This paper addresses a long standing problem, the counting of the microstates of supersymmetric asymptotically AdS black holes in terms of a holographically dual field theory. We focus on a class of asymptotically AdS 4 static black holes preserving two real supercharges which are dual to a topologically twisted deformation of the ABJM theory. We evaluate in the large N limit the topologically twisted index of the ABJM theory and we show that it correctly reproduces the entropy of the AdS 4 black holes. An extremization of the index with respect to a set of chemical potentials is required. We interpret it as the selection of the exact R-symmetry of the superconformal quantum mechanics describing the horizon of the black hole.

We provide a general formula for the partition function of three-dimensional N = 2 gauge theories placed on S 2 × S 1 with a topological twist along S 2 , which can be interpreted as an index for chiral states of the theories immersed in background magnetic fields. The result is expressed as a sum over magnetic fluxes of the residues of a meromorphic form which is a function of the scalar zero-modes. The partition function depends on a collection of background magnetic fluxes and fugacities for the global symmetries. We illustrate our formula in many examples of 3d Yang-Mills-Chern-Simons theories with matter, including Aharony and Giveon-Kutasov dualities. Finally, our formula generalizes to Ω-backgrounds, as well as two-dimensional theories on S 2 and four-dimensional theories on S 2 × T 2 . In particular this provides an alternative way to compute genus-zero A-model topological amplitudes and Gromov-Witten invariants.

We apply localization techniques to compute the partition function of a two-dimensional N = (2, 2) R-symmetric theory of vector and chiral multiplets on S 2 . The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet-Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. As applications we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

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.

Let R be an E 2 ring spectrum with zero odd dimensional homo-topy groups. Every map of ring spectra M U → R is represented by a map of E 2 ring spectra. If 2 is invertible in π 0 R, then every map of ring spectra M SO → R is represented by a map of E 2 ring spectra.

We describe configurations of 5-branes and 7-branes which realize, when compactified on a circle, new isolated four-dimensional N = 2 superconformal field theories recently constructed by Gaiotto. Our diagrammatic method allows to easily count the dimensions of Coulomb and Higgs branches, with the help of a generalized s-rule. We furthermore show that superconformal field theories with E 6,7,8 flavor symmetry can be analyzed in a uniform manner in this framework; in particular we realize these theories at infinitely strongly-coupled limits of quiver theories with SU gauge groups.

We apply c-extremization, whose proof we review in full detail, to study two-dimensional N=(0,2) superconformal field theories arising from the low-energy dynamics of D3-branes wrapped on Riemann surfaces and M5-branes wrapped on four-manifolds. We compute the exact central charges of these theories using anomalies and c-extremization. In all cases we also construct AdS_3 supergravity solutions of type IIB and eleven-dimensional supergravity, which are holographic duals to the field theories at large N, and exactly reproduce the central charges computed via c-extremization.Comment: 72 pages, 5 figures. This paper is a sequel to arXiv:1211.4030. v2: minor modifications and updated reference

We study Seiberg-like dualities in three dimensional N = 2 supersymmetric theories, emphasizing Chern-Simons terms for the global symmetry group, which affect contact terms in two-point functions of global currents and are essential to the duality map. We introduce new Seiberg-like dualities for Yang-Mills-Chern-Simons theories with unitary gauge groups with arbitrary numbers of matter fields in the fundamental and antifundamental representations. These dualities are derived from Aharony duality by real mass deformations. They allow to initiate the systematic study of Seiberg-like dualities in Chern-Simons quivers. We also comment on known Seiberg-like dualities for symplectic and orthogonal gauge groups and extend the latter to the Yang-Mills case. We check our proposals by showing that the localized partition functions on the squashed S 3 match between dual descriptions.