Abstract
Supersymmetric localization provides exact results that should match QFT computations in some regularization scheme. The agreement is particularly subtle in three dimensions where complex answers from localization procedure sometimes arise. We investigate this problem by studying the expectation value of the 1/6 BPS Wilson loop in planar ABJ(M) theory at three loops in perturbation theory. We reproduce the corresponding term in the localization result and argue that it originates entirely from a non-trivial framing of the circular contour. Contrary to pure Chern-Simons theory, we point out that for ABJ(M) the framing phase is a non-trivial function of the couplings and that it potentially receives contributions from vertex-like diagrams. Finally, we briefly discuss the intimate link between the exact framing factor and the Bremsstrahlung function of the 1/2-BPS cusp.
Highlights
Contributions come only from diagrams containing collapsible propagators [2, 3], that is propagators joining two points on the contour that can get together; 2) In Landau gauge where these calculations are performed, the gauge propagator is one-loop exact
The agreement is subtle in three dimensions where complex answers from localization procedure sometimes arise. We investigate this problem by studying the expectation value of the 1/6 BPS Wilson loop in planar ABJ(M) theory at three loops in perturbation theory
In this paper we are going to investigate this problem by studying the bosonic 1/6 BPS Wilson loop in ABJ(M) theory [12,13,14] in the planar limit
Summary
As extensively discussed in literature [1,2,3], in pure Chern-Simons theory the vacuum expectation value of Wilson loop operators on close paths Γ. The structure of the framing factor in susy and non-susy pure CS theories heavily relies on the fact that in Landau gauge these theories are all-loop finite and in dimensional reduction scheme not even finite corrections to the vector propagator seem to arise [4, 6] (statement (1) above) This implies that the 1/k effect coming from the exchange of a tree-level propagator, eventually exponentiated by summing all order diagrams as, is the only possible source of framing. We are going to prove that this is the case, being this contribution associated to a higher-order correction to the framing function coming from a non-vanishing finite two-loop correction to the gauge propagator. To give support to this expectation we compute the first non-trivial contribution to Πe(p, λi), that is the two-loop gauge propagator
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