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 S on ℝ3× S1. 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 ex- pected symmetry properties of the line defect expectation value and reproduces the results derived using the AGT correspondence.
Highlights
’t Hooft defects are an important tool for understanding non-perturbative phenomena in quantum field theory
In this paper we study the contribution of monopole bubbling to the expectation value of supersymmetric ’t Hooft defects in Lagrangian theories of class S on R3 × S1
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
Summary
’t Hooft defects are an important tool for understanding non-perturbative phenomena in quantum field theory. Where HP is the Hilbert space of the 4D theory in the presence of the ’t Hooft defect L[P,0], F is the fermion number, β is the radius of the thermal circle, H is the Hamiltonian, Θ = (θe, θm) is the vector of electric and magnetic theta angles and Q = (γe, γm) is the vector of asymptotic electric and magnetic charges.2 This index can be computed as the path integral of the 4D theory on R3 × S1 with appropriate boundary conditions by using localization [33]. We will explain the origin of these problems with the standard localization procedure applied to the computation of Zmono(P, v) and describe how to correct it by studying the example of type A1 theories of class S with G = SU(2) The subtleties of this computation will come from trying to apply localization to a theory with a continuum of states coming from non-compact directions in field space with finite potential energy at infinity. We will show that the localization computation for the Witten index misses the contribution of BPS states along these non-compact directions and that by adding their contribution to the Witten index, we reproduce the AGT results
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