A q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. Wilson lines, surface defects, etc., and the charged excitations have q spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries (q = 0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a subgroup). They can also have 't Hooft anomalies, which prevent us from gauging them, but lead to 't Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.

We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N = 2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U (N ), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.1 The factors of 2 appearing the determinants did not appear in the original version of this paper, but were found later by Drukker, Marino,and Putrov [5]. 2 We have absorbed the Chern-Simons level k into "Tr", see the next section for more details.-2 -them as a perturbative expansion in the 't Hooft coupling, we find agreement with a perturbative calculation done in field theory [10,11,12], which provides a check of our result. It is possible that the matrix model could be solved exactly in the large N limit using a saddle point approximation [13], as we will briefly mention at the end of the paper. AcknowledgmentsThis work was supported in part by DOE grant DE-FG02-92ER40701. A.K. would like to thank Lev Rozansky and Alexei Borodin for useful discussions. I.Y. would also like to thank Joseph Marsano, John Schwarz, Ketan Vyas and Ofer Aharony for their input.

Many examples of low-energy dualities have been found in supersymmetric gauge theories with four supercharges, both in four and in three space-time dimensions. In these dualities, two theories that are different at high energies have the same low-energy limit. In this paper we clarify the relation between the dualities in four and in three dimensions. We show that every four dimensional duality gives rise to a three dimensional duality between theories that are similar, but not identical, to the dimensional reductions of the four dimensional dual gauge theories to three dimensions. From these specific three dimensional dualities one can flow to many other low-energy dualities, including known three dimensional dualities and many new ones. We discuss in detail the case of three dimensional SU(N_c) supersymmetric QCD theories, showing how to derive new duals for these theories from the four dimensional duality.Comment: 84 pages, 3 figures, harvmac. v2: added an appendix on the reduction of the 4d index to the 3d partition function, added references, minor corrections and change

This is the foreword to the special volume on localization techniques in quantum field theory. The summary of individual chapters is given and their interrelation is discussed.

We test several conjectural dualities between strongly coupled superconformal field theories in three dimensions by computing their exact partition functions on a three sphere as a function of Fayet-Iliopoulos and mass parameters. The calculation is carried out using localization of the path integral and the matrix model previously derived for superconformal N = 2 gauge theories. We verify that the partition functions of quiver theories related by mirror symmetry agree provided the mass parameters and the Fayet-Iliopoulos parameters are exchanged, as predicted. We carry out a similar calculation for the mirror of N = 8 super-Yang-Mills theory and show that its partition function agrees with that of the ABJM theory at unit Chern-Simons level. This provides a nonperturbative test of the conjectural equivalence of the two theories in the conformal limit.