---
title: 1/24 BPS Wilson Loops in ABJ(M) Theory
url: https://www.emergentmind.com/topics/1-24-bps-wilson-loops
type: topic
---

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

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 [2211.16501].

## 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 $\{\alpha_i, \bar\alpha^i\}_{i=1,2}$ and $\{\beta^j, \bar\beta_j\}_{j=3,4}$. The BPS property is ensured by a particular choice of these couplings so that they preserve the single supercharge

\[
\mathcal{Q} = (Q_{12}^+ - i S_{12}^+) + (Q_{34+} - i S_{34+}).
\]

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 $\alpha,\beta$ 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 $\bar\alpha^i\alpha_i=1$ or $\beta^j\bar\beta_j=-1$ leads to 1/2 BPS loops with enhanced $SU(3)$ symmetry, recovering the maximal supersymmetric circular Wilson loops.
  
The interpolation is achieved by varying $(\alpha_i,\beta^j)$ 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 $\Psi(\tau)$ with action:

\[
S_{\rm eff} = S_{\rm ABJM} + \int d\tau \, \operatorname{Tr} [\, \bar{\Psi}(\partial_\tau + i \mathcal{L}(\tau)) \Psi \,].
\]

The expectation value of the Wilson loop is given by

\[
\langle W \rangle = \langle \mathrm{sTr} \mathcal{P} e^{-i\oint \mathcal{L}} \rangle = \frac{1}{2} \langle \operatorname{Tr} \Psi(2\pi) \bar\Psi(0) \rangle_{1D}.
\]

This formalism enables the computation of the vacuum expectation value up to two loops in the 't Hooft coupling $g=\sqrt{2\pi/k}$. At one loop, $A(\alpha,\beta)=0$ by planarity, so the leading non-trivial correction is order $g^4$, manifesting the nontrivial dynamics and parameter dependence of 1/24 BPS loops [2211.16501].

## 4. Renormalization Group Flows and $\beta$-Functions

The couplings $\{\alpha_i, \bar\alpha^i, \beta^j, \bar\beta_j\}$ in the loop superconnection run under the RG with nontrivial $\beta$-functions:

\[
\begin{align*}
\beta_{\alpha_k} &= \frac{g^2}{4\pi}(N_1 + N_2)\big( \bar\alpha^i\alpha_i + \beta^j\bar\beta_j - 1 \big)\alpha_k, \\
\beta_{\beta^\ell} &= \frac{g^2}{4\pi}(N_1 + N_2)\big( \bar\alpha^i\alpha_i + \beta^j\bar\beta_j + 1 \big)\beta^\ell.
\end{align*}
\]

These $\beta$-functions are marginally relevant deformations away from the 1/6 BPS bosonic fixed point $(\alpha=\beta=0)$. The RG flows interpolate between
- a UV fixed point corresponding to the 1/6 BPS bosonic loop (preserving four supercharges, $su(1,1|1)$ algebra),
- to IR fixed points on the circle $\bar\alpha^i\alpha_i + \beta^j\bar\beta_j = 1$ or $-1$, corresponding to enhanced BPS loops (up to 1/2 BPS, $su(1,1|3)$ 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:

\[
\partial_x \ln \langle W \rangle = 2 \kappa \beta_x,
\]
where $\kappa = \pi g^2 N_1 N_2 /(N_1+N_2)$, and $x$ is one of the real interpolation parameters. Since $\beta_x<0$ along the flow from $x=0$ (UV) to $x=1$ (IR), the expectation value monotonically decreases, $g_{UV}>g_{IR}$, interpreting $g=\langle W \rangle_{fixed\;pt}$ as the defect sphere partition function.

At each fixed point, the Wilson line defines a defect SCFT:
- For 1/6 BPS: $su(1,1|1)$ algebra, with field insertions in $U(N_1|N_2)$ and $su(1,1|1)$.
- For 1/2 BPS: $su(1,1|3)$ algebra.
Local defect operators and their spectrum are encoded in the 1D theory. The small-$x$ perturbation away from the UV fixed point has anomalous dimension $\gamma=-(g^2/4\pi)(N_1+N_2)<0$, 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 $\alpha,\beta=0$                   | 1/6          | Bosonic Wilson loop     |
| Only $\alpha$ or only $\beta$ nonzero  | 1/6          | Fermionic type I/II     |
| $\bar\alpha^i\alpha_i=1$ (type I)      | 1/2          | 1/2 BPS fermionic loop  |
| $\beta^j\bar\beta_j=-1$ (type II)      | 1/2          | 1/2 BPS fermionic loop  |
| Generic $\alpha,\beta$                 | 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 [1207.0611], culminating in explicit Airy function expressions. For the 1/2 BPS Wilson loop with winding $m$:

\[
W_m^{1/2}(N,k) = \frac{1}{4}\csc{\Big(\frac{2\pi m}{k}\Big)} \frac{\operatorname{Ai}\left(C^{-1/3}(N - k/24 - (6m+1)/(3k))\right)}{
\operatorname{Ai}\left(C^{-1/3}(N - k/24 - 1/(3k))\right)}
\]

with $C=2/(\pi^2 k)$ and $\operatorname{Ai}$ 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 $g$-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 [2211.16501][1207.0611].

Source: https://www.emergentmind.com/topics/1-24-bps-wilson-loops