Loop operators and S-duality from curves on Riemann surfaces

By N. Drukker, D. Morrison, T. Okuda. 2009

160 citations recorded by the source corpus.

We study Wilson-'t Hooft loop operators in a class of N=2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn's classification of homotopy classes of non-self-intersecting curves on an associated Riemann surface--the same surface which characterizes the gauge theory. Our analysis provides an explicit prediction for the action of S-duality on loop operators in these theories which we check against the known duality transformation in several examples.

Subfields: Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/8bef3d6e1f42cd8071c2afdb287cdbc6ee4f28bf

How researchers see this paper

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In Mathematical Physics

1. D. Gaiotto
2. Sergei Gukov
3. Tudor Dimofte
4. E. Witten
5. Yuji Tachikawa
6. J. Teschner
7. Abhijit Gadde
8. N. Drukker

In Nuclear and High Energy Physics

1. D. Gaiotto
2. Sergei Gukov
3. Yuji Tachikawa
4. Tudor Dimofte
5. E. Witten
6. N. Seiberg
7. L. Rastelli
8. Abhijit Gadde

Citations in the corpus

• On the Defect Group of a 6D SCFT (2015, methodology): “A notable example of this type are the 4D N = 2 theories of class S [7–10] obtained by compactifications of the 6D (2, 0) theories on Riemann surfaces [11–15].”
• Chern-Simons theory and S-duality (2011, background): “From the viewpoint of the four-dimensional N = 2 gauge theory — constructed by compactifying the 6d five-brane theory on C — the operators Wγ can be identified either as half-BPS UV line operators [59] or, via passing to a double cover of C, as BPS charges in the four-dimensional N = 2 supersymmetry”
• Gauge theory loop operators and Liouville theory (2009, methodology): “First based on the comparison with the gauge theory in [25], we expect non-intersecting loops to provide a complete basis for loop operators.”
• Framed BPS states (2010, unknown): “Along the way, in Section 9, we extend the classification of line operators for the conformal A1 theories (proposed in [15]) to asymptotically free theories.”
• Gauge Theories Labelled by Three-Manifolds (2011, background): “From a different perspective, the classical external edge coordinates xE = exp(XE) on a triangulated geodesic boundary C = ∂M also correspond to vevs of line operators in the four-dimensional N = 2 theory T [C, su(2)]; and the quantized x̂E = exp(X̂E) correspond to the quantum line operators themsel”
• Reading between the lines of four-dimensional gauge theories (2013, background): “3 Similar conditions are present also for field theories with no known Lagrangian, such as the so-called class S theories, where the line operators still have a similar classification [8,7].”
• Hitchin systems, N = 2 gauge theories and W-gravity (2009, background): “o higher differentials. Let us remark that once one recognizes the Toda theory/W-gravity to be the theory quantizing the extended Teichmu¨ller space of Hitchin, one can extend the analysis of [2] and [10] to the higher rank case. In [2] it was shown that the Liouville dual of loop operators of the f”
• The omega deformation, branes, integrability and Liouville theory (2010, background): “2, and hence the half-BPS operators studied in [57] have analogs in our situation.”
• Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry (2009, background): “The authors of [10] propose a suggestive identification between the set of line operator charges and the set of (homotopy classes of) closed non-selfintersecting curves in C.”
• Five-dimensional AGT conjecture and the deformed Virasoro algebra (2009, background): “The AGT relation is also closely related with the current active subjects such as loop operators [11]–[13], wall crossing [14]–[16], integrability structures (see, e.”
• S-duality and 2d topological QFT (2009, background): “A striking correspondence between the Nekrasov’s instanton partition function [4] of the 4d theory and Liouville field theory on Σ has been conjectured in [5] and further explored in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].”
• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems (2009, unknown): “September 2009”
• The Quiver Matrix Model and 2d-4d Conformal Connection (2009, background): “al strings, etc. Recent progress has been triggered by the construction of a large class of N = 2 superconformal SU(n) “generalized quiver” gauge theories in four dimensions by Gaiotto [1]. (See also [2, 3, 4, 5, 6]). Subsequently an interesting conjecture has been made by Alday, Gaiotto and Tachika”
• Affine SL(2) Conformal Blocks from 4d Gauge Theories (2010, background): “This construction led to the classification of the loop operators in four dimensions in terms of the loops on C2 [8] and the evaluation of their expectation values [6, 9].”
• The virtue of defects in 4D gauge theories and 2D CFTs (2010, background): “The same symmetries (Q and J +R) can be preserved by supersymmetric Wilson-’t Hooft loops [32, 3, 4], by surface operators [3] and by domain walls.”