Four Dimensional $\mathbf{\mathcal{N}=4}$ SYM and the Swampland

TL;DR

The paper proves a universal bound on the number of massless modes for quantum gravity theories with 16 supercharges in Minkowski space: the gauge group rank satisfies $r_G \le 26-d$. This bound, derived from a detailed analysis of BPS strings, anomaly inflow on their worldsheet, and a strong form of the distance conjecture, implies that 4d ${\cal N}=4$ SYM with rank $>22$ cannot couple consistently to ${\cal N}=4$ supergravity, placing such theories in the swampland. It also argues that non-chiral theories enjoy T-duality on a circle, connects winding and momentum modes in a dual description, and aligns with known string constructions, offering evidence for the string lamppost principle. Overall, the work strengthens the SLP by showing that a clear, testable bound on massless content arises from quantum gravity consistency conditions and that string theory constructions saturate or realize these bounds.

Abstract

We consider supergravity theories with 16 supercharges in Minkowski space with dimensions $d>3$. We argue that there is an upper bound on the number of massless modes in such theories depending on $d$. In particular we show that the rank of the gauge symmetry group $G$ in $d$ dimensions is bounded by $r_G\leq 26-d$. This in particular demonstrates that 4 dimensional ${\cal N}=4$ SYM theories with rank bigger than 22, despite being consistent and indeed finite before coupling to gravity, cannot be consistently coupled to ${\cal N}=4$ supergravity in Minkowski space and belong to the swampland. Our argument is based on the swampland conditions of completeness of spectrum of defects as well as a strong form of the distance conjecture and relies on unitarity as well as supersymmetry of the worldsheet theory of BPS strings. The results are compatible with known string constructions and provide further evidence for the string lamppost principle (SLP): that string theory lamppost seems to capture ${\it all}$ consistent quantum gravitational theories.