1/24 BPS Wilson Loops in ABJ(M) Theory

• 1/24 BPS Wilson loops are supersymmetric line operators in ABJ(M) theory constructed to preserve only one of the 24 supercharges.
• They interpolate between 1/6 and 1/2 BPS configurations by adjusting complex coupling parameters, offering a controlled setting to study defect RG flows.
• The framework leverages perturbative expansions and Fermi gas techniques to yield precise predictions, supporting tests of defect g-theorems in one-dimensional SCFTs.

A 1/24 BPS Wilson loop in ABJ(M) theory is a supersymmetric Wilson loop operator defined along the Euclidean circle, generically preserving a single real supercharge—corresponding to the minimal possible fraction (1/24) of the total 24 supercharges in the $\mathcal{N}=6$ superconformal algebra. These operators interpolate continuously between known 1/6-BPS and 1/2-BPS Wilson loops and serve as prototypes for studying defect RG flows and dynamical enhancement and breaking of supersymmetry in the space of line operators (Castiglioni et al., 2022).

1. Definition and Structure of 1/24 BPS Wilson Loops

The 1/24 BPS Wilson loop in ABJ(M) theory is constructed as the supertrace of a superconnection $\mathcal{L}(\tau)$ in the fundamental representation of $U(N_1|N_2)$, integrated over the unit circle $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$ with $\tau\in[0,2\pi]$:

$$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$

The superconnection has bosonic diagonal blocks coupled to the gauge fields $A_\mu$, $\hat{A}_\mu$ and scalar matter, while the off-diagonal blocks contain specific $\tau$-dependent couplings to bifundamental fermions. The generic form reads

$$\mathcal{L}(\tau) = \begin{pmatrix} A_\mu \dot{x}^\mu\! -\! (2\pi i/k) M_J{}^I C_I \bar C^J & \bar{f}(\tau) \ f(\tau) & \hat{A}_\mu \dot{x}^\mu\! -\! (2\pi i/k) M_J{}^I \bar C^J C_I \end{pmatrix}$$

with explicit dependence on eight complex parameters $\mathcal{L}(\tau)$0 and $\mathcal{L}(\tau)$1. The BPS property is ensured by a particular choice of these couplings so that they preserve the single supercharge

$\mathcal{L}(\tau)$2

For generic parameter values, only this supercharge is preserved—yielding a 1/24 BPS line operator. Special limits enhance the preserved symmetry, connecting to known 1/6 and 1/2 BPS Wilson loops.

2. Supersymmetry Interpolation and Enhanced Fixed Points

The parameter space of the 1/24 BPS Wilson loops forms a family interpolating between distinct supersymmetric loop operators:

• 1/6 BPS bosonic and fermionic loops: Setting all $\mathcal{L}(\tau)$3 to zero or restricting activation to a single block renders the loop purely bosonic or enhances the preserved supercharges to 1/6 BPS.
• 1/2 BPS fermionic loops: Imposing the additional normalization $\mathcal{L}(\tau)$4 or $\mathcal{L}(\tau)$5 leads to 1/2 BPS loops with enhanced $\mathcal{L}(\tau)$6 symmetry, recovering the maximal supersymmetric circular Wilson loops.

The interpolation is achieved by varying $\mathcal{L}(\tau)$7 continuously. The full parameter space exhibits RG flows between these fixed points (see Section 4).

3. 1D Effective Theory and Perturbative Expansion

A one-dimensional auxiliary quantum field theory is constructed on the Wilson loop contour, introducing an off-diagonal Grassmann-odd supermatrix field $\mathcal{L}(\tau)$8 with action:

$\mathcal{L}(\tau)$9

The expectation value of the Wilson loop is given by

$U(N_1|N_2)$0

This formalism enables the computation of the vacuum expectation value up to two loops in the 't Hooft coupling $U(N_1|N_2)$1. At one loop, $U(N_1|N_2)$2 by planarity, so the leading non-trivial correction is order $U(N_1|N_2)$3, manifesting the nontrivial dynamics and parameter dependence of 1/24 BPS loops (Castiglioni et al., 2022).

4. Renormalization Group Flows and $U(N_1|N_2)$4-Functions

The couplings $U(N_1|N_2)$5 in the loop superconnection run under the RG with nontrivial $U(N_1|N_2)$6-functions:

$U(N_1|N_2)$7

These $U(N_1|N_2)$8-functions are marginally relevant deformations away from the 1/6 BPS bosonic fixed point $U(N_1|N_2)$9. The RG flows interpolate between

• a UV fixed point corresponding to the 1/6 BPS bosonic loop (preserving four supercharges, $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$0 algebra),
• to IR fixed points on the circle $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$1 or $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$2, corresponding to enhanced BPS loops (up to 1/2 BPS, $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$3 algebra).

Along generic directions in the parameter space, only one supercharge is preserved (genuine 1/24 BPS line), yielding what are termed "enriched RG flows," with at least one supercharge preserved along the flow.

5. g-Theorem and Defect SCFT Structure

A one-dimensional g-theorem is established for these BPS defect RG flows, relating the expectation values of Wilson loops at UV and IR fixed points:

$x^\mu(\tau) = (0,\cos\tau,\sin\tau)$4

where $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$5, and $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$6 is one of the real interpolation parameters. Since $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$7 along the flow from $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$8 (UV) to $x^\mu(\tau) = (0,\cos\tau,\sin\tau)$9 (IR), the expectation value monotonically decreases, $\tau\in[0,2\pi]$0, interpreting $\tau\in[0,2\pi]$1 as the defect sphere partition function.

At each fixed point, the Wilson line defines a defect SCFT:

• For 1/6 BPS: $\tau\in[0,2\pi]$2 algebra, with field insertions in $\tau\in[0,2\pi]$3 and $\tau\in[0,2\pi]$4.
• For 1/2 BPS: $\tau\in[0,2\pi]$5 algebra. Local defect operators and their spectrum are encoded in the 1D theory. The small-$\tau\in[0,2\pi]$6 perturbation away from the UV fixed point has anomalous dimension $\tau\in[0,2\pi]$7, consistent with marginal relevance.

6. Relation to 1/6 and 1/2 BPS Wilson Loops and Exact Results

The classification of 1/24 BPS Wilson loops reveals that special regions in parameter space enhance the preserved supersymmetry:

Parameter regime	BPS fraction	Interpretation
All $\tau\in[0,2\pi]$8	1/6	Bosonic Wilson loop
Only $\tau\in[0,2\pi]$9 or only $$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$0 nonzero	1/6	Fermionic type I/II
$$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$1 (type I)	1/2	1/2 BPS fermionic loop
$$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$2 (type II)	1/2	1/2 BPS fermionic loop
Generic $$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$3	1/24	Minimal BPS Wilson loop

The vacuum expectation value of 1/6 and 1/2 BPS circular Wilson loops is computed via the Fermi gas approach to ABJM/ABJ matrix models (Klemm et al., 2012), culminating in explicit Airy function expressions. For the 1/2 BPS Wilson loop with winding $$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$4:

$$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$5

with $$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$6 and $$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$7 the Airy function.

7. Significance and Applications

The construction and analysis of 1/24 BPS Wilson loops establish a continuous family of supersymmetric line operators in ABJ(M) theory, realizing a broad spectrum of preserved supersymmetry fractions. These loops provide tractable models for studying RG flows on defect operators, BPS enhancement/splitting phenomena, and the strong coupling behavior of nontrivial line operators. Their study also leads to rigorous tests of conjectured defect $$W[C] = \mathrm{sTr} \, \mathcal{P} \exp\left(-i \oint d\tau \mathcal{L}(\tau)\right).$$8-theorems, explicit computation of operator expectation values via supermatrix models and Fermi gas methods, and a better understanding of the operator spectrum of 1D SCFTs localized on Wilson loop contours (Castiglioni et al., 2022, Klemm et al., 2012).