Framed defects in ABJ(M)

We investigate the role of framing in a family of 1/24 BPS Wilson loops in ABJ(M) theory, which define flows between 1/6 BPS and the 1/2 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 RG flow and the g-theorem. We obtain a non-trivial identity between the one-point function of the defect stress tensor and a Q-exact correlator, which establishes a direct link between scale invariance, superconformal invariance and framing, and clarifies the deep connection between scale and cohomological anomalies. Finally, we propose a holographic interpretation of framing at strong coupling, identifying it with a coupling to the background B-field in the dual string theory.

💡 Research Summary

The paper investigates how framing – a topological regularisation choice intrinsic to three‑dimensional Chern‑Simons‑matter theories – influences a family of 1/24‑BPS Wilson loops in ABJ(M) theory. These loops depend on eight complex parameters (α_i, β_j) and interpolate continuously between the well‑studied 1/6‑BPS (bosonic) and 1/2‑BPS (fermionic) superconformal fixed points. At the classical level the whole family is cohomologically equivalent: they differ by Q‑exact deformations under a common supercharge Q. The authors ask whether this equivalence survives quantum corrections once framing is taken into account.

Section 2 reviews the construction of the interpolating loops and the notion of framing. In pure Chern‑Simons theory the expectation value of a Wilson loop is a knot invariant, but regularisation of short‑distance singularities requires a point‑splitting that introduces a nearby, non‑intersecting contour. The linking number between the original and displaced contour defines the framing number f, which appears as a phase factor. When matter fields are present, the framing dependence becomes more intricate: higher‑loop corrections can generate non‑trivial f‑dependent contributions beyond a simple overall phase.

In Section 3 the authors perform an explicit two‑loop perturbative computation of the expectation value of the 1/24‑BPS loop for arbitrary framing f and arbitrary values of the deformation parameters. The main findings are:

1. Framing = 1 is special. At f = 1 all members of the interpolating family have the same expectation value, independent of the α, β parameters. This value matches the exact result for the bosonic 1/6‑BPS loop obtained from supersymmetric localisation, confirming that localisation implicitly computes observables at framing 1.

2. Generic framing breaks cohomological equivalence. For f ≠ 1 the expectation value acquires a complex phase that cannot be removed by taking the modulus, except for the 1/2‑BPS case. The two‑node structure of the superconnection leads to two terms with a relative f‑dependent phase, producing interference effects. Consequently, the quantum expectation values differ among the interpolating loops, signalling a failure of the classical Q‑exact equivalence.

Section 4 studies correlation functions of local operators inserted on the defect. The authors analyse both bosonic and fermionic defect sectors and compute the one‑point function of the defect stress‑tensor T_{μν}. They discover a non‑trivial Ward identity (eqs. 4.17 and 4.23) that relates ⟨T⟩ to a Q‑exact correlator on the defect. Crucially, ⟨T⟩ vanishes exactly at f = 1, where supersymmetry is restored, while it is non‑zero for other framings. This provides a concrete link between scale invariance, superconformal invariance, cohomological triviality, and the choice of framing.

Section 5 addresses the g‑theorem for defect conformal field theories, which states that the defect entropy (the g‑function) should decrease along RG flows. The authors show that framing modifies the defect’s normal‑bundle transformation under conformal maps, generating additional contributions to the Ward identities. As a result, the g‑function can decrease (f < 1), stay constant (f = 1), or even increase (f > 1) along the RG flow, thereby violating the standard g‑theorem unless the framing is fixed to one.

In Section 6 the authors turn to the strong‑coupling regime using the AdS/CFT correspondence. Wilson loops in ABJ(M) are dual to fundamental strings whose world‑sheet ends on an AdS₂ minimal surface inside AdS₄ × CP³. The authors propose that the framing number corresponds to a background Kalb‑Ramond B‑field coupling on the string world‑sheet. When the B‑field is tuned such that the effective framing equals one, the string solution preserves the same supersymmetries as the field‑theory loop, matching the localisation result. This holographic picture offers a geometric interpretation of framing as a background flux rather than a purely regularisation artefact.

The conclusions summarise the achievements: (i) a full two‑loop perturbative evaluation of the interpolating Wilson loops at arbitrary framing; (ii) identification of framing = 1 as the supersymmetry‑preserving prescription that enforces quantum cohomological equivalence; (iii) derivation of Ward identities linking the defect stress‑tensor to Q‑exact correlators, valid only at f = 1; (iv) demonstration that generic framing obstructs a direct application of the g‑theorem; and (v) a holographic identification of framing with a B‑field coupling at strong coupling. The work clarifies the pivotal role of framing in the quantum dynamics of ABJ(M) Wilson loops, bridges perturbative field‑theory results with exact localisation and holography, and opens avenues for studying framing effects in other supersymmetric defect setups.