Gauge theory loop operators and Liouville theory

Abstract

We propose a correspondence between loop operators in a family of four dimensional N = 2 gauge theories on S 4 -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 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 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.

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

“…The theory on D 2 × S 1 has a boundary condition labelled by a vacuum, which can be constructed in the UV using our exceptional Dirichlet boundary conditions from section 4. 3d mirror symmetry exchanges Higgs and Coulomb branches, and is found 1 The idea that Omega backgrounds [16][17][18][19] are related to quantization arose in [20] and many related works, including [21][22][23][24][25][26][27]; cf. the recent review [28].…”

Affine sl(N)conformal blocks from $ \mathcal{N} = 2 $ SU(N) gauge theories

“…In this case, since |0, p; j only has non-zero inner product with its conjugate: 12) our proposal leads to…”

“…Wilson and 't Hooft loops), and three-dimensional (domain wall) defects. Such defects have been considered in [11][12][13], and [14], respectively.…”

Wilson loops and chiral correlators on squashed spheres

“…An interesting spacetime to consider is that of a squashed sphere [15]. A motivation for considering four-dimensional gauge theories on squashed spheres comes from the AGT correspondence that relates the partition function of N = 2 supersymmetric theories to correlators in a dual two-dimensional CFT with the squeezing parameter 1 / 2 parametrizing the central charge of the CFT [16,17,19]. In [18], it was proposed that the expectation value of a circular Wilson loop oriented along, let us say the first plane, on the squashed sphere is given by the insertion of the phase tr e 2πia 1…”