Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter

Abstract: We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N=2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.

• The paper introduces localization to reduce complex Wilson-loop path integrals to tractable non-Gaussian matrix models.
• It computes explicit Wilson loop expectation values for models like ABJM and U(N) Chern-Simons, aligning with known perturbative outcomes.
• The results simplify analyses in strong coupling and bolster duality conjectures within the AdS/CFT framework.

Overview of "Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter"

This paper by Kapustin, Willett, and Yaakov addresses the computation of supersymmetric Wilson loops in Chern-Simons theories with matter using localization techniques. The authors leverage the property that these theories reduce the path integral to a non-Gaussian matrix model, thereby enabling the calculation of Wilson loop expectation values. The study focuses on $\mathcal{N}=2$ superconformal Chern-Simons theories, though the localization approach presumes the presence of superconformal symmetry.

Central to this investigation is the reduction of Chern-Simons theories to matrix models, which provides a route for calculating Wilson loop expectation values. This is performed explicitly for theories like the ABJM model, with gauge group $U(N)$, showing alignment with existing perturbative calculations. The methods outlined are also applicable to other theories, such as Gaiotto-Witten and BLG.

Key Results

• Localization and Reduction: The paper adopts localization to simplify the computation of Wilson loops in superconformal Chern-Simons theories with matter. This reduction turns the path-integral into a matrix integral, greatly simplifying calculations compared to the original, more complex framework.
• Representation and Matrix Models: The Wilson loops studied preserve a complex supersymmetry, enabling their localization and computation within a non-Gaussian matrix model framework. The authors present explicit results for pure $U(N)$ Chern-Simons theory, finding consistency with known results. A similar result is derived for the ABJM theories, confirming earlier perturbative results.
• Matrix Integral for Partition Function: The resultant matrix model for a partition function in Chern-Simons theory, with gauge group $G$ and chiral multiplets in representation $R \oplus R^*$, is given by:

$$Z = \frac{1}{|\mathcal{W}|}\int da \; e^{-4i \pi^2 \text{Tr } a^2} \frac{\text{det}_{Ad} (2 \sinh( \pi a))}{\text{det}_{R} (2 \cosh( \pi a))}$$

Implications and Future Directions

The results have significant implications for both the practical and theoretical exploration of superconformal supersymmetric gauge theories. Practically, the results provide a more tractable method for calculating Wilson loops, circumventing complex direct perturbative methods, especially in strongly coupled regimes. Theoretically, they offer substantial support to duality conjectures in AdS/CFT correspondence, as the computations provide thorough tests of dualities between supersymmetric gauge theories and string/M-theory.

In future exploration, these techniques can be applied to more complex manifolds, potentially establishing even broader classes of dualities. Moreover, extending these results to larger $N$ limits may yield insights into string theory in the AdS context. Further computational improvements could also lead to better approximation methods or perhaps exact solutions using saddle point techniques.

Speculation on AI Application: With increasing capabilities, AI can significantly enhance simulation and computation in superconformal theories, potentially automating the matrix model reduction or even executing large-scale simulations to explore further holographic correspondences.