Gauge theory loop operators and Liouville theory

By N. Drukker, Joaquim Gomis, T. Okuda, J. Teschner. 2009

268 citations recorded by the source corpus.

We propose a correspondence between loop operators in a family of four dimensional $$ \mathcal{N} $$ = 2 gauge theories on S4 — including Wilson, ‘t Hooft and dyonic operators — and Liouville theory loop operators on a Riemann surface. This extends the beautiful relation between the partition function of these $$ \mathcal{N} $$ = 2 gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa. We show that the computation of these Liouville correlators with the insertion of a Liouville loop operator reproduces Pestun’s formula capturing the expectation value of a Wilson loop operator in the corresponding gauge theory. We prove that our definition of Liouville loop operators is invariant under modular transformations, which given our correspondence, implies the conjectured action of S-duality on the gauge theory loop operators. Our computations in Liouville theory make an explicit prediction for the exact expectation value of ’t Hooft and dyonic loop operators in these $$ \mathcal{N} $$ = 2 gauge theories. The Liouville loop operators are also found to admit a simple geometric interpretation within quantum Teichmüller theory as the quantum operators representing the length of geodesics. We study the algebra of Liouville loop operators and show that it gives evidence for our proposal as well as providing definite predictions for the operator product expansion of loop operators in gauge theory.

Subfields: Applied Mathematics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/5cd418302cdda1517dc3843108226e29379e324f

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Citations in the corpus

Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I (2010, methodology): “Knowing the modular transformation properties of the Liouville conformal blocks [57, 74, 75, 78] now allows us to investigate and test the S-duality conjectures in these gauge theories, as illustrated in [2,11].”
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.g., [17, 18] and references therein) and so on.”
Chern-Simons theory and S-duality (2011, methodology): “In other words, for Wilson and ’t Hooft loops we have q 1 4W1/2H1/2− q− 1 4H1/2W1/2 = −(q 1 2 − q− 12 )W (1,1) 1/2 [28, 80].”
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].”
Bootstrapping the superconformal index with surface defects (2012, background): “25) in [72], which displays operator inserting a ’t Hooft operator in the N = 2∗ theory.”
The virtue of defects in 4D gauge theories and 2D CFTs (2010, methodology): “Explicit computation of the monodromy acquired by the V1,2j+1(z) degenerate field defining the Liouville theory Verlinde loop operator using the CFT fusion matrices confirmed that [3, 4] 〈Wj〉Tg,n(A1) = 〈Lj(p)〉Cg,n .”
3-Manifolds and 3d Indices (2011, methodology): “Crucially, they can be computed explicitly through a laborious algorithm [36, 37], sidestepping the difficult localization calculations of gauge theory.”
Hitchin systems, N = 2 gauge theories and W-gravity (2009, background): “Let us remark that once one recognizes the Toda theory/W-gra vity to be the theory quantizing the extended Teichmüller space of Hitchin, one can extend t he analysis of [3] and [11] to the higher rank case.”
Framed BPS states (2010, unknown): “For example, [57] argued that, in the case when the ADE group is A1 and C has only regular punctures, surface operators along non-self-intersecting paths on C carrying the fundamental representation of SU(2) suffice to describe all UV gauge theory operators: no higher representations seem to be need”
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”