Some exact solutions for maximally symmetric topological defects in Anti de Sitter space

Journal of High Energy Physics, Mar 2018

Orlando Alvarez, Matthew Haddad

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP03%282018%29012.pdf

Some exact solutions for maximally symmetric topological defects in Anti de Sitter space

HJE Some exact solutions for maximally symmetric topological defects in Anti de Sitter space Orlando Alvarez 0 1 Matthew Haddad 0 1 Field Theories in Lower Dimensions 0 1320 Campo Sano Ave , Coral Gables, FL 33146 , U.S.A 1 Department of Physics, University of Miami We obtain exact analytical solutions for a class of SO(l) Higgs eld theories in a non-dynamic background n-dimensional anti de Sitter space. These nite transverse energy solutions are maximally symmetric p-dimensional topological defects where n = (p + 1) + l. The radius of curvature of anti de Sitter space provides an extra length scale that allows us to study the equations of motion in a limit where the masses of the Higgs eld and the massive vector bosons are both vanishing. We call this the double BPS limit. In anti de Sitter space, the equations of motion depend on both p and l. The exact analytical solutions are expressed in terms of standard special functions. The known exact analytical solutions are for kink-like defects (p = 0; 1; 2; : : : ; l = 1), vortex-like defects (p = 1; 2; 3; l = 2), and the 't Hooft-Polyakov monopole (p = 0; l = 3). A bonus is that the double BPS limit automatically gives a maximally symmetric classical glueball type solution. In certain cases where we did not nd an analytic solution, we present numerical solutions to the equations of motion. The asymptotically exponentially increasing volume with distance of anti de Sitter space imposes di erent constraints than those found in the study of defects in Minkowski space. Solitons Monopoles and Instantons; Field Theories in Higher Dimensions 1 Introduction Defects in constant curvature spaces The Darboux frame and the spherically symmetric ansatz Finite transverse energy constraints 4.1 Double well potential model with the double BPS limit 5.1 5.2 5.3 5.4 6.1 6.2 4.1.1 4.1.2 4.1.3 4.1.4 5.2.1 5.2.2 5.2.3 5.2.4 The double BPS kink-like defects (l = 1) The double BPS vortex-like defects (l = 2) Case of q = 1 (no acceptable solution) Case of q = 2 Case of q = 3 Static spherically symmetric classical glueballs 2 6= 0 and m2A # 0 Flat space equations of motion Conclusions 2 3 4 6 7 8 A Maximally symmetric submanifolds of maximally symmetric spaces A.1 Totally geodesic submanifolds A.2 Examples of totally geodesic submanifolds A.3 Intrinsically at submanifolds B Relation to the work of Lugo, Moreno and Schaposnik { i { HJEP03(218) In this article we obtain exact analytical solutions for a class of SO(l) Higgs eld theories in a non-dynamic background n-dimensional anti de Sitter space AdSn. These eld theories admit maximally symmetric p-dimensional topological defects. The world brane of a maximally symmetric p-defect is a q = p + 1 dimensional timelike submanifold q AdSq that is isometrically embedded in AdSn; it is the gauge invariant set corresponding to the zero locus of the Higgs eld. The value of l is determined by n = q + l. The search for a maximally symmetric defect solution to the equations of motion requires the Lorentzian submanifold q to admit the largest possible group of isometries. For q-dimensional manifolds, this Lie group has dimension 12 q(q + 1). The choices of this q-dimensional manifold are Minkowski space Mq, anti de Sitter space AdSq, and de Sitter space dSq. In this article we mostly discuss the anti de Sitter cases. We show that the Minkowski and de Sitter cases do not give a maximally symmetric solution. The anti de Sitter case gives a maximally symmetric solution when the isometric embedding AdSq ,! AdSn is totally geodesic. In our defect considerations we assume that when we refer any of these maximally symmetric manifolds, we are implicitly considering the simply connected universal covering space. We need that p 0 or equivalently that q 1. The mathematical reason is that the formalism we employ requires an Euclidean signature for the metric of the normal tangent space (T space is the whole tangent space of AdSn at that point, which has Minkowski signature, and our formulas do not apply directly.1 We are restricted to AdSn with dimensionality n = q + l 2. We classify AdS topological defects by their degrees of longitudinal and transverse freedom using the tuple notation (q; l). We develop a universal formalism that allows us to study all values of (q; l). We only need to study in detail three types of defects: kink defects, vortices, and monopoles. Our parlance is that a defect with l = 1 (with one transverse dimension) is kink-like, a defect with l = 2 is vortex-like, and a defect with l = 3 is hedgehog-like or monopole-like. For a review of Minkowski space kinks (l = 1) look in [1, Chapter 6]. The Nielsen-Olesen vortex [2] has l = 2, and p = 1 or equivalently q = 2. The 't Hooft-Polyakov monopole [3, 4] has l = 3, and p = 0 or equivalently q = 1. 1The q = 0 case corresponds to instantons, and the embedding manifold is not AdSn but the Euclidean signature negative constant curvature hyperbolic space Hn. { 1 { Line vortices in AdS4 were discussed in reference [7], and line vortices in dS4 in [8]. HJEP03(218) For l 4, the transverse energy of these spherically symmetric solutions in AdSn diverges. For a comprehensive review of topological solitons we recommend the book by Manton and Sutcli e [5]. We do not discuss electrically charged defects such as the Julia-Zee dyon [6]. There is a major di erence between the study of the equations of motion for maximally symmetric p-defects in Minkowski space Mn and in anti de Sitter space AdSn. In Minkowski space, the equations of motion only depend on the transverse dimensionality l, and the study of the solutions is independent of the dimensionality p of the defect. This is not the case for p-defects in anti de Sitter space where the equations of motion depend on both p and l. This forces us to do a case by case analysis as we vary p and l, e.g., see gure 4. In the rst reference, approximate analytic solutions for vortices in AdS4 were found after applying some simplifying approximations to the equations of motion. Vortex holography played a strong part of the discussions in these papers. The study of magnetic monopoles in AdS4, the case with (q; l) = (1; 3), has been around for a while. The earliest work we are familiar with are papers authored by Lugo and Schaposnik [9], and Lugo, Moreno and Schaposnik [10]. We collectively refer to these two articles as LMS. In appendix B we discuss how their work is related to ours. Numerical axially symmetric monopole solutions in AdS4 are explored in [11]. No exact analytic solutions were found in these references. Approximate analytic and numerical methods are used to discuss multi-monopoles and multi-monopole walls and their importance in the AdS/CFT correspondence in [12] and in [13]. Atiyah [14] had earlier discussed magnetic monopoles in Euclidean hyperbolic 3-space H3 by exploiting the conformal invariance of the self-dual Yang-Mills equations. His 4-dimensional manifold was S1 H3 with the Euclidean signature product metric. This manifold is conformal to Euclidean space E4. He used the observation due to Bogomolny that the self-dual Yang-Mills equations applied to a time independent SU(2) gauge eld are equivalent to the Bogomolny equations.2 The product manifold Atiyah uses is not the Lorentzian manifold AdS4, and his monopoles are not AdS4 monopoles. In the 't Hooft-Polyakov monopole in Minkowski space M4 there are two independent length scales determined by the mass of the Higgs scalar m and by the mass of the massive vector meson mA. A good way to see this is to observe that after an appropriate rescaling of the elds, the action for the SO( 3 ) Georgi-Glashow model may be schematically written as I = 2 0 Z m2 ' 2 1 2 2 2 ; 1 m2A here 0 is the vacuum expectation value of the scalar eld. For example, you expect solutions of the classical equations of motion for the rescaled dimensionless scalar eld ' to depend on mA and on the dimensionless ratio m =mA: ' = '(mAx; m =mA). Prasad and Sommer eld [15] discovered an exact solution to the equations of motion by considering a non-trivial limit of the equations of motion in which m # 0, keeping mA xed, and enforcing an appropriate topological asymptotic boundary condition of the Higgs eld ' 2The methods that fail in the AdS4 discussion in appendix C will work positively in Atiyah's scenario. { 2 { |k| = 1/ρ2 m2φ S P -B S N i l e n Flat space equations e n i l S P B e l b u o D BPS line m2A 2. We explore the parameter space for analytic maximally symmetric solutions line. In section 6 we show that for parameter values in the pink plane m2A # 0 there is an analytic solution for the gauge eld, but the scalar eld has to be studied numerically. The gray plane # 0 is where Lugo, Moreno and Schaposnik looked for BPS monopole solutions in AdS4, see at in nity.3 The net e ect is that the equations of motion only depend on one length scale 1=mA that controls the asymptotic behavior along with the correct boundary conditions imposed manually. The dimensionless parameter m =mA # 0 in the Prasad and Sommer eld limit. The Prasad-Sommer eld solution satis es the rst order equations of Bogomolny [16] that guarantee a solution with a saturated lower bound on the energy [16, 17]. It is known that the mass of the monopole is given by M = (4 F satis es F (0) = 1 and F (1) 1:787, see [5, p. 255]. In this paper we also discuss the Prasad and Sommer eld limit m # 0 for other values of q and l. We refer to this as the limit of Bogomolny, and Prasad and Sommer eld (BPS) even though there may be no Bogomolny equations. The BPS limit was studied in LMS for the case of monopoles in AdS4. In studying the equations of motion for defects in AdSn, we encounter an additional length scale , the radius of curvature4 of AdSn. Now there are three independent length scales 1=m , 1=mA, and . This leads to a three dimensional parameter space, see gure 1, 3This is the same as letting the '4 self coupling # 0 while maintaining the asymptotic boundary 4The radius of curvature is de ned by solution of the vacuum Einstein equations R = jkj 1=2 where k < 0 is the sectional curvature; AdSn is a 1 g R + g 2 20=mA) F (m2 =m2A) where the function that can be explored for exact solutions of the equations of motion. The appearance of this additional length scale was already noticed in [9]. The scalar eld solution ' of the equations of motion depends on and on two dimensionless quantities m and mA : ' = '(x= ; m ; mA ). We attempt to extend the methods of Bogomolny, and Prasad and Sommer eld, and look for regions in the parameter space where we might nd exact solutions. We consider a limit for the equations of motion for a maximally symmetric defect where 6= 0 is xed, but m # 0 and mA # 0. The net e ect is that the equations of motion only depend on one length scale that controls the asymptotic behavior along with the correct boundary conditions imposed manually. N.B. This is a very delicate limit because the action is singular in this limit but the equations of motion are not. We take the limiting equations of motion as the starting point in our analysis, and we abandon the action. These limiting equations are consistent and do not follow from an action principle. This is analogous to consistent equations of motions such as the selfdual Yang-Mills equations, or the self-dual equations of motion for the 4-form in type IIB supergravity which are not derivable from an action. The limiting equations of motion partially decouple. The one for the gauge eld is completely decoupled from the Higgs eld and can be solved independently. The gauge eld solution can then be inserted into the Higgs eld equation of motion which is now linear. In addition, this \double BPS limit" preserves the nonlinear interactions of the gauge elds. The vanishing of the dimensionless parameters, m and mA , leads to exact analytic solutions in a variety of cases, see gure 5 for admissible pairs (q; l). For example, there are exact analytic solutions for kink defects (q; l) = (q; 1) given by (5.8), for the Nielsen-Olesen vortex line (q; l) = (2; 2) given by (5.12), and the 't Hooft-Polyakov monopole (q; l) = (1; 3) given by (5.19) and (5.22). We note that the transverse size of these p-defects in the double BPS limit is comparable to the radius of curvature of AdSn. These double BPS equations of motion are a rst step in a perturbative expansion of the full equations of motion where the small parameters are m and mA . In this sense, we can make contact with the action again. We have not studied the stability of these double BPS solutions in anti de Sitter space. Our only attempt at trying to prove stability was to look for a Bogomolny type bound. This bound is used to establish the stability of the Prasad-Sommer eld solution [16, 17] in Minkowski space. The curvature of AdSn invalidates some of the Minkowski space arguments as explained in appendix C. We show that in the monopole case there are no rst order Bogomolny type equations that imply the equations of motion. Additionally, there is a partial Bogomolny bound relating the energy to the magnetic charge when the magnetic charge density is non-negative. In principle, the underlying symmetries of AdSq and AdSn should greatly aid in the stability analysis of linear perturbations of the equations of motion around these double BPS solutions. The organization of this article is the following: a formalism is developed to study maximally symmetric p-defects in AdSn in sections 2 and 3, leading to the general maximally symmetric equations of motion (3.15) for all values of (q; l). One useful feature of { 4 { HJEP03(218) our formalism is that the normal radial coordinate we employ is always the physical distance from the defect world brane no matter what (q; l) pair we are studying. We found it convenient to use an orthonormal frame because the Pythagorean theorem automatically organized the calculation for us. For example, the transverse energy functional consists of ve positive semi-de nite summands, therefore niteness of transverse energy follows if each summand is nite. The theorem that requires q to be totally geodesic is relegated to mathematical appendix A. The nite transverse energy constraints are discussed in section 4, and summarized in gure 4. In section 5 we discuss the double BPS limit and the exact solutions we found. In section 6, we brie y explore numerically a portion of the parameter space mA # 0 where the gauge eld is explicitly known but the scalar eld has to be studied numerically. In appendix B we relate our work to the work of Lugo, Moreno and Schaposnik. Finally in appendix C we restrict to magnetic monopoles in AdS4 in the BPS limit. We discuss that the BPS equations do not imply the equations of motion. We show the existence of a partial bound on the mass if the magnetic charge density is non-negative. 2 Defects in constant curvature spaces isometry group of Minkowski space Mn, and Mn We use a SO(l) gauged Higgs type eld theory as the model for a topological defect. We rst discuss the de nition of a maximally symmetric p-dimensional defect in Minkowski space and subsequently generalize the notion to a Lorentzian constant curvature space. A p-dimensional maximally symmetric defect in Mn is a topologically stable solution to the equations of motion that is invariant with respect to the action of the subgroup P(q) SO(l) P(n) where q = p + 1 and q + l = n. Here P(n) is the Poincare group, the P(n)= SO(1; n 1). The world brane (time evolution of the defect brane) for the defect is the q-dimensional manifold the symmetry group of the solution is P(q) SO(l) we know that the world brane the defect is a timelike q-plane. Notice that q is intrinsically at, and that the invariance group of the solution implies that the defect is static for any choice of time direction in Let M n be a Lorentzian manifold with constant sectional curvature k and with isometry group5 Isom0(M n) of dimension 12 n(n + 1). A p-dimensional maximally symmetric defect in M n is a topologically stable solution to the equations of motion that is invariant with respect to the action of the subgroup Isom0( q ) SO(l) Isom0(M n) where q is a maximally symmetric q-dimensional Lorentzian submanifold. A maximally symmetric q . q is a constant curvature manifold with dim Isom0( q ) = 12 q(q + 1). Here q is the world brane of the defect core.6 Finding such a q M n requires an appropriate generalization of choosing a plane. The correct \ atness" notion that leads to a maximally symmetric defect is to require q to be a totally geodesic submanifold. An embedded submanifold q of a general manifold M n is said to be totally geodesic if every geodesic (with respect to the induced metric) on q is also a geodesic on M n. Next, we see how this is related to the di erential geometric data. Let D and DM be the Levi-Civita connections on the respective manifolds. A Darboux frame is an orthonormal 5Isom0(N ) is the connected component to the identity of the isometry group of the manifold N . 6By the core of the defect we mean the region in spacetime where the energy density is concentrated. q. Since q for { 5 { frame adapted to the orthogonal decomposition T M = T + (T a submanifold neighborhood of the point , we consider an orthonormal framing (e^a; n^i), where the e^a are tangential to and the n^i are normal to . We use the index convention that latin indices from the beginning of the alphabet a; b; c; d run from 1; 2; : : : ; q and latin indices from the middle of the alphabet i; j; k; : : : take l = n q values from q + 1; : : : ; n. Let u = ua^ea and v = vb^eb be vector elds tangent to q, then the two connections are related by DuM v = Du v uavb Kabi n^i. The symmetric tensor Kabi is called the second fundamental form or the extrinsic curvatures. If u is a tangent vector on then DuM u = Du u uaub Kabin^i. In a totally geodesic submanifold, we would have that DuM u = 0 and Du u = 0 for all geodesics on . This is only possible if the extrinsic curvatures Kabi = 0. Summarizing, totally geodesic submanifolds are those where the extrinsic curvatures vanish. From the viewpoint of standard General Relativity, totally geodesic submanifolds are very desirable because if a test mass in is given an initial velocity tangential to then its motion will remain in . As shown in detail in appendix A, a totally geodesic q-dimensional submanifold q of AdSn is a constant curvature Lorentzian submanifold with the same sectional curvature k as M n and with a at normal bundle (T )?. These are the only submanifolds that admit the possibility of nding a maximally symmetric solution, see remark 1 in the appendix A.3. 3 The Darboux frame and the spherically symmetric ansatz To work out the equations of motion for our maximally symmetric defect, it is convenient to use a coordinate system adapted to the geometry of the problem, i.e., an analog of spherical coordinates. The construction is based on the method of Cartan discussed in our previous paper [18]. Let q be a Lorentzian q-dimensional submanifold of AdSn. If then in a submanifold neighborhood of the point consider a Darboux frame (e^a; n^i). The dual Darboux coframe is denoted by ('a; 'i). Choose a geodesic of AdSn starting at 2 q with initial normal velocity = i n^i, and go a distance k k along the geodesic to a point x 2 AdSn. The coordinates of the point x are ( ; i). Formally, this is the exponential map exp : 2 (T )? 7! x 2 AdSn. Note that 'i = d i. Cartan's idea is to extend the orthonormal Darboux coframe by parallel transporting it along the normal geodesics; in this way you construct an orthonormal coframe at ( ; ) denoted by (#a; #i). In our previous paper we showed that j Kabj 5' b The velocity of a geodesic is constant, and in this way we denote the orthogonal projector along the velocity vector by (PL)ij = ular to the velocity by (PT )ij = ij 2 i j i j = = k k2, and the orthogonal projector perpendick k . Using these, it is easy to write down the orthogonal decomposition of the extended Darboux frame component #i = #iL + #iT where #iL = PL(D )i ; # iT = PT (D )i : { 6 { In the above (D )i = d i + !ij j , where !ij is the connection on the normal bundle (T )?. The formulas above are general. We are interested in the Lorentzian case where AdSq q ,! M n is a totally geodesic submanifold. As discussed previously, we know that Kabi = 0 and that the normal bundle (T )? is at, see appendix A. We can always locally trivialize the normal bundle so we set !ij = 0, i.e., the normal part of the Darboux frame n^i is parallel along q. In summary we have k k PT (d )i (3.1a) (3.1b) k k k k !2 2 ds2Sl 1 5 ; (3.2) 3 q AdSq, 1)-sphere. The metric on AdSn is given by ds2AdSn = gaMb #a coordinates we see that 2 ds2AdSn = cosh2 j j k k ds2AdSq + 4d 2 + where ds2AdSq = gaMb 'a 'b is the standard constant curvature metric on = k k is the radial distance, and ds2Sl 1 is the round metric on the unit (l The part of the metric in the square brackets is the pullback of the metric on AdSn to (T )? via the exponential map exp . Said di erently, this is the induced metric on exp (T )?, the image of the normal tangent space under the exponential map. This induced metric is isometric to the standard metric on Euclidean hyperbolic space Hl, see the discussion associated with eq. (3.5). We re-emphasize that = k k is the physical distance from a point 2 q to the point ( ; ) 2 M n. We will use this physical distance to measure the behavior of our elds as you move away from the defect world brane. A simple model that has a topological p-defect in AdSn is a Higgs model with an SO(l) gauge symmetry, l 2, where n = (p + 1) + l = q + l. The model has a scalar eld I that transforms under the vector representation of SO(l). The uppercase latin indices I; J; K; : : : from the middle of the alphabet will take values from 1 to l. We are looking for a maximally symmetric p-defect that is invariant under the action of Isom0(AdSq) SO(l), and therefore our elds do not depend on the coordinates on the normal coordinates i . A connection that is compatible with the symmetries is AIaJ = AaJI = 0, and AIJ = j AJI with covariant derivative Dj I = @j I + AIJ j j J and curvature Fij = @iAj @j Ai+[Ai; Aj ]. The equations of motion are obtained by extremizing a of AdSq and only depend the action I = R Mn L, where L is the Higgs model Lagrangian density. We are looking for p-defect solutions that are maximally symmetric with symmetry group Isom0( ) SO(l) and under these conditions the action for the Higgs type model is Ispherically sym = E ? Z q ; (3.3) where is the volume element on . The transverse energy7 E? in a local orthonormal 7E? is the tension of the associated p-brane. { 7 { frame for (T M )? is given by iq " sinh jkj1=2 The volume element formula is a special case of a result from our previous paper [18]. We implicitly assumed spherical symmetry for the Lagrangian density to do the angular integrals. Here U is the potential, g is the gauge coupling constant, and Vl 1 is the (l 1)volume of the unit sphere Sl 1. We emphasize to the reader the presence of the hyperbolic cosine factor that would not be there in the case of M n = Mn. The origin of this hyperbolic cosine factor is the metric (3.2). You can make a mistake in uenced by the familiarity of working in Minkowski space where the transverse energy would be obtained by using the pullback metric to (T )? via the exponential map. In this case for p-defects in AdSn, the metric on the normal tangent space is the metric on Euclidean hyperbolic space Hl k k jkj1=2 k k !2 2 ds2Sl 1 Said di erently, removing the hyperbolic cosine term in (3.4) leads to incorrect equations of motion. The equations of motion arise from varying the action (3.3). Note that the in Minkowski space Mn. It is convenient to de ne the Jacobian factor k ! 0 limit of (3.4) is the familiar transverse energy for a spherically symmetric p-defect J ( ) = Vl 1 [cosh ( = )]q [ sinh ( = )]l 1 ; where the \radius of curvature" of AdSn is = 1=jkj1=2. The SO(l) spherically symmetric ansatz we employ is a generalization and slight variant of the original one used by 't Hooft [3] and by Polyakov [4] in the SO( 3 ) Georgi-Glashow model. The ansatz is (3.5) (3.6) I k k I ( ) = ( ) and AIJ ( ) = f ( ) I d J J d I ; (3.7) where and f are functions only of the radius . We choose the potential U to be of the general symmetry breaking form such as the one shown in gure 2. The form chosen for !1! is a hedgehog type ansatz with ( ) 0 corresponding to the topological winding number 1 solutions. We want the world brane of the p-defect to be a gauge invariant set thus we require ( = 0) = 0 which implies (0) = 0. The form for A is motivated by the abelian constant curvature spherically symmetric ansatz8 and by the identi cation of the SO(l) Lie algebra associated with the gauge group to the one associated with the rotational isometries of (T )?. In this article we only discuss situations where both the functions ( ) and f ( ) are non-trivial functions of that describe localized defects. We 8The expected behavior near the world brane is constant eld strength so we expect f (0) 6= 0. { 8 { 1 2 = k k note that expressions (3.4) and (3.7) are actually valid for l = 1 if you interpret V0 = 2 since S0 E1 consists of two points f 1; +1g. Ansatz (3.7) for l = 1 is the odd parity kink (domain wall) solution with an automatically vanishing gauge eld. A brief computation and implementing (3.1) gives D I = d I + AIJ J = 0( ) (PLd )I + = 0( ) (PL#)I + 1 1 2 we exploit that in the orthonormal polar coframe, the longitudinal direction is the radial direction, and that PT and PL are orthogonal projectors to conclude kD k2 = 0( )2 + (l 1) 1 Next we compute the curvature by using di erential forms F IJ = dAIJ + AIK The computation is a bit more involved but greatly simpli es by using the orthogonal projectors: (3.9) ^ AKJ . F IJ = f ( ) 1 (d )IT ^ (d ) J T = f ( ) 1 + + { 9 { f0 ν = 0 ν = ∞ HJEP03(218) This expression for E? is valid for l Equation (3.12) may be simpli ed by introducing the auxiliary function h de ned by 2 1 2 f ( ) : and f as a function of radial distance for a topological defect. The region with the gradient shade in yellow represents the core of the defect. Using the properties of the orthonormal polar coframe we conclude kF IJ k2 = (l 1)(l The norm on 2-forms is normalized by observing that in an orthonormal coordinate system in E2, a constant U(1) eld strength is given by F = f d 1 ^ d 2 and kF k2 = f 2. When we write kF IJ k2 we mean sum over all I and J . For example in an SO(2) gauge theory, you have kF IJ k2 = kF 12k2 + kF 21k2 = 2kF 12 2 k . In general when summing over the spacetime indices i, j you obtain FiIjJ FiIjJ = 2 kF IJ k2. Note that the Yang-Mills lagrangian is quadratic in f in the abelian SO(2) case. Collating all the terms, we have that the transverse energy E ? of the defect is E (" + + 1 " 1 g2 2 1 4 (l 1 2 (l (l 1) 1 1) The expected defect boundary conditions on h are h( ) eq. (4.2). In terms of and h, the transverse energy E? looks like: E [ sinh( = )] h0( )2 With the defect boundary conditions our ansatz does not admit pure gauge solutions. 2 then setting F IJ = 0 in (3.11) leads to f ( ) = 0 or f ( ) = 4= 2, algebraic forms for f ( ) that are incompatible with the defect boundary If we restrict to the case of l = 2, then in polar coordinates ( ; ') for the normal bundle we would have that A12 = f ( ) 12 2 d' !1 ! d' and thus we conclude that we have nontrivial holonomy because if we integrate along the circle at in nity we have H A12 = 2 . This tells us that the total vortex ux is 2 in our normalization, that may be related to the conventional normalization by A = gAconv. In the above discussion, the action leads to the standard Laplacian in the equations of motion. The situation is the same for the theory with the conformal Laplacian, which is obtained by adding to Ispherically sym a term of the type cRMn 2 where c is some xed constant. Since the scalar curvature RMn is a constant, that term can be absorbed into the potential energy function U as a correction to the quadratic term in . We still choose the potential U to be of the general symmetry breaking form such as the one shown in Minimizing E?, we can derive the equations of motion for the defect. The equation of !0 ! 0, see (3.15a) tanh( = ) + coth( = ) (l (l Finite transverse energy constraints We would like the p-defects in AdSn to have nite transverse energy (3.14). To achieve this, we have to study the convergence of the integral for E ? in the two asymptotic limits gure 2. motion for is Similarly, the h equation of motion is 0 = = 1 d J d d 2 d 2 dU d + + 0 = d2h d 2 J d d q (l q + (l 1) h( )2 ( ) is more important. If the transverse energy diverges in the hope for a defect. A divergence in the integral as ! 1 limit then there is no ! 0 may be resolved by considering a di erent ultraviolet completion of the model. For example, you could add higher derivative terms to the action analogous to what is done in the 4D Skyrme model [19] to stabilize the model. In this section, we consider both limits within the context of expression (3.14) ( ) we choose is ( ) see gure 2, tells us that as !+1 U ( ) 12 m2 ( we require that the case ! 1 behavior is tricky, and has to be carefully analyzed on a case by case basis: 3. We are interested in solutions that satisfy the boundary conditions 0 ! 0, and h( ) ! 0, both exponentially in as ? integrand (3.14) consists of ve positive semi-de nite summands. A solution requires that the integral of each summand converge. These conditions impose growth rates on the elds h and . We observe that the growth rates that lead to convergent E ? are not necessarily the growth rates given by the equations of motion. We are interested in topological defect solutions with nite transverse energy so the boundary condition that ! + 0 for the winding number +1 defect. The form of the potential, 0)2. We would like for the length scale given by the radius of curvature ! +1 we are near a quadratic minimum and we have that to dominate the Compton wavelength 1=m of the scalar eld, so in our later applications 1=m even though this does not enter into the convergence analysis. Note 1=m essentially reduces to the at space case. For all l 1, we note that requiring that the scalar eld decays exponentially ( ) 0 = O e 21 (n 1) = = 21 + as ! 1 for some term9 in (3.14). For the future we note that for topological defects in AdSn, the asymptotic and h depend not only on the transverse dimensionality l but also on the dimension p = q 1 of the defect, see (4.6) and (4.10). This is di erent than the situation > 0 guarantees the convergence of the 0( )2 term and the U ( ) of the familiar defects in Mn. To have a gauge theory, we need l 2 which implies that n = q + l 3. In the same spirit, we analyze the asymptotic behavior of summands two and four in (3.14) that are associated with the kinetic energy of the gauge eld h. Those integrals will converge if Finiteness of E ? imposes strong constraints if l elds' self interactions contribute to the energy via the term 3 because the non-abelian gauge Vl 1 (l 1)(l 4g2 2) Z 1 h( )2 1 9Convergence of the integral R +1 ( ) d is guaranteed if there exists > 0 such that ( ) = O(1= 1+ ). In fact the integral will converge with the weaker condition ( ) = O 1= (ln )1+ . (4.1) (4.2) (4.3) This integral will converge if n < 5. Because l 3 and q 1, we see that there is exactly one case where the integral converges, namely q = 1 with l = 3. This is the soliton case, a 0-defect, which is the 't Hooft-Polyakov monopole in AdS4. The n = 5 case where the integral is linearly divergent may be salvageable via some unknown method (see the at space discussion below). There you would have an SO(4) soliton (p = 0), and an SO( 3 ) line defect (p = 1). The situation for the existence SO(l) nonabelian topological defects for n 6 appears to be quite dire because of the exponentially divergent energy. The results are di erent in the at space case, i.e., of (4.3) are given by d e(n 5) = : (4.4) = 1. Here, the asymptotics !1 Z 1 in eq. (3.14). We know that h( ) determined from the asymptotics of the hyperbolic functions ! 0 and the convergence of the integral may be Z 1 d This integral converges if l = 3 for all values of q. Thus SO( 3 ) topological p-defects are energetically allowed for all values of p 0, note that n 4. These are 't Hooft-Polyakov p-defects that have nite energy. The l = 4 case has logarithmically divergent energy and may be salvageable by modifying the model in some way. An analogy is the XY-model10 where the vortex has a logarithmically divergent energy that can be tamed by adding an abelian gauge eld. Another approach to nite energy in the XY model is to just have the scalar eld but restrict the eld con gurations to the topological sector with zero net winding number. The consequence of this is that the eld decays faster at in nity and nite energy. In this way you can have nite energy vortex anti-vortex pairs. The vortices have an ultraviolet energy divergence near the world brane. This is usually remedied by imposing a nite radius cuto in the core, or including higher derivative terms n s (4.5) (4.6) such as in the Skyrme model. 4.1.1 Asymptotic behavior of If we look at (3.15a) we note that as 10The XY model is the SO(2) nonlinear sigma model in E2, i.e., n = 2 at space. 0 = d 2 + m2 ; m = n 2 1 + m2 + n 2 ; to the other terms because h is exponentially small and 1= sinh2( = ) is also exponentially small.11 In this case, the asymptotic behavior of the equation of motion (3.15a) is ! 1 the term containing h2 is very small compared where = behavior is ( ) 0 0 = e m where . The solution to the equation of motion with the correct asymptotic and > 0 is a constant. This decay behavior leads to convergence in the relevant transverse energy integral summands if q always satis ed in our models because n 2. The contribution to the transverse energy density in the asymptotic region from the purely scalar eld part is u ( ) = 1 2 d d 2 + U ( ) 1 2 m2 + m2 The important result here is that the pure part contribution to the transverse energy integral converges for all values of n 2. The topological kink (l = 1) exists for all q 1. 4.1.2 As where Asymptotic equation of motion for h ! 1 with our boundary conditions, the asymptotic equation of motion is 0 = d2h n d + m2A h( ) ; where m2A = g2 20. The exponentially decaying solution to this equation is h( ) = h e To have a gauge eld, we need l 2 which implies that n 3. The relevant transverse energy terms will converge if fourth terms of (3.14): To compute the contribution of h to the energy density, we consider the second and = n 2 q [(n 3 + s n 2 3)=2 ]2 + m2A > 0 according to criterion (4.2). The asymptotic energy density for the gauge eld is uh( ) = l 1 g2 [ sinh( = )]2 2 1 h0( )2 + 2 1 g2 ( )2 h( )2 : uh( ) 4(l 1) e 2 = 1 g2 2 2 2 + m2A h2 e 2 ; and the associated integrand is !+1 ! Vl 1(Sl 1) l 1 2n 1 e(n 1) = 2(l 1) e 2 = g2 2 2 + m2A h2 e 2 = Vl 1(Sl 1) (l 1) 2n 2 g2 q 4m2A 2 + (n 3)2 = (4.9) (4.10) (4.11) (4.12) (4.13) The l = 2 case is the abelian vortex and we see that in (3.14) that the non-abelian selfinteraction term vanishes identically. The asymptotic energy density integrand for the gauge eld is Notice that the h part contribution to the transverse energy integral converges for all values of q 2. Therefore, the topological abelian vortex localized at a totally geodesic AdSq ,! AdSq+2 is energetically possible, i.e., vortex line, vortex sheet, etc. . . For q = 1, the vortex soliton requires mA > 0. The asymptotic behavior of h if l 3 Expression (4.13) is the contribution from the second and fourth summands of (3.14). We note that since n 4 this contribution to the transverse energy converges even if mA = 0. The only concern is the fth summand and we have already addressed it in section 4.1 in the derivation of (4.4). The conclusion is that we do not expect solutions if n 5 for l = 0. We look for Frobenius solutions of the form 1 + O( 2) . There are two real Frobenius indices, + and , with + > . In both the and h cases, + there is a solution of the form = l 2 Z. From the theorem of Fuchs, we know that + 1 + O( 2) . The solution involving may also have a logarithm. This solution does not satisfy the desired boundary conditions or the nite transverse energy constraint. From now on we only consider the + solution. The boundary conditions as the ! 0 behavior of (3.15a). For ! 0 are ( ) ! 0 and h( ) ! +1. First, we examine , the equation of motion is approximately and h, we will encounter two second order linear (4.15) (4.16) we look at l l 1 d d + (l First, we discuss the l = 1 case where we immediately see that the solution behaves like ( ) a + b . We are interested in setting a = 0 because that solution vanishes at = 0 and has an odd extension to < 0 corresponding to a kink localized at the origin. Next, 2, where a Frobenius type solution of the form ( ) = C 1 + O( 2) + = 1 or = 1 l. Thus we conclude that the small 2 to discuss the h equation of motion. We de ne h~( ) = h( ) ! 0 behavior of (3.15b). From (3.14) we see that we have to + O( 3 ) for l boundary condition becomes h~(0) = 0. For is approximately 0 = 2 d2h~ d 2 + (l 3) dh~ d and jh~( )j 2(l 2) h~( ) : 1, the equation of motion 7 6 5 4 3 2 1 HJEP03(218) 12 f (0) 2 + O( 4). ( ) h( ) !0 ! !0 Near region: Far region: solid circles. The light red region contains disallowed values of (q; l) due to the divergence of the transverse energy arising from the nonabelian interactions of the gauge eld, see eq. (4.4). The open squares represent values where the E exponential. Remember that n = q + l and that q 1 and l ? divergence is linear, and the open circles where it is The Frobenius indices are + = 2 and ary condition h~(0) = 0 we learn that h~( ) = = 2 Implementing the bound12 f (0) 2 + O( 4), or equivalently 1. h( ) = 1 is given by To summarize, the asymptotic behaviors of the elds for this kind of topological defect 1 2 3 4 5 6 7 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 1 2 near( ) = 0(0) + O 3 ; ! hnear( ) = 1 f (0) 2 + O 4 ; ( ) h( ) !1 !1 ! far( ) = 0 ! hfar( ) = h e e m ; ; q n = 8 l l l l 1 2 1 2 (4.17) (4.18) (4.19) (4.20) In deriving the asymptotic behavior we explicitly assumed that the radius of curvature satis ed 0 < < 1. Taking a limit such as (1= ) # 0 may be delicate and we have to be very careful. There are no issues with the at space limit (1= ) # 0 if m > 0 and mA > 0. On the other hand, you cannot directly apply the asymptotic formulas above when you approach the BPS line in gure 1. This is a delicate limit in which you have (1= ) # 0 and m # 0. The reason is that the assumptions leading to eq. (4.19) are now invalid. The solution (7.2) of the at space BPS equations lead to a Coulombic tail asymptotic behavior ( ) ! 0 + O(1= ) because the at space eld is massless. You have to be equally careful when you approach the NS-BPS line. The behavior along the double BPS line is safe because the radius of curvature remains non-zero and governs the asymptotic behavior. Double well potential model with the double BPS limit In this section, we will explore a method of obtaining exact solutions to the equations of motion for some of these defects. We study the equations of motion for a model with potential function U ( ) = 1 8 2 It is convenient to rescale to dimensionless variables via the mass of the Higgs boson is m2 = 20, and (l ! , ! 0 . In at space, 1) vector bosons acquire mass via the Higgs mechanism with value m2A = g2 20. The transverse energy (3.14) may be written as Z 1 0 E ? = Vl 1 l 2 2 0 d [cosh ]q [sinh ]l 1 2 (l equations of motion are We note that [ ] = M 1 = L, [ 0] = M (n 2)=2, therefore the prefactor of the integral n 2 20 = q, and we see that [E?] = M q = M=Lp as required. The 1 2 (m )2( 2 1) = d2h (l Note that the equations of motion for a p-defect in AdSn depend on both q and l. The equations of motion (7.1) for a p-defect in Mn depend only on l. d d dh d (5.1) (5.2) (5.3a) (5.3b) In the Minkowski space Mn version of the Higgs model, there are two independent length scales 1=m and 1=mA that enter the equations of motion. We are studying topological defects in AdSn and there is automatically an independent third length scale , the radius of curvature of AdSn. The spherically symmetric equations of motion (5.3) depend on three mass scales m2 , m2A, and 1= 2 if l 2. The radius of curvature is implicit in (5.3) because the coordinate that appears in those equations of motion is actually the dimensionless radial distance = . From the form of the equations of motion there are various parameter limits that can be studied, see gure 1. The limit 1= 2 # 0 corresponds to the Minkowski at space equations of motion. This is the yellow planar region in the gure. At a boundary of this region is the BPS line corresponding to solutions where m2 # 0 but m2A 6= 0. There is also a very delicate non-standard BPS (NS-BPS) like limit in at space where you consider m2A # 0 with m2 6= 0; there are no solutions with our ansatz because the h- eld does not satisfy the two point boundary conditions at = 0 and = 1. This NS-BPS limit is denoted by the dashed green line. LMS studied the gray planar region given by m2 # 0, and they were interested in the BPS line boundary but did not consider the double BPS line, see appendix B. We will nd explicit analytic solutions with 1= 2 6= 0 but m2 # 0 and m2A # 0 along what we call the double BPS limit. Finally in the limit m2A # 0 we have the pink planar region where we have an analytic form for the h- eld, and a numerical solution for the - eld which are brie y discussed in section 6. At this point, it is worth the e ort to be more explicit in the limit we are taking, as it will appear in much of what is to follow. As previously mentioned, this limit is when m2 # 0 and m2A # 0, and can only be taken when there is a third length scale for the physics. In our case, we have one given by the radius of curvature of AdSn. It is also important to understand the way that this limit is taken. Up until this point, we have been using the action of a topological defect embedded in AdSn, this is what leads to the transverse energy integral (5.2) and the equations of motion (5.3). What we do is take the double BPS limit in the equations of motion, allowing the masses to fall towards zero and partially decoupling the ODEs, see (5.4) below. As remarked in the Introduction, the action is singular in this limit but the equations of motion are not. The double BPS equations of motion do not follow from any action. The solutions that we obtain from this method are a good point to start in a perturbative analysis of the full equations of motion that do follow from the action. In this double BPS limit, the equations of motion become 0 = 0 = d 2 d 2 (q tanh + (l 1) coth ) (q tanh + (l 3) coth ) d d dh d + + (l 1) [sinh ]2 h( )2 ( ) ; (l 2) [sinh ]2 h( )2 Note that the h equation of motion (5.4b) has decoupled, yet it is nonlinear. Thus we have a standalone second order ODE for h, whose solution can be inserted into eq. (5.4a) to obtain a standalone linear second order ODE for . If you are interested in pure Yang-Mills theory then you can just study equation (5.4b) for h and ignore the equation of motion. In solving the above we impose two point boundary conditions: (0) = 0, (1) = 1; 2.0 1.5 where the normalization factor is given by A plot of the solution is given in gure 8. 5.2.4 In the case of q = 4, we found an exact solution in terms of con uent Heun functions, see the NIST Digital Library of Mathematical Functions [20, x 31.12]. This (q; l) = (4; 2) solution is not very illuminating because of the unfamiliarity of the con uent Heun functions. For the case q 5 we were unable to nd exact solutions to the linear di erential equation (5.11), but we constructed numerical solutions for many q 2. There are well known subtleties in trying to construct numerical solutions because the natural initial conditions (0) = 0 and 0(0) = 1 are numerically unstable due to the regular singular point at = 0. To get around this, we compute the terms of the regular power series solution ps to (5.11) to O( 7) with initial conditions (0) = 0 and 0(0) = 1: + 1 24 6q 3 + 60q2 960 10q + 3 5 + 4480q3 + 420q2 1036q 322560 You can verify that the above agrees with the power series expansion of the exact solution (5.12a) for the case q = 2, and (5.14a) for the case q = 3. The idea is to replace the numerically unstable initial conditions at = 0 with nearby initial conditions at where 0 < sion: ( ) = n 1. The initial conditions are set by the truncated power series expanps( ) and 0( ) = 0ps( ). Typically we chose from 10 8 to 10 6. The numerical solution num thus obtained will not have the correct normalization at = 1 but since we have a linear ODE we know that the correct normalized solution will be norm( ) = num( )= num(1). The di erence between the exact solution in q = 2 and the numerical solution is plotted in gure 9. The normalized numerical solutions for q = 2; 3; 4; 8; 24 are shown in gure 10. Difference between exact and numerical solutions for q=2 normalized numerical solution for the scalar eld with initial data speci ed at = 1 10 6 and determined by the truncated power series solution. HJEP03(218) and at and h elds in the vortex case l = 2 for various values of q. The elds are the solid curves, and the h elds are the dotted curves. Remember that the h solution is exact, is obtained numerically. The numerical solutions are normalized to the correct behavior Double BPS hedgehog-like defects (l = 3) The discussion about the niteness of the energy in section 4.1.4 tells us that we only have to consider the case of a soliton (p = 0 or equivalently q = 1) embedded in AdS4. The equations of motion become 0 = 0 = d 2 d 2 d2h d 2 (tanh + tanh dh d + 1 d d 2 2 0.8 0.6 You can check that the solution to (5.18) is and we are left with an uncoupled linear ODE for : 0 = (tanh This ODE admits even and odd solutions, and it has a regular singular point at function, and we ignore the With our initial conditions (0) = 0 and 0(0) = 1, we nd the power series solution The exact normalized solution is given by the odd function norm( ) = N(1;3) sinh 1 sech4 2F1 2 + 2 1 p2; 2 2 1 p2; 5 ; tanh2 2 where N(1;3) = 3 sin p = 2 p It is easy to verify that the power series expansion (5.21), derived directly from ODE (5.20), agrees with the power series expansion of (5.22) if you remove the normalization factor N(1;3) in order that both functions have derivative 1 at the spherically symmetric ansatz is discussed in appendix C. = 0. The magnetic charge for Surprisingly, the exact scalar eld pro les for (q; l) = (2; 2) and (q; l) = (1; 3) are very similar, see gure 12. (5.19) (5.20) (5.22) (5.23) ν 5 ρ Difference between the exact solutions for 2+2 and 1+3 1 2 3 4 n = 4 with (q; l) = (2; 2) and (q; l) = (1; 3). In both cases, the eld satis es the same two point boundary conditions, also eq. (4.6) tells us that both solutions decay exponentially at the same rate e 3 = . It is not surprising that both solutions will be similar but the 1% discrepancy is a bit surprising. Static spherically symmetric classical glueballs Note added after submission to arXiv: on the day our manuscript appeared on the arXiv, there was also a manuscript by Ivanova, Lechtenfeld and Popov [23] where they study nite action and nite energy solutions of Yang-Mills theory in AdS4. This work is based on an earlier paper [24]. We did not realize at the time of submission that our double BPS limit monopole solution automatically gives a static spherically symmetric solution of the SO( 3 ) Yang-Mills equations in AdS4 with nite energy. It should have been obvious. The reason is that in the double BPS limit, eq. (5.18) does not depend on the scalar eld . This equation is the Yang-Mills equation. All we have to do is ignore the scalar eld and its equation of motion (5.17). Equivalently, you can look at the transverse energy (5.2) and ignore the scalar eld . We know the solution is h( ) = sech , see (5.19). The mass of this static spherically symmetric Yang-Mills solution is E ? = 3 2 2 g2 : (5.24) This solution is a classical glueball in AdS4. We do not know if this solution is stable. Also, this solution goes away in the at space limit ! 1 in accordance with known theorems. We have not explored whether there is a generalization to AdS4 of the Minkowski space theorems of Coleman [25], and Coleman and Smarr [26] that constrain the existence and the properties of glueballs. It is not immediately apparent how to relate the static Yang-Mills solution in [23, section 6] to the static solution presented here. Another observation is that the discussion of the double BPS limit in the SO(2) case in section 5.2 provides exact maximally symmetric purely electromagnetic p-defects associated with an embedding AdSq ,! AdSn given by eq. (5.10). The transverse energy density of this solution is given by of the defect. 6 where q 2: (5.25) It is interesting that the transverse energy density is proportional to the dimensionality p Here we brie y discuss solutions to the spherically symmetric ansatz in the parameter plane de ned by mA # 0 with 1= 2 6= 0, see the pinkish plane in gure 1. The equations of motion in this limit become 1 2 (q tanh + (l [sinh ] + + Equation (6.1b) already appeared as eq. (5.4b) in section 5 where we analyzed its solutions in detail. There we learned that the solution is given by h( ) = 1=(cosh )q 1 if l = 2 and q 2, see (5.10). For l = 3, there is only a q = 1 solution h( ) = 1= cosh , see eq. (5.19). The strategy is to insert the known h solution into ODE (6.1a), and look for a solution that satis es the two point boundary conditions (0) = 0 and (1) = 1. The analysis is more complicated than in the double BPS limit because the eld now satis es a non-linear ODE; the solution can be determined numerically. We use a modi cation of the power series technique discussed in section 5.2.4 to get around the numerically unstable initial conditions at = 0. 6.1 Vortex-like defects (l = 2) 0(0) arbitrary is given to O( 7) by The power series solution to the equation of motion with initial condition (0) = 0 and 1.0 in the mA # 0 limit with the Higgs eld Compton wavelength 1=m solutions are compared to the exact double BPS solution. = 2 , and 1=m = =2. The where = m . Because the ODE is nonlinear, we see nonlinear behavior on the initial condition 0(0) beginning at O( 5). We replace the initial conditions at = 0 with those given by the power series at a nearby point = 1 method" where we vary 0(0) numerically until we nd a solution with approximately the correct asymptotic behavior (1) = 1, see gure 13 and gure 14. A comparison between the numerical solutions and the exact double BPS solution is given in gure 15. 6.2 Monopole defects (q; l) = (1; 3) In this case, the power series solution to the equation of motion with initial condition (0) = 0 and 0(0) arbitrary is given to O( 7) by + 1 604800 7 + 5 700 0(0) + 3 4 0(0) + 60 2 0(0)3 + 72 2 0(0) 58232 0(0) 5 6 0(0) 940 4 0(0)3 230 4 0(0) 9480 2 0(0)3 5636 2 0(0) where, once again, = m . We see a similar nonlinear behavior to that of the previous case. We use the initial conditions provided by our power series at a su ciently nearby point. Here we use = 1 10 8. See gure 16 for a comparison between our numerical solution and the double BPS solution when 1=m = =2, and gure 17 for the relative di erence between these two solutions. 1.4 1.2 ϕ'(0)=2.44176761 ϕ'(0)=2.44176762 Double BPS in the mA # 0 limit with the Higgs eld Compton wavelength 1=m = =5. As expected, the decay behavior of the eld is dominated by the Compton wavelength. As a comparison, the curve in green is the exact double BPS solution where the Higgs eld has in nite Compton wavelength and the length scale is set by the radius of curvature . The shooting method is very sensitive to the initial condition 0(0) due to the potential presence of exponential growth terms; compare the dotted blue curve with the dashed orange curve. Numerical methods become more nicky as m increases. Normalized Relative Difference (mϕρ = 1/2) Normalized Relative Difference (mϕρ = 2) 0.20 0.15 0.10 0.05 10 ρ ν = 2 4 6 8 (a) Normalized relative di erence for m 1=2. The two solutions di er by up to 1%. 2 4 6 8 (b) Normalized relative di erence for m The two solutions di er by up to 20%. ν 10 ρ = 2. exact double BPS solution. Numerical experiments in the mA # 0 limit indicate that if m then the numerical solution is well approximated by the exact double BPS solution. If m . 1 > 1 then the numerical solution begins to di er from the exact double BPS one, e.g., see gure 14. 1.0 0.8 space in the mA # 0 limit with the Higgs eld Compton wavelength 1=m = =2. The solution is compared to the exact double BPS monopole solution. The initial derivative of the numerical solution is 0(0) = 1:424467087. Normalized Relative Difference (mϕρ = 2) 0.10 0.05 2 4 6 8 ν 10 ρ is the exact double BPS solution. With m = 2, the normalized relative di erence is less than 20%. For completeness, the at space equations of motion for spherically symmetric defects are given by taking the " 1 limit of eqs. (5.3): These equations are well known and have been thoroughly studied over the past 40 years. HJEP03(218) The Prasad-Sommer eld monopole solution (l = 3) is given by 1 2 m2 ( 2 1) = m2A ( )2 h( ) = ( ) = coth mA 1 mA 2 d 2 d 1) 2) h( )2 ( ) ; h( )2 (7.1a) (7.1b) (7.2) = 1 + O e 2mA : = 2mA e mA + O e 3mA The scalar eld has a 1= Coulomb tail because m = 0. 8 In this article we studied the equations of motion for maximally symmetric p-defects in AdSn. In the double BPS limit the radius of curvature is the only length scale that appears in the equations of motion, and we were able to nd exact analytic solutions in many cases. We also saw that the radial exponential increase in volume in AdSn plays a crucial role, and requires a case by case study for admissible values of (q; l). The method we advocate in this paper is part of a broader strategy to study solutions of the Yang-Mills Higgs system in AdSn. The solutions found in this double BPS limit are the rst step in a perturbative expansion in small parameters m and mA for the full equations of motion. A Maximally symmetric submanifolds of maximally symmetric spaces In this section we derive necessary conditions satis ed by a maximally symmetric submanifold of a maximally symmetric manifold. Any local orthonormal coframe on a constant curvature manifold M n with associated Levi-Civita connection ! = will satisfy the Cartan structural equations for a manifold of constant sectional curvature k: d! ^ ! + k ^ (A.1a) (A.1b) These constant dim Isom0(M n) = curvature manifolds are maximally symmetric spaces with 12 n(n + 1), where Isom0(M n) is the component of the isometry group of M n that is connected to the identity. In fact, eqs. (A.1) are the Maurer-Cartan equations for a Lie group. !ab ^ !ij ^ !ai ^ j + !ai ^ i a ; ; !ac ^ !cb + !ai ^ !bi + k a !ab ^ !bi !aj ^ !ji + k a !ik ^ !kj + !ai ^ !aj + k i ^ ^ ^ i j ; ; : 0 = !ai ^ !ab ^ a !ac ^ !cb + !ai ^ !bi + k a b !ab ^ !bi !aj ^ !ji ; !ik ^ !kj + !ai ^ !aj : d a = d i = d!ai = d!ij = d a = d!ab = d!ai = d!ij = (A.2a) (A.2b) (A.2c) (A.2d) (A.2e) (A.3a) (A.3b) (A.3c) (A.3d) (A.3e) (A.4) Next we use the Cartan structural equations and a bit of the theory of exterior differential systems [22], mostly the Frobenius theorem for integrability of a Pfa an system of equations. Assume q is an isometrically embedded q-submanifold of the constant curvature M n. If we use the index conventions that latin indices from the beginning of the alphabet a; b; c; d run from 1; 2; : : : ; q and latin indices from the middle of the alphabet i; j; k; : : : take l = n q values from q + 1; : : : ; n then the structural equations in an orthonormal coframe adapted to the tangent bundle of the submanifold may be written as The submanifold q in this adapted coframe is given by the exterior di erential system i = 0 and consequently d i = 0. Using these conditions in (A.2) we nd Applying Cartan's Lemma to (A.3b), we conclude that on q we have !ai = Kabi b ; where Kabi = Kbai are tensors on damental form. When restricted to q called the extrinsic curvatures or the second fun q, the Cartan structural equation for the intrinsic curvature is d!ab + !ac ^ !cb = 12 Rabcd c ^ embedding in a space of constant curvature: . Thus we obtain the Gauss equation for Rabcd = KaciKbdi KadiKbci + k gac gbd gad gbc ; (A.5) where gab is the induced metric on q due to the isometric embedding. A.1 Totally geodesic submanifolds A submanifold q of a general manifold M n is said to be totally geodesic if every geodesic (with respect to the induced metric) on q is also a geodesic on M n. If D and DM are respectively the Levi-Civita connections on the respective manifolds then the de nition of the second fundamental form (extrinsic curvatures) tells us that for the tangent vector eld X to a curve on we have DXM X = DX X Ki(X; X) n^i, where fn^ig is a local orthonormal frame for the normal bundle (T )?. The de nition of totally geodesic implies that Ki(X; X) = 0 for all X( ) 2 T and so we conclude that Kabi = 0. It is easy to show that totally geodesics submanifolds exist by constructing an example directly. Fix a point x 2 M and a q-dimensional vector subspace Vx TxM . Consider all the geodesics in M n that begin at x with initial velocity in Vx. The locus of all these geodesics12 is a q-dimensional totally geodesic submanifold q M n. From the viewpoint of standard General Relativity, totally geodesic submanifolds are very desirable because if a test mass in is given an initial velocity tangential to then its motion will be restricted to . Next we ask what are the totally geodesic submanifolds of a constant curvature manifold M n. The totally geodesic condition !ai = 0 means that equations (A.3) restricted to q become d a = !ab ^ !ac ^ !cb + k a b (A.6a) (A.6b) (A.6c) HJEP03(218) Thus we have derived necessary conditions for submanifold q to be a totally geodesic submanifold of the constant curvature manifold M n. The rst two equations in (A.6) are the Cartan structural equations for a q-dimensional constant curvature manifold with the same constant sectional curvature k as M n. These totally geodesic submanifolds are maximally symmetric spaces with dim Isom0( q) = 12 q(q + 1). Equation (A.6c) is the statement that the connection on the normal bundle (T )? is at. q A.2 Examples of totally geodesic submanifolds First we consider the example of the sphere of radius a, M n = Sn. The main observation a is that the equatorial sphere San 1 also has radius a. A geodesic in San 1 is a great circle in Sn 1, and this great circle is also a great circle in Sn. Thus the equatorial Sn 1 is a a a a totally geodesic submanifold of Sn. Repeating the argument we see that the equatorial a sphere Sn 2 of San 1 is a totally geodesic submanifold of San 1 and consequently a totally a geodesic submanifold of San. We can repeat this argument until get down to the equatorial S1. Thus we have shown that there exists an \equatorial" Saq that is a totally geodesic a submanifold of San. To extend the result above to arbitrary k and Euclidean or Minkowski signature for the metric, we repeat the discussion in the previous paragraph in terms of equations. The n-sphere of radius a is the set of points in En+1 that satis es the equation (x1)2 + (x2)2 + +(xn)2 +(xn+1)2 = a2. The isometry group of Sn is SO(n+1), the stability group of the North Pole (0; 0; : : : ; a) is SO(n), and Sn SO(n + 1)= SO(n). A polar (n 1)-sphere13 has radius a and contains the point (0; : : : ; 0; a), the North Pole. It is obtained by adjoining to the spherical constraint equation an additional equation xn = 0. To see that this \polar" to Sn and is the unit normal vector eld to the polar Sn 1. This vector is parallel in En+1 and thus DXSn (@=@xn) = 0 where the vector eld X is tangent to Sn 1, and DSn is the connection induced on Sn from the Euclidean structure of En+1. Therefore, the extrinsic 12The geodesics should be short in an appropriate sense. 13We replace equatorial great spheres by polar great spheres to simplify the equations. curvature vanishes. You can inductively repeat this argument. A polar (n 2) sphere containing (0; : : : ; 0; a) is obtained by adjoining an additional constraint xn 1 = 0. You can either do an inductive argument or note that the extrinsic curvature vanishes again way down to a polar S1. This sequence of \polar spheres" is a collection of totally geodesic submanifolds of Sn. You can move these totally geodesic spheres to other locations by using a transformation in the isometry group Isom0(Sn) = SO(n + 1). The argument for the vanishing of the extrinsic curvatures is identical in the three cases below and we skip it. The geodesic submanifolds that we construct can be moved by using the appropriate isometry group. For hyperbolic space Hn (Euclidean signature and k < 0), we consider the connected set of points that satis es (x1)2 + (x2)2 + + (xn)2 (xn+1)2 = a2 in Rn+1 with the signature of the Rn+1 metric being (+; +; : : : ; +; ) and contains the \North Pole" (0; : : : ; 0; a). The isometry group of Hn is SO(n; 1), the stability group of the North Pole is SO(n), and thus Hn SO(n; 1)= SO(n). The \polar" Hn 1 is obtained by adjoining the constraint xn = 0. The \polar" Hn 2 is obtained by adjoining the additional constraint xn 1 = 0. This argument can be repeated until you get down to a \polar" H1 Hq constructed this way are all totally geodesic submanifolds of Hn. R. The If we are looking for manifolds of Lorentzian signature then we get de Sitter space dSn (k > 0) or anti de Sitter space AdSn (k < 0). For de Sitter space dSn, we consider the connected set of points that satis es (x1)2 + (x2)2 + + (xn)2 + (xn+1)2 = a2 in Rn+1 with signature of the Rn+1 metric being ( ; +; : : : ; +; +) and contains the point (0; : : : ; 0; a). The isometry group of dSn is SO(n; 1), the stability group of the North Pole is SO(n 1; 1), and dSn SO(n; 1)= SO(n 1; 1). The \polar" dSn 1 is obtained by adjoining the constraint xn = 0. The \polar" dSn 2 is obtained by adjoining the additional constraint xn 1 = 0. This argument can be repeated until you get down to a \polar" dS1. The dSq constructed this way are all totally geodesic submanifolds of dSn. For anti de Sitter space AdSn, we consider the connected set of points that satis es (x1)2 + (x2)2 + + (xn)2 (xn+1)2 = a2 in Rn+1 with signature of the Rn+1 metric being ( ; +; : : : ; +; ) and contains the point (0; : : : ; 0; a).14 The isometry group of AdSn is SO(n 1; 2)= SO(n 1; 2), the stability group of the North Pole is SO(n 1; 1), and AdSn SO(n 1; 1). The \polar" AdSn 1 is obtained by adjoining the constraint xn = 0. The \polar" AdSn 2 is obtained by adjoining the additional constraint xn 1 = 0. This argument can be repeated until you get down to a \polar" AdS1. AdSq constructed this way are all totally geodesic submanifolds of AdSn. We remind the reader that we always have in mind the simply connected universal cover of the quadric hypersurface in question. A.3 Intrinsically at submanifolds Next we show that there are intrinsically at submanifolds that can be isometrically embedded in negative constant curvature spaces. These manifolds are not totally geodesic 14In this model for anti de Sitter space, AdS1 is a closed timelike curve. In fact for any n 1, this model has a closed timelike curve, and for this reason we always implicitly assume we are working in the simply connected universal covering space. The universal cover of AdS1 is a timelike line. submanifolds. We impose the intrinsic atness condition Rabcd = 0. We also require a special form for the extrinsic curvature tensor Kabi = hab Ci, where hab is the at metric on . Note that Ci is the mean curvature vector. The reason for this form is that we require the extrinsic curvature to be compatible with the isometries of the vector space Inserting these conditions into (A.3), we obtain d a = !ab ^ b 0 = kCk2 + k a DCi d!ij = a = 0 ; q . (A.7a) (A.7b) (A.7c) (A.7d) (A.8) (A.9) To satisfy the structural equations above, we require from (A.7b) that k = Thus, we have our rst result that the space M n must have constant negative curvature15 kCk2 since we are restricting our analysis in this paper to k 6= 0. Next, we observe that since the f ag are linearly independent, eq. (A.7c) implies that 0 = DCi = dCi + !ij Cj . Thus the mean curvature vector C = Ci n^i is covariantly constant and we have a preferred direction at normal bundle (T )?, see condition (A.7d). If M n has Lorentzian signature then the maximal possible symmetry group of a defect is P(q) SO(l 1) as opposed to the AdSq case where you can have SO(2; q 1) SO(l). There is no spherically symmetric Mq. Said di erently, the \transverse part of the Lagrangian" is Remark 1. If AdSq is not a totally geodesic submanifold then there is a non-zero mean curvature vector that gives a preferred normal direction. The Gauss equation in this case becomes k q = kCk2 + kMn . The symmetry group of the p-defect is maximally Isom0( q 1). There are no maximally symmetric solutions in this case. You can verify that the world branes in references [7, 9, 10] are totally geodesic subIt is easy to discuss a basic example that is commonly used in physics. Consider AdSn q Mn 1, i.e., q = n 1. The metric on AdSn is written in manifolds. with embedded submanifold the \upper half space" form If we restrict to the q = n 1 dimensional submanifold then the induced metric on it, 1 y 0 ds2 = 2 gaMb d a d b ds2AdSn = gaMb d a d b + (dy)2 y2 with y > 0 : q de ned by y = y0, y0 constant, is a rescaled Minkowski metric. The mean curvature vector is parallel to the unit normal to In . Any vector subspace of this n 1 is automatically a at submanifold of AdSn. gure 18 we compare a totally geodesic constant curvature submanifold and a zero curvature submanifold of AdSn in the \upper half space" representation. 15If M n has Lorentzian signature and q is a timelike submanifold then (T M )? has Euclidean signature and kCk2 H~ = h0H(x) 0 = 1=(g 0 ) 2 LMS LMS h0 r0 r 0x2 their K(x) x = mAr, [x] = M 0 x = (mA ) sinh( = ) r= = sinh( = ) !1 ! 2 1 e = !1 ! 21 (mA )e = K 6= 0 Mn−1 K = 0 AdSn−1 AdSn conformal boundary AdSn in the upper half space representation with metric (A.8). Choose a point 2 AdSn 1 and consider all the geodesics beginning at with spacelike initial velocity tangential to AdSn 1 . The locus of all these geodesics is a submanifold of the totally geodesic AdSn 1 represented by the blue curve. A physical deformation (motion slower than the speed of light) of AdSn 1 into Mn 1 would take an in nite amount of time because the asymptotic parts of both manifolds are in nitely far apart. The distances between the corresponding arrowheads in Mn 1 and in AdSn 1 are in nite. This article 2 g I 0 0 = 1=(mA ) 2 r2= 2 h( ) =3 < 0 and jkj = 1= 2 = 1=r02. Since we use brane, we will use for the LMS asymptotic decay index to avoid confusion. for the radial distance from the defect world B Relation to the work of Lugo, Moreno and Schaposnik This is an attempt at relating the work of Lugo, Moreno and Schaposnik (LMS) with ours. They choose a di erent coordinate system for AdS4 where the metric is 1 + r2= 2 dt2 + 1 + r2= 2 dr2 + r2 d 2 + sin2 d'2 : It is important to notice that r is not the radial distance. You can verify that \t-axis", is a totally geodesic submanifold. A notational dictionary is in table 2. A rst observation is that we have a di erent viewpoint on what is BPS. Their philosophy is described in the paragraph following eq. (4.32) in reference [9]. They assume that (B.1) 1, the 1/ρ2 n i l S P B l b u o D BPS line m2A ν• = 1 ν• = 23 ν• = 2 ν• = 3 ν• = ∞ gure 1, corresponds to the generalized BPS limit where m2 # 0. We have reinterpreted the results of LMS as the dashed straight lines m2A = + 1)= 2 in this plane, where LMS discovered the existence of a power series solution if is a positive half integer. LMS also studied other values of numerically in [10]. The BPS line corresponds to formally setting line corresponds to formally setting = 1, which is the same as sending " 1. The double BPS = 0, which is the same as sending mA # 0. The work of LMS seems to indicate that solutions along the dashed lines are special but we have not explored this. Note that those dashed lines interpolate between the two lines where we have exact analytic solutions. LMS found reasonable numerical solutions using a relaxation method with starting point the BPS solution; this is not surprising in light of this gure. On the other hand, this gure suggests that in some sense, the double BPS solutions are far away from the Prasad-Sommer eld solutions. setting LMS = 0 means that there is no potential and the asymptotic value of the Higgs eld can oat to some non-zero value H~ (1) as you solve the equations of motion. This is natural because in their numerical work they are using a relaxation method [21] taking the exact BPS solution as a starting point. They also postulated a power series expansion in 1=x about x = 1, see eqs. (B.3), and discovered that the expansion is well behaved if the asymptotic value of the Higgs eld takes a very speci c form H~ (1) 2 1 (g )2 1 2 2 2 + 1) where ; 1; ; 2; ; 3; : : : ; (B.2) see [10, eq. (29)]. Here speci es the decay behavior of the K eld (B.3b). For non half integer values of , the power series may have to be generalized to include logarithms. We interpret the BPS solution as a limiting solution in the limit # 0. This means that the potential (5.1) governs the asymptotic behavior and requires that k k ! 0 as ! 1. Since the asymptotic value 0 is arbitrary in our formalism, we can try to compare results by setting H~ (1) = 0, and we nd (g 0)2 = m2A = + 1)= 2. Next we corroborate this identi cation by analyzing the asymptotic behavior of the elds. H~ (x) = H~ (1) + K(x) = Kx ++11 1 + O 1=x2 h0H3 + h( ) ! h e e 3 = (B.3a) Notice that the asymptotic behavior of the Higgs eld agrees in both our computations e = . To make a connection with the second equation we note that (4.10) says = 12 + q 14 + (mA )2, and thus we conclude that + 1 = . This relates the decay exponent to the at space mass mA and the radius of curvature via the equation + 1) = (mA )2. These ideas are summarized in gure 19. since x that C C.1 At 2 Theorem we nd where S2 ,! (T lim !1 RB3 The lack of Bogomolny equations and a partial bound The magnetic ux In this appendix we make an unsuccessful attempt to determine the stability of our double BPS monopole solution. We analyze whether ours is a minimum energy static solution. We discuss various topics related to static magnetic monopoles in AdS4. We address whether # 0 Prasad-Sommer eld limit [15] there is an analog of the rst order Bogomolny equations [16] that gives solutions of the equations of motion. More precisely, we show that the arguments [16, 17] that lead to the Bogomolny bound in the M4 theory do not generalize to the AdS4 case. In this section we ignore the potential term because we are always implicitly taking the BPS limit. The value of g is generic. First, we compute the magnetic ux. The metric on M = AdS4 may be written in the form cosh2 jkj1=2k k d 2 + d 2 + 2 ds2S2 ; (C.1) where the coordinate is the proper time along the 1-dimensional defect world line 1 = AdS1. The restriction of the AdS4 metric to the normal tangent space (T )? is given by ds2AdS4 (T M)? = d 2 + jkj1=2k k !2 2 ds2S2 : 1, the normal tangent space (T )? is a three three dimensional vector space. as a consequence of the Bianchi identity DBI = 0. If we x De ne the magnetic 2-form by BI = 12 IJK F JK and observe that d Z (T )? D I Z (T )? d I BI = lim I BI ; )? is the 2-sphere of radius . Here we used the de nition R(T )? , where B3 ,! (T )? is the 3-ball of radius and S2 = @B3. As shown by 2 Z !1 S2 (C.2) (C.3) = (C.4) (C.5) (C.6) 't Hooft [3] and Polyakov [4], the SO(2) unbroken gauge symmetry magnetic eld 2-form is I BI = 0 and thus we expect that, for a monopole, the abelian magnetic ux Fm, which is a measure of the magnetic charge, is given by: Thus, we conclude that Z lim !1 S2 Z 1 (T )? 0 I BI = 0 Fm : D I ^ B I = Fm : The integrand in the left hand side of the equation may be interpreted as the magnetic charge density. Note that the conventionally normalized magnetic eld Bconv is related to our magnetic eld by B = g Bconv, and one has to be careful in analyzing the g # 0 limit. The topological ux quantization condition should hold for all values of g. Next, we want to see how the above computation works within our ansatz for the spherically symmetric hedgehog because this illuminates why there are no Bogomolny equations in this case. Using eq. (3.10) we obtain Z S2 I BI = Z S2 ( )f ( ) 1 sinh( = ) 2 S2 ; where S2 is the area 2-form on S2 , ! (T )?. Note that S2 has area 4 2 sinh2( = ). Due to its spherical symmetry, the integral is easily computed and we obtain the result found by 't Hooft and Polyakov: Z S2 2 ( )f ( ) 1 = 4 ( ) 1 h( )2 !1! 4 0 ; (C.7) where we used the asymptotic boundary conditions. With our normalization, the magnetic ux is Fm = 4 , which is the correct topological quantization condition for the case where the SO( 3 ) gauge group breaks down to SO(2) via a vector representation scalar eld.16 Note the cancellation of the hyperbolic sine factors, which was necessary to obtain a nite non-zero magnetic ux. Next, we discuss a couple of attempts at trying to obtain the Bogomolny bound and ?? ! = s( 1)k(n k)!. the associated Bogomolny equations. We implicitly assume that we are always taking the Prasad-Sommer eld limit # 0, and maintaining the appropriate boundary conditions. It is computationally useful to use the Hodge inner product. We give a brief review to establish conventions. Let V be a vector space with inner product, either Euclidean or Lorentzian signature, dim V = n. The normalized volume element determined by an orthonormal basis is . If ! is a k-form, it is convenient to de ne a norm by k!k?2 = ! 1 k ! 1 k =k!. With this normalization each independent term in the k-form only contributes once in the summation. A pointwise bilinear product, h ; i, and the Hodge dual operator ? on k-forms are de ned by h!; !i = ! ^ ?! = k!k?2 . If s = +1 in Euclidean 1 in Minkowski signature then please note that k?!k?2 = s k!k?2, and 16Often in the literature, the model considered is a SU(2) gauge theory with an adjoint representation scalar eld that breaks the symmetry group down to U(1). In this case, the magnetic ux quantization condition is Fm = 2 . ux to the energy functional 2 In this section, we x and we restrict to the three dimensional normal tangent space )?. In particular, the metric on (T )? is given by (C.2), and the Hodge duality operator refers to Hodge duality with respect to this metric. The strategy in this section is to begin with the correct expression for the magnetic ux and to try to get to the energy functional whose variation gives the equations of motion. In this appendix, we have to use Hodge duality on the normal tangent space (T and on the full manifold AdS4. For this reason we introduce a special notation: on the normal tangent space (T )? with metric (C.2), we denote the Hodge duality operation by ~, and the associated norm by k k~. 2 We observe that 2 0 Fm=g = D I ^ (~~ BI )=g Z 2 2 Z Z Z (T )? (T )? (T )? (T )? D I ^ BI =g = D I ; ~BI =g 2 Z (T )? (T )? D I ~BI =g; D I ~BI =g (T )? kD k2~ + kBk2~ =g2 (T )? : Here (T )? = ~1 is the volume element on (T )?. Rearranging terms we nd 2 (T )? kD k2~ + 1 g2 kBk2~ (T )? = + 0 Fm=g : D 1 g 2 (T )? (C.8) This is the expression [16, 17] that is used to show in M4 that a solution to the Bogomolny equations D I = ~BI =g is an absolute minimum of the left hand side of (C.8). Unfortunately, this is not what we need because the functional that has to be minimized to obtain the equations of motion is not the left hand side of (C.8) but the transverse energy functional (3.4), which in Hodge star notation is E ? = cosh( = ) kD k2~ + 1 g2 kBk2~ (T )? : (C.9) The hyperbolic cosine factor in the previous equation is necessary to obtain the correct equations of motion. The static equations of motion are obtained by restricting the 4dimensional action, constructed with the AdS4 metric, to time translationally invariant eld con gurations. Note that @=@ is a Killing vector since the metric components in (C.1) are independent of . Equation (C.8) is what you would get for an instanton solution in Euclidean hyperbolic 3-space H3. Solutions to the BPS equations would be absolute minima of the equations of motion in the Prasad-Sommer eld limit, see also the discussion in Atiyah [14]. The Bogomolny argument works in a situation where the spacetime is a product manifold M 4 = R V 3 with the product metric ds2M = d 2 + ds2V . It also works in the Euclidean signature instanton case where the manifold M 3 is Euclidean hyperbolic 3-space, ibid. and footnote 1. C.3 From the energy functional to the magnetic ux In this section, we begin with the correct functional that has to be minimized to obtain the static equations of motion and we try to see if we can nd a bound for this functional that is related to the magnetic ux . From now on, the metric is the AdS4 Lorentzian metric (C.1) with associated inner product h ; i. The Hodge duality operation is with respect to this metric, the AdS4 volume element is denoted by = ?1, and the normalized timelike 1-form is ^0 = cosh( = ) d . The action for our static con guration is I = kD k?2 + kBk?2 =g2 E d : Next we rewrite the action as Z AdS4 hD ; D i + hB; Bi =g2 2I = Z : To try to get Bogomolny type equations, we observe that ^0 and that ?B is also a timelike 2-form. Note that ^0 is orthogonal to D ^ D is a timelike 2-form, and thus, we can rewrite the above equation as 2I = Z AdS4 Z Z Z Z AdS4 AdS4 AdS4 AdS4 D^0 ^ D ^0 ^ D ^0 ^ D ^0 ^ D D^0 ^ D ; ^0 ^ D E ?B=g; ^0 ^ D ?B=g ?B=g ?B=g 2 2 2 + 2 2 Z Z AdS4 AdS4 Z d ^0 ^ D ^0 ^ D h?B; ?Bi =g2 Z D^0 ^ D ; ?B=gE AdS4 ^ (?? B=g) ^ B=g Z (T )? cosh( = ) D ^ B Using the time translational invariance we can write the above as E ? = + cosh( = ) D ^ B : (C.10) Next we simplify the integral over AdS4. Let (^0; ^1; ^2; ^3) be the adapted orthonormal coframe for AdS4 we constructed early in the manuscript. On the normal tangent space )? with metric (C.2), we denoted the Hodge duality operation by ~, and the associated norm by k k~. It is easy to see that ? ^0 ^ 2 ^i = ~ ^i by using 0123 = 1. We observe (C.11) (C.12) that rewrite (C.10) as = cosh( = ) d ^ (T )?, and we can E?( ) = (T )? (T )? ^ B : If we set = 1, i.e., k = 0, then (C.11) leads to the standard Bogomolny type argument for a lower bound on the mass of 4 =g. The bound is saturated by eld con gurations that satisfy the Bogomolny equations D = ~B=g. If 0 < < 1 and k k2~ is a positive de nite norm, then this equation implies a bound ^ B : Z E This bound does not appear very useful because the bound is eld con guration dependent. You can verify the eld dependency by computing the variation of the integral with respect to a variation of the scalar eld. If ( ) is a variation of the scalar eld with compact support, then you obtain Z (T )? cosh( = ) D I Z (T )? d ^ BI ( I ) : Note that the right hand side of the equation above vanishes for the topological nature of the magnetic ux (C.5), which is invariant under deformations of = 1, and it exempli es the eld con guration. C.4 A bound on the mass for non-negative magnetic charge density We previously mentioned that the left hand side of magnetic ux equation (C.5) may be interpreted as the magnetic charge density. Here we prove a theorem that states that if the magnetic charge density is non-negative in the Prasad-Sommer eld limit and if 0 < then the mass satis es the strict inequality E ( ) > 4 0=g. ? A 3-form on (T )? is said to be non-negative if there exists a function f : (T con gurations, and let C+ = f (T )?. Let C = f( ; A)g be the space of admissible eld C be the subset of eld con gurations where the 3-form D I BI = d I BI is non-negative. These are the con gurations with non-negative magnetic charge density. Since cosh( = ) > 1 for > 0 and 0 < < 1, the use of (C.12), (C.5) and (C.7) leads to the conclusion that if we restrict to eld con gurations in C+, then there is a lower bound provided by magnetic ux quantization < 1, )? ! E?( ) C+ ^ BI = g 0 : Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References (2002) 389 [hep-th/0105134] [INSPIRE]. (2002) 329 [hep-th/0203003] [INSPIRE]. 467 (1999) 43 [hep-th/9909226] [INSPIRE]. B 473 (2000) 35 [hep-th/9911209] [INSPIRE]. [5] N. Manton and P. Sutcli e, Topological solitons, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge U.K. (2004). [6] B. Julia and A. Zee, Poles with both magnetic and electric charges in nonabelian gauge theory, Phys. Rev. D 11 (1975) 2227 [INSPIRE]. [7] M.H. Dehghani, A.M. Ghezelbash and R.B. Mann, Vortex holography, Nucl. Phys. B 625 [8] A.M. Ghezelbash and R.B. Mann, Vortices in de Sitter space-times, Phys. Lett. B 537 [9] A.R. Lugo and F.A. Schaposnik, Monopole and dyon solutions in AdS space, Phys. Lett. B [10] A.R. Lugo, E.F. Moreno and F.A. Schaposnik, Monopole solutions in AdS space, Phys. Lett. [11] E. Radu and D.H. Tchrakian, New axially symmetric Yang-Mills-Higgs solutions with negative cosmological constant, Phys. Rev. D 71 (2005) 064002 [hep-th/0411084] [INSPIRE]. [12] S. Bolognesi and D. Tong, Monopoles and holography, JHEP 01 (2011) 153 [arXiv:1010.4178] [INSPIRE]. [13] P. Sutcli e, Monopoles in AdS, JHEP 08 (2011) 032 [arXiv:1104.1888] [INSPIRE]. [14] M. Atiyah, Magnetic monopoles in hyperbolic spaces, in Michael Atiyah collected works. Vol. 5: gauge theories, M. Atiyah ed., Oxford University Press, Oxford U.K. (1988). [15] M.K. Prasad and C.M. Sommer eld, An exact classical solution for the 't Hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett. 35 (1975) 760 [INSPIRE]. [16] E.B. Bogomolny, Stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976) 449 [INSPIRE]. [17] S.R. Coleman, S.J. Parke, A. Neveu and C.M. Sommer eld, Can one dent a dyon?, Phys. Rev. D 15 (1977) 544 [INSPIRE]. (2017) 033 [arXiv:1701.04449] [INSPIRE]. Release 1.0.14 (2016). University Press, Cambridge, U.K. (1992). [19] T.H.R. Skyrme, A uni ed eld theory of mesons and baryons, Nucl. Phys. 31 (1962) 556. [20] NIST Digital Library of Mathematical Functions | x31.12, http://dlmf.nist.gov/31.12, [21] W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical recipes in C, Cambridge [18] O. Alvarez and M. Haddad, Emergent gravity in spaces of constant curvature, JHEP 03 equations on de Sitter dS4 and Anti-de Sitter AdS4 spaces, JHEP 11 (2017) 017 Commun. Math. Phys. 56 (1977) 1 [INSPIRE]. d [cosh ( = )]q [ sinh ( = )]l 1 1 0( )2 + 1 0( )2 + 1) h( )2 ( )2 d [cosh ( = )]q [ sinh ( = )]l 1 3 dh = 1 solution may be taken to be an odd 2 solution that may be taken to be an even function . 3) dh 0 . [1] S. Coleman , Aspects of symmetry, Cambridge University Press, Cambridge U.K. ( 1988 ). [2] H.B. Nielsen and P. Olesen , Vortex line models for dual strings , Nucl. Phys. B 61 ( 1973 ) 45 [3] G. 't Hooft, Magnetic monopoles in uni ed gauge theories , Nucl. Phys. B 79 ( 1974 ) 276 [4] A.M. Polyakov , Particle spectrum in the quantum eld theory , JETP Lett . 20 ( 1974 ) 194 Publications volume 18 , Springer, Germany ( 1991 ). [23] T.A. Ivanova , O. Lechtenfeld and A.D. Popov , Finite-action solutions of Yang-Mills [24] T.A. Ivanova , O. Lechtenfeld and A.D. Popov , Solutions to Yang-Mills equations on four-dimensional de Sitter space , Phys. Rev. Lett . 119 ( 2017 ) 061601 [arXiv: 1704 .07456] [25] S.R. Coleman , There are no classical glueballs , Commun. Math. Phys. 55 ( 1977 ) 113


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP03%282018%29012.pdf

Orlando Alvarez, Matthew Haddad. Some exact solutions for maximally symmetric topological defects in Anti de Sitter space, Journal of High Energy Physics, 2018, 12, DOI: 10.1007/JHEP03(2018)012