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We study the structure of the non-perturbative free energy of a one-parameterclass of little string theories (LSTs) of A-type in the so-called unrefinedlimit. These theories are engineered by $N$ M5-branes probing a transverse flatspace. By analysing a number of examples, we observe a pattern which suggeststo write the free energy in a fashion that resembles a decomposition intohigher-point functions which can be presented in a graphical way reminiscent ofsums of (effective) Feynman diagrams: to leading order in the instantonparameter of the LST, the $N$ external states are given either by thefundamental building blocks of the theory with $N=1$, or the function thatgoverns the counting of BPS states of a single M5-brane coupling to oneM2-brane on either side. These states are connected via an effective couplingfunction which encodes the details of the gauge algebra of the LST and which inits simplest (non-trivial) form is captured by the scalar Greens function onthe torus. More complicated incarnations of this function show certainsimilarities with so-called modular graph functions, which have appeared in thestudy of Feynman amplitudes in string- and field theory. Finally, similarstructures continue to exist at higher instanton orders, which, however, alsocontain contributions that can be understood as the action of (Hecke) operatorson the leading instanton result.

From Little String Free Energies Towards Modular Graph Functions