Abstract
Gauge theories in the presence of codimension two vortex defects are known to be related to the theories on orbifolds. By using this relation we study the localized path integrals of 2D \( \mathcal{N}=\left(2,2\right) \) SUSY gauge theories with point-like vortex defects. We present a formula for the correlation functions of vortex defects inserted at the north and the south poles of squashed spheres. For Abelian gauge theories the correlators are locally constant as functions of the parameters of the defect, but exhibit discontinuity at some threshold values determined from the R-charges of the matter multiplets. For non-Abelian gauge groups the correlators depend non-trivially on the types of gauge symmetry breaking due to the defects.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
S. Gukov, Surface Operators, arXiv:1412.7127 [INSPIRE].
C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
M. Taki, Surface Operator, Bubbling Calabi-Yau and AGT Relation, JHEP 07 (2011) 047 [arXiv:1007.2524] [INSPIRE].
H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String, Adv. Theor. Math. Phys. 16 (2012) 725 [arXiv:1008.0574] [INSPIRE].
C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, Affine sl(N) conformal blocks from \( \mathcal{N}=2 \) SU(N) gauge theories, JHEP 01 (2011) 045 [arXiv:1008.1412] [INSPIRE].
N. Wyllard, W-algebras and surface operators in N = 2 gauge theories, J. Phys. A 44 (2011) 155401 [arXiv:1011.0289] [INSPIRE].
N. Wyllard, Instanton partition functions in N = 2 SU(N) gauge theories with a general surface operator and their W-algebra duals, JHEP 02 (2011) 114 [arXiv:1012.1355] [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].
H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, The Liouville side of the Vortex, JHEP 09 (2011) 096 [arXiv:1107.2787] [INSPIRE].
Y. Nakayama, 4D and 2D superconformal index with surface operator, JHEP 08 (2011) 084 [arXiv:1105.4883] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [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].
A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
M. Bullimore, H.-C. Kim and P. Koroteev, Defects and Quantum Seiberg-Witten Geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].
J. Gomis and B. Le Floch, M2-brane surface operators and gauge theory dualities in Toda, arXiv:1407.1852 [INSPIRE].
S. Nawata, Givental J-functions, Quantum integrable systems, AGT relation with surface operator, arXiv:1408.4132 [INSPIRE].
D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, arXiv:1412.2781 [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].
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].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
E. Frenkel, S. Gukov and J. Teschner, Surface Operators and Separation of Variables, JHEP 01 (2016) 179 [arXiv:1506.07508] [INSPIRE].
J. Gomis and B. Le Floch, ’t Hooft Operators in Gauge Theory from Toda CFT, JHEP 11 (2011) 114 [arXiv:1008.4139] [INSPIRE].
J. Gomis, T. Okuda and V. Pestun, Exact Results for ’t Hooft Loops in Gauge Theories on S 4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].
D. Gang, E. Koh and K. Lee, Line Operator Index on S 1 × S 3, JHEP 05 (2012) 007 [arXiv:1201.5539] [INSPIRE].
T. Okuda, Line operators in supersymmetric gauge theories and the 2d-4d relation, arXiv:1412.7126 [INSPIRE].
B. Assel and J. Gomis, Mirror Symmetry And Loop Operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].
Y. Ito, T. Okuda and M. Taki, Line operators on S 1 × R 3 and quantization of the Hitchin moduli space, JHEP 04 (2012) 010 [arXiv:1111.4221] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for supersymmetric abelian vortex loops in 2+1 dimensions, JHEP 06 (2013) 099 [arXiv:1211.2861] [INSPIRE].
N. Drukker, T. Okuda and F. Passerini, Exact results for vortex loop operators in 3d supersymmetric theories, JHEP 07 (2014) 137 [arXiv:1211.3409] [INSPIRE].
I. Biswas, Parabolic bundles as orbifold bundles, Duke Math. J. 88 (1997) 305.
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].
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].
J. Gomis and S. Lee, Exact Kähler Potential from Gauge Theory and Mirror Symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].
T.T. Wu and C.N. Yang, Dirac Monopole Without Strings: Monopole Harmonics, Nucl. Phys. B 107 (1976) 365 [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].
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: 1507.07650
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
Hosomichi, K. Orbifolds, defects and sphere partition function. J. High Energ. Phys. 2016, 155 (2016). https://doi.org/10.1007/JHEP02(2016)155
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2016)155