Abstract
We use supersymmetric localization techniques to study the low-energy dynamics of BPS vortex-strings in four-dimensional \( \mathcal{N}=2 \) theories. We focus on theories with SU(Nc) × U(1) gauge group and Nf hypermultiplets, all in the fundamental representation of SU(Nc) but with general U(1) charges. Recently, we proposed a condition that determines whether the low-energy string dynamics is captured by a two-dimensional worldsheet theory that decouples from the bulk [1]. For strings for which this decoupling applies, we propose a prescription for extracting the two-sphere partition function of the string worldsheet theory from the four-ellipsoid partition function of the parent theory. We obtain a general formula for the worldsheet two-sphere partition function in terms of the parameters of the four-dimensional theory and identify \( \mathcal{N}=\left(2,2\right) \) GLSMs that possess these partition functions in a large class of examples. In these examples, the weak coupling regime of the four-dimensional theory is mapped to the weak coupling regime of the worldsheet theory. In addition, we study the classical string zero-modes in flat space and obtain predictions for the worldsheet spectra, which agree with the low-energy spectra of the GLSMs obtained in the localization analysis. For Nf = 2Nc = 4, we discuss the map between string worldsheet theories under four-dimensional S-duality and use our prescription to study examples in which the weak coupling regime of the four-dimensional theory is mapped to the strong coupling regime of the worldsheet theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Gerchkovitz and A. Karasik, Vortex-strings in \( \mathcal{N}=2 \) SQCD and bulk-string decoupling, JHEP 02 (2018) 091 [arXiv:1710.02203] [INSPIRE].
N. Dorey, The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms, JHEP 11 (1998) 005 [hep-th/9806056] [INSPIRE].
A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].
R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, NonAbelian superconductors: Vortices and confinement in N = 2 SQCD, Nucl. Phys. B 673 (2003) 187 [hep-th/0307287] [INSPIRE].
M. Shifman and A. Yung, NonAbelian string junctions as confined monopoles, Phys. Rev. D 70 (2004) 045004 [hep-th/0403149] [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, Non-Abelian semilocal strings in N = 2 supersymmetric QCD, Phys. Rev. D 73 (2006) 125012 [hep-th/0603134] [INSPIRE].
M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Moduli space of non-Abelian vortices, Phys. Rev. Lett. 96 (2006) 161601 [hep-th/0511088] [INSPIRE].
R. Auzzi, M. Shifman and A. Yung, Composite non-Abelian flux tubes in N = 2 SQCD, Phys. Rev. D 73 (2006) 105012 [Erratum ibid. D 76 (2007) 109901] [hep-th/0511150] [INSPIRE].
M. Eto et al., Constructing non-Abelian vortices with arbitrary gauge groups, AIP Conf. Proc. 1078 (2009) 483 [INSPIRE].
L. Ferretti, S.B. Gudnason and K. Konishi, Non-Abelian vortices and monopoles in SO(N) theories, Nucl. Phys. B 789 (2008) 84 [arXiv:0706.3854] [INSPIRE].
N. Dorey, T.J. Hollowood and D. Tong, The BPS spectra of gauge theories in two-dimensions and four-dimensions, JHEP 05 (1999) 006 [hep-th/9902134] [INSPIRE].
M. Eto et al., On the moduli space of semilocal strings and lumps, Phys. Rev. D 76 (2007) 105002 [arXiv:0704.2218] [INSPIRE].
M. Eto et al., Non-Abelian Vortices of Higher Winding Numbers, Phys. Rev. D 74 (2006) 065021 [hep-th/0607070] [INSPIRE].
M. Shifman and A. Yung, Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories, Rev. Mod. Phys. 79 (2007) 1139 [hep-th/0703267] [INSPIRE].
D. Tong, TASI lectures on solitons: Instantons, monopoles, vortices and kinks, in Theoretical Advanced Study Institute in Elementary Particle Physics: Many Dimensions of String Theory (TASI 2005), Boulder, Colorado, June 5-July 1, 2005 (2005) [hep-th/0509216] [INSPIRE].
D. Tong, Quantum Vortex Strings: A Review, Annals Phys. 324 (2009) 30 [arXiv:0809.5060] [INSPIRE].
M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Solitons in the Higgs phase: The Moduli matrix approach, J. Phys. A 39 (2006) R315 [hep-th/0602170] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].
N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact Results in D = 2 Supersymmetric Gauge Theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].
F. Benini and S. Cremonesi, Partition Functions of \( \mathcal{N}=\left(2,2\right) \) Gauge Theories on S 2 and Vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].
H.-Y. Chen and T.-H. Tsai, On Higgs branch localization of Seiberg-Witten theories on an ellipsoid, PTEP 2016 (2016) 013B09 [arXiv:1506.04390] [INSPIRE].
Y. Pan and W. Peelaers, Ellipsoid partition function from Seiberg-Witten monopoles, JHEP 10 (2015) 183 [arXiv:1508.07329] [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].
T. Fujimori, T. Kimura, M. Nitta and K. Ohashi, 2d partition function in Ω-background and vortex/instanton correspondence, JHEP 12 (2015) 110 [arXiv:1509.08630] [INSPIRE].
J. Gomis, B. Le Floch, Y. Pan and W. Peelaers, Intersecting Surface Defects and Two-Dimensional CFT, Phys. Rev. D 96 (2017) 045003 [arXiv:1610.03501] [INSPIRE].
Y. Pan and W. Peelaers, Intersecting Surface Defects and Instanton Partition Functions, JHEP 07 (2017) 073 [arXiv:1612.04839] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [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].
N.S. Manton and J.M. Speight, Asymptotic interactions of critically coupled vortices, Commun. Math. Phys. 236 (2003) 535 [hep-th/0205307] [INSPIRE].
C.H. Taubes, Arbitrary N: Vortex Solutions to the First Order Landau-Ginzburg Equations, Commun. Math. Phys. 72 (1980) 277 [INSPIRE].
D. Tong, Monopoles in the Higgs phase, Phys. Rev. D 69 (2004) 065003 [hep-th/0307302] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
J. Gomis and S. Lee, Exact Kähler Potential from Gauge Theory and Mirror Symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].
J. Gomis and B. Le Floch, M2-brane surface operators and gauge theory dualities in Toda, JHEP 04 (2016) 183 [arXiv:1407.1852] [INSPIRE].
E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere Partition Functions and the Zamolodchikov Metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [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].
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: 1711.03561
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gerchkovitz, E., Karasik, A. New vortex-string worldsheet theories from supersymmetric localization. J. High Energ. Phys. 2019, 90 (2019). https://doi.org/10.1007/JHEP03(2019)090
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2019)090