NSF-ITP-99-104

RI-8-99, EFI-99-41

hep-th/9909110

Little String Theory in a Double Scaling Limit

Amit Giveon and David Kutasov

Institute for Theoretical Physics

University of California, Santa Barbara, CA 93106, USA

Racah Institute of Physics, The Hebrew University

Jerusalem 91904, Israel

Department of Physics, University of Chicago

5640 S. Ellis Av., Chicago, IL 60637, USA

A double scaling limit can be defined in string theory on a Calabi-Yau (CY) manifold by approaching a point in moduli space where the CY space develops an isolated singularity and at the same time taking the string coupling to zero, while keeping a particular combination of the two parameters fixed. This leads to a decoupled theory without gravity which has a weak coupling expansion, and can be studied using a holographically dual non-critical superstring description. The usual “Little String Theory” corresponds to the strong coupling limit of this theory. We use holography to compute two and three point functions in weakly coupled double scaled little string theory, and study the spectrum of the theory in various dimensions. We find a discrete spectrum of masses which exhibits Hagedorn growth.

9/99

1. The double scaling limit

In the last few years it became clear that consistency of string theory implies the existence of new non-local theories without gravity in six or less dimensions. One way of defining these “little string theories” (LST’s) is to consider string dynamics in vacua which contain -branes, in the decoupling limit [1]. In this limit, fluctuations in the bulk of spacetime decouple, and one is left with the dynamics of modes which live on the fivebranes. For example, taking this limit in type IIA or IIB string theory in flat spacetime with fivebranes leads to maximally supersymmetric six dimensional theories (sixteen supercharges) with or supersymmetry, respectively.

LST’s do not contain gravity but have other features that are reminiscent of critical string theories. In particular, they have a Hagedorn density of states, and upon compactification on tori they exhibit T-duality. There are some apparent differences as well: (1) Unlike critical strings, for which it is believed that one can only study on-shell physics (e.g. the S-matrix), LST’s are expected to have well defined off-shell Green’s functions. (2) Critical string theory in flat non-compact spacetime has a fundamental scale, the string scale , and a dimensionless coupling, . For small , the theory is weakly coupled. LST’s, as defined above, have a scale , and are further labeled by the number of fivebranes . They appear to be inherently strongly coupled. One might be tempted to associate with a coupling in LST, but there is no known sense in which the theory becomes weakly coupled at large (it does become weakly coupled at large for sufficiently low energies). Nevertheless, we will see that one can in fact define LST’s which are arbitrarily weakly coupled, so in this respect LST is actually similar to critical string theory.

Another definition of LST, which will be more useful for our purposes, involves the study of string vacua of the form , where is a CY manifold with an isolated singularity, in the decoupling limit . Since in this limit all non-trivial physics is localized near the singularity, to study the resulting LST one can replace the CY manifold by its form near the singular point.

In the maximally supersymmetric, six dimensional case of [1], and the possible singularities have an ADE classification. The theory on -branes mentioned above corresponds to an singularity, near which the manifold looks like an ALE space – the singular, non-compact hypersurface

in . The two descriptions of six dimensional LST given above are related by T-duality [2,3]. The theory on type IIA -branes is equivalent to the IIB theory on times the ALE space (1.1), and vice versa.

An interesting generalization of the above procedure is the following. Let be one of the moduli of CFT, and be the value of for which the surface develops an isolated singularity. The standard LST limit is , followed by . Instead, one can consider a double scaling limit, i.e. take and at the same time, keeping some combination, , fixed.

To describe this limit more precisely it is convenient to blow up the vicinity of the singularity and replace the surface by the ALE space (1.1) describing the region near the singularity (at ). One way of smoothing the singularity is to replace (1.1) by

where can be thought of as one of the moduli of the underlying CFT ( in the previous notation). As we will see later, string theory on times the surface (1.2) depends on , only through the scaling parameter . The double scaling limit corresponds to , with held fixed. Note that: (1) The string coupling expansion corresponds in this case to an expansion in powers of . (2) The original definition of LST corresponds to ( first, then ), and hence gives rise to a strongly coupled theory. (3) While in string theory on one actually has to take the double scaling limit to decouple the LST from gravity and bulk string theory, replacing the surface by (1.2) automatically implements the limit. In particular, does not have to be small in (1.2); in fact, the LST becomes weakly coupled at large . (4) As emphasized in [4], the double scaling limit defined above is qualitatively very similar to the one used in the study of two dimensional critical string theory [5], and one can borrow much of the intuition developed there. (5) String theory on resolved ALE spaces was considered in some very interesting pre-holography papers [6,2]. We will comment later on the relation of these papers to our work.

The deformation (1.2) has a natural interpretation in terms of
the theory of -branes^{†}^{†} We thank O. Aharony and T.
Banks for discussions of this issue.. Rewriting (1.2) as

and performing the T-duality of [2,3], we find a system of fivebranes distributed uniformly on a circle of radius , which is related to as in (1.3). In type IIB string theory, the low energy theory on -branes is SYM theory with sixteen supercharges, and (1.3) corresponds to a point in the moduli space of vacua of the gauge theory, in which a complex scalar field in the adjoint representation of the gauge group has a v.e.v.

The double scaling limit with fixed has a nice interpretation in this language. At the point (1.4) in moduli space, the gauge symmetry is broken to , and the mass of the off-diagonal gluons (“W-bosons”) is (since W-bosons correspond to -branes stretched between the fivebranes).

Thus, the double scaling limit is simply the decoupling limit of [1] with fixed W-boson masses. The scaling parameter is the scale of these masses (in string units). The double scaled theory is weakly coupled for energies well below . For one expects to recover the strongly coupled LST at the origin [1]. A similar story (with slightly different details) can be told for type IIA fivebranes.

The above discussion can be generalized to higher dimensional Calabi-Yau spaces. One again studies the limit where the CY manifold develops an isolated singularity and blows up the region near the singularity to define the double scaling limit. There are many possible singularities that can appear on CY manifolds. We will restrict our attention to quasi-homogenous hypersurface singularities, in which the vicinity of the singularity is (as in (1.1)) a hypersurface in , where is a quasi-homogenous polynomial with weight one under , i.e.

for some set of positive weights . The fact that the singularity is isolated implies that is transverse, i.e. the only point at which all derivatives vanish is the origin, . The double scaling limit gives rise in this case to a dimensional non-gravitational theory. It involves string theory on times the non-compact manifold

We will see below that the combination of and on which physics in this background depends is the scaling variable

where

We will restrict to . The resulting LST is weakly coupled for large and we will study it in this limit.

2. The holographic description

The construction described in the previous section is useful for establishing the existence of LST’s, but it does not provide efficient techniques for studying them in detail. To go further, we will use a holographic description of these theories. This description, which is a generalization of the AdS/CFT correspondence [7], was proposed in [8] (see also [9,10,11,12,13,14]) and was generalized to the case of general hypersurface singularities (1.6) in [4]. We will next briefly summarize it.

The claim is that holography relates the decoupled dynamics at the singular point in string theory on times the space (1.5) (“the boundary theory”) to a “bulk theory,” string theory on

where is the real line, labeled by , along which the dilaton varies linearly,

and is a Landau-Ginzburg SCFT of chiral superfields with superpotential (1.5). It is easy to check [4] that consistency of string propagation on (2.1) requires a relation between (1.8) and (2.2):

We will label the in (2.1) by ; its radius is determined by the GSO projection to be . The GSO projection further acts as an orbifold by a shift along the circle, , and (roughly) as fermion number on the rest of the theory (see [4] for the details).

For the six dimensional, maximally supersymmetric case (1.2), the superpotential is , and the LG model is an minimal model, which can be thought of as the coset SCFT . The GSO projection acts as a orbifold on , turning it into . The backgound (2.1) thus becomes , recovering the results of [15].

The theory described by (2.1) is singular. The string coupling blows up as ; the theory runs to strong coupling. This is not surprising in view of the earlier discussion, since (2.1) is dual to the singular background (1.6) with , or (1.7) , which as we mentioned is strongly coupled. One would like to turn on in (1.6) and study the dual theory for large , where it is weakly coupled. Turning on in (1.6) corresponds [16,4] in the dual theory (2.1) to adding to the worldsheet Lagrangian the superpotential term

Here , are worldsheet superfields whose bosonic components are the scalar fields , discussed above. A few comments about the interaction (2.4) are in order: (1) This is the Liouville interaction. It preserves worldsheet superconformal symmetry and hence spacetime supersymmetry. At the same time it prevents the dilaton from running to strong coupling. (2) Even before we add (2.4) to the Lagrangian, the fact that this operator is in the spectrum of the theory implies that the radius of is an integer multiple of . An analysis of the rest of the spectrum shows [4] that this radius is equal to . (3) We can now derive (1.7). Standard scaling arguments (see e.g. [5]) imply that the perturbative expansion in the vacuum (2.1), (2.2), (2.4) is a series in . Using (2.3), this leads to (1.7).

It is interesting to compare our discussion to the work of [6,2]. In these papers it was pointed out that the sigma-model whose target space is the non-compact manifold (1.6) can be alternatively described by a Landau-Ginzburg model with superpotential

where is an additional chiral superfield, and

The description (2.5) is natural since the equation in dimensional weighted projective space satisfies the CY condition with , and if we can set it to one and recover the space (1.6). It was also argued in [6,2] that the patch with should not appear since in the LG phase the superpotential (2.5) pushes to large values.

The first term in the superpotential (2.5) appears to be ill defined: the corresponding potential does not have a minimum at a finite value of , and in general is non-integer, which makes non single valued. In [6,2] it was proposed to interpret it as an coset SCFT at level , and arguments were presented to support this interpretation. Geometrically, this coset corresponds to a semi-infinite cigar [17], with the string coupling going to zero far from the tip and approaching an arbitrary finite value at the tip ().

Comparing (2.1), (2.4) with (2.5) we see that our description of (1.6) is very similar, but appears to be different from that of [6,2]. For the two are identical: both contain a LG model with superpotential , and an infinite cylinder with a linear dilaton along . When is turned on, the two descriptions naively disagree. In our case, the strong coupling singularity at is cut off by a superpotential (2.4), whereas in [6,2] it is eliminated by changing the topology of the cylinder to that of the semi-infinite cigar.

Since both descriptions are quite well motivated, we would like to propose that they are in fact equivalent, i.e. that Liouville (with a cosmological constant ) and (with at the tip of the cigar) are isomorphic SCFT’s. They are related by strong-weak coupling duality on the worldsheet (possibly, a kind of T-duality). We have not proven this statement but would like to offer the following comments in its support: (1) The matching of the central charges of the two theories relates the slope of the linear dilaton (2.2) to the level of . The relation, which follows from (2.3), (2.6), is . Thus, if the two are equivalent, the relation between them is indeed a strong-weak coupling duality. Liouville is weakly coupled for large Liouville central charge, i.e. large , while is weakly coupled for large (when the curvature of the cigar goes to zero everywhere). (2) Recall that the radius of the (labeled by ) in Liouville is . The asymptotic radius of the Euclidean cigar is [17]. The two are indeed inverses of each other, , as appropriate for T-dual theories. (3) In Liouville, the cosmological term breaks translation invariance in . In contrast, in translation invariance around the cigar is not broken, but winding number is not a good symmetry, since winding modes around the cigar can move to the tip and contract to a point. This again is consistent with an interpretation of the relation between the two theories as T-duality, which exchanges winding and momentum modes. In both theories, as the respective symmetries are restored. (4) A closely related duality to the one advocated here appeared in unpublished work by V. A. Fateev, A. B. Zamolodchikov and Al. B. Zamolodchikov. It relates the bosonic CFT to a bosonic analog of our with the superpotential (2.4) replaced by a bosonic potential of the form .

Finally, we should remark on the possible relation of the duality proposed above to the standard T-duality with respect to the Abelian isometry of the cigar CFT [18]. The latter takes the cigar into a trumpet, which seems different from the Liouville background at small . However, it might be that non-perturbative worldsheet effects modify the trumpet background. The Abelian T-duality of an ALE space, mentioned previously, is an example where the naive dual background is expected to receive worldsheet instanton corrections which modify it to the (near-horizon geometry of the) -brane background (more generally, the Abelian dual of a Kaluza-Klein monopole is expected to receive worldsheet instanton corrections which modify it to an H-monopole [19]).

3. Correlation functions and spectrum

In this section we will use the holographic description of [8,4] to compute correlation functions in weakly coupled double scaled little string theory. Recall that, in general, holography relates observables in the non-gravitational “boundary theory” to non-normalizable on-shell states in the “bulk theory.” A large class of such observables corresponds to non-normalizable vertex operators (i.e. fundamental string states) in the bulk theory, and we will focus on those here.

To get started, we need to decide which of the two T-dual descriptions of the bulk theory to use, the or Liouville one. As explained in section 2, the two are supposed to be equivalent, but each is more appropriate in a different region of parameter space. For large it is better to use , while for small the Liouville description provides a better qualitative guide to the physics. The transition between the two descriptions occurs at (corresponding to the self dual radius ).

In our context, lies in the range^{†}^{†} The lower bound corresponds
to pure Liouville or superconformal ,
where . .
For example, in the maximally supersymmetric six dimensional
case we have , the number of fivebranes, and it can become
arbitrarily large, but is bounded from below by two.
Therefore, it is natural to use the variables
to describe the observables and compute correlation functions.
We next recall a few facts about CFT.

3.1. CFT

A convenient description of the coset CFT is obtained by starting with and gauging the null diagonal obtained from of and the extra (and similarly for the other worldsheet chirality). Realizing the extra in terms of a canonically normalized free scalar field , we gauge the axial symmetry

which leads to the cigar geometry. Observables in the coset theory are obtained from observables in CFT by imposing invariance under (3.1) (and identifying gauge equivalent observables). Thus, one can use results about correlation functions in CFT to study the coset.

Observables in CFT^{†}^{†} More precisely, we will
be discussing the Euclidean analog of the group
manifold, . (see e.g. [20] for
a recent discussion) are obtained by studying functions
on the manifold and then applying to them the generators
of the current algebra. A convenient basis for such
functions is

where are coordinates on (the metric is ) and ) are auxiliary variables introduced in [21]. In the quantum CFT on , the are primaries of the full current algebra; their scaling dimensions are

For future reference we record their behavior for large (see [20] for a more detailed discussion),

For studying the coset it is convenient to “Fourier transform” the fields and define the mode operators

where is the volume of the boundary of . The OPE algebra of the mode operators with the currents is

and similarly for .

A set of observables in the coset theory is obtained by coupling to ,

Note that is not charged under (3.1) and hence is physical. Its scaling dimensions are

where run over a set of momenta and windings consistent with the fact that the radius of is [17]:

One can think of as the momentum around the cigar (which is conserved) and as the winding (which is not).

Not all values of are allowed in this theory. Non-normalizable vertex operators correspond to real . Furthermore, unitarity implies that only operators with are kept [22]. Unitarity also implies that all scaling dimensions of Virasoro primaries in the CFT should be positive (except for the identity); in particular, (3.8) must satisfy .

At large positive the form of the observables (3.7) simplifies. Using (3.4), (3.5), (3.7) we find that

As explained in [23], in this region the theory simplifies and the target space can be thought of as a cylinder, , labeled by and . The powers of , in (3.10) can be dropped and go over to the standard observables in a linear dilaton vacuum.

We will be interested below in a supersymmetric version of the above coset CFT. This leads to a few small modifications in the analysis. If we denote by the total level of the current algebra on , then the level of the bosonic part of the algebra is (with the fermions contributing units). Thus, the scaling dimension of the primaries (3.2) is shifted by and the unitarity bound of [22] becomes

Since the current that we are modding out by still has level , formulae like (3.1) remain unchanged. Thus, in the superconformal coset we have observables given by (3.7) (with , satisfying (3.9)) whose scaling dimensions are

Positivity of the scaling dimensions (3.12) implies in this case the constraint

Two and three point functions of the primaries (3.2) in CFT were computed in [24]. The two point function is

where is a numerical factor which can be found in [24]. Since we will be primarily interested in the analytic structure of the two point function, it is sufficient to note that does not have poles or zeroes for finite and . We have written (3.14) in a notation suited for application to the supersymmetric case, i.e. the level of the bosonic implied in (3.14) is . The two point function (3.14) does not have any singularities for all satisfying the inequality

As approaches the lower bound in (3.15), the two point function goes to zero, while when it approaches the upper bound it diverges.

Clearly, for physical applications (e.g. for string theory on ) both types of singularities are undesirable. For example, outside of the range (3.15) states in the spacetime CFT of string theory on have negative norm. Fortunately, the lower bound in (3.15) is precisely the condition for non-normalizability of the vertex operators discussed above, while the upper bound almost coincides with the unitarity constraint (3.11). In fact, (3.15) is slightly stronger. Note that this is not in contradiction with [22]. In these papers it is assumed that the current algebra primary state has positive norm, and one asks whether there are any negative norm states in the current algebra block obtained by acting on the primary with creation operators (after modding out ). The two point function (3.14) instead probes the question whether the norm of the primaries changes sign as one varies . Thus, below we will impose (3.15) on the observables.

Using (3.14) one can now compute the two point functions of observables in the coset theory (3.7). The first step is to transform from the “local fields” (3.2) to the modes (3.5). To simplify the formulae we will set , i.e. restrict to pure winding modes around the cigar (3.9). One finds

(here and below we supress the dependence).

To compute the two point function of coset observables (3.7) we multiply (3.16) by the two point function of the exponential in . This does not change the result; hence we find

3.2. Two point functions in double scaled LST

To use the results of the previous subsection for calculating two point functions in double scaled little string theory we have to construct BRST invariant observables in string theory on

The analysis is very similar to that of [4,16]. We will focus on (NS,NS) sector fields, since the rest of the observables can be obtained from these by applying the spacetime supercharges.

The lowest lying states are the “tachyons,” described by the picture vertex operators

where () is the momentum along and , are the left and right moving bosonized superghost fields. The mass-shell condition and GSO projection lead to the following physical state constraints:

Another set of observables corresponds to gravitons, whose picture vertex operators have the form

where are the worldsheet fermions on , is the polarization tensor, and the physical state constraints are

There are also transversality conditions on which we will not specify.

The most general (NS,NS) sector observable is a linear combination of vertex operators of the following form:

where is a polynomail in , , , and their derivatives; and are its total left and right-moving scaling dimensions, respectively. Similarly, is a polynomial in , , their worldsheet superpartners , and their derivatives, with total scaling dimensions . is a vertex operator from the sector of (3.18). BRST invariance and the GSO projection require (3.23) to be a bottom component of a worldsheet superfield, and in addition to satisfy

where and are the left-moving scaling dimension and charge of , respectively, and is the left-moving fermion number of (3.23) (similar relations hold for the right-movers). Moreover, the operator (3.23) must be invariant under the symmetry (3.1). Note also that is not in general integer, unlike (3.9) with . The reason for that is that the GSO projection acts as an orbifold on (see [4]), coupling it to the rest of the background (3.18).

We are now ready to compute correlation functions of the observables (3.19), (3.21), (3.23). Recall that correlation functions of non-normalizable on-shell vertex operators in the bulk theory are interpreted in the boundary theory as off-shell Green’s functions [7,8]. Thus, their analytic structure provides information about the spectrum and interactions of the boundary theory. The analytic structure of the two point function, for example, contains information about the spectrum of the boundary theory. We will next study it in our case.

For the general vertex operator (3.23), the only non-trivial part of the two point function is the worldsheet correlation function . This is the only source of singularities of the amplitude, and thus we will focus on it. As a first example, consider the two point function of the tachyon field (3.19),

where , satisfy the constraints (3.13), (3.15), (3.20). Note that the from (3.17) has disappeared; it cancelled against a zero mode of the ghosts which normally would make the two point function vanish in string theory.

The two point function (3.25) has a series of poles, which we interpret as contributions of on-shell states in LST which are created from the vacuum by the operator . We next analyze these poles.

One series of poles corresponds to

It is easy to check that these are single poles. The residues of the poles all have the same sign, in agreement with the expected unitarity of LST. The corresponding masses of excitations are determined by the first line in (3.20):

For the smallest posssible value of , (recall the second line in (3.20) and (3.26)), we find (using (3.15)) states with , with masses

The lowest lying state is massless; it is followed by a finite series of excited states. More generally, we have

The range of is obtained from (3.15). Note that Lorentz invariance of the LST implies that all the states that can be created from the vacuum by the tachyon field (and whose masses are given by (3.29)) have spin zero.

The poles of (3.25) which are
related to (3.26) by are not independent
of those discussed above. All other poles violate
one or more of the bounds^{†}^{†} Note,
for example, that if we imposed the constraint (3.11) rather
than (3.15), we would be lead to the rather unpleasant
prediction that LST has tachyons. The pole of (3.25) at ,
leads (3.20) to , which
is in general negative. Furthermore, if is integer (3.25) has a double pole at that value of , which seems difficult
to interpret in LST. (3.13), (3.15).

The massless state corresponding to in (3.29) has a
natural interpretation in the fivebrane theory. We will describe
it in the six dimensional maximally supersymmetric case, but
the discussion can probably be extended to general singularities
(1.6). As we saw in section 1, six dimensional double scaled
type IIB LST describes the dynamics of fivebranes at the point
(1.4) in the Coulomb branch. Due to the Higgs mechanism, the
only massless states one expects to find in this theory are
supermultiplets corresponding to the Cartan subalgebra of .
The massless state with in (3.28) belongs to one of these
supermultiplets^{†}^{†} It is sufficient to exhibit any one of the members
of the supermultiplet, since the supersymmetry structure, worked out
in [4], then guarantees the appearance of the rest of the supermultiplet,
with the right quantum numbers..

It is not difficult to exhibit the other massless multiplets. For this, we need to examine the two point functions of tachyons (3.19) which have a non-trivial wave function in the minimal model denoted by in (3.18). In our case the superpotential is (1.1)

corresponding to an minimal model. The chiral worldsheet operators can be written as , with . A natural generalization of the tachyon vertex (3.19) is

where

which is simply (3.24) for this particular case. The tachyon (3.19) is in fact . The two point function of is proportional to that of , (3.25), and has poles giving rise to a spectrum similar to (3.29). It is easy to check that there is a physical pole at , , , which gives rise to a massless state in six dimensional LST. There are thus precisely massless states, corresponding to , in agreement with the above expectations from gauge theory.

Moving on to the two point function of the graviton, we see that in this case with . The two point function again has poles for satisfying (3.26), which using (3.22) corresponds to the mass spectrum

All the states that the bulk graviton (3.21) couples to are
massive^{†}^{†} The two point function of the graviton provides an example of the
importance of the constraint (3.13). Ignoring it, one finds a series
of poles of (3.17) at , , ,
which, if present, would correspond to tachyons in LST with masses which
can be read off (3.22), ..
This is consistent with the fact that the boundary
theory is not a theory of gravity – the spectrum does not
contain a massless spin two particle.

Consider, finally, the two point function of the most general on-shell vertex operator (3.23). Let be the total excitation level corresponding to this vertex operator. As before, the poles correspond to (3.26), which in this case leads to the spectrum

where is defined as in the first line of (3.29), . It runs over the range implied by the second line in (3.24), , subject to the unitarity constraint

which generalizes (3.13) to operators which have a non-trivial projection in .

3.3. Three point functions

Three point couplings between the states (3.34) can be obtained from the three point function in the usual way (by taking the external legs on-shell and computing the residue of the resulting poles). To find the three point functions in the double scaled LST one follows the same steps as in the computation of the two point functions. We will only present the procedure schematically, leaving the details for future work. First, one needs the three point functions of the primaries (3.2) in the CFT. Those are given in [24]:

where

does not depend on , and its explicit form is given in [24]. To compute the three point functions in the coset theory we need to transform the fields , , to the modes (3.5); this leads to integrals of hypergeometric functions depending on . Finally, the three point functions in the spacetime theory are obtained by following the same steps as in subsection 3.2.

4. Discussion

By defining Little String Theory in a double scaling limit (see section 2), we were able to study the theory at weak coupling, (1.7). We used this description to compute correlation functions in the theory to leading order in .

One of the main results of this paper is the calculation of the two point functions of observables in double scaled LST ((3.25) and its generalization to other vertex operators), and the corresponding spectrum of masses (3.34).

The two point functions we computed have a relatively simple
analytic structure. They exhibit a
series of single poles which we interpreted as arising from
single particle states in LST, created from the vacuum by
applying different operators^{†}^{†} As we pointed out, a nice
consistency check is that the residues of all poles in the
two point functions have the same sign, which is
necessary for unitarity of LST.. This might come as a surprise,
since in general one would expect two point functions in a
non-trivial theory such as LST to be complicated functions
of the momenta. The eigenstates of the Hamiltonian need not
be interpretable in terms of free particles, and one would
expect a complicated structure containing resonances, thresholds
(branch cuts) and “brown muck.”

Of course, crucial to the simplicity of the two point functions is the fact that the theory is weakly coupled. As mentioned above, we only computed the two point function to leading order in , and it is reasonable to expect the theory to be free in this limit. A good analogy is confining gauge theory in four dimensions. For finite the spectrum of the theory is complicated and, correspondingly, the two point functions of gauge invariant observables are expected to have a complicated analytic structure. As the theory simplifies and becomes essentially a free field theory of the bound states (glueballs). The two point functions of observables like are expected to simplify in the limit and have an analytic structure similar to the one found here.

Another interesting fact about the two point functions is that while the full spectrum of masses we find is rather rich, most operators do not couple to most of the states! There are two aspects to this, one that seems natural, and a second one that looks more surprising.

The natural one is the following. Even if the spectrum of LST at large consists of free particles with the masses (3.34), one would in general expect that in the absence of special symmetries, generic observables can create from the vacuum both single and multi-particle states. The latter would contribute branch cuts in the two point functions. The fact that we only find poles means that the observables that we study, on-shell vertex operators in the bulk theory, only couple to single particle states in LST. This seems natural, since they are single string states in the bulk, and we are performing an expansion in . In the gauge theory analogy mentioned above, fundamental string vertex operators are analogs of single trace operators in large Yang-Mills theory.

More surprisingly, we find that even within the set of single particle operators/states, most operators do not couple to most states. The tachyon operators (3.19) only couple to states (3.34) with , the gravitons (3.21) couple to states with , etc. This is not expected to happen in general. E.g. in the large gauge theory example the spectrum is believed to exhibit Hagedorn growth, and one expects that all single trace operators have a non-zero overlap with all states which have the same quantum numbers (e.g. the same spin). For example, the operator mentioned above, should be able to create from the vacuum all states with spin zero.

The fact that in LST we find an infinite number of additional selection rules suggests that the weakly coupled theory has a very large symmetry group. It would be nice to understand it better.

It is natural to suggest that in LST there is a state-operator correspondence, namely, that each operator in the bulk can create a particular one particle state in the boundary theory. For example, a tachyon vertex operator (3.19) with given , (or , ) can create from the vacuum a unique state, with mass given by (3.29). If this proposal is correct, it is easy to compute the degeneracy of states with any mass (3.34). Since the mass spectrum (3.34) is very similar to that of the bulk theory, and the state-operator correspondence suggests that the degeneracies are the same as well, one finds a Hagedorn density of states

where , the inverse Hagedorn temperature, is the same as that of the bulk theory. We emphasize that this last conclusion assumes the state-operator correspondence, and should be examined more carefully.

Note that all the poles of the two point functions that give rise to the masses (3.34) occur for values of that satisfy (3.26) (or, equivalently, its mirror image related by ). Algebraically, these are values for which the underlying representations are one sided. They contain a state, , which is annihilated by (or , annihilated by ). These representations are sometimes refered to as the principal discrete series. A natural question is why does the principal discrete series play a special role in the analysis, and how one can think of the resulting spectrum of the LST.

We would like to make two comments regarding these issues. The first is that the appearance of the principal discrete series is not particularly surprising if we compare to the better understood case of string theory on [7,25]. In that case, the holographically dual spacetime theory is a two dimensional CFT and the operators (3.2) correspond to local operators in this theory.

While there are well known subtleties [26,24] with the state-operator correspondence in CFT on , the spacetime CFT is a more or less conventional unitary CFT, and in particular it does have a state-operator correspondence. The state corresponding to a given operator ( is a vertex operator which depends on the rest of the string background; see [25] for the details) is obtained by acting with on the vacuum. In radial quantization the ‘in’ vacuum corresponds to , hence incoming states correspond to .

Expanding in modes, as in (3.5), we see that the mode that creates the above state in the spacetime CFT has . Modes with larger must kill the vacuum, while those with smaller do not contribute in the limit . Similarly, there are states created by acting on the vacuum with . The modes that create these states have . The modes (3.26) discussed in section 3 are related to them by ; they create states when acting on the ‘out’ vacuum at .

Thus, at least in string theory on the modes (3.26) corresponding to the principal discrete series certainly do create physical states in the spacetime theory. It is perhaps not surprising that something very similar seems to be going on in the case.

The second comment concerns the physical interpretation of the spectrum (3.34). It is known [27] that the dynamics of winding modes on the cigar is equivalent by T-duality to the motion of a particle in a potential which is attractive for small and goes to a constant at large . The spectrum contains bound states corresponding to the principal discrete series, and a continuum above a gap, which corresponds to the normalizable states with . The discrete states can presumably be thought of in the original variables as some sort of bound states whose wave function is supported near the tip of the cigar. The states with make wave packets that can live arbitrarily far from the tip of the cigar and have an arbitrary momentum (hence the continuum). It is easy to verify using (3.24), (3.34) that the continuum starts right above the highest mass discrete state (in a sector with given , ), as expected from the form of the potential for the winding modes [27].

One can ask whether the normalizable states with
should be included in the LST. For this to be the case, it seems to us
necessary that there are some observables in the theory which
can create these states from the vacuum. The class of observables
analyzed in this paper, the non-normalizable fundamental string operators,
does not seem to couple to these states. If no other observables do
either, we would be inclined to believe that these modes decouple
from the LST dynamics^{†}^{†} This issue is further confused by the
fact [24,26] that in correlation functions of non-normalizable
operators in CFT on one is typically instructed to sum over
all the normalizable states in internal channels. Nevertheless, it seems
that the singularity structure of the resulting correlation functions
can be interpreted purely in terms of states bound to the tip of the cigar..

Returning to the spectrum (4.1), our conclusion that the density of states of the LST grows exponentially with energy, with the same Hagedorn temperature as that of the bulk theory, seems to be at odds with the discussion of [28], where it was argued that the Hagedorn temperature is in fact lower by (roughly) a factor of (and thus the density of states grows more rapidly with energy than (4.1)).

Our attitude to this apparent discrepancy is the following. The analysis of [28] applies in our language to energies , while we computed the spectrum of excitations with energies . We find a Hagedorn growth of the density of states (4.1) with a smaller than that of [28] in the range . It is reasonable to expect that the density of states with is much higher and agrees with [28].

The regime is strongly coupled in our description and is difficult to study directly. As has been learned in many examples in the last few years, typically only states that are protected by supersymmetry can be reliably counted at strong coupling. Perhaps D-brane states which preserve some of the supersymmetry in the background (3.18) would resolve the discrepancy (see [29] for a recent discussion).

Finally, some of the vacua of LST that our discussion applies to correspond to theories on fivebranes wrapped around Riemann surfaces, which are relevant for describing strongly coupled gauge theories using branes [4]. We hope that our results will help improve this understanding.

Acknowledgements: We thank O. Aharony, T. Banks, O. Pelc, M. Porrati, J. Teschner, C. Vafa and A. B. Zamolodchikov for useful discussions. We also thank the ITP in Santa Barbara for hospitality during the course of this work. D.K. thanks the HET group at Rutgers University for hospitality. This research was supported in part by NSF grant #PHY94-07194. The work of A.G. is supported in part by the Israel Academy of Sciences and Humanities – Centers of Excellence Program, and by BSF – American-Israel Bi-National Science Foundation. D.K. is supported in part by DOE grant #DE-FG02-90ER40560.

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