Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds

We study supersymmetric gauge theories with an R-symmetry, defined on non-compact, hyperbolic, Riemannian three-manifolds, focusing on the case of a supersymmetry-preserving quotient of Euclidean AdS$_3$. We compute the exact partition function in these theories, using the method of localization, thus reducing the problem to the computation of one-loop determinants around a supersymmetric locus. We evaluate the one-loop determinants employing three different techniques: an index theorem, the method of pairing of eigenvalues, and the heat kernel method. Along the way, we discuss aspects of supersymmetry in manifolds with a conformal boundary, including supersymmetric actions and boundary conditions.Comment: v3:79p, minor clarifications and references adde

Holographic renormalization and supersymmetry

Holographic renormalization is a systematic procedure for regulating divergences in observables in asymptotically locally AdS spacetimes. For dual boundary field theories which are supersymmetric it is natural to ask whether this defines a supersymmetric renormalization scheme. Recent results in localization have brought this question into sharp focus: rigid supersymmetry on a curved boundary requires specific geometric structures, and general arguments imply that BPS observables, such as the partition function, are invariant under certain deformations of these structures. One can then ask if the dual holographic observables are similarly invariant. We study this question in minimal N = 2 gauged supergravity in four and five dimensions. In four dimensions we show that holographic renormalization precisely reproduces the expected field theory results. In five dimensions we find that no choice of standard holographic counterterms is compatible with supersymmetry, which leads us to introduce novel finite boundary terms. For a class of solutions satisfying certain topological assumptions we provide some independent tests of these new boundary terms, in particular showing that they reproduce the expected VEVs of conserved charges.Comment: 70 pages; corrected typo

A topologically twisted index for three-dimensional supersymmetric theories

We provide a general formula for the partition function of three-dimensional (formula presented) gauge theories placed on S2 7S1 with a topological twist along S2, 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 \u3a9-backgrounds, as well as two-dimensional theories on S2 and four-dimensional theories on S2 7 T2. In particular this provides an alternative way to compute genus-zero A-model topological amplitudes and Gromov-Witten invariants

Gravity duals of supersymmetric gauge theories on three-manifolds

We study gravity duals to a broad class of N=2 supersymmetric gauge theories defined on a general class of three-manifold geometries. The gravity backgrounds are based on Euclidean self-dual solutions to four-dimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions defined on the four-ball the gravitational free energy depends only on the supersymmetric Killing vector, finding a simple closed formula when the solution has U(1) x U(1) symmetry. Our result agrees with the large N limit of the free energy of the dual gauge theory, computed using localization. This constitutes an exact check of the gauge/gravity correspondence for a very broad class of gauge theories with a large N limit, defined on a general class of background three-manifold geometries.Comment: 74 pages, 2 figures; v2: minor change

Dynamique des populations et mosaïque forestière en hêtraie-sapinière naturelle préservée. une comparaison entre Vosges et Carpates

A comparison of population dynamics and forest architecture has been carried in two well preserved beech-fir forests of Europe, in the Carpates and in the Vosges. The two reserves present marked differences in climate, soils and local environment. We show that beech dominates all the strata. In a situation of valley bottom (Carpates), its shady canopy explains growth patterns of understorey. Beech and fir present alternating phases of suppression and re-growth. Seedlings and saplings grow preferentially at the edges of adult canopies and in gaps. In the Vosges, slope induces imbricated crowns, and lateral light inputs are higher than in the Carpates. However regeneration is less dense because of less favourable soils and climate, and located preferentially in the neighbourhood of gaps. Both forests present a fine-grained architecture of eco-units, which increase resistance to disturbancesUne recherche comparative de sylvigénèse en hêtraie-sapinière a été entreprise dans deux réserves forestières intégrales, présentant des différences écologiques marquées dans les Hautes Vosges et les Carpates orientales. Les données recueillies concernent la structure des populations de ligneux (hêtre, sapin, épicéa, érable sycomore) et l'organisation architecturale des mosaiques forestières, par deux profils architecturaux, qui illustrent un aspect des situations sylvigénétiques rencontrées dans ces forêts de montagnes. Les principaux résultats démontrent que le hêtre domine les strates des deux forêts. En situation de fond de vallon (Carpates), l'ombrage qui résulte d'une canopée dense impose aux arbustes dominés (jeunes hêtres, mais surtout sapins) une croissance en palier parfois très longue, qui explique les situations préférentielles des semis en bordure des éco-unités matures ou dans les trouées. En situation de pente (Vosges), l'architecture en pile d'assiette des dominants, qui résulte de l'importance des apports lumineux latéraux, raccourcit cette attente. Mais les conditions plus rudes rencontrées dans cette réserve diminuent les densités à tous les étages. L'architecture des deux réserves montre une mosaïque fine d'éco-unités à différentes hauteurs, composées de plusieurs espèces ligneuses atteignant parfois des dimensions impressionnantes. L'ensemble est très stable, en dépit des aléas climatiques

Higgs branch localization in three dimensions

We show that the supersymmetric partition function of three-dimensional N =2 R-symmetric Chern-Simons-matter theories on the squashed S 3 and on S 2 7 S 1 can be computed with the so-called Higgs branch localization method, alternative to the more standard Coulomb branch localization. For theories that could be completely Higgsed by Fayet-Iliopoulos terms, the path integral is dominated by BPS vortex strings sitting at two circles in the geometry. In this way, the partition function directly takes the form of a sum, over a finite number of points on the classical Coulomb branch, of a vortex-string times an antivortex-string partition functions. \ua9 2014 The Author(s)