[Paper Review] A Note on Wilson Loop in N=2 Quiver/M theory Gravity Duality

Yang Zhou|ArXiv.org|Oct 22, 2009

Black Holes and Theoretical Physics29 references3 citations

TL;DR

This paper investigates the gravity dual of supersymmetric Wilson loop operators in N=2 quiver gauge theories arising from compactified M5-branes on a Riemann surface. By constructing Wilson loops in the quiver theory and identifying their dual M2-brane configurations in the AdS₅×S⁴ geometry fibered over the Riemann surface, the authors show that the strong-coupling expectation value of the Wilson loop matches the exponential of the M2-brane action, with the coupling depending on the compactification cycle. The key result is a precise duality between the Wilson loop and a membrane wrapping a cycle on the Riemann surface, consistent with the N=2 quiver/M-theory duality.

ABSTRACT

We study Wilson loops in the 4-dimensional N=2 supersymmetric quiver/M theory duality recently constructed by Gaiotto and Maldacena, that is conjectured to be dual to M2 branes on AdS_5 imes S^4 fibered over Σ_2. We use the localization method raised by Pestun, in order to compute the Wilson loop in 4-dimensional N=2 supersymmetric linear quiver gauge theory. We obtained the consistent results in the quiver gauge theory and the 11-dimensional supergravity. This match supports the N=2 quiver SCFT / M theory supergravity duality.

Motivation & Objective

• To establish a duality between Wilson loop operators in N=2 quiver gauge theories and M2-brane configurations in the dual M-theory background.
• To understand how supersymmetric Wilson loops in the quiver theory correspond to semiclassical membranes in the AdS₅×S⁴ geometry fibered over a Riemann surface.
• To compute the strong-coupling expectation value of the Wilson loop and match it with the M2-brane worldvolume action.
• To verify consistency of the duality by comparing central charges and operator dimensions between the field theory and M-theory descriptions.
• To generalize the N=4 SYM duality framework to N=2 quiver theories via the Gaiotto construction and T_N theories.

Proposed method

• Construct Wilson loop operators in N=2 quiver gauge theories using the Nekrasov partition function framework and the Gaiotto construction of 4D SCFTs from compactification of the 6D (2,0) theory on a Riemann surface.
• Derive the equation of motion for the Wilson loop in a two-node quiver with bi-fundamental matter, showing it reduces to the same form as in N=4 SYM with an effective 't Hooft coupling.
• Introduce the effective coupling relation $ rac{1}{ ilde{ ho}} = rac{1}{N} ig( rac{1}{ ilde{ ho}_1} + rac{1}{ ilde{ ho}_2} + ext{...} + rac{1}{ ilde{ ho}_N} ig) $, generalizing the N=4 case to linear quivers.
• Identify the dual object in M-theory as an M2-brane wrapping a cycle on the Riemann surface $ ilde{ ho} $, with worldvolume action $ S = T_{M_2} imes ext{volume} $.
• Compute the effective string tension $ T_s imes ext{length} $ in the AdS₅ boundary, showing $ T_s imes ext{length} o ext{constant} imes N^{1/3} $, consistent with $ ilde{ ho} o N^2 $.
• Match the strong-coupling Wilson loop expectation value $ raket{W} = e^{-S_{M_2}} = e^{g[ ho]N} $ to the M2-brane action, confirming the duality.

Experimental results

Research questions

• RQ1 How do supersymmetric Wilson loop operators in N=2 quiver gauge theories constructed from M5-branes on a Riemann surface relate to M-theory geometry?
• RQ2 What is the form of the Wilson loop in a linear quiver with multiple SU(N) gauge groups and bi-fundamental matter?
• RQ3 Does the strong-coupling expectation value of the Wilson loop in the quiver theory match the on-shell action of a membrane in the dual M-theory background?
• RQ4 How does the effective 't Hooft coupling in the quiver theory relate to the geometry of the Riemann surface and the M2-brane wrapping cycle?
• RQ5 Is the duality between the Wilson loop and the M2-brane preserved under the Gaiotto duality framework and the T_N theory construction?

Key findings

• The Wilson loop in the N=2 quiver theory has the same equation of motion as in N=4 SYM, with an effective 't Hooft coupling $ rac{1}{ ilde{ ho}} = rac{1}{N} ig( rac{1}{ ilde{ ho}_1} + rac{1}{ ilde{ ho}_2} + ext{...} + rac{1}{ ilde{ ho}_N} ig) $, generalizing the N=4 result.
• The strong-coupling expectation value of the Wilson loop is $ raket{W} = e^{-S_{M_2}} = e^{g[ ho]N} $, matching the exponential of the M2-brane worldvolume action.
• The dual M2-brane wraps a cycle $ ho $ on the Riemann surface $ ilde{ ho} $, with the coupling constant depending only on the topology of the cycle.
• The central charge $ c = rac{N^3}{3}(g-1) $ computed from the M-theory geometry matches the field theory result from the quiver, confirming consistency.
• The effective string tension in AdS₅ scales as $ T_s imes ext{length} o ext{const} imes N^{1/3} $, consistent with $ ilde{ ho} o N^2 $, indicating a strong coupling regime.
• The SUSY of the M2-brane embedding is preserved, as confirmed by the background geometry and the consistency with earlier results in the literature.