Curved Dyonic Domain Walls in Four Dimensions

In this short paper we show the existence of solitonic solutions of four dimensional ungauged N=1 supergravity coupled to arbitrary vector and chiral multiplets whose Ricci scalar curvature is constant. The Ricci scalar of spacetimes indeed depends on the σ-model, namely the complex scalars and their first derivative. Then, we give two explicit models, namely static domain walls and static spherical symmetric black holes which are related to our previous works.


INTRODUCTION
Topological defects such as domain wall solutions of supergravity have acquired a large interest due to their duality with renormalization group (RG) flows described by a beta function of field theory in the context of AdS/CFT correspondence [1] .In particular there has been a lot of study considering these solutions which preserve some fraction of supersymmetry in five dimensional supergravity theory.
Inspired by the development of supergravity theory it is of interest to extend the case to more general theory without supersymmetry.In this short paper, we study solitonic solutions of Einstein-Maxwell-Higgs in four dimensions again inspired by four dimensional N=1 supergravity with general gauge-scalar couplings.In particular, we consider a class of solutions called charged domain wall.If the electric and magnetic charges are included in the setup, then the domain walls interpolate two ground states of constant scalar curvature which are not Einstein.These generalize our previous results in neutral domain walls discussed [2][3][4][5][6] in which they connect two Einstein spaces, particularly anti-de Sitter spaces.
This paper provides our preliminary results on curved dyonic domain walls of four dimensional Einstein-Maxwell-Higgs theory.The "dyonic" means that the solitonic object has both electric and magnetic charges.Moreover, in the theory we should turn on the function called scalar potential of the theory in order to have a domain wall solution.
The structure of the paper can be mentioned as follows.In section 2 we state our general results on charged domain walls.In section 3 we discuss shortly some aspects of four dimensional Einstein-Maxwell-Higgs theory inspired by four dimensional N=1 supergravity coupled to vector and chiral multiplets.Then we discuss field equations of motions of the theory in section 4. Section 5 provides discussion on charged domain walls in the asymptotic regions.

METHOD CURVED DOMAIN WALLS
Now we turn our attention to consider some geometrical aspects of domain walls in (pseudo)-Riemannian geometry.In general, the domain wall ansatz metric can be written down as '' 3 '2

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'' which implies that Ricci scalar is given by where ' da a du . The quantities ˆab R and R are the components of Ricci tensor and Ricci scalar in three dimensions, respectively.
Firstly, let us consider a case when the four-manifold is Einstein, where , 0,...3 and 4 is real.
Theorem 1 Suppose we have the condition (5) which follows that the three dimensional hypersurface is also Einstein with constant 3 .Then, there are only two cases: 1.For 4 0 , then where 00 , AB are real constants, Since domain wall interpolates between ground states of different "cosmological" constant in the asymptotic limit, both cases are possible are possible.In the context of field theory the results in Theorem 1 describe an uncharged domain wall [2][3][4][5][6] .
Next we turn to a case of constant Ricci scalar Theorem 2 Suppose we have the condition ( 6) such that the three dimensional hypersurface is of a constant scalar curvature with constant 3 .Then, we also two cases: 1.For 4 0 , then 2

EINSTEIN-MAXWELL-HIGGS THEORY
In this section we give a short description of Einstein-Maxwell-Higgs theory in four dimensions.This theory is inspired by bosonic parts of N=1 supergravity coupled to arbitrary vector and chiral multiplets in four dimensions.For interested reader, the complete N=1 supergravity can be found, for example, in [7] .
Let us now discuss the ingredients of Einstein-Maxwell-Higgs theory in four dimensions.This theory consists of a gravity coupled to v n vector and c n real scalar fields.Furthermore, the Lagrangian of the theory has the form where , 1,..., c i j n and , 1,..., v n .The real scalars i z span a Riemannian manifold with metric ij g .The quantity F is an Abelian field strength of A , and  F mn L is a Hodge dual of F .The functions and are real gauge couplings which depend on the scalars i z .

The real function ()
Vz is referred to as the scalar potential.

EQUATIONS OF MOTIONS
Let us first discuss the equations of motions of the fields which can be obtained by varying the action related to the Lagrangian ( 7) with respect to g , A , and i z .Then, we have three equations.First, the Einstein field equation where the energy-momentum tensor T is given by Second, the gauge field equation of motions is together with the Bianchi identities 0 F . (11) Third, the scalar field equation of motions Taking the trace of (12) we finally get NM where 1,3 M is a solution of ( 6)-(10) describing a four dimensional curved spacetime.This will be discussed in detail in the following sections.
dx dx du , (1) where , 0,1, 2 ab and au is the warp factor.The components of Ricci tensor have the form '2 2 Theorem 1, the results in Theorem 2 describe a charged domain walls in the asymptotic regions interpolating between the spaces of constant scalar curvature which are not Einstein.
the Ricci scalar of the spacetimes depends only on the dynamics of the real scalars i z .The Ricci scalar (11) becomes a constant if the scalars i z are frozen or in other words 0