Abstract
We study half-BPS surface operators in supersymmetric gauge theories in four and five dimensions following two different approaches. In the first approach we analyze the chiral ring equations for certain quiver theories in two and three dimensions, coupled respectively to four- and five-dimensional gauge theories. The chiral ring equations, which arise from extremizing a twisted chiral superpotential, are solved as power series in the infrared scales of the quiver theories. In the second approach we use equivariant localization and obtain the twisted chiral superpotential as a function of the Coulomb moduli of the four- and five-dimensional gauge theories, and find a perfect match with the results obtained from the chiral ring equations. In the five-dimensional case this match is achieved after solving a number of subtleties in the localization formulas which amounts to choosing a particular residue prescription in the integrals that yield the Nekrasov-like partition functions for ramified instantons. We also comment on the necessity of including Chern-Simons terms in order to match the superpotentials obtained from dual quiver descriptions of a given surface operator.
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References
S. Gukov, Surface operators, in New dualities of supersymmetric gauge theories, J. Teschner ed., Springer Germany (2016), arXiv:1412.7127.
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
S. Gukov and E. Witten, Rigid surface operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [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].
S. Nawata, Givental J-functions, quantum integrable systems, AGT relation with surface operator, Adv. Theor. Math. Phys. 19 (2015) 1277 [arXiv:1408.4132] [INSPIRE].
S.K. Ashok, M. Billó, E. Dell’Aquila, M. Frau, R.R. John and A. Lerda, Modular and duality properties of surface operators in N = 2∗ gauge theories, JHEP 07 (2017) 068 [arXiv:1702.02833] [INSPIRE].
A. Gorsky, B. Le Floch, A. Milekhin and N. Sopenko, Surface defects and instanton-vortex interaction, Nucl. Phys. B 920 (2017) 122 [arXiv:1702.03330] [INSPIRE].
D. Gaiotto, Surface operators in N = 2 4d gauge theories, JHEP 11 (2012) 090 [arXiv:0911.1316] [INSPIRE].
J. Gomis and B. Le Floch, M 2-brane surface operators and gauge theory dualities in Toda, JHEP 04 (2016) 183 [arXiv:1407.1852] [INSPIRE].
D. Gaiotto, S. Gukov and N. Seiberg, Surface defects and resolvents, JHEP 09 (2013) 070 [arXiv:1307.2578] [INSPIRE].
D. Gaiotto and P. Koroteev, On three dimensional quiver gauge theories and integrability, JHEP 05 (2013) 126 [arXiv:1304.0779] [INSPIRE].
F. Benini, D.S. Park and P. Zhao, Cluster algebras from dualities of 2d \( \mathcal{N} \) = (2, 2) quiver gauge theories, Commun. Math. Phys. 340 (2015) 47 [arXiv:1406.2699] [INSPIRE].
C. Closset, S. Cremonesi and D.S. Park, The equivariant A-twist and gauged linear σ-models on the two-sphere, JHEP 06 (2015) 076 [arXiv:1504.06308] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in the proceedings of the 16th International Congress on Mathematical Physics (ICMP09), August 3-8, Prague, Czech Republic (2009), arXiv:0908.4052 [INSPIRE].
A. Hanany and K. Hori, Branes and N = 2 theories in two-dimensions, Nucl. Phys. B 513 (1998) 119 [hep-th/9707192] [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
F. Cachazo, M.R. Douglas, N. Seiberg and E. Witten, Chiral rings and anomalies in supersymmetric gauge theory, JHEP 12 (2002) 071 [hep-th/0211170] [INSPIRE].
E. D’Hoker, I.M. Krichever and D.H. Phong, The effective prepotential of N = 2 supersymmetric SU(N c) gauge theories, Nucl. Phys. B 489 (1997) 179 [hep-th/9609041] [INSPIRE].
S.G. Naculich, H.J. Schnitzer and N. Wyllard, The N = 2 U(N ) gauge theory prepotential and periods from a perturbative matrix model calculation, Nucl. Phys. B 651 (2003) 106 [hep-th/0211123] [INSPIRE].
N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998) 323 [hep-th/9609219] [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. Wijnholt, Five-dimensional gauge theories and unitary matrix models, hep-th/0401025 [INSPIRE].
L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4D gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [INSPIRE].
V. Mehta and C. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980) 205.
I. Biswas, Parabolic bundles as orbifold bundles, Duke Math. J. 88 (1997) 305.
B. Feigin, M. Finkelberg, A. Negut and R. Leonid, Yangians and cohomology rings of Laumon spaces, arXiv:0812.4656.
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].
G.W. Moore, N. Nekrasov and S. Shatashvili, D particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000) 77 [hep-th/9803265] [INSPIRE].
M. Billó, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential in \( \mathcal{N} \) = 2⋆ theories (I): the ADE algebras, JHEP 11 (2015) 024 [arXiv:1507.07709] [INSPIRE].
M. Billó, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential of \( \mathcal{N} \) =2⋆ theories (II): the non-simply laced algebras, JHEP 11 (2015) 026 [arXiv:1507.08027] [INSPIRE].
H. Awata et al., Localization with a surface operator, irregular conformal blocks and open topological string, Adv. Theor. Math. Phys. 16 (2012) 725 [arXiv:1008.0574] [INSPIRE].
L.C. Jeffrey and F. . Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291.
D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, JHEP 10 (2016) 012 [arXiv:1412.2781] [INSPIRE].
M. Bullimore, H.-C. Kim and P. Koroteev, Defects and quantum Seiberg-Witten geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
H.-Y. Chen, T.J. Hollowood and P. Zhao, A 5d/3d duality from relativistic integrable system, JHEP 07 (2012) 139 [arXiv:1205.4230] [INSPIRE].
K.K. Kozlowski and J. Teschner, TBA for the Toda chain, arXiv:1006.2906 [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
A. Sciarappa, Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory, JHEP 10 (2017) 116 [arXiv:1706.05142] [INSPIRE].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, in From fields to strings, M. Shifman ed., Worlds Scientific, Singapore (2005), hep-th/0302191 [INSPIRE].
R. Flume, F. Fucito, J.F. Morales and R. Poghossian, Matone’s relation in the presence of gravitational couplings, JHEP 04 (2004) 008 [hep-th/0403057] [INSPIRE].
S.K. Ashok et al., Chiral observables and S-duality in N = 2∗ U(N ) gauge theories, JHEP 11 (2016) 020 [arXiv:1607.08327] [INSPIRE].
G.V. Dunne, Aspects of Chern-Simons theory, in the proceedings of Topological Aspects of Low-dimensional Systems, July 7-31, Les Houches, France (1998), hep-th/9902115 [INSPIRE].
D. Tong, Dynamics of N = 2 supersymmetric Chern-Simons theories, JHEP 07 (2000) 019 [hep-th/0005186] [INSPIRE].
M. Aganagic, K. Hori, A. Karch and D. Tong, Mirror symmetry in (2 + 1)-dimensions and (1 + 1)-dimensions, JHEP 07 (2001) 022 [hep-th/0105075] [INSPIRE].
O. Aharony and D. Fleischer, IR dualities in general 3d supersymmetric SU(N ) QCD theories, JHEP 02 (2015) 162 [arXiv:1411.5475] [INSPIRE].
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Ashok, S.K., Billò, M., Dell’Aquila, E. et al. Surface operators, chiral rings and localization in \( \mathcal{N} \) =2 gauge theories. J. High Energ. Phys. 2017, 137 (2017). https://doi.org/10.1007/JHEP11(2017)137
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DOI: https://doi.org/10.1007/JHEP11(2017)137