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Waves, boosted branes and BPS states in M-theory

Certain type II string non-threshold BPS bound states are shown to be related to non-static backgrounds in 11-dimensional theory. The 11-d counterpart of the bound state of NS-NS and R-R type IIB strings wound around a circle is a pure gravitational wave propagating along a generic cycle of 2-torus. The extremal (q_1,q_2) string with non-vanishing momentum along the circle (or infinitely boosted black string) corresponds in D=11 to a 2-brane wrapped around 2-torus with momentum flow along the (q_1,q_2) cycle. Applying duality transformations to the string-string solution we find type IIA background representing a bound state of 2-brane and 0-brane. Its lift to 11 dimensions is simply a 2-brane finitely boosted in transverse direction. This 11-d solution interpolates between a static 2-brane (zero boost) and a gravitational wave in 11-th dimension (infinite boost). Similar interpretations are given for various bound states involving 5-branes. Relations between transversely boosted M-branes and 1/2 supersymmetric non-threshold bound states 2+0 and 5+0 complement relations between M-branes with momentum in longitudinal direction and 1/4 supersymmetric threshold bound states 1+0 and 4+0. In the second part of the paper we establish the correspondence between the BPS states of type IIB strings on a circle and oscillating states of a fundamental supermembrane wrapped around a 2-torus. We show that the (q_1,q_2) string spectrum is reproduced by the membrane BPS spectrum, determined using a certain limit. This supports the picture suggested by Schwarz.


Introduction
In view of recent suggestions that D = 11 supergravity may be a low-energy effective field theory of a more fundamental 'M-theory', it is important to clarify further how different p-branes and their BPS bound state configurations in D = 10 string theories can be understood from eleven-dimensional perspective (for recent reviews and refs. see [1,2,3]). At the level of classical configurations of the effective field theory, the threshold (i.e. zero binding energy) BPS bound states of p-branes (corresponding, in particular, to superpositions of D-branes [4,5,6] with p − p ′ = 4, 8 [7][8][9][10]) originate from combinations of intersecting 2-branes and 5-branes in 11 dimensions [11][12][13][14][15][16].
As for non-threshold BPS bound states with non-zero binding energy (typically, with M 1+2 = M 2 1 + M 2 2 ) which cannot be viewed as n weakly coupled p-branes at equilibrium (and are not described by n independent harmonic functions), their 11-dimensional origin was not much discussed in the past. The basic example of such D = 10 configuration is the (q 1 , q 2 ) bound state of the NS-NS and R-R strings in type IIB theory [17,18,1,4], from which various other similar bound states (corresponding, in particular, to superpositions of D-branes with p − p ′ = 2, 6 [7,9,19]) may be constructed by applying T and SL(2, Z) dualities.
It was suggested in [17,1] from the consideration of the nine-dimensional 0-brane spectra that the (q 1 , q 2 ) string-string type IIB bound states should be related to the states of 2-brane wrapped around a 2-torus in D = 11. One of the aims of the present paper is to extend and complete what was done in [17,1] at two different levels. At the 'macroscopic' classical-solution level, we shall explicitly identify the D = 11 background which has the (q 1 , q 2 ) string solution of [17] as T -dual of its dimensional reduction. We shall also determine the D = 11 counterparts of several type IIA non-threshold bound state backgrounds related to the string one by T and SL(2, Z) dualities. At the microscopic quantum-state level, we shall demonstrate that the mass spectrum of BPS states of (q 1 , q 2 ) string indeed matches the spectrum of the corresponding oscillating states of wrapped membrane not only for the zero-mode parts of the masses, as was already shown in [17,1], but also including the oscillator parts. Section 2 will be devoted to a discussion of classical type II D = 10 solutions corresponding to various non-threshold BPS bound states of branes and their D = 11 counterparts. We shall first determine (in Section 2.1) which D = 11 solution is related, by dimensional reduction and T -duality, to the (q 1 , q 2 ) string type IIB background constructed in [17] by applying SL(2, R) duality transformation to the fundamental (1, 0) string solution of [20]. In the general case of the (q 1 , q 2 ) string with momentum (which is a straightforward generalization of the solution of [17] and preserves 1/4 of maximal supersymmetry) the corresponding D = 11 solution is the extremal limit of a 2-brane [21] wrapped around a 2-torus and infinitely 'boosted' [12,22] along generic (q 1 , q 2 ) cycle of the torus. In agreement with the interpretation given in [17], the momentum (or the charge of the 'boost' harmonic function) of the (q 1 , q 2 ) string is the winding number (charge) of the 2-brane, while the winding number of the (q 1 , q 2 ) string is the momentum of the 2-brane boosted along the (q 1 , q 2 ) cycle. The counterpart of the zero-momentum string is thus a zero-charge limit of 2-brane boosted to the speed of light, or simply a gravitational wave along the cycle of 2-torus.
For arbitrary values of the charges, this 1/4 supersymmetric D = 11 solution can be thought of as a bound state of a 2-brane and a gravitational wave (or 2+ ր, where the direction of the arrow indicates a 'diagonal' direction of the momentum flow on 2torus). In analogy with the fundamental string case [20,23,24], it should represent, at macroscopic level, an excited BPS state of the membrane with the momentum being carried by oscillations propagating along the (q 1 , q 2 ) cycle of the torus. Its dimensional reduction is type IIA solution representing a bound state of a fundamental string and a 0-brane with a boost along the string, 1 + 0 →. The T -duality in the string direction relates this to our starting point, type IIB bound state of the wave, R-R string and fundamental string, In general, bound states of type IIB branes, constructed by applying SL(2, R) duality transformations, can be also obtained from the 11-dimensional theory by starting with an appropriate M-brane configurations and boosting them along a non-trivial (q 1 , q 2 ) cycle of the 2-torus. This is in agreement with the observation [17,25] that SL(2, Z) duality of type IIB theory has its origin in the toroidal SL(2, Z) in D = 11 (in the case of D = 11 backgrounds with at least two isometries which we shall be considering here this was originally understood in [26]). We shall illustrate this further in Section 2.2 on several examples, starting with the (q 1 , q 2 ) bound state of NS-NS and R-R 5-branes [17,4]. The 1/4 supersymmetric 1 NS + 1 R + ↑ and 5 NS + 5 R + ↑ configurations are the special cases of the non-threshold 1/8 supersymmetric configuration (1 + 5) NS + (1 + 5) R + ↑ which is an SL(2, Z) rotation of the threshold bound state of the NS-NS string and solitonic 5-brane [27] or of its SL(2, Z) dual R-R version [28]. Its D = 11 counterpart is shown to be the intersection of the 2-brane and 5-brane over a string [12] with a momentum flow along generic 'diagonal' direction of the 2-brane torus, i.e. 5⊥2+ ր.
In Section 2.3 we shall address the question of D = 11 interpretation of other nonthreshold type II BPS bound states related to the (q 1 , q 2 ) string by the duality transformations. We shall demonstrate that the type IIA bound state of a 2-brane and a 0-brane is just a dimensional reduction of the standard extremal D = 11 2-brane solution finitely (v < 1) boosted along the isometric 11-th dimension which is transverse to the brane, or 2 →. Analogous surprisingly simple interpretation (which is not obvious at the level of the type IIA backgrounds) applies also to the 5+0 bound state. The relations between boosted branes in D = 11 and 'p-brane + 0-brane' bound states in D = 10 can be summarized as follows: (a) 1/4 supersymmetric threshold bound states: 2+ ↑ −→ 1 + 0, 5+ ↑ −→ 4 + 0; (b) 1/2 supersymmetric non-threshold bound states: 2 → −→ 2 + 0, 5 → −→ 5 + 0.
We shall discuss two natural generalizations, 2⊥2 → and 5⊥2 →. In particular, 2⊥1 + 0 solution (T -dual to 1 NS + 1 R + ↑) is the reduction of the D = 11 configuration 2⊥2 → in which one of the two 2-branes which intersect over a point [11] is finitely boosted in a direction of the other 2-brane orthogonal to it. We shall also comment on the existence of several different 11-dimensional solutions which reduce to D = 10 solutions related by T and SL(2, Z) dualities.
In Section 3 we shall address the relation between the type IIB strings and D = 11 membrane at the 'microscopic' level, by explicitly identifying the quantum supermembrane states which correspond to the BPS spectrum of the bound states of type IIB strings. These excited membrane states have oscillations only along one (momentum) direction, in agreement with what is also implied by the macroscopic effective field theory picture. As was shown in [17,1], the zero-mode part of the mass of BPS states of type IIB (q 1 , q 2 ) string on a circle is in perfect agreement with the zero-mode part of the mass of the fundamental supermembrane states wrapped around a 2-torus with a momentum along the (q 1 , q 2 ) cycle of the torus. Our aim will be to check that the oscillator parts in the masses also agree. We shall argue that in order to determine the relevant BPS spectrum of the supermembrane is sufficient to calculate it in a certain region of the coupling parameters. Solving the resulting gaussian theory, we shall find that its oscillating membrane BPS states do, indeed, have the same masses as the BPS states of type IIB strings.

Boosted branes and non-threshold BPS bound states: classical solutions
Gravitational waves carrying linear momentum seem to play as important role in 11dimensional theory as 2-brane and 5-brane solutions. This may not be unexpected since plane fronted waves are known to preserve supersymmetry [29,30,31]; they are related, in D = 10 theory, to the fundamental string backgrounds by T -duality [32,33]; and a single plane fronted wave (Schwarzschild background boosted to the speed of light) is the D = 11 image of the 0-brane of D = 10 type IIA theory [29,34]. Nevertheless, it is a bit surprising (though, in fact, implicit in [17]) that the D = 11 solution that corresponds to a rather complicated-looking (q 1 , q 2 ) string background of type IIB theory turns out to be related to a dimensional reduction of the pure gravitational wave propagating along the (q 1 , q 2 ) cycle of 2-torus in M 11 = T 2 × M 9 which describes a zero-charge limit of 2-brane boosted to the speed of light. Wrapping a (non-zero charge) 2-brane around T 2 corresponds to adding a momentum (wave) to the type IIB (q 1 , q 2 ) string [17]. As in the case of the D = 10 fundamental string [24,23], here the wave may be represented by transverse oscillations of the 2-brane propagating along the cycle (i.e. in one circular direction) and carrying the corresponding momentum. This suggests that, as in the string-theory case, there should exist the corresponding BPS states of the quantum supermembrane. This issue will be addressed in Section 3, while below we shall discuss only the classical effective field theory solutions. In what follows we shall consider the 11-dimensional space M 11 = T 2 × M 9 with the isometric rectangular 2-torus coordinates (y 1 , y 2 ) having periods (2πR 1 , 2πR 2 ) (the case of more general T 2 is treated similarly). The gravitational wave propagating in y 1 direction (i = 1, ..., 8) is special as this pp-wave can be represented [29] as the extremal (infinite boost, zero mass) limit of the boosted Schwarzschild solution, or, more precisely, since y 1 , y 2 are isometries, of a black 2-brane of zero charge, The boost parameter β is related to the Kaluza-Klein electric charge Q, i.e. the momentum along the boost direction y 1 . In the extremal limit µ → 0, β → ∞, Q =fixed, the metric (2.2) takes the form (2.1) (Q = Q, K =K) with W = K − 1 = Q r 6 . The 1/2 supersymmetric wave can be superposed with 1/2 supersymmetric extremal 2-brane (this can be shown, e.g., by boosting the black 2-brane [35] and taking the extremal limit). The resulting 1/4 supersymmetric background is [12,22] More generally, W (r) in (2.4) can be replaced by any solution W (u, x) of the Laplace equation ∂ 2 x W = 0. In particular, as in the string case [24,23], choosing W = f i (u)x i corresponds to adding a wave of membrane oscillations propagating along y 1 (f i is related to the profile of oscillations). The macroscopic properties of such solution are the same as of the solution with W = Q r 6 with Q being proportional to the asymptotic value of momentum carried by the wave (∼ [ duf i (u)] 2 ).
It is important to stress that one is able to construct a supersymmetric BPS background by making an infinite boost in one isometric direction only (boosting in y 1 and y 2 directions with two independent parameters β a and taking the limit µ → 0, β a → ∞ does not give a new extremal solution). Similar observation applies in the general p-brane case and suggests that the BPS states of p-branes are always 'string-like', i.e. have oscillations in one direction only. This should also follow directly from the BPS requirement and supersymmetry algebra (cf. [36]).
Dimensionally reducing the '2-brane plus wave' (or 2+ ↑) solution (2.4) to D = 10 along the 'spectator' direction y 2 we get the 'fundamental string plus wave' (or 1+ ↑) background. Under the T -duality in y 1 , it transforms into the same type of solution of type IIB theory 1 NS + ↑ (with H 2 ↔ K = 1 + W , i.e. Q ↔ Q). If instead we reduce along the boost direction y 1 , we get the type IIA solution, which can be interpreted as a bound state of the fundamental string and the 0-brane, 1 + 0. The T -dual solution of type IIB theory is the boosted R-R string, or 1 R + ↑ . This suggests that reducing to D = 10 along some 'mixed' direction we should get a more general class of solutions which will interpolate between the two special cases 1+ ↑ and 1 + 0, or, when T -duality transformed into type IIB theory, between 1 NS + ↑ and 1 R + ↑. This is equivalent to putting the wave (i.e. momentum) along some generic cycle of the 2-torus.
Let us now show that the D = 11 counterpart of the original zero-momentum (q 1 , q 2 ) string solution of type IIB theory [17] is the limit of (2.7) in which the 2-brane charge is set equal to zero, 3 Q = 0, H 2 = 1, i.e. a zero-charge membrane boosted to the speed of light, or simply a gravitational wave travelling along a cycle of T 2 (in complete analogy with boosted string case [29,32].
Rewriting the resulting metric as one can easily read off the type IIA solution which follows upon the dimensional reduction along y 2 ≡ y 11 direction. The resulting 1/2 supersymmetric type IIA background describes just a 0-brane finitely boosted to a velocity v = cos θ ≤ 1 in the isometric direction y 1 (we shall denote this configuration as 0 →) Note that −t 2 +ỹ 2 1 = −t 2 +y 2 1 and also that the coefficient of the boosted 0-brane harmonic functionK depends on the boost parameter. When q 2 = 0 (infinite boost limit, sin θ → 0) we get just a plane gravitational wave in y 1 direction; when q 1 = 0 we get a static 0-brane in the space with one compact isometric direction, or 0 1 .
T -duality along y 1 converts the boosted 0-brane into the string-string bound state of [17]. Indeed, the R-R vector field originating from off-diagonal (11, µ) components of the metric has both time-like and internal spatial parts, which transform under T -duality into the ty 1 component of the R-R 2-tensor B (2) µν and the R-R scalar χ of type IIB theory. Using the T -duality transformation rules, or directly, the relation between a D = 11 background with two isometries and the corresponding type IIB background given in [26] we find exactly the type IIB (q 1 , q 2 ) string solution obtained in [17] by applying the SL(2, R) transformation to the fundamental string of [20] 4 The special cases are the NS-NS string (q 1 = 1, q 2 = 0, i.e. θ = 0,K = 1), and the R-R string (q 1 = 0, q 2 = 1, i.e. θ = π 2 ,K = K). This is the background in the case of the simplest 2-torus τ = i or to the trivial vacuum [17]. The total momentum of the wave is just the tension of the (q 1 , q 2 ) string. Generalization of the correspondence between (2.9) and (2.13) to the case of an arbitrary 2-torus is straightforward. In particular, in the case of the rectangular torus with different radii of y 1 and y 2 , i.e. τ 2 = R 1 /R 2 , one should change q 2 → R 1 R 2 q 2 in the above expressions and thus replace the (q 1 , q 2 ) string tension factor (in the string . All works in a similar way if we add back the 2-brane, i.e. switch on its charge, starting with (2.7) with H 2 = 1, and reduce down to D = 10 along y 2 . The D = 11 solution (2.7) then leads to a D = 10 type IIA background which can be interpreted as an effective field theory representation of a 1/4 supersymmetric non-threshold bound state of a fundamental string and a 0-brane boosted along the string direction, i.e. 1 + 0 →. T -duality along y 1 transforms it into the boosted version of the type IIB (q 1 , q 2 ) string which differs from (2.13) only by an extra wave term in the metric 14) The 2-brane charge Q (equal to c 0 where w 0 is an integer winding number) thus becomes the momentum of the (q 1 , q 2 ) string. As in the case of the fundamental NS-NS string [24,23], the momentum flow along (q 1 , q 2 ) string has also a microscopic representation in terms of right-moving string oscillations, i.e. there exists a similar solution which is supported by an oscillating string source and has the same asymptotic value of momentum.
This provides an explicit classical-solution realization of the observations in [17] which were based on the expected form of the D = 9 0-brane spectrum. 4 We shall always use string-frame metric in D = 10. Here y 1 is the coordinate dual to y 1 in (2.9).

Type IIB bound states of strings and 5-branes from 2⊥5 in D = 11
Analogous considerations apply in general to type IIB solutions obtained by making an SL(2, Z) transformation on a 'generating' background with one or more compact isometries. Since according to [26,17] the action of SL(2, Z) should correspond to a modular transformation of the compactification 2-torus in D = 11, such SL(2, Z) families of solutions should originate from the D = 11 counterpart of the generating solution that is boosted along generic cycle of the 2-torus. 5 Let us consider the SL(2, Z) bound states of NS-NS and R-R 5-branes in type IIB theory [17,4] and explain their 11-dimensional origin. To get an idea of what kind of D = 11 backgrounds we should expect to find, it is useful to consider the special cases, describing a type IIB solution, its possible T -dual images in type IIA theory, and D = 11 solutions from which the latter follow upon dimensional reduction: (1) solitonic NS-NS 5-brane of IIB theory with a momentum flow along one of its directions 5 NS + ↑ is T -dual to: (a) bound state of 5-brane and a wave 5+ ↑ in type IIA theory (if duality is performed along one of 4 'unboosted' directions), which is the type IIA image of boosted 11-dimensional 5-brane reduced along an isometric coordinate orthogonal to it, 6 (5+ ↑) 1 ; (b) bound state of 5-brane and fundamental string 5 + 1 (if duality is performed along the boosted direction), which is a reduction of the D = 11 5⊥2 solution describing an orthogonal intersection of 5-brane and 2-brane over a string [12].
Let us first ignore the momentum flow and consider the (q 1 , q 2 ) bound state 5 NS + 5 R obtained by applying the SL(2, R) rotation to 5 NS of [40,41]. Its T -dual can be interpreted as a 1/2 supersymmetric non-threshold bound state of the solitonic 5-brane and R-R 4brane in type IIA theory, i.e. 5 + 4. The image of the latter in D = 11 is 5 1 , i.e. the 5-brane in M 11 = S 1 y 2 × M 10 , reduced down to D = 10 along the mixture of the isometric direction y 2 transverse to the 5-brane, and a 5-brane direction y 1 (which will be orthogonal to the 4-brane within the type IIA 5-brane). These two directions form the 2-torus related to type IIB SL(2, Z) duality which is the analogue of T 2 in the previous section.
The D = 11 5 1 background can be viewed as the limiting case of the 5⊥2 solution when the charge of 2-brane is set equal to zero (with y 2 being the 2-brane direction orthogonal to 5-brane). The reduction is done along the (q 1 , q 2 ) cycle of the 2-torus around which the 2-brane is wrapped. When q 1 = 0, q 2 = 1 we get simply 4 1 (the reduction goes along a 5-brane direction y 1 common to 5 and 2). When q 1 = 1, q 2 = 0 we get the type IIA 5-brane (the reduction goes along the 2-brane direction y 2 transverse to the 5-brane).
Lifting the latter to D = 11 along y 11 ≡ y 2 and setting first the momentum to zero (i.e. V = 0) we find the following metric which reduces to (2.9) when there are no 5-branes H 5 = 1, H 1 = K. This can be re-written simply as the 'rotated' 5 1 -brane where y ′ 1 , y ′ 2 are the rotated coordinates (2.6). As expected, the same turns out to be true for non-vanishing momentum: the background one finds can be put into the form (cf. (2.7); y 2 ≡ y 11 ) This is just a 'rotated' version of 5⊥2+ ↑, i.e. of the intersection of the 2-brane and 5-brane boosted along the common string [12,13] where the momentum now flows along the (q 1 , q 2 ) cycle of the (y 1 , y 2 )-torus around which the 2-brane is wrapped. Special cases of the corresponding type IIA and type IIB solutions include (depending on direction of reduction): 2⊥5 −→ 2⊥4 in IIA, related by T -duality to (1 + 5) R in IIB; 2⊥5 −→ 1⊥4 in IIA, related by T -duality to 5 R + ↑ in IIB; 2⊥5 −→ 1 + 5 in IIA, related by T -duality to 5 NS + ↑ or (1 + 5) NS in IIB. Reduction down to D = 5 gives a family of regular extremal black hole solutions related to the simplest NS-NS [27] and R-R [28] ones by U-duality. Reduction along the four 5-brane directions gives a family of solitonic strings in D = 6.
We conclude that the 5⊥2+ ր background (2.18) provides a unified D = 11 description of various 1-brane and 5-brane bound state configurations of type IIB theory. 7

2 + 0 and 5 + 0 bound states as transversely boosted M-branes
The type IIA 1/4 supersymmetric threshold bound states of a string and 0-brane, 1 + 0, and of 4-brane and 0-brane, 4 + 0, have a simple origin in D = 11: the corresponding solutions are readily obtained by dimensional reduction from extremal M-brane configurations with momentum along the brane 2+ ↑ and 5+ ↑, or black M-branes infinitely boosted in one direction parallel to the brane [12,42]. 8 The existence of 1 NS + 1 R type IIB bound state suggests by T and SL(2, Z) duality the existence of other 1/2 supersymmetric non-threshold type IIA bound states that can be interpreted as 2 + 0, 5 + 0, etc. 9 In view of the above identification of the D = 11 counterpart of the string-string bound state background it is of interest (in particular, in connection with [44]) to enquire about the D = 11 origin of such bound states. As we shall demonstrate below, the corresponding classical solutions are indeed very simple: they 7 In addition to the special cases of (1 + 5) N S + (1 + 5) R + ↑ there are also 1/2 supersymmetric non-threshold bound state of the type 1 N S + 5 R , i.e. the fundamental string lying on R-R 5-brane (and its S-dual 1 R +5 N S ). Such solution can be constructed by T -duality in 4 transverse directions from (1 N S + 1 R ) 4 . Its T -dual (along the string direction) counterpart in IIA theory is 4-brane finitely boosted in transverse dimension, or 4 →. 8 For a discussion of alternative D = 11 embeddings of 1 + 0 and 4 + 0 and also the embedding of 8 + 0 see [16]. 9 The existence of related D-brane bound states was noted in [7]. For a discussion of 2 + 0 bound state in D-brane representation see [19]. The corresponding classical solutions cannot be obtained by formal T -duality transformations [43] of the standard R-R p-brane backgrounds.
are just a coordinate (Lorentz) transformation of the extremal M-brane background and describe an M-brane finitely boosted in the direction transverse to its internal space. 10 Dimensional reduction along this direction gives 2 + 0 and 5 + 0 type IIA solutions. These solutions interpolate between the standard static M-brane (zero boost) and plane wave (infinite boost). As will follow from the relation to the 1 NS + 1 R solution, the coefficient in the harmonic function of the transversely boosted brane depends on the boost parameter and goes to zero in the limit of the infinite boost when the D = 11 solution becomes just a plane wave (and the corresponding type IIA solution becomes a 0-brane). We shall start with the explicit construction of 2+0 solution by duality transformations from 1 NS + 1 R background of [17] and then lift the result to D = 11. Assuming that the harmonic function W in (2.13) does not depend on two of the transverse coordinates (to be denoted as y 2 , y 3 ), i.e. starting with (2.13) with two extra isometries, (1 NS + 1 R ) 2 , we may apply the T -duality in these isometric directions to obtain a type IIB solution which has an interpretation of a bound state of a fundamental string and a 3-brane, 1 NS + 3, where K andK have the same form as in (2.10), and we do not list other nonvanishing components of the 4-tensor determined by self-duality of its field strength. This background interpolates between the fundamental string 1 2 (q 1 = 1, q 2 = 0, i.e. θ = 0,K = 1) and the 3-brane (q 1 = 0, q 2 = 1, i.e. θ = π 2 , K =K). The simple SL(2, Z) transformation which interchanges the NS-NS and R-R strings and leaves 3-brane invariant then gives similar 1 R + 3 background with the same values of D 4 , χ and interchanged values of B (1) and B (2) . The final step is to apply T -duality in y 1 direction This background can be interpreted as (2 + 0) 1 since it interpolates between the 0-brane (q 2 = 0,K = 1) and the 2-brane wrapped around y 2 , y 3 (q 1 = 0, K =K) in the space with one extra isometry (y 1 ). As y 1 does not play any special role, the restriction of the extra isometry can be relaxed (i.e. y 1 can be added to x s ) obtaining the 2 + 0 background which is spherically symmetric in all 7 transverse coordinates. This type IIA solution can now be lifted to D = 11 (cf. (2.7),(2.9)) This can be re-written simply as This is just the 2-brane (2.4) with harmonic function H 2 =H 2 boosted to a subluminal velocity v = cos θ ≤ 1 in the isometric transverse direction y 11 (this background will be denoted as 2 →). This matches perfectly with the 2 +0 interpretation of the corresponding type IIA solution! The background (2.23) interpolates between the standard extremal 2-brane and a plane wave: for q 1 = 1, q 2 = 0 (i.e. infinite boost v = 1, sin θ = 0,H 2 =K = 1) we get (by carefully taking the limit sin θ → 0) the gravitational wave along y 11 direction which reduces to 0-brane in D = 10, while for q 1 = 0, q 2 = 1 (i.e. zero boost v = 0, sin θ = 1, H 2 =K = 1 + Q r 6 , Q ≡ Q) we find just the static 2 1 M-brane which reduces to type IIA 2-brane. For general v (2.23) represents a 2-brane with transverse momentum (in the y 11 direction). The energy is given by the usual relativistic expression with the first (second) term being zero in the infinite (zero) boost limit. This gives a 'kinematic' interpretation to the mass formula of a non-threshold BPS bound state. 11 Let us note also that there exists a more direct way of relating the static 2-brane in the D = 11 space with an isometry along y 11 (i.e. 2 1 ) and the type IIB string-string solution (2.13): one is to reduce down to D = 10 along the 'rotated' direction of the (y 3 , y 11 ) torus, i.e. to wrap the 2-brane around its generic cycle, obtaining the 2 + 1 non-threshold bound state of type IIA theory, and then to make T -duality along y 3 . The resulting IIB background is (1 NS + 1 R ) 1 , i.e. the string-string bound state in the space with an isometry along the y 3 direction orthogonal to the string direction y 2 .
The above discussion admits the following generalization. If we start with the boosted version (2.14) of the type IIB (q 1 , q 2 ) string and apply duality transformations we find the following sequence of 1/4 supersymmetric non-threshold bound states: The last one is a generalization of the 1/4 supersymmetric threshold configuration 2⊥1 of 2-brane orthogonally intersected by a fundamental string over a point where an extra 0-brane is now placed. Recalling that 2⊥1 is the dimensional reduction of 2⊥2 in D = 11 [11,12] it is clear that 2⊥1 + 0 should be a dimensional reduction of 2⊥2 →, where one 2-brane is finitely boosted in the transverse direction which is a longitudinal direction of the other 2-brane, i.e.
wheret,ỹ 11 andH 2(1) are defined in a similar way as above. y 2 , y 3 and y 1 , y 11 are the coordinates of the two 2-branes, and the first brane is boosted along y 11 direction of the second. In the infinite boost limit (sin θ = 0,H 2(1) = 1) this becomes 2+ ↑ which reduces down to 1 + 0 in D = 10. Another chain of duality transformations leads to the type IIA bound state of a solitonic 5-brane and a 0-brane. Its image in D = 11 is simply a 5-brane finitely boosted in the transverse isometric direction, i.e. 5 →. The corresponding background is the obvious analogue of (2.23). Starting with (1 NS + 1 R + ↑) 4 we arrive at 1/4 supersymmetric type IIA solution 5 + 1 + 0 describing the bound state of a 5-brane with a fundamental string and a 0-brane. This is a non-threshold bound state, except for the special cases 1 + 0 and 5 + 1 which are at threshold. Its D = 11 image is 2⊥5 →, i.e. the orthogonal intersection of 5-brane and 2-brane over a string with 5-brane finitely boosted along the direction of 2-brane orthogonal to it. In the limit of the infinite boost the 5-brane charge goes to zero and this configuration becomes just the longitudinally boosted 2-brane, 2+ ↑ (which reduces to 1 + 0). The corresponding solution is the direct analogue of the above expressions (cf. (2.18)) where the 2-brane coordinates are y 1 and y 11 .
There is one more example of this dichotomy: 2 + 0 is T -dual to 4 + 2 which in turn results upon dimensional reduction [39] from another D = 11 solution which interpolates between the 2-brane and 5-brane backgrounds. It can be interpreted as a 2-brane lying within a 5-brane [46] (see also [47,39]). In the notation used here it is given by (cf. (2.18),(2.22)) 12
Since for θ = 0 this background reduces to the 2-brane (K = 1, K = H 2 ) and for θ = π/2 to the 5-brane (K =K = H 5 ), and in view of the close analogy with similar solutions discussed above, it can be interpreted as describing their 1/2 supersymmetric non-threshold bound state 2 + 5. Reducing to D = 10 along a 5-brane coordinate one finds the 4 + 2 solution, while the reduction along a 2-brane coordinate gives a similar bound state of a 4-brane and a fundamental string 4 + 1. The latter, (4 + 1) 1 , is T -dual to a non-threshold type IIB bound state of a R-R 5-brane and a fundamental string, 5 R + 1 NS , which thus has 5 + 2 bound state as its D = 11 counterpart.
It remains an interesting question whether the above D = 11 solutions with boosts are somehow connected to this static solution in an intrinsically 11-dimensional way (which goes beyond the connection by string dualities of dimensionally reduced backgrounds with KK modes dropped out). This may give a hint about a relation between M-branes and waves, similar to the T -duality relation between fundamental strings and waves. 13 The D = 11 solutions discussed above have many generalizations. One may combine the threshold bound state intersecting solutions with longitudinal momentum along common string direction [11][12][13][14][15] with boosts of branes in transverse directions. One may also combine intersection solutions with static non-threshold 2 + 5 bound states of [46] (some of such solutions were discussed in [48]). 14

Correspondence between microscopic BPS states of type IIB strings and quantum supermembrane
It was suggested in [17,1] that the spectrum of BPS states of type IIB (q 1 , q 2 ) string on a circle should be in correspondence with the BPS spectrum of the fundamental supermembrane wrapped around a 2-torus with a momentum along the (q 1 , q 2 ) cycle of the torus. What was shown in [17] is that the zero-mode parts of the spectra do match. Since the states in question are, in general, oscillating states it is important to check that the oscillator parts in the masses (and the constraints) also agree.
As was argued in [17], the BPS spectrum of (q 1 , q 2 ) string should be the same as that of perturbative (1, 0) string provided the tension is rescaled by q 2 1 + e −2φ 0 q 2 2 . The problem which we address below is how to determine the oscillator part of the quantum supermembrane BPS spectrum, given that this is a complicated interacting theory (assuming it is well defined as a quantum theory in the first place).
For a membrane wrapped around the torus, the light-cone gauge Hamiltonian H turns out to contain a 'free' (quadratic) H 0 part and an interaction part H int . The theory can be solved in a special limit of large torus area in which the interaction term H int drops out. Our main assumption will be that the masses of the BPS states of wrapped supermembrane do not receive corrections in decreasing the torus area while keeping fixed the torus modular parameters and the D = 10 string tension T 2 = 1/2πα ′ (in this supersymmetric problem one expects that masses of BPS states with given charges should not receive quantum corrections depending on continuous parameters like radii of the 2-torus, i.e. masses should remain the same as for H int = 0). Solving the gaussian theory with H = H 0 we find that 13 Just like D = 10 solutions with at least one isometry are related by T -duality, D = 11 solutions with at least two isometries are related by similar symmetry transformations [26] of supergravity actions. In string theory, T -duality is not only a symmetry of the leading term in the effective action but is present to all orders in string perturbation theory as it has its origin in 2-d duality of world-sheet action. The question is about analogous exact symmetry of M-theory. 14 We expect that one of such solutions reduces to 4 + 2⊥2 + 0 bound state discussed in D-brane approach in [19].
the resulting oscillating membrane BPS states do, indeed, have the same masses as the BPS states of type IIB strings. This provides strong support for the suggestion of [17]. It should be emphasized that this correspondence between the BPS states of (q 1 , q 2 ) string and wrapped membrane with 'quadratic' Hamiltonian is quite non-trivial, as the 'quadratic' approximation reproduces not only the perturbative NS-NS (1, 0) part of string spectrum but also the R-R (0, 1) and the general (q 1 , q 2 ) parts as well. It is remarkable that these (non-perturbative) bound states of type IIB superstring theory admit an interpretation in terms of excitations of a fundamental supermembrane with certain values of Kaluza-Klein momentum and winding number.
This complements the relation between the (q 1 , q 2 ) string and D = 11 membrane with momentum flow along the (q 1 , q 2 ) cycle of 2-torus established at the macroscopic effective field theory level in Section 2. In the case of the fundamental type II strings the classical 'string+wave' solutions are in correspondence with microscopic BPS states [20,23,24], with momentum being carried by chiral (right-moving) string oscillations. This suggests that the 'membrane+wave' solution (2.7) (more precisely, its oscillating generalization mentioned in Section 2) should also correspond to microscopic membrane BPS states with longitudinal momentum carried by membrane oscillations propagating along one direction -the (q 1 , q 2 ) cycle of the torus. Indeed, as shown below, these are the membrane states whose masses coincide with the masses of the BPS states of (q 1 , q 2 ) string.

Supermembrane theory on R
The membrane states investigated here are excitations of a membrane with non-trivial winding number around the target-space torus. The spectrum of the light-cone membrane Hamiltonian in this case is discrete (an earlier study of the spectrum on R 9 × S 1 × S 1 is in ref. [49]).
Let X 1 and X 2 represent the compact coordinates, with periods 2πR 1 and 2πR 2 . X 2 will be viewed as an extra 'eleventh' coordinate absent in superstring theory. Let σ, ρ ∈ [0, 2π) be the world-volume spatial dimensions. The transverse, single-valued coordinates X i , i = 3, ..., 10, and the canonical momenta P i can be expanded in a complete set of functions on the torus 15 The canonical commutation relations imply For a membrane wrapped around the rectangular target-space torus so that one has

5)
The winding number that counts how many times the toroidal membrane is wrapped around the target-space torus is A membrane with w 0 = 0 is topologically protected against usual supermembrane instabilities [50]. It is convenient to choose the light-cone gauge to remove the single-valued partX 1 of X 1 , i.e. X + = x + + α ′ p + τ , X ± ≡ (X 0 ±X 1 )/ √ 2 . The standard light-cone Hamiltonian of the supermembrane is given by [51,52] where a = 1, 2, ..., 10 andX 1 = 0. Here θ α (α = 1, ..., 16) is a real SO(9) spinor. The (mass) 2 operator M 2 = 2p + p − − p 2 a is given by 2H − p 2 a , where p a is the center-of-mass momentum of the membrane, p a = dσdρP a . For simplicity of presentation, in what follows we will omit the fermionic terms in the formulae (their inclusion is straightforward, see, e.g., [49,53]).
The Hamiltonian has a residual symmetry associated with area-preserving diffeomorphisms, which can be fixed by setting (3.10) One can split P 2 (σ, ρ) = P C 2 +P 2 , where P C 2 = P C 2 (ρ) belongs to the (Cartan) subspace generated by e imρ , andP 2 to the complement. The local constraints can be solved forP 2 in terms ofX 2 and the transverse coordinates and momenta X i , P i .P 2 can be ignored in the problem of BPS masses we are interested here (in the large radius limit considered belowP 2 gives subleading contributions of order O( 1 R 2 ), see ref. [53]). Separating the winding contributions, we can put the Hamiltonian in the form where i = 3, ..., 10 and a, b = 2, ..., 10, and X 2 = X C 2 . As discussed in [53], the spectrum of this Hamiltonian is discrete provided w 0 = w 1 w 2 = 0 (for w 0 = 0 one or both terms in the quadratic part of the potential vanish, flat directions remain, leading to a continuous spectrum). Inserting the expansions (3.1), (3.5) and (3.6), we obtain 14) g has the interpretation of type IIA string coupling. The operator α ′ H 0 is the generator of translations in the world-volume time τ . For the center-of-mass coordinate X a (0,0) one obtains the simple time dependence X a (0,0) (τ ) = x a + α ′ p a τ . The system described by (3.14), (3.15) can be represented as a direct product of the classical zero-mode system (which is known explicitly for all values of R 1 , R 2 ) and a complicated interacting system of oscillation modes. The latter can be solved exactly in the special limit R 1 , R 2 → ∞, T 3 → 0 with α ′ and τ 2 fixed since then the interaction term drops out. 16 In this limit g 2 → ∞, and the Hamiltonian becomes that of an infinite set of harmonic oscillators labelled by (k, m). In this supersymmetric model, one may argue that since the quantum corrections to the masses of BPS states should not depend on continuous parameters, their values should thus be the same as in this limiting case, i.e. in the case when the interaction term drops out. This is what we shall assume below.
Let us introduce the standard creation and annihilation operators satisfying

Matching the membrane and type IIB string BPS spectra
The nine-dimensional membrane mass operator is given by where H is given by (3.25). Our aim is now to compare the BPS part of this membrane spectrum with the type IIB (q 1 , q 2 ) string BPS spectrum given in [17], where the correspondence between the spectra was established at the zero-mode level. We shall show that it extends to the oscillator level as well.
From the ten-dimensional string-theory point of view, the Kaluza-Klein momentum p 2 = l 2 /R 2 corresponds to the Ramond-Ramond charge. Let us first consider the perturbative or (1, 0) string states with l 2 = 0, the winding number l 1 , the Kaluza-Klein momentum w 0 /R ′ 1 and the mass (note that under T -duality relating IIA and IIB spectra to the vacuum. For these states, ω k0 = w 2 k, and (cf. (3.25), (3.28)) H = w 2 N + σ = w 1 w 2 l 1 , so that (3.34) This coincides with the masses of perturbative string BPS states obtained upon setting N L = 0 in (3.32). The mass formula for the free string states of the non-perturbative type IIB string with charges (q 1 , q 2 ) given in [17] is where l 1 = nq 1 , l 2 = nq 2 , with q 1 , q 2 being co-prime, and T (q 1 ,q 2 ) = T 2 n l 2 1 + τ 2 2 l 2 2 , T 2 = (2πα ′ ) −1 , τ 2 = Eq. (3.35) should be exact for BPS states [17]. The case of NS-NS string (3.35) corresponds to l 2 = 0, q 1 = 1, q 2 = 0. In the case of the R-R string l 1 = 0, q 1 = 0, q 2 = 1 one finds that for BPS states with N L = 0 (3.38) In the supermembrane spectrum (3.31) the minimal mass for given charges is obtained for states with N − ρ = N − σ = 0. For the counterpart of the R-R states with l 1 = 0, the constraints (3.27) are similar to (3.33) N + σ = 0, N + ρ = l 2 w 2 . N + σ is a sum of positive definite terms, so it is equal to zero only if each term in it vanishes. The most general physical state satisfying N + σ = N − σ = 0 is thus obtained by acting by α i (0,−m) on the vacuum. On these states, N + ρ and H take the form (see (3.25)) so that 40) in agreement with the type IIB string expression (3.38). The previous two examples of comparison with the (1, 0) and (0, 1) string spectra illuminate the fact that the BPS states in the supermembrane spectrum have excitations only in one direction -along the momentum vector. This is also what is suggested by correspondence between the (q 1 , q 2 ) string and wrapped membrane with momentum along (q 1 , q 2 ) cycle at the level of the classical solutions discussed in Section 2. Let us now identify the relevant oscillation states in the supermembrane spectrum for generic (q 1 , q 2 ). Performing the rotation as in (2.6),(2.5), y 1 = cos θ X 1 + sin θ X 2 , y 2 = − sin θ X 1 + cos θ X 2 , where, for generic R 1 , R 2 , q 2 → R 1 R 2 q 2 , i.e. tan θ = q 2 R 1 q 1 R 2 , we may align the momentum with the direction y 1 . The map between the target-space torus and the toroidal membrane surface is given by the zero-mode part y 0 1 = w 1 R 1 cos θ σ + w 2 R 2 sin θ ρ , y 0 2 = −w 1 R 1 sin θ σ + w 2 R 2 cos θ ρ . (3.41)