Generating Geodesic Flows and Supergravity Solutions
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tion 2 Branes as Geodesics on Moduli 2.1 From p-branes to (-1)-branes . 2.2 (−1)-brane Geometries . . . . 2.3 Geodesic Curves . . . . . . . . 3 The 3.1 3.2 3.3 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 5 7 8 10 10 12 17 17 20 21 22 22 25 26 28 30 31 32 32 33 35 37 44 46 48 50 53
∗Institute for Theoretical Physics, K.U. Leuven,
Thomasvr@itf.fys.kuleuven.ac.be
Abstract
We consider the geodesic motion on the symmetric moduli spaces that arise after timelike and spacelike reductions of supergravity theories. The geodesics correpond to timelike respectively spacelike p-brane solutions when they are lifted over a pdimensional flat space. In particular, we consider the problem of constructing the minimal generating solution : A geodesic with the minimal number of free parameters such that all other geodesics are generated through isometries. We give an intrinsic characterization of this solution in a wide class of orbits for various supergravities in different dimensions. We apply our method to three cases: (i) Einstein vacuum solutions, (ii) extreme and non-extreme D = 4 black holes in N = 8 supergravity and their relation to N = 2 STU black holes and (iii) Euclidean wormholes in D ≥ 3. In case (iii) we present an easy and general criterium for the existence of regular wormholes for a given scalar coset.
2
1
Introduction
Over the years much effort has been put into the investigation of (non-)BPS solutions to (matter-coupled) supergravity theories. The relevance of these solutions relies on the fact that they provide crucial information about the underlying string theories and their dualities. In particular, we focus on supergravity solutions that have the structure of a p-brane. In general two different kinds of p-brane solutions are considered: timelike p-branes that are related to the string theory D-branes [1] (or M-branes) or spacelike pbranes (known as S-branes) who are conjectured to describe time-dependent phenomena in string theory [2]. Timelike p-branes have a Lorentzian worldvolume and are stationary solutions whereas spacelike p-branes have a Euclidean worldvolume and are explicitly timedependent. In view of the above it is important to find new solutions. One way to do this is to develop new solution-generating techniques. These techniques are often based on reducing the p-brane solution over the brane worldvolume to obtain a corresponding (−1)-brane solution. It turns out that the dynamics of these (−1)-brane solutions is described by a geodesic motion on the moduli space that follows from this reduction [3, 4]. This has led to the study of the geodesic solutions and, more general, the study of the integrability of the geodesic equations on symmetric spaces. Most of the focus has been on the geodesic curves that correspond to time-dependent supergravity solutions [5–14]. We consider the problem of defining, in an intrinsic, model-independent way, the most general geodesic that corresponds both to time-dependent and stationary supergravity solutions. In order to achieve this we use the isometry group of the moduli space to construct the geodesic with the minimal number of free parameters such that all other geodesics can now be obtained by an isometry rotation of this particular solution. We call this solution the minimal generating solution. This method is closely related to the compensator-algorithm developed in [5]. In our approach there is an important difference between the Riemannian and pseudoRiemannian moduli spaces. The generating geodesic in the Riemannian case was shown to be carried by the dilatons only [12]. The pseudo-Riemannian case turns out to be richer. The aim of this paper is to extend the discussion to the pseudo-Riemannian case. One of the main results, derived in this paper, is the derivation of a theorem, see (3.68), valid for a wide class of orbits, defined by a diagonalizable generator Q of the geodesic, that characterizes the geodesic generating solution in terms of the group-theoretical properties of the corresponding moduli space. Our theorem applies to all supergravities with symmetric scalar manifolds. This includes all theories with more then 8 supercharges and applies to an interesting subset of theories with 8 and less supercharges. We show that the generating solution can be found in a suitable sub-manifold of the original scalar manifold defining a consistent truncation of the theory. We also make general comments which apply to all orbits, including thus the cases in which Q is not diagonalizable. To illustrate our methods we consider three different classes of solutions. We first focus on a class of vacuum Einstein solutions. The application of our theorem to this case reproduces some well-known and some less known solutions. We next consider stationary black hole solutions in four-dimensional supergravity. For that we reduce the four-dimensional black hole solutions, via a timelike reduction, to three dimensions, where they become 3
∗Institute for Theoretical Physics, K.U. Leuven,
Thomasvr@itf.fys.kuleuven.ac.be
Abstract
We consider the geodesic motion on the symmetric moduli spaces that arise after timelike and spacelike reductions of supergravity theories. The geodesics correpond to timelike respectively spacelike p-brane solutions when they are lifted over a pdimensional flat space. In particular, we consider the problem of constructing the minimal generating solution : A geodesic with the minimal number of free parameters such that all other geodesics are generated through isometries. We give an intrinsic characterization of this solution in a wide class of orbits for various supergravities in different dimensions. We apply our method to three cases: (i) Einstein vacuum solutions, (ii) extreme and non-extreme D = 4 black holes in N = 8 supergravity and their relation to N = 2 STU black holes and (iii) Euclidean wormholes in D ≥ 3. In case (iii) we present an easy and general criterium for the existence of regular wormholes for a given scalar coset.
2
1
Introduction
Over the years much effort has been put into the investigation of (non-)BPS solutions to (matter-coupled) supergravity theories. The relevance of these solutions relies on the fact that they provide crucial information about the underlying string theories and their dualities. In particular, we focus on supergravity solutions that have the structure of a p-brane. In general two different kinds of p-brane solutions are considered: timelike p-branes that are related to the string theory D-branes [1] (or M-branes) or spacelike pbranes (known as S-branes) who are conjectured to describe time-dependent phenomena in string theory [2]. Timelike p-branes have a Lorentzian worldvolume and are stationary solutions whereas spacelike p-branes have a Euclidean worldvolume and are explicitly timedependent. In view of the above it is important to find new solutions. One way to do this is to develop new solution-generating techniques. These techniques are often based on reducing the p-brane solution over the brane worldvolume to obtain a corresponding (−1)-brane solution. It turns out that the dynamics of these (−1)-brane solutions is described by a geodesic motion on the moduli space that follows from this reduction [3, 4]. This has led to the study of the geodesic solutions and, more general, the study of the integrability of the geodesic equations on symmetric spaces. Most of the focus has been on the geodesic curves that correspond to time-dependent supergravity solutions [5–14]. We consider the problem of defining, in an intrinsic, model-independent way, the most general geodesic that corresponds both to time-dependent and stationary supergravity solutions. In order to achieve this we use the isometry group of the moduli space to construct the geodesic with the minimal number of free parameters such that all other geodesics can now be obtained by an isometry rotation of this particular solution. We call this solution the minimal generating solution. This method is closely related to the compensator-algorithm developed in [5]. In our approach there is an important difference between the Riemannian and pseudoRiemannian moduli spaces. The generating geodesic in the Riemannian case was shown to be carried by the dilatons only [12]. The pseudo-Riemannian case turns out to be richer. The aim of this paper is to extend the discussion to the pseudo-Riemannian case. One of the main results, derived in this paper, is the derivation of a theorem, see (3.68), valid for a wide class of orbits, defined by a diagonalizable generator Q of the geodesic, that characterizes the geodesic generating solution in terms of the group-theoretical properties of the corresponding moduli space. Our theorem applies to all supergravities with symmetric scalar manifolds. This includes all theories with more then 8 supercharges and applies to an interesting subset of theories with 8 and less supercharges. We show that the generating solution can be found in a suitable sub-manifold of the original scalar manifold defining a consistent truncation of the theory. We also make general comments which apply to all orbits, including thus the cases in which Q is not diagonalizable. To illustrate our methods we consider three different classes of solutions. We first focus on a class of vacuum Einstein solutions. The application of our theorem to this case reproduces some well-known and some less known solutions. We next consider stationary black hole solutions in four-dimensional supergravity. For that we reduce the four-dimensional black hole solutions, via a timelike reduction, to three dimensions, where they become 3
