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

TL;DR

This paper establishes an upper bound on the rank of gauge groups in supergravity theories with 16 supercharges in dimensions greater than three, showing that certain 4D ${ m extbf{ extit{N}=4}}$ SYM theories cannot be coupled to gravity and are in the swampland.

Contribution

It introduces a bound on the gauge group rank in supergravity theories with 16 supercharges, linking swampland conditions to constraints on 4D ${ m extbf{ extit{N}=4}}$ SYM theories.

Findings

Bound on gauge group rank: $r_G \,\leq\, 26 - d$.

4D ${\rm \textbf{\textit{N}=4}}$ SYM with rank > 22 cannot couple to supergravity.

Supports the string lamppost principle by aligning swampland constraints with string theory.

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.