Wilson loops in ABJM theory reloaded

1. Problem Statement

The paper addresses the computation of supersymmetric Wilson loops in the ABJM theory (a 3D $\mathcal{N}=6$ superconformal Chern-Simons-matter theory with gauge group $U(N)_k \times U(N)_{-k}$). Specifically, it targets the 1/6 BPS Wilson loop ($W_{1/6}$) and the 1/2 BPS Wilson loop ($W_{1/2}$) with arbitrary winding number $n$.

Key Challenges:

• Parameter Dependence: Exact expressions are required for arbitrary values of the rank $N$ and the Chern-Simons level $k$.
• Semiclassical Failure: Previous attempts using the Fermi gas approach combined with a semiclassical (WKB) expansion successfully reproduced the partition function $Z(N,k)$ but failed to match high-precision numerical data for the Wilson loops when the winding number $n \geq 2$.
• Singularity Issues: The localization integral defining the Wilson loop is only well-defined for $k > 2n$. The semiclassical expansion, which assumes small $k$ (strong coupling), encounters singularities at $k=2n$ that cannot be analytically continued to the physical integer values of $k$ without encountering obstructions.
• Discrepancy: There was a longstanding unexplained discrepancy between numerical results and the semiclassical predictions for the perturbative part of the Wilson loop.

2. Methodology

The authors develop a new, exact technique that avoids the semiclassical approximation entirely. The methodology relies on the Fermi gas formalism combined with the Tracy-Widom approach to integral operators.

Core Steps:

1. Operator Representation:

   • The partition function $\Xi(z, k)$ (grand canonical) is represented as a Fredholm determinant of an integral operator $\rho$: $\Xi = \det(1 + z\rho)$.
   • The Wilson loop generating function $W_n(z, k)$ is expressed as a trace involving the resolvent of this operator: $W_n = \text{tr}\left( \frac{z\rho}{1+z\rho} U^n \right)$, where $U$ is a holonomy operator.

2. Introduction of the $\psi$-function:

   • An auxiliary function $\psi(x|z) = \langle E | (1 - z\rho)^{-1} | x \rangle$ is introduced. This function satisfies a Baxter equation (a finite-difference equation):
     $$\psi(x + i\pi k | z) + \psi(x - i\pi k | z) = z \chi(x) \psi(x|z)$$
     where $\chi(x) = 1/(2\cosh(x/2))$.

3. Derivation of Exact Relations:

   • Wronskian Relation: A relation connecting $\psi(x|z)$ and $\psi(x|-z)$ is derived, involving the free energy $F(z) = \log(\Xi(z)/\Xi(-z))$.
   • Shift Relation: By deforming integration contours in the complex plane, the authors derive a relation expressing $\psi(x + 2\pi i | z)$ as a linear combination of $\psi(x|z)$ and $\psi(x|-z)$. This allows the evaluation of $\psi$ at shifted arguments (crucial for the Wilson loop integral).
   • Parity Relation: A symmetry relation $\psi(-x|z) = e^{-x/k + F(z)} \psi(x|-z)$ is established.

4. Quantization Conditions:

   • By analyzing the consistency of the shift relations and the Baxter equation, the authors derive quantization conditions that relate the phase $\Phi(z)$ of the function $\psi$ at specific imaginary points (e.g., $x = \pm i\pi$) to the free energy $F(z)$. These conditions differ for even and odd $k$.

5. Recurrence Relations:

   • The integral defining the Wilson loop is decomposed into terms involving potentials $V_n(z)$ and integrals $U_n(z), I_n(y|z)$.
   • A system of recurrence relations is derived that allows these integrals to be computed entirely in terms of the special values of $\psi$ (which are fixed by the quantization conditions) and the free energy $F(z)$.

3. Key Contributions

• Exact Closed-Form Expressions: The paper provides exact expressions for the generating functions of Wilson loops $W_n(z, k)$ in terms of the free energy $F(z)$ and a phase function $\Phi(z)$ determined by algebraic/quantization conditions.
• Resolution of the Discrepancy: The method proves that the semiclassical expansion fails for $n \geq 2$ because the analytic continuation required for the WKB approximation is obstructed by singularities at $k=2n$. The new exact method bypasses this, matching numerical data perfectly.
• Specific Results for Low $n$:
  • For $n=1$ and $k=4$, the 1/2 BPS Wilson loop is given by a remarkably simple formula:
    $$W_{1/2}^{n=1}(z) = \frac{1}{2} \sqrt{z - 4 + 4e^{-2F(z)}}$$
  • General relations for $n=1, 2$ and arbitrary $k$ are derived, linking $W_{1/2}^{n=2}$ to $W_{1/2}^{n=1}$ and $F(z)$.
• Recovery of Known Limits:
  • Weak Coupling ($k \to \infty$): The exact results reproduce the known weak-coupling expansion.
  • Strong Coupling/Large $N$ ($N \to \infty$): The results match high-precision numerical data, confirming the validity of the approach in the non-perturbative regime.

4. Results

• Generating Function Formula: The Wilson loop generating function is expressed as:
  $$W_n(z, k) = \frac{i n^{-1}}{\sin(2\pi n/k)} \left[ A_n(z) + i z \partial_z \Phi(z) B_n(z) + i z \partial_z F(z) C_n(z) \right]$$
  where $A_n, B_n, C_n$ are real functions of $z$ constructed from $F(z)$ and $\Phi(z)$.
• Quantization of Phase: The phase $\Phi(z)$ is not arbitrary but is constrained by polynomial equations involving $F(z)$ (e.g., for $k=1, 2, 3, 4$).
• Large $N$ Behavior: In the limit of large chemical potential $\mu$ (large $N$), the authors derive the coefficient $C_n(k)$ in the asymptotic expansion of the Wilson loop:
  $$C_n(k) = -\sum_{j=1}^n \cot\left(\frac{2\pi j}{k}\right)$$
  This matches the numerical fitting results from previous literature, resolving the previous mismatch.

5. Significance

• Theoretical Breakthrough: This work provides the first exact, non-perturbative method for computing supersymmetric Wilson loops in ABJM theory for arbitrary $N$ and $k$ without relying on semiclassical approximations or conjectured dualities to topological strings.
• Clarification of Semiclassical Limits: It rigorously explains why the semiclassical approach fails for Wilson loops with $n \geq 2$ (due to the breakdown of analyticity in the complex plane for finite $k$), distinguishing the behavior of Wilson loops from the partition function.
• Bridge to Numerics: The derived exact formulas provide a powerful tool to compute Wilson loops efficiently, serving as an analytical counterpart to high-precision numerical simulations.
• Future Applications: The framework (Tracy-Widom approach applied to Fermi gas resolvents) is generalizable. The authors suggest it can be applied to latitude Wilson loops, Wilson loops with real masses, and deformed spheres ($S^3$ squashing), potentially solving similar non-perturbative problems in other supersymmetric gauge theories.

In summary, the paper successfully "reloads" the computation of Wilson loops by replacing approximate semiclassical methods with an exact operator-based formalism, resolving a decade-long discrepancy between theory and numerics in the ABJM model.