagi, Tao Probing the End of the World, Prog. Theor. Exp. Phys. 2015 (2015) 083B02 [arXiv:1504.03672] [INSPIRE].
J.J. Heckman, D.R. Morrison and C. Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [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].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].
A. Braverman, M. Finkelberg and H. Nakajima, Instanton moduli spaces and -algebras, arXiv:1406.2381 [INSPIRE].
S. Kim, K.-M. Lee and S. Lee, Dyonic Instantons in 5-dim Yang-Mills Chern-Simons Theories, JHEP 08 (2008) 064 [arXiv:0804.1207] [INSPIRE].
B. Collie and D. Tong, Instantons, Fermions and Chern-Simons Terms, JHEP 07 (2008) 015 [arXiv:0804.1772] [INSPIRE].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, Discrete θ and the 5d superconformal index, JHEP 01 (2014) 079 [arXiv:1310.2150] [INSPIRE].
O. Aharony, M. Berkooz, S. Kachru and E. Silverstein, Matrix description of (1, 0) theories in six-dimensions, Phys. Lett. B 420 (1998) 55 [hep-th/9709118] [INSPIRE].
A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, hep-th/0302191 [INSPIRE].
H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP 12 (2011) 031 [arXiv:1110.2175] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1506.03871
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Gaiotto, D., Kim, HC. Duality walls and defects in 5d \( \mathcal{N}=1 \) theories. J. High Energ. Phys. 2017, 19 (2017). https://doi.org/10.1007/JHEP01(2017)019
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2017)019