Abstract
We consider topological defect networks with junctions in A N − 1 Toda CFT and the connection to supersymmetric loop operators in \( \mathcal{N}=2 \) theories of class \( \mathcal{S} \) on a four-sphere. Correlation functions in the presence of topological defect networks are computed by exploiting the monodromy of conformal blocks, generalising the notion of a Verlinde operator. Concentrating on a class of topological defects in A 2 Toda theory, we find that the Verlinde operators generate an algebra whose structure is determined by a set of generalised skein relations that encode the representation theory of a quantum group. In the second half of the paper, we explore the dictionary between topological defect networks and supersymmetric loop operators in the \( \mathcal{N}={2}^{*} \) theory by comparing to exact localisation computations. In this context, the the generalised skein relations are related to the operator product expansion of loop operators.
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References
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [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 [Addendum ibid. 1210 (2012) 051] [arXiv:1206.6359] [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].
N. Wyllard, A N − 1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP 09 (2009) 031 [arXiv:0907.2593] [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].
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].
V.B. Petkova, On the crossing relation in the presence of defects, JHEP 04 (2010) 061 [arXiv:0912.5535] [INSPIRE].
J. Gomis and B. Le Floch, ’t Hooft Operators in Gauge Theory from Toda CFT, JHEP 11 (2011) 114 [arXiv:1008.4139] [INSPIRE].
F. Passerini, Gauge Theory Wilson Loops and Conformal Toda Field Theory, JHEP 03 (2010) 125 [arXiv:1003.1151] [INSPIRE].
D. Xie, Higher laminations, webs and N = 2 line operators, arXiv:1304.2390 [INSPIRE].
G. Kuperberg, Spiders for rank 2 lie algebras, Commun. Math. Phys. 180 (1996) 109 [q-alg/9712003].
H.L. Verlinde, Conformal Field Theory, 2-D Quantum Gravity and Quantization of Teichmüller Space, Nucl. Phys. B 337 (1990) 652 [INSPIRE].
D. Gaiotto and E. Witten, Knot Invariants from Four-Dimensional Gauge Theory, Adv. Theor. Math. Phys. 16 (2012) 935 [arXiv:1106.4789] [INSPIRE].
D. Kim, Graphical calculus on representations of quantum lie algebras, J. Knot Theor. Ramif. 15 (2006) 453 [math.QA/0310143].
S. Morrison, A diagrammatic category for the representation theory of u q (sl n ), arXiv:0704.1503.
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].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
V.A. Fateev and A.V. Litvinov, On differential equation on four-point correlation function in the Conformal Toda Field Theory, JETP Lett. 81 (2005) 594 [hep-th/0505120] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory I, JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory II, JHEP 01 (2009) 033 [arXiv:0810.3020] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
T. Okuda and V. Pestun, On the instantons and the hypermultiplet mass of N = 2* super Yang-Mills on S 4, JHEP 03 (2012) 017 [arXiv:1004.1222] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
Y. Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories, JHEP 05 (2014) 020 [arXiv:1309.0697] [INSPIRE].
D. Xie, Aspects of line operators of class S theories, arXiv:1312.3371 [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].
T. Dimofte, D. Gaiotto and R. van der Veen, RG Domain Walls and Hybrid Triangulations, arXiv:1304.6721 [INSPIRE].
M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, JHEP 1410 (2014) 62 [arXiv:1401.3379] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
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Bullimore, M. Defect networks and supersymmetric loop operators. J. High Energ. Phys. 2015, 66 (2015). https://doi.org/10.1007/JHEP02(2015)066
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DOI: https://doi.org/10.1007/JHEP02(2015)066