Framing and localization in Chern-Simons theories with matter

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.

The quantum 1/2 BPS Wilson loop in N = 4 $$ \mathcal{N}=4 $$ Chern-Simons-matter theories

“…We stress that such an occurrence shares the same ilk of that recently uncovered at three loops for the 1/6 BPS WL in the ABJM model in [13] and mentioned at the beginning of this section. In that situation an analogous 1PR diagram contributes, along with other diagrams, to reproduce the three loop imaginary term of the localization weak coupling expansion.…”

“…We evaluate them using DRED prescription in D = 3 − 2 . This regularization has been already proved to be consistent with supersymmetry in three-dimensional CS theories [1,13,[19][20][21][22][23].…”

“…We find that at third order a non-vanishing, purely imaginary correction appears. Comparing it with a perturbative calculation done at non-vanishing framing, we prove that this contribution corresponds to a loop correction to the framing factor of the Wilson loop due to interacting matter [13]. Therefore, we expect no three-loop corrections to the expectation value of the actual 1/2 BPS operator when computed in ordinary perturbation theory at framing zero.…”

“…For CS theories coupled to matter the identification of framing contributions in WL expectation values computed with localization and their perturbative origin is less clear. This issue has been recently analyzed in [13] for the 1/6 BPS WL in the ABJ(M) model. There, it has been shown that starting from three loops matter interactions induce nontrivial perturbative corrections to the one-loop framing factor in (4.12), reproducing the localization prediction at third order.…”

“…We now apply the procedure of [13] to N = 4 CS-matter theory under investigation to provide a perturbative explanation of the imaginary terms in localization results (4.8) and (4.11) as coming from framing. In order to do so, we focus on the bosonic 1/4 BPS WL W + 1/4 , whose framing contributions are easier to understand perturbatively.…”