CENTRE DE PHYSIQUE THEORIQUE - CNRS - Luminy, Case 907. F{13288 Marseille Cedex 9 - France. Unite Propre de Recherche Chern-Simons Vortices

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1 hep-ph/94059 CENTRE DE PHYSIQUE THEORIQUE - CNRS - Luminy, Case 907 F{388 Marseille Cedex 9 - France Unite Propre de Recherche 706 Conformal Properties of Chern-Simons Vortices in External Fields C. DUVAL ( )P. A. HORV ATHY ( )L.PALLA ( 3 ) Abstract. The construction and the symmetries of Chern-Simons vortices in harmonic and uniform magnetic force backgrounds found by Ezawa, Hotta and Iwazaki, and by Jackiw and Pi are generalized using the non-relativistic Kaluza-Klein-type framework presented in our previous paper. All Schrodinger-symmetric backgrounds are determined. April 994 CPT-94/P.308 anonymous ftp or gopher : cpt.univ-mrs.fr ( ) and Departement dephysique, Universite d'aix-marseille II, UFR de Luminy, Case 907, F{388 MARSEILLE Cedex 09 (France), duval@cpt.univ-mrs.fr. ( ) Permanent address : Departement de Mathematiques, Universite detours, Parc de Grandmont, F{3700 TOURS (France), horvathy@univ-tours.fr ( 3 ) Permanent address : Institute for Theoretical Physics, Eotvos University, H{088 BUDAPEST, Puskin u. 5-7 (Hungary), palla@ludens.elte.hu

2 Duval, Horvathy, Palla. Introduction The construction of static, non-relativistic Chern-Simons solitons [] was recently generalized to time-dependent solutions, yielding vortices in a constant external magnetic eld, B [-4]. Putting! = B=, the equation to be solved is (:) i D! t! = D ~!!! (We use units where e = m = ). Here the covariant derivative means!: (:) D! i(a! ) ia ( =0;;), where A is a vector potential for the constant magnetic eld, A 0 =0, A i = ijx j B! ij x j (i; j =;) and (A! ) is the vector potential of Chern-Simons electromagnetism i.e. its eld strength is required to satisfy the eld-current identity (:3) B! i A j! =! and E ia t A i! = ij J j! with! =!! and ~ J! =(=i)[ ~ D!!!( ~ D!!) ]. These equations can be solved [-4] by applying a coordinate transformation to a solution,, of the problem with! =0 studied in Ref. [], according to (:4) with!(t; ~x) = cos!t exp (A! ) @ i! r tan!t! Nt ; exp i N!t ( ~ X;T); tan!t (:5) T =! ; X ~ = R(!t) ~x: cos!t Here N = R d ~x is the vortex number and R() is the matrix of a rotation by angle in the plane. (The pre-factor exp[in!t=] and the extra [(!=)Nt] are absent from the corresponding formula of Ezawa et al. []). A similar construction works in a harmonic background [4]. In a previous paper [5] non-relativistic Chern-Simons theory in + dimensions was obtained by reduction from an appropriate (3 + )-dimensional Lorentz manifold. As an application, we reproduced the results in Ref. []. Here we show that the above generalizations arise by reduction from suitable curved spaces, that they all share the (extended) Schrodinger symmetry of the model in [], and we determine all background elds which have this property.

3 Time-dependent Chern-Simons::: 3. Chern-Simons theory in Bargmann space ( + )-dimensional non-relativistic Chern-Simons theory can be lifted to `Bargmann space' i.e. to a 4-dimensional Lorentz manifold (M;g) endowed with a covariantly constant null vector [5]. The theory is described by a massless non-linear wave equation (:) n D D R 6 + where D = r ia ( =0;;;3), r is the metric-covariant derivative and R denotes the scalar curvature. The scalar eld and the `electromagnetic' eld strength, f =@ [ a ], are related by the identity o =0; (:) f = p g j ; j = i [ (D ) (D ) ] : Eq. (.) [but not (.)] can be obtained from variation of the `partial' action (:3) S = Z (D ) D + R 6 j p j j j4 gd 4 x: M The quotient of M by the integral curves of is non-relativistic space-time we denote by Q. A Bargmann space admits local coordinates (t; ~x; s) such that (t; ~x) label Q and s. When supplemented by the equivariance condition (:4) D = i ; our theory projects to a non-relativistic non-linear Schrodinger/Chern-Simons theory on the ( + )-dimensional manifold Q. The eld strength f is clearly the lift of a closed two-form F on Q. So, the vector potential may bechosen as a =(A ;0) with A s-independent. In this gauge, (t; ~x) =e is (t; ~x) is then a function on Q. A symmetry is a transformation of M which interchanges the solutions of the coupled system. Each -preserving conformal transformation is a symmetry and the variational derivative =S=g provides us with a conserved, traceless and symmetric energymomentum tensor. The version of Noether's theorem proved in [5] says that for any -preserving conformal vectoreld (X ) on Bargmann space, the quantity Z (:5) Q X = X p d ~x t (where the `transverse space' t is a space-like -surface t = const: and ij is the metric induced on it by g ) is a constant of the motion. The conserved quantities are conveniently calculated using the formula [5] (:6) = i [ (D ) (D ) R ] 6 6 j j +(D ) D + j j4 :

4 4 Duval, Horvathy, Palla For example, M can be at Minkowski space with metric d X ~ +dt ds, where X ~ R and S and T are light-cone coordinates. This is the Bargmann space of a free, nonrelativistic particle [6]. The system of equations (.,,4) projects in this case to that of Ref. []; the -preserving conformal transformations form the (extended) planar Schrodinger group, consisting of the Galilei group with generators J (rotation), H (time translation), G ~ (boosts), ~P (space translations), augmented with dilatation, D, and expansion, K, and centrally extended by `vertical' translation, N. With a slight abuse of notation, the associated conserved quantities are denoted by the same symbols. (Explicit formul are listed in [] and [5]). Applying any symmetry transformation to a solution of the eld equations yields another solution. For example, a boost or an expansion applied to the static solution 0( X)ofJackiw ~ and Pi produces time-dependent solutions. Using the formul in [6] we nd (:7) (T; ~ X)= ( kt exp i h X ~ ~ b + T ~ b + k ( X ~ + ~ bt ) kt i ) 0( ~ X + ~ bt kt ); which is the same as in []. Now we present some new results. The most general `Bargmann' manifold was found long ago by Brinkmann [7]: (:8) g ij dx i dx j +dt ds + ~ Ad~x +A 0 dt ; A 0 = U where the `transverse' metric g ij as well as the `vector potential' A ~ and the `scalar' potential U are functions of t and ~x only. Clearly, s isacovariantly constant null vector. The null geodesics of this metric describe particle motion in curved transverse space in an external electromagnetic elds E ~ t ~A ru ~ and B = r ~ A ~ [6]. Consider now a Chern-Simons vector potential (a! ) = (A! ) ; 0 in the background (.8). [The subscript ( )! refers to an external-eld problem]. Using that the only nonvanishing components of the metric (.8) are g ij ;g is = A i, g ss =U+A i A i,g st =,we nd that the integrand in the partial action (.3) is (:9) D! ~ h D! ~ + i (D! ) t (D! ) t i + R 6 j j j j4 ; where the covariant derivative (D! ) means (.) with vector potential A. Thus, including the `vector-potential' components into the metric (.8) results, after reduction, simply in modifying the covariant derivative D in `empty' space (A = 0) according to D! (D! ) : The associated equation of motion is hence Eq. (.). (This conclusion can also be reached directly by studying the wave equation (.)). Let now ' denote a conformal Bargmann dieomorphism between two Bargmann spaces i.e. let ' :(M;g;)! (M 0 ;g 0 ; 0 ) be such that '? g 0 = gand 0 = '?. Such a

5 Time-dependent Chern-Simons::: 5 mapping projects to a dieomorphism of the quotients, Q and Q 0 we denote by. Then the same proof as in Ref. [5] allows one to show that if (a 0 ; 0 ) is a solution of the eld equations on M 0, then (:0) a =('? a 0 ) and ='? 0 is a solution of the analogous equations on M. Locally we have'(t; ~x; s) =(t 0 ;~x 0 ;s 0 ) with (t 0 ;~x 0 )=(t; ~x) and s 0 = s +(t; ~x) so that ='? 0 reduces to (:) (t; ~x) =(t)e i(t;~x) 0 (t 0 ;~x 0 ); A =? A 0 (=0;;): Note that ' takes a -preserving conformal transformation of (M;g;)into a 0 -preserving conformal transformation of (M 0 ;g 0 ; 0 ). Conformally related Bargmann spaces have therefore isomorphic symmetry groups. The associated conserved quantities can be related by comparing the expressions in Eq. (.6). Note rst that, for as in Eq. (.0), D =('? D 0 0 )+ r 0 '? 0. Using R = '? R 0 +6 r 0 r0 and = 0 as well as = (t) and that g t is non-vanishing only for = s one nds hence that = '? 0 0 : But p = '?p 0. Therefore, the conserved quantity (.5) associated to X =(X )on (M;g;) and to X 0 = '? X on (M 0 ;g 0 ; 0 ) coincide, (:) Q X = '? Q 0 X 0: The labels of the generators are, however, dierent (see the examples below). 3. Flat examples Consider now the Lorentz metric (3:) d~x osc +dt osc ds osc! r osc dt osc where ~x osc R, r osc = j~x osc j and! is a constant. Its null geodesics correspond to a non-relativistic, spinless particle in an oscillator background [6,9]. Requiring equivariance (.4), the wave equation (.) reduces to ( ) ~D (3:) i@ tosc osc = +! r osc osc osc osc ( ~ D = i ~ A, = =), which describes Chern-Simons vortices in a harmonic force background, studied in Ref. [3]. The clue is that the mapping '(t osc ;~x osc ;s osc )=(T; ~ X;S) [9], where (3:3) T = tan!t osc ; ~ ~x osc!r X = osc ; S = s osc tan!t osc! cos!t osc

6 6 Duval, Horvathy, Palla carries the oscillator metric (3.) conformally into the free form, d ~ X +dt ds, with conformal factor (t osc )=jcos!t osc j such that sosc S. Our formula lifts the coordinate transformation of Ref. [4] to Bargmann space. A solution in the harmonic background can be obtained by Eq. (.). A subtlety arises, though: the mapping (3.3) is many-to-one: it maps each `open strip' (3:4) I j = (~x osc ;t osc ;s osc ) (j )<!t osc < (j + ) ; where j = 0;;::: corresponding to a half oscillator-period onto the full Minkowski space. Application of (.) with an `empty-space' solution yields, in each I j, a solution, (j) osc. However, at the contact points t j (j +=)(=!), these elds may not match. For example, for the `empty-space' solution obtained by an expansion, Eq. (.7) with ~ b =0;k6=0, (3:5) lim t osc!t j 0 (j) osc =( ) j+! k e i! k r osc 0(! k ~x) = lim t osc!t j +0 (j+) osc : Then continuity is restored by including the `Maslov' phase correction [0]: (3:6) 8 >< (A osc ) 0 (t osc ;~x osc )= >: osc(t osc ;~x osc )=( ) j cos!t osc ~A osc (t osc ;~x osc )= i! exp r tan osc!t osc (T; ~ X) cos!t osc A0 (T; ~ X)!sin!t osc ~x osc ~ A(T; ~ X) ; cos!t osc ~ A(T; ~ X); where j is as in (3.4). Eq. (3.6) extends the result in Ref. [4] from jt osc j <=!to any t osc. (j) For the static solution in [] or for that obtained from it by a boost, lim tosc!t j osc =0 and the inclusion of the correction factor is not mandatory. Chern-Simons theory in the oscillator-metric has again a Schrodinger symmetry, whose generators are related to those in `empty' space as (3:7) 8 J osc = J ; H osc = H +! >< K; (C osc ) = H! Ki!D ; ( ~ P osc ) = Pi! ~ ~ G ; >: N osc = N : The oscillator-hamiltonian, H osc, is hence a combination of the Hamiltonian and of the expansion valid for! = 0, etc. The generators H osc and (C osc ) span o(; ) and the

7 Time-dependent Chern-Simons::: 7 ( ~ P osc ) generate the two-dimensional Heisenberg algebra [9]. Eq. (3.7) adds (C osc ) and ( ~ P osc ) to the J osc and H osc in Ref. [3]. Consider next the metric (3:8) d~x +dt hds + ijbx j dx ii where ~x R and B is a constant. Its null geodesics describe a charged particle in a uniform magnetic eld in the plane [6]. Again, when imposing equivariance, Eq. (.) reduces precisely to Eq. (.) with = = and covariant derivative D! given as in Eq. (.). The metric (3.8) is readily transformed into an oscillator metric (3.): the mapping '(t; ~x; s) =(t osc ;~x osc ;s osc ) given by (3:9) t osc = t; x i osc = x i cos!t + i j xj sin!t; s osc = s which amounts to switching to a rotating frame with angular velocity! = B= takes the `constant B-metric' (3.8) into the oscillator metric (3.). The vertical sosc s are permuted. Thus, the time-dependent rotation (3.9) followed by the transformation (3.3), which projects to the coordinate transformation (.5) of Refs. [] and [3], carries conformally the constant-b metric (3.8) into the! = 0-metric. It carries therefore the `empty' space solution e is with as in (.7) into that in a uniform magnetic eld background according to Eq. (.0). Taking into account the equivariance, we get the formul of [] i.e. (.4) without the N -terms but multiplied with the Maslov factor ( ) j. (The N -term arises due to a subsequent gauge transformation required by the gauge xing in [3]). It also allows to `export' the Schrodinger symmetry to non-relativistic Chern-Simons theory in the constant magnetic eld background. The [rather complicated] generators, listed in Ref. [], are readily obtained using Eq. (.). For example, time-translation t! t + in the B-background amounts to a time translation for the oscillator with parameter plus a rotation with angle!. Hence H B = H osc!j = H +! K!J: Similarly, a space translation for B amounts, in `empty' space, to a space translations and a boost, followed by a rotation, yielding PB i = Pi +! ij G j, etc. 4. Conformally at Bargmann spaces All our preceding results apply to any Bargmann space which can be conformally mapped into Minkowski space in a -preserving way. Now we describe these `Schrodingerconformally at' spaces. In D = n + > 3 dimensions, conformal atness is guaranteed by the vanishing of the conformal Weyl tensor (4:) C = R 4 D [ [ R] ] + (D )(D ) [ [ ] ] R:

8 8 Duval, Horvathy, Palla Now R 0 for a Bargmann space, which implies some extra conditions on the curvature. Inserting the identity R =0into C = 0, using the identity R 0 (R R ), we nd0= R R +R=(D ). Contracting again with and using that is null, we end up with R = 0. Hence the scalar curvature vanishes, R = 0. Then the previous equation yields [ R ] = 0 and thus R = for some vector eld. Using the symmetry of the Ricci tensor, R [] =0,we nd that = % for some function %. We nally get the consistency relation (4:) R = % : The Bianchi identities (r R = 0 since R = 0) % = 0, i.e. % is a function on spacetime Q. The conformal Schrodinger-Weyl tensor is hence of the form (4:3) C = R 4 D %[ [ ] ] : It is noteworthy that Eq. (4.) is the Newton-Cartan eld equation with %=(4G) as the matter density of the sources. It follows from Eq. (4.) that the transverse Ricci tensor of a Schrodinger-conformal at Bargmann metric necessarily vanishes, R ij = 0 for each t. The transverse space is hence (locally) at and we can choose g ij = g ij (t). Then a change of coordinates (t; ~x; s)! (t; G(t) = ~x; s) where G(t) i j = g ij(t) [which brings in a uniform magnetic eld and/or an oscillator to the metric], casts our Bargmann metric into the form (.8) with g ij = ij while remains unchanged. The non-zero components of the Weyl tensor of the general D = 4 Brinkmann metric (.8) are found as C xyxt = C ytts = xb; C xyyt =+C xtts = yb; (4:4) C xtxt (@ y A x A x ) A y B; y U; C ytyt (@ y A x A x ) A x y U; C xtyt (@ x A y +@ y A x )+@ y U (A 4 x@ x A y )B: Then Schrodinger-conformal atness requires 8 < A i = ijb(t)x j + a i ; r~a=0; t ~a=0; (4:5) : U(t; ~x) = C(t)r + F(t)~x+K(t): ~ (Note, en passant, that (4.) automatically holds: the only non-vanishing component of the Ricci tensor is R tt t ( ~ r ~ A) B U).

9 Time-dependent Chern-Simons::: 9 This Schrodinger-conformally at metric hence allows one to describe a uniform magnetic eld B(t), an attractive [C(t)=! (t)] or repulsive[c(t)=! (t)] isotropic oscillator and a uniform force eld F ~ (t) in the plane whichmay all depend arbitrarily on time. It also includes a curlfree vector potential ~a(~x) that can be gauged away if the transverse space is simply connected: a i i f and the coordinate transformation (t; ~x; s)! (t; ~x; s + f) results in the `gauge' transformation A i!a i f= B ij x j.however, if space is not simply connected, we can also include an external Aharonov-Bohm-type vector potential. Being conformally related, all these metrics share the symmetries of at Bargmann space: for example, if the transverse space is R we get the full Schrodinger symmetry; for R nf0gthe symmetry is reduced rather to o() o(; ) R, just like for a magnetic vortex []. The case of a constant electric eld is quite amusing. Its metric, d~x +dtds F ~ ~xdt, can be brought to the free form by switching to an accelerated coordinate system, (4:6) ~ X = ~x + ~ Ft ; T=t; S = s ~ F ~xt 6 ~ F t 3 : This example (chosen by Einstein to illustrate the equivalence principle) also shows that the action of the Schrodinger group e.g. a rotation looks quite dierent in the inertial and in the moving frames. Let us nally mention that the above results admit a `gauge theoretic' interpretation. In conformal (Lorentz) geometry, the Weyl tensor C arises as part of the o(n +;)- valued curvature of a Cartan connection for a D = n + dimensional base manifold. The Schrodinger-conformal geometry in this dimension can be viewed as a reduction of the standard conformal geometry to the Schrodinger subgroup Sch(n +;) O(n +;) [3]. The curvature of the reduced Cartan connection then denes the Schrodinger-Weyl tensor which is thus characterized by (4:7) C =0; a property coming from the previous embedding and strictly equivalent to Eqs. (4.,3). 5. Conclusion Our `non-relativistic Kaluza-Klein' approach provides a unied view on the various vortex constructions in external elds, explains the common origin of the large symmetries, and allows us to describe all such spaces. We have also pointed out that the formula (.4) may require a slight modication for times larger then a half oscillator-period. Acknowledgement. We are indebted to Professor R. Jackiw for stimulating discussions. L. P. would like to thank Tours University for the hospitality extended to him and to the Hungarian National Science and Research Foundation (Grant No. 77) for a partial nancial support.

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