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JHEP06(2017)123

Published for SISSA by Springer

Received: April 24, 2017

Accepted: June 15, 2017

Published: June 23, 2017

A Riemann-Hilbert approach to rotating attractors

M.C. Cˆamara,

G.L. Cardoso,

T. Mohaupt

and S. Nampuri

Center for Mathematical Analysis, Geometry and Dynamical Systems,

Department of Mathematics, Instituto Superior T´ecnico,

Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Department of Mathematical Sciences,

University of Liverpool, Peach Street, Liverpool L69 7ZL, U.K.

E-mail: ccamara@math.tecnico.ulisboa.pt,

gcardoso@math.tecnico.ulisboa.pt, Thomas.Mohaupt@liverpool.ac.uk,

nampuri@gmail.com

Abstract: We construct rotating extremal black hole and attractor solutions in gravity

theories by solving a Riemann-Hilbert problem associated with the Breitenlohner-Maison

linear system. By employing a vectorial Riemann-Hilbert factorization method we ex-

plicitly factorize the corresponding monodromy matrices, which have second order poles

in the spectral parameter. In the underrotating case we identify elements of the Geroch

group which implement Harrison-type transformations which map the attractor geometries

to interpolating rotating black hole solutions. The factorization method we use yields an

explicit solution to the linear system, from which we do not only obtain the spacetime

solution, but also an explicit expression for the master potential encoding the potentials of

the inﬁnitely many conserved currents which make this sector of gravity integrable.

Keywords: 2D Gravity, Black Holes in String Theory, Integrable Field Theories, Sigma

Models

ArXiv ePrint: 1703.10366

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP06(2017)123

JHEP06(2017)123

1 Introduction

Exploring the space of solutions is an essential part of deepening our understanding of

gravity. While a large variety of methods is available, a full classiﬁcation of solutions is

far out of reach. Studies therefore concentrate on special classes of solutions which exhibit

symmetries. Black holes, i.e. spacetimes with event horizons, are a particularly important

class of solutions as they provide a laboratory for testing ideas about quantum gravity.

Birkhoﬀ’s theorem and its generalizations provide a complete classiﬁcation of stationary

axisymmetric black holes in Einstein and Einstein-Maxwell theory. Among matter coupled

gravitational theories, eﬀective supergravity theories arising from string theory are partic-

ularly important, as they have a potential UV completion. Therefore their stationary black

hole solutions have been studied extensively. The subclass of static, spherically symmetric

extremal black holes exhibits special properties: the near horizon geometry contains an

AdS

-factor, and the attractor mechanism [1–3] forces scalar ﬁelds to take speciﬁc values

at the horizon. This eﬀectively halfs the number of degrees of freedom, and allows to ﬁnd

extremal black holes by solving ﬁrst order gradient ﬂow equations. While the attractor

mechanism generalises to rotating, stationary axisymmetric extremal black holes [4], ﬁnd-

ing explicit solutions becomes much harder, as the solutions depend non-trivially on two

variables and one has to solve PDEs rather than ODEs. Therefore one needs to explore

alternative systematic methods for constructing rotating solutions. One such method was

pioneered in the seminal work of Breitenlohner and Maison [5], and has since then been

explored further by various authors, including [6–9]. It is based on the observation that

the stationary axisymmetric sector of four-dimensional gravity is integrable, and that the

problem of solving the Einstein equations can be replaced by solving a linear system de-

pending on an additional variable, the ‘spectral parameter.’ Solving the linear system is in

turn equivalent to solving a Riemann-Hilbert (R-H) problem. Apart from four-dimensional

pure gravity, the method can be applied to matter-coupled and higher-dimensional gravi-

tational theories, subject to two conditions: (i) solutions must admit as many commuting

isometries as needed for the consistent reduction to a two-dimensional theory, (ii) the eﬀec-

tive two-dimensional theory must be a scalar sigma model with a symmetric target space,

coupled to two-dimensional gravity. Thus, so far, the method is restricted to actions with

no more than two derivatives, and without a cosmological constant or scalar potential.

1.1 Concepts

We now sketch the working philosophy and concepts of the R-H formulation. Firstly, by im-

posing suﬃciently many commuting isometries the theory is reduced to a two-dimensional

theory, which can have Euclidean or Minkowski signature. Since we are interested in sta-

tionary solutions, we focus on the Euclidean case. The two-dimensional eﬀective action

contains an Einstein-Hilbert term and a scalar sigma model, whose target space is assumed

to be a symmetric space G/H. The essential part of solving the equations of motion is

solving the scalar equation of motion, which can be shown to be the integrability condition

of an auxiliary linear system, depending on an additional variable, the spectral parameter

τ ∈ C [5]. The linear system and the spectral parameter are related to the extension of

– 1 –

JHEP06(2017)123

the manifest rigid G-symmetry of the two-dimensional theory, to a hidden rigid symme-

try under an inﬁnite-dimensional group

G, which is known as the Geroch group [5, 6].

Quantities depending on the spectral parameter can be interpreted as elements of

G or

of its Lie algebra

g. Moreover, the linear system can be interpreted as relating a master

current J, which incorporates the inﬁnitely many conserved currents of the theory, to an

associated master potential X, by J = ?dX. To each solution of the linear system, one

can associate a so-called monodromy matrix M(ω), which depends on another spectral

parameter ω ∈ C. Due to the coupling to gravity, the spectral parameter τ is a function of

the two-dimensional spacetime, and is subject to a diﬀerential equation. The other spectral

parameter ω ∈ C arises as an integration constant of this equation and is independent of

spacetime. The relation between τ and ω deﬁnes an algebraic curve, called the spectral

curve, which has the two-dimensional spacetime coordinates x as parameters. Given the

monodromy matrix the solution of the linear system can be recovered by solving a R-H

factorization problem M(ω) = M

−

(τ, x)M

(τ, x), where M

have certain analyticity prop-

erties, to be reviewed in due course. For our purposes it is essential that the factorization

is of the speciﬁc, so-called canonical form given above. This implies that the solution of

the equations of motion can be extracted from M

−

, while M

encodes the inﬁnite set of

conserved currents.

1.2 Methodology

We will restrict ourselves to three gravitational theories: pure four-dimensonal gravity, four-

dimensional Einstein-Maxwell theory and the four-dimensional Einstein-Maxwell-Dilaton

theory obtained by reducing pure ﬁve-dimensional gravity on a circle. Having converted

the equations of motion to a R-H problem, one faces two problems: proving the existence

and possibly uniqueness of a solution, and obtaining the solution explicitly. Most of the lit-

erature is focussed on existence, and employs factorization algorithms that quickly become

very cumbersome when applied in practice. Moreover [5] and the subsequent literature,

including [6–9], impose a particular ansatz which requires the spacetime to be asymptoti-

cally ﬂat and the monodromy matrix to have only ﬁrst order poles in ω. These are severe

limitations as they exclude extremal solutions, and the attractors solutions which are their

near-horizon limits.

We will present a diﬀerent method, where an explicit factorization of the monodromy

matrix is obtained by solving auxiliary vectorial R-H problems by making systematic use

of Liouville’s theorem of complex analysis. A main advantage of our method is that it

is completely general, and in particular does not require that the underlying spacetime

solution is asymptotically ﬂat, or that the monodromy matrix has only ﬁrst order poles

in ω. The practicability of the method is demonstrated by a variety of examples where

we obtain the factorizations of monodromy matrices explicitly, that is by specifying the

factors M

as matrices of rational functions, with all parameters expressed in terms of

physical quantities such as mass, angular momentum, and charges. We have also included

the proofs of various theorems for the sake of clarity and completeness.

We also study the action of the Geroch group on spacetime solutions, solutions of

the linear system, and the associated monodromy matrices. Our formulation of the linear

– 2 –

JHEP06(2017)123

system makes the relation of its variables to the inﬁnitely many conserved currents and

associated potentials transparent. This allows to not only use the ﬁnite-dimensional group

G but the full Geroch group

G to generate new solutions from seed solutions.

1.3 Results

We present a method for factorizing monodromy matrices, and apply it to a variety of ex-

amples with ﬁrst and second order poles in ω, and with ﬂat as well as non-ﬂat asymptotics.

We observe that while non-extremal black holes have monodromy matrices with ﬁrst order

poles associated to them, extremal black holes and attractor geometries have monodromy

matrices with second order poles. This solves the long-standing problem of constructing

extremal black hole solutions within the R-H formulation. We remark that the factorization

problem for second order poles is diﬀerent from the one for ﬁrst order poles, and cannot

be obtained by taking a limit where ﬁrst order poles coalesce. This can be understood by

drawing an analogy with what happens when performing an additive decomposition of a

rational function into a sum of terms with one pole. A decomposition involving terms with

double poles can never be viewed as a limiting case of another decomposition involving

only terms with simple poles. We also observe that, when using our universal method in

the context of rational monodromy matrices, factorization problems are actually easier to

solve for second order poles than for ﬁrst order poles. This is how it should be, since solving

the Einstein equations is easier for extremal than for non-extremal solutions.

From the explicit R-H factorizations we perform we obtain the following backgrounds

which have hitherto been inaccessible:

1. Static attractor geometries for four-dimensional Einstein-Maxwell and ﬁve-dimen-

sional pure Einstein theory reduced on a circle. The double poles of the mon-

odromy matrices correspond to the double zeros of the temporal metric warp factor

at the horizon.

2. Rotating attractor geometries [4], which are the near-horizon limits of extremal ro-

tating black holes. The monodromy matrices again exhibit double poles. In the

underrotating case the monodromy matrix can be chosen triangular, which simpliﬁes

factorization. In fact, the monodromy matrix takes the same form as for the static

attractor, the diﬀerence being just the value of certain coeﬃcients. Therefore ob-

taining underrotating solutions is as diﬃcult as obtaining non-rotating solutions, and

it is manifest that underrotating extremal solutions, which are distinguished by the

absence of an ergo-region from overrotating solutions, are generated by ‘spinning up’

non-rotating charged extremal solutions. In the overrotating case the monodromy

matrix has a diﬀerent form which indicates that such solutions, which have an er-

goregion and generalize the Kerr solution, form a separate branch in the space of

stationary solutions. Despite that the monodromy matrix cannot be brought to tri-

angular form, we still achieve to factorize it, and present the explicit solution for the

uncharged subcase.

– 3 –

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