Boundary States in IIA Plane-Wave Background

Transcription

1 hep-th/368 Boundary States in IIA Plane-Wave Background arxiv:hep-th/368v 7 Jun 3 Yeonjung Kim, a and Jaemo Park b a epartment of Physics, KAIST, Taejon 35-7, Korea b epartment of Physics, POSTECH, Pohang , Korea Abstract We work out boundary states for Type IIA string theory on a plane wave background. By directly utilizing the channel duality, the induced conditions from the open string boundary conditions are imposed on the boundary states. The resulting boundary states correctly reproduce the partition functions of the open string theory for pp and p p cases where p branes are half BPS brane if located at the origin of the plane wave background. geni@muon.kaist.ac.kr jaemo@physics.postech.ac.kr

2 Introduction Recently there have been great interests on the string theory on the plane wave background[]. Initially Metsaev worked out Type IIB string theory on the plane wave background in the lightcone gauge[, 3]. Subsequently the Type IIB string theory on the plane wave background attracted great interests in relation to the correspondence with N=4 Supersymmetric Yang-Mills theories[4]. In a series of paper[5, 6, 7, 8, 9], simple Type IIA string theory on the plane wave background have been studied. The background for the string theory is obtained by compactifying the -dimensional plane wave background on a circle and going to the small radius limit. The resulting string theory has many nice features. It admits light cone gauge where the string theory spectrum is that of the free massive theory as happens in Type IIB theory. Furthermore the worldsheet enjoys (4,4) worldsheet supersymmetry. The structure of supersymmetry is simpler in the sense that the supersymmetry commutes with the Hamiltonian so that all members of the same supermultiplet has the same mass. The various / BPS -brane states were analyzed in the lightcone gauge, which gives consistent results with the BPS branes in the matrix model. Subsequently, covariant analysis was carried out for the -brane spectrums, which agrees with the previous results[]. In this letter, we initiated the study of boundary states of Type IIA string theories. Partly we were motivated by the works in Type IIB side. In [], the boundary states for Type IIB theories were worked out and were shown to exhibit interesting channel duality between open string and closed string. In [], more general cases were studied. In their developments, the lightcone supersymmetries, kinematical as well as dynamical, have played a crucial role in the construction of the boundary states. However in the construction of the boundary states in the more general conformal field theories[5, 6], the supersymmetries are not important. And many of the theories considered do not have the spacetime supersymmetries. Rather, by directly utilizing the channel duality, one can first construct the boundary states, and then figure out the open string spectrums after the modular transformation and vice versa. In the simplest example of the flat space, starting from the open string boundary conditions, one can obtain the conditions to be imposed on the boundary states by simply interchanging the role of σ and τ of the worldsheet coordinates[3, 4]. Indeed Michishita worked out Type IIB boundary states along this line of idea and obtained the consistent results with the previous works whenever the comparison is available[8]. Also this approach was adopted in [9]. Here we adopt the same philosophy to obtain the boundary states in Type IIA string theory on the plane wave background. The resulting boundary states give the correct open string partition function after the modular transformation. In this letter we just work out the boundary states for pp and p p case where

3 p branes are half BPS at the origin of the plane wave. Obviously there are more general cases to be studied. The detailed explorations of the boundary states of the IIA theory will appear elsewhere[]. An interesting fact for these p branes is that they do not have the dynamical supersymmetries away from the origin of the plane wave background []. Similar phenomenon occurs in Type IIB cases as well [, ]. As we finish the draft, we are aware of the work[7] where Type IIA boundary states are constructed following the approach of [, ]. Free string theory and -branes in IIA plane-wave background We closely follow the notation of [7] to describe the Type IIA string theory on the plane wave background. The background is given by ds = dx + dx A(x I )(dx + ) + Σ 8 i=dx i dx i, (.) F +3 = µ, F +4 = µ (.) 3 and A(x I ) = Σ 4 µ i= 9 (Xi ) + Σ 8 µ i =5 ) (.3) 36 (Xi where we definex ± = (X ± X 9 ). Throughout this letter, we use the convention that unprimed coordinates denote,,3,4 directions while primed coordinates denote 5,6,7,8 directions. Also unprimed quantities are associated with,,3,4 directions and primed coordinates are related to 5,6,7,8 directions. For the worldsheet coordinates, we use ± = ( τ ± σ ). The worldsheet action for the closed string is given by S LC = d σ( 4π µ X + µ X + µx i µ X i + m 9 Σ4 i= Xi X i + m 36 Σ4 i = X Xi i Σ b=± ( iψ b + ψ b iψ b ψ b) + im 3 ψ +γ 4 ψ im 3 ψ γ 4 ψ +) (.4) where m = p + µ and γ i are 8 8 matrices satisfying {γ i, γ j } = δ ij. The sign of subscript ψ A ± denotes the eigenvalue of γ34 while the superscript A =, denotes the eigenvalue of γ 9. The theory of interest contains two supermultiplets (X i, ψ, ψ +) and (X i, ψ +, ψ ) of (4,4) worldsheet supersymmetry with the masses m and m respectively. The fermions of the 3 6 first supermultiplet has γ 349 eigenvalue of while those of the second has the eigenvalue of. The mode expansion for the bosonic coordinates is given by X i = i (a i ω e iω τ a i eiω τ ) + i {e iωnτ (a i n einσ + ã i n e inσ ) ωn e iωnτ (a i n e inσ + ã i n einσ )} (.5)

4 where we have ω n = ( m 3 ) + n with n and a i a i The commutation relation is given by ω p i i ω, (.6) x i ω p i + i ω. x i [a i n, aj m ] = δij δ nm, n, m (.7) [ã i n, ãj m ] = δij δ nm, n, m >. (.8) And the mode expansion for the primed coordinates is given by X i = i (a i e iω τ a i eiω τ ) + i {e iω n τ (a i ω n einσ + ã i n e inσ ) n ω e iω n τ (a i n e inσ + ã i n einσ )} (.9) where we have ω n = ( m 6 ) + n with n and a i a i The commutation relation is given by ω p i i ω x i, (.) ω p i ω + i x i. The bosonic part of the lightcone Hamiltonian H bc is [a i n, aj m ] = δi j δ nm, n, m (.) [ã i n, ã j m ] = δ i j δ nm, n, m >. (.) H bc = π dσ π p + ( P i P i + σx i σ X i + (m 3 ) X i X i + P i P i + σx i σ X i + (m 6 ) X i X i ) (.3) with P i = τ X i and P i = τ X i. In terms of the oscillators p + H bc = ω a i a i + ω a i a i + ω n (a i n a i n + ã i n ã i n) + ω n(a i n a i n + ã i n ã i n) + H bc (.4) 3

5 where H bc is the zero-point energy contribution. Throughout the letter, repeated indices i and i are assumed to be summed over,,3,4 and 5,6,7,8 respectively. The fermionic mode expansion is given by ψ = (χe iω τ + χ e iω τ ) + c n {e iωnτ ( ψ n e inσ i ω n n ω γ 4 ψ n e inσ ) +e iωnτ ( ψ n e inσ + i ω n n γ 4 ψ n ω einσ )} (.5) where c n = / + ( ωn n ω ) and with ψ + = (+iγ4 χe iω τ iγ 4 χ e iω τ ) + χ = ψ iγ 4 ψ, χ = ψ + iγ 4 ψ (.6) c n {e iωnτ (ψ n e inσ + i ω n n ω γ 4 ψn e inσ ) +e iωnτ (ψ n einσ i ω n n ω γ 4 ψ n e inσ )}. (.7) The anticommutation relations are {ψn a, ψb m } = δab δ nm, n, m > { ψ n, a b ψ m} = δ ab δ nm, n, m > {χ a, χ b } = δ ab. (.8) The mode expansion of the superpartners for X i is ψ + = (χ e iω τ + χ e iω τ ) + +e iω nτ ( ψ n e inσ i ω n n where c n = / + ( ω n n ) and ω ω c n{e iω nτ ( ψ ne inσ + i ω n n γ 4 ψ ω ne inσ ) γ 4 ψ n einσ )} (.9) with ψ = χ = ψ + iγ 4 ψ, χ = ψ iγ 4 ψ (.) ( iγ4 χ e iω τ + iγ 4 χ e iω τ ) + ω c n nτ (ψ {e iω n e inσ i ω n n γ 4 ω ψ n e inσ ) +e iω n τ (ψ n einσ + i ω n n γ 4 ψ n e inσ )}. (.) 4

6 The anticommutation relation is given by {ψ a n, ψ b m } = δ ab δ nm, n, m > (.) { ψ a n, ψ b m } = δab δ nm, n, m > {χ a, χ b } = δ ab. (.3) The fermionic contribution to the Hamiltonian is p + H fc = ω χ χ + ω χ χ + ω n (ψ n ψ n + ψ n ψ n ) + ω n (ψ n ψ n + ψ n ψ n ) + H fc.(.4) For the closed string, the zero point energy contribution is zero, i.e., H bc + H fc =. Now we discuss the Hamiltonian of the open strings and their mode expansions. First we deal with the bosonic part of the Hamitonian H b = π p + π dσ( P i P i + σx i σ X i + ω X i X i + P i P i + σx i σ X i + ω X i X i ) (.5) with P i = τ X i and P i = τ X i. For the Neumann boundary condition, X i N = i (a i N ω e iωτ a i N e iωτ ) + i (a i N n e iωnτ a i N n eiωnτ )(e inσ + e inσ ), (.6) ωn with the commutation relation being [a i N n, a j N m ] = δ nmδ i Nj N, i N, j N. (.7) The contribution to the Hamiltonian is p + H b = ω n a i N n a i N n + H bn. (.8) For the irichlet boundary conditions with x i (σ = ) = x i, x i (σ = π) = x i X i with the commutation relation = xi (e ωσ e ωσ ) x i (e ω(σ π) e ω(σ π) ) e πω e πω +i (a i n e iωnτ + a i n eiωnτ )(e inσ e inσ ) (.9) ωn [a i n, a j m ] = δ nmδ i j, i, j >. (.3) 5

7 The contribution to the Hamiltonian is p + H = ω (e πω + e πω )((x i ) + (x i ) 4x i x i 4π e πω e πω + For the x i ) ω n a i n a i n + H bo. (.3) directions, we have the similar expressions, but we should replace unprimed quantities by primed quantities, e.g. ω n by ω n. The total bosonic Hamiltonian is given by p + H bo = (ω n a i n a i n + ω n a i n a i n + ω n a i N n a i N n + ω n a i N n a i N n ) + ω (e πω + e πω )(( x ) + ( x ) 4x x ) 4π e πω e πω + ω (e πω + e πω )(( x ) + ( x ) 4x 4π e πω e πω For the fermion, we impose the boundary conditions x ) + H b. (.3) ψ ± σ=,π = ηωψ σ=,π. (.33) Here η = is pp case, which means that open string ends on a p brane at one end and ends on another p brane at the other end. And η = denotes p p case where p means an anti p brane. In this letter we only deal with pp and p p cases, but certainly more general possibilities exist, which will be explored elsewhere. Let us discuss η = + case first. In order for the p brane to have the supersymmetry at the origin of the plane wave background, Ω should satisfy γ 4 Ωγ 4 Ω = {Ω, γ 9 } = {Ω, γ 34 } =. (.34) The actual form of the Ω depends on the dimension of the world volume., 4, 6, 8 branes were shown to have the supersymmetry at the origin and the expressions of Ω are tabulated at [7]. However the detailed form of Ω is not important for the subsequent discussions. One interesting point is that depending on the position, the number of the supersymmetries to be preserved by the -brane is different. The -branes away from the origin have no dynamical supersymmetry as shown in the covariant analysis[]. From the conditions above, one obtain the conditions for the open string modes ψ n = Ωψ n, ψ n = Ωψ n, (.35) for all n. To obtain the open string mode expansion one should replace ψ n, ψ n by the above conditions in the closed string expressions (.5), (.7), (.9),(.). The fermionic 6

8 contribution to the Hamiltonian is p + H fo = (ω n ψ n ψ n + ω n ψ n ψ n ) +( i ω ψ γ 4 Ωψ i ω ψ γ 4 Ωψ ) + H f. (.36) For η =, the zero point energy contribution for p brane is given by H b + H f = ω n N + ω n N with n N +n N = p where n N is the number of Neumann directions among,,3,4 directions and n N is the number of Neumann directions among 5,6,7,8 directions. For the p p, we have η = and the fermion modes have half-integer modings in worldsheet coordinates σ and the energy eigenvalues are given by ω n ( m 3 ) + (n ), ω n ( m 6 ) + (n ) and p + H f = ( ω n ψ n ψ n + ω n ψ n ψ ) + H n f (.37) where {ψ a, ψ b } = δ n n nm ab and the similar relation holds for ψ s. The value of zero point energy H b + H f will be given shortly after introducing suitable quantities. Now one can evaluate the cylinder amplitude for the open string Z O = dt t dp + dp Tre πt( p + p +p + H). (.38) where H = H bo + H fo of (.3) and (.36). For the evaluation, it is convenient to define f (m) (q) = q m ( q m ) / f (m) (q) = q m ( + q m ) / f (m) 3 (q) = q m f (m) 4 (q) = q m ( + q ( q where q = e πt and m, m are defined as m = (π) m = (π) p= ( q n +m ) ( + q n +m ) (n ) +m ) (n ) +m ) (.39) e ps e π m s ( ) p p= The modular transformation property is proved to be[] e ps e π m s (.4) f (m) (q) = f ( ˆm) ( q), f (m) (q) = f ( ˆm) 4 ( q), f (m) 3 (q) = f ( ˆm) 3 ( q) (.4) 7

9 where ˆm = mt and q = e π t following []. The result is With this expression, the cylinder amplitude is easily evaluated Z O = dt t (sinhπtω ) n N (sinhπtω ) n N f ω A (q)4 f ω A (q)4 f ω (q) 4 f ω (q) 4 exp πt( ω (e πω + e πω )(( x ) + ( x ) ) 4x x 4π e πω e πω ) exp πt( ω (e πω + e πω )(( x ) + ( x ) ) 4x 4π e πω e πω x ) (.4) where A= for pp and A=4 for p p. The last two lines of (.4) are coming from the irichlet directions. Using the result of [], one can see that zero energy contribution for p p is n N ω +4 ω 4 ω + n N ω +4 ω 4 ω. After the modular transformation, with t = l, ˆω = ω t, we have Z C = dl(sinhπ ˆω ) n N (sinhπ ˆω ) f ˆω n N B ( q)4 ω f ˆ B ( q)4 f ˆω ( q)4 ω f ˆ ( q)4 exp ( ˆω (e l ωˆ π + e lωˆ π )(( x ) + ( x ) ) 4x x e lωˆ π e l ˆ ω π ) exp ( ˆω (e l ω ˆ π + e l ω ˆ π )(( x ) + ( x ) ) 4x x ) (.43) e l ω ˆ π e l ω ˆ π with q = e 4πl. This will be compared with the overlap of the boundary states in the next section. 3 Boundary state construction As explained in [], if we have the usual lightcone gauge in the open string we have to take the nonstandard lightcone gauge in the closed string channel where the role of X + and p + are reversed. In this case, X + H c = p + H bc + p + H fc (3.) where p + H bc and p + H fc are given by (.4) and (.4) respectively with replacing m by ˆm mt. This change of the mass parameter is due to the conformal transformation needed going from the open string channel to the closed string channel []. If we have a channel duality between closed strings and open strings we can obtain the condition for the boundary states from the boundary conditions for the open strings by interchanging the role of σ and τ coordinates of the world sheets. For the Neumann boundary conditions for a bosonic coordinate X, σ X = at σ =, the corresponding condition for the boundary states is τ X = at τ =. This gives the 8

10 condition whose solution is a n + ã n =, ã n + a n =, a + a = (3.) exp( n a nã n (a ) ). (3.3) Here is the vacuum which is annihilated by a n, ã n, a. Its CPT conjugate state is given by exp( n a n ã n a ) (3.4) For the irichlet conditions, we impose σ X = at τ = with x i = x i. We have whose solution is a n ã n =, ã n a n = (n ), a a = i ω x (3.5) exp( (a i ω x ) + a nã n ) (3.6) n For the evaluation of the overlap of the boundary conditions we need exp( (a ω + i x ) )e πlω a a exp( (a i ω x ) ) = ( e 4πlω ) / exp( ω x + x x x e πlω ) e 4πlω = exp( ω (x + x )) ( e 4πlω ) / exp( ω (e πlω + e πlω )(x + x ) 4x x e πlω e πlω ) (3.7) Thus the total bosonic boundary state is given by exp( exp( (ai i a i N n ãi N n (ai N ) a i N n ã i N n (ai N ) ) ω x i ) + exp( (ai ω i x i ) + a i n ã i n ) a i n ã i n ) (3.8) where the bosonic vacuum is annihilated by all lowering operators corresponding to the raising operators appearing in the expression. Now the boundary condition for the fermions at the open string channel is translated into the conditions at the closed string channel ψ ± σ= = ηψ σ= (3.9) ψ ± τ= = iηψ τ=. (3.) 9

11 From ψ = iηψ+ at τ =, we obtain χ = ηωγ 4 χ, χ = ηωγ 4 χ. (3.) which are equivalent if we use γ 4 Ωγ 4 Ω = Ωγ 4 Ωγ 4 =, which can be proved by using the fact that Ω and γ 4 are either commute or anticommute. For nonzero modes we have ψ n = iηωψ n, ψ n = iηω T ψ n (3.) which lead to the same boundary state. From the above conditions we obtain exp( ηχ Ωγ 4 χ + iη ψ n Ωψ n (3.3) where the fermionic vacuum state is annihilated by ψ n, ψ n and χ. From ψ+ = iηψ at τ = gives χ = ηωγ 4 χ, χ = ηωγ 4 χ. (3.4) and ψ n = iηωψ n, ψ n = iηωt ψ n. (3.5) The total fermionic boundary state is given by exp( ηχ Ωγ 4 χ ηχ Ωγ 4 χ + iη ψ n Ωψ n + iη ψ n Ωψ n (3.6) where the fermionic vacuum is annihilated by all of the corresponding lowering operators for the raising operators appeared in the expressions. Boundary states at other τ = τ are generated by the closed string Hamiltonian acting on boundary states at τ =, i.e, B, τ = τ = e ix+ H cτ B, τ =. Let B, η be the boundary states corresponding to the -branes located at x with η appearing in the definition of the fermion boundary states and B, η be an analogous states with different values of x and η. Let B, η be the CPT conjugate states for B, η. The evaluation of the overlap of the boundary state leads to Z C = = dl B, η e πlx+ H C B, η dl ( ηη e 4πlω ) ( ηη ) 4 ( ηη e 4πlωn e 4πlω n ) 4 ( e 4πlω ) ( ) e 4πlωn 4 ( n) e 4πlω 4 exp( ˆω (( x ) + ( x ) )) exp ( ˆω (e l ωˆ π + e lωˆ π )(( x ) + ( x ) ) 4x x ) e lωˆ π e l ˆω π exp( ˆω (( x ) + ( x ) )) exp ( ˆω (e l ω ˆ π + e l ω ˆ π )(( x ) + ( x ) ) 4x x ). e l ˆω π e l ω ˆ π

12 For the pp boundary condition we have ηη = and p p boundary condition we have ηη =. In the process of the calculation we crucially use the fact γ 4 Ωγ 4 Ω =. Since Ω commutes with γ 349, two supermultiplets can be separately dealt with. One can check that the above result is coincident with the open string results up to overall normalization factor which can be absorbed into the normalization factor of the boundary state. This provides a consistency check for the boundary state construction carried out here. Acknowledgments We greatly appreciate Y. Michishita for explaining his results on the Type IIB boundary states to us. The work of J. P. is supported by the Korea Research Foundation (KRF) Grant KRF--7-C. References [] M. Blau, J. Figueroa-O Farrill, C. Hull and G. Papadopoulos, Penrose limits and maximal supersymmetry, hep-th/8, Class. Quant. Grav. 9 () L87; M. Blau, J. Figueroa-O Farrill and G. Papadopoulos, Penrose limits, supergravity and brane dynamics, hep-th/. [] R. R. Metsaev, Type IIB Green Schwarz superstring in plane wave Ramond Ramond background, hep-th/44, Nucl. Phys. B65 () 7. [3] R. R. Metsaev and A. A. Tseytlin, Exactly solvable model of superstring in plane wave Ramond-Ramond background, hep-th/9. [4]. Berenstein, J. Maldacena and H. Nastase, Strings in flat space and pp waves from N=4 super Yang Mills, hep-th/, JHEP 4 () 3. [5] S. Hyun and H. Shin, N=(4,4) Type IIA String Theory on PP-Wave Background, JHEP () 7, hep-th/874. [6] K. Sugiyama and K. Yoshida, Type IIA String and Matrix String on PP-wave, Nucl. Phys. B644 () 8, hep-th/89. [7] S. Hyun and H. Shin, Solvable N=(4,4) Type IIA String Theory in Plane-Wave Background and -Branes, hep-th/58. [8] S. Hyun and H. Shin, Branes from Matrix Theory in PP-Wave Background, Phys. Lett. B543 () 5, hep-th/69.

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