Modular interpolating functions for N=4 SYM

TL;DR

The paper addresses non-perturbative information for anomalous dimensions in ${\cal N}=4$ SYM by constructing interpolating functions that are invariant under the full modular group. It introduces a family of interpolants built from real Eisenstein series $E_s(\tau)$ that reproduce perturbative data up to $m$ loops while automatically incorporating S-duality via $PSL(2,\mathbb{Z})$ invariance. Explicit results for leading-twist anomalous dimensions and structure constants are presented up to four and three loops, respectively, with convergence observed as $m$ grows and a tendency for maxima to occur at the duality-invariant point $\tau_3 = e^{i\pi/3}$. The work also analyzes level-crossing between leading and sub-leading twist operators, finding that crossings are likely for $N$ above a small threshold and that the intercept region near $\tau_3$ governs the maximum anomalous dimensions. Overall, the modular interpolations provide a non-perturbative, symmetry-respecting framework to study observables in ${\cal N}=4$ SYM and yield predictions consistent with bootstrap constraints at specific duality points.

Abstract

We construct interpolating functions fully compatible with S-duality. We then consider the problem of resumming perturbative expansions for anomalous dimensions of low twist non-protected operators in N=4 super Yang-Mills theory. When the rank of the gauge group is small, the interpolations suggest that anomalous dimensions of leading twist operators take their maximum value at the point $τ=\exp(iπ/3)$. For fixed spin and large enough rank, there is a level-crossing region, where the anomalous dimension of the leading twist operator reaches its maximum and then bounces back.

Figures (5)

• Figure 1: Interpolating functions for anomalous dimensions of leading-twist operators with spin zero and two, for $SU(2)$ and $m=1,2,3,4$. In both cases, the upper line corresponds to $m=1$, the second line to $m=3$, the third line to $m=2$ and the lower line to $m=4$. We have also indicated the forbidden region with gray and the corner value with a continuous horizontal line.
• Figure 2: Interpolating functions for structure constants of operators with spin zero and two, for $SU(2)$ and $m=1,2,3$. The lower line corresponds to $m=1$, the line in the middle to $m=2$ and the upper line to $m=3$.
• Figure 3: Dimensions of leading and sub-leading operators of spin zero, for several values of $N$. The interpolating functions actually suggest that crossing does occur for $N$ bigger than four/five.
• Figure 4: Interpolating functions for $n=3/2$ (lower curve), $n=2$ (middle curve) and $n=5/2$ (upper curve), for spin zero and two.
• Figure 5: $\gamma_{toy-pert}(g)$ for $\alpha=1/2$ (dashed line) vs the modular invariant interpolating function (solid line) for $m=40$.