HUTP-01/A011, UCSD-PTH-01-03 hep-th/0103067

A Large Duality via a Geometric Transition

F. Cachazo, K. Intriligator and C. Vafa

Jefferson Physical Laboratory

Harvard University

Cambridge, MA 02138, USA

UCSD Physics Department

9500 Gilman Drive

La Jolla, CA 92093

We propose a large dual of 4d, supersymmetric, Yang-Mills with adjoint field and arbitrary superpotential . The field theory is geometrically engineered via D-branes partially wrapped over certain cycles of a non-trivial Calabi-Yau geometry. The large , or low-energy, dual arises from a geometric transition of the Calabi-Yau, where the branes have disappeared and have been replaced by suitable fluxes. This duality yields highly non-trivial exact results for the gauge theory. The predictions indeed agree with expected results in cases where it is possible to use standard techniques for analyzing the strongly coupled, supersymmetric gauge theories. Moreover, the proposed large dual provides a simpler and more unified approach for obtaining exact results for this class of supersymmetric gauge theories.

March 2001

1. Introduction

Partially wrapping D-branes over non-trivial cycles of non-compact geometries yields large classes of interesting gauge theories, depending on the choice of geometry. It has also been suggested in [1,2] that D-branes, wrapped over cycles, have a dual description (in a suitable regime of parameters) involving transitions in geometry, where the D-branes have disappeared and have been replaced by fluxes. This duality can be reformulated and explained as a geometric flop in the context of M-theory propagating on holonomy manifolds [3,4]. In this paper, we use these ideas to propose a new class of dualities.

The simplest case, which will be the main focus of this paper, corresponds to an supersymmetric gauge theory with adjoint chiral superfield and tree-level superpotential

where the gauge group can be either or , depending on whether or not we treat as a Lagrange multiplier imposing tracelessness of . For simplicity, we generally refer to , with the understanding that the can be obtained by imposing the Lagrange multiplier condition. Without the superpotential (1.1), the theory would be super-Yang-Mills. The theory with superpotential (1.1) arises [5] by wrapping type IIB D5 branes on special cycles of certain Calabi-Yau geometries; the choice of and the parameters are given by the geometry. Using the corresponding geometric transition, we construct a dual theory without the D-branes, but with suitable fluxes. There is also a mirror IIA description, involving D6 branes wrapping 3 cycles. The IIB description is simpler, in that there are no worldsheet instanton corrections to the superpotential. However, the IIA perspective is useful for explaining the origin of these dualities, as they are related to geometric flop transitions in M-theory on holonomy geometries [3].

The classical theory with superpotential (1.1) has many vacua, where the eigenvalues of are various roots of

In the vacuum where classically , the gauge group is broken as

In the geometric construction [5], this is seen because we can wrap D5 branes on any of choices of . Such a vacuum exists for any partition of .

Applying the proposal of [1,2] to each , a transition occurs where we are instead left with s. As we discuss, the non-compact Calabi-Yau geometry is now given by the following surface in :

with the degree polynomial (1.2) and a degree polynomial. As for any Calabi-Yau, we can form an integral basis of 3-cycles, and , which form a symplectic pairing

with the periods of the Calabi-Yau given by the integral of the holomorphic 3-form over these cycles. In the present case (1.4), we have , with the cycles compact and the non-compact. We denote the periods as

with the prepotential. is a cutoff needed to regulate the divergent integrals; this is actually an infrared cutoff in the geometry integral, which will naturally be identified with the ultraviolet cutoff of the 4d QFT. Using (1.6), the polynomial in (1.4) is to be solved for in terms of the periods .

As in [2], the dual theory obtained after the transitions to the geometry (1.4) has a superpotential due to fluxes through the 3-cycles of (1.4):

with 3-form flux through and 3-form flux through [6,7]. If not for the superpotential (1.7), the dual theory would yield a 4d, supersymmetric, gauge theory, with the the chiral superfields in the vector multiplets. In terms of this field theory, the superpotential (1.7) corresponds to breaking to by adding electric and magnetic Fayet-Iliopoulous terms [8]. There will be supersymmetric vacua, with the massive and thus fixed to some particular , but with the gauge fields left massless. In the applications we consider, all , the bare gauge coupling of the gauge theory; this combines in a natural way with , replacing the cutoff with the physical scale of the gauge theory.

The duality proposal, generalizing that of [2], is that these gauge fields coincide with those of the original theory (1.3) after the get a mass gap and confine. In particular, the exact quantum effective gauge couplings of the remaining massless gauge fields should be given by the prepotential of the above dual, , evaluated at . Further, as in [2], the are to be identified with the “glueball” chiral superfields , whose first component is the gaugino bilinear. Finally, we claim that the superpotential (1.7) is the exact quantum effective superpotential of the low-energy theory with superpotential (1.1), in the vacuum with the Higgsing (1.3).

Note that the dual theory only knows about the values of the via the coefficients appearing in (1.7). In particular, the and are completely independent of the , depending only on and the parameters via (1.4). Upon adding (1.7) to the dual theory, one obtains which are complicated functions of the , , and . Integrating out the gives the exact quantum 1PI effective superpotential of the original theory.

The geometric transition leads to a new duality, which can be stated in purely field theory terms: the theory with adjoint and superpotential (1.1) is dual to a theory and superpotential (1.7). This duality is reminiscent of that of [9].

The above duality makes some highly non-trivial predictions for the exact gauge couplings and the exact effective superpotential . This allows us to check the duality, by comparing with the exact results which can (at least in principle) be obtained for these quantities via a direct field theory analysis. The above quantities can be exactly obtained (again, at least in principle) by viewing the theory with adjoint and superpotential (1.1) as a deformation of , and using the known exact results for field theories. We find perfect agreement between these results, which is a highly non-trivial check of our proposed duality.

The organization of this paper is as follows: In section 2 we review the large duality of [2] for Yang-Mills theory, and briefly discuss the extension to include massive flavors in the fundamental of . In section 3, we discuss how to geometrically engineer the general theory with adjoint and superpotential (1.1). In section 4 we propose the large dual of these theories via the transition in the CY geometry where s are blown down, s are blown up, and the branes have been replaced with fluxes. In section 5 we analyze the theory with adjoint and superpotential using standard supersymmetric field theory tools. In section we specialize these results to the case of the cubic superpotential. In section 7 we analyze the proposed large duals and show how the leading order computation of gauge theory based on gaugino condensate follows from monodromies of the geometry. In section 8 we specialize to the cubic superpotential and compute exact results for the quantum corrected superpotential using the proposed dual. We find perfect agreement with the results based on a direct gauge theory analysis. In appendix A we present the details of the analysis for one of the field theory examples, and in appendix B we discuss the series expansion for computing the periods for the case of cubic superpotential.

2. Review of the large N duality for Yang-Mills

Consider type IIA string theory on a non-compact Calabi-Yau threefold of , i.e. the conifold, with defining equation given by

and consider wrapping D6 branes on the , with the unwrapped dimensions filling the Minkowski spacetime. This gives rise to a 4d pure Yang-Mills theory. The duality proposed in [2], which was motivated by embedding the large topological string duality of [1] into superstrings, states that in the large limit this theory is equivalent to type IIA strings propagating on the blow up of the conifold. This is a geometry involving a rigid sphere , where the normal bundle to the in the CY is given by a bundle over it (i.e. two copies of the spinor bundle over the sphere). The branes have disappeared and have been replaced by an RR flux through and an NS flux on the dual four cycle [2]. This duality has been embedded into M-theory, where it admits a purely geometric interpretation [3,4]. The gauge theory decouples from the bulk in the limit where the size of the blowup sphere is small. The size is fixed in terms of the units of flux, and the appropriate decoupling limit is large . gets identified [2] with the glueball superfield of the theory, so its expectation value corresponds to gaugino condensation in the theory.

As noted in [2] one can also consider the mirror description of this geometry, which is simpler to work with (as the worldsheet instanton corrections to spacetime superpotential are absent). This corresponds to switching from IIA to IIB theory and reversing the arrow of transition: the original theory is obtained from type IIB branes wrapped around the in the blown up conifold geometry and, in the large limit, this is equivalent to type IIB on the deformed conifold background:

The deformation parameter will, again, be identified with the glueball superfield. Rather than the original D5 branes, there are now units of RR flux through , and also some NS flux through the non-compact cycle dual to . This mirror description is related to a particular limit of the large duality proposed in [10] and [11].

The value of the modulus is fixed [2] by the fluxes, and this is captured by a superpotential for , whose first component is proportional to . Specializing (1.5) and (1.6) to the conifold, we have a single compact 3-cycle , and a single dual, non-compact 3-cycle . The period of the holomorphic 3-form is . There are units of RR flux through , and the NS flux through ; is identified with the bare coupling of the 4d gauge theory.

The holomorphic three-form is given by

The 3-cycles and can be viewed as 2-spheres spanned by a real subspace of fibered over , as in [12,13,14], and integrating over the fiber yields a one-form in the -plane:

The A-cycle, projected to the x-plane, becomes an interval between . Thus the A-period is given by:

The B-period can be viewed as an integral from to infinity. However this integral is divergent, and thus must be cutoff to regulate the infinity. Giving dimension , has dimension 3/2, so we put the cutoff at where has mass dimension 1:

Note that, under , , shifting the period by an period. Using the fact that we have units of RR flux through and units of NS flux through the B-cycle, we find the superpotential [2]:

is related to the bare coupling constant of the gauge theory by . The coefficient of in the above superpotential is given by

which is the geometric analog of the running of the coupling. depends on in such a way that the above quantity is finite as :

which is exactly the expected running of the coupling constant for the 4d Yang-Mills theory. The upshot is to replace the cutoff in the above expression with the scale of the gauge theory, which we will denote by . We thus have for the superpotential

(the linear term is a matter of convention and defines what one means by the physical scale ). This is indeed the superpotential of [15] for the massive glueball . Integrating out via leads to the supersymmetric vacua of supersymmetric Yang-Mills:

2.1. Gauge Theoretic Reformulation of the duality

We can formulate the above large duality in purely gauge theoretic terms. The conifold geometry without the fluxes corresponds to an gauge theory with a charged hypermultiplet [16]. Turning on fluxes is equivalent to adding electric and magnetic Fayet-Iliopoulous superpotential terms, which softly break to . The vector multiplet consists of a neutral chiral superfield and an photon. The photon is left massless, and is to be identified with the overall of the original theory. The chiral superfield gets a mass, and is to be identified with the massive glueball chiral superfield of the theory.

The identification of the of the dual theory with the is consistent with the fact that minimization of the superpotential gives rise to

where we used the special geometry to connect the periods of the B-cycles with the coupling constant of the . Note that the coupling of the theory is as it should be where is the bare coupling of the theory and is identified with times the identity matrix in adjoint. In fact the “charged hypermultiplet” of the is nothing but the baryon field of the original theory. To see this note that before turning on the RR flux on , wrapping a brane around it gives a charged hypermultiplet. Turning on the RR flux, induces units of fundamental charge on it, as noted in the context of AdS/CFT correspondence in [17,18,19]. After turning on the flux the field is not allowed by itself, i.e., it is attached to fundamental strings going off to infinity. Thus after the FI deformations of the superpotential it is slightly misleading to think of the theory as having a fundamental hypermultiplet. In that context one can simply view this as an effective theory with the SW geometry as would have been the case with a fundamental hypermultiplet.

2.2. Adding Massive Fields

As discussed in [2], we can also consider adding some quark chiral superfields, in the fundamental representation of . In the type IIB description this is done by taking a D5 brane wrapping a holomorphic 2-cycle not intersecting the , but separated by a distance , where is proportional to the mass of the hypermultiplet, as the matter comes from strings stretching between the non-compact brane and the branes wrapped on . If denote the bundle over , the 2-cycle is the curve over a point on . Passing this through the conifold transition, which in these coordinates is given by

and rewriting it by a change of variables in the form

we have a D5 brane wrapping a 2-cycle given by and . Since here has dimension , and should be proportional to the mass , we identify . As discussed in [20] such a D-brane gives rise to an additional spacetime superpotential

This gives the running of the mass parameter with the cutoff . We define the renormalized mass by . Generalizing to any number of matter fields in the fundamental representation, with mass matrix , we find

Integrating out via yields the correct field theory result:

Figure 1: Location of the branch cut in the x-plane. Contours of integration of the different periods of the geometry including those coming from massive fields.

3. Geometric engineering =1 theories with adjoint and superpotential

The Yang-Mills theory of the previous section can be regarded as a special case of the more general theory with adjoint and superpotential as in (1.1),

For , the adjoint gets a mass and we recover the case reviewed in the previous section. We here review the geometric construction of [5] for general .

For , the 4d field theory would be pure Yang-Mills system. To geometrically engineer that, all we need is a in a Calabi-Yau manifold for which the normal bundle is (i.e. it has the same normal geometry as if the were in a ). If we wrap D5 branes around the we obtain an gauge theory in the uncompactified worldvolume of the D5 brane. The adjoint scalar gets identified with the deformations of the brane in the direction, normal to the .

To describe the geometry in more detail, let denote the coordinate in the north patch of and in the south patch. Let denote the coordinate of direction in the north and south patches respectively, and let denote the coordinates of in the north and south patches respectively. Then we have

There is a continuous family of s, labeled by arbitrary , at . Each of the D5 branes can wrap a at any value of . In the gauge theory living in the unwrapped directions, this freedom to choose any for each brane corresponds to moving along the Coulomb branch, with the of each brane corresponding to an eigenvalue of the adjoint field .

This connection between and the Coulomb branch moduli makes it clear how the geometry must be deformed to obtain the theory with superpotential (3.1). Rather than having the , with coordinate and at the point , for arbitrary , it should exist only for particular values of , namely the values where . This is the case if (3.2) is deformed to

which is indeed only compatible with at the choices of where . Note that now we can distribute the D5-branes among the vacua , i.e. branes wrapping the corresponding at . This gives a geometric realization of the breaking of .

4. Large N Duality Proposal

We now obtain the large dual of the theory with adjoint and superpotential by considering the geometric transition where each of the ’s have shrunk and have been replaced by a finite size . As already mentioned, the sizes of the s will correspond to the non-zero gaugino condensation expectation values in the factors of non-Abelian gauge groups in (1.3). The needed blow-down of the of the geometry of the previous section has been discussed in [21] and we will review it here. We start with the defining equation (3.3). Its blowdown can be obtained by the change of variables as follows: define , , , ; using (3.3), these satisfy

By completing the square involving and and redefining the variables slightly we obtain the equation

This geometry is singular, even for a generic ; near each critical point of it has the standard conifold singularity. The large dual follows from desingularizing the geometry (4.1), allowing the s to have finite size, rather than zero size as in (4.1).

4.1. Desingularization of the Geometry

Consider the most general desingularization of (4.1), subject to the restriction of [13] that the deformation be a normalizable mode. For the case at hand, as is a polynomial of degree , the most general desingularization of (4.1) subject to the normalizability restriction is to add a polynomial of degree in [14], giving the geometry

Under this deformation, each of the critical points (1.2) (where ) splits into two, which we denote as and .

As in the case of the conifold, the period integrals of the holomorphic three-form over the and cycles can be written as integrals of an effective one-form over projections of the cycles to the plane. As in the conifold case, the non-trivial 3-cycles have simple projections to the plane. The one-form is given by doing the integral over the fiber cycles (corresponding to the coordinates on the surface (3.3)); this gives

Therefore, the periods of the holomorphic three-form over the 3-cycles of (4.2), which are compact 3-spheres, are given by,

where the sign depends on the orientation; the periods over the dual cycles are

The map between the coefficients in and the can thus be obtained by direct computation, and can then be solved for as particular functions .

Figure 2: Geometry before and after introducing the deformation . The choice of branch cuts and integration contours for the different periods is also shown. Dashed lines are paths on the lower sheet.

As we already mentioned, the values of are mapped under the duality to the glueball fields for the non-Abelian factors in (1.3). (The can be defined in a gauge invariant way.) Just as with the case of pure Yang-Mills, the of the dual theory will become massive and obtain particular expectation values thanks to a superpotential , with the expectation values determined from finding the critical points of . The dual superpotential arises from the non-zero fluxes left after the transition.

Rather than having D-branes, as present before the transition, the above deformed geometry will have units of flux through the -th cycle . In addition, there is an flux through each of the dual non-compact cycles, with given in terms of the bare coupling constant of the original 4d field theory. We thus have the superpotential, given in terms of the and periods (1.6) as

This depends on the coefficients of the classical superpotential (1.1) of the original theory with adjoint by way of the geometry (4.2). is a function of the , or equivalently the unknown parameters in . The supersymmetric vacua have fixed , obtained by solving

These will depend on the , the parameters entering in the original and thus on the geometry (4.2), and , the integral infrared cutoff.

In the classical limit, where we set the to zero, and thus , the period of the one-form (4.3) gives

Then the dual superpotential is (ignoring the irrelevant constant ). This indeed matches with the classical superpotential of the original theory, given by simply evaluating the superpotential (1.1) in the vacuum with breaking (1.3), where eigenvalues of the field take eigenvalue .

4.2. Aspects of the gauge fields

The dual theory obtained after the transition is an gauge theory, broken to by the superpotential (4.6). The , which are in the same multiplet as the , get masses and frozen to particular by . On the other hand, the gauge fields remain massless. The couplings of these ’s can be determined from or the prepotential , with , of the geometry under consideration:

The couplings (4.9) should be evaluated at the obtained from (4.7).

Note that (4.7) and (4.9) imply

We identify the the i-th block field strength with the generator in which is times the identity matrix in the -th block and zero elsewhere. In this way the correspond to field strengths of the ’s coming from the and the will corresponds to the overall . Thus the above equation is consistent with the fact that the overall is a linear combination of the ’s with coefficients given by , together with the fact that the bare coupling constant of the overall should be the same as that of the original theory, as the is decoupled. Moreover it is consistent with the fact that there is no coupling between the field strength of this overall with the other . Thus extremizing the superpotential is equivalent to this structure for the gauge coupling constants of the factors.

One can also relate the coupling constants of the factors to the period matrix of the hyperelliptic curve

To see that, from (4.9) we will have to compute the period integrals of about the cycles of the hyperelliptic curve, where . As we will discuss in section 7 the coefficient of of is proportional to the sum of ’s and thus considering gives rise to a linear combination of

with , a basis of the holomorphic one-forms on the hyperelliptic curve. Thus can be identified with the period matrix of the hyperelliptic curve.

4.3. Gauge theoretic reformulation

Just as in the case of we can reformulate this duality in terms of a duality of two gauge systems: We start with pure Yang-Mills theory for gauge group and deform it by the superpotential of degree in the scalar field, breaking the into factors . The gaugino bilinear together with the forms an multiplet. One considers a dual multiplet containing softly broken to by a superpotential term. Note that the we have proposed is of the form that appears in an theory with a gauge group with some matter fields (whose structure is dictated by the superpotential). In fact the dual system we have been considering is of the type studied in [22] and was connected to a type IIB description considered here in [14]. In such a formulation the decoupling of the overall from the other ’s occurs as in (4.10), consistent with the minimization of the superpotential.

5. Field theory analysis

We now analyze the strong coupling dynamics of the theory, with adjoint and superpotential (1.1), in the vacuum with the classical breaking (1.3). In the quantum theory, each super Yang-Mills in (1.3) generally confines, with supersymmetric vacua. The vacua correspond to -th roots of unity phases of the gaugino condensate , with the glueball chiral superfield. The in (1.3) are free, and therefore remain unconfined and present in the low energy theory. The vacua can also have more interesting behavior. For example, in with a cubic superpotential for but no quadratic mass term, the vacuum is at the non-trivial conformal field theory point of [22].

The low energy theory contains an effective superpotential which gives the chiral superfield expectation values via [23]

can often be obtained exactly, thanks to its holomorphic dependence on and [24]. In the present case, we’ll discuss how can indeed, in principle, be obtained exactly via the curves [25,26,27]; in practice, however, the result is quite difficult to obtain.

5.1. Approximate via naive integrating in

The effective superpotential can often be obtained exactly via starting from the low-energy effective theory and “integrating in” the massive matter fields [23,28]. As discussed in [28], for this procedure to give an exact answer, one must be able to argue that the scale matching relations are known exactly and that a possible additional unknown contribution to the superpotential necessarily vanishes. Our theory with adjoint and superpotential (1.1), does not admit this kind of symmetry and limits arguments needed to prove the naive scale matching relations and as exact statements. So naive integrating in need not give the exact answer for ; nevertheless, it is still useful here for obtaining an approximate answer.

To illustrate how naive integrating in can fail to give the exact answer in the theory with adjoint , consider the vacuum where classically , leaving unbroken. Such a vacuum exists for any tree level superpotential (1.1). The mass of in this vacuum is , independent of the other . The low energy theory is pure Yang-Mills and the dynamical scale of this theory is related to that of the original high energy theory by matching the running gauge coupling at the threshold scale , giving . The low-energy theory has vacua with gaugino condensation and low-energy superpotential

Using (5.1) one could use this to try to find the in this vacuum, but the answer would be incorrect for with . The exact answer can be found from deforming the curve following [29], as reviewed in the next subsection. The exact effective superpotential is found from this to be

The term coincides with (5.2), so both give the same , but (5.2) gives all other , whereas (5.3) gives higher .

The terms in (5.3) which are missing from (5.2) are weighted by , which should be small as compared with the leading term . The reason is that the higher appear irrelevant in the original description, so their required UV cutoff should be larger than the dynamical scale in order for the theory to be well-defined, i.e. the should be small. So the lesson is that naive integrating in here needn’t give the exact answer, but it does generally give the leading term or terms.

On the other hand, naive integrating in actually does give the exact answer for in the vacua where [30]. In fact, the exact curve of the entire theory can be re-derived via “integrating in” in the vacua [30].

We now outline the naive integrating-in procedure for the general vacuum (1.3). The low-energy SYM with gauge group (1.3) leads to a low-energy superpotential via gaugino condensation in each of the decoupled, non-abelian groups:

The term is simply the value of the classical superpotential (1.1), evaluated in the classical vacuum:

with the defined in (1.2). As in (5.1), .

The dynamical scale entering in (5.4) is that of the low-energy theory, which is related to the scale of the high-energy theory by matching the running gauge coupling across two thresholds: that of the massive W-bosons, and that of the mass of the field in the vacuum. The classical masses of the W-bosons which are charged under are . The mass of the adjoint is classically . The scale of the low-energy is thus obtained by naive threshold matching to be

It will be useful in what follows to also integrate in the glueball fields :

The are massive, with supersymmetric vacua , and integrating out the leads back to (5.4).

The final result of naive integrating in is thus expressed in terms of the as

The quantum term in (5.8), coming from gaugino condensation, is to be omitted when ; e.g. in the case of [30], where and all other . The result (5.8) happens to be exact when no but, as emphasized above, (5.8) is only an approximation to the exact answer in the more general case, where some .

5.2. The exact via deforming the results

In this subsection, we obtain the exact 1PI generating function by deforming the exact solution [25,26,27] of the theory by the (1.1). The large duality proposal of section 4 gives the exact superpotential as (4.6), with the glueball fields included. (As verified in section 7, the naive integrating in result (5.8) is indeed an approximation to this exact result; generally there is an infinite series expansion of corrections to the naive formula (5.8).) Upon integrating out the massive from (4.6), one obtains , which we will verify indeed agrees with the field theory result obtained in this subsection. Our (4.6), however, contains the additional information about the glueball fields . Although the are massive, this additional information about their superpotential is physical; for example between the different vacua gives the BPS tension of the associated domain walls. Perhaps there’s also a way to exactly integrate in the in the context of the deformed field theory, though this is not presently known.

The theory deformed by only has unbroken supersymmetry on submanifolds of the Coulomb branch, where there are additional massless fields besides the . The additional massless fields are the magnetic monopoles or dyons, which become massless on some particular submanifolds [25]. Near a point with massless monopoles, the superpotential is

and the supersymmetric vacua are at those satisfying

the first equations are for all and the second for all (with for ). The value of the superpotential (5.9) in this vacuum is simply

with the solution of , where the monopoles are massless. The explicit monopole masses on the Coulomb branch can be obtained via the appropriate periods of the one-form [26],

but this will not be needed here.

In the vacuum (1.3), there are massless photons, whereas the original theory had massless photons. So the vacuum (1.3) must have mutually local magnetic monopoles being massless and getting an expectation value as in (5.10), for . It can indeed be shown from (5.10) that if the highest Casimir with nonzero in is , as in (1.1), then the supersymmetric vacuum necessarily has at least mutually local monopoles condensed. (More than condensed monopoles correspond to those classical vacua in (1.3) where some , and thus there are fewer than photons left massless.) The vacuum obtained from integrating out as in (5.10), will give some values of the which are determined in terms of the .

Solving for the supersymmetric vacua as in (5.10), is equivalent to minimizing , subject to the constraint that lie on the the codimension subspace of the Coulomb branch where at least mutually local monopoles or dyons are massless. This is just a matter of replacing the monopoles with Lagrange multipliers, imposing that the lie in the subspace with massless monopoles; i.e. we integrate out the with , with the monopole masses on the Coulomb branch and Lagrange multipliers, and the . The resulting will be some fixed value, depending on the and , giving finally .

Recall that the curve of the theory is