= USC-95/019 CERN-TH. 95-167 hep-th/9507032 \date=June 1995 \titlepage\titleHyperelliptic curves for Supersymmetric Yang-Mills

Joseph A. Minahan^{†}^{†}⋆
\addressDepartment of Physics, University of Southern California, Los Angeles, CA 90089-0484 USA
\andauthor
Dennis Nemeschansky^{†}^{†}† . On leave from Physics Department, University of
Southern California, Los Angeles, CA , USA.
\addressTheory Division, CERNCH-1211, Geneva 23, Switzerland

In this paper we discuss the hyperelliptic curve for super Yang-Mills with six flavors of hypermultiplets. We start with a generic genus two surface and construct the curve in terms of genus two theta functions. From this one can construct the curve for . This curve is explicitly dual under a subgroup of which is not isomorphic to . We then proceed to construct the curve for the general theory and discuss the duality properties of the theory. The results given here differ from those given previously. \endpage

N. Seiberg and E. Witten, hep-th/9407087, Nucl. Phys. 426 (1994) 19. \REF\SWN. Seiberg and E. Witten, hep-th/9408099, Nucl. Phys. B431 (1994) 484. \REF\KLTYA. Klemm, W. Lerche, S. Theisen and S. Yankielowicz, hep-th/9411048, Phys. Lett. B344 (1995) 169. \REF\AFP.C. Argyres and A.E. Faraggi, hep-th/9411057, Phys. Rev. Lett. 73 (1995) 3931. \REF\HOA. Hanany and Y. Oz, hep-th/9505075, On the Quantum Moduli Space of Vacua of N=2 Supersymmetric Gauge Theories. \REF\APSP. C. Argyres, M. R. Plesser and A. D. Shapere, hep-th/9505100, The Coulomb Phase of N=2 Supersymmetric QCD. \REF\FKH. M. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag (1980), New York. \REF\ClemensC. H. Clemens, A Scrapbook of Complex Curve Theory, Plenum Press (1980), New York.

supersymmetric Yang-Mills theories have been shown to have a very rich structure. In particular, it has been argued that the exact effective action can be found from a relatively simple complex curve. This curve basically contains all of the low energy physics, in that singularities of the curve describe situations where light particles become massless.

The original work in this direction discussed gauge theories with hypermultiplets transforming in the fundamental represention of [\SWI,\SW]. This work was then extended to first theories with [\KLTY,\AF], and then most recently to theories with [\HO,\APS]. In particular the authors in [\APS] derive a curve for a general theory with . The curves for can then be found by taking appropriate limits.

However, we believe that the final curve presented in [\APS] is not correct. The authors claim that the curve for general can be constructed out of standard toroidal theta functions. The basic argument is that by taking the appropriate limit, one can reduce an theory with flavors to an theory with flavors. This they argue leads to constraints on the structure of the curve. Then, to get the final form of the curve, they take another limit, reducing the theory to an theory with flavors. They then compare coefficients that are functions of the coupling and then inductively determine the coefficients for all theories starting from .

While these reductions of gauge groups make sense for weak coupling, we see no reason why they should be exact for strong coupling. In fact, one can see that there are troubles just with the reduction of with to with . The resulting form of the curve in [\APS] is actually inconsistent with the result of Seiberg and Witten[\SW]. has a discrete global parity symmetry that should be present even in strong coupling. One can compute the discriminant of the curve in [\SW] and find that it is invariant under the parity transformation. But this is not true for the curve in [\APS], nor can the symmetry be restored by a redefinition of the curve variables and . This then casts some doubt on the whole reduction scheme.

In this paper we argue how to construct the general curve for with , and we explicitly construct it for . We show that for the case, the class of invariant theories is generated by a subgroup of that is not . We also show that this subgroup generates a different fundamental region than the usual region. We find that the curve depends on genus two theta functions, but the period matrix, which is the argument of the theta functions, is restricted to a particular form. This will restrict the number of independent theta functions. The construction begins with the special case of . In this case, there is only one scale in the theory, so the classical coupling is the full quantum coupling. The curve is found by explicitly constructing a genus two curve with known periods. Then we allow for nonzero and , which breaks the conformal invariance of the theory. Making some mild assumptions about the possible terms that can appear in the curve and comparing to the semi-classical duality symmetry of allows one to find the final SU(3) curve. The result found here differs from that in [\APS].

We begin with a basic discussion of curves for hyperelliptic surfaces with genus (c.f. [\FK]). While there is still no proof that the curve for has to be hyperelliptic, it has so far been succesful at reproducing known behavior at weak coupling.

The advantage of hyperelliptic surfaces is their great simplification. The generic hyperelliptic curve for a genus surface is of the form

where the are the branch points (Weierstrass points) of the surface. There is also associated with this curve a Jacobian variety which is the dimensional complex plane modded out by the lattice generated by the pair . is the dimensional identity matrix and is the period matrix. These lattice elements are found from the curve by integrating a canonical set of one forms around the noncontractible loops of the curves. The usual basis is

If the surface is hyperelliptic, then there is a simple map of the curve into . Namely, the integrals over the one forms between branch points are given by the halfway points of the lattice, or the half-periods. This then suggests that there should be a map between the half-periods of and the branch points.

Suppose that the branch points are ordered and let be a map . We can choose and then define for any point on to be the integral over the normalized canonical set of one forms from to . Then per our previous arguments, is a half period in . Which half periods is determined by the homology basis one chooses. We choose the basis such that the integral of the canonical set of one forms around the branch points and gives where is the column in and the integrals around the points and give , where is the column in . Note that this particular choice of integration paths and have the canonical intersection matrices , Therefore, we find

and

where the form of this last integral is determined by the intersection matrices. We can now determine at the branch points. It satisfies

and

Next consider the -functions, which are maps of into the complex numbers. The -functions with characters are defined as

The -functions have nice transformation properties under the modular transformations . The transformations are generated by the set

where is an element of and is a symmetric matrix with integer entries. Then for a generic matrix of the form , transforms as

In particular, note that for the transformation of the form , . Hence we have that

We can use the theta functions to find a map of the half periods into the branch points. First note that for . In fact the generate the vector of Riemann constants for the -functions. If we add to in the argument of the -function, then we find that the -function vanishes at . Likewise, if we add to the argument, then we find that the -function vanishes at . Hence consider the function

We see that double zeros appear in the numerator at and they appear in the denominator at . Hence is a meromorphic function with a double zero at and a double pole at . Hence, the curve describing this surface is given by

where and the function is , with a constant. We are also free to fix the point to any value, so we choose . This then determines . Hence the values of the branch points are given by

Actually, we have to be a little careful, since some of these expressions involve zero divided by zero. In these cases, one should instead consider differentials of these functions. Then there should exist analogs of Jacobi’s triple product identities to express the derivatives of theta functions in terms of the other theta functions.

But we do not really need such identities. Instead, one can shift the -functions by different odd periods, such that the new function still has a double pole at and a double zero at , but still has , and is also now well defined at the point in question. For instance, in the genus 2 case, one finds using \bpeq that, and are given by

However is not well defined in \bpeq. However if we instead shift by and , we obtain a new function that is well defined at all points except . But it must be the same function as before, assuming that it is normalized such that . Doing this, one finds that

If we redefine and such that no terms appear in the denominator of , we find the generic quintic equation for a surface of genus 2

The advantage of writing the curve this way is that the discriminant is a modular form. The discriminant is defined as

where is the leading coefficient in the polynomial in . Written this way, is invariant under the transformations , . Using the class of identities[\Clemens]

plus those identities that can be generated by modular transformations on \thid, one can easily show that the discriminant for \gtwocurve is

where the product is over the 10 even genus two -functions This construction of the discriminant generalizes to higher genus hyperelliptic surfaces. However, one needs to be mindful that the identities similar to \thid on surfaces with are only true for period matrices that are compatible with hyperelliptics. For , not all Riemann surfaces are hyperelliptic. In fact, for these higher genus surfaces, one will see identities arise when constructing the hyperelliptic curve. The values one finds for the branch points depends on what divisor one chooses to begin the construction. But in the end, the same result should appear, even though one will find that the branch points depend on different combinations of -functions. Hence there must be identities between these different combinations.

We now wish to compare the period matrix of the surface with the matrix of couplings arising from the gauge theory. For the classical gauge theory, non-zero expectation values of , where

generically break the gauge theory to . A convenient basis for the generators are , where is the generator which has . In this case, we have a matrix of couplings that satisfies

up to a subgroup of transformations, that act on in the same way that they act on . is given by

where is the coupling.

Quantum effects will change the general form of in \couplings for generic values of and , where are the masses of the hypermultiplets. However, consider the case where and for . If the number of hypermultiplets is , then there is only one scale in the theory, namely . This means that the -function is zero and hence the coupling matrix does not change from its classical value.

If maintains its classical value, then it also has some extra symmetries that are missing for generic quantum values. In particular, there are a class of transformations that leave invariant. These transformations are generated by the matrix

where is a particular matrix that satisfies . Then satisfies . is basically a generator for Weyl reflections. Since the couplings are not running, and they started out equal, is invariant under the reflections. However, once for , or for some , then the couplings will differ. The Weyl reflections then rotate these couplings into each other.

So now let us suppose that we have a surface where the period matrix is invariant under the transformations generated by \Rgen. Let us concentrate on the special case of . It is straightforward to find the matrix that leaves invariant, where is

One finds that is the appropriate matrix. For this particular , there must be additional identities among the even -functions. Using \thtrans one finds the identities

is invariant under the transformation. With these identities, one can rewrite the genus 2 curve \gtwocurve as

Actually, using the identities in \thid we can reduce this even more. Setting , and , we can rewrite \gtwonew as

The equation in \gtwo describes a hyperelliptic surface with a symmetry. We now wish to compare this to the curve that one would expect for the theory with six massless hypermultiplets and all gauge invariant expectation values zero except for . The curves constructed are in sextic form, so we want to find a transformation of the quintic in \gtwo into this more usual form. Generic arguments give the form of the sextic to be

where is given in \suncoupling. The curve is written in this form in order to compare with previous results, but we will find that it is more useful to express the curve in a slightly modified form. To proceed, we set , the period matrix for \gtwo equal to , the matrix of couplings. Hence, there should be an transformation such that \gtwo is transformed into an equation of the form

Under the transformation , , the branch points are transformed to . Clearly, we should choose the transformation such that three sets of branch points are equal up to a cube root of unity, and the other three points are also equal up to a cube root of unity. A transformation that accomplishes this is given by

where . is chosen such that the equation can be written as

where we have now inserted a scale into the equation. (This will change the discriminant by a factor of ). Written in this form, one finds

Using the identity

which follow from the identities in \thid, can be further simplified to

In order to compare with the result in [\APS], we can rescale such that the curve is in the form

where is given by

Note that this function is different from the one presented in [\APS] and also conflicts with the conjecture in [\HO].

Let us examine the transformation properties of , and . First consider a rotation of the angle such that . Hence . This should leave the theory invariant. As far as the period matrix is concerned, we have

The terms on the diagonal transform all -functions into themselves, but the off-diagonal pieces transform , and . Clearly is invariant under this.

Next consider the dual transformations where . This corresponds to a transformation of the coupling matrix

In other words, the dual theory has coupling . This transformation takes , and . In fact, we also learn something else. Since and are both modular forms of weight 24, then clearly is also a modular form, which is also clear from \rval. By inspecting the explicit form for , one sees that also picks up a minus sign under the transformation.

These two sets of transformations generate a subgroup of and it is this subgroup under which the theory is invariant. The two generators are

and their action on is and . We stress that this subgroup is not . For instance, this group has , but . Note that generates the transformation (up to factors involving )

This group also has an interesting fundamental region. It is bounded by and . The region is an orbifold with three singularities of order , and . This is similar to the fundamental region for , except that in this case, the point is at infinity, while in the case the point is at infinity.

Now let us examine the weak coupling behavior of . In the limit the -functions have the limits , and . Hence, in this limit behaves as . The argument of the exponent has the form expected from instanton contributions. However, the coefficient in front of the exponent differs from the one presented in [\APS].

Before proceeding with the more generic case of nonzero let us consider the classical spectrum for the massless theory. This will give a better understanding as to why has the transformations stated above, as well as giving us clues to what the final form of the curve should be. To this end, let the orthogonal generators of the cartan subalgebra be

Hence the charges of the hypermultiplets with this basis are given by

where is the charge. If the components of the adjoint scalar are and , then the masses of the hypermultiplets are given by

The charges of the monopoles are given by and . Their masses are

Classically, the charges and masses of the hypermultiplets and monopoles are mapped into each other under the dual transformation and , . It is easy to see that the transformation of corresponds to the transformation . The gauge invariant quantities and are given by

The transformation of and leads to the transformations and .

Consider now the curve when is nonzero, keeping the hypermultiplets massless. By general arguments we expect the form of the curve to be

where is a function to be determined and and are the functions in \rsvals. The theory should be invariant under shifts of the angle, which are generated by . Since and are clearly invariant under this symmetry, then should be as well. Moreover, the theory should be invariant under all transformations that are conjugate to , in particular the transformation . Up to an overall factor, this transformation maps into itself and exchanges with . Since has modular weight 12 under , then in order for \nzu to have nice modular properties, should have weight 4 under this transformation. Finally, in order to have the correct weak coupling behavior, the leading order behavior of should be as , where . There are two functions constructed from the -functions that have these properties,

thus, we expect to be either or , or perhaps a linear combination of the two.

However, there is a significant difference between and . Under , the functions transform as

Since transforms as , we see that the duality properties depend on whether is or . If , then \nzu is clearly dual under , since the extra factors can be reabsorbed into with no change in or . We are also free to make a complex rotation and implement the transformation and along with the dual transformation. However, if , then the transformation is dual, if in addition we have the transformation

The second case is much closer to the classical dual transformation, the only difference being that it is missing a shift in by . Thus it appears that is the better choice. This also has the advantage of avoiding -functions in the denominator’s of the coefficients. Given this selection, one then learns that the true quantum duality transformations are given by \qudu.

We can also show that the classical duality cannot be the true quantum duality. The discriminant in \nzu is given by

If and have the classical dual transformations, then there is no choice of that leaves the form of the discriminant invariant. It is not even possible to find a that leaves the piece inside the square brackets invariant.

Up to now, we have been expressing the curve in terms of -functions, but all coefficients in \nzu can be written as polynomials of the two functions and . Given the form of , this is perhaps more convenient, with the curve now given by

In fact one can simplify \nzug even more by absorbing a factor of into , leaving the curve

where .

The situation becomes more complex when the bare masses of the hypermultiplets are no longer zero. The curve should be chosen such that at weak coupling, the discriminant is zero if

We also expect that the curve can be written as a polynomial in the functions and (or ) and that there exists a one-form which has residues that are proportional to [\SW]. A curve satisfying these properties is given by

where is to be determined and where and are shifted values of and , where the shifts depend on .

It was argued in [\APS] that the one-form is given by

where

and is a constant which is adjusted so that the residues have the proper values. Clearly, has poles at , with the residues proportional to . There is also a pole at . The sum of all the residues, including the one at infinity, must be zero in order that the integral of is zero along any contractible loop over the genus two surface. It is easy to see that this puts no restrictions on and , but sets a condition on , which one can show satisfies

Hence the massive curve is given by

Unlike the case, the curve has no parity symmetry to help fix the final form of and , although even for there is still some ambiguity in these shifts. Since there does not appear to be any symmetry gained by shifting and , one might consider setting them to and . Another possibility is to choose and such that the curve reduces to

after a shift in . This second choice appears better in that it has the simplest -duality symmetry.

Under , transforms as , and (we have also included a complex rotation) and a factor of is absorbed into . Clearly it is necessary to transform as well in order to keep the form of the curve invariant. A simple calculation shows that the curve in \masscurveII is invariant if transforms as . Hence the duality symmetry seems to be much simpler than the case, where there was a complicated triality symmetry that goes along with the dual transformations[\SW].

While the simple dual structure in \masscurveII is not necessarily a proof that this is the correct curve, we can at least show that it is consistent with other behavior. If we let , keeping fixed, then the theory reduces to with five flavors. The curve, after rescaling is

We can then let such that is kept fixed. The curve is now

Notice that there is no dependence inside the parentheses, even though shifting by violates no symmetries. This result is sensible when one considers the reduction of to . Classically, is reduced to by letting , and become large, scaling them such that , , and as , with the quadratic casimir and the masses. We expect this behavior to persist in weak coupling.

Under an transformation, one can map the cubic curve in [\SW] to the quartic

where and are standard theta functions. The curve in \fourcurve reduces to \sutwofour if , , and , where

Now if
\fourcurve had a shift in of a constant multiplied by ,
we would have had to subtract this term right back off again in order to
reduce it to the curve. Since is of order ,
this term will be of the same order as and hence would imply
that there is really no regime where the classical reduction of
to can take place.
This seems unlikely^{†}^{†}† The authors in [\HO] argue that such
shifts should appear in order to have a singularity structure with
multiplicities when , saying that there is
a possible global symmetry . But
it is not clear where such a symmetry could come from.. In fact
we can make this argument for any curve with , that is,
the scale only appears in front of the product
. The one exception is , where the curve is

This shift in is necessary to preserve parity. But of course there is no other nonabelian gauge group that can reduce to, so this is not a contradiction to the previous argument.

Notice that and reduce to and in the weak coupling limit, but that there is some contribution of the masses to , and that these contributions show up in strong coupling. This is a scenario that was not actually allowed for in [\APS], but must actually occur in order to reduce to . In fact the curve in [\APS] is not consistent with parity.

This constraint on the curve still does not uniquely pick the shifts in and , although it does rule out some possible shifts. For example, a shift of the form is not allowed, since the shifted term would survive down to . On the other hand, a term of the form survives down to but not . Of course these shifts do not actually alter the monodromies, they just shift the singularities to different values of and .

The generalization to higher is possible, but certainly much messier. It is still true that the number of independent theta functions is less than the number of even theta functions. There are also a host of identities that allow one to greatly simplify the curve. We hope to address this and other issues in a subsequent paper.

J.A.M. thanks Nick Warner for many helpful discussions. D.N would like to thank S. Yankielowicz for discussions. This research was supported in part by D.O.E. grant DE-FG03-84ER-40168.