Abstract
We demonstrate explicitly that the vacuum expectation values (vevs) of BPS line operators in 4d N=2 super Yang–Mills theory compactified on a circle, computed by localization techniques, can be expanded in terms of Darboux coordinates as proposed by Gaiotto, Moore, and Neitzke [1]. However, we need to augment the expressions for Darboux coordinates with additional monopole bubbling contributions to obtain a precise match. Using D-brane realization of these singular BPS line operators, we derive and incorporate the monopole bubbling contributions as well as predict the degeneracies of framed BPS states contributing to the line operator vevs in the limit of vanishing simultaneous spatial and R-symmetry rotation fugacity parameter.
Highlights
To match the Darboux coordinate expansion with the complete localization computation, further refinement is needed to incorporate the so-called “monopole bubbling” effect
By reducing supersymmetry to N = 2, we will obtain a generalization of the Darboux coordinates including the factors due to this monopole bubbling effect
We will perform the match in the limit where the fugacity parameter for simultaneous spatial and R-symmetry rotations λ vanishes, which is analogous to the limit of deformation parameters 1,2 → 0 in the Nekrasov instanton partitions defined on Ω background [5, 6] in order to recover the underlying Seiberg-Witten curves
Summary
We will study 4d N = 2 supersymmetric gauge theories on R3 × S1, parameterized by Cartesian coordinates: xμ = (xi, τ ), (μ = 1, 2, 3, 4, i = 1, 2, 3) with τ ∼ τ + 2πR. Following [3, 8, 10] (for recent surveys, see [9, 11, 12]), we review here the relevant details about the line operators and BPS states in such compactified theories. We will review the Darboux coordinates, which give the metric on their Coulomb branch. This will serve to fix the notations and terminology used in the rest of the note
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