I will explain how to compute the Hilbert series that counts chiral operators parametrising the Coulomb branch of the moduli space of vacua of three-dimensional N=4 supersymmetric gauge theories. The formula has applications to Physics, such as to the study of infrared dualities in three dimensions and of string backgrounds with eight supercharges, as well as to Mathematics, where it provides a new way to characterise HyperKaehler cones as algebraic varieties.  I will then show how the Coulomb branch of certain 3d N=4 generalised quiver gauge theories can be used to compute the Hilbert series of the moduli space of k G-instantons, for any simple Lie group G. The construction is alternative to the ADHM construction that is only available for classical groups G.

I will review the localisation of 2d N = (2, 2) supersymmetric gauge theories with a vector U(1) R-symmetry on the round 2-sphere, leading to a matrix integral expression for the partition function, in the so called "Coulomb branch" representation. For gauge theories which are completely Higgsed in the presence of a Fayet-Iliopoulos term, an alternative localization scheme yields the "Higgs branch" representation of the same partition function as the product of vortex times antivortex partition functions, weighted by semiclassical factors and summed over isolated vacua on the Higgs branch. Finally, I will briefly overview the potential of these exact results for the study of type II string compactifications and supersymmetric gauge theories in various dimensions. Based on arXiv:1206.2356 with F. Benini.

I will review recent progress in understanding the 3d N=2 superconformal field theories which describe the low energy dynamics of M2-branes probing toric CY_4 cones in M-theory, based on a KK reduction of M-theory to type IIA string theory. A key role in field theory is played by 't Hooft monopole operators, disorder operators which encode the geometric details of the reduction.

Following the work of Aharony, Bergman, Jafferis and Maldacena, a great effort has been made to understand the 3d conformal gauge theories describing M2-branes at conical singularities. Special attention has been directed to N=2 supersymmetric AdS4/CFT3 pairs arising from M2-branes at toric CY 4-folds, where powerful algebraic techniques are available. After reviewing some results concerning brane tilings, the relation between the toric CY3 and CY4 moduli spaces of 4d and 3d toric quiver gauge theories, and M-theory/type IIA duality, I will explain how flavours naturally arise in this framework, and how their matter content and superpotential couplings to bifundamental fields are encoded in geometric data. Quantum effects are crucial in computing the moduli spaces of such flavoured quiver gauge theories. Any quiver Chern-Simons (CS) theory of the kind considered in the AdS4/CFT3 literature so far can be obtained by coupling a 3d gauge theory with the same quiver diagram but no CS terms to chiral flavours, and integrating them out via a real mass term.