We consider supersymmetric AdS 3 × Y 7 and AdS 2 × Y 9 solutions of type IIB and D = 11 supergravity, respectively, that are holographically dual to SCFTs with (0, 2) supersymmetry in two dimensions and N = 2 supersymmetry in one dimension. The geometry of Y 2n+1 , which can be defined for n ≥ 3, shares many similarities with Sasaki-Einstein geometry, including the existence of a canonical R-symmetry Killing vector, but there are also some crucial differences. We show that the R-symmetry Killing vector may be determined by extremizing a function that depends only on certain global, topological data. In particular, assuming it exists, for n = 3 one can compute the central charge of an AdS 3 × Y 7 solution without knowing its explicit form. We interpret this as a geometric dual of cextremization in (0, 2) SCFTs. For the case of AdS 2 × Y 9 solutions we show that the extremal problem can be used to obtain properties of the dual quantum mechanics, including obtaining the entropy of a class of supersymmetric black holes in AdS 4 . We also study many specific examples of the type AdS 3 × T 2 × Y 5 , including a new family of explicit supergravity solutions. In addition we discuss the possibility that the (0, 2) SCFTs dual to these solutions can arise from the compactification on T 2 of certain d = 4 quiver gauge theories associated with five-dimensional Sasaki-Einstein metrics and, surprisingly, come to a negative conclusion. † On leave at the Galileo Galilei Institute, Largo Enrico Fermi, 2, 50125 Firenze, Italy.the five-form flux is properly quantized. With this set-up, the link of the cone still has an R-symmetry vector which, moreover, foliates the link with a transversely conformal Kähler metric. A main result of this paper is to show that the central charge c R of the dual SCFT can be obtained by extremizing a specific functional that depends on the space of R-symmetry vectors as well as the basic cohomology class of the transverse Kähler form.As we shall see, these complex cone geometries and the related extremal problem can be formulated for all the geometries Y 2n+1 with n ≥ 3 introduced in [9]. In particular, the results are also applicable to a class of supersymmetric AdS 2 × Y 9 solutions of eleven-dimensional supergravity, with only electric four-form flux, introduced in [10],

We construct supersymmetric AdS 3 solutions in F-theory, that is Type IIB supergravity with varying axio-dilaton, which are holographically dual to 2d N = (0, 4) superconformal field theories with small superconformal algebra. In F-theory these arise from D3-branes wrapped on curves in the base of an elliptically fibered Calabi-Yau threefold Y 3 and correspond to self-dual strings in the 6d N = (1, 0) theory obtained from F-theory on Y 3 . The non-trivial fibration over the wrapped curves implies a varying coupling of the N = 4 Super-Yang-Mills theory on the D3-branes. We compute the holographic central charges and show that these agree with the field theory and with the anomalies of self-dual strings in 6d. We complement our analysis with a discussion of the dual M-theory solutions and a comparison of the central charges.

We continue to develop the program initiated in [1] of studying supersymmetric AdS 3 backgrounds of F-theory and their holographic dual 2d superconformal field theories, which are dimensional reductions of theories with varying coupling. Imposing 2d N = (0, 2) supersymmetry, we derive the general conditions on the geometry for Type IIB AdS 3 solutions with varying axio-dilaton and five-form flux. Locally the compact part of spacetime takes the form of a circle fibration over an eight-fold Y τ 8 , which is elliptically fibered over a base M 6 . We construct two classes of solutions given in terms of a product ansatz M 6 = Σ × M 4 , where Σ is a complex curve and M 4 is locally a Kähler surface. In the first class M 4 is globally a Kähler surface and we take the elliptic fibration to vary non-trivially over either of these two factors, where in both cases the metrics on the total space of the elliptic fibrations are not Ricci-flat. In the second class the metric on the total space of the elliptic fibration over either curve or surface are Ricci-flat. This results in solutions of the type AdS 3 × K3 × M τ 5 , dual to 2d (0, 2) SCFTs, and AdS 3 × S 3 /Γ × CY 3 , dual to 2d (0, 4) SCFTs, respectively. In all cases we compute the charges for the dual field theories with varying coupling and find agreement with the holographic results. We also show that solutions with enhanced 2d N = (2, 2) supersymmetry must have constant axio-dilaton. Allowing the internal geometry to be non-compact leads to the most general class of Type IIB AdS 5 solutions with varying axio-dilaton, i.e. F-theoretic solutions, that are dual to 4d N = 1 SCFTs.In [1] this approach was initiated by studying the F-theory solutions dual to 2d (0, 4) SCFTs, which were shown to be of the typeis an elliptically fibered Calabi-Yau three-fold, and the complex structure of the elliptic fiber E τ is identified with the axio-dilaton τ IIB . This provides a generalization of the known solutions with (4, 4) supersymmetry where Y 6 = Y 4 × E τ with Y 4 = K3 or T 4 and the axio-dilaton is constant. The discrete subgoup Γ of SU(2) can be modded out, whilst retaining (0, 4) supersymmetry. In fact, these solutions were shown to be the most general ones dual to 2d (0, 4) SCFTs, supported by five-form flux.The dual field theories are closely related to the MSW string [9], and have a dual description in terms of D3-branes wrapped on a curve inside the base Kähler B 2 of the elliptically fibered Calabi-Yau three-fold: C ⊂ B 4 ⊂ Y τ 6 . The varying axio-dilaton induces a varying coupling of the 4d N = 4 Super-Yang Mills theory on the D3-brane, along the curve C. The resulting theory is supersymmetric when a particular topological twist, the so-called topological duality twist [3,4], is applied along the curve C [5, 6, 10]. The dual M-theory setup is the MSW string wrapped on an elliptic surface, and a dual M-theory solution confirms the F-theoretic results in [1], including the holographic comparison of the central charges to leading and subleading orders. In this context t...

We classify the necessary and sufficient conditions to obtain the near-horizon geometry of extremal supersymmetric rotating black holes embedded in 11d supergravity which are associated to rotating M2-branes. Such rotating black holes admit an AdS2 near-horizon geometry which is fibered by the transverse spacetime directions. In this paper we allow for the most general fibration over AdS2 with a flux configuration permitting rotating M2-branes. Using G-structure techniques we rewrite the conditions for supersymmetry in terms of differential equations on an eight-dimensional balanced space. The 9d compact internal space is a U(1)-fibration over this 8d base. The geometry is constrained by a master equation reminiscent of the one found in the non-rotating case. We give a Lagrangian from which the equations of motion may be derived, and show how the asymptotically AdS4 electrically charged Kerr-Newman black hole in 4d $$ \mathcal{N} $$ N = 2 supergravity is embedded in the classification. In addition, we present the conditions for the near-horizon geometry of rotating black strings in Type IIB by using dualities with the 11d setup.

We construct $$ \mathcal{N} $$ N = (2, 2) supersymmetric AdS3 solutions of type IIB supergravity, dual to twisted compactifications of 4d $$ \mathcal{N} $$ N = 4 super-Yang-Mills on Riemann surfaces. We consider both theories with a regular topological twist, and a twist involving the isometry group of the Riemann surface. These solutions are interpreted as the near-horizon of black strings asymptoting to AdS5× S5. As evidence for the proposed duality we compute the central charge of the gravity solutions and show that it agrees with the field theory result.