Abstract: The partition function on the three-sphere of many superconformal field theories (SCFTs)(including the ABJM theory) can be reduced to a matrix integral by using localization techniques. This makes possible to study this quantity in the strong coupling, large N regime, by studying the associated matrix models. In this talk I will review these developments, which lead to precision tests of the AdS/CFT correspondence, and explain a new method to study a large class of matrix models appearing in this context. The method is based on reformulating the model as a one-dimensional ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian, and leads to a simple picture of the strongly coupled dynamics of these theories (including the famous 3/2 scaling of the number of degrees of freedom). Moreover, simple semiclassical calculations in the Fermi gas make possible to determine the full 1/N asymptotics of the partition function.
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