Exact Results and Holography of Wilson Loops in N=2 Superconformal (Quiver) Gauge Theories

TL;DR

This paper computes exact circular Wilson loop expectation values in ${\cal N}=2$ superconformal (quiver) gauge theories using localization to a non-Gaussian one-matrix model, and analyzes their large-\(N\), large-\(\lambda\) behavior. For the $A_1$ theory, the Wilson loop exhibits non-exponential growth, while in the \(\hat{A}_1\) theory the untwisted sector grows exponentially like in ${\cal N}=4$ SYM, but the twisted sector shows non-analytic dependence on the difference of the two 't Hooft couplings due to worldsheet instantons. The holographic interpretation attributes these features to string-scale geometry and infinite worldsheet instanton resummations in dual backgrounds, suggesting noncritical string duals for the ${\cal N}=2$ theories and extending the analysis to general \(\hat{A}_{k-1}\) quivers. Overall, the work connects exact gauge-theory results from localization to intricate holographic pictures, highlighting how worldsheet effects shape Wilson-loop scaling beyond the familiar AdS/CFT paradigm.

Abstract

Using localization, matrix model and saddle-point techniques, we determine exact behavior of circular Wilson loop in N=2 superconformal (quiver) gauge theories. Focusing at planar and large `t Hooft couling limits, we compare its asymptotic behavior with well-known exponential growth of Wilson loop in N=4 super Yang-Mills theory. For theory with gauge group SU(N) coupled to 2N fundamental hypermultiplets, we find that Wilson loop exhibits non-exponential growth -- at most, it can grow a power of `t Hooft coupling. For theory with gauge group SU(N) x SU(N) and bifundamental hypermultiplets, there are two Wilson loops associated with two gauge groups. We find Wilson loop in untwisted sector grows exponentially large as in N=4 super Yang-Mills theory. We then find Wilson loop in twisted sector exhibits non-analytic behavior with respect to difference of two `t Hooft coupling constants. By letting one gauge coupling constant hierarchically larger/smaller than the other, we show that Wilson loops in the second type theory interpolate to Wilson loop in the first type theory. We infer implications of these findings from holographic dual description in terms of minimal surface of dual string worldsheet. We suggest intuitive interpretation that in both type theories holographic dual background must involve string scale geometry even at planar and large `t Hooft coupling limit and that new results found in the gauge theory side are attributable to worldsheet instantons and infinite resummation therein. Our interpretation also indicate that holographic dual of these gauge theories is provided by certain non-critical string theories.

Figures (6)

• Figure 1: Quiver diagram of ${\cal N}=2$ superconformal gauge theories under study: (a) $\hat{A}_0$ theory with $G$ = SU$(N)$ and one adjoint hypermultiplet, (b) $A_1$ theory with $G$=SU($N$) and $2N$ fundamental hypermultiplets, (c) $\hat{A}_1$ theory with $G=$SU($N) \times$ SU($N$) and $2N$ bifundamental hypermultiplets. The $A_1$ theory is obtainable from $\hat{A}_1$ theory by tuning ratio of coupling constants to 0 or $\infty$. See sections 3 and 4 for explanations.
• Figure 2: Typical distribution of the eigenvalue density $\rho$.
• Figure 3: Dependence of twisted sector Wilson loops on the parameter $B$. It shows discontinuity at $B=0$, resulting in non-analytic behavior of the Wilson loops to both gauge couplings.
• Figure 4: Response of gauge theories to external color charge source. (a) For $A_1$ theory, an external color charge in fundamental representation of the gauge group is screened by the $N_f = 2 N_c$ flavors of massless matter fields, which are in fundamental representation (blue arrow). (b) For $\hat{A}_1$ theory, an external color charge in fundamental representation of the first gauge group is screened by the massless matter fields. As the matter fields are in bi-fundamental representations (black and white arrows), color charge in the second gauge group is regenerated and anti-screened. The process repeats between the two gauge groups and leads the theory to exhibit Coulomb behavior.
• Figure 5: Semiclassical Wilson loop in brane configuration of ${\cal N}=2$ superconformal gauge theories under study: (a) $\hat{A}_1$ theory with $G=$SU($N) \times$ SU($N$) and $2N$ bifundamental hypermultiplets. $N$ D4-branes stretch between two widely separated NS5-branes on a circle. The F1 (fundamental string) ending on or emanating from D4-brane represent static charges. On D4-branes, having finite gauge coupling, conservation of the F1 flux is manifestly. (b) $A_1$ theory with $G$=SU($N$) and $2N$ fundamental hypermultiplets. The $A_1$ theory is obtained from $\hat{A}_1$ in (a) by approaching the two NS5-branes. The flux is leaked into the coincident NS5-branes and run along their worldvolumes. On D4-branes, having vanishing gauge coupling, conservation of the F1 flux is not manifest.