A Crack in the Conformal Window

TL;DR

The paper applies $F$-maximization to ${\cal N}=2$ 3d $U(N_c)$ gauge theories with $N_f$ flavors in the Veneziano limit to map the IR R-symmetry and sphere free energy, revealing a crack in the conformal window at $x_c\approx1.45$ where monopole operators saturate the unitarity bound. Below this threshold, an emergent global symmetry and the Aharony dual description enable the exact determination of $\Delta$ and $F$ for $1\le x< x_c$, while adding a Chern-Simons term with level $k$ removes the crack and yields distinct $\kappa=|k|/N_c$ regimes for meson dimensions. The work also analyzes meson scaling dimensions across these regimes, including the special case $\kappa=1$, and outlines how dualities (Aharony and Giveon-Kutasov) govern the IR fixed points and phase structure. Overall, the results illuminate the detailed IR dynamics, dual descriptions, and the role of monopoles in 3d ${\cal N}=2$ gauge theories, with implications for conformal windows and RG flows in related models.

Abstract

In {\cal N} = 2 superconformal three-dimensional field theory the R-symmetry is determined by locally maximizing the free energy F on the three-sphere. Using F-maximization, we study the {\cal N} = 2 supersymmetric U(N_c) gauge theory coupled to N_f pairs of fundamental and anti-fundamental superfields in the Veneziano large N_c limit, where x = N_f / N_c is kept fixed. This theory has a superconformal window 1 \leq x \leq \infty, while for x < 1 supersymmetry is broken. As we reduce x we find "a crack in the superconformal window" - a critical value x_c \approx 1.45 where the monopole operators reach the unitarity bound. To continue the theory to x < x_c we assume that the monopoles become free fields, leading to an accidental global symmetry. Using the Aharony dual description of the theory for x < x_c allows us to determine the R-charges and F for 1 \leq x < x_c. Adding a Chern-Simons term removes the transition at x_c. In these more general theories we study the scaling dimensions of meson operators as functions of x and κ= |k| / N_c. We find that there is an interesting transition in behavior at κ= 1.

Figures (12)

• Figure 1: $\Delta$ as a function of $x = {N_f \over N_c}$ in the Veneziano limit. The black points were computed using the saddle point method (method 1, section \ref{['Method1']}) and the orange boxes were computed by extrapolating from the small $N_c$ numerical results (method 2, section \ref{['Method2']}). The dotted red curve is the convergence bound $1 - {1 \over x}$; we find that $\Delta$ meets the converge bound at the critical value $x_c \approx 1.45$. The smooth black curves at large and small values of $x$ are the analytic approximations \ref{['DeltaX']} and \ref{['DeltaXMag']}, respectively. In the region right of the red curve we use the electric formulation of the theory, and in the region left of the red curve we use the magnetic formulation modified by decoupling the fields $V_\pm$. The right plot is a zoomed in version of the left one and includes only the numerical results from method 1.
• Figure 2: $F/N_c^2$ in the Veneziano limit as a function of $x = {N_f \over N_c}$. The free energy decreases monotonically as a function of $x$, consistent with the $F$-theorem. The black points were computed numerically using the saddle point method (method 1, section \ref{['Method1']}). The upper orange curve is the analytic approximation \ref{['FNcVeneziano']} and the lower orange linear approximation at smaller $x$ is given in \ref{['FNcVenezianoD']}.
• Figure 3: "The crack in the conformal window." The superconformal window in the $U(N_c)$ gauge theory with $N_f$ flavors is above the dotted red line $N_f = N_c$. The electric description has no emergent global symmetries above the black curve. The "standard" magnetic localization prescription works between the black and orange curves. These curves were calculated using the $1/N_f$ expansion result for $\Delta$ in \ref{['DeltaFinal']}.
• Figure 4: $F/N_f^2$ in the Veneziano limit as a function of $N_c / N_f$. The quantity is peaked at the "crack in the conformal window", $N_c / N_f \approx 1 / 1.45$. The black points were computed numerically using the saddle point method (method 1, section \ref{['Method1']}). The left orange curve is calculated from the analytic approximation \ref{['FNcVeneziano']}, and the right orange curve at larger values of $N_c / N_f$ is calculated from \ref{['FNcVenezianoD']}.
• Figure 5: $\Delta$ as a function of $x = {N_f \over N_c}$ in the Veneziano limit at various value of $\kappa = \left\lvert k \right\rvert / N_c$. The black, brown, and orange points correspond to $\kappa = 0.01, \, 0.4, \, 0.9$, respectively. The points were computed numerically using the saddle point method, described in section \ref{['Method1']}. The smooth curves at larger values of $x$ come from the analytic approximation to $\Delta$ in \ref{['DeltaXCS']}. The linear approximations at small $x$ were plotted using the analytic approximation \ref{['DeltaXCSMag']}.