't Hooft Defects and Wall Crossing in SQM

Abstract

In this paper we study the contribution of monopole bubbling to the expectation value of supersymmetric 't Hooft defects in Lagrangian theories of class $\mathcal{S}$ on ${\mathbb{R}}^3\times S^1$. This can be understood as the Witten index of an SQM living on the world volume of the 't Hooft defect that couples to the bulk 4D theory. The computation of this Witten index has many subtleties originating from a continuous spectrum of scattering states along the non-compact vacuum branches. We find that even after properly dealing with the spectral asymmetry, the standard localization result for the 't Hooft defect does not agree with the result obtained from the AGT correspondence. In this paper we will explicitly show that one must correct the localization result by adding an extra term to the standard Jeffrey-Kirwan residue formula. This extra term accounts for the contribution of ground states localized along the non-compact branches. This extra term restores both the expected symmetry properties of the line defect expectation value and reproduces the results derived using the AGT correspondence.

arXiv comment61 Pages plus Appendices

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• A slow review of the AGT correspondence
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• 't Hooft lines of ADE-type and Topological Quivers
• ABCD of 't Hooft operators
• New ${\text{AdS}}_2/{\text{CFT}}_1$ pairs from ${\text{AdS}}_3$ and monopole bubbling
• Wall-crossing and operator ordering for 't Hooft operators in N=2 gauge theories
• Quantum Geometry and $\theta$-Angle in Five-Dimensional Super Yang-Mills
• Quantized Coulomb Branches, Monopole Bubbling and Wall-Crossing Phenomena in 3d $\mathcal{N}=4$ Theories
• S-duality of boundary lines in $\mathcal{N}=4$ SYM theories and supersymmetric indices
• $\mathcal{N}=4$ line defect correlators of type BCD
• The quantum UV-IR map for line defects in $\mathfrak{gl}(3)$-type class $S$ theories