Group Theory and the CFT Distance Conjecture: $\mathcal{N}=2$ Tensionless Strings Have No (Co)Weight
We perform a systematic survey of the Hagedorn behaviour at infinite-distance points in the conformal manifold of four-dimensional large-$N$ $\mathcal{N}=2$ Superconformal Field Theories admitting a Lagrangian description. Many properties of these theories can be understood in terms of the Lie algebra encoding the shape of their quiver. We find that in the overall-free limit, the Hagedorn temperature is determined by the largest eigenvalue of an affine or finite Cartan adjacency matrix. This defines two types of universality classes of theories sharing the same high-energy exponential growth of states characteristic of string-like spectra. The first and largest is the affine case, corresponding to orbifold and orientifold projections of $\mathcal{N}=4$ super-Yang-Mills, and all share the same temperature. The others fall into universality classes following an ADE classification with a temperature set by the dual Coxeter number, and can be obtained by deforming the affine case. Our results apply to all large-$N$ $\mathcal{N}=2$ quivers with any classical gauge symmetry, including those with matter charged beyond bifundamental representations. We further discuss the string-theoretic construction of these theories and some of the holographic implications, as well as how our methods extend to broad families of theories with less supersymmetry. We also consider limits where only part of the theory becomes free, and find lower and upper bounds on the exponential rate predicted by the CFT Distance Conjecture. Both these bounds and the Hagedorn temperature are set by the same eigenvalue, and when the quiver has a single gauge node the lower bound is saturated, giving a natural explanation for the three universality classes recently found in the literature.