Paper detail

TBA type equations and tropical curves

We revisit the wall-crossing behaviour of solutions to the Thermodynamic Bethe Ansatz type equations arising in a class of three-dimensional field theories, expressed as sums of "instanton corrections". We explain how to attach to an instanton correction at a critical value a set of (combinatorial types of) tropical curves in R^2 of fixed degree, which determines its jump to leading order. We show that a weighted sum over all such curves is in fact a tropical count. This goes through to the q-deformed setting. Our construction can be regarded as a formal mirror symmetric statement in the framework proposed by Gaiotto, Moore and Neitzke.