String Theory from Maximal Supersymmetry

TL;DR

This work analyzes non-gravitational, planar 4d EFTs that reduce to $N=4$ SYM at leading order and derives highly nontrivial nonlinear relations among the 4-point Wilson coefficients $a_{k,q}$ by enforcing $N=4$ SUSY, $SU(4)$ R-symmetry, and tree-level factorization on 4-, 5-, and 6-point amplitudes. By combining these nonlinear constraints with S-matrix bootstrap assumptions (unitarity, analyticity, Regge behavior) and a mass gap, the authors show the allowed region for the coefficients becomes non-convex but numerically converges to the open-string Veneziano amplitude values, parameterized by the string tension $\alpha'$ and the gap $M_{\text{gap}}$. They further reveal that the nonlinear SUSY constraints imply string monodromy order-by-order in the low-energy expansion, and they rule out several potential UV completions that might seem consistent at 4-point level, such as the infinite spin tower or isolated massive multiplet exchanges. The results indicate that supersymmetry, R-symmetry, and positivity suffice to single out the open-string UV completion at tree level, highlighting the power of higher-point amplitudes to constrain EFT data beyond causality or swampland considerations. Taken together, these findings strongly restrict the space of consistent quantum field theories and suggest intriguing extensions to gravity and AdS/CFT contexts.

Abstract

We explore the space of non-gravitational, maximally supersymmetric, planar 4d effective field theories (EFTs) that have $\mathcal{N}=4$ super Yang-Mills (SYM) at leading order. We show that in the weakly-coupled regime, highly non-trivial nonlinear constraints on the 4-point Wilson coefficients follow from enforcing $\mathcal{N}=4$ supersymmetry and $SU(4)$ R-symmetry together with the requirement of standard tree-level factorization on the massless poles of the 4-, 5-, and 6-point EFT scattering amplitudes. Additionally, when these novel constraints are combined with positivity, the resulting bounds on the 4-point Wilson coefficients converge to the values of the open string Veneziano amplitude. Our results strongly suggest that supersymmetry, R-symmetry, and positivity are sufficient to single out this unique UV completion at tree level. Our findings, moreover, highlight the power of higher-point amplitudes in constraining EFT data and imply that the space of consistent quantum field theories is even more restricted than previously suggested by causality or swampland-based approaches.

Figures (3)

• Figure 1: 4d positivity bounds. Dark Gray: The universal positivity constraints restrict the lowest-order Wilson coefficients to lie in the convex regions bounded by the dark gray curves. In the plot on the left, the curves are the exact bounds (\ref{['unibounds']}); on the right, the universal bounds are obtained numerically with SDPB for $k_\text{max}=7$. Red: When the nonlinear constraints (\ref{['eq:nonlinear_constraints_akq']}) arising from SUSY together with higher-point tree-level factorization in the EFT are imposed, the positivity bounds reduce the allowed regions to those shown in red. On the right, the constraints (\ref{['eq:nonlinear_constraints_akq']}) require $a_{2,1}/a_{0,0}=(1/4)a_{2,0}/a_{0,0}$, which directly restricts the allowed space to a line. However, it also imposes an upper bound on $a_{2,0}/a_{0,0}$: at $k_\text{max}=7$ it is $0.65802$, which can be compared with the string value $\zeta_4/\zeta_{2} \approx 0.657974\dots$. On the left, the red bounds are obtained numerically with SDPB at $k_\text{max}=7$: the upper and lower bound on $a_{1,0}/a_{0,0}$ for given $a_{2,0}/a_{0,0}$ differ by less than $5 \times 10^{-4}$. This difference is not visible in the plot, hence the allowed region on the red appears as a curve. Figure \ref{['fig:kmax345']} illustrates how the bounds for lower values of $k_\text{max}$ converge to $k_\text{max}=7$ bounds shown here.
• Figure 2: The nonlinear SUSY constraints reduce the convex universal region bounded by the dark gray curves to the smaller non-convex regions shown on the left for $k_\text{max}=3$ (light blue) $4$ (dark blue), and $5$ (green). The small black dot indicates the Veneziano values of the Wilson coefficients for $\alpha' M_\text{gap}^2 = 1$. On the right, we zoom in on the region near the top of the $k_\text{max}=5$ region on the left and show the bounds for $k_\text{max}=5$ (green), $k_\text{max}=6$ (orange), and $k_\text{max}=7$ (red). The black dashed curve is the open string with $\alpha' M_\text{gap}^2 \le 1$. The bootstrap is done in 4d.
• Figure 3: Chew–Frautschi plots generated by SDPB when $a_{1,0}/a_{0,0}$ is maximized for $a_{2,0}/a_{0,0}$ fixed to the value of the string, $\zeta_4/\zeta_2$, assuming the leading Regge trajectory to have slope 1. The nonlinear SUSY relations are imposed for $k\le 7$ while the regular crossing null constraints are imposed to $k_\text{max}=12$. The plot on the left shows the spectrum resulting from having 4d Legendre polynomials in the partial wave expansion used to the derive dispersion relations (\ref{['eq:dispersiveSUSY']}). On the right, the $D=4$ Legendre polynomials are replaced by the $D=10$ Gegenbauer polynomials; this corresponds to bootstrapping the $D=10$ amplitude with the external states restricted to a $D=4$ subspace, but the massive exchanged states having spins in 10d representations. For $D=4$ and $D=10$, respectively, the maximal value of $a_{1,0}/a_{0,0}$ is within $9 \cdot 10^{-6}$ and $4 \cdot 10^{-7}$, respectively, of the string value. In both cases, the bootstrap finds multiple states along the leading Regge trajectory and a few along the daughter trajectories (indicated with dashed lines), but much less convincingly so, especially in $D=4$. One feature worth noting is that the $D=4$ spectrum includes the spin-0 state at mass-squared $3M_\text{gap}^2 = 3/\alpha'$, but it is absent in the $D=10$ spectrum (see indication by the orange arrow in both plots). This is exactly the situation for the string spectrum: the string is critical in $D=10$, where the coupling of the $3M_\text{gap}^2$ scalar is exactly zero, but in sub-critical dimensions it is non-vanishing (and positive). It is encouraging that this key feature of the string spectrum is captured by the SDPB spectrum.