Framed defects in ABJ and ABJM theories

We investigate the role of framing in a family of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>1</a:mn> <a:mo>/</a:mo> <a:mn>24</a:mn> </a:math> Bogomol’nyi–Prasad–Sommerfield (BPS) Wilson loops in Aharony-Bergman-Jafferis-(Maldacena) theory, which define flows between <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mn>1</c:mn> <c:mo>/</c:mo> <c:mn>6</c:mn> </c:math> BPS and the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mn>1</e:mn> <e:mo>/</e:mo> <e:mn>2</e:mn> </e:math> BPS superconformal fixed points. We analyze in perturbation theory how framing affects both the expectation values of these operators and the correlation functions of local insertions on the defect, as well as its interplay with renormalization group flow and the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>g</g:mi> </g:math> -theorem. We obtain a nontrivial identity between the one-point function of the defect stress tensor and a <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi mathvariant="script">Q</i:mi> </i:math> -exact correlator, which establishes a direct link between scale, superconformal invariance, cohomological equivalence and framing. This analysis confirms that framing one preserves supersymmetry and cohomological equivalence of operators at the quantum level, establishing a precise match with localization predictions. Finally, we propose a holographic interpretation of framing at strong coupling, identifying it with a coupling to the background <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>B</l:mi> </l:math> -field in the dual string theory.