[Paper Review] Wilson loops in $\mathcal{N}=4$ $SO(N)$ SYM and D-Branes in $AdS_5 imes \mathbb{RP}^5$

Simone Giombi, Bendeguz Offertaler|arXiv (Cornell University)|Jun 18, 2020

Black Holes and Theoretical Physics67 references4 citations

TL;DR

This paper studies half-BPS circular Wilson loops in $χ=4$ $SO(N)$ SYM theory using supersymmetric localization, deriving exact matrix model expressions for their expectation values. It establishes quantitative duality with type IIB string theory on $AdS_5 \times \mathbb{RP}^5$, showing exact agreement between the strong coupling limit of the spinor representation loop and the classical D5-brane action wrapping $\mathbb{RP}^4$, and confirms the bremsstrahlung function via D-brane fluctuations.

ABSTRACT

We study the half-BPS circular Wilson loop in ${\cal N}=4$ super Yang-Mills with orthogonal gauge group. By supersymmetric localization, its expectation value can be computed exactly from a matrix integral over the Lie algebra of $SO(N)$. We focus on the large $N$ limit and present some simple quantitative tests of the duality with type IIB string theory in $AdS_5 imes \mathbb{RP}^5$. In particular, we show that the strong coupling limit of the expectation value of the Wilson loop in the spinor representation of the gauge group precisely matches the classical action of the dual string theory object, which is expected to be a D5-brane wrapping a $\mathbb{RP}^4$ subspace of $\mathbb{RP}^5$. We also briefly discuss the large $N$, large $λ$ limits of the $SO(N)$ Wilson loop in the symmetric/antisymmetric representations and their D3/D5-brane duals. Finally, we use the D5-brane description to extract the leading strong coupling behavior of the "bremsstrahlung function" associated to a spinor probe charge, or equivalently the normalization of the two-point function of the displacement operator on the spinor Wilson loop, and obtain agreement with the localization prediction.

Motivation & Objective

• To test the AdS/CFT duality between $\mathcal{N}=4$ $SO(N)$ SYM and type IIB string theory on $AdS_5 \times \mathbb{RP}^5$ using Wilson loops.
• To compute the exact expectation value of the half-BPS Wilson loop in the spinor representation via supersymmetric localization.
• To compare the strong coupling limit of the Wilson loop with the classical action of the dual D5-brane in $AdS_5 \times \mathbb{RP}^5$.
• To extract the bremsstrahlung function for the spinor defect CFT from D-brane fluctuations and match it with localization predictions.
• To explore the dual D-brane description of large-rank symmetric/antisymmetric representations and their string theory counterparts.

Proposed method

• Apply supersymmetric localization to reduce the path integral of the Wilson loop in $SO(N)$ SYM to a matrix integral over the Lie algebra of $SO(N)$.
• Compute the large $N$ saddle point of the $SO(N)$ matrix model to access the strong coupling regime.
• Evaluate the Wilson loop in the spinor representation using the matrix model and take the large $N$ limit.
• Identify the dual string theory object as a D5-brane wrapping $\mathbb{RP}^4 \subset \mathbb{RP}^5$ in $AdS_5 \times \mathbb{RP}^5$.
• Compute the classical on-shell action of the D5-brane, including boundary terms, and compare it with the localization result.
• Analyze transverse fluctuations of the D5-brane to compute the two-point function of the displacement operator and extract the bremsstrahlung function.

Experimental results

Research questions

• RQ1 Does the strong coupling limit of the Wilson loop in the spinor representation of $SO(N)$ SYM match the classical action of a D5-brane in $AdS_5 \times \mathbb{RP}^5$?
• RQ2 How does the $SO(N)$ gauge theory differ from $SU(N)$ in the AdS/CFT dictionary, particularly regarding brane duals and $1/N$ expansion?
• RQ3 What is the leading strong coupling behavior of the bremsstrahlung function for a spinor defect in the $SO(N)$ theory?
• RQ4 How do D-brane configurations in $AdS_5 \times \mathbb{RP}^5$ realize the duals of large-rank symmetric and antisymmetric representations?
• RQ5 Can the D5-brane description reproduce the localization prediction for the normalization of the displacement operator two-point function?

Key findings

• The strong coupling limit of the Wilson loop in the spinor representation of $SO(N)$ SYM precisely matches the classical D5-brane action in $AdS_5 \times \mathbb{RP}^5$ wrapping $\mathbb{RP}^4$, with the action given by $S_{D5} = -\frac{2N}{3\pi}\sqrt{\frac{\lambda}{2}}\sin^3\theta_k$.
• The D5-brane dual to the antisymmetric Wilson loop in $SO(N)$ theory wraps an $S^4$ at angle $\theta_k$ satisfying $\theta_k - \sin\theta_k\cos\theta_k = \pi k / N$, consistent with the $SO(N)$ rank constraint.
• The bremsstrahlung function for the spinor defect CFT is extracted from D5-brane fluctuations and matches the localization prediction, confirming the normalization of the displacement operator two-point function.
• The $SO(N)$ case differs from $SU(N)$ in the string theory dual by a replacement $\sqrt{\lambda} \to \sqrt{\lambda/2}$ in the D-brane action, reflecting the reduced $R$-symmetry space $\mathbb{RP}^5$.
• The D3-brane and D5-brane duals for large-rank symmetric/antisymmetric representations in $SO(N)$ SYM are consistent with the $AdS_5 \times \mathbb{RP}^5$ background, with flux quantization matching gauge theory representation labels.
• The equatorial case $\theta_k = \pi/2$ ($k = N/2$) is a subtle special case due to $\mathbb{Z}_2$ identification, suggesting possible double wrapping or decomposition into self-dual representations.