Wilson loops in 3-dimensional N=6 supersymmetric Chern-Simons Theory and their string theory duals (0809.2787v4)

Abstract: We study Wilson loops in the three-dimensional N=6 supersymmetric Chern-Simons theory recently constructed by Aharony, Bergman, Jafferis and Maldacena, that is conjectured to be dual to type IIA string theory on AdS_4 x CP^3. We construct loop operators in the Chern-Simons theory which preserve 1/6 of the supercharges and calculate their expectation value up to 2-loop order at weak coupling. The expectation value at strong coupling is found by constructing the string theory duals of these operators. For low dimensional representations these are fundamental strings, for high dimensional representations these are D2-branes and D6-branes. In support of this identification we demonstrate that these string theory solutions match the symmetries, charges and the preserved supersymmetries of their Chern-Simons theory counterparts.

• The paper constructs Wilson loop operators in ABJM theory that preserve 1/6 supersymmetry using path-ordered exponentials and scalar field interactions.
• It performs a two-loop perturbative analysis to evaluate circular Wilson loops, showing a notable simplification in the bosonic and fermionic contributions.
• The research establishes a duality with type IIA string theory by mapping Wilson loops to fundamental strings and D-branes, reinforcing the gauge/string correspondence.

Wilson Loops in 3D Supersymmetric Chern-Simons Theory

The paper by Drukker, Plefka, and Young focuses on the investigation of Wilson loops within the framework of three-dimensional $\mathcal{N} = 6$ supersymmetric Chern-Simons theory. This theory, developed by Aharony, Bergman, Jafferis, and Maldacena (ABJM), is conjectured to correlate with type IIA string theory on the background $AdS_4 \times CP^3$. The paper characterizes loop operators in the Chern-Simons theory that preserve $1/6$ of supersymmetry and scrutinizes their dual descriptions in the string theory setup, specifically exploring their dynamics both at weak and strong coupling.

Main Contributions

1. Wilson Loop Operators Construction: The authors construct Wilson loop operators in the $\mathcal{N} = 6$ Chern-Simons-matter theory that preserve specific supersymmetries. These loops are defined as path-ordered exponentials of gauge connections and additional scalar field interactions. They preserve $1/6$ of the available supercharges when the path describes a straight line or a circle.
2. Perturbative Analysis: The paper explores the perturbative evaluation of these Wilson loops at weak coupling, calculating their expectation values up to the two-loop order. This assessment shows that for circular Wilson loops, the contribution from the bosonic and fermionic fields simplifies under certain conditions.
3. String Theory Duals: At strong coupling, the dual description of these Wilson loops in type IIA string theory is considered. For fundamental representations, the loops correspond to fundamental strings, while for higher dimensional representations, D-branes (specifically D2-branes and D6-branes) are the relevant constructs. The analysis confirms that these string configurations preserve the same symmetries and charges as their Chern-Simons counterparts.
4. Supersymmetry and Symmetries: The work rigorously tests the invariance of these operators under the symmetries of the theory, ensuring their compatibility with the predicted AdS/CFT dualities. The configurations being studied maintain specific unbroken $SU(2) \times SU(2)$ subgroup symmetries, which is consistent across both gauge and string theory modalities.

Implications and Future Directions

This research opens up several avenues for further work in understanding gauge/gravity dualities in lower dimensions and contributes to our knowledge of supersymmetric gauge theories. The correspondence between gauge theory loops and string theory branes enriches our understanding of the dynamics of these systems at both weak and strong coupling limits.

The results have implications for the theoretical understanding of the AdS/CFT correspondence in three-dimensional settings, offering a basis for exploring similar constructions in other lower-dimensional or less symmetric cases. This could lead to more comprehensive indices for understanding non-perturbative phenomena like confinement and chiral symmetry breaking.

Speculations on Future Developments

Rapid advances in theoretical approaches and computational methods suggest potential developments in:

• Exact Results: Extending the perturbative calculations to exact results for Wilson loops, perhaps through an identification of suitable integrable structures or localization techniques, could enhance the predictive power of the model.
• Gauge/String Dualities: The elucidation of more dual pairs involving different boundary conformal field theories or bulk geometries, potentially including scenarios beyond $AdS_4 \times CP^3$, could broaden the understanding of holographic principles in three dimensions.
• Connections to Supergravity: Further exploration into the higher-order corrections and the consistency of these models within a supergravity framework would firm up the theoretical foundations and potential applications in string/M-theory contexts.

Overall, this paper highlights critical connections between gauge theories in three dimensions and their corresponding string theoretic duals, contributing significantly to the burgeoning field of holographic dualities.