BPS Domain Wall from Toric Hypermultiplet

Transcription

1 Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 8, HIKARI Ltd, BPS Domain Wall from Toric Hypermultiplet Rio N. Wijaya 1, Fiki T. Akbar 1, Jusak S. Kosasih 1 and Bobby E. Gunara 1,2 1 Theoretical Physics Lab, Department of Physics Institute of Technology Bandung Jl. Ganesha 10 Bandung Indonesia 2 Indonesia Center for Theoretical and Mathematical Physics (ICTMP Copyright c 2018 Rio N. Wijaya, Fiki T. Akbar, Jusak S. Kosasih and Bobby E. Gunara. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we present the gradient flow equations of BPS domain walls of five dimensional gauged N = 2 supergravity coupled to a hypermultiplet where the hypermultiplet scalar manifold is chosen to be the most general selfdual Einstein spaces admitting torus symmetry. An interesting result is that the gradient flows describe the supersymmetric flows and the spin- 1 2 fermion masses on the walls which is on the upper half plane. For further analysis, we provide stability condition for the flow equation obtained for BPS domain walls. Keywords: Domain Walls, Flow equation, Selfdual Einstein 1 Introduction For the past decade, gauged N = 2 supergravity in five dimensions has been one of the main interest to study because of several development. A special class of gauged minimal supergravity admitting a BPS domain wall solution [1] is a result of Calabi-Yau compactification of Horava-Witten theory[2]. BPS domain wall solution can also be identified with four dimensional N = 1 theory of a strongly-coupled heterotic compactification[3].

2 370 Rio N. Wijaya, Fiki T. Akbar, Jusak S. Kosasih and Bobby E. Gunara Extension of Randall-Sundrum scenario[4] to a supersymmetric case is another development to note for. One of the succesful extension in the context of five dimensional gauged N = 2 supergravity is shown in [5]. In relation to minimal Calabi-Yau compactification of M-theory, this model is chosen to be couples with universal hypermultiplet. In [6], a general case of the model has been studied. The result is N = 2 supergravity coupled to selfdual Einstein metric with torus symmetry[7]. It is found that the supersymmetric flows is on the two dimensional hyperbolic space which is also upper-half plane. This paper will derive the gradient flow equations for a model coupled to arbitrary hypermultiplet. We will then apply these to the model studied in [6]. We find that on the walls one of the gradient flows describe the supersymmetric flows, while the other gives the fermion masses. This paper is organized as follows. Section 2 will discuss some basic theory about N = 2 supergravity in five dimensions. In this section, we also discuss gauged N = 2 supergravity coupled to hypermultiplets. The gradient flows for this model are also discussed. Section 3 we review some basic propertis of selfdual Einstein metric with torus symmetry. We discuss the general condition of BPS solutions in section 4. Section 5 will provide a calculation about stability condition for the flow equation. The Final section will conclude our discussion with some added topic about our future works. 2 Five Dimensional N = 2 Supergravity 2.1 The Multiplets The general coupling of n V vector and n H hypermultiplets to supergravity in five dimensions has been studied in [8, 9]. 1 The main ingredients of such theory are 2 a gravitational multiplet (e∵, ψˆµi, A 0ˆµ, â, ˆµ = 0,..., 4 ; i = 1, 2 where e∵, ψˆµi and A0ˆµ are the fünfbein, the gravitinos, and the graviphoton respectively. n V vector multiplets (A xˆµ, λ x i, φ x, x = 1,..., n V, where A xˆµ, λx i and φ x denote the gauge fields, the gauginos, and real vector multiplet scalar fields respectively. 1 Our convention follows rather closely these references. 2 Tensor multiplets are omitted for simplicity.

3 BPS domain wall from toric hypermultiplet 371 n H hypermultiplets (ζ A, q X, A = 1,.., 2n H ; X, Y = 1,..., 4n H where ζ A and q X are the hyperinos and the real hypermultiplet scalar fields respectively. The N = 2 supersymmetry demands that the scalar fields (φ x, q X parametrize a Riemannian manifold M of real dimensions (n V + 4n H which is factorized as[10] M = M V M H (1 where M V is an n V -dimensional real manifold which is called very special real manifold and parametrized by the scalar fields φ x while M H denotes a quaternionic Kähler manifold of real dimension 4n H parametrized by the hyperscalars q X. Thus, we can simplify our problem by considering only pure N = 2 supergravity coupled to hypermultiplets. 2.2 Quaternionic Kähler Geometry In the case where n H > 1, we have that the real dimension of M H is greater than 4. The hypermultiplet scalar manifold M H is a quaternionic Kähler manifold [1] on which it admits a triplet of almost complex structures (J 1, J 2, J 3 satisfied the USp(2 algebra. The holonomy of this manifold belongs to a subgroup of USp(2 USp(2n H. This follows the existence of a pair (ω i Xj, ωa XB which denotes the USp(2 and USp(2n H connections respectively where the flat indices i, j = 1, 2 and A, B = 1,..., 2n H. The metric tensor g XY is then endowed on this manifold and can be written down in terms of the vielbeins f X ia as 3 g XY = ɛ ij C AB f ia X f jb Y, (2 where ɛ ij and C AB are the USp(2 and Usp(2n H invariant tensors respectively. From the above properties we can conclude that the Riemannian curvature can be decomposed into the following: R XY W Z f X iaf Y jb = ɛ ij R XY AB + C AB R XY ij, (3 where R XY AB denotes the USp(2n H curvature, while R XY ij is the USp(2 curvature that has form R XY ij = κ ( f XiC f C Y j f Y ic f C Xj, (4 3 These vielbeins obey also a stronger versions of eq.(2, see [8, 11].

4 372 Rio N. Wijaya, Fiki T. Akbar, Jusak S. Kosasih and Bobby E. Gunara where we choose κ = 1. Furthermore, using (3, we can show that this manifold is Einstein. For n H = 1 the definition of quaternionic Kähler manifold corresponds to the notion of oriented Riemannian four dimensional spaces. Thus, it has an additional property, i.e. it is self dual beside Einstein. We will discuss this in section Gauged N = 2 Lagrangian and Gradient Flow Equations In this subsection we write some terms of gauged N = 2 Lagrangian with gauging of the isometries of the quaternionic Kähler manifold and the supersymmetry transformation of the fermions which are useful for our analysis. The full Lagrangian and transformation laws of the fields are given in [8]. The Lagrangian up to four fermion terms is given by e 1 5d N=2 Lbosonic = 1 2 R 1 4 F 0ˆµˆνF 0ˆµˆν 1 2 g XY Dˆµ q X D ˆµ q Y e 1ɛˆµˆν ˆρˆσˆτ F 0ˆµˆνF 0ˆρˆσA 0ˆτ where 1 4 ψ iˆν Γˆν ˆµˆρ Dˆµ ψ iˆρ + i ζ A Γˆµ Dˆµ ζ A + S ij ψiˆµ Γˆµˆν ψ jˆν + 2iN ia ψ iˆνγˆν ζ A + M AB ζa ζ B V(q, Dˆµ q X = ˆµ q X + k X 0 (qa 0ˆµ, (6 and k X 0 (q are the Killing vectors of the quaternionic manifold. Let us discuss in detail the quantities in fermion mass-like terms. The mass of the gravitinos S ij is defined as (5 S ij i 6 8 P r (σ r ij, (7 where (σ r ij = ɛ ik (σ r k j with (σ r k j r = 1, 2, 3 are the standard Pauli matrices and the prepotentials P r are determined by the Killing vectors of the quaternionic manifold k0 X (q via k X 0 (q = 4 3 RrY X D Y P r, (8 where D Y P r X P r + 2ɛ rst ωx s P t and ωx s is the USp(2 connection. Using some geometrical properties of the quaternionic Kähler manifold the mixing quantity N ia and the hyperino mass M AB is related to the gravitino mass via

5 BPS domain wall from toric hypermultiplet 373 the gradient flow equations 4 X S ij = 1 ( fxia Nj A + f XjA Ni A 4 f XiA X N jb = 4 C AB Sij + ɛ ij M AB,, (9 where X is a general covariant derivative. Furthermore, the scalar potential related to the gravitino mass S ij and the mixing quantity N ia has the form V(φ, q = 4P r P r + 2N ia N ia. (10 The supersymmetry transformations of the fermions up to vector terms are given by δψˆµi = Dˆµ (ωɛ i + i γˆµ P ij ɛ j, 6 δζ A = i (11 2 f ix(γ A ˆµ ˆµ q X ɛ i + Ni A ɛ i, where Dˆµ (ω is the covariant derivative with respect to the Lorentzian spin connection and USp(2 connection, while P ij ip r (σ r ij. 3 Toric Selfdual Einstein Spaces For 4-dimensional manifold, a quaternionic Kähler manifold is equivalent to selfdual Einstein manifold. This property is required in order to classify the selfdual structures with T 2 isometry in four dimensions. A complete classification of selfdual four dimensional manifold admitting T 2 isometry has been explicitly constructed in [14]. The Einstein condition further simplify that the description is related to solutions of a second order of linear differential equtions as we will see below. As derived in [7], the most general toric selfdual Einstein metric is given by 5 [ 1 2 ds 2 = 4ρ 2 (F ρ + Fη 2 ] [( (dρ 2 F 2 +dη 2 F 2ρFρ α 2ρF η β ] 2 + [ 2ρFη α + (F + 2ρF ρ β] 2 + F 2( F 2 4ρ 2 (Fρ 2 + Fη 2, (12 where α = ρ dφ and β = (dϕ + η dφ. The function F (ρ, η satisfies the Laplace equation in two dimensional hyperbolic space spanned by (ρ, η ρ 2 (F ρρ + F ηη = 3 F, ( These gradient flows have also been studied in general matter coupled four dimensional N = 2 supergravity, see [12] and further, in mathematical literature [13]. 5 This selfdual Einstein space also includes homogeneous and inhomogeneous quaternionic Kähler geometries.

6 374 Rio N. Wijaya, Fiki T. Akbar, Jusak S. Kosasih and Bobby E. Gunara with F ρρ = 2 F and F ρ 2 ηη = 2 F. One takes ρ > 0 and η R while (φ, ϕ η 2 are periodic coordinates. Furthermore, it has positive scalar curvature if f satisfies F 2 > 4ρ 2 (Fρ 2 + Fη 2 and negative if F 2 < 4ρ 2 (Fρ 2 + Fη 2. Clearly that the metric (12 has T 2 isometry along the periodic coordinates (φ, ϕ. From (12 the vielbein can be defined as 6 f A i = F 2 4ρ 2 (F 2 ρ + F 2 η + (σ 2 A i (σ 0 A i dρ + F 2 4ρ 2 (F 2 ρ + F 2 η (σ 1 A i dη 2ρF 2ρF [(F 2ρF ρ α 2ρF η β] F 2[ F 2 4ρ 2 (F 2ρ + F 2η ] + (σ3 A [ 2ρF η α + (F + 2ρF ρ β] i F 2[ F 2 4ρ 2 (F 2ρ + F 2η ], (14 where (σ 0 A i is the identity matrix while (σ r A i, r = 1, 2, 3 are the standard Pauli matrices. The hyperscalars q X = (ρ, η, φ, ϕ. The USp(2 connections for the case at hand are [7] ω 1 = 1 ( F η dρ + ( 1 ρf 2 F + ρf ρ dη, ρ ω 2 = F dφ, ω3 = 1 (η dφ + dϕ, ρf (15 from which one can further compute the U Sp(2 curvature R 1 = 1 [( 1 dρ dη F 2 4 F 2 + ρ 2 (Fρ 2 + Fη 2 ρ [ 2 R 2 = 1 ( F ρf 2 η dϕ dρ + ρf ρ + ηf η F 2 ( F ρ + F ( dϕ dη + ρf η ηf ρ ηf 2ρ 2ρ [ R 3 = 1 ( F ρf 2 η dϕ dη ρf ρ + ηf η F 2 ] + dφ dϕ dφ dρ, ] dφ dη, dφ dη ( F ρ + F ( dϕ dρ + ρf η ηf ρ ηf ] dφ dρ. 2ρ 2ρ (16 6 Since we are dealing with four dimensional quaternionic Kähler manifold, the index A = 1, 2.

7 BPS domain wall from toric hypermultiplet Gradient Flows on BPS Domain Walls 4.1 Conditions for BPS Domain Walls In this subsection we provide necessary and sufficient conditions for BPS domain walls which preserves half of the original supersymmetries of the N = 2 supergravity. For that purpose we first introduce the domain wall ansatz for the metric ds 2 = a 2 (x 4 η µν dx µ dx ν + (dx 4 2, (17 where µ, ν = 0, 1, 2, 3, with respects four-dimensional Poincaré invariance, and in the model the scalars allow to vary along the fifth direction x 4. Furthermore, in order to have a residual supersymmetry on the walls the supersymmety transformations (11 have to vanish for some Killing spinor parameter ɛ i. Let us first consider general case i.e., for arbitrary hypermultiplet. When all vectors and fermions vanish, the transformations (11 on the background (17 become [5] ( 1 da δψ µi = µ ɛ i + γ µ 2a dx γ 4ɛ 4 i + i P ij ɛ j, 6 δψ 4i = 4 ɛ i dqx dx 4 ωj Xi ɛ j + i γ 4 P ij ɛ j, (18 6 δζ A = fix ( A i2 dqx γ4 dx 2 R rxy D 4 Y P r, 6 where we have assumed that the scalars q X depend only on x 4. In order to have a residual supersymmetry on the walls, all of the equations in (18 must be set to zero. The second equation allows us to assume that the Killing spinor ɛ i is only x 4 dependent, while the first equation gives a projector equation [5] i da a dx γ 4ɛ 4 i = 2 P ij ɛ j, (19 6 which leads us to an equation relating the warp factor a(x 4 and a real superpotential W = P 3 r P r : 2 d lna = ±W. (20 dx4 The third and fourth equations in (18 give rise to the conditions for the vector multiplet- and the hypermultiplet scalars as follow [5] dq X dx 4 = 3gXY Y W. (21 The equations (20 and (21 describe our supersymmetric flow on domain wall and called the BPS equations which solve the full set of equations of motion

8 376 Rio N. Wijaya, Fiki T. Akbar, Jusak S. Kosasih and Bobby E. Gunara and further, by using them one obtains the form of the scalar potential from Lagrangian (5 for gravitational stability [15]: V = 6W gxy X W Y W. ( BPS Gradient Flows From Toric Selfdual Einstein Spaces Let us first choose the isometry of the metric (12 generated by k 0 = φ g ϕ, (23 where we have normalized k 0 such that the prefactor of φ equals one. We can then obtain the triplet of prepotential [6] P 1 = 0, P 2 = 1 ρ 2 F, P 3 = 1 η g, (24 2 ρ F which can be used to compute the real superpotential [ ] 1 (η g2 W = ρ +. (25 6F 2 ρ Moreover, the scalar potential (22 particularly become [6] V = 1 (η g2 18 ρ (ρ 2 F F 2 ρ [F 2 4ρ 2 (Fρ 2 + Fη 2 ] [( ρw 2 + ( η W 2 ]. (26 and the supersymmetric flow equations (20 and (21 simplify as dφ dx = dϕ 4 dx = 0, 4 dρ dx = 12 ρ 2 F 2 4 [F 2 4ρ 2 (Fρ 2 + Fη 2 ] ρw, dη dx 4 = 12 ρ 2 F 2 [F 2 4ρ 2 (F 2 ρ + F 2 η ] ηw. (27 We see that the T 2 isometry implies that we have two dimensional problem instead of four dimensions with respect to the metric ds 2 = on the upper half-plane. [ 1 4ρ 2 (F 2 ρ + F 2 η F 2 ] (dρ 2 + dη 2, (28

9 BPS domain wall from toric hypermultiplet 377 Now we turn to discuss the gradient flows (9 on this BPS domain walls. We trace the first equation in (9 with S ij, i.e. S ij X S ij = 1 2 S ij ( f XiA N A j, (29 then using (8 and further, evaluated on the walls, we regain the last two equations in (33. Thus, the first equation in (9 describes the supersymmetric flows on the walls. Finally we consider the second equation in (9. First, we contract it with the USp(2 invariant tensor C AB and then, using some geometrical properties of quaternionic i.e., the USp(2 algebra of its triplet of the almost complex structure J r and D 2 P r = 4 P r, [D X, D Y ] P r = 2ɛ rst R s XY P t, (30 one then finds that this satisfies trivially. Second, similar way as before but using another USp(2 invariant tensor ɛ ij results M AB = ɛ ij f XiA X N jb, (31 which is the hyperino mass. Evaluating on the walls one has M AB = 2 ( 4ρ 2 F 2 2 [ [(σ ɛ 1 6 F 2 4ρ 2 (Fρ 2 + Fη 2 ij ia (σ 1 jb (σ 0 ia (σ 0 jb] R r ρηd ρ D η P r ] + (σ 0 ia (σ 1 jb R r XY DρP 2 r (σ 1 ia (σ 0 jb R r XY DηP 2 r. (32 5 Stability of Flow Equations Looking at the flow equation (33, we define new variables to calculate the stability X(ρ, η = dρ dx = 12 ρ 2 F 2 4 [F 2 4ρ 2 (Fρ 2 + Fη 2 ] ρw, Y (ρ, η = dη dx 4 = 12 ρ 2 F 2 [F 2 4ρ 2 (F 2 ρ + F 2 η ] ηw. (33 Note that the critical point ρ W = 0 is reached when X(ρ, η = Y (ρ, η = 0. The hessian matrix is then consist of the first derivative of X and Y with respect to ρ and η. We have ( ρ X Hess = ρ Y (34 η X η Y

10 378 Rio N. Wijaya, Fiki T. Akbar, Jusak S. Kosasih and Bobby E. Gunara as the Hessian matrix. Calculating the first derivative of X and Y, which consist of second derivative of W, we have the Hessian matrix in critical point is ( 2 Hess = K(ρ, η ρ W ρ η W ρ η W ηw 2, (35 where K(ρ, η = 12ρ 2 F 2 /[F 2 4ρ 2 (F 2 ρ + F 2 η ]. To calculate the stability of the superpotential, we need to find the eigenvalue of this matrix. The eigenvalue are the solutions of the quadratic equation λ 2 ± K(W ρρ + W ηη λ + K 2 (W ρρ W ηη W 2 ρη = 0. (36 There are two cases in which we can calculate the solutions which depends on the sign of the second term. If the sign is positive, the eigenvalue becomes λ 1,2 = 1 2 K(W ρρ + W ηη ± 1 2 K (W ρρ W ηη 2 + 4W 2 ρη. (37 We need both eigenvalue to be positive for the stable condition. We asign the subscript 1 to the positive sign of the second term, and subscript 2 to the negative sign. We will see that the two conditions for the eigenvalues bounded the value of K(W ρρ + W ηη from K (W ρρ W ηη 2 + 4Wρη 2 to K (W ρρ W ηη 2 + 4Wρη. 2 In other words, the stability condition is given by K (W ρρ W ηη 2 + 4Wρη 2 < K(W ρρ + W ηη < K (W ρρ W ηη 2 + 4Wρη. 2 (38 If the sign of the second term in quadratic equation is negative, we have the eigenvalues are given by λ 1,2 = 1 2 K(W ρρ + W ηη ± 1 2 K (W ρρ W ηη 2 + 4W 2 ρη. (39 Same as the previous set of eigenvalues, we need both eigenvalue to be positive for a stable system. We assign the subscript 1 to the positive sign of the second term, and subscript 2 to the negative sign. We see that the second term is always positive. This means that if λ 2 > 0, λ 1 automatically greater than zero. Thus, we only need λ 2 greater than zero for the system to be stable. For this eigenvalue, simple calculation shows that the stability conditions is given by 6 Conclusions W ρρ W ηη > 4W 2 ρη. (40 The derivation of gradient flow equations for five dimensional N = 2 supergravity coupled to arbitrary hypermultiplet has been done, resulting in equation (9. Some of the properties of BPS domain walls, particularly for four dimensional flat spacetime are discussed after equation (17.

11 BPS domain wall from toric hypermultiplet 379 We have shown by using equation (19 that the domain walls we obtain preserve half of the supercharges of their parenthal theory in five dimensions. A collection of BPS equations are derived in (20 and (21. These equations are the solutions of the equations of motion and ensure the potential (22 for gravitational stability. Our results show that the the gradient flows (9 describe the supersymmetric flows on walls and also, generate the hyperino mass. The final section derives the stability condition for the fake superpotential. As we have seen, the sign on the flow equation give rise to two different stability conditions. One is given by equation (38, and the other is given by equation (40 Note that we do not derive the effective theory on BPS domain walls which is N = 1 supersymmetric theory in four dimensions described by the left over massless fields. This however might be carried out in a different way. The fifth coordinate x 4 is compactified to singular spaces S 1 /Z 2 and the resulting theory is effectively also N = 1 supersymmetric theory [16]. Acknowledgements. This work is supported by P3MI ITB References [1] A. Lukas, Burt A. Ovrut, K. S. Stelle, Daniel Waldram, The Universe as a Domain Wall, Phys. Rev. D, 59 (1999, no. 8, [2] P. Horava and E. Witten, Heterotic and Type I String Dynamics from Eleven Dimensions, Nucl. Phys. B, 460 (1996, [3] E. Witten, Strong coupling expansion of Calabi-Yau compactification, Nucl. Phys. B, 471 (1996, [4] L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett., 83 (1999, [5] A. Ceresole, Gianguido DallAgata, Renata Kallosh, Antoine Van Proeyen, Hypermultiplets, Domain Walls and Supersymmetric Attractors, Phys. Rev. D, 64 ( [6] L. Anguelova and C. I. Lazaroiu, Domain walls of N=2 supergravity in five dimensions from hypermultiplet moduli spaces, Journal of High Energy

12 380 Rio N. Wijaya, Fiki T. Akbar, Jusak S. Kosasih and Bobby E. Gunara Physics, 2002 (2002, [7] D. M. J. Calderbank and H. Pedersen, Selfdual Einstein Metrics with Torus Symmetry, J. Diff. Geom., 60 (2002, [8] A. Ceresole and G. Dall agata, General Matter Coupbled N=2, D=5 Gauged Supergravity, Nucl. Phys. B, 585 (2000, [9] J. Ellis and M. Gunaydin and M. Zagermann, Options for Gauge Groups in Five-Dimensional Supergravity, Journal of High Energy Physics, 2001 (2001, [10] G. Sierra, N = 2 Maxwell Matter Einstein Supergravities in d=5,4, aand 3, Phys. Lett. B, 157 (1985, [11] J. Bagger and E. Witten, Matter Couplings in N=2 Supergravity, Nucl. Phys. B, 22 (2001. [12] R. D Auria and S. Ferrara, On Fermion Masses, Gradient Flows and Potential in Supersymmetric Theories, Journal of High Energy Physics, 2001 (2001, [13] D. V. Alekseevsky, Flows on quaternionic Kahler and very special real manifolds, Comm. Math. Phys., 238 (2003, [14] D. D. Joyce, Explicit Construction of Selfdual 4-Manifolds, Duke Math. Jour., 77 (1995, [15] K. Behrndt, Carl Herrmann, Jan Louis, Steven Thomas, Domain Walls in Five Dimensional Supergravity with Non-Trivial Hypermultiplets, Journal of High Energy Physics, 2001 (2001, 11 [16] F. P. Zen, Bobby E Gunara, Arianto, Hari Zainuddin, On Orbifold Compactifircation of N=2 Supergravity in Five Dimensions, Journal of High Energy Physics, 2005 (2005, Received: March 14, 2018; Published: December 18, 2018