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Mayer-Cluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz

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In arXiv:0908.4052, Nekrasov and Shatashvili pointed out that the N=2 instanton partition function in a special limit of the Omega-deformation parameters is characterized by certain thermodynamic Bethe ansatz (TBA) like equations.In this talk I will present an explicit derivation of this fact as well as generalizations to quiver gauge theories. The TBA equations derived entirely within gauge theory have been proposed to encode the spectrum of a large class of quantum integrable systems. I will conclude with some remarks on this correspondence.