AbstractsMathematics

The basic objective of this thesis is to apply techniques of supersymmetric localization for studying Mirror Symmetry in N = 4 gauge theories in three space-time dimensions. We start by discussing an M-theory/Type IIA description of Mirror Symmetry for affine quiver gauge theories and present a neat classification of various infinite families of dual theories that arise in this fashion. We then perform an extremely non-trivial check of Mirror Symmetry for a large class of affine A and D type quivers and their truncated versions using partition function on a round sphere as a function of hypermultiplet masses and FI parameters. The computation allows one to explicitly derive the 'mirror map'  – a linear map relating masses of a theory with FI parameters of the dual and vice-versa. In addition, we demonstrate that there exists a one-to-one correspondence between the various 'building blocks' of the partition function of a given theory and String Theory objects in the Type IIB description of such theories. In the next part of the thesis, we introduce the idea that a very large class of dual quiver gauge theories may be systematically generated starting from dual pairs consisting only of simple linear quivers by appropriately gauging avor symmetries on one side of the duality. The gauging operation on one side of the duality results in a corresponding 'ungauging' on the other side. We show that this program can be carried out to obtain various known as well as new families of mirror pairs  – including affine quivers of A-D-E type, star-shaped quivers and quivers with symplectic gauge groups. S³ partition function turns out to be a particularly convenient tool to implement the gauging procedure since it easily generalizes to any arbitrary shape of the quiver as well as arbitrary rank of the gauge group at a given node in the quiver. Additional checks on the new mirror pairs obtained in this fashion are performed using Hilbert Series computation on the Coulomb and Higgs branches of the dual theories in question. Remarkably, we find that our prescription gives a straightforward recipe to construct 'good' duals in certain cases where naive mirror duals obtained via S-duality from Type IIB brane construction are 'bad' in the Gaiotto-Witten sense. Advisors/Committee Members: Distler, Jacques (advisor), Fischler, Willy (committee member), Freed, Dan (committee member), Kaplunovsky, Vadim (committee member), Paban, Sonia (committee member).