Strings in pp-wave background and background B-field from membrane and its symplectic quantization
aa r X i v : . [ h e p - t h ] N ov Strings in pp-wave background and background B-field frommembrane and its symplectic quantization
Sunandan Gangopadhyay ∗ S. N. Bose National Centre for Basic Sciences,JD Block, Sector III, Salt Lake, Kolkata-700098, India
November 6, 2018
Abstract
The symplectic quantization technique is applied to open free membrane and strings in pp-wave background and background gauge field obtained by compactifying the open membranein the presence of a background anti-symmetric 3–form field. In both cases, first the Poissonbrackets among the Fourier modes are obtained and then the Poisson brackets among themembrane(string) coordinates are computed. The full noncommutative phase-space struc-ture is reproduced in case of strings in pp-wave background and background gauge field.We feel that this method of obtaining the Poisson algebra is more elegant than previousapproaches discussed in the literature.
Keywords:
Membranes, pp-wave, Noncommutativity
PACS:
It has been discovered recently that there exists a new maximal supersymmetric IIB supergravitybackground, namely pp-wave Ramond-Ramond (RR) background [1]. It consists of a plane-wavemetric supported by a homogeneous RR 5–form flux ds = 2 dX + dX − − µ X I X I ( dX + ) + dX I dX I , I = 1 , ..., F +1234 = F +5678 = 2 µ . (1)The background has 32 symmetries and is related (by a special limit [2]) to the AdS × S background [1, 3, 4]. Remarkably, string theory in this background is exactly solvable [5].Solvability in this context means that it is possible to find exact solutions of the classical stringequations of motion, perform a canonical quantization and then determine the Hamiltonianoperator.On the other hand, the theory of membranes has also been studied extensively over the lastdecade [6]. In a recent paper [7], it has been shown that one can obtain the action of an infinitenumber of massive strings in the pp-wave background by compactifying the bosonic membraneaction. Some properties of closed and open strings in this background has also been investigatedin this paper.Noncommutativity is another area which has attracted a lot of attention in the past few years[8] owing to the inspiration of superstring theories. It is a well known result now that open ∗ [email protected] An open membrane is a two dimensional object which sweeps out a three dimensional world-volume parametrized by τ , σ and σ . These parameters can be collectively referred to as ξ i ,( i = 0 , , S = Z dτ L = − πα ′ Z d ξ (cid:16) g ab η µν ∂ a X µ ∂ b X µ − (cid:17) ; ( a, b = 0 , ,
2) (2)where, L = − πα ′ Z π Z π dσ dσ (cid:16) g ab η µν ∂ a X µ ∂ b X µ − (cid:17) (3)is the Lagrangian and g ab = diag ( − , + , +), η µν = diag ( − , + , ..., +). The final term ( −
1) standsfor the cosmological constant and does not appear in the string theory action.The variation of (2) gives the equation of motion( ∂ − ∂ − ∂ ) X µ ( τ, σ , σ ) = 0 (4)and two types of BC(s). They are the Dirichlet BC(s) δX µ ( τ, σ , σ ) | σ =0 ,π = 0 δX µ ( τ, σ , σ ) | σ =0 ,π = 0 (5)and the Neumann BC(s) ∂ X µ ( τ, σ , σ ) | σ =0 ,π = 0 ∂ X µ ( τ, σ , σ ) | σ =0 ,π = 0 . (6)2he canonically conjugate momenta Π µ to X µ reads (we set 2 πα ′ = 1 for convenience and weshall recover it whenever neccessary):Π µ ( τ, σ , σ ) = δS P δ ( ∂ X µ ( τ, σ , σ )) = η µν ∂ X ν . (7)In order to quantize consistently, we need well-defined PB(s) among the canonical variables X µ ( τ, σ , σ ) and Π µ ( τ, σ , σ ) which read: { X µ ( τ, σ , σ ) , Π ν ( τ, σ ′ , σ ′ ) } = δ µν δ ( σ − σ ′ ) δ ( σ − σ ′ ) (8) { X µ ( τ, σ , σ ) , X ν ( τ, σ ′ , σ ′ ) } = { Π µ ( τ, σ , σ ) , Π ν ( τ, σ ′ , σ ′ ) } = 0 . (9)The Hamiltonian is obtained by means of Legendre transformation H = Z π Z π dσ dσ Π µ ∂ X µ − L = 12 Z π Z π dσ dσ [ η µν (Π µ Π ν + ∂ X µ ∂ X ν + ∂ X µ ∂ X ν ) −
1] (10)and the time-evolution of X µ ( τ, σ , σ ), Π µ ( τ, σ , σ ) is governed by ∂ X µ ( τ, σ , σ ) = { X µ ( τ, σ , σ ) , H } (11) ∂ Π µ ( τ, σ , σ ) = { Π µ ( τ, σ , σ ) , H } . (12)However, at the open membrane end points, the PB(s) (8) are not compatible with the BC(s)(5, 6). This implies that the basic PB(s) must be modified in order to make them consistentwith the BC(s). In the rest of the paper, we shall work with the Neumann BC(s) (6).The solution to the equations of motion (4) compatible with the BC(s) (6) read X µ ( τ, σ , σ ) = x µ + p µ τ + i ∞ X n =1 √ n (cid:16) α µn e inτ − α µ † n e − inτ (cid:17) cos( nσ )+ i ∞ X m =1 √ m (cid:16) α µ m e imτ − α µ † m e − imτ (cid:17) cos( mσ )+ i ∞ X n,m =1 ( n + m ) − / (cid:16) α µnm e i √ n + m τ − α µ † nm e − i √ n + m τ (cid:17) cos( nσ ) cos( mσ )= x µ + p µ τ + i ∞ X n =1 √ n (cid:16) α µn ( τ ) − α µ † n ( τ ) (cid:17) cos( nσ )+ i ∞ X m =1 √ m (cid:16) α µ m ( τ ) − α µ † m ( τ ) (cid:17) cos( mσ )+ i ∞ X n,m =1 ( n + m ) − / (cid:16) α µnm ( τ ) − α µ † nm ( τ ) (cid:17) cos( nσ ) cos( mσ ) (13)where we have defined α µn ( τ ) = α µn e inτ , α µ † n ( τ ) = α µ † n e − inτ , and so on.The canonically conjugate momenta (7) expressed in terms of the Fourier components read3 µ ( τ, σ , σ ) = η µν " p ν − ∞ X n =1 √ n (cid:16) α νn ( τ ) + α ν † n ( τ ) (cid:17) cos( nσ ) − ∞ X m =1 √ m (cid:16) α ν m ( τ ) + α ν † m ( τ ) (cid:17) cos( mσ ) − ∞ X n,m =1 ( n + m ) / (cid:16) α νnm ( τ ) + α ν † nm ( τ ) (cid:17) cos( nσ ) cos( mσ ) . (14)We shall now use the FJ method [24] to obtain the PB(s) between the Fourier components. Theidea is to write a Lagrangian in the first-order form L = a n ( ξ ) ∂ ξ n − H (15)where ξ n stand for all the canonical variables and a n ( ξ ) can be read directly from the inverse ofthe matrix f mn = ∂a n ( ξ ) ∂ξ m − ∂a m ( ξ ) ∂ξ n (16)provided the inverse of f mn exists. The first order form of the Lagrangian (3) reads: L = Z π Z π dσ dσ Π µ ∂ X µ − H . (17)Substituting (13) and (14) in the above equation yields L = π η µν " p µ p ν − i ∞ X n =1 (cid:16) α νn ( τ ) + α ν † n ( τ ) (cid:17) (cid:16) ˙ α µn ( τ ) − ˙ α µ † n ( τ ) (cid:17) − i ∞ X m =1 (cid:16) α ν m ( τ ) + α ν † m ( τ ) (cid:17) (cid:16) ˙ α µ m ( τ ) − ˙ α µ † m ( τ ) (cid:17) − i ∞ X n,m =1 (cid:16) α νnm ( τ ) + α ν † nm ( τ ) (cid:17) (cid:16) ˙ α µnm ( τ ) − ˙ α µ † nm ( τ ) (cid:17) − H (18)where, H = π η µν " p µ p ν + ∞ X n =1 n (cid:16) α µ † n α νn + α ν † n α µn (cid:17) + ∞ X m =1 m (cid:16) α µ † m α ν m + α ν † m α µ m (cid:17) + 12 ∞ X n,m =1 ( n + m ) / (cid:16) α µ † nm α νnm + α ν † nm α µnm (cid:17) − π τ .It is now easy to read from the above first order form of the Lagrangian (18) three sets ofvariables ξ n = ( α µn ( τ ) , α µ † n ( τ )), ξ m = ( α µ m ( τ ) , α µ † m ( τ )), ξ nm = ( α µnm ( τ ) , α µ † nm ( τ )), and their4orresponding one forms a n ( ξ ) = − iπ η µν [( α νn ( τ ) + α ν † n ( τ )) , − ( α νn ( τ ) + α ν † n ( τ ))], a m ( ξ ) = − iπ η µν [( α ν m ( τ )+ α ν † m ( τ )) , − ( α ν m ( τ )+ α ν † m ( τ ))], and a nm ( ξ ) = − iπ η µν [( α νnm ( τ )+ α ν † nm ( τ )) , − ( α νnm ( τ )+ α ν † nm ( τ ))]. The matrix f for these three sets of variables can now be computed using (16) andthe result is f = (cid:18) B − B (cid:19) (20)in which 0 is a null matrix and B is a diagonal matrix B µνnn ′ = iπ η µν δ nn ′ ; ( n, n ′ = 1 , , ... ) f or α µn modes (21) B µνmm ′ = iπ η µν δ mm ′ ; ( m, m ′ = 1 , , ... ) f or α µ m modes (22) B µνnn ′ mm ′ = iπ η µν δ nn ′ δ mm ′ ; ( n, n ′ , m, m ′ = 1 , , ... ) f or α µnm modes. (23)The inverse of the matrix f can be easily obtained and reads f − = (cid:18) − B B (cid:19) . (24)Hence, according to FJ method, the non-trivial PB(s) are given by { α µn ( τ ) , α ν † n ′ ( τ ) } = iπ η µν δ nn ′ { α µ m ( τ ) , α ν † m ′ ( τ ) } = iπ η µν δ mm ′ { α µnm ( τ ) , α ν † n ′ m ′ ( τ ) } = 2 iπ η µν δ nn ′ δ mm ′ (25)which further reduces to { α µn , α ν † n ′ } = iπ η µν δ nn ′ { α µ m , α ν † m ′ } = iπ η µν δ mm ′ { α µnm , α ν † n ′ m ′ } = 2 iπ η µν δ nn ′ δ mm ′ . (26)Now substituting (13) and (19) in (11) leads to the PB(s) among the zero modes: { x µ , p ν } = 1 π η µν . (27)With the above results in hand, the PB(s) among the canonical variables X µ ( τ, σ , σ ) andΠ µ ( τ, σ , σ ) read: { X µ ( τ, σ , σ ) , X ν ( τ, σ ′ , σ ′ ) } = { Π µ ( τ, σ , σ ) , Π ν ( τ, σ ′ , σ ′ ) } = 0 (28) { X µ ( τ, σ , σ ) , Π ν ( τ, σ ′ , σ ′ ) } = δ µν ∆ + ( σ , σ ′ ) ∆ + ( σ , σ ′ ) (29)where, ∆ + ( σ, σ ′ ) = 1 π X n =0 cos( nσ ) cos( nσ ′ ) (30)satisfies the usual properties of the delta function in the interval [0 , π ] [16].Note that the above symplectic structure is consistent with the Neumann BC(s) (6).5 Open membrane in the constant three-form field background
The Polyakov action of a membrane in the presence of a background anti-symmetric three-formfield A µνρ reads: S = − πα ′ Z d ξ (cid:20)(cid:16) g ab η µν ∂ a X µ ∂ b X µ − (cid:17) + 13 ǫ abc A µνρ ∂ a X µ ∂ b X ν ∂ c X ρ (cid:21) . (31)where ǫ abc is anti-symmetric in all the indices.The variation of the above action leads to the equations of motion (4) and the BC(s)( ∂ X µ − A µνρ ∂ X ν ∂ X ρ ) | σ =0 ,π = 0 (32)( ∂ X µ + A µνρ ∂ X ν ∂ X ρ ) | σ =0 ,π = 0 . (33)The canonically conjugate momenta Π µ to X µ is given by:Π µ ( τ, σ , σ ) = η µν ∂ X ν − A µνρ ∂ X ν ∂ X ρ . (34)Using Π µ , the BC(s) (32), (33) can be expressed in terms of the phase-space variables as: h(cid:16) η µκ − A µνρ A ν κβ ∂ X β ∂ X ρ (cid:17) ∂ X κ − A µνρ Π ν ∂ X ρ i | σ =0 ,π = 0 (35) h(cid:16) η µβ + A µνρ A νκβ ∂ X κ ∂ X ρ (cid:17) ∂ X β + A µνρ Π ν ∂ X ρ i | σ =0 ,π = 0 . (36)The above form of the BC(s) indicates that it is problematic to find exact solutions to theequations of motion (4). So we study the low energy limit where the membrane goes to stringtheory in the limit of small radius for the cylindrical membrane.To do this, we take the σ – direction of the membrane to be wrapped around a circle with radiusR. We choose further the gauge fixing condition [7, 18] X = σ ; (0 ≤ σ ≤ πR ) . (37)Now substituting the Fourier expansion of the world-volume fields X µ ( τ, σ , σ ) ( µ = 2): X µ ( τ, σ , σ ) = + ∞ X n = −∞ X µn ( τ, σ ) e inσ /R ; X µ − n ( τ, σ ) = X µ † n ( τ, σ ) (38)in the action (31) and using (37), we obtain (recovering the 2 πα ′ factor): S = 12 ˜ α ′ + ∞ X n = −∞ Z dτ dσ (cid:16) ∂ X µn ( τ, σ ) ∂ X − nµ ( τ, σ ) − ∂ X µn ( τ, σ ) ∂ X − nµ ( τ, σ ) − m n X µn ( τ, σ ) X − nµ ( τ, σ ) − A µν ∂ X µn ( τ, σ ) ∂ X ν − n ( τ, σ ) (cid:17) + 12 ˜ α ′ X n = m =0 ,n = − m i ( n + m ) R A µνρ =2 ∂ X µn ( τ, σ ) ∂ X νm ( τ, σ ) X ρ =2 − ( n + m ) ( τ, σ ); m n = | n | R , ˜ α ′ = α ′ /R = 12 ˜ α ′ S + X n =0 S n + X n = m =0 ,n = − m i ( n + m ) R Z dτ dσ A µνρ =2 ∂ X µn ( τ, σ ) ∂ X νm ( τ, σ ) X ρ =2 − ( n + m ) ( τ, σ ) (39) Note that if the σ – direction of the membrane is wrapped around a circle of radius R, then the X -directionis also compact on the same circle. S = 12 ˜ α ′ Z dτ dσ h ˙ X µ ( τ, σ ) ˙ X µ ( τ, σ ) − ∂ X µ ( τ, σ ) ∂ X µ ( τ, σ ) − B µν ˙ X µ ( τ, σ ) ∂ X ν ( τ, σ ) i (40)is the usual low energy string theory action in the presence of background gauge field A µν = B µν and S n = 12 ˜ α ′ Z dτ dσ h ∂ X µn ( τ, σ ) ∂ X − nµ ( τ, σ ) − ∂ X µn ( τ, σ ) ∂ X − nµ ( τ, σ ) − m n X µn ( τ, σ ) X − nµ ( τ, σ ) − B µν ∂ X µn ( τ, σ ) ∂ X ν − n ( τ, σ ) i (41)is the action of massive strings in pp-wave background and background gauge field B µν [25].Clearly, from (39), (40) and (41) we observe that the last term contains modes which are higherin energy than the first two terms. Hence in the low energy limit, we only consider the firsttwo terms in the action (39). The symplectic quantization of the usual string theory action (40)leads to the well-known noncommutativity at the end points of the string [23]. In this paper,we shall carry out the symplectic quantization of massive strings in pp-wave background andbackground gauge field B µν .The variation of S n gives the equation of motion (cid:16) ∂ − ∂ + m n (cid:17) X µn ( τ, σ ) = 0 (42)and the BC(s) (cid:16) ∂ X µn ( τ, σ ) − B µν ∂ X νn ( τ, σ ) (cid:17) | σ =0 ,π = 0 . (43)The canonically conjugate momenta Π nµ ( τ, σ ) to X µn ( τ, σ ) reads:Π nµ ( τ, σ ) = η µν ∂ X ν − n ( τ, σ ) − B µν ∂ X ν − n ( τ, σ ) . (44)Using Π nµ ( τ, σ ), the above BC can be expressed in terms of phase-space variables as: h M µρ ∂ X ρn ( τ, σ ) − B µρ Π ρ − n ( τ, σ ) i | σ =0 ,π = 0 (45)where, M µρ = ( δ µρ − B µν B νρ ) .The solution to the equations of motion (42) compatible with the above BC(s) read [25] X µn ( τ, σ ) = X µ (0) n ( τ, σ ) + X µ (1) n ( τ, σ ) (46)where, X µ (0) n ( τ, σ ) = (cid:20) x µ n cos( ˜ m n τ ) + p µ n sin( ˜ m n τ )˜ m n (cid:21) cosh( ˜ m n Bσ ) − B µν [ − p ν n cos( ˜ m n τ ) + ˜ m n x ν n sin( ˜ m n τ )] sinh( ˜ m n Bσ )˜ m n B (47)is the “zero mode” part (i.e. the modes with the lowest frequency) and X µ (1) n ( τ, σ ) = X l =0 ω ln e − iω ln τ (cid:18) ia µln cos( lσ ) + ω ln l B µν a νln sin( lσ ) (cid:19) . (48) We consider the case in which the B –field takes the form B µν = (cid:18) B − B (cid:19) . B is the eigen-value of the matrix B µν and the frequencies are defined by: ω ln = sgn ( l ) q l + m n ; ˜ m n = m n √ B . (49)Using X µ † n ( τ, σ ) = X µ − n ( τ, σ ), we find that the Fourier modes of the massive strings satisfythe relations x µ † n = x µ , − n ; p µ † n = p µ , − n ; a µ † l,n = a µ − l, − n . (50)The canonically conjugate momenta Π (1) nµ ( τ, σ ) to X µ (1) n ( τ, σ ) expressed in terms of the Fouriermodes read:Π µ (1) n ( τ, σ ) = X l> M µρ h a ρl, − n e − iω ln τ + a ρ − l, − n e iω ln τ i cos( lσ )+ iB µρ X l> (cid:18) lω ln − ω ln l (cid:19) h a ρl, − n e − iω ln τ − a ρ − l, − n e iω ln τ i sin( lσ ) . (51)Now we apply the FJ procedure to obtain the algebra between the non-zero Fourier modes. Todo this, we once again write down the “non-zero mode” part of the Lagrangian in a first orderform L (1) n = Z π dσ Π (1) nµ ∂ X µ (1) n − H (1) n (52)which on substitution of the mode expansions (48) and (51) read L (1) n = iπ η µκ X l> (cid:20) ω ln Q µρ ( ln ) (cid:16) a ρl, − n e − iω ln τ + a ρ − l, − n e iω ln τ (cid:17) × ∂ (cid:16) a κl,n e − iω ln τ − a κ − l,n e iω ln τ (cid:17) + ... i − H (1) n (53)where, Q µρ ( ln ) = ( δ µρ − ω ln l B µν B νρ ) and the ‘...’ represent terms which do not play a role inthe determination of the symplectic structure. The explicit form of H (1) n is also not required forobtaining the PB(s) between the modes.As before, one can again read a set of variables ξ ln and the corresponding canonical one-form a nl ( ξ ). The matrix (16) once again reads the same as (20) with the diagonal matrix B being: B µνlnl ′ n ′ = − iπω ln sgn ( l ) Q µν ( ln ) δ n + n ′ , δ l,l ′ . (54)Now from the inverse of f which reads the same as (24), it is easy to read the PB(s) among the“non-zero” Fourier modes using the FJ technique. They are: { a µln ( τ ) , a νl ′ , − n ′ ( τ ) } = − iπ sgn ( l ) ω ln ( Q − ln ) ) µν δ n + n ′ , δ l,l ′ . (55)The PB(s) among the zero modes x µ n and p µ n can be determined from the evolution equations(11, 12) and read: { x µ n , p ν n ′ } = π ˜ m n ¯ Bπ tanh( π ˜ m n ¯ B ) ( M − ) µν δ n + n ′ , . (56)where M µν has been defined earlier. Now since the BC(s) are valid on the boundaries, it isnatural to demand that { X µn ( τ, σ ) , X νn ′ ( τ, σ ′ ) } = 0 ; f or σ , σ ′ ∈ (0 , π ) (57)8rom which we get: { x µ n , x ν n ′ } = − ( BM − ) µν δ n + n ′ , (58) { p µ n , p ν n ′ } = − ˜ m n ( BM − ) µν δ n + n ′ , . (59)With the above results (55, 56, 58, 59) in hand, and after some lengthy calculation, we obtainthe following PB(s): { X µn ( τ, σ ) , Π n ′ ν ( τ, σ ′ ) } = δ µν δ n + n ′ , ∆ + ( σ , σ ′ ) (60) { X µn ( τ, σ ) , X µn ′ ( τ, σ ′ ) } = δ n + n ′ , ( BM − ) µν × , σ = σ ′ = 0 − , σ = σ ′ = π , otherwise . (61) { Π nµ ( τ, σ ) , Π n ′ ν ( τ, σ ′ ) } = δ n + n ′ , m n B µν × +1 , σ = σ ′ = 0 − , σ = σ ′ = π , otherwise . (62)which agrees with [25]. Note that in the B → In this paper, we employ the Faddeev-Jackiw symplectic formalism to study the problem ofopen free membrane and strings in pp-wave background and background gauge field obtainedby compactification of interacting membrane. The starting point is the solutions to the classicalmembrane(string) equations of motion. It is then observed that one can find the PB(s) amongthe Fourier modes first, using which the PB(s) among the original variables can be obtained.This idea was first proposed in [9] where the authors used the time-independent symplectic form([26], [25]) to fix the PB(s) among the Fourier components. In this paper, we follow a slightlydifferent method (due to [23]) to obtain the symplectic structure among the Fourier modes. Thesolutions of the membrane(string) equations of motion are substituted into the Lagrangian andthen integration over the spatial variables is carried out to cast the Lagrangian in a first orderform involving the Fourier modes from which the PB(s) (among the Fourier modes) can be easilyread off. Finally, using this algebra we obtain a noncommutative phase-space structure. Ourresults agree with the previous work in the literature [25].
Acknowledgements
The author would like to thank the referee for useful comments.
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