Rotating Central Charge Membranes
Pedro D. Alvarez, Maria Pilar Garcia del Moral, Joselen M. Peña, Reginaldo Prado
RRotating Central Charge Membranes
P. ´Alvarez ,a , M.P. Garc´ıa del Moral ,b , J.M. Pe˜na ,c , R. Prado ,d Departamento de F´ısica, Universidad de Antofagasta, Aptdo 02800, Chile.E-mail: a [email protected]; b [email protected]; c [email protected] d [email protected] Abstract.
In this work we obtain dynamical solutions of the bosonic sector of thesupermembrane theory with central charges formulated on M × T , denoted by MIM2. Thetheory with this condition corresponds to a supermembrane with a C − flux. This sector of theM2-brane is very interesting since classically is stable as it does not contain string-like spikeswith zero energy and at quantum level has a purely discrete supersymmetric spectrum. We findrotating solutions of the MIM2 equations of motion fulfilling all of the constraints. By showingthat the MIM2 mass operator, contains the mass operator discussed in [1], then we show that therotating solutions previously found in the aforementioned work that also satisfy the topologicalcentral charge condition, are solutions of the MIM2. Finally, we find new distinctive rotatingmembrane solutions that include the presence of a non-vanishing symplectic gauge connectiondefined on its worldvolume.
1. Introduction
In this work we obtain the first classical solutions of the equations of motion associated tothe bosonic sector of the supermembrane with central charges formulated in the Light ConeGauge (LCG) on M × T . This theory has interesting properties since it has a purely discretesupersymmetric spectrum and then it can represent part of the microscopical degrees of freedomof the M-theory. In the literature there have been several studies of solutions of the membranetheory like for example [2, 3] in the context of matrix models, or like [4–7] when it is formulatedon a G2 or AdS backgrounds, among others.Rotating membrane solutions have always interest since they can provide a preliminarsignal for interpreting membranes as extended spinning particles if they become well-definedat quantum level and a proper background is chosen. They can be also be considered sourcesof supergravity solutions in eleven dimensions that can describe charged rotating black holes inlower dimensions, like for example [8, 9] or they can even describe spinning solitonic solutions.Solitons have also been used in the context of rotating systems like boson stars and Q-ballstars [10].In [11] it was shown that spherical and toroidal membrane solutions could be obtained fromthe membrane when it is embedded on spherical backgrounds. In [12] new M2-brane solutionsin the LCG were generalized and in [1] the authors found M2-brane rotating solutions on thesame background M × T modelling spinning membranes. In our case we are mainly focused inthe study of rotating MIM2 solutions on M × T formulated in the LCG. We will compare ourresults with those of [1] adding the topological central charge. The central charge conditionon the background considered is a geometrical condition imposed on the wrapping of the a r X i v : . [ h e p - t h ] J a n embrane around the compactified target-space that induces the presence of monopoles over theworldvolume and generates a central charge in the supersymmetric algebra. The compactifiedsupermembrane formulated with central charges was shown in [13] to have a purely discretespectrum with finite multiplicity from [0 , + ∞ ). Furthermore this theory is equivalent or ’dual’to a toroidally compactified supermembrane on a flat metric with a constant C − induced fluxbackground. For all of these reasons it becomes an interesting sector to be characterized. Thedynamical equations from the supermembrane theory are a system of coupled non-linear partialequations highly constrained where it is non-trivial to obtain analytical solutions. Moreover,it is known that those equations may admit have soliton solutions when they are formulatedon certain backgrounds [14]. Indeed the topological condition that we impose represents theexistence of monopoles over the worldvolume of the membrane.
2. The M2-brane with central charge
We will consider the bosonic sector of the supermembrane with central charges formulated on atarget space M × T . We will shorten it by MIM2 since it represents a supermembrane minimallyimmersed in the background. We define the embedding maps X m ( τ, σ, ρ ) with m = 2 , ..., X r ( τ, σ, ρ ) : Σ → T , where r = 9 ,
10, those describingthe embedding on the compactified torus. The coordinates of the supermembrane worldvolumeare denoted by ( σ, ρ ). The maps X r satisfy the winding condition (cid:73) C s dX r = R r m sr , (1)with m sr winding numbers and R r the torus radii. The MIM2 is subject to the central chargecondition (cid:90) Σ dX r ∧ dX s = (cid:15) rs nA, n ∈ Z / { } . (2)This condition implies that the one-forms associated to the embedding map of the compactsector, can be decomposed by a Hodge decomposition dX r ( σ, ρ, τ ) = dX rh ( σ, ρ ) + dA r ( σ, ρ, τ ),with dX rh = R r m sr d (cid:98) X s ( σ, ρ ) a closed one form defined in terms of the harmonic forms d (cid:98) X s and dA r an exact one-form. It was shown that the integer n associated to the central charge conditionis n = det W where W is the winding matrix. The central charge condition characterizes thenontrivial principle U (1) bundle defined on the membrane worldvolume. The LCG Hamiltonianof the theory corresponds to [15] H MIM = T − / (cid:90) Σ d σ √ W (cid:104) (cid:16) P m √ W (cid:17) + 12 (cid:16) P r √ W (cid:17) (cid:105) + T − / (cid:90) Σ d σ √ W (cid:104) T { X m , X n } + T D r X m ) + T F rs ) (cid:105) , (3) where Σ is a Riemann surface, T is the tension of the surface of the membrane and W isthe determinant of the induced spatial part of the foliated metric on the membrane. { A, B } = √ W (cid:15) ab ∂ a A∂ b B ; ( a, b = σ, ρ ) is the symplectic bracket. The canonical momentum associated tothe scalar fields are P m y P r . There exists a new dynamical degree of freedom associated to aone-form dA that transforms as a symplectic connection under symplectomorphisms. Associatedto it, there exists a symplectic derivative and symplectic curvature defined by D r X m = D r X m + { A r , X m } , F rs = D r A s − D s A r + { A r , A s } , (4) respectively. There, D r is a symplectic connection, with D r • = 2 πm ur θ uv R r (cid:15) ˜ r ˜ s √ W ∂ ˜ r (cid:98) X v ∂ ˜ s • , r index is fixed and θ uv ( u, v = 9 ,
10) is a matrix that has relation with the monodromy of theheory when formulated on a torus bundle [15]. The constraint of the theory associated to theAPD is: D r (cid:16) P r √ W (cid:17) + (cid:26) X m , P m √ W (cid:27) = 0 . (5)From (3) we derive the equations of motion for the dynamical fields X m ( τ, σ, ρ ) and A r ( τ, σ, ρ )¨ X m ( σ, ρ, τ ) = − {{ X n , X m } , X n } − {D r X m , X r } , (6)¨ A r ( σ, ρ, τ ) = − { X n , D r X m } − { X s , F rs } . (7)In order to find the admissible M2-brane solutions, the system of equations (2), (5), (6) and (7)must be solved.
3. BRR-like rotating membrane solutions from the MIM2
In [1], the authors obtained a set of rotating membrane solutions when a membrane iscompactified on M × T . They found rotating solutions associated to the following ansatz X = κτZ a ( τ, σ, ρ ) = X a − + iX a = r a e iβ a ( τ,σ,ρ ) , a = 1 , . . . , X r ( τ, σ, ρ ) = R r ( n r σ + m r ρ ) + q r τ, r = 9 . , where r a ( σ, ρ, τ ) but we will assume to be constant, β a ( τ, σ, ρ ) = ω a τ + k a σ + l a ρ, with ω arotation frequency, k a , l a integers associated to the Fourier modes and q r integers parametrizingthe KK modes.In the following, we will show that the mass operator of the MIM2 contains the BRR massoperator, and hence, contains the set of BRR solutions. In [15] the mass operator of MIM2 wasfound. It has contributions from the central charge, from Kaluza-Klein (KK) momentum andfrom the Hamiltonian: M = T [(2 πR ) n ( Im ˜ τ )] + (cid:18) m | q ˜ τ − p | R [ I m ˜ τ ] (cid:19) + T / H MIM . (9)In order to reproduce the BRR mass operator M BRR from the MIM2 theory in the LCG, weassume the following ansatz: A r ( τ, σ, ρ ) = constant, X ( τ, σ, ρ ) = constant ,Z a ( τ, σ, ρ ) = r a e iβ a ( τ,σ,ρ ) , with a = 1 , , X r ( τ, σ, ρ ) = R r ( n r σ + m r ρ ) + q r τ + A r ( τ, σ, ρ ) , r = 9 , . (10)The KK momentum contribution can be rewritten as (cid:18) m | q ˜ τ − p | R [ I m ˜ τ ] (cid:19) = ˜ n R + ˜ n R = ( P kk ) + ( P kk ) , ˜ n = mp, ˜ n = mq , (11)with ˜ τ = ˜ τ + i ˜ τ the Teichmuller parameter of the T being ˜ τ = 0 and R ≡ R , R ≡ R Im ˜ τ . In the other hand, the central charge contribution, can be written T [(2 πR ) n ( Im ˜ τ )] = T π R R ( n m − m m ) , (12)where we have used that the central charge corresponds to the determinant of the windingnumbers, and the membrane tensions have been identified.inally, the angular momentum defined in [1] was identified as J a = (cid:90) dσρ δSδ ˙ β a = 4 π T r a ω a , (13)for the ansatz (8). By considering the ansatz (10) and using the equation of motion associatedto the Z a complexified embedding maps, to obtain the value for the frequency ω c , it can beexpressed as ω c − (cid:88) a =1 r a ( k a l c − l a k c ) − (cid:88) r =9 , R r ( n r l c − m r k c ) = 0 . (14)It is then possible to reproduce the mass operator associated to the energy of the system obtainedin [1], E = 2(4 π T ) / J a ω a + (4 π T ) R R ( n m − n m ) + ˜ n R + ˜ n R . (15)Since the MIM2 is formulated in the LCG, one plane less is observed. Due to the fact that,for the ansatz considered, both of the mass operators coincide and their associated equationsof motion are also reproduced. One can realize that to reproduce [1] results we have frozen thedynamical degree of freedom associated to the gauge symplectic connection, this also restrictsthe APD constraint value to the one used by [1]. Hence those rotating solutions that also satisfythe central charge condition are also solutions of the MIM2, whenever they satisfy (2).
4. New rotating membrane solutions
In the following we will consider new rotating solutions to the MIM2 theory that includes thepresence of a dynamical symplectic gauge connection. Due to the complexity of the equations,we consider the ansatz (10) but now allowing the simplest non trivial dependence of the gaugefield, A r ( σ, τ ). The equations of motion become reduced to: − ω c + (cid:88) a =1 r a ( l c k a − k c l a ) + (cid:88) r =9 , R r ( m r k c − n r l c ) + l c (cid:88) r =9 , ( ∂ σ A r ) − l c (cid:88) r =9 . R r ( m r k c − n r l c ) ∂ σ A r + i l c (cid:88) r =9 , R r m r ( ∂ σ ∂ σ A r ) = 0 , (16)for the case of Z c with c = 1 , ,
3, and¨ A s = − (cid:88) r =9 , ( R r m r ) ( ∂ σ ∂ σ A s ) + (cid:88) r =9 , ( R r m r ) ( R s m s ) ( ∂ σ ∂ σ A r ) − (cid:88) a =1 r a l a ( ∂ σ ∂ σ A s ) , (17)for the case of A s with s = 9 ,
10. The APD constrain acquires this simple expression (cid:80) r =9 , (cid:16) ∂ σ ˙ A r ∂ ρ ˆ X r (cid:17) = 0 and the central charge condition must also be satisfied (2). Wepropose the following ansatz for the gauge field A ( σ, τ ) = S ( σ ) + T ( τ ). Then, from (16) theequation becomes C + C S (cid:48) + C S (cid:48) = C S (cid:48) − C S (cid:48) ≡ ± λ , (18)where S (cid:48) ≡ ∂ σ S , and C = − ω + R ( m k − n l ) + R ( m k − n l ) , C = 2 (cid:96)R ( m k − n (cid:96) ) ,C = 2 (cid:96)R ( m k − n (cid:96) ) , C = (cid:96) . (19)Then, depending on the value of λ , there are two possibilities: igure 1. A r ( σ, τ ) with s r = 1, and τ and B r fixed. • If λ (cid:54) = 0, the solution is S ( σ ) = 1 l (cid:18) R ( m k − n (cid:96) ) ± (cid:113) R ( m k − n l ) + λ (cid:19) σ + B . (20) S ( σ ) = 1 l (cid:18) R ( m k − n (cid:96) ) ± (cid:113) ω + λ − R ( m k − n l ) (cid:19) σ + B . (21)Such that, when λ < < | λ | ≤ ω . • If λ = 0 there is solution if ω (cid:62) R ( m k − n l ) S ( σ ) = 0 or S ( σ ) = 2 l R ( m k − n l ) σ + B , (22) S ( σ ) = 1 l (cid:18) R ( m k − n l ) ± (cid:113) ω − R ( m a k − n (cid:96) ) (cid:19) σ + B . (23)Next, we introduce the solution S r ( σ ) = s r σ + B r with s r , a function of ( m, n, l, k, R, ω ), inthe equation of motion (17), this implies¨ T r ( τ ) = 0 , ⇒ T r ( τ ) = a r τ + b r . (24)Since the gauge field A r ( τ, σ ) is single-valued, then we must redefine S ( σ ) as a periodic functionto become well-defined. See figure 1. With these results, we can write the solution for the gaugefield as A r = s r σ + a r τ + B r − i (2 πs r ) , with σ ∈ [2 πi, π ( i + 1) ) i ∈ Z , (25)This solution is a string-like configuration and hence one can ask whether it introduces classicalinstabilities. In [16] it was shown that the MIM2 does not contain classically any flat directionat zero cost energy. Indeed it can be explicitly checked that ( F rs ) (cid:54) = 0. Hence, the completesolution for Z a and A r represents new admissible rotating solutions including all the fields ofthe bosonic sector of the MIM2 theory.We have also analyzed other possibilities for the ansatz of A : For example if A r ( σ, τ ) = a r στ , a r = constant , admits a solution but with the central charge equal to zero n = 0. For the ansatz A r ( σ, τ ) = a r f r ( σ ) τ , the equations of motion only allow the trivial solution f r ( σ ) = 0. We find(25) as the only admissible string-like solution for A r ( σ, τ ). . Conclusions We have characterized for first time solutions of the classical equations of the bosonic sectorof the supermembrane theory with central charges (MIM2), a sector of the M-theory withgood quantum properties. We find that they admit rotating membrane solutions. The MIM2classically does not posses string-like configurations and it is stable at quantum level. Hence thedynamics of the solutions describe an stable extended object. The solutions of the equations ofmotion are highly constraint because of the APD diffeomorphims and the topological conditionassociated to the presence of a monopole contribution. In spite of this, we obtain that for theparticular ansatz considered in [1], when it is formulated in the LCG, it is contained in the massoperator of the MIM2. Hence the rotating membrane solutions found in [1] that also satisfythe central charge condition, are all MIM2 solutions. Furthermore we also find new rotatingsolutions that include the presence of a gauge symplectic field defined on the worldvolume. Inthis work we discuss the simplest configuration that corresponds A r ( σ, τ ). We obtain admissiblestring like configurations that are linear in the spatial and time variables and satisfy all of theequations of motion and constraints. It seems that stability arguments of the solutions andthe central charge condition avoids that more general solutions like A ( σ, τ ) = αf ( σ ) g ( τ ) to beallowed. The gauge field A represents an admissible field since it implies the existence of anon-vanishing contribution to the M2-brane with central charges mass operator with F (cid:54) = 0.This work is part of larger study in which we also intend to characterize the existence of solitonicsolutions on the supermembrane with fluxes [17].
6. Acknowledgements
The authors are very grateful to A. Restuccia for helpful discussions. R. P. y J.M.P. thank tothe projects ANT1956 y ANT1955 of the U. Antofagasta. All the authors want to thank toSEM18-02 project of the U. Antofagasta. MPGM, R. Prado thank to the international ICTPproject NT08 for kind support.
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