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Hanany-Witten effect and SL(2,Z) dualities in matrix models

We provide tests of dualities for three-dimensional N=4 quiver SCFTs with brane realizations in IIB string theory, by matching their exact partition functions on $S^3$. The dualities are generated by SL(2,Z) transformations and Hanany-Witten 5-brane moves. These contain mirror symmetry as well as dualities identifiying fixed points of Yang-Mills quivers and Chern-Simons theories. The partition function is given by a matrix model, that can be nicely rearranged into a sequence of factors mimicking the brane realization. Identities obeyed by these elementary factors can be used to match the partition functions of dual theories, providing tests for the full web of dualities. In particular we are able to check mirror symmetry for linear and circular quivers with gauge nodes of arbitrary ranks. Our analysis also leads to a proof of a conjectured formula evaluating the matrix models of linear quiver theories.