hep-th/9603003, RU-96-12, IASSNS-HEP-96/19

Comments on String Dynamics in Six Dimensions

N. Seiberg

Department of Physics and Astronomy

Rutgers University, Piscataway, NJ 08855-0849, USA

and

E. Witten

Institute for Advanced Study

Princeton, NJ 08540, USA

We discuss the singularities in the moduli space of string compactifications to six dimensions with supersymmetry. Such singularities arise from either massless particles or non-critical tensionless strings. The points with tensionless strings are sometimes phase transition points between different phases of the theory. These results appear to connect all known supersymmetric six-dimensional vacua.

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1. Introduction

In this note we will study some aspects of the dynamics of supersymmetric compactifications of string theory to six space-time dimensions. There are four kinds of supersymmetry algebras in six dimensions, labeled by the number of supersymmetries of each chirality:

1. The minimal supersymmetry algebra, or (0,1), is generated by two spinors of the same chirality.

2. The chiral (0,2) algebra is generated by four spinors of the same chirality.

3. The non-chiral (1,1) algebra is generated by two spinors of one chirality and two spinors of the opposite chirality.

4. The maximal supersymmetry algebra, (2,2), is based on four spinors of one chirality and four spinors of the opposite chirality.

The simplest object to study is the moduli space of vacua of these theories. These are labeled by the expectation values of massless scalars. A lot of information about these spaces can be obtained without a detailed knowledge of the microscopic theory. The constraints following from supersymmetry and anomaly considerations turn out to be very powerful. Together with some information from string theory, these constraints lead to a beautiful and coherent description of the (2,2), (0,2) and (1,1) compactifications. In particular, all known compactifications of string theory (or M-theory) with these supersymmetries appear to be on the same moduli space of vacua.

In this note we will focus on theories, one of the questions being whether likewise they are all connected. models appear in compactifications of the heterotic or type I theories on K3 as well as in other examples. Such theories were studied recently in [1-9]. The massless representations of supersymmetry with low spins are labeled by their representations 1. gravity multiplet: 2. tensor multiplet: 3. vector multiplet:

There are two kinds of scalars – those from the hypermultiplets and those from the tensor multiplets. The moduli space of hypermultiplets is similar to that of the analogous supersymmetric theory in four dimension obtained by compactification on a two-torus. It is given by a quaternionic manifold. Singularities in this space can at least in many cases be associated with massive vector multiplets becoming massless thus leading to new massless vectors multiplets and massless hypermultiplets at that point. At such singularities the unbroken gauge symmetry is enhanced. This is the only dynamics of six-dimensional hypermultiplets that can be seen classically, and therefore the only kind that is possible in the quantum theory as long as the physics is free in the infrared (and therefore described at long distances by an effective classical theory). Upon reduction to five or fewer dimensions, there are additional possibilities, since below six dimensions the vector multiplet contains scalars, and a “Coulomb branch,” parametrized by the expectation values of these scalars, can emanate from a singular point in hypermultiplet moduli space.

There is, however, in six dimensions something that deserves to be called a Coulomb branch: it is parametrized by the expectation values of the scalars in the tensor multiplets. Upon reduction to five or four dimensions these multiplets become identical to vector multiplets, so this factor in the moduli space reduces below six dimensions to an ordinary Coulomb branch. One of the points of this note is the identification of the singularities in the six-dimensional tensor multiplet Coulomb branch with singularities associated to tensionless strings. As in four dimensions there can be Higgs branches emanating from these singularities.

In order to explore the Coulomb branch, we should first discuss some properties of the tensor fields. In general, theories in six dimensions have one gravity multiplet and any number of tensor multiplets, say . The gravity multiplet has a massless state in the (0,1) of while every tensor multiplet has a massless state in (1,0). These are two-form gauge fields with a one-form gauge invariance . The corresponding field strengths are constrained to be self-dual for the field in the gravity multiplet and anti-self-dual for those in tensor multiplets. Except for , there is no known Lagrangian description of these theories.

Since the tensor multiplets include gauge fields, the reparametrizations
of the Coulomb branch are quite restricted. Non-linear transformations
of the tensor fields would not preserve gauge invariance, and
hence their scalar superpartners
are also not naturally subjected to arbitrary
nonlinear transformations. The low energy
theory of the two-forms has a
duality group, which is a symmetry of the low
energy supergravity^{†}^{†} See
[10]
for review and references. when accompanied by suitable
transformations of the scalars. These transformations of the scalars
are the only natural ones to consider. Note that these are not
necessarily symmetries of the theory but merely the most general
freedom in the parametrization of the field space, preserving the
general form of the low energy supergravity.

As in SUSY in four dimensions, the metric on the hypermultiplet moduli space is independent of the scalars in the tensor multiplet. Also, as in four dimensions, the metric on the Coulomb branch and the kinetic terms for the vector and tensor fields are independent of the scalars in the hypermultiplets. They can depend only on the scalars in tensor multiplets.

Given these rules, there is no supersymmetric coupling that can be described in classical field theory which could lead to masses for massless particles by changing the scalars in the tensor multiplets. This means that such a process cannot occur as long as the physics is infrared-free. Conversely, it is impossible for massive particles to become massless at a particular point on the Coulomb branch, by any mechanism that can be described at low energies by a free field theory. Therefore, singularities in the Coulomb branch are more subtle. They necessarily involve the possibility that at the singular point, the physics is non-free in the infrared, a seemingly rather exotic possibility since conventional six-dimensional quantum field theories are believed to be infrared-free. We will argue that the singular points on the Coulomb branch are associated with strings that are becoming tensionless. These are non-critical string theories; non-critical here simply means that the graviton is not a mode of the string, so that the non-trivial infrared physics associated with the Coulomb branch singularities is occurring in flat six-dimensional Minkowski space. For this reason also, these non-trivial infrared fixed points very possibly may obey the general axioms of quantum field theory, for instance possessing a local energy-momentum tensor and other local fields, though apparently not being realized as Lagrangian field theories in any natural way. (This last point has been stressed to us by L. Susskind.)

It is helpful to have a concrete example in mind. Consider the simple case with , where it is easy to describe the system in terms of a Lagrangian. The free Lagrangian in the Einstein frame is

where is the coupling constant. Here and
are linear combinations of gauge and gravitational
Chern-Simons three-forms, and the four-forms and
can be interpreted as electric and magnetic
sources^{†}^{†} A similar Lagrangian in four dimensions with a one
form describes the coupling of Maxwell fields to electric and magnetic
sources.. This interpretation becomes obvious from the equation of
motion and the Bianchi identities for , which read

The charged objects are strings whose electric and magnetic charges are defined as

where is a four manifold transverse to the string. Note that for non-zero charges, or should decay sufficiently slowly at infinity. In this case the duality group is . The interesting duality transformation expresses the Lagrangian (1.1) in terms of the dual field strength . This duality transformation exchanges the two equations in (1.2) and inverts the coupling constant. Since it reverses the sign of the anti-self-dual tensor, by supersymmetry it should also reverse the sign of the scalar in that multiplet. Therefore, we can identify in (1.1) as that scalar.

The supersymmetry algebra, because of its chiral nature, can be extended by a central charge that transforms as a Lorentz-vector but does not admit Lorentz-scalar central charges. Of course, a Lorentz-vector central charge would be carried by a string, not a particle. This is in keeping with the fact that the supergravity multiplet contains a two-form, and not an ordinary gauge field; the two-form naturally couples to a string, not a particle. A standard BPS argument leads to an inequality for the tension of any string that couples electrically or magnetically to the field:

Here and are the electric and magnetic charges of the string and and depend on the coordinates on the moduli space. As in the analogous four-dimensional case, and can depend only on the scalars in the tensor multiplets and not on the scalars in the hypermultiplets. In the example (1.1) above and . Such BPS saturated strings were studied in [11].

As we move on the moduli space the tensions of the strings can vary. A string labeled by can become tensionless at the point where and lead to a singularity in the moduli space. The situation is very similar to the analogous one in in four dimensions. There masses of particles (rather than tensions of strings) are controlled by a BPS formula on the Coulomb branch. A singularity in the Coulomb branch is associated with massless particles. At the vicinity of this singular point we can focus on the string which becomes tensionless and on the unique tensor multiplet which couples to it. The other modes of the theory are either much heavier or decouple from that string. Then an effective Lagrangian like (1.1) gives a good description of the massless modes in the vicinity of the singularity. The singularity is at with . Furthermore, only the anti-self-dual field couples to the non-critical string which becomes tensionless there while the self-dual field decouples.

To make this discussion of the vicinity of the singular point more precise, we should define an appropriate scaling limit in which one sees the non-trivial infrared fixed point. In the previous discussion we have set the Planck scale to one, but now we will be more explicit and will denote it by . Then, the appropriate scaling limit is combined with holding fixed. In this limit all the massive modes, whose mass is of order , and most of the massless modes decouple. (If this transition occurs at weak string coupling, there are also modes whose mass is much smaller than . They decouple because of the small string coupling.) We are left with the modes of the non-critical string which becomes tensionless at the singularity (its tension is finite in the scaling limit) and one tensor multiplet whose scalar vanishes at the critical point. There might also be additional “relevant” massless fields in some cases. This scaling theory has an accidental duality symmetry (which might be a symmetry of the full theory) under which and . It is crucial that the gravity multiplet also decouples in this limit. This follows from the fact that in the limit gravity decouples from all finite mass (or finite tension) objects. Alternatively, it follows from the fact that the self dual field which is in the gravity multiplet decouples.

As in in four dimensions, one can conceive of the possibility of Higgs branches emanating from the singular point. On the Higgs branch some tensor multiplets are “Higgsed.” Even without knowing the details of the microscopic theory, anomaly considerations constrain the transition from the Coulomb branch to the Higgs branch. Denoting the number of tensor multiplets by , the number of hypermultiplets by and the number of vector multiplets by , anomaly cancelation implies

Therefore, if is unchanged and exactly one tensor is Higgsed, there should be 29 more hypermultiplets on the Higgs branch than on the Coulomb branch. Other transitions where the value of also changes are also conceivable.

In section two we will discuss compactifications of the type I and heterotic strings to six dimensions while in section three we will discuss the heterotic string. In section four we will examine compactifications of M theory on which do not correspond to perturbative vacua of the heterotic string. Using these compactifications we will interpolate between different vacua of the heterotic string. In section five we will discuss the strong coupling transition and will mention some vacua of string theory which do not have a dilaton.

2. Compactifications of the theory on K3

When the type I theory is compactified on a K3 surface the low energy theory has and the Lagrangian is of the form (1.1). For Type I, the scalar in the tensor multiplet is essentially the volume of K3 measured in the type I metric

In particular is independent of the ten dimensional dilaton .

The hypermultiplets include fields which are charged under the gauge symmetries and also some moduli of K3. In particular, one combination of and transforms in a hypermultiplet. Straightforward dimensional reduction shows that the appropriate combination is , the effective six-dimensional string coupling constant.

The coupling of the two-form

is correlated with the gauge kinetic terms [12]

through

and labels different gauge groups. We recognize the two linear combinations of Chern-Simons three-forms, and as and in (1.1). Instantons in the gauge fields are localized in four dimensions and hence they are strings in six dimensions. We see that an instanton in the gauge group labeled by with instanton number leads to a string with charges . Since the instanton field strength is self-dual in four dimensions, it preserves half the supersymmetries and hence this string is BPS saturated. The tension of this string is, from the central charge formula, . Note that this value of the tension can also be deduced from the value of the action of this configuration; it comes from the coefficient of the gauge kinetic term. If the charges are such that the tension can vanish, at some point, the gauge coupling diverges at that point. This is the transition point discussed in [1]. Here we identify it as associated with tensionless non-critical strings.

To some extent, the fact that the gauge coupling goes to infinity enables one to get an intuitive picture of how infrared non-trivial physics comes about. In six-dimensional Yang-Mills theory, the bare gauge coupling has dimensions of length. The theory is believed to be infrared-free, meaning that it is weakly coupled at distances much bigger than . As , the length scale above which weak coupling prevails becomes greater and greater, and at the critical point of the tensor moduli space, and there is no such length scale: the physics is non-trivial in the infrared.

Consider for example the special point in the moduli space with gauge symmetry . For we have and , and for and . Since , an instanton is purely electric . From a ten dimensional point of view, the gauge fields exist on a five-brane which fills the non-compact dimensions. Instantons on such branes were studied in [13], where it was shown that when they become small they can leave the brane as “elementary” strings. This is consistent with the fact that they are purely electric and therefore can be identified with the dual heterotic string. An instanton has (an overall factor of 2 appeared because of the index of the vector representation of ). In ten dimensions the small instanton is a Dirichlet five-brane. The six-dimensional instanton is obtained by wrapping this five-brane on K3 to yield a string in six dimensions. We see that for this string becomes tensionless and leads to a singularity. Note that this is not a strong coupling singularity. It happens at a point where the sigma model is strongly coupled (the sigma model coupling is of order one there) but the string coupling constant can be arbitrarily small.

An interesting subtlety has to be mentioned here. These two strings have several collective coordinates. One of them is associated with the width of the string – the size of the instanton. This collective coordinate is non-compact and its quantization leads to a continuum. Therefore, there is no gap in the spectrum of the string. It might be that this continuum of states can be interpreted as a brane in its ground state surrounded by soft non-Abelian gauge bosons. However, as explained in [13], the world-sheet theory of the instanton has another branch where there is clearly a gap in the spectrum. It is in this branch that the string can be identified as the elementary heterotic string. The other strings may also have such alternative branches.

Under heterotic - Type I duality, the Type I model just discussed can be described as an heterotic string, for which the singularity discussed above occurs at strong coupling. The scalar in the tensor multiplet is given in the heterotic string description by

where is the volume of K3 in the heterotic string units and is the ten dimensional heterotic dilaton. The various strings discussed in the type I compactification are easily identified in the heterotic theory. The purely electric string is the fundamental heterotic string while the dyonic string is a solitonic five-brane wrapped over the K3.

3. Compactifications of the heterotic string on K3

Compactifications of the heterotic string on K3 also give models in six dimensions. These models are associated with superconformal field theories. They are labeled, at least initially, by the instanton numbers and in the two factors. In this context the total instanton number

Classically, smooth instantons with or 3 do not exist. Therefore, we have the following possibilities:

1. , . At the generic point in the moduli space the instantons completely break the first and the unbroken gauge group is the second . At a special point (the standard embedding) the first is broken to . At that point the theory has 10 hypermultiplets in the of as well as 65 gauge invariant hypermultiplets and a single tensor multiplet () which includes the dilaton. It is easy to see that equation (1.5) is satisfied and the anomaly polynomial factorizes as it should [14,15]

with for the factor and for the factor. At more generic points on the moduli space can be Higgsed to a subgroup.

2. with . The instantons break the two factors. At special points, where all the instantons in each are at an subgroup, the unbroken gauge group is . There are also half hypermultiplets in , half hypermultiplets in as well 62 gauge invariant hypermultiplet and a tensor multiplet. Again, equation (1.5) is satisfied and the anomaly polynomial factorizes

with for the two ’s. At more generic points the two groups can be Higgsed. To break completely we need at least 10 instantons. Therefore in the theories with and the gauge symmetry can be completely broken.

As in (2.3), the anomaly polynomial (3.3) allows us to find the gauge kinetic terms. They are proportional to

where and are the field strengths of subgroups of the the first and the second factors. Assuming, without loss of generality, that , the gauge coupling of the first is always finite while the other gauge coupling diverges at with

At that point the string associated with small instantons in that gauge group becomes tensionless. We will return to this singularity below.

It can actually be shown using -duality that the model is equivalent to the heterotic string with standard embedding of the spin connection in the gauge group [16], and similarly the model is equivalent to that discussed in [2]. It also seems likely, given results in [8], that the and models are related purely at the level of conformal field theory, by some sort of -duality.

4. Compactifications of M-theory on

The eleven dimensional description (M-theory) naturally leads to additional supersymmetric vacua which do not have a perturbative heterotic string theory interpretations. Some of them were discussed in [3] and have a perturbative type II interpretation [4]. Others, which we will focus on below, were mentioned in [1]. The common fact about these vacua is that they have more than one tensor multiplet, .

Consider a compactification of M-theory on . The gauge fields of the two factors are on two different “end of the world” nine-branes [17]. We can now embed instantons in one factor and instantons in the other. However, there is an extra possibility not seen in the perturbative heterotic compactifications: we can also add five-branes which are located at points on , and fill the non-compact six dimensions. The location of such a five-brane on is labeled by five real parameters. These parameters form a hypermultiplet (the coordinate on K3) and a tensor multiplet (the coordinate in ). Therefore, together with the tensor multiplet which includes the dilaton (the distance between the two nine branes), such vacua have tensor multiplets. As these five-branes are sources for the antisymmetric gauge fields, the equation is now replaced by

In other words, an instanton from one of the nine-branes can be replaced by a five-brane [1].

We can easily extend the anomaly considerations to this more general case (see also the discussion in [18]):

1. . Here the unbroken group is and there are tensor multiplets (24 from the branes and one which includes the dilaton) and 44 hypermultiplets (24 from the locations of the branes and 20 moduli of the K3 metric). This spectrum satisfies (1.5).

2. , . At special points in the moduli space the instantons break the gauge group to . There are also tensor multiplets and half hypermultiplets in and gauge invariant hypermultiplets ( from the locations of the five-branes, from the moduli of the gauge bundle and 20 from the moduli of the metric). Again, (1.5) is satisfied. At more generic points on the moduli space the symmetry can be Higgsed.

3. . At special points the gauge symmetry is with tensors, half hypermultiplets in , half hypermultiplets in and hypermultiplets ( from the locations of the 5 branes, deformations of the two gauge bundles and 20 metric moduli). As before, (1.5) is satisfied and at more generic points the gauge symmetry can be Higgsed down.

Equation (1.5) is a necessary condition for cancelling the anomaly. Other conditions come from examining the anomaly eight form. In all of these cases it is

with for an unbroken and for an unbroken ( for subgroups of the first factor and for the second). In the special cases with it coincides with (3.3). The form of (4.2) shows that it is a sum of terms associated with the two different boundaries. Note that for the anomaly does not have to factorize because there are more two-forms to cancel it [12].

These new vacua of string theory which are not perturbative heterotic vacua allow phase transitions between different values of . The vacua with given initial values of can be thought of as the Higgs branch. As an instanton shrinks, a singularity is found. That singularity can be interpreted as resulting from a five-brane stuck to the end of the world nine-brane. It is very plausible that another phase is obtained when the five-brane moves away from the boundary. Arguments for this have been given in [19]. Since we gain a tensor multiplet when the five-brane leaves the boundary, the new phase can be interpreted as a Coulomb branch. This phase transition is, therefore, rather like what was discussed in general terms in the introduction. On the Coulomb branch there is a BPS saturated string which is an eleven-dimensional two-brane which stretches from the five-brane to the boundary (we know from [17] that two-branes can end on the boundary and from [20] that they can end on five-branes). At the singularity this string becomes tensionless. After being emitted to the bulk, the five-brane could possibly travel to the other end of and be absorbed on the other boundary. So in M-theory, assuming such transitions are really possible, all values of are connected.

Other singularities in the moduli space occur when the number of tensors is ; i.e. when there are at least two five-branes in the bulk. These singularities correspond to two five-branes approaching each other. Again, this is a singularity in the Coulomb branch which is associated with non-critical strings. This non-critical string was first discussed in [21] where it appeared near a singularity of the IIB theory compactified on K3 yielding a (0,2) supersymmetric theory in six dimensions. This singularity was interpreted as arising from two five-branes approaching each other in [20,22]. In [3], this singularity appeared in a theory with only (0,1) supersymmetry like the theories discussed here.

The scaling theory associated with this non-critical string, which we will call the non-critical Type II string, is different from that of the small instanton. One way to see that is to note that this non-critical string has twice as many supersymmetries as the other – the low energy scaling theory around the singularity has accidental supersymmetry. It has world-sheet supersymmetry (as is clear from its origin in Type II) in contrast to the world-sheet supersymmetry of the string related to small instantons. Another difference between the two cases is that the singularity associated with the non-critical Type II string is of real codimension five while the other is of real codimension one. One needs to tune a hypermultiplet (the distance between the two five-branes in K3) and the scalar in a tensor multiplet (the distance between the two five-branes in ) to find the non-critical Type II string. This is consistent with the (0,2) space-time supersymmetry of this string; one needs to tune a tensor multiplet of (0,2) supersymmetry, which includes five real scalars, to find the string; the five scalars form in space-time supersymmetry a hypermultiplet and part of a tensor multiplet. This might seem in contradiction with the fact mentioned in the introduction that the BPS bound is independent of hypermultiplets. Why is it then that we have to tune a hypermultiplet to find the tensionless string? The answer is that this string is BPS-saturated in (0,2) supersymmetry but it is not BPS-saturated in (0,1) supersymmetry. Therefore, when the scalar in the tensor multiplet is tuned to a point where the BPS bound vanishes but the hypermultiplet is at a generic point, this string has positive tension. However, by tuning also the hypermultiplet it can become tensionless.

One can show that, unlike the case of the small instanton, there is no Higgs branch emanating from this singularity. One way to see that is to consider the type IIB string compactified on K3 where the same non-critical string appears. In this case anomaly considerations fix the number of (0,2) tensor multiplets and there is no branch where this number is reduced.

The singularity in the moduli space associated with this transition is the orbifold singularity . The identification by reflects Bose symmetry of the two five-branes. Therefore, when one adjusts the parameters to see make the tension of this string vanish, one sees an enhanced symmetry of the physics. We return to this point in the next section.

5. First Look At The Strong Coupling Singularities

We have mentioned above other kinds of singularities in the Coulomb branch – those associated with the diverging gauge coupling. In this section we are going to take a first look at them. For simplicity we limit the presentation to the case of one tensor multiplet and to compactifications which have a perturbative heterotic string description. They are labeled by the two instanton numbers , and we will assume without loss of generality that . Then the gauge coupling of a subgroup of the second diverges at the value of satisfying (3.5). At that point instantons in that group lead to tensionless strings and complicated dynamics can arise. Independent of what this dynamics is, it is clear that the gauge bosons in the first are spectators which do not participate in it. This is obvious, for instance, in the M-theory description, where the two ’s are supported on K3’s that can be arbitrarily far apart (while making one of them larger), only one of which is affected by the singularity.

What kind of string can arise at the strong coupling singularity? For , there is a generic unbroken gauge group , whose coupling goes to infinity at . Therefore, in this case, there must be a singularity at for generic values of the other fields. The fact that the string becomes tensionless upon adjusting and nothing else strongly suggests that it is BPS-saturated, with a tension controlled purely by . The instanton in the strongly coupled gauge group will indeed do the job if there is nothing else. For , there is no value of at which gauge couplings diverge [1], so one has no reason to expect a singularity to arise on adjusting for generic values of the other fields; the existence of such a singularity would really contradict the duality of [1]. For , such general arguments do not make it clear what happens at .

If a singularity does arise at , can one continue beyond it in space? The obvious intuitive idea, analogous to symmetry in conformal field theory, is that the physics at might be isomorphic to that at , related by a symmetry. This is, however, possible only if is the self-dual value of the dilaton, that is if , since the only transformation of the dilaton that leaves invariant the low energy supergravity is . A look at (3.5) shows that if and only if . Thus, only at this value of the instanton number is the naive idea of finding isomorphic physics beyond the strong coupling singularity conceivable.

Note further that it has been argued from F-theory [8] that
the and models are actually the same (equivalent on
the nose, not just connected by a series of phase transitions). The
theory does indeed have a symmetry, so if
they are equivalent, the theory must have one also. Thus,
the continuation beyond the singularity must hold in this case. Since
the theory does not have a strong coupling singularity at
for generic values of the hypermultiplets, the same must be
true at ; as we noted two paragraphs ago, there is no
contradiction here, since the theory has generically no
unbroken gauge group. On the other hand, since upon adjusting some
additional parameters the theory does have unbroken gauge
symmetries whose couplings diverge at , it must be that it
develops a tensionless non-critical string upon adjusting
together with some other parameters. Since more parameters than
are involved, this string must not be BPS-saturated at generic
values of the parameters, so it must carry a and not just
world-sheet supersymmetry. In fact, from F-theory one can
see [16] that the relevant string at the strong coupling point of
the model is the non-critical
Type II string of [21], which also
entered above in discussing models with more than one tensor
multiplet; as we discussed above, to make this string tensionless, one
must adjust one hypermultiplet as well as one tensor
multiplet.^{†}^{†} The key point in getting the non-critical
Type II string here
from F-theory is that the F-theory description of this model
involves [8] compactification on the Hirzebruch surface , and the singularity arises when an exceptional two-sphere of
self-intersection number collapses; the resulting singularity
looks just like Type IIB theory at an singularity, which
gives the Type II string as in [21]. We explained above
that no Higgs branch emanates from a singularity of this nature.
As remarked at the end of the last section, a symmetry
appears at a point at which the non-critical Type II string becomes
tensionless. In the case under discussion, this is
the strong-weak coupling symmetry of the
model.

For further insight, we look at the structure of the gauge kinetic energy for , as was first done in [7]. Using (3.4), the gauge kinetic terms are

The fields are expected to be spectators in the strong coupling transition, for reasons given above. The fact that their coupling is invariant under is, as noted in [7], compatible with the existence of a symmetry under which they are spectators. The gauge fields have a coupling that diverges at the self-dual point and are definitely not spectators. (The gauge fields are only massless if some parameters in addition to are adjusted, to restore a gauge symmetry that generically is spontaneously broken. To make sense of the singularity in the gauge coupling, the parameters in question must include the extra hypermultiplet that must be adjusted to make the non-critical Type II string tensionless; this can be verified using F-theory.) It must be that the formula (5.1), which holds in the region containing the perturbative heterotic string of , should be modified so that the coefficient of is really . We do not understand very well how this comes about, though since the gauge fields involved are not spectators there does not appear to be a contradiction.

The fact that the gauge coupling diverges at is related to the fact that there is no term in the anomaly eight-form; this in turn means that the gauge fields couple to hypermultiplets with the same quadratic Casimir operator as that of the vector multiplet. But, upon reduction to four dimensions, that is the condition for vanishing beta function! This seems like a rather interesting way to generate a large class of finite models from string theory. The symmetry of , and the behavior near the strong coupling singularity, are thus very plausibly related, upon toroidal compactification to four dimensions, to interesting behavior of finite models.

Could there be for any values of a Higgs branch emanating from the strong coupling singularity? Since the number of tensor multiplets is for perturbative heterotic strings, a Higgs branch would have , no tensor multiplets at all. It would be a branch without a dilaton, something one would like to find in four dimensions! Since in such a branch, there is only the self-dual two-form in the supergravity multiplet, anomaly cancelation requires [12] that the anomaly eight-form should not only factorize but should be a perfect square. Since the gauge fields in the first are spectators in such a transition, they should survive in the Higgs branch. So we can test for the occurrence of such a branch by looking at the anomaly form including the Riemann tensor and (but not the fields, which might get masses on the Higgs branch). It is easy to find, using (1.5), that the relevant anomaly is

This is a perfect square only for , and thus . For other values of there can be no Higgs branch.

We do not know the interpretation of this result for , but for one can show using F-theory that the Higgs branch does indeed exist [16]. In fact, the Higgs branch in this case is simply F-theory on (which has no tensor multiplets as is clear from [6]). The transition to the Higgs branch is made by blowing down the exceptional curve in the Hirzebruch surface (the right surface for as explained in [8]) to go to . The non-critical string is made by wrapping a Type IIB three-brane around , and carries a rank eight current algebra, strongly suggesting that it coincides with the string seen in M-theory when a five-brane approaches the boundary; that string also carries a rank eight current algebra, and seems to have a Higgs branch (small instanton), as discussed in [19] and above. Actually, the M-theory transition where a five-brane enters or leaves the boundary can be seen in F-theory by blowing up and blowing down points; by a sequence of such blow-ups and blow-downs, to give transitions from to , one can see the M-theory process in which an instanton is emitted at one boundary as a five-brane and then reabsorbed at the other boundary, changing and .

Acknowledgements

We would like to thank J. Polchinski, S. Shenker, and L. Susskind for discussions. This work was supported in part by DOE grant #DE-FG05-90ER40559 and in part by NSF grant #PHY95-13835.

References

[1][email protected] Duff, R. Minasian and E. Witten, hep-th/9601036. [2][email protected] Gimon and J. Polchinski, hep-th/9601038. [3][email protected] Sen, hep-th/9602010. [4][email protected] Dabholkar and J. Park, hep-th/9602030. [5][email protected] Witten, hep-th/9602070. [6][email protected] Vafa, hep-th/9602022. [7][email protected] Aldazabal, A. Font, L. E. Ibanez, and F. Quevedo, hep-th/9602097. [8][email protected] Morrison and C. Vafa, hep-th/9602114. [9][email protected] Aspinwall and M. Gross, hep-th/9602118. [10][email protected] Salam and E. Sezgin, eds., Supergravities In Diverse Dimensions (World Scientific, 1989). [11][email protected] Duff, S. Ferrara, R.R. Khuri and J. Rahmfeld, hep-th/9506057. [12][email protected] Sagnotti, Phys. Lett. 294B (1992) 196, hep-th/9210127. [13][email protected] Douglas, hep-th/9602098. [14][email protected] Erler, J. Math. Phys. 35 (1994) 1819, hep-th/9511030. [15][email protected] Schwarz, hep-th/9512053. [16][email protected] Witten, to appear. [17][email protected] Horava and E. Witten, hep-th/9510209. [18][email protected] Witten, hep-th/9602070. [19][email protected] Ganor and A. Hanany, hep-th/9602120. [20][email protected] Strominger, hep-th/9512059. [21][email protected] Witten, hep-th/9507121. [22][email protected] Witten, hep-th/9512219.