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A geometrical viewpoint on the ‘Deformation Method’ V. I. Afonsoa and Diego Julio Cirilo-Lombardob,c aUnidade Acadêmica de F´ısica, UFCG, Campina Grande, Para´ıba, Brazil. b National Institute of Plasma Physics (INFIP), Consejo Nacional de Investigaciones Cient´ıficas y Tecnicas (CONICET), Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Cuidad Universitaria, Buenos Aires 1428, Argentina. c Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research, 141980 Dubna, Russia 5 At the classical level, redefinitions of the field content of a Lagrangian allow to rewrite an inter- 1 actingmodelonaflattarget spaceintheformofafreefieldmodel(nopotentialterm)onacurved 0 target space. In the present work, we explicitly show that the idea of the ‘deformation method’ 2 introduced in [1], can be interpreted in a very simple geometrical picture, in terms of a correspon- dencebetween themetrics of thecurvedtarget spaces that arise in thefree versions of themodels. p Tothataim,weshowexplicitrelations betweenthemapfunctionlinkingthefieldsandthemetrics e S of the free models. Also, the geometrical viewpoint puts in clear evidence the limitations of the method,restricted unavoidably tomodels of a single scalar field. However, by considering complex 2 andevenquaternionicfieldmodels, theapplicability ofthedeformation procedurecanbeextended to systems with a content of two and four (constrained) real fields, respectively. To illustrate more ] possibilities, we also consider some supersymmetric models. In particular, a geometrical relation h betweentheflatMinkowskianmetricandaFubini-Studyspacemetricisexplicitlyconstructed. This t - widening of the range of applicability of the method derives from the pure geometrical viewpoint p hereexplored. e h [ PACSnumbers: Keywords: TopologicalDefects,DeformationMethod,Quaternions,Fubini-Studyspace. 2 v 6 1. Introduction. the free versions of the models. The general idea of the 9 The ‘deformation method’ –introduced in [1] and ex- mappings that can be constructed is synthesized in the 9 ploredinseveralworks[2]–consistinamappingbetween diagram (1). There it is shown, on the left side, ‘the 3 two classical scalar field models supporting BPS states, original’ ( ) and the ‘deformed’ (˜) interacting models 1. which links the corresponding solution spaces by a map relatedbyLthedeformationfunctionLf. Therightsidecol- 0 (configuration space diffeomorphism) f dubbed ‘defor- umn shows the corresponding relation between the free 14 mation function’. This is possible whenever there exist versions of the models (Lg and L˜g respectively), which : a specific relation between the potentials governing the are alsoconnected by f, but now seenas a mapbetween v dynamics ofthe fields. That relationcan be obtained by the target spaces metrics g and g˜. i X constructionand,inthatsense,themethodresultsmuch [V(φ)] [(g(θ)] more efficient in finding new defect-like solutions than, g r L ←→ L a for instance, the Rajaraman’s ‘trial orbit method’ of [7]. f f l l As it is well known, the holonomy group of a semi- ˜ [V˜(φ˜)] ˜ [g˜(θ˜)] g˜ Riemannianmanifold(M,g)atapointp( M)isdefined L ←→ L ∈ asthegroupofparalleltransportsalongloopsbasedonp. In principle, the relationhere proposed (as well as the It provides a powerful tool to study the geometric struc- originaldeformationmethod of [1]) applies only to mod- ture of the manifold and to discuss, for instance, the ex- els of a single scalar field. However,considering complex istence of parallel sections of geometric vector bundles. –as in [5, 9]– and quaternionic [8] field models, the pro- In his seminal paper [3], Berger gave a list of possible cedureapplicabilitycanbe enlargedtocomprisesystems holonomygroupsofsimplyconnected(semi-)Riemannian with acontentoftwo andfour (despite constrained)real manifoldsundertheassumptionthatthegroupactsirre- field components, respectively. ducibly on the tangent space at p. Precisely, such prop- The results of the present work provide a deeper un- erties of these manifolds allow, at the classical level, to derstandingofthedeformationmethodandofferamech- mapmodels comprisinginteractingfields onaflattarget anism to map geometries of interest in a wide range of space, to free fields models on a curved target space. contexts like Sugra, D-branes, Braneworlds, etc. Themainobjectiveofthepresentworkistoshowthat As an illustration of the possibilities of application of deformationmethod canbe reinterpretedas asimple ge- themethod,afterreviewingthedeformationprocedurein ometrical map between the metrics of target spaces of a generalcomplex case in Section II we apply, in Section 2 III,thenewgeometricalinterpretationtoa‘Quaternionic The BPS kink solutions for this system are obtained Wess-Zumino model’, developed in a previous work [8]. from the solutions of the undeformed first order equa- Then, in Section IV, we analyze the possibilities of by- tions, by simply taking the inverse map [5], that is, passing the severe limitations of the method in its stan- ϕ˜K(x)=f 1(ϕK(x)). (4) dard form by exploring the purely geometric viewpoint − (nodefect-typesolutionsinvolved). Inparticular,wecon- 2.2. Free model. The free action corresponding to sider supersymmetric models and work out the very im- themodel(1)requiresthe introductionofaKählerman- portant case of the Fubini-Study space. The summary ifold with metric g zz¯ and discussion of our results is left to Section VI. 2. Deformation procedure. = 1 dxng θ˙zθ¯˙z¯, (5) g zz¯ The dynamics of an interacting complex scalar field ϕ I 2Z onabi-dimensionalMinkowskispacetimeisdescribedby where θ˙z =dθz/dx. Variation of this action leads to the a Lagrangianof the form field equations = 1∂ ϕ¯∂µϕ V(ϕ,ϕ¯), (1) L 2 µ − θ¨w+gwz¯ g θ˙w+g θ¯˙w¯ g θ¯˙w¯ θ˙z =0 (6) zz¯,w zz¯,w¯ zw¯,z¯ where the bar stands for complex conjugation and h − i V(ϕ,ϕ¯)isascalarpotential. Aswearemainlyconcerned Beingg themetricofaKha¨lermanifold,thelasttwo zw withthestudyandgenerationofdefect-likesolutions,we terms in (6) cancel, as in complex coordinates g =g zz¯ z¯z will consider here potentials showing spontaneous sym- and g =g =0. Therefore, we obtain zz z¯z¯ metry breaking. g θ¨z +g θ˙wθ˙z =0 (7) In order to obtain physical (finite energy) solutions, zz¯ zz¯,w the asymptotic conditions: ϕ v and ϕ˙ 0 with | | → | | → Wecanalsodroptheindicesandputtheexpressioninthe V′(v) = 0, must be achieved as x → ±∞ (the dot simpler form θ¨+∂θlngθ˙2 = 0, which can be integrated stands for derivation with respect to the spatial coordi- to obtain the relation nate, ϕ˙ =dϕ/dx, etc.). Thatis,solutionsconnectpoints of the vacua manifold (the set of minima of V). Also, in θ˙θ¯˙ = g 1(θ,θ¯), (8) 0 − C the cases of interest (bosonic sector of supersymmetric with C an integration constant. theories), the potential V can be put in terms of a ‘su- 0 C ∈ perpotential’ W (with V(ϕ) = 1W (ϕ)W (ϕ)). Under If the interacting and the free models are to describe | | 2 ′ ′ the same physics, their field equations should match af- such conditions, the static [15] second order variational ter making explicit the relationbetween the fields φ and field equation, ϕ¨¯ 2∂ V = 0, can be integrated once to − ϕ θ. Now, putting (2) in the form ϕ˙ϕ˜˙ = V(ϕ,ϕ) V , obtain the first order equations − 0 and comparing with (8), we can establish the relations ϕ˙ = eiαW (ϕ), ϕ¯˙ = e iαW (ϕ). (2) between the curved (free) and the flat (interacting) ver- ′ − ′ sions Here, eiα corresponds to the U(1) degeneracy on the choosingof the superpotential, andV =V(v), the ‘clas- θ˙2+θ˙2 = ϕ˙2+ϕ˙2 (9) 0 1 2 1 2 sical vacuum value’ (value of the degenerate minima of g 1(θ,θ¯) = V(ϕ,ϕ¯) V , (10) 0 − 0 the scalar potential) was set to zero for simplicity. C − wherewehavemadeexplicittherealandimaginarycom- 2.1. ‘Standard’ deformation. The deformation ponents of the fields. The simplest solution arises by procedure for complex field models was explored in Ref. taking θ =ϕ + and θ =ϕ + or, simply, 1 1 1 2 2 2 [5]. The deformed Lagrangian density has the general C C form ˜=(1/2)∂µϕ˜∂ ϕ˜ V˜(ϕ˜,ϕ˜), and describes the dy- θ =ϕ+ (11) µ L − C namics of the new complex field ϕ˜ = ϕ˜ +iϕ˜ , related 1 2 with C, an arbitrary constant. to the original field ϕ by a (at least) meromorphic func- C ∈ tion f, such that ϕ = f(ϕ˜) = f (ϕ˜ ,ϕ˜ )+if (ϕ˜ ,ϕ˜ ), 2.3. ‘Geometric map’ deformation. Now, start- 1 1 2 2 1 2 which, naturally, must fulfill Cauchy-Riemann condi- ing with the free versionof the complex deformed model tioTnsh,e∂dϕ˜y1nfa1m=ic∂sϕ˜g2fo2vearnnded∂bϕ˜y2f1 =an−d∂ϕ˜˜1afr2e. different, but l(aL˜tgion=s an21ag˜lzoz¯gθõ˙zuθ¯˜˙sz¯)t,of(r9o)manpdre(s1c0ri)pftoiornth(e3)d,eafnordmuesdinfigelrdes- wecanrelateV˜(ϕ˜,ϕ¯˜)andW˜(ϕ˜L)tothLeoriginalmodelby θãnd ϕ˜, we can write –see [5]– V˜(ϕ˜,ϕ¯˜)= V(ff(ϕ˜()ϕ˜,)f2(ϕ˜)) = 12W˜ ′(ϕ˜)W˜ ′(ϕ˜) (3) C˜0g˜−1(θ˜,θ¯˜)= V(f|(fϕ˜′±(ϕ˜),±f)(|2ϕ˜±)) − |f′(Vϕ˜0±)|2 (12) | ′ | or, using (9), The correspondingfirst-orderequations for the new field ϕ˜ read exaclty as (2), but now with the superpotential C˜ g˜ 1(θ˜,θ¯˜)= C0g−1(θ,θ¯)+V0 V0 . (13) W˜ determined by (3). 0 − f (ϕ˜)2 − f (ϕ˜ )2 | ′ | | ′ ± | 3 Notethat,dependingonthevaluesofthepotentialatde preserve the invariance mentioned above, leading to a minima, these expressions can annihilate giving rise to breakingof the full symmetry of the theory atthe quan- singular points in the target space, as we will see later. tum (perturbative) level. Finally, using relation(11) and letting constants aside 3.1 Topological sectors and singularities. An- for simplicity, we obtain an ODE for the function f otherimportantpoint to remarkhereis that, asstressed before, the asymptotic values of the field solutions f′(ϕ˜)f′(ϕ˜)=|f′(ϕ˜)|2 =g˜(θ˜,θ¯˜)/g(θ,θ¯), (14) lead to singular points of the target space metric, as g 1(φK(x) v ) V(v ) V = 0 and, naturally, − 0 which implements the map between the metrics of the a simi±lar be→havi±or s→hould ±be−expected of the ‘free de- free complex models. formed model’s metric g˜ [17]. Thus, while degenerated 3. Geometrical Interpretation. minima of the originalpotential are mapped to degener- Sigmamodelscanbegenericallydescribedbyanaction atedminimaofthedeformedpotential,inthe freemodel of the form [16] case, they are all mapped to the same singular point of themetric. Thisissoastheydependonthevaluesofthe S = dφG (φ) dφ (15) potentialcalculatedattheminimawhich,byassumption, ij Z ∗ are degenerated. Summarizing, topological sectors of the field models Classicallyandgeometricallyspeaking,wecanthink G ij have their counterpart as disconnected regions on the as a metric on the target space (T). In such direction, target space, separated by singularities. This is a di- relation(14)canbeunderstoodasadifferentialequation rectconsequenceofthe factthatwearedealingwithsin- determining the simplest map f connecting two given gle field models. This fact may result relevant for some metrics g and g˜, describing the target spaces geometries supergravitytheories, which demand such a type of con- of two distinct free field models. That is, structions – see, for instance, [10]. 3.2. Toy example: Sine-Gordon as φ4 modelde- df[g(f)]1/2 = dφ˜[g˜(φ˜)]1/2. (16) Z ±Z formation. Before attacking the quaternionic case, let us illustrate how the mechanism works by analyzing a This expression reminds us the invariance of the action trivial example. Consider the two very well known real S under Diff(T) : φ φ′(φ) Gij(φ) Gi′j′(φ′). scalar field models φ4 and sine-Gordon. As shown in → ⇒ → This is reminiscent ofthe determinantalcharacterof the [4], these two models can be linked by the deformation Sigma model as measure, and expresses that a Sigma method. This is accomplished by simply taking the φ4 model is defined by an equivalence class of metrics. model, V(φ) = 1(1 φ2)2, which supports two Z kink Theotherimportantingredientistheinvarianceunder solutions φK(x)2= −tanh(x) and applying the def2orma- reparameterization implied by (16), due to the square tion functio±n f(φ˜)±= sin(φ˜). Using prescription (3), we root (e.g. Nambu-Goto/Barbashov-Chernikovaction). immediately arrive to Taking a common canonical basis χ in the target space (where the functional quantities have explicit de- V(f(φ˜)) 1(1 sin(φ˜)2)2 pendence) we have g˜(χ) = g(f(χ))[f′]2 and g˜(χ) = V(φ˜)= [f (φ˜)]2 = 2 c−os2(φ˜) = 21cos2(φ˜), (18) g(φ˜(χ))[φ˜]2. So, immediately, we obtain ′ ′ whichisthesine-Gordonpotential[18]. Thesine-Gordon g(φ˜) [f ]2 df 2 solutions are then obtained by inverting the f function, ′ = = (17) g(f) [φ˜]2 (cid:20)dφ˜(cid:21) namely ′ Intheoriginalprescriptionofthemethod,thefunction φ˜k(x)=fk−1(φK(x))=±arcsin(tanh(x))+kπ, (19) f connectsthe fieldsolutions(andpotentials)oftwodif- where k Z specifies the branch [19] of the inverse of f. ferent models (φ = f(φ˜)). In our new approach, f con- Note t∈hat, while the kink solutions of the φ4 model nects different metrics related by the equivalence class connectthesingletopologicalsectordefinedbythepairof of diffeomorphisms in the target plus reparametrization degeneratedminimaofthe potential(v =+1andv = + invariance. This puts in evidence the reason why the 1), the sine-Gordon model presents an infinite num−ber − originalprocedure workedin allthe casestreated in sev- of topological solutions, one for each topological sector, eral articles: it can be achieved only when considering connectingadjacentminima ((k 1/2)π and(k+1/2)π) − just a single scalar field. The freedom reduces then, to – see Fig. 1 in [4]. the choice of the field of numbers over which this single Let us now consider the free versions of the models. quantity is defined (namely, reals, complex or quater- From relations (11) and (13) we have, for the φ4 model nions). Notice also that, from a perturbative quantum me- gλ( φ+C1)=C0(1 φ2)−2, (20) ± − chanical point of view, G (X) is sometimes thought as ij and for the sine-Gordon model, the sum of an infinite number of coupling constants: G (X) = G0 + G1 Xk +... These constants do not g˜ ( φ˜+C˜ )=C˜ cos 2(φ˜), (21) ij ij ij,k sG ± 1 0 − 4 with θ = φ+C and θ˜= φ˜+C˜ , respectively. The corresponding vacuum (minima) manifold is de- 1 1 ± ± Now, from (16), the deformation function connecting scribed by the set of the N-roots of n in the field of the these free models is given by quaternions, i.e. S2 spheres. The first order equation, Πq =W (q), for our QWZ potential (24) reads ′ df g(f)= df = dφ˜ = dφ˜ g˜(φ˜) Z p Z 1−f2 ±Z cos(φ˜) ±Z q dq =n qN =n (q ikq )N (25) f = sin(φ˜) (22) dx − − 0− k ⇒ b This expression, with x identified below, arises from the That is, by using the ‘geometric picture’ we obtain ex- relationbetweentheleftregularsuperpotentialW(q)and actlythesamemappingfunctionthatconnectstheinter- the BPS conditions [8]. actingversionofthemodelsgivenin[4]. Notealsointhe solution above the clear interplay between the minima 4.2. Case N = 2. New BPS solution (Non- of the potential and the singularities at the geometrical commutative base space). We now present here a new level or, better, between the topological sectors and the BPS quaternionic solution (not worked out in [8]), for causally disconnected regions of the target space deter- the case N = 2, with the fields on a quaternionic base mined by the singularities. manifold(non-commutativespacetimeequivalent)astar- get space. This is obtained by identifying the spacetime 4. Quaternionic Field Model Deformation. spatialcoordinatexwithoneofthecomplexdirectionsof In a previous work [8], we have shown genuine BPS thequaternionicmanifold. Inthiscasewetakex i X3 solutionsforaquaternionicWess-Zuminomodel(QWZ), → 3 (i.e. x SU(2)) and, consequently, Π = i3∂ . As in the context of hyper-Kähler structures. Even when ∈ − 3b a consequence of this choice, the spacetime assumes the thisisastandardchoice,wefoundvalidtoworkwiththe b structure S O(3) S SU(2). QWZ model as it serves as the basic prototype for any 1 1 ⊗ ∼ ⊗ The potential for the N = 2 case of our QWZ model analysisinvolvinghypercomplexquaternionicstructures, takes the form appearingin severalareasofmodern theoreticalphysics. In general, domain walls are co-dimension one solu- V = 1(n q2)(n q2) (26) tions, that separate the spacetime into regions corre- 2 − − sponding to different vacua. In the simplest case, a do- = 12n2−n(q02−q32)+ 21(q02+q32)2 mainwallissupportedby agaugepotentialthatcouples Fig. 1 shows the form of potential (26) as a function of to its world volume and the field strength of this gauge two of the quaternionic components (q and q ). potential is dual to a cosmological constant. In a more 0 3 generalsetting,with nontrivialcouplingsto scalarfields, thiscosmologicalconstantappearsasanextremumofthe potentialterm in the Lagrangian. The resulting solution describes then a flow towards an extremum and, if the potential possessesseveralextrema, the solutionmayin- terpolate between them. In the present section we will analyzeadomainwall-likesolution,analogoustotheone obtained in [12], developing a connection between solutions obtained in our previous work [8]. 4.1. Hyper-Kähler domain wall solution. As starting point, consider a single quaternionic field gov- erned by the Generalized Quaternionic Lagrangian density (GQL) of the form – see [8] – Figure 1: Quaternionic potential V(q) (n=1). = 1ΠqΠq 1 W (q)2, (23) L 2 − 2| ′ | The first order equation (25) takes the form wheretheCauchy-FueteroperatorisgivenbyΠ i0∂ 0 ≡ − iwb1h∂i1le−ii0b2∂=2 −Iiba3n∂d3, iwkit(hk∂=0 ≡1,∂2/,3∂)x0obaenyd t∂hke≡sta∂nb/d∂axrkd, ddXq3 =n−(q02−qi2−2ikqkq0), n∈Z. (27) b quaternionic algebra. Einstein’s summation convention b b Breakingequation(27)intoitsscalarandvectorialparts, is adopted while the prime indicates derivative with re- we obtain the system of equations spect to the argument of the function. The quaternionic Wess-Zumino superpotential takes dq0 = 2q q , dq1 = 2q q the form dX3 − 0 3 dX3 − 0 2  (28) W′(q)=n−qN =n−(q0+ikqk)N, (24)  ddXq33 = n−q02+qi2, ddXq23 = 2q0q1  where N Z. In general, n H, but thbrough this work Theseequationsarecoupledinpairs,sowecanobtain we will ta∈ke n C or in its s∈ubfields. an explicit solution by just putting q = q = 0. The 2 1 ∈ 5 system (28) reduces then to can switch between hyperbolic and trigonometric char- acters. In the context of quaternionic BPS structures dq0 = 2q q , dq3 =n q2+q2. (29) developed in [8], hyperbolic (negative n) case of solu- dX − 3 0 dX − 0 3 tion(31)canbeinterpretedasadomainwallcenteredat 3 3 X = β /2, showing the typical defect behavior [20] – Aninterestingresultarisesifwenotethatsystem(29) see3|fi0guren2. Naturally,this defect-like behavioris lostin can be reduced to a Liouville type equation. In fact, the trigonometric case. making the substitution q0 e2α = Y2, and writing As a result of the geometrical structure of the quater- ddXY3 ≡Y′, we obtain the ODE≡: Y′′ =Y5−nY, that can nionicsolution(31),thepotential(26)fortheN =2case be integratedto [Y 2 1Y6+nY2] =0, from which we of our QWZ model takes the form ′ − 3 ′ obtain the ‘energy equation’, V[q(X )]=2n2Sech2(Φ) 4Sech2(Φ) 3 . (32) 3 1 − dY C+ 13Y6−nY2 −2 =±dX3. (30) (cid:0) (cid:1) (cid:2) (cid:3) Forthe sakeofillustration, letus particularizethe pa- The above implicit relation is integrable but not invert- rameters n and c in order to get β = 0. Then, the 0 n ibleingeneral. However,intheimportantparticularcase profile of the potential (32) in terms of the base space C = 0, that is, the momentum map, on-shell or surface variable X take the different forms depicted in Fig.3. 3 constraint, the equation is fully solvable and invertible. Thus,weobtainanon-commutativeN =2BPSsolution of the form q(X )=(√3n)Sech(Φ)+i3(i√n)Tanh(Φ)) (31) 3 b where, Φ = 2i√nX3 βn, and βn = 2Ln 3neic0√n , − (cid:16) (cid:17) whereiistheimaginaryunitofthefieldofcomplexnum- bers C and c an arbitrary (complex) constant. 0 (a)Casen<0 (a)Hyperbolic(defect-like)case(n<0) (b)Casen>0 (b)Trigonometriccase(n>0) (c)Case n>0 Figure 2: Quaternionic solution. (a) n=−1, c0 =ln(3); (b) n=1, c0 =iln(3). Figure3: SpatialprofileofquaternionicpotentialV[q(x)]. (a) ‘Invertedvolcano’potential(n=−1,−0.75and−0.25). Figs. Itisworthnotingherethat,dependingonthevaluesof (b)and(c)showtwodifferentscalesofthepatternappearing the constants c0 and n (and therefore βn), solution (31) in the n>0 (trigonometric) case. 6 4.3. Quaternionic model deformation. In com- adjusting the coefficients taking 2i√n = µ and y = 0 plete analogy with the complex case considered in pre- iβ /(2√n), and then identifying n vious sections, our generalized quaternionic Lagrangian 1 (23), can be explicitly connected to a free Sigma model. X3 y and Q3 Θ= in−2q3, (42) ↔ ↔ − (a) Free quaternionic model. As stated in [8], the con- Now the metric determinants explicit the distinct curva- nection (harmonic map) between the models can be re- tures of the solutions duced to the relation V =[det(g )] 1 U 1, (33) 4n 1 ab − − g = , g = . (43) ≡ ANS −1 Θ2 2n2(1 Θ2)(1 4Θ2) where gab is the metric associated to the hyper-Kähler − − − manifold(targetspace)ofthefreemodelandU isrelated Then, to the line element as shown below. Thus, the metric g(Θ) 1 determinant in the N =2 case reads F′(Θ)2 = = − (44) g (Θ) 8n3(1 4Θ2) ANS det(g ) = 1 n (q2 q2) 2+2q2q2 −1 (34) − ab h2(cid:2) − 0 − 3 (cid:3) 0 3i from which we have = Cosh2(Φ) (35) √2 2n2 1−4Tanh2(Φ) F (Θ)=−8n3/2 Lnh√n(cid:16)2Θ+p4Θ2−1(cid:17)i+C1. (45) (cid:0) (cid:1) whereinthelastequalitywehavemadeexplicitthebase In orderto get the function mapping one solutioninto space coordinate via solution (31) (Φ=2i√nX β ). the other, that is, f such that Θ = f(Q ), we use once 3 n 3 − (b) Geometry of the wall: the line element. Inorderto againrelation(17). Then, under conditions (42), we can make the corresponding geometrical analysis, let us in- write the derivative of the deformation function as troduce the obvious transformation Θ = TanhΦ, which g (Q (X )) allows passing from hyperbolic/trigonometricto polyno- f′(Q3)2 = ANS 3 3 (46) g(f(Q )) mial expressions. Then solution (31) and potential (32) 3 take the form = 8n3 1−f(Q3)2 1−4f(Q3)2 . − (cid:0) 1(cid:1)(cid:0)Q2 (cid:1) q(Θ) = i0(√3n) 1 Θ2+i3(i√n)Θ (36) − 3 − V[Θ] = 2n2 1 pΘ2 1 4Θ2 =[det(g )] 1(37) Integratingthisexpressionweget,besidesthetrivialcon- b b ab − − − stant solutions f(Q )= 1, 1, (cid:0) (cid:1)(cid:0) (cid:1) 3 ± ±2 Now, taking into account the specific form of the line element in the hyper-Kähler and quaternionic cases we f(Q )=i√nSn √8n3Ln √nΞ +c , (47) 3 1 can write h (cid:0) (cid:1) i ds2 = U−1dq0 dq0+Udq3 dq3 (38) where Ξ ≡ Q3 + Q23−1. This is the function map- = 12n2(1−⊗Θ2)Θ2hU−1i⊗0⊗i0+ 13−ΘΘ22U i3⊗i3i ip√inngf(tQhe3)s)oolubttiaoinnsepdofbythoeurtwgoeommeotdreiclsv(eqr3sio=n oif√tnhΘe de=- formation recipe. b b b b Therefore, we have g = 24n4Θ2 1 Θ2 2 1 4Θ2 , 5. Fubini-Study space deformation. 00 − − − and g =2 1 Θ2 / 1 4Θ2 . (cid:0) (cid:1) (cid:0) (cid:1) Let us now extrapolate the results obtained up to this 33 − − (c) ‘Geom(cid:0)etric ma(cid:1)p’(cid:0)and defo(cid:1)rmation. Asafinalillus- pointtoacasenotinvolvingdefect-likesolutions,thatis, inwhichwearenotworriedaboutconnectingtheasymp- tration, let us propose the problem of finding a relation totic values of the solutions of the originalanddeformed (deformationfunction)connectingourhyper-Kählerwall models. In particular, in connection with our previous solution(31)tothe solutionobtainedby Arai,Nitta and results[8],weanalyzetheFubini-Study (FS)spacecase. Sakai (ANS) in [12] –see also [13]–, where the geometry specified by metric determinant One important (and usually disregarded) rôle of the Fubini-Study metric arises in the context of Quantum g det(g ) = µ2 (39) Mechanics. The Hilbert space description of quantum ANS ≡ ab |ANS 1−Q23 theory is generically complex and, in order to compute is related to the solution the transition probabilities, a (complex) scalar product is defined. This scalar product determines an Euclidean Q (y)=Tanh(µ(y+y )), (40) 3 0 geometry. However, another geometry also emerges in this setting: the Fubini-Study geometry, which arises as One way to compare (40) to our wall solution follows. q (X )=(i√n)Θ=(i√n)Tanh(Φ) , (41) As the superposition principle is assumed to be valid 3 3 in quantum processes, the theory must be linear, and where Φ 2i√nX β , is to write both expressions twolineardependentvectorsrepresentthesamephysical 3 n ≡ − in the same coordinate basis. This is accomplished by state. Restricting to unit vector reduces the redundancy 7 to phase factors. Consequently, two curves on the unit The simplestcasewecanconsiderto obtaina solution vectors space –a curve could be, for instance a piece of a for this equation is the ‘momentum map’ (L = 0 or µ solution of the Schrödinger equation– differing only in α=const.). Inthatcase,aftercovariantelimination,we a phase, describe the same set of physical states. In arrive to the solution this context, the ‘Fubini-Study length’ corresponds to the minimal length a curve of states can assume. The r= Re ic sn(iwx,c /c ) , (56) + − { − − } requirement of the minimal length on the set of vector states induces a geometry, represented by the ‘Fubini- where wx = w xµ, c = 1 [1+2c] and c and w µ µ Study metric’ [14]. Imposing the condition that the Eu- are scalar and vector±constan±tsp, respectively. The mo- clidean and the Fubini-Study lengths coincide piecewise mentum map L = 0 introduces a degeneration of the µ inthecurve(paralleltransportcondition)definesthege- solution ϕ˜ = ϕ˜ = r, which results real, continuous and ometric(orBerry)phase,whichcorrespondstothediffer- periodical, showing the typical behavior of the sn – see ence between an arbitrary solution states (closed) curve Fig. 4. andtheoneobtainedimposingtheminimallengthcondi- tion. Moreprecisely,itsinitialandfinalpoints willdiffer r in a phase factor: the ‘geometric phase’. 0.5 5.1 FS space from a complex field model. The spaces described above are realized by Kähler Manifolds -4 -2 2 4 6 x (complex manifold M with a Hermitian metric h and -0.5 a fundamental closed 2-form ω), with group structure CPn =S2n+1/S1.TheHermitianmetriccomponentsare given by Figure 4: r(x)solution with α=const (‘momentum map’). (1+ z 2)δ z z h h ∂ ∂ = | | ij − i j. (48) In the general case (Lµ = 0), the norm of the scalar ij ≡ i j (1+ z 2)2 field takes the form 6 (cid:0) (cid:1) | | Then the Kähler potential (hij =∂i∂¯jK(z,z¯)) reads r = Re sn iλwx, λ− λ sn2 iλwx,λ− 12 − (cid:26) (cid:16) λ+(cid:17)h +− (cid:16) λ+(cid:17)i (cid:27) K(z,z¯)=ln(1+δ zizj). (49) (57) ij where λ 1 L Lµ/4 andλ= λ /2. This solution, µ + Consider now the Lagrangiandensity depicted±in≡Fi±g. 5(a), is periodicpwith compact support onthe spacetimecoordinates(we can,forinstance,iden- 1 2 ˜ = ∂ ϕ˜∂µϕ˜ 1+ ϕ˜2 (50) tify x with the radial coordinate of a cylindrical space- µ LK 2 −(cid:16) | | (cid:17) time). The phase of the general solution can also be explic- Fromthe formofthe potentialterm andrelation(10), it itly determined however, the expression is cumbersome isstraightforwardtoconcludethatthetargetspaceofthe and we will just mention here that it presents a highly freeversionofthemodelcanberelatedtoaFubini-Study oscillating behavior near the origin of coordinates, and space with a Kha¨ler potential as in (49). stabilizes quickly to α L Lµ as moving away (See Letusfirstanalyzethemodel(50). Puttingthefieldin µ → its polar representation, ϕ˜ reiα, the Lagrangian takes Fig. 5(b)). p ≡ the form Α 1.0 ˜ = 1 (∂ r∂µr)+(∂ α∂µα)r2 1+r2 2. (51) 6r LK 2 µ µ − 5 0.5 (cid:2) (cid:3) (cid:0) (cid:1) 4 The corresponding Euler-Lagrangeequations are 3 -0.05 0.05 x 2 -0.5 ∂µ r2∂ α = 0 (52) 1 µ -5 5 x -1.0 ∂µ∂ r (∂ α∂µα)r+4r(cid:0)1+r2(cid:1) = 0 (53) (a) (b) µ µ − (cid:0) (cid:1) Equation (52) above expresses the existence of a con- Figure 5: Lµ 6= 0 solution. (a) Norm r(x); (b) Highly served quantity oscilating phaseα(x). L r2∂ α. (54) µ ≡ µ 5.2. Fubini-Study space from a flat space. We have shown an explicit complex field solution in connec- Substituting L in (53), we have µ tion with a Fubini-Study space. Consider now a one di- L Lµ mensional (n = 1) flat complex space, described by a ∂µ∂µr− µr3 +4r 1+r2 =0 (55) flat metric gflat. From relation (10), the corresponding (cid:0) (cid:1) 8 Lagrangiandensity with a potential reads [21] consistently implemented only in the case of single-field models. This is due, mainly, to the very restrictive con- g−1 straintimposed by preservationofBPS conditions (first flat 1 1 order equations). = ∂ ϕ∂µϕ V(ϕ,ϕ)= ∂ ϕ∂µϕ m2, (58) LK 2 µ − 2 µ −z}|{ As a way to delimitate the widest applicability of the procedure, systems with multiple field components were That is, V(ϕ,ϕ¯) = m2 = const. Then, using the prece- considered. Namely, real (R), complex (C) and also dentresults,wecanexplicitlylinkmodel(58)toaFubini- quaternionic (H) one-dimensionalfield models were ana- Studyspace,whichcorrespondstothetargetspaceofthe lyzed. Allcaseswhereillustratedbyconstructingexplicit free version of the Lagrangian(50). That is, taking maps between different associated geometries. The presentworkis partofa largerinvestigation,cen- gf−l1at ↔V =m2 and g˜F−S1 ↔V˜ =(1+|ϕ˜|2)2 (59) tering efforts in finding and studying new quaternionic BPS solutions. In that direction, a novel quaternionic Now, as stated in (14), the function f connecting ϕ and domain wall solution was obtained here by directly per- ϕ˜ satisfies forming a geometrical map. In constructing domain wall solutions, the method dϕ 2 g˜ m2 |f′(ϕ˜)|2 =(cid:12)(cid:12)dϕ˜(cid:12)(cid:12) = gfFlaSt = (1+|ϕ˜|2)2, (60) slahtoewdsinhigthhelytrsaunitsafobrlemdatuieontso. tHheowkeinvder,ofougreogmeoemtreietsricrea-l (cid:12) (cid:12) approach put in clear evidence the strong limitations of (cid:12) (cid:12) that is, dϕ = mdϕ˜/(1+iϕ˜)2, which directly leads to the deformation procedure as a tool for generating solu- relation tions, as it is the simplest harmonic map between the m m metrics (of the geometrical Lagrangians associated to ϕ=f(ϕ˜)= = (ϕ˜+i) (61) the mapped dynamical models). Thus, in more involved ϕ˜ i (1+ ϕ˜2) − | | geometries-field models, the use of pure geometrical ap- Relation(61)abovecanbe easilycheckagainstthe re- proach to the procedure here presented is mandatory in sultsofprevioussectionsbyreconstructingthe deformed order to preserve BPS conditions, SUSY structure or potential applying (3) to get gauge transformations, as was clearly illustrated by the mapping of the Fubini-Study geometry in Section 5. V(f(ϕ˜),f(ϕ˜)) m2 Note also that we have avoidedthe problem of preser- = =(1+ ϕ˜2)2 =V˜(ϕ˜,ϕ¯˜) f (ϕ˜)2 m2/(1+ ϕ˜2)2 | | vation of the Noether charges (e.g. the U(1)-charge in | ′ | | | the complex case) arising when proposing the simplest Thus, at least for the present case, the procedure can be possible mappings between the interacting and the free applied in a purely geometric context, i.e. when there models, by working all the examples in the momentum are no defect solutions associated. map. Differently from the ‘standard’ mapping of references 6. Concluding Remarks. [1, 2], which does not assures the preservation of such In the present article we have considered a technique symmetries,thegeometricalapproachcould,inprinciple, for generating solutions, known as deformation method. be extended to a N = 2 gauged supergravity theory in Differently from all the previous works on this subject, four dimensions. This should be possible given that a herewe analyzethe procedureunder ageometricalpoint hyper-Kähler manifold is a quaternionic space in H of view, in order to get a better understanding of the M which, as shown in [8], and also in Sec. 4.1, stable BPS meaning and limitations of the method. Thus, the main solutions can be consistently constructed. ideaofouranalysiswasdevelopedworkingwiththemet- Another interesting potential feature of the geometri- ric fields of the equivalent free models, connecting differ- caltreatmentofthemappingsshownthroughthepresent ent solutions and also generating new ones. work,is the possibility of establishing a parallelbetween Two main results are to be remarked: First, we have effective models with modified kinetic term (‘k-fields’), shown through this paper that the ‘deformation proce- ofinterestto the dark energy-matterproblem,andmod- dure’ is nothing but the simplest geometrical map be- elsdescribedbyLagrangiansofthestandardform. These tween free Lagrangians in curved spacetimes, equivalent ideaisunderexplorationandwillbepresentedelsewhere. totherespectiveflatspaceLagrangianswithapotential. Second,asputinevidencebythegeometricaltreatment, 7. Aknowledgments. DJCL would like to thank weconcludethatthedeformationmethodisunavoidable CONICET (Argentina) and JINR (Russia), and VIA limited to systems with higher symmetries and can be thanks PROCAD/CAPES (Brasil). [1] D.Bazeia,L.LosanoandJ.M.C.Malbouisson,Phys.Rev. [2] C.A. Almeida, D. Bazeia, L. Losano, J.M.C. Malbouis- D66(2002) 101701. son,Phys.Rev.D69(2004)067702;V.Afonso,D.Bazeia, 9 F.A.Brito,JHEP0608(2006) 073;D.Bazeia, L.Losano 1634(2005)[Yad.Fiz.68,1698(2005)][hep-th/0401102]. (ParaibaU.),Phys.Rev.D73(2006)025016; V.I.Afonso, [13] M. Arai, M. Nitta and N. Sakai, hep-th/0401084. D. Bazeia, M.A. Gonzalez Leon, L. Losano, J. Mateos [14] A. Uhlmann and B. Crell, Geometry of State Spaces, Guilarte,Phys.Rev.D76(2007)025010; D.Bazeia, M.A. Lect. NotesPhys. 768, 1-60 (2009). GonzalezLeon,L.Losano,J.MateosGuilarte,Phys.Rev. [15] Non-staticdefect-likesolutionsforscalarfieldmodelsare D73 (2006) 105008. obtainedbysimplyapplyingaLorentzboostonthestatic [3] Surlesgroupesd’holonomiehomogènesdevariétésàcon- ones. nexion affine et des variétés riemanniennes, M. Berger, [16] InthecaseoftheBorn-Infeldtheoreticalframework,S = Bull. Soc. Math. France 83, 279 (1995). R pG(φ), where the determinant of the metric of the [4] D. Bazeia, L. Losano, J.M.C. Malbouisson, R. Menezes, target space is the wedge product of the Maurer-Cartan Physica D237 (2008) 937-946. forms. [5] V.I. Afonso, D. Bazeia, M.A. Gonzalez Leoń, L. Losano [17] Note, however, that the behavior of f′ could perhaps and J. Mateos Guilarte, Nucl.Phys. B810 (2009) 427- affectthedivergenceorderofg˜,ascanbeseenfrom(14). 459. [18] This corresponding to the more general expression [6] M. K. Prasad and C. M. Sommerfield, Phys. Rev. Lett. V(φ)=αcos(βφ)+|α|,fixingtheparameterstotheval- 35, 760 (1975); E. B. Bogomol’nyi, Sov.J. Nucl. Phys. ues α=1/4, β=2. [7] R.Rajaraman, Phys.Rev.Lett. 42, 200 (1979). [19] Again, this result is consistent with the general form o [8] V.I. Afonso, D. Bazeia and D.J. Cirilo-Lombardo, the sine-Gordon kink solutions, φn(x) = 2arctan(ex)+ Phys.Lett.B713 (2012) 104-109. (2n+1)π, taking the specific values of α= 1/4, β =2, 2 [9] V.I. Afonso, D. Bazeia, M.A. Gonzalez Leoń, L. Losano and n=k−1, as can beeasily checked. and J. Mateos Guilarte, Phys.Lett. B662 (2008) 75-79. [20] Note that the pure imaginary profile of the scalar com- [10] C.M.Hull,G.Papadopoulos,P.K.Townsend,Phys.Lett. ponent of the solution (q0(X3)) is represented in Fig. 2, B316 (1993) 291-297. in thesame graphic than thereal q3(X3) component. [11] D.j. Cirilo-Lombardo and V.I.Afonso, work in progress. [21] As before, we are considering thesimplest C0=1 case. [12] M. Arai, M. Nitta and N. Sakai, Phys. Atom. Nucl. 68,