Melvin Space-times in Supergravity

We consider Melvin-like cosmological and static solutions for the theories of ${\cal N}=2$, $D=4$ supergravity coupled to vector multiplets. We analyze the equations of motion and give some explicit solutions with one scalar and two gauge fields. Generalized Melvin solutions with four charges are also constructed for an embedding of a truncated ${\cal N}=8$ supergravity theory. Our results are then extended to supergravity theories with the scalar manifolds $SL(N, R)/SO(N, R)$. It is shown that solutions with $N$ charges only exist for $N=8$, $6$ and $5$ corresponding to theories with space-time dimensions $D=4$, $5$ and $7$.


Introduction
Recently an active area of research has been the study of supersymmetric gravitational backgrounds in supergravity theories with various space-times dimensions and signatures [1].Our present work will only focus on non-supersymmetric time-dependent and static solutions in some supergravity theories.Time dependent solutions in string theory and their relevance to questions in cosmology have been considered by many authors (see for example [2][3][4][5][6][7] and references therein).
Many years ago, an interesting class of vacuum solutions for Einstein gravity depending on one variable was constructed by Kasner [8].Related solutions were also found by several authors [9][10][11][12].The Kasner metric can be generalized to all space-time signatures and dimensions [13].The D-dimensional Kasner vacuum solution can take the form where ǫ 0 , ǫ i = ±1.The so-called Kasner exponents satisfy Four-dimensional charged Kasner universes of the form (1.1) with fixed exponents, as solutions admitting Killing spinors, were considered in [14].Generalized Melvin-fluxtubes, domain walls and cosmologies for Einstein-Maxwell-dilaton theories and the correspondence among them were explored in [15].These generalized solutions were obtained by applying a solution generating technique to seed Levi-Civita and Kasner space-times.The results of [15] were later extended to gravitational theories with a dilaton and an arbitrary rank antisymmetric tensor in [16].There, explicit solutions were constructed for the various D = 11 supergravity and type II supergravity theories constructed by Hull [17].Moreover, Melvin space-times were studied in [18] for the ungauged five-dimensional N = 2 supergravity theory.The equations of the very special geometry underlying the structure of the five dimensional theories turned out to be extremely useful in the analysis of the equations of motion and in the construction of the solutions.Our present work shall mainly extend the results of [15,16,18] and deal with nonsupersymmetric Melvin-like cosmological and static solutions of the theories of ungauged four-dimensional N = 2 supergravity theories coupled to vector multiplets in arbitrary space-time signature [19].In these theories, the geometry of the scalar fields is a direct product of the geometry of vector multiplets scalars and that of the hypermultiplets scalars.In the present work, the hypermultiplets are ignored and only the scalars of vector multiplets are kept.For the detailed study of the extension of special geometry to theories with Euclidean and neutral signatures, we refer the reader to [20][21][22][23][24].The vector multiplet sector of four-dimensional supergravity in arbitrary space-time signature has been considered in [25] via the reduction of Hull's eleven-dimensional supergravity theories on Calabi-Yau threefolds, followed by a reduction on spacelike and timelike circles.Moreover, four-dimensional N = 2 supergravity coupled to vector and hypermultiplets in signatures (0, 4), (1,3) and (2, 2) were obtained via the compactification of type-II string theories with signatures (0, 10), (1,9) and (2,8) on Calabi-Yau threefolds [26].
Our work is planned as follows.In the next section we briefly present some of the basic properties of the ungauged four-dimensional N = 2 supergravity theories and their equations of motion for the metric, gauge and scalar fields.In section three, we perform an analysis of the equations of motion and derive our solutions.Section four contains some explicit solutions for two inequivalent Lorentzian supergravity models where the scalar manifold is given by SL(2, R)/SO(2) and for an embedding of a truncated N = 8 supergravity.In section five, we present two classes of solutions for supergravity theories where the scalars lie in the coset SL(N, R)/SO(N, R).It is demonstrated that solutions with N charges exist only for N = 8, 6 and 5 corresponding to space-time dimensions D = 4, 5 and 7. Our results are summarized in section six.

4D Supergravity
The Lagrangian of the general theory of ungauged N = 2, D = 4 supergravity theories can be given by The theory has n + 1 gauge fields A I , (F I = dA I ) and X I are functions of n complex scalar fields z a .The details of the Lorentzian four-dimensional supergravity theories and their formulation in terms of special geometry can be found in [19].Our analysis is not restricted to Lorentzian theories and is valid for all space-time signatures.The parameter α takes the values ±1.Roughly speaking, the theories of Euclidean and (2, 2) signature can be obtained by replacing the complex structure with a paracomplex structure [20][21][22][23][24].
To use a unified description, we define i ǫ which satisfies i 2 ǫ = ǫ and īǫ = −i ǫ .Here ǫ = 1 for the theories with Euclidean and neutral signature and ǫ = −1 for the Lorentzian theory.We note the relation where g a b = ∂ a ∂bK is the Kähler metric and K is the Kähler potential of the supergravity theory.
In a formulation of special geometry, one relates the coordinates X I to the covariantly holomorphic sections obeying the constraint by The Kähler potential is given by We also have the relations In cases where the N = 2 supergravity models can be described in terms of a holomorphic homogeneous prepotential F = F (X I ) of degree two, we have Here we list the following useful relations The scalar and gauge couplings appearing in the Lagrangian are given by where and with the notation (NX) The gauge fields equations of motion derived from (2.1) are given by The field equations of the scalars z a and za are The Einstein equations of motion are given by (2.14)

Solutions
We consider solutions of the form where U is a function of τ.We obtain from (2.11) for a non-vanishing where q I are constants.The non-vanishing components of the Ricci tensor for the metric (3.1) are given by Using (3.2) and (3.3) and the Einstein equations of motion (2.14), we obtain, for real scalars, the two equations We now employ the relations of special geometry in the analysis of the equations (3.4) and (3.5).For real X I , the prepotential F and all its derivatives are purely imaginary.In this case, we obtain from (2.9) the following relations Using (2.8) and (3.6), we obtain the relation As an ansatz for our solution we take then we obtain from (3.7) and special geometry the relations Using these relations in (3.4) and (3.5), we finally obtain The first equation in (3.10) can be solved by with constants A I and B I. The second equation then reduces to the algebraic condition 12) It remains to analyze the scalar equations of motion.After some calculation employing the equations of special geometry, the scalar equations of motion reduce to the algebraic condition A dual solution can be obtained where we have a non-vanishing F I xy = p I .In this case, X I are imaginary.The analysis of Einstein equations of motion then gives Again we take the solution Special geometry relations give the following equations Note that in this case the relations (2.9) for imaginary X I imply Consequently, using (3.16) and (3.17), the equations (3.14) reduce to The first equation in (3.18) admits the solution which upon plugging in the second equation of (3.18) gives the algebraic condition Again the analysis of the scalar equation gives, after some calculation involving special geometry relations, the algebraic condition

Examples
Consider the N = 2 supergravity model with a Lorentzian signature and with the prepotential F = −iX 0 X 1 .This corresponds to a model where the scalar manifold is given by SL(2, R)/SO (2).For this model we obtain from (3.11) the solution Using (3.8) and the algebraic conditions (3.12) and (3.13) (with ǫ = α = −ǫ 3 = −1), we finally arrive at the metric (4.2) Using (3.2) and the second equation in (3.6), we obtain for the gauge fields 3) The dual solution with F 0 xy = p 0 and F 1 xy = p 1 has the scalar fields given by and the metric as in (4.2) with q 0 and q 1 replaced by p 0 and p 1 .These solutions can be referred to as generalized Melvin cosmologies [15].Similarly we can also construct Melvin domain wall solutions (4.5) Taking z as an angle coordinate gives Melvin fluxtubes with two charges.The original Melvin fluxtube [27] can be obtained using our formalism with F = −i (X 0 ) 2 and the exponents a = b = 0 and c = 1.
As another example we consider solutions of N = 8, SO(8) supergravity [28] by focusing on the U(1) 4 Cartan subgroup.Anti-de Sitter black holes solutions of the gauged version of this theory were considered in [29,30].The resulting model can be embedded in an N = 2 supergravity model with the following prepotential Note that the previous two models considered are also consistent truncation of N = 8, SO(8) supergravity.Using (3.6) and (4.8) we obtain (4.9) Using our analysis we obtain the generalized Melvin cosmological solutions The scalars and gauge fields are given by Similarly one can also obtain generalized Melvin domain wall and fluxtube solutions.

SL(N, R)/SO(N, R) coset models
In this section we consider solutions to D-dimensional supergravity theories with scalar fields parameterizing the space SL(N, R)/SO(N, R).These theories can be described by the following Lagrangian The gauge kinetic term metric given by (5. 2) The scalars are described by X I subject to the condition The scalars ϕ can be described in terms of X I by Domain wall and charged time-dependent solutions for the gauged versions of these theories were considered in [31] and [32].We start by considering solutions of the form For these solutions, the gauge fields two-form is given by where P I are constants.The analysis of the equations of motion derived from (5.1) gives the solution provided the conditions (5.12) are satisfied.The analysis of the scalar equations reveals no further conditions.The condition (5.12) was also obtained in the study of domain wall solutions and S-branes [31,32].As both space-time dimensions and N must be integers, it is evident that our solutions are only valid for the space-time dimensions D = 4, 5 and 7 corresponding to the cases N = 8, 6 and 5.
A second class of solutions, with the condition (5.12), can be obtained with the metric where (5.14) and 1) . (5.15)

Summary
We have considered solutions depending on one variable for the theories of four-dimensional N = 2 supergravity theories with vector multiplets.Depending on the signature of the theory, our charged solutions can describe time-dependent cosmological or static solutions.These solutions which are labelled as Melvin space-times can be thought of as charged generalizations of Kasner spaces.We found explicit solutions for specific models in N = 2 supergravity with two charges.Solutions with four charges for a truncation of N = 8 supergravity theory that can be embedded in N = 2 supergravity were also presented.Moreover, solutions with N charges for N = 8, 6 and 5, corresponding to supergravity theories with space-time dimensions D = 4, 5 and 7 and SL(N, R)/SO(N, R) scalar manifolds were also found.
It is well known that Melvin fluxtubes can be generated from Minkowski space-time as a seed solution [33].Generalised Melvin solutions in Einstein-Maxwell theory were constructed using these techniques in [15] with Kasner space being the seed solutions.The generating techniques were generalized to dilaton gravity in [34] and to gravity with a cosmological constant in [35].In our present analysis, we started with the vacuum Kasner solution as a seed solution and found solutions with non-trivial scalar and gauge fields through an explicit analysis of the equations of motion.It would be of interest to study generalized Melvin solutions in gauged supergravity theories in various dimensions.We hope to report on this in a future publication.