String propagation in backgrounds with curved space-time

Nuclear Physics B348 (1991) 89-107 North-Holland

STRING PROPAGATION IN BACKGROUNDS WITH CURVED SPACE-TIME* Itzhak BARS**

School of Natural Sciences, The Institute for Adcanced Study, Princeton, NJ 08540, USA Dennis NEMESCHANSKY

Physics Department, Unicersity of Southern Cahfornia, Los Angeles, CA 90089-0484, USA Received 21 May 1990

We study propagation of bosonic strings and superstrings in background metrics with curved space-time using non-compact current algebras. We discuss 2 + 1 dimensional (super)strings on the SO(2,1) group manifold. Using a coset model we construct a ( d - 1 ) + 1 d i m e ~ de Sitter (super)strings for d = 2, 3, 4 . . . 10 or 26. We also discuss the poss~ility of a discrepancy. in counting the number of time (i.e. non-compact) coordinates for the classical string v e ~ the quantum string. We illustrate this with several examples. We show evidence that a string or superstring propagating in a background with curved time cannot be excited to arbitrari~y high states and that it is unitary only in low lying states. We offer a possible interpretation of this phenomenon.

1. Introduction

In this paper we consider string theories propagating in background metrics with curved space as well as curved time. We consider a class of models described by non-compact current algebras for which certain exact statements can be made about the spectrum. In particular, we are interested in the question of whether a consistent model can be constructed by demanding conformal invariance. We address the conditions that are required for the no-ghost theorem of covariantly quantized strings to be valid when we consider a curved time coordinate. Previous treatments of string propagation in curved backgrounds with curved time have been perturbative. Conformal invariance has 0een achieved by requiring that the beta-function vanish [1]. In contrast, in the class of models we discuss, conformal invariance is treated exactly as in any current algebra model. * Research supported in part by the U.S. Department of Energy under Grant No. DE-FG0384ER40168 and DE-FG02-90ER40542. ** On leave of absence from the University of Southern California. 0550-3213/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

I. Bars, D. Nemeschansky / String propagation

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In order to express our idea we begin our discussion by analyzing a simple toy model. Later we generalize in various directions, including many dimensions, various non-compact groups and supersymmetry. We also raise the question of how to count the number of time coordinates in the classical versus the quantum string.

2. A toy model on SO(2,1) As a prototype of the models we discuss in this paper let us consider a (2 + 1)-dimensional example with the string coordinate X~(T, tr),/~ = 0, 1,2 taking values on the group manifold SO(2, l). See also ref. [2] for related work on this example. The string action is the W e s s - Z u m i n o - W i t t e n model with a central extension k. When viewed as a classical sigma model this corresponds to a string propagating on a (2 + l)-dimensional curved space-time. The background metric G~,(X) and the antisymmetric tensor field B ~ ( X ) are determined by the parametrization chosen for the group manifold in terms of the coordinate X ~. The dynamics is most conveniently expressed in terms of the left- and right-moving currents that describe the symmetries of the theory. The stress tensor splits into left- and right-moving parts and each one is a quadratic form in the currents. The left- or right-moving currents satisfy the SO(2, 1) K a c - M o o d y algebra

"

k

re. AJi,+,,,+n-~r I 6,,+m.O ,

/.t=O, 1,2,

(1)

where Eu''a is the Levi-Civita symbol in three dimensions with e °~2 = + 1 and r/u~ is the three-dimensional Minkowski metric with signature ( - l , 1, 1). Since 7/u~ is proportional to the Killing metric of SO(2, l) we use it to lower and raise the indices of the structure constants: eu~a = Eu~Pr/oa. The e n e r ~ - m o m e n t u m tensor has the form T ( z ) - ~Lnz n-E, where the Virasoro generator L,, is 'F~/.t t,

/.t

1,

L,, = ( k - 2) Y'-" J-,,,L, +,,,'.

(2a)

The commutation relations of the Virasoro generators L n and the currents J,~ have the form = -mL"+,,,, ¢

[L,,,Lm]=(n-m)L,,+m where the central charge is c = 3 k / ( k - 2).

+ --~(n3-n)6,,+m,

(2b)

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In covariant quantization, in addition to the original degrees of freedom X~(tr, ~'), we need to introduce the ghost fields b and c. The theory is conformal invariant provided the BRST operator is nilpotent which is equivalent to having c = 26. The physical states are determined by requiring that the BRST operator annihilates them. For ghost-free states this is equivalent to the Virasoro conditions ( L o - 1)[phys) = 0,

L,[phys) = 0,

n >f 1.

(3)

If we choose k = 5 2 / 2 3 we satisfy the central charge condition c = 26. It remains to find out the states that satisfy eq. (3). In the flat case the Virasoro conditions eliminate all negative norm states from the spectrum for all d < 26 [3] and require the mass operator M 2= _p2 (analog of quadratic Casimir) to have certain quantized eigenvalues, M 2 = _ 1,0, 1,2, . . . . In the curved case our problem is to find the analogous statement. We shall see that negative norm states remain even after we satisfy eq. (3). Before we plunge into algebra let us first establish some parallels between the curved case and the flat case in order to see what we might expect. In the flat case the Virasoro algebra is constructed directly from oscillators Ol,t~ as follows: d-!

!

L,, = ~ , , ,

Y'. :Ol~,,Ol~,+m',

(4)

g=O

with c = d < 26. The oscillators satisfy the algebra [ #

v]

pin " Olin

--" IITIPV~n +m.O "

where 7/u~ is the Minkowski metric in d dimensions with signature ( - 1 , 1 , . . . , 1). The origin of negative norm states in the flat case is due to the negative norm oscillators associated with the time direction in eq. (4). The counting of the negative norm oscillators is precisely equal to the number of Virasoro constraints. 0 Hence, if we were able to solve the constraints explicitly we could solve for the a n oscillators in terms of the remaining positive norm oscillators and quantize the theory only in terms of the positive normed ones. This is indeed the case when the string is quantized non-covariantly, say in the light-cone gauge. Therefore, in a covariant quantization, we may reasonably expect that all negative norm states are eliminated by eq. (3). This is demonstrated by the no-ghost theorem [3]. We may regard the flat case as the k ~ 0o limit of a curved current algebra, where we first renormalize the currents as a~ = 2 v f ~ J~ (for n ~= 0) and then take the limit with a~ fixed. The non-abelian pieces in the current algebra become negligible and drop out, leaving the abelian K a c - M o o d y algebra of (4), except for the zero modes that remain non-abelian. For finite k, w~ may regard the K a c - M o o d y algebra of (1) as a non-abelian generalization of oscillators. The

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I. Bars, D. Nemeschansky / String propagation

central extension of eq. (1) determines the negative versus positive norm properties of the non-abelian oscillators. However, one must remember that the non-compact nature of the structure constants affects the norm of a state. As we see below this will complicate the issue. Let us first study the states at level zero. The zero modes of the currents J~ generate a SO(2, 1) Lie algebra. They replace the momentum operator p~ = a 0~ of the flat case. To obtain positive norm states at level zero it is clear that we have to restrict ourselves to unitary representations Ij, m ) of SO(2, 1). What about higher levels? Can we use the Virasoro constraints L,, to eliminate the negative norm states generated by time like currents j,0 as we did in the flat case? As we shall see, the situation is more complicated. The physical states have a positive norm only for limited values of j, while most values of j that satisfy the Virasoro conditions correspond to a negative norm. This is unlike the flat case for which all states that satisfy the Virasoro conditions have positive (or zero) norm. At first sight it seems surprising to find that we do not have a no-ghost theorem for the non-compact current algebra. Perhaps we can make this result more intuitive by reminding the reader that a similar situation occurs when only the space coordinates are curved. If we consider a compact current algebra associated with the group G, it is known that its states can have positive norm only in specific unitary representations which depend on the central extension k [4]. Therefore the curvature associated with a non-zero k limits the possible admissible representations of G, thus requiring that the unitary theory contain a finite number of primary fields. If we let k ~ ~ we recover the flat case with d = dim G. In this case there are an infinite number of primary fields. The zero modes J~ remain curved, but this is not the issue. Thus, for curved time coordinates we should also expect some limitation on the admissible representations as long as k remains finite. In the following analysis we find the conditions on j. Let us analyze the Hilbert space of the SO(2, 1) K a c - M o o d y algebra and impose the Virasoro conditions. The Hilbert space of the SO(2, 1) K a c - M o o d y algebra has been discussed in a different context in refs. [5,6], where the unitarity of the parafermionic theory SO(2, 1 ) / U ( 1 ) w a s analyzed. Our problem here is different in that the Virasoro constraints will give rise to a different set of states than the U(1) projection of the SO(2, 1) K a c - M o o d y algebra. However, some of the analysis of refs. [5, 6] will be useful in discussing our problem. As we saw earlier the primary states of SO(2, 1) are labeled Ij, m). An excited state at level / satisfies the Virasoro condition L 0 = 1 if

j ( j + 1) (k-2)

+/=I.

(5)

Above we have used that the L 0 eigenvalue of the primary state ]j, m ) is given by the Casimir j(j + 1). In order to apply the remaining L,, conditions we consider each excited level l separately.

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93

At level zero we have the primary state lj, m ) with J ~ l j m ) = O, J~ljm)

n >1 1,

= 1 / m ( m + 1) - j ( j

J°ljm ) = mljm ), + 1)Ij, m +_ 1),

(6)

where Jr+ = J~ + / / 2 . The first equation in (6) shows that all the states at level zero are annihilated by Ln, n >f 1. This is analogous to the flat case in which the state Ip ~) is annihilated by the oscillators. We find that at level zero eq. (5) gives - - j ( j + 1) = 6 / 2 3 .

(7)

Above we have used k - 2 = 6 / 2 3 so that c = 26. In order to have positive norm states at level zero, we need that the generators J~ be hermitian and that the states lj, m ) and J ~ l j , m ) have positive norm. From eq. (6) it is easy to see that these conditions are satisfied if

m(m ÷ 1) - j ( j + 1) >/0.

(8a)

This is equivalent to the condition that defines a unitary representation of SO(2, 1). The only unitary representation with positive - j ( j + 1)> ~i is the principal series l ( j = - ~ + ½ip). From eq. (7) we find !

J = - 3 +- i ~ / ~ .

(8b)

In our case the Casimir j ( j + 1) plays the role of the mass squared operator and the label m replaces the angular directions of the momentum. Thus, the states at level zero are analogous to the tachyon of the flat case, with mass M 2 = _ 1. Moving to the first excited level (! = 1 ) w e find, using eq. (5), the condition j ( j + 1)= 0. This is analogous to the zero mass level of a flat string M 2 = 0. The two solutions j = - 1,0 correspond to the following unitary representations of SO(2,1): the one-dimensional trivial representation for which j - - 1 , m = 0, the positive discrete series representation j = 0 for which m = 1,2,3,... and the negative discrete series representation j = 0 for which m = - 1, - 2, - 3, . . . . An arbitrary state at level one ( l - 1) has the form ao J °_l l j m )

+ a +J+_! Ij, m - 1) + a _ J = l l j , m + 1)

(9)

where the a0, a ± are coefficients to be determined by the Virasoro conditions. In

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I. Bars, D. Nemeschansky / String propagation

eq. ( 9 ) w e of course restrict ourselves to values of j and m determined above. Since we are looking at a state at level one, eq. (6) shows that the Virasoro L,, with n >/2 annihilate all the states (9). Applying L~ to the state (9) gives no condition on the a's when j = - 1 . For j = 0 w e find a 0 = - a + v / 1 - 1/m -a_~/1 + 1 / m . Using eq. ( 1 ) w e can compute the norms of the states. For j = - 1 the states J + ~ l - 1,0) have a positive norm, but the state J_°~l- 1,0) has a negative norm even though it satisfies all of the Virasoro conditions. For j = 0 the states of the form (9) with the given solution for a 0, but arbitrary a_+ have either positive or zero norm. These states behave like the vector state of the fiat theory. Thus, we have found that at the first excited level j = - 1 is not admissible, but j = 0 is. At the second excited level, eq. (5) gives j(j + 1 ) = 6/23. This requires that we have either the positive or negative discrete representation with a positive j. In this case we have j = - ~ + i ~/47/92 and m = j + l , j + 2 , . , . or m = - ( j + l ) , - ( j + 2), . . . . This is analogous to the first massive levels M 2 = 1 of the flat string. The states at level two are constructed by applying linear combinations of operators J~- ~ J "-~ and ju-2 on the state Ijm ). Let us consider the state 14,) =JS iJ-llj, j + 1).

(lO)

At level two this is the only state with eigenvalue Jt° = j - 1. Therefore it cannot mix with any other states at level two. Using eqs. (2) and (6) it is easy to see that the state (10) satisfies all the Virasoro conditions. From eq. (1) we see that the norm of 14,) is

(4"14") ~ ( k - 2 j -

2 ) ( k - 2j - 1).

(11)

With the above values of j, k this expression is negative. Thus, again, even though the conformal invariance condition is satisfied we cannot admit this value of j. At the third excited level j(j + 1)= 2 ( k - 2) = 12/23, which means that we again have only the discrete series. In fact, at all excited levels we can have only the discrete series. Following our analysis of level two let us consider the state (J-l)31j, j + 1) which again has unique L0, j o quantum numbers and hence cannot mix with any other state at level three. This state satisfies all the Virasoro conditions. The norm is proportional to (k-

2j- 2)(k-

2j-

1)(k-

2j).

(12)

This is also negative for the given values of k and j. Extending the arguments to higher levels we find that the generalization of the state (10) (J-~)tlj, j + 1), with j(j + 1)= ( 1 - 1)(52/23) satisfies the Virasoro conditions. The sign of the norm now depends on the level. For example, we have a

L Bars, D. Nemeschansky / String propagation

95

negative norm state at levels 2 < l ~< 4, 11 < 1 < 17, 27 < 1 ~<37, etc. and therefore the corresponding values of j determined by (5) are eliminated*. So far we have analyzed only a small subset of the Hilbert space. There is no reason to expect that there does not exist additional negative norm states that would eliminate other values of j. One can quite easily argue that this is indeed the case. Let us start by labeling the states of the SO(2,1) Kac-Moody algebra in terms of the parafermionic theory SO(2,1)/U(1). Then every state in the SO(2, t) theory can be written as a product of a state in the parafermionic theory and the U(1) theory,

S 0 ( 2 , 1) =

SO(2,1)

x U(I).

The U(1) theory clearly contains non-unitary states. From the commutation rela0 \ tions ( 1 ) w e see that the state J-,,lX/, where IX) is a primary state in the U(1) theory, has a negative norm. The parafermionic theory on the other hand is unitary if 2(j + 1 ) < k for j in the discrete series. There is no condition on j in the principal or supplementary series [5, 6]. A simple counting argument shows that the Virasoro conditions (3) are sufficient to eliminate all the states generated by j o- - n " However, according to simple counting, there is no room for the additional negative norm states of the parafermionic theory. Therefore, we must limit the values of j by the unitarity conditions of the parafermionic theory. With our value of k = 52/23, we can only allow j + 1 < 26/23 in the discrete representations plus any continuous representations that satisfy the condition L 0 = 1. This rough argument allows us to conclude that there are no additional values of j that can satisfy both unitariq," and the Virasoro conditions, in agreement with our findings above. So far we have seen that, on the basis of unitarity, the principal series representation with j = 2' -+ ivti/92 and the discrete series representations with j - 0 are permissible at levels l = 0, 1 respectively. Thus, the Virasoro conditions by themselves are insufficient to insure the unitarity of the (2 + 1)-dimensional theory * We may ask whether in this analysis it was necessary to take only unitary representations at level zero. Why not admit both unitary and non-unitary representations at level zero when requiring that the excited states at level ! have positive norm? (We thank T. Banks for raising the question.) The answer is that for most of the excited states the value of j is not a half-integer, so that there are no additional non-unitary or unitary representations beyond the ones we have already considered. However, even if there were additional non-unitary representations that may be considered for some of the levels, our results do not change, although we may admit additional bad states associated with the extra non-unitary representations. This is because the excited states associated with an irreducible representation at level zero are orthogonal to the excited states associated with another irreducible representation. The orthogonality property would prevent taking linear combinations of bad excited states, coming from unitary and non-unitary representations at level zero to form positive norm states. Thus, admitting non-unitary representations at level zero would only invite more trouble.

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I. Bars, D. Nemeschanslo, / String propagation

when time is curved, while they were sufficient for the flat string in d < 26 O~raensions. Presumably, the curved (2 + 1)-dimensional string based on the ~ ( 2 , 1) W Z W model, whose dynamics is described by current algebra and the Virasoro constraints, can exist only in the physical states described above. This is the equivalent of only two mass levels, as opposed to an infinite number of excited mass levels for the flat string. Thus, it appears that a string whose time coordinate is curved cannot fluctuate to arbitrarily excited states!

3. Relaxing the condition on k

Now we relax the condition on k by introducing more compact dimensions. In this case the stress tensor has the form T = Tsot2. !~+ T ' ,

(13)

where T' describes a unitary conformal field theory that commutes with the SO(2, 1) currents. The states of this theory can be written as a linear combination of the tensor product ]qJ) ® ]q~'),

(14)

where i~) is a state in the SO(2, 1) theory and [qJ') is a state in the conformal field theory defined by T'. Since we assumed that T' defines a unitary theory the state I~') has a positive norm. Imposing the Virasoro conditions on our theory we find that the central charge c' of the theory defined by the stress tensor T' satisfies 3k c= (k-2)

+c'=26.

(15a)

The mass shell condition L 0 = 1 gives

- j ( j + 1) +I+A=

k-2

1,

(15b)

where A is the T' conformal dimension of the state I~,'). In eq. ( 1 5 b ) w e have assumed that Iq/) is an excited state at level l. From eq. (15)we can solve for k in terms of c', 52

--

C r

k = 23 - c' "

(15c)

If c' ~< 23 we see that k is always larger than 5 2 / 2 3 of sect. 2. Keeping c' and A

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97

fixed we find that at level 1, !

j= -~ + 1/(l+zi-

1)(k-

2) + 1 / 4 .

(15d)

Since both factors (k - 2) and ( a + 1 - 1) are now larger than the correspondh-lg values in sect. 2, we expect to get discrete series representations with values of j that are larger than those of the corresponding values at the same levels L As before let us consider the state (J2~)taj,j + 1) of the SO(2,1) theory. If we take a tensor product of this state with a primary state of the conformal field theory defined by the stress tensor T', we see that the Virasoro conditions are satisfied by L,, = L s°t2" t) + L~, since each term gives zero. The norm of these states is proportional to

(k-2j-2)(k-2j-

1)(k-2j)...(k-2j+l-3).

(16)

Just as in sect. 2, we find that this form can be either positive or negative depending on the level 1 and j. However, for sufficiently high level l, there are always negative norm states. The conformal field theory defined by the stress tensor T' has the effect that the negative norm states will occur at higher levels compared to the previous example. In fact, as k -~ ~ or c' ~ 23, the negative norm states get pushed out to infinity, producing a positive def'mite Hflbert space. For a general state we can again use our rough argument given at the end of sect. 2. The parafermionic theory SO(2,1)/U(1) is unitary provided we take discrete representations that satisfy

2 ( j + 1) <

5 2 - 2c' 23 - c'

.

(17)

We cannot go beyond this range because there will be more negative norm states than there are Virasoro conditions. This is in agreement with the analysis above. We have shown that as long as the time coordinate remains curved in the (2 + 1)-dimensional subspace of the string, Virasoro conditions by themselves are insufficient to insure the no ghost theorem. This was, however, not the case for a string propagating in flat time. The additional conditions of unitarity demand that only certain representations of the non-compact current algebra are admissible. This means that there are fewer "mass" levels available to strings propagating in a curved time. Note that this is true for all models T'. These models are continuously connected to the d = 26 flat theory by the limiting procedure c' -~ 23.

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4. Curving time and space in higher dimensions In this section we describe current algebra models that represent a string propagating in (d - 1) + 1 dimensions. Using the coset method [7] we consider the case of SO(d - 1,2)_, (18)

SO( d - 1, 1) _k

where ( - k) is the level of the K a c - M o o d y algebra. We call these models de Sitter strings in d dimensions and we consider d - 2,3, 4, . . . . First, let us justify that this model is a string with one time coordinate and d - 1 space coordinates. We label the currents of S O ( d - 1 , 2 ) by j,u~,j,~, with V. = 0, 1 , . . . , ( d - 1). The commutation rules of the currents include a central extension proportional to k. Any state in the Hilbert space of the S O ( d - 1,2)_ k theory can be written as a linear combination of tensor products I ~ ) ® [~'), where I~) is a state in the coset model S O ( d - 1 , 2 ) _ J S O ( d - 1 , 1 ) _ k and [$') is a state in S O ( d - 1 , 1 ) _ k. We can identify the Hilbert space of the coset model (18) by considering primary states IXi) of the S O ( d - 1, 1)_, K a c - M o o d y algebra. According to the decomposition above we see that

[7/) = )-"~Ai[gli)

®lxi)

(19)

i

is a state in the Hilbert space of the SO(d - 1,2)_ k theory. For a state given by eq. (19) we have J,~"IT/) = )-".A,[$~)

®J,~,vlXi)

=0,

n >/1.

(20)

i

Above we have used that Ixi> is a primary state of the S O ( d - 1, 1)_ k theory. We have found that the states IT/> in S O ( d - 1,2)_ k that satisfy J,,UvlT/> = 0 generate the Hilbert space of the coset model (18). The important point to notice is that the non-abelian "oscillators" associated with the Lorentz subgroup have been set equal to zero on the states of interest. Thus, the currents j,u are the ones that play the essential role in the construction of the Hilbert space of the coset model and we have one time-like current and (d-l) space-like currents. In the limit k ~ ~ we have one negative norm oscillator which is identified with the time direction and ( d - 1 ) positive norm oscillators that are identified with the space directions. In this limit the coset space theory is described by the Hilbert space of a flat string in ( d - 1 ) + 1 dimensions, except for the zero modes. These considerations justify our view that the de Sitter strings defined above through the coset model (18) correspond to strings propagating in curved backgrounds in ( d - l ) + 1 dimensions.

I. Bars, D. Nemeschansky / String propagation

99

Another way of seeing that the de Sitter strings contain a single time coordinate, at least at the classical level, is to adopt the formalism of ref. [8]. It was argued there that any coset G / H theory is equivalent to a gauged sigma model with H the gauge group. The gauge fields are non-propagating. After choosing the unitary gauge, the counting of propagating degrees of freedom confirms our view expressed above concerning the de Sitter strings. The physical states are selected by imposing the Virasoro constraints. For the coset model the Virasoro generators L~/n and central charge co~ n satisfy I_,,G,= L~ + L~/n and c 6 = c H + c6/n. Since our states satisfy eq. (2u) they are automatically annihilated by L~n = 0, n >/1. Therefore, imposing the -n- na / n , n > ~ l conditions is equivalent to demanding /_.~ = 0, n >i l, on the states that already satisfy (20). We also need to impose the mass shell condition with L~/H= I with CG/n = 26. For d >/4 we find c=

d ( d + 1)k 2 ( k - d + 1)

-

d ( d - 1)k

2 ( k - d + 2)

= 26,

CV-,.,)

Lo=-k_d+ 1+ k_d+2+l=l,

(21)

where C(2d-l'2) and C[ d-l"i) are the eigenvalues of the quadratic Casimir operators for S O ( d - 1,2) and S O ( d - 1,1) respectively. For d = 2 we have the coset model SO(l, 2)/SO(1, 1) with 3k

c= (k-2)

1=26,

Lo=

j ( j + 1) m2 k-2 +~+/=1"

(22)

Here we have labeled the state in SO(2, l) by Ij, m ). For d = 3 we can use the fact that SO(2, 2)_ k ~ SO(2, 1)_k ® SO(2, 1)_k. The diagonal subgroup is SO(2, 1)_ 2k at level ( - 2 k ) and the coset model is SO(2,2)/SO(2, 1)with

c=

3k 2 (k-

1)(k-E)

=26,

L0= -

Ji( Jl + 1 ) + J2( J2 + 1) k-2

+

j( j + 1) 2k-2

+ / = 1. (23)

Here Jl, J2 label the Casimirs of SO(2, 2), while j labels the Casimir of the diagonal subgroup SO(2, 1). The d = 3 case can be slightly generalized by taking different levels kl, k 2 for the two SO(2,1) factors. In this case the level of the diagonal subgroup is k I + k 2. In order to satisfy the condition c = 26 on the central charge the level k will depend on the dimension d. For example k = 9 / 4 for d = 2, k -- 2.48 or 0.91 for

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I. Bars, D. Nemeschansky / String propagation

d = 3 and k -- 4.08 or 1.74 for d = 4, etc. Since c = 26 is a quadratic equation for k, there are two solutions for each d >/3. For both branches, k increases as d increases from 3 to 26, and k ~ ~ as d ~ 26. We note that we have a c - 26 theory even in two dimensions. The d = 3 theory is different from the d = 3 theory based on the SO(2,1) group manifold we analyzed in sect. 2. It is easy to analyze the d -- 2 theory since the representation theory of SO(l, 2) is well understood. Let K~, K 2 be the non-compact generators and J3 be the compact generator. In the coset model S O ( 1 , 2 ) / S O ( I , I) we have to diagonalize the non-compact generator K 2. This generator is related to the compact generator by a non-unitary SO(1,2) rotation K2=exp(-K:r/2)(-iJa)exp(Kl~'/2). The rotation angle irr/2 is purely imaginary. This non-unitary transformation relates the Hilbert space ljm) in which the compact generator is diagonal to the Hilbert space in which the non-compact generator is diagonal. The eigenvalues of K 2 are purely imaginary and quantized as ira. This can also be deduced from the commutation rules of SO(1,2) by defining ladder-like operators (J3 -+ K~) that shift the eigenvalues of K 2 by (q-i). This requires a non-trivial definition of dot product with respect to which K 2 is hermitian. It turns out that such a dot product is ( ~ , x ) = (~lexp(rrK~)lx), where (~bl is the naive bra obtained by conjugating the ket l~b). However, this is precisely equivalent to the dot product in the basis in which -/3 is diagonal. To perform the analysis of the SO(1,2)/SO(1, 1) theory, we make use of the SO(1,2)/U(1) analysis of refs. [5,6] in which the compact generator is diagonal. These remarks imply that the states will have a positive norm if 2(j + 1) < k when j describes a discrete series representation. There is no restriction on j if it describes either the supplementary series or principal series. Applying these conditions on the solutions of eq. (22)we find that there are only a limited number of states available for strings propagating in a curved (1 + 1)dimensional space-time. This result is similar to the conclusion reached in sect. 3 when we considered strings propagating on the group manifold SO(2,1). We have performed a detailed analysis of eq. (23) for d = 3, with similar conclusions. We have not analyzed the remaining d >i 4 cases in detail, however we expect that our general intuitive arguments of sect. 2 should apply. We expect that for all our de Sitter strings, except when d = 26, we have only a finite number of unitary states that satisfy conformal invariance implemented through the Virasoro conditions. The number of such acceptable states increase as d increases and becomes infinite when d = 26. It appears that when time is curved, the string cannot be excited to arbitrarily high modes.

5. Supersymmetry and curved time in various dimensions Our SO(2, 1) and de Sitter strings can be generalized to N = 1 supersymmetric models. Corresponding to every current J " we introduce a free fermion O"(z). In the supersymmetric case we impose the super Virasoro constraints L,, G,, n >I 1

L Bars, D. Nemeschansky / String propagation

101

on physical states. Furthermore, superconformal invariance requires L o = 1 / 2 , c = 15. In the case of SO(2,1) the free fermions are in the adjoint representation. Using results from ref. [6] we see that the stress tensor of the fermions can be written in a Sugawara form corresponding to the current algebra SO(2,1) 2. This allows us to embed the full theory in S 0 ( 2 , 1 ) _ k ® S0(2,1)2 with central charge 3k 3 c = k-"~-2 + 2 = 15.

(24)

Solving for k we find k = 18/7. In the de Sitter theories the fermions carry the labels of the coset G / H rest, Ring in a system with N = 1 supersymmetry [9]. Since the model we are considering has zero torsion we can use a conformal embedding to rewrite the supersymmetric G / H theory in the form G_t, ® Hg-h/H-k+g-h, where g,h are the Coxeter numbers of the groups G , H respectively [6]. Applying this to the de Sitter superstrings we find d=2

SO(l,2)_t, ® SO(l, 1)2 SO(l, l)-k+2 S0(2, 1)_, ® S0(2,1)_, ® S0(2,1)2

d=3 d~4

S0(2,1)-2k+2 SO(d-

1,2)_, ® SO(d-

SO(l, 1)-k+i

1,1),

.

(25)

These forms allow us to easily compute the central charges and the dimensions of the primary fields. The central charges for the supersymmetric G / H are given by c = 3k dim(G/H)/2(kg) [6]. For the de Sitter strings we fred d=2 d=3 d >/4

3k c= ~ = 15---,k =5-2 ' k-2 9k c= = 1 5 ~ k =--~, 2 ( k - 2) c=

3kd 2(k-d+

1)

= 15 ~ k =

10( d - 1) (lO-d)

(26)

In particular, we see that k = 5 for d = 4, and k is infinite when d = 10. The analysis of these models proceeds as in the previous sections. Without going into details, we expect that again, as long as the time coordinate remains curved for d < 10, there are only a finite number of states with positive norm that satisg, all the Virasoro conditions.

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We may also consider N = 2 supersymmetric coset models. In this case we have a Kiihler manifold and the free fermions carry the same labels as the coset G / H [9]. To carry out the analysis we generalize the observations of ref. [6] on non-compact groups. As in the de Sitter theories we can rewrite the fermionic part as a K a c - M o o d y theory with the symmetry group H. As an example we consider the group G = SU(n + p , m ) ~ and subgroup H = S U ( n ) x S U ( p , m ) x U(I). The N = 2 model may now be written as the coset model S U ( n + p, m ) k x S U ( n ) p+,,, x SU( p, m ) . x U ( 1 ) S U ( n ) k ÷p +,,, × SU(

+,, x U(1)

(27)

A trivial calculation gives the central charge 3kn( p + m ) c= k+n+m+p'

(28)

as well as dimensions of the various states. Requiring c = 15 will obviously put conditions on k, n, m, p. For example, if we set n = m = 1, p = 9, k = 11 we have a c = 15 model. Another example of a model with c = 15 is obtained if we have k = 1 = m, p = 9, n = 11. Note that k = 11 and k = 1 are special because in this case we have free fermionic representations. If we consider SU(10, l)~ it is easy to see that it can be rewritten in terms of free fermions in the adjoint representation [6]. The SU(10, 1)~ model can be rewritten in terms of free fermions in the fundamental representation. For these N = 2 models we do not have a classical argument that shows us a priori that we are dealing with a single time dimension. In fact, if we examine currents and fermions that belong to the G / H of eq. (27)we see that, for k > 0 we have np positive norm complex currents and fermions. The number of negative norm complex currents and fermions is nm. For k < 0 the sign of these norms are interchanged. In our examples above we had n = m = 1 and k > 0 and therefore we find classically one complex time coordinate along with the corresponding negative norm fermion. However, as we shall see below, when we have a conformal embedding the counting of time coordinates may be different from the naive one at the classical level.

6. How many time-like dimensions? So far we assumed that the dimension of space and time is determined by a classical argument that involved a simple counting of compact and non-compact dimensions of the currents and the sign of the central extension. This may not always be the case as can be seen by considering some examples of conformal embeddings. This is because conformal embeddings project out states. The counting of the time-like dimensions in the remaining Hiibert space may be quite different.

I. Bars, D. Nemeschansky / String propagation

103

As an example, let us consider the tensor products of minimal N = 2 models that involve only space coordinates. For these models the dimension of the space is related to the central charge. For instance, let us consider the tensor product of five minimal N = 2 models at level three. Each of these models has a central charge 9 / 5 , giving a total central charge of nine. This model has been analyzed in great detail and it is known that it corresponds to a superstdng propagating o n a six-dimensional Calabi-Yau manifold. On the other hand, we can also represent each minimal model as a tensor product of a parafermionic theory SU(2)3/U(I} and a U(1} theory. It is, however, not correct to say that each minimal model h a three dimensions and therefore the compact part has dimension ten. In ~ n t i n g the total dimension we have taken into account the fact that we have a su~rs)~nmetric theory. Another trivial example with only space dimensions is SU(n)~. Since th~ mode! has n 2 - 1 currents a naive counting would indicate that we have n 2 - t space dimensions. However, we can rewrite this theory in terms of n free complex fermions or using the Frenkel-Kac construction in terms of n free bosons whose momenta lie on the root lattice of the group SU(n). We now see that the counting is quite different. Of course, what is going on in this example is that the Hdlbert space has been greatly reduced by throwing away aH the zero norm states and thek descendants. A similar situation occurs for all compact groups at integer level k. We must bear in mind, however, that non-compact groups do not always have zero norm states [5, 6]. Let us now give an example of conformal embedding for a non-compact group. Consider the Lorentz group in 2d dimensions at level 1, S O ( 2 d - I, 1)~. The currents are labelled by J~,~(z), with / ~ , u = 0 , 1 , 2 , . . . , 2 d - 1 . Naive counting would suggest that we have 2 d - 1 non-compact or time-like dimensions and ( 2 d - 1 ) ( d - 1) compact or spacelike dimensions. Since we are looking at a theory at level one, we can represent the currents in terms of 2d free majorana fermions: J ~ = (1/v/2)O~O~. In the case of SO(2d - 1, 1) one of the fermions has a negative norm whereas the remaining ( 2 d - 1) fermions have a positive norm. The stress tensor has the Sugawara form

r(z) J,d-"(z) ~

~

(29)

When we represent the currents in terms of the free fermions we must at the same time restrict the Hilbert space to the representations that corresponds to the fermions. In general this is much smaller than the one allowed by a more general representation of the currents. How many time-like coordinates do we really have in this Hilbert space? We answer this question by bosonizing the free fermions. Two positive norm fermions make one positive norm boson, while one negative norm fermion with one positive norm fermion make one negative norm boson. The bosonization formulas for two

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I. Bars, D. Nemeschansky / String propagation

positive norm fermions

~2,~/3 are

well known:

e ix('') ÷ e-i.~-(z)

~2( z ) ="

~

eix(z) _ e - i X ( z )

",

~3(z) = :

iv~

"'

(30)

where x(z) is a positive norm boson with propagator (x(z)x(w)) = - l o g ( z - w). Similarly, we find the bosonization formula for one positive norm fermion ~ and one negative norm fermion ~0, e t(=) + e - t ( z )

,~(z) ="

~

et(-) _ e-ttz}

",

$o(Z) = "

v~-

""

(31)

In eq. (31) t(z) is a negative norm boson whose propagator is (t(z)t(w))= l o # z - w). We therefore see that if the S O ( 2 d - 1,1)~ theory is represented in terms of free fermions we have only one time-like coordinate. Using the above example we can construct a theory with c = 26 by taking SO(51, 1) v From the arguments above we see that this theory has one time-like dimension and ~ space-like dimensions. This is quite different from the counting suggested on naive grounds. Furthermore, this is not a flat string theory since it must be constrained to the Hilbert space generated by the fermions in the Neveu-Schwarz and Ramond sectors. For the current algebra this means that at level zero we are restricted to scalar, vector and spinor representations of S O ( 2 d - 1, 1). The dimension of the states of the theory is given by

L°-

1 C2(R ) 2 2d-1 +1"

(32)

where C2(R) iS the quadratic Casimir operator of the representation and the positive integer I is the level. The factor of 1 / 2 in front is due to the normalization of the currents that produces the level k = 1 in SO(2d - 1, 1)k. For the representations we are considering the Casimirs have the following values: C2(scalar) = 0 ,

C2(vector ) = 2 d -

C2(spinor) = d ( 2 d - 1)

1 '

8

(33) "

The physical states must satisfy the conditions L 0 = 1, L n = 0, n >t 1. In the above example with c = d = 26 we see that for the scalar representation we have a physical state at level one, J~_~[scalar). In the vector representation we cannot find l a physical state since L 0 = 3 ÷ l is never an integer. Similarly, in the spinor representations we do not have any physical states because d is not a multiple of 16. It is remarkable that in the entire Hilbert space there are such few states that satisfy the Virasoro condition. This is unlike our previous examples in which there were an infinite number of such states. Since all our states fill finite-dimensional

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105

representations of a non-compact group we see that among the states that satisfy the Virasoro conditions there are negative norm states. We can allow other values of d if we take a tensor product with another conformal theory described by a stress tensor T'. As in sect. 3 the T' theory must be unitary and furthermore the central charge must be c ' = 2 6 - d, so that the total central charge is c = 26. Such a model still has a single time-like dimension. In this case it is possible to find a finite number of positive norm states. ~ e s e states can be constructed by taking a direct product [scalar) × [A'= 1), where the state 0scalar) is the level-zero state described above and IA' = 1) is any dimensionone Virasoro primary state in the T' theory. This means that in the unitary sector S O ( 2 d - 1 , 1 ) ~ is just a bystander except for contributing its share to c = 26. However, as we have found in all other models, the unacceptable negative norm states are present although finite in number. They have the form /~_~[scalar) × [A' = 0), nvector) ×gA' = 3), i i s p i n o r s ) x i A ' = 1 - d / 1 6 ) , where the [A') that appear, if they exist, must be Virasoro primaries in the T' theory. There are other models based on other groups that would lend t h e ~ | v e s to conformal embeddings of the type we described above. For example SU(I, t) 2 which is equivalent to SO(2,1)~, with c = 3 / 2 , can be represented in terms of three free fermions in the adjoint representation. If we combine this with an additional theory with c ' = 24½ we can construct the type of model considered in sect. 3. In this model one negative and one positive norm fermion together are equivalent to one negative norm boson or one time-like dimension. Similarly, in a Frenkel-Kac construction, SU(2, 1)~ with c = 2 can be described by two bosons one with positive and one with negative norm. Their momenta must lie on the root lattice and hence it contains a single curved time-like coordinate. This model can be combined with a unitary theory with c ' = 24 to give a total central charge c = 26. As before all these models, however, contain negative norm states. The fact that the models described above could be rewritten explicitly in terms of a single time coordinate has shown us that the naive counting of time-like dimensions may be misleading if the model includes a projection of the I-filbert space. Furthermore, in the specific models above the correct time-like dimension is obtained by multiplying c with the ratio of the number of non-compact currents to the number of total currents. Suppose we try to generalize this observation to a more general model with central charge c and signature ( - , . . . , - , + , . . . , +), where the number of naive time-like dimensions is t and the number of naive space-like dimensions is s. For k > 0 the signature is determined by the compact and non-compact generators in the space. For k < 0 the signature is reversed. Here we are not restricting ourselves to just group manifolds we also allow coset models. Then the number of time-like dimension is given by c

t

t+s

.

(34)

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L Bars, D. Nemeschansky / String propagation

This is an "experimental" formula that applies to our examples above. Without justification we may try to conjecture that it applies to other coset models. Then for the N = 2 models of eqs. (27) and (28) with c = 15 and p = 9m we get a single time-like coordinate since ¢

2nm

2(p+m)n

3 = ~,

(35)

where we have requested 3 / 2 because of supersymmetry. It is not clear whether any such formula could apply for k < 0 since in this case there are no zero norm states [5, 6] and we have no good understanding of whether it is possible to project out states from the Hilbert space. The reason that it is interesting to wonder about this possibility is that it may explain the difficulties we encountered in the previous sections. If we did not have a single time-like coordinate in the quantum version of the theory then it would explain why the Virasoro constraints could not remove all the negative norms from the theory.

7. C o n c l u s i o n s

We have investigated models whose classical action describes string propagation in curved space-time with a single time-like coordinate and many space-like coordinates. We have mainly concentrated on backgrounds described by current algebra because of the exact treatment of conformal invariance afforded by these models. Such string backgrounds are generally both time and space dependent. We have shown evidence that a bosonic string or a superstring propagating in a background with curved time cannot be excited to arbitrarily high states. Although there are an infinite number of high excitations which satisfy the Virasoro or the super Virasoro constraints most of the highly excited levels contain states with a negative norm. A possible interpretation of this phenomenon is that the quantum theory may be describing a different time-like dimension than the classical theory. However, barring this uncertain possibility, an alternative possibility is that the theory remains consistent within the unitary sector allowing the string or superstring to live only in some low excitations. That is, if a string evolves into this type of space-time configuration it may propagate only in the allowed unitary states. Whether this is the right interpretation may have to await further understanding of interactions for such string states. If the string cannot remain as a consistent theory only within the unitary sector, then it could mean that the time coordinate refuses to curve in string theory. The phenomena we have uncovered may have some relation to possible singularities in string theory. They may also have implications for understanding string propagation and interactions under conditions similar to those in the early Universe and may lead to some new points of view in that area.

L Bars, D. Nemeschansky / String propagation

107

We would like to thank T. Banks, D. Gross, C. Vafa, and E. Witten for discussions.

Note added After this paper was submitted for publication we became aware of a paper by Petropoulos [10] which overlaps with some of our discussion in sect. 3. We have also been made aware that the supersymmetric SU(1,1) current algebra model is related to the Friedman-Gibbons electrovac solution [11]. References [1] C. Callan, D. Friedan, E. Martinec and M. Perry,, Nuci. Phys. B262 (1985) 593 [2] J. Balog, L. O'Raifeartaigh, P. For~acs and A. WipL Nucl. Phys. B325 (1989) 225 [3] R. Brower, Phys. Rev. D6 (1972) 1655; P. Goddard and C.B. Thorn, Phys. Lett. B40 (1972) 235; C. Thorn, in Vertex operators in mathematics and physics, ed. J. Lepowslq¢, S. Mandelsta~-n and I.M. Singer (Springer-Verlag, New York) p. 411 [4] D. Gepner and E. Witten, Nucl. Phys. B278 (1986) 493 [5] L. Dixon, J. Lykken and M. Peskin, Nucl. Phys. B325 (1989) 329 [6] I. Bars, Nucl. Phys. B334 (1990) 125 [7] M.B. Halpern, Phys. Rev. D4 (1971) 2398; P. Goddard, A. Kent and D. Olive, Phys. Lett. B152 (1985) 88 [8] K. Bardakci, E. Rabinovici and B. Saering, Nucl. Phys. B299 (1988) 151; H.J. Schnitzer, Nucl. Phys. B324 (1989)412; D. Karabali, Q-Han Park, H.J. Schnitzer and Z. Yang, Phys. Lett. B216 (1989) 307 [91 Y. Kazama and H. Suzuki, Nuci. Phys. B321 (1989) 232 [10] P.M.S. Petropoulos, Phys. Lett. B236 (1990) 151 [11] I. Antoniadis, C. Bachas and A. Sagnotti, Phys. Lett. B235 (1990) 255