CERN-TH/95-231 HUTP-95/A032 hep-th/9508155

Nonperturbative Results on the Point Particle Limit of

N=2 Heterotic String Compactifications

Shamit Kachru, Albrecht Klemm, Wolfgang Lerche,

Peter Mayr and Cumrun Vafa

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138

Theory Division, CERN, 1211 Geneva 23, Switzerland

Using heterotic/type II string duality, we obtain exact nonperturbative results for the point particle limit () of some particular four dimensional, supersymmetric compactifications of heterotic strings. This allows us to recover recent exact nonperturbative results on gauge theory directly from tree-level type II string theory, which provides a highly non-trivial, quantitative check on the proposed string duality. We also investigate to what extent the relevant singular limits of Calabi-Yau manifolds are related to the Riemann surfaces that underlie rigid gauge theory.

CERN-TH/95-231 August 1995

Recently there have been exciting developments in understanding
non-perturbative aspects of string theory through conjectured string
dualities [1]. In particular, the geometry of moduli spaces of
and supersymmetric string vacua is getting better
understood. Since for the geometry of the moduli space is
uncorrected even non-perturbatively, the cases are much more
interesting, as far as shedding light on non-perturbative dynamics of
string theory is concerned. This is also mirrored in the interesting
dynamics of the field theories
[2,3,4,5,6]. In this paper we
show how some of the exact results on the quantum moduli space of
certain string vacua [7] can reproduce in the point particle limit (where
) the exact field theory results of
[2].^{†}^{†} The question of going to the point particle limit has been
addressed before in [8,9,10,11,12].
This provides a truly nonperturbative, quantitative check on the
proposed heterotic/type II string duality.^{†}^{†} It also
substantiates the conjectures that the local analogs of the
Seiberg-Witten Riemann surfaces are given by Calabi-Yau manifolds
[8,9] and that space-time Yang-Mills instanton
effects can be described in terms of world-sheet instantons of type
II string theory [4].

We will concentrate on the two main models studied in [7], for which there have already been many non-trivial checks in perturbation theory [13,14,15]. We will first study in some detail the rank three model of [7], which we will call model A, and then discuss how our results generalize to the second main model of [7] (the rank four model, which we will call model B). More details, especially concerning the string and gravitational contributions to the exact nonperturbative effective action, will appear in a subsequent paper [16].

1. Description of Model A

Model A has two equivalent descriptions: We can view it as the heterotic string compactified on , where we choose the and moduli of the two-torus to be equal so that there is an extra gauge symmetry. We also choose the second Chern class of the gauge bundle to be , giving a total of that equals the second Chern class of ; this is required for world-sheet anomaly cancellations. This model has hypermultiplets whose scalars characterize the geometry of with the corresponding bundles on it, and vector multiplets whose scalars give the modulus of the two-torus and the dilaton/axion field . The dual description of this model is given by a type IIB (or type IIA) string compactification on a Calabi-Yau manifold (or its mirror), with defining polynomial

where are coordinates of and where we mod out by all phase symmetries that preserve the holomorphic three-form. This Calabi-Yau manifold has and , giving rise to vector multiplets (whose scalar expectation values correspond to and above) and 129 hypermultiplets (including the type II string dilaton). If we wish to study the moduli space of vector multiplets, tree-level type II string theory is exact, whereas if we wish to study the moduli space of hypermultiplets, the tree level of the heterotic side is exact. In this paper we consider the moduli space of vector multiplets, and so we study the classical moduli space of the type II side spanned by and in the above defining equation.

The classical moduli space of this model has been studied in great detail in [17,18]. It is convenient to introduce the variables

According to the identification of [7], the and fields of the heterotic side should be identified (in the large /weak coupling regime) with the special coordinates corresponding to and , respectively. In particular, for large one has:

This identification was in part motivated by the fact that at the perturbative heterotic model develops an gauge symmetry. The existence of the gauge symmetry of heterotic strings is reflected by the existence of the conifold locus of , which is given by

For weak coupling, , there is a double singularity at (corresponding to ). Moreover, for finite coupling corresponding to finite , there are two singular loci for , in line with the field theory results of [2] where one has two singular points in the moduli space associated with massless monopoles/dyons.

2. What to Expect when Gravity is Turned Off?

It would be a very non-trivial test of all these ideas if we could show that in the limit of turning off gravitational/stringy effects, we would reproduce the results of [2], where the quantum moduli space of pure Yang-Mills theory with gauge group has been studied. This corresponds to considering the point particle limit of strings obtained by taking . To this end note that the variable that vanishes at the point should, to leading order, be identified with . To make this dimensionally correct, we must have

Note that as , the full -plane is mapped to an infinitesimal neighborhood of . This in particular means that the effect of the modular geometry of is being turned off in this limit, as one expects. Furthermore, in order for the scale of the theory to satisfy , we should tune the string coupling constant (which is defined naturally at the string scale) to be infinitesimally small. Taking into account the running of the gauge coupling constant, we should take, to leading order in :

Thus, by dimensional transmutation the coupling constant of , , can be traded with the scale , at which the gauge theory becomes strongly coupled. Note that the conifold locus (1.4) in the limit goes to

which is the expected behaviour.

Let us recall that supergravity moduli are characterized by a prepotential , which in our case is a function of . Using the axionic shift symmetry, it is easy to see that it has an expansion of the form [8]

where corresponds to one-loop string corrections and where is the contribution from the -th stringy instanton sector. This expansion is most convenient when we are dealing with large . Since we are interested in near , it is more convenient to shift to [19], and consider another expansion of given by

where is some polynomial of first degree in , and where we have chosen . We now consider turning off gravity by taking the limit . Note that since both and the variable defined in [2] are good special coordinates and are proportional to leading order, they have to be identified via

This is consistent^{†}^{†} To be precise, in order to recover the
conventional definition of in relation to , note that there is
a proportionality constant in this equation related to the second
order expansion coefficient of the -function near . We can
avoid this by rescaling in the definition of (2.1) and redefining
in such a way that
is invariant. This will have no effect on the equations below. with
(2.1), and also correctly translates the modular transformation to the Weyl transformation
[9,20,21,12].
Now using (2.2) and (2.6) we reexpand (2.5) and get

In order to recover the results of Seiberg and Witten as , we must find for large :

where are the instanton coefficients of [2] in the weak coupling regime. In other words, let be the prepotential obtained in [2]. Then, if behaves as above, we would have

where is a quadratic polynomial of first degree in and second degree in . In order to compare the above result with the periods of that we will determine in this limit below, it is useful to recover, using special geometry, the periods from the prepotential given in (2.9). Since we are working in a non-homogeneous basis, the -type periods can be taken to be (proportional to)

while the corresponding -type periods are given by

This simplifies due to the remarkable fact (whose full physical significance remains to be uncovered) proven in [22] that

This is crucial for us in obtaining the rigid theory in the limit of turning off gravity. Using (2.12) we find that the periods must be certain linear combinations of the periods with . Thus, up to linear combinations, we should get the following 6 periods

We will verify below that the periods of the Calabi-Yau manifold in the limit are indeed given by linear combinations of the above six periods.

3. Geometrical Characterization of the Appearance of

Before going on in the next section to solve the Picard-Fuchs equations and obtain the six periods that we expect to emerge in the point particle limit, we would like to give a geometrical idea of how the two most interesting periods, namely the rigid periods of [2], appear. Given the fact that in [2] were periods of a meromorphic one-form on a torus, it is important to see, by a means more transparent than direct computation, how this geometrical structure is encoded in the Calabi-Yau manifold in the vicinity of , .

As we approach the conifold locus in moduli space, some three-cycle is shrinking to zero size. We expect that the computation of is only affected by integrals localized in the neighborhood of the collapsing cycle. Therefore, we should try to understand the appearance of the rigid periods by approaching , in moduli space and by simultaneously rescaling variables to “blow up” a neighborhood of the singular locus on the manifold .

From the above we know that as we approach the point of interest,
the moduli of scale as^{†}^{†} In the following, we use the
dimensionless variable .

If we want to keep the conifold singularity at a finite point in our rescaled variables, fixing it turns out that the unique choice of rescaling is , . Then, if we in addition define , and rescale by irrelevant numerical factors, we find that the defining equation of can be rewritten as

Here we have simply used that as , the defining polynomial can be written as .

As goes to zero, the leading singularity is described by . However, note that this is itself a singular space! It is quite clear that the periods that we are interested in are governed by the subleading piece (3.3), which smooths out the singularity in (3.2) for finite . More concretely, we are suggesting that the periods related to three-cycles that are not collapsing as we approach should be controlled by the leading term in the -expansion. On the other hand, the -dependent periods, , are governed by the sub-leading term, , which is the first -dependent term in the -expansion. Therefore, in order to study the periods , it should be enough to focus on the variation of with . Furthermore, since we are interested in the leading behavior in the expansion, we can solve for in (3.2) at the singularity. This will be more fully justified below.

Thus solving for the singular locus in , we find , , and substituting this into , we see that the manifold whose Hodge variation must give is the curve

We notice that this curve is very similar to the following torus in (in the patch where ),

which underlies the rigid gauge theory [2]. Note, in
particular, that the two curves share the same discriminant locus,
. One may in fact view (3.4) as a triple cover of the
Seiberg-Witten torus.^{†}^{†}
A related point was made in [11] where it is observed that a triple
cover of
(3.5) (for ) appears as the locus in
.

Obtaining the geometrical torus, or more precisely a geometrical object with equivalent Hodge variation, is not quite enough to yield the periods and . Another ingredient in [2] was the choice of a particular meromorphic one-form , whose periods are and . The meromorphic one-form has no residue and satisfies , where is the holomorphic one-form on the torus. How does this emerge for us? In order to address that and make our discussion of the limit above somewhat more rigorous, we use the definition of periods given in [23,24]. That is, we write the periods of as

where is given by the defining polynomial in weighted projective space, and are a basis for an appropriate class of cycles [23]. In our case, . We can now reformulate what we were doing before: In the limit as the leading contribution to the integral comes from going to the saddle points of (this is of course nothing but the stationary phase method). Thus we have to find the ‘minima of the action’, which means solving

The minima of the action are not isolated, because gives a singular manifold. Shifting

we find that

It is easy to see that for fixed the ‘mass matrix’ for
has nonvanishing determinant. Thus the ‘fields’
for generic are infinitely
massive^{†}^{†} This is true for all except – it may
be that
the other four periods are related to this contribution. as
. We can therefore integrate them out, and
this results, in leading order, in substituting in , which is precisely what we did above. From the
gaussian integration we get in addition a factor of , so
that

In order to make contact with geometry, we are at liberty to add an extra -independent integral, since multiplying ’s by overall constants is irrelevant. We thus choose

(the choice of the term in front of the measure will be explained momentarily). In the computations of the periods we are free to make any birational transformation, so we choose , , and obtain (after irrelevant rescaling of variables)

Note that the choice of above makes the factor in front scale invariant, and it is thus a function on the resulting elliptic curve. We then find that the above period is the same as that of the following meromorphic one-form (by going to the patch and rewriting , ):

Note that this meromorphic one-form has only second order poles, so its periods make sense, and also that

which is a necessary requirement [2]. In fact, these two ingredients fix up to an exact form. After the transformation [25] which brings the quartic torus (3.7) and the cubic torus of [2] into Weierstrass form one can show that differs from the meromorphic one form of [2] only by an exact piece. Thus computed from the periods of agree with those of [2].

4. Specialization of Picard–Fuchs Operators

In the previous section we have seen how two of the periods of reproduce the expected point particle limit periods . In this section we will show, by solving the Picard–Fuchs equations, that all six periods have indeed the expected form (2.13).

Solutions of the relevant Picard-Fuchs equations have been computed in [18], and provide the instanton-corrected period vector in the large complex structure limit (). However, for comparison with the rigid Yang-Mills theory, we need to expand the periods around (), a point which is outside the radius of convergence of the solutions given in [18]. Therefore, we will make a variable transformation to solve the Picard–Fuchs system directly in variables centered at . This is the point of tangency between the conifold (monopole) locus, , and the weak-coupling line, (see [17] for details of the moduli space).

In order to obtain appropriate solutions in form of ascending power series, partly multiplied by logarithms, we have to be careful in the choice of variables. A proper way to do that is to blow up twice the point of tangency by inserting s. This leads to divisors with only normal crossings, and the associated variables will automatically lead to solutions of the desired form. More precisely, as shown in Fig.1, the blow-up introduces two exceptional divisors, and . The latter can be associated with the Yang-Mills quantum moduli space, which is given by the plane. It intersects with the other divisors at the points and , corresponding to the semi-classical limit, the massless monopole points and the orbifold point, respectively.

Fig.1 The double blow-up of the intersection of the conifold locus with leads to three divisor crossings and thus to three canonical pairs of expansion variables . They describe the physical regimes of the Seiberg-Witten theory at , and , respectively.

To recover the rigid periods , we consider for the time being the specialization of the Picard–Fuchs system only in the semi-classical regime, , which corresponds to the intersection of with ; the other two regimes can be treated in a completely analogous way. The appropriate variables are and ; in particular, is the variable which we will send to zero when we turn off gravity. After transforming the Picard–Fuchs system [17,18] to these variables, we find four solutions with index and two solutions with index , where the index is defined by the lowest powers of the variables (modulo integers) that appear in the solution. From the monodromy around it is clear that the solutions related to the rigid periods must be those with index . Specifically, we find for the first Picard–Fuchs operator in the limit the following leading pieces:

Note that this vanishes identically when applied to a function of the form . On the other hand, after rescaling the solutions by (which is motivated by the form of the solutions given below), the second PF operator becomes:

For , this becomes precisely the PF operator of the rigid theory [26] that has as solutions
! ^{†}^{†} From the form of solutions given below one can see that
there are no logarithms of in the relevant solutions, so that
the dependence in cannot cancel out when acting on
them. Note also that is obtained from the differential
operator that acts on the ordinary torus periods
by
. Note, in addition, that
for , (modulo an exact form),
confirming our choice of meromorphic one-form in (3.7).

Having thus explicitly shown that the Seiberg-Witten periods appear as solutions of the Calabi-Yau Picard-Fuchs system in the limit , it remains to verify that the structure of the full Calabi–Yau period vector is consistent with our physics expectations. Indeed, the leading terms of the six solutions in the limit are given by:

in perfect accordance with (2.13) ! (We intend to present the precise linear combinations that correspond to the geometric periods elsewhere [16].) Note that the appearance of odd powers of signals the breaking of the discrete -symmetry of the rigid Yang-Mills theory to . This is due to the string winding modes, which break this symmetry already in string perturbation theory.

5. Calabi–Yau Monodromies and the Heterotic Duality Group

In the previous section we have solved the Picard–Fuchs equations near . We would now like to find the monodromies of the Calabi-Yau manifold, which represent non-perturbative quantum symmetries from the viewpoint of the heterotic string. In particular, we will show that certain monodromies reproduce the monodromies of the quantum Yang-Mills theory and thus underlie the Riemann-Hilbert problem whose solution is given by the Seiberg-Witten periods, .

The calculation of the monodromy generators is completely analogous to that of the octic discussed in [17], and we refer to this paper for details and notation. In summary, the monodromy group is generated by three elements, denoted by ,, and , which are obtained by loops in the moduli space around the identification singularity , the strong coupling singularity and the conifold singularity , respectively.

The period vector in an integral basis is determined by the holomorphic prepotential as . By fixing the integral basis as in [27,17,28], we find in the large complex structure limit the following prepotential:

where and are inhomogeneous special coordinates,
defined as quotients of the periods . Here, is the unique power series
solution in the domain around , while ,
are the unique solutions of the form
, . More
precisely, via mirror symmetry and are the complexified
parameters of the Kähler classes and that generate
the Kähler cone and that correspond to the divisors in the linear
system of degree
two monomials and degree one monomials. As a consequence, the cubic
part^{†}^{†} The constants are related to topological invariants
[28] by and [27] , i.e. in this case
, and . of
is fixed by the classical intersection numbers and is given by
.

The Kähler structure parameters can be related to the heterotic
moduli by . We now identify the
generators that correspond to the semi-classical heterotic duality
symmetries, namely to T-duality and to the dilaton shift. The shift
generators are immediately determined by the
large complex structure limit to be , whereas a generator respecting the weak
coupling limit and acting as on the
semi-classical period vector is given by
. A generator that acts on the special
coordinates in a particularly interesting^{†}^{†} As was pointed out in
[13], this transformation leads to a symmetry that is quite
mysterious from the point of view of heterotic strings, if we choose
the following, alternative identification: . For
this identification, simply acts as an exchange of ! way is given by .

An explicit matrix representation of the generators of the duality group in the large complex structure basis is given by:

To make contact with the results of [2], we make a further symplectic change of basis to the string frame introduced in [9], which is characterized by a semi-classical period vector of the form

In this basis, the monodromies and read

Note that contains the monodromy [2] at in an unexpectedly simple way: the non-trivial matrix acting on the 3rd and 6th entry of the period vector is precisely of the rigid gauge theory, in a basis with periods . In fact, we can do better and determine also the strong coupling (monopole) monodromies from the fact that the monodromy around the conifold locus is . In this way, we find , with , where and are matrices with the monopole monodromies and in the entries as the only non-trivial elements.

The identification of the semi-classical heterotic string monodromies, as well as of non-perturbative monodromies of the field theory limit as part of the Calabi–Yau monodromies, provide a non-trivial check on the type II-heterotic string duality. More importantly, we get a prediction for the non-perturbative duality group of the heterotic theory: it is generated by and , subject to certain relations that are implied by the large complex structure limit [17,28], and by the Van–Kampen relations. Specifically, one can show that

It is easy to see that the PF solutions of the previous section are compatible with the above monodromies.

6. Discussion of Model B

We would like to discuss how Model specializes to the rigid , and Yang-Mills theories, respectively, near the appropriate points in moduli space; we will mainly be concerned with the point, but will also briefly discuss the two other cases.

On the heterotic side, Model B is obtained by compactifying the heterotic string on with the second Chern class of the gauge bundle chosen to be . This model has three vector multiplets containing , and 244 hypermultiplets. Note that in perturbation theory we get in this model an enhanced gauge symmetry on the line , an gauge symmetry at the point , and an gauge symmetry at the point [29,20,21,12].

In the dual type II string framework [7], the defining polynomial of the Calabi-Yau manifold is: