Special geometry, black holes and Euclidean supersymmetry

We review recent developments in special geometry and explain its role in the theory of supersymmetric black holes. To make this article self-contained, a short introduction to black holes is given, with emphasis on the laws of black hole mechanics and black hole entropy. We also summarize the existing results on the para-complex version of special geometry, which occurs in Euclidean supersymmetry. The role of real coordinates in special geometry is illustrated, and we briefly indicate how Euclidean supersymmetry can be used to study stationary black hole solutions via dimensional reduction over time. This article is an updated and substantially extended version of the previous review article `New developments in special geometry', hep-th/0602171.Comment: 39 pages, Contribution to the Handbook on Pseudo-Riemannian Geometry and Supersymmtry, ed. V. Corte

Duality and black hole partition functions

Supersymmetric black holes provide an excellent theoretical laboratory to test ideas about quantum gravity in general and black hole entropy in particular. When four-dimensional supergravity is interpreted as the low-energy approximation of ten-dimensional string theory or eleven-dimensional M-theory, one has a microscopic description of the black hole which allows one to count microstates and to compare the result to the macroscopic (geometrical) entropy. Recently it has been conjectured that there is a very direct connection between the partition function of the topological string and a partition function for supersymmetric black holes. We review this idea and propose a modification which makes it compatible with electric-magnetic duality.Comment: 5 pages, based on talk given at the 11th Marcel Grossmann meeting, Berlin, July 23-29, 200

Effective Actions near Singularities

We study the heterotic string compactified on K3 x T^2 near the line T=U, where the effective action becomes singular due to an SU(2) gauge symmetry enhancement. By `integrating in' the light W^\pm vector multiplets we derive a quantum corrected effective action which is manifestly SU(2) invariant and non-singular. This effective action is found to be consistent with a residual SL(2,Z) quantum symmetry on the line T=U. In an appropriate decompactification limit, we recover the known SU(2) invariant action in five dimensions.Comment: 33 pages, LaTeX. v2: cosmetic correction on titlepage. v3: references and note adde

Supergravity on R4 x S1/Z2 and singular Calabi-Yaus

We discuss the moduli space singularities that are generally present in five-dimensional vector-coupled supergravity on a spactime of the form R4 x S1/Z2, with vector fields surviving on the Z2 fixed planes. The framework of supergravity is necessarily ambiguous when it comes to the non-singular embedding theory, so we focus on those models coming from Calabi-Yau three-folds with wrapped membranes.Comment: 13 p

Wald's entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling

The Bekenstein-Hawking entropy of black holes in Einstein's theory of gravity is equal to a quarter of the horizon area in units of Newton's constant. Wald has proposed that in general theories of gravity the entropy of stationary black holes with bifurcate Killing horizons is a Noether charge which is in general different from the Bekenstein-Hawking entropy. We show that the Noether charge entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling on the horizon defined by the coefficient of the kinetic term of specific graviton polarizations on the horizon. We present several explicit examples of static spherically symmetric black holes.Comment: 20 pages ; added clarifications, explanations, new section on the choice of polarizations, results unchanged; replaced with published versio

Moduli Stabilization with the String Higgs Effect

We review the notion of the Higgs effect in the context of string theory. We find that by including this effect in time dependent backgrounds, one is led to a natural mechanism for stabilizing moduli at points of enhanced gauge symmetry. We consider this mechanism for the case of the radion (size of the extra dimensions) and find that as decompactification of the large spatial dimensions takes place the radion will remain stabilized at the self dual radius. We discuss how this mechanism can be incorporated into models of string cosmology and brane inflation to resolve some outstanding problems. We also address some issues regarding which string states should be included when constructing low energy actions in string cosmology.Comment: 20 pages, references added, typos correcte

Special Geometry of Euclidean Supersymmetry III: the local r-map, instantons and black holes

We define and study projective special para-Kahler manifolds and show that they appear as target manifolds when reducing five-dimensional vector multiplets coupled to supergravity with respect to time. The dimensional reductions with respect to time and space are carried out in a uniform way using an epsilon-complex notation. We explain the relation of our formalism to other formalisms of special geometry used in the literature. In the second part of the paper we investigate instanton solutions and their dimensional lifting to black holes. We show that the instanton action, which can be defined after dualising axions into tensor fields, agrees with the ADM mass of the corresponding black hole. The relation between actions via Wick rotation, Hodge dualisation and analytic continuation of axions is discussed.Comment: 72 pages, 2 figure

ASK/PSK-correspondence and the r-map

We formulate a correspondence between affine and projective special K\"ahler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special K\"ahler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special K\"ahler manifolds in the image of the supergravity r-map. The above one-parameter deformations are interpreted as perturbative $\alpha'$-corrections in heterotic and type-II string compactifications with $N=2$ supersymmetry. Also affine special K\"ahler manifolds with quadratic prepotential are mapped to one-parameter families of projective special K\"ahler manifolds with quadratic prepotential. We show that the completeness of the deformed supergravity r-map metric depends solely on the (well-understood) completeness of the undeformed metric and the sign of the deformation parameter

Non-extremal black holes from the generalised r-map

We review the timelike dimensional reduction of a class of five-dimensional theories that generalises 5D, N = 2 supergravity coupled to vector multiplets. As an application we construct instanton solutions to the four-dimensional Euclidean theory, and investigate the criteria for solutions to lift to static non-extremal black holes in five dimensions. We focus specifically on two classes of models: STU-like models, and models with a block diagonal target space metric. For STU-like models the second order equations of motion of the four-dimensional theory can be solved explicitly, and we obtain the general solution. For block diagonal models we find a restricted class of solutions, where the number of independent scalar fields depends on the number of blocks. When lifting these solutions to five dimensions we show, by explicit calculation, that one obtains static non-extremal black holes with scalar fields that take finite values on the horizon only if the number of integration constants reduces by exactly half.Comment: 22 pages. Based on talk by OV at "Black Objects in Supergravity School" (BOSS2011), INFN, Frascati, Italy, 9-13 May, 201