Abstract
Starting with the superconformal indices for 4d \( \mathcal{N}=2 \) and \( \mathcal{N}=1 \) supersymmetric gauge theories, that are related by superpotential deformation, we perform the contour integrations and isolate the residue contributions which can be attributed to the surface defects. These defects can be interpreted as the IR limit of dynamical vortices. Given the 2d \( \mathcal{N}=\left(2,2\right) \) and \( \mathcal{N}=\left(0,2\right) \) world sheet theories for these surface defects, we then verify this statement by explicitly computing their elliptic genera and identifying their fugacity parameters through superconformal algebras. We show them precisely match, hence extend previous prescriptions for inserting surface defects into 4d supersymmetric partition functions to \( \mathcal{N}=1 \) setting. We also study how 4d \( \mathcal{N}=1 \) IR dualities descend into the \( \mathcal{N}=\left(0,2\right) \) world sheet theories of their surface defects, and extend the \( \mathcal{N}=\left(2,2\right) \) triality observed earlier to other \( \mathcal{N}=\left(0,2\right) \) surface defects unrelated to dynamical vortices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
D. Kutasov and A. Schwimmer, On duality in supersymmetric Yang-Mills theory, Phys. Lett. B 354 (1995) 315 [hep-th/9505004] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
D. Gaiotto, Surface Operators in N = 2 4d Gauge Theories, JHEP 11 (2012) 090 [arXiv:0911.1316] [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
D. Gaiotto, S. Gukov and N. Seiberg, Surface Defects and Resolvents, JHEP 09 (2013) 070 [arXiv:1307.2578] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
A. Hanany and D. Tong, Vortex strings and four-dimensional gauge dynamics, JHEP 04 (2004) 066 [hep-th/0403158] [INSPIRE].
M. Shifman and A. Yung, NonAbelian string junctions as confined monopoles, Phys. Rev. D 70 (2004) 045004 [hep-th/0403149] [INSPIRE].
M. Edalati and D. Tong, Heterotic Vortex Strings, JHEP 05 (2007) 005 [hep-th/0703045] [INSPIRE].
D. Tong, The Quantum Dynamics of Heterotic Vortex Strings, JHEP 09 (2007) 022 [hep-th/0703235] [INSPIRE].
D. Tong, Quantum Vortex Strings: A Review, Annals Phys. 324 (2009) 30 [arXiv:0809.5060] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [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].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, arXiv:1211.1986 [INSPIRE].
H.-Y. Chen, H.-Y. Chen and J.-K. Ho, Connecting Mirror Symmetry in 3d and 2d via Localization, arXiv:1312.2361 [INSPIRE].
F. Benini and W. Peelaers, Higgs branch localization in three dimensions, JHEP 05 (2014) 030 [arXiv:1312.6078] [INSPIRE].
M. Fujitsuka, M. Honda and Y. Yoshida, Higgs branch localization of 3d N = 2 theories, arXiv:1312.3627 [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, (0, 2) trialities, JHEP 03 (2014) 076 [arXiv:1310.0818] [INSPIRE].
F.A. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic Hypergeometry of Supersymmetric Dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [INSPIRE].
N. Dorey, S. Lee and T.J. Hollowood, Quantization of Integrable Systems and a 2d/4d Duality, JHEP 10 (2011) 077 [arXiv:1103.5726] [INSPIRE].
H.-Y. Chen, N. Dorey, T.J. Hollowood and S. Lee, A New 2d/4d Duality via Integrability, JHEP 09 (2011) 040 [arXiv:1104.3021] [INSPIRE].
L.F. Alday, M. Bullimore, M. Fluder and L. Hollands, Surface defects, the superconformal index and q-deformed Yang-Mills, JHEP 10 (2013) 018 [arXiv:1303.4460] [INSPIRE].
M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, arXiv:1401.3379 [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, arXiv:1308.4896 [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Walls, Lines and Spectral Dualities in 3d Gauge Theories, JHEP 05 (2014) 047 [arXiv:1302.0015] [INSPIRE].
W. Peelaers, Higgs branch localization of \( \mathcal{N}=1 \) theories on S 3 × S 1, JHEP 08 (2014) 060 [arXiv:1403.2711] [INSPIRE].
Y. Yoshida, Factorization of 4d N = 1 superconformal index, arXiv:1403.0891 [INSPIRE].
K. Hori and D. Tong, Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N = (2,2) Theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, On the Superconformal Index of N = 1 IR Fixed Points: A Holographic Check, JHEP 03 (2011) 041 [arXiv:1011.5278] [INSPIRE].
P.C. Argyres, M.R. Plesser and N. Seiberg, The Moduli space of vacua of N = 2 SUSY QCD and duality in N = 1 SUSY QCD, Nucl. Phys. B 471 (1996) 159 [hep-th/9603042] [INSPIRE].
S. Bolognesi, A Coincidence Problem: How to Flow from N = 2 SQCD to N = 1 SQCD, JHEP 11 (2008) 029 [arXiv:0807.2456] [INSPIRE].
K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
C. Beem and A. Gadde, The N = 1 superconformal index for class S fixed points, JHEP 04 (2014) 036 [arXiv:1212.1467] [INSPIRE].
V.P. Spiridonov, Essay on the theory of elliptic hypergeometric functions, Russ. Math. Surv. 63 (2008) 405 [arXiv:0805.3135].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [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: 1407.4587
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
Chen, HY., Chen, HY. Heterotic surface defects and dualities from 2d/4d indices. J. High Energ. Phys. 2014, 4 (2014). https://doi.org/10.1007/JHEP10(2014)004
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2014)004