1/2 BPS Maldacena–Wilson Loop in ABJM

• The 1/2 BPS Maldacena–Wilson loop is a codimension-two vortex operator defined by scalar singularities and specific gauge holonomy in ABJM theory.
• It preserves half of the supersymmetry (12 out of 24 supercharges) through BPS equations and branch cut structures sensitive to the Chern–Simons level.
• Its holographic dual is a probe M2–brane in AdS4×S7/ℤk, enabling precision tests of the AdS4/CFT3 correspondence via semiclassical analysis.

The 1/2 BPS Maldacena–Wilson loop is a class of codimension-two non-local operators in three-dimensional supersymmetric Chern–Simons–matter (notably ABJM) theories that are defined by imposing a vortex-like singularity in a complex scalar (and a specific holonomy in the gauge connection) along a one-dimensional curve. These operators are half–BPS, preserving 12 out of 24 supercharges, and provide a three-dimensional analog of surface operators in four-dimensional $\mathcal{N}=4$ super Yang–Mills. On the holographic side, such loop operators correspond to probe M2–brane solutions in $AdS_4\times S^7/\mathbb{Z}_k$ that terminate on the prescribed curve at the conformal boundary, with precise matching between the gauge–theory and gravity parameters.

1. Definition and Construction in Gauge Theory

A 1/2 BPS vortex Maldacena–Wilson loop in the $U(1)\times U(1)$ (and analogously for nonabelian $U(N)\times U(N)$) ABJM theory is characterized by a singular profile for one of the bifundamental complex scalar fields $C$, and a compatible background for the gauge field $A$. In a holomorphic local coordinate $z$ transverse to the loop, the BPS conditions and scaling dimension (dimension 1/2 for $C$) constrain the singularity to be

$$C(z) = \frac{\sqrt{\beta}}{z^{1/2}}, \quad A_z = -i\frac{\alpha}{2k z}$$

where $\beta>0$ determines the strength of the scalar singularity, $\alpha$ is an angular parameter (mod $2\pi$ due to large gauge invariance), and $k$ is the Chern–Simons level. This configuration satisfies the BPS equations $D_{\bar z}C=0$ and preserves 12 supercharges by specifying the appropriate vanishing of supersymmetry variations of the fermions (see (0810.4344), eqs. (2.11), (2.19)).

A key property is that the vortex profile enforces a branch cut: encircling the loop, the scalar $C$ acquires a $(-1)$ monodromy, rendering the basic configuration single-valued only for even $k$. Odd $k$ necessitates either a "doubling" of the vortex configuration or the introduction of nonabelian twists to maintain consistency of all gauge-invariant observables.

2. Preserved Supersymmetry and BPS Equations

The vortex loop preserves half of the $\mathcal{N}=6$ supersymmetry (12/24). The explicit counting arises from analyzing the solution to the BPS equations and the reality conditions on the supersymmetry variations. Upon decomposing the supercharges by helicity in the $(z, \bar z)$-plane, three of the Poincaré supercharges are preserved (eq. (2.11)), and by leveraging the reality structure of the spinors, this extends to six real Poincaré and six conformal supercharges preserved out of the total 24. This ensures the "half–BPS" character of the operator and singles out its protected status.

3. Parameters and Monodromy Structure

The fundamental data specifying the 1/2 BPS loop are $(\alpha, \beta)$:

• $\beta$ controls the amplitude of the codimension-two scalar singularity.
• $\alpha$ sets the gauge field holonomy around the loop, which is an angular variable due to large gauge transformations. Conformal invariance and the scaling dimension of $C$ rigorously fix the exponent of the singularity. The branch-cut structure, and thus the global consistency conditions, depend delicately on the parity of $k$. For $k$ odd, a doubling of the operator or suitable nonabelian structure is required.

4. Holographic M2–brane Dual Description

The $AdS_4/CFT_3$ correspondence identifies the 1/2 BPS vortex loop in ABJM theory with an M2–brane in $AdS_4\times S^7/\mathbb{Z}_k$ whose worldvolume is $AdS_2\times S^1$: it ends on the prescribed curve at the boundary and wraps an internal $S^1$ in $S^7/\mathbb{Z}_k$ (eqs. (3.16)–(3.18)). The mapping is explicit:

• The AdS radial position $u$ (determined by $\sinh u$) is related to $\beta$ as

$$\sinh u = \frac{\sqrt{\beta}}{\pi/(2\lambda)}, \quad \lambda = N/k$$

• The internal phase $\xi$ along the $S^1$ is proportional to $2\pi\alpha$ (eq. (3.24)).
• The preserved supersymmetry on the brane matches exactly the subset defined by the gauge theory BPS equations.

The classical expectation value is

$$\langle V_\text{loop}\rangle = \exp\left[-S_{M2}\right] = \exp\left[k\pi\frac{\sqrt{2\lambda}}{2} \right]$$

signaling non-trivial dependence on the 't Hooft coupling at strong coupling.

5. Physical and Mathematical Interpretation

The 1/2 BPS Maldacena–Wilson ("vortex") loop operator is not an ordinary order operator but a "disorder" (singularity-imposing) operator: it specifies codimension-two boundary conditions for both scalar and gauge fields along a curve. It generalizes the familiar Wilson loop by imposing a geometric (singular) profile rather than an explicit insertion. These operators provide a non-local probe of gauge-dynamics and the AdS$_4$/CFT$_3$ correspondence.

Such operators form a precise three-dimensional analog of surface operators in $\mathcal{N}=4$ SYM: both have complementary descriptions in terms of probe branes and in bubbling geometries, admit semiclassical computation of their expectation values and correlation functions, and enable direct tests of the duality across regimes.

6. Summary Table: Key Defining Data

Parameter	Gauge Theory Role	M2–brane Dual
$\alpha$	Gauge field holonomy	Internal phase $\xi\sim2\pi\alpha$ in $S^7/\mathbb{Z}_k$
$\beta$	Strength of scalar singularity	Radial position $u$, $\sinh u\propto\sqrt{\beta}$
Curve $C$	Support of the operator, typically a straight line or circle	Endpoint of M2–brane boundary on $AdS_4$

Further, the preserved 12 supercharges are matched exactly in the probe brane analysis.

7. Significance and Extensions

The construction of the 1/2 BPS Maldacena–Wilson (vortex) loop in ABJM and related Chern–Simons–matter theories provides a robust non-local observable whose physical and holographic characteristics allow for precision tests of the duality, parameter matching, and semiclassical cross-comparisons of observables. The explicit bridge between disorder operators in field theory and brane excitations in supergravity underlines the universality of codimension-two defects in supersymmetric gauge/string dualities (0810.4344). The full set of correlation functions and their semiclassical expansions, as well as possible bubbling supergravity solutions, provide a platform for further developments in understanding defects, non-local observables, and the landscape of BPS sectors in lower-dimensional gauge theories.