Resummation and S-duality in N=4 SYM

TL;DR

The paper addresses the challenge of resumming perturbative anomalous dimensions in N=4 SYM under S-duality constraints by introducing symmetry-respecting interpolating functions invariant under finite-order PSL(2,Z) subgroups. These interpolations, including symmetric Padé and fractional-power schemes, are matched to perturbative data and applied to low-twist operators such as Konishi across SU(N) with N=2,3,4, yielding finite-coupling predictions at duality-invariant points like τ=i and τ=e^{iπ/3}. The results are broadly compatible with conformal bootstrap bounds, with indications of bound saturation at self-dual points for low N, and reveal a striking, narrow linear relation in the space of lowest operator dimensions as the conformal manifold is traversed. This suggests a robust structural constraint on N=4 SYM across the conformal manifold and demonstrates the value of duality-guided interpolation for non-perturbative insights.

Abstract

We consider the problem of resumming the perturbative expansions for anomalous dimensions of low twist, non-BPS operators in four dimensional N=4 supersymmetric Yang-Mills theories. The requirement of S-duality invariance imposes considerable restrictions on any such resummation. We introduce several prescriptions that produce interpolating functions on the upper half plane that are compatible with a subgroup of the full duality group. These lead to predictions for the anomalous dimensions at all points in the fundamental domain of the complex gauge coupling, and in particular at the duality-invariant values τ=i and τ=exp(iπ/3). For low-rank gauge groups, the predictions are compatible with the bounds derived by conformal bootstrap methods for these anomalous dimensions; within numerical errors, they are in good agreement with the conjecture that said bounds are saturated at a duality-invariant point. We also find that the anomalous dimensions of the lowest twist operators lie within an extremely narrow window around a straight line as we vary the moduli of the theory over the two dimensional fundamental domain.

Figures (10)

• Figure 1: The upper half plane is tessellated by images of the fundamental domain of PSL$(2,\mathbb{Z})$. The $\mathbb{Z}_2$-invariant interpolating functions defined here are well-suited to describe anomalous dimensions in two copies of the fundamental domain, shown as shaded in the figure. The solid line is $\theta=0$, and represents the best case for the $\mathbb{Z}_2$-invariant interpolating function.
• Figure 2: The $\mathbb{Z}_3$ invariant interpolation is particularly well suited to describe anomalous dimensions along the bold segments in the above figure. Because the method is not invariant under $y\leftrightarrow -y$, its accuracy is sure to degenerate for, e.g., $y<0$. The regions where the best behavior is expected are again shaded.
• Figure 3: Interpolations of the Konishi anomalous dimensions for gauge group $SU(2)$. The different plots depict the results of the (left) $\mathbb{Z}_2$ invariant and (right) $\mathbb{Z}_3$ invariant resummation schemes, evaluated as a function of $g$ with (top) $\theta=0$ and (bottom) $\theta=\pi$. We show interpolations defined using (short-dashed) two loops, (long-dashed) three loops, and (solid) four loops in perturbation theory. Red and orange lines correspond to Padé approximants with integral and half-integral powers, respectively. Blue and green lines represent FPP interpolations with integral and half-integral powers. As described at the end of appendix \ref{['sa']}, some of these graphs coincide. The two horizontal lines correspond to the upper bound (top line) and the best estimate based on a corner value (bottom line) obtained from the numerical bootstrap results of 1304.1803. See §\ref{['s5']} for a more detailed description of these bounds.
• Figure 4: Interpolations of the Konishi anomalous dimensions for gauge group $SU(3)$. The different plots depict the results of the (left) $\mathbb{Z}_2$ invariant and (right) $\mathbb{Z}_3$ invariant resummation schemes, evaluated as a function of $g$ with (top) $\theta=0$ and (bottom) $\theta=\pi$.
• Figure 5: Interpolations of the Konishi anomalous dimensions for gauge group $SU(4)$. The different plots depict the results of the (left) $\mathbb{Z}_2$ invariant and (right) $\mathbb{Z}_3$ invariant resummation schemes, evaluated as a function of $g$ with (top) $\theta=0$ and (bottom) $\theta=\pi$.