N = 1 dualities of SO and USp gauge theories and T-duality of string theory

Extending recent work on SU gauge theory, we engineer local string models for N = 1 four-dimensional SO and USp gauge theories coupled to matter in the fundamental. The local models are type IIB orientifolds with D7 branes on a curved orientifold 7-plane, and matter realized by adding D3 branes on the orientifold plane. The Higgs branches of the SO and USp theories can be matched with the moduli spaces of SO and USp instantons on the compact four-dimensional part of the D7 branes worldvolume. The R-charge of the gauge theories is identiﬁed with a U (1) symmetry on the worldvolume of an Euclidean D3 brane instanton. We argue that the quantum ﬁeld theory dualities of these gauge theories arise from T-dualities of type IIB strings exchanging D7 and D3 charges. A crucial role is played by the induced D3 charge of D7 branes and an orientifold 7-plane, both partially compactiﬁed on a Z 2 orbifold of K 3.


Introduction
It has become increasingly clear over the last year that many results in supersymmetric gauge field theories can be derived rather efficiently using string theory.In particular Tduality symmetries of perturbative strings has emerged as an extremely powerful tool in this connection.For the case of N = 4 supersymmetric field theories strong-weak coupling duality is argued to arise from T -duality [1][2]3 .The non-perturbative physics of N = 2 supersymmetric field theories can be derived from mirror symmetry [3], a symmetry that amounts to T -duality of type II strings [4][5] [6].In many of these works the basic idea is building a local string model for the gauge field theory.This is done by isolating the part of the compactification data which is relevant for field theory questions and taking the limit where gravity is turned off.In the resulting local model the compactification data is replaced by the relevant non-compact piece of the internal space.
N = 1 supersymmetric theories and their dualities, have also begun to be understood in the context of string theories.In particular, N = 1 pure Yang-Mills was engineered in [7] in the context of F-theory compactification on elliptically fibered Calabi-Yau fourfolds.This amounts to considering a space S of complex dimension two (a codimension one subspace of the base), over which the elliptic fibration acquires an ADE singularity.Moreover it was shown that (to avoid adjoint matter) the space S must satisfy h 2,0 (S) = h 1,0 (S) = 0.More recently, and for the case of SU (N c ), this was extended to include N f fundamentals and anti-fundamentals [8], by adding N f D3 branes filling four-dimensional spacetime and bringing them close to S. It was checked that this local string model does indeed reproduce some of the well known results concerning these gauge theories.Higgs branches of these gauge theories were correctly identified with the moduli spaces of SU (N c ) instantons on S. Nonperturbative generation of superpotentials, and quantum corrections to moduli spaces of the field theories were seen to arise when expected through the effects of euclidean three branes in the local models.Moreover, it was argued that both the quantum field theory dualities between N = 2 supersymmetric SU (N c ) gauge theory with N f flavors and SU (N f − N c ) gauge theory with N f flavors [9], and the corresponding N = 1 dualities, that hold upon addition of some neutral matter [10], arise in the string models as T -duality transformations of type IIB.This makes sense even though one is discussing F-theory vacua which generically have no T-duality symmetries.This is because in the context of SU gauge groups one can realize the F-theory local model by D7 branes of a fixed perturbative type IIB string [11] for which the T-duality symmetry applies.The T-duality transformation roughly speaking inverts the volume of S. This, in particular, exchanges D3 and D7 brane charges.Due to an induced D3 brane charge on the curved D7 worldvolume, this symmetry ends up exchanging N It is natural to ask if the results of [8] can be extended to other groups and in particular to SO and U Sp groups.In fact a similar story should repeat at least for SO, because Ftheory backgrounds with SO gauge symmetry can be viewed as type IIB orientifolds [12] for which T-dualities apply and can shed light on N = 1 dualities.Similarly, since the difference between the SO and U Sp theories at the string perturbative level is the choice of the sign for the diagrams with odd number of crosscaps (as we will review in section 2) the arguments work with equal ease for the U Sp theories.In the present paper we will extend the results of [8] to the case of N = 1, four dimensional SO(N c ) gauge field theories with 2N f flavors of matter in the fundamental, and to the case of U Sp(N c ) gauge theory with 2N f flavors in the fundamental. 4In particular we show that the Higgs moduli is mapped to the instanton moduli space on S. Moreover we argue that the gauge symmetry duality is mapped to T-duality which exchanges SO(N c ) with SO(2N f − N c + 4).The dual group differs from the naive expectation SO(2N f ), due to the contribution of induced D charges by D7 branes and the orientifold.As in the SU case the curved worldvolume of the D7 brane induces −N c /2 units of D3 charge (accounting for the −N c shift).The orientifold 7-plane carries D7 charge, as is familiar, and due to its curvature induces D3 charge (giving rise to a net +4 shift), as will be discussed at length below.Similar statements apply for U Sp(N c ) for which the dual group is U Sp(2N f − N c − 4).These results are in accord with field theory dualities for SO(N c ) discussed in Refs.[10] [13], and field theory dualities for U Sp(N c ) discussed in Ref. [14].These field theory dualities can be 'derived' by flowing from the N = 2 versions of the above gauge theories [15] [16].In fact the stringy realization of these N = 1, SO and U Sp dualities also starts from an N = 2 situation, as was the case for SU [8].
Even though the action of T-duality on D-brane charges is relatively simple, the action of T-duality on curved spaces with wrapped D-branes has not been investigated before.This is an important subject to study further, perhaps using techniques familiar from mirror symmetry, in order to shed further light on field theory dualities.Nevertheless, the fact that N = 1 dualities for SU, SO, and U Sp gauge theories with fundamental matter 4 In our conventions U Sp(N ), with N necessarily even, is the compact gauge group defined by the set of N × N complex matrices that are both unitary ( belong to U (N, C)) and symplectic (belong to Sp(N, C)).The corresponding Lie algebra is the real Lie algebra usp(N ) defined by the set of N ×N antihermitian matrices A that in addition satisfy the symplectic condition J A = −A t J with J = 0 I N/2 −I N/2 0 .The usp(N ) algebra is of rank N/2, and real dimension N (N + 1)/2.
The fundamental representation of this algebra is of dimension N and it is pseudoreal, and the adjoint representation appears in the symmetric part of the product of two fundamentals.
can all be derived in a simple and unified way suggests strongly that our assumptions about T-dualities are valid.

The local model for SO and U Sp groups
As mentioned in the previous section, local models of F-theory compactified on elliptic CY 4 's which admit a perturbative string description are a natural setup to gain insight into N = 1 dualities.In particular for the SO models we can replace the F-theory description with a type IIB orientifold.The type IIB orientifold description also allows us to easily construct the U Sp series.In this section we discuss the local models we need in detail.
We start by constructing a local model for type IIB string on a Calabi-Yau threefold.The local model for the Calabi-Yau threefold is the total space of a complex 2-surface K together with its canonical line bundle.This gives a non-compact threefold with c 1 = 0. We then consider an orientifold of the above model by letting the space reflection part act trivially on K but as an inversion on the line bundle over K.The zero section of the line bundle, which can be identified with K, times the Minkowski space M 4 can be viewed as an orientifold 7-plane, or more briefly, an O7 plane.
We now put N c /2 physical D7 branes on the O7 plane. 5Depending on the choice of weight factor for string diagrams involving crosscaps the branes give rise to SO(N c ) or U Sp(N c ) gauge theory.To see this recall that in open strings we start with N c Chan-Paton factors.In oriented strings this gives in the open string sector a U (N c ) gauge symmetry.For non-orientable strings, however, we have to take into account the action of orientifolding on the open string sector.This amounts to symmetrization or anti-symmetrization of the Chan-Paton indices, leading to U Sp(N c ) or SO(N c ) respectively.The difference between these two is the choice of the sign for the action of the orientifold (twist) operator Ω on the open string sector.The two possible choices of sign correspond, for the case of the open string one loop vacuum graphs, to either adding or subtracting the diagrams for the annulus and the Moebius strip.Since the Moebius strip is a disk with a crosscap, the net effect is to weight string diagrams having no external open strings with a weight factor (−1) c where c denotes the number of crosscaps [17] 6 .We learn that to exchange SO and U Sp all that needs to be done is to change the sign of the contribution of the crosscap.In particular a diagram with a single crosscap comes with opposite signs in SO(N ) and U Sp(N ) theories.This fact will be used below. 5If the branes are moved away from the orientifold plane, branes will have images under the space reflection.The total number of branes would be N c , out of which N c /2 branes are called physical because their positions can be adjusted at will.While the total number of Chan-Paton indices is N c , the number of physical branes coincides with the rank of the group.
6 This is a well-defined weight factor since the number of crosscaps is well defined except for the fact that one can trade three crosscaps for a single crosscap plus a handle.Thus any surface can be said to have either two, one or no crosscaps.Only surfaces without crosscaps are orientable.
If K = T 4 one finds N = 4 gauge theory in four dimensions.If K = K3 one finds N = 2, and if K is generic, one finds N = 1 gauge theory in four dimensions, with h 2,0 (K) + h 1,0 (K) matter multiplets in the adjoint of the gauge group [7].We will mainly consider in the latter case the situation with h 2,0 = h 1,0 = 0.The generalization when they are non-vanishing is exactly as in [8] and will not be repeated here.
If we wish to obtain matter in the fundamental representation of the gauge group of the seven branes we have to bring in some D3 branes filling the spacetime, and put them on K, where they appear as points.This case was studied in the context of D5 branes and D9 branes in [18][19] a case equivalent to ours by T-duality.In the case of SO(N c ) if we bring in N f D3 branes and put them all at the same point on K we obtain an N = 2, U Sp(2N f ) symmetry in four dimensions, with extra matter in the antisymmetric representation N f (2N f − 1), as well as half-hypermultiplets in the mixed representation (N c , 2N f ).If we choose the other sign factor for diagrams with odd number of crosscaps, the roles of SO and U Sp are exchanged.In particular, we get U Sp(N c ) from D7 branes and the extra sector we obtain from D3 branes is an N = 2 system with SO(2N f ) gauge symmetry.Moreover, in addition to the mixed matter half-hypermultiplets in (N c , 2N f ), we get matter in the symmetric representation of SO(2N f ), i.e., in the N f (2N f + 1) dimensional representation.
Summarizing, the two gauge groups we have engineered are and the matter content is given as follows.For K = T 4 , in N = 2 hypermultiplets where the top sign refers to SO(N c ) × U Sp(2N f ) and the lower sign refers to U Sp(N c ) × SO(2N f ).The last N = 2 hypermultiplet arises by noting that the N = 4 theory has an extra adjoint hypermultiplet in the N = 2 terminology.For K = K3, we find in N = 2 hypermultiplets i.e. the adjoint is lost due to having less supersymmetry preserved on the D7 worldvolume.For K = S, in terms of four-dimensional N = 1 chiral multiplets we get where the factors of 2 arise from writing an N = 2 hypermultiplet as two N = 1 chiral multiplets, and the last representation arises from writing the N = 2 vector multiplet on D3 branes in terms of an adjoint N = 1 chiral multiplet.

Matching of Higgs branches to Instanton moduli spaces
For concreteness let us first consider the case with SO(N c ) gauge group for the D7 branes.The Higgs phase of the gauge theory models arise in the local string models as follows.The N f D3 branes, living inside the D7 branes, fill four dimensional Minkowski space and appear as N f points in K.When these N f points coincide the U Sp(2N f ) symmetry is unbroken and the D3 branes can be viewed as N f coinciding zero size instantons of SO(N c ).Each three brane contributes an instanton number of +1, giving a total instanton number k = N f .In this section we test our construction of local models by checking that the dimensionality of the Higgs branches of the field theories match the dimensionality of the corresponding moduli space of instantons on K in the local string model.This can be viewed as an extension of Ref. [18].The fact that not only the dimension but also the local structure of the instanton moduli space matches the Higgs description follows from [18] for the N = 2 case.For the N = 1 case presumably an extension of the N = 2 result works; such extension was established for N = 1 supersymmetric SU (N ) gauge theories in [8].
We begin by recalling the dimensionalities of moduli spaces of instantons.Consider a gauge theory based on the gauge group G and living on a four dimensional manifold K.The moduli space M k (G, K) of instantons with total instanton number k has complex dimension where c 2 (G) is the dual Coxeter number of the group G and χ(K) and σ(K) denote the Euler characteristic and the signature of K respectively.As mentioned earlier, we have three cases of interest: K = T 4 , K3 and S. For T 4 we have χ + σ = 0, for K3 we have χ + σ = 8, and for S we have χ + σ = 4.We therefore write We also list, for easy reference, the dual Coxeter numbers of the groups we will be dealing with We can now consider the Higgs branches of the supersymmetric field theories.In N = 2 four dimensional gauge theory we count the (real) dimensionality of the Higgs branch M H using the equation Consider first the case of T 4 , which gives rise to the gauge group SO(N c ) × U Sp(2N f ), with the matter representations listed in (2.2).We find in agreement with the expectation from the first equation in (3.2), for the case of G = SO(N c ). Had the U Sp(2N f ) part of the gauge symmetry been completely broken down to [U Sp(2)] N f by pulling the D3 branes apart on K, the dimensionality of the Higgs branch in the resulting theory would have still been the same.
For the case when K = K3 we still have an N = 2 gauge field theory.We must now use the matter representations listed in (2.3) which now give in agreement with the expectation from the second equation in (3.2), for the case of G = SO(N c ).Finally, we consider the case when K = S. Let us first note that the last representation listed in (2.4) which arises from an N = 2 vector multiplet comes with a superpotential term W induced from N = 2 supersymmetry.Solving the dW = 0 constraint implies that this field appears as a constraint on the Higgs branch and thus acts to decrease the dimension of Higgs branch, rather than add to it.We thus find (counting in chiral multiplets which correspond to computing 1  2 dim R M H ) in agreement with the third equation in (3.2), for the case of G = SO(N c ) (note that one equation refers to complex dimension while the other refers to real dimension).
For the D7 branes giving U Sp the story is similar.The main difference is that now the instanton number for U Sp is related to N f by k = 2N f .This is in agreement with [20] where it was explained that the k-instanton moduli space of U Sp theories is governed by an SO(k) gauge theory.Indeed, in our case the SO group induced by the three branes is SO(2N f ).To understand this counting explicitly note that we can have integral or half-integral D-brane charges for the SO group.Only pairs of half-integral D-branes can be moved off the orientifold plane.In the case we have been considering we have brought D3 branes onto the orientifold plane O7, and thus each one will count as two instantons of zero size on top of each other.Once they are on O7 they can split into two half-integral D3 charges, each of which would correspond to one zero-size instanton of U Sp(N c ).Note in particular that for one instanton of zero size in U Sp(N c ) we would get an SO(1) group corresponding to a half D3 brane stuck on the orientifold plane.The absence of gauge group in this case implies the absence of a scalar degree of freedom which can move the D3 brane off the orientifold plane.We thus conclude that for N f D3 branes the instanton number for U Sp(N c ) is k = 2N f .It is now straightforward to repeat the above analysis for the U Sp D7 branes and show that the dimensions for the instanton moduli spaces match the dimensions of the Higgs branches.

R-charge and Instanton Corrections
It was shown in [8] that point instantons of SU (N c ) supersymmetric QCD can be identified with Euclidean D3 branes wrapped around S. Moreover, following [21] an Rcharge Q was defined so that instantons contribute to the superpotential if there is a charge violation ∆Q = 1.In this section we will extend these considerations to the local models for SO and U Sp gauge theories.With this end in mind, let us first review the calculation of charge violation in the SU (N c ) local model of [8].
We have an Euclidean D3 brane wrapping S and we assume we are in the Higgs branch, namely, there are SU (N c ) background gauge fields on S of instanton number N f .The total charge violation arises from additive contributions from two sectors.The first sector, arising from open strings stretched from the D7 branes to the Euclidean D3 brane, represents matter in the fundamental of SU (N c ), living on S and interacting with the background SU (N c ) gauge fields.The charge violation in this sector is given by the index, in the fundamental representation, of the twisted Dirac operator ∂ A , with A denoting the SU (N c ) background gauge fields.This index is given by The second sector arises from open strings stretched from the Euclidean D3 brane onto itself, and corresponds to a twisted N = 4 supersymmetric U (1) theory on the Euclidean D3 brane.Certain aspects of the field theory on this Euclidean instanton have been studied recently in [22].The contribution to ∆Q from this sector is zero, because the fermions in the hypermultiplet and vector multiplet carry opposite Q charge7 and they have equal number of zero modes, i.e. from this sector we have The total charge violation is thus given by ∆Q = N c − N f .One expects superpotential generation for ∆Q = 1, or equivalently, for N f = N c − 1.This expectation is confirmed by field theory analysis.Now we come to the case of SO and U Sp gauge theories.Let us consider the SO case first.The SO(N c ) background gauge fields on S define a background with instanton number N f , and we have an Euclidean D3 brane wrapped around S. Once again, we have two sectors contributing to charge violation.We thus find that the total violation is given by We thus expect that for ∆Q = 1, i.e. when the number of flavors 2N f is given by 2N f = N c − 3 we have a superpotential generated.Moreover, as in [8], one can see that the superpotential must have a first order pole.This is in accord with field theory expectations [13].The situation, however, is more complicated than in the SU case, because for ∆Q = 1 we now have an unbroken SO(3) gauge theory with non-trivial infrared dynamics and gaugino condensates will play a role in the creation of the superpotential.Similarly, for ∆Q = 0, corresponding to 2N f = N c − 2, there could be a quantum correction to the moduli space.As it turns out, in this case there are instanton corrections to the coupling of the unbroken U (1) on the moduli space.These have been computed exactly in [13] and it would be interesting to see how the above Euclidean D3 instanton (and multi-instantons) reproduce those results. 8he situation for the U Sp(N c ) gauge group is similar, modulo a small twist, as in the previous section.If we consider one Euclidean D3 brane wrapped around S, as noted in the previous section, it corresponds to two zero-size instantons on top of each other.Since this is most similar to the case considered above let us first compute the Q charge violation for two instantons and at the end divide the result by two to obtain the charge violation for one Euclidean D3 instanton.For two instantons we have an SO(2) gauge group living on the Euclidean D3 brane.The R-charge violation from the mixed sector is the same as before.But the violation from the D3-D3 sector is different, because now we have three hypermultiplets (symmetric representation of SO(2)) and one vector multiplet (adjoint of SO( 2)) and we find Thus the net violation of the Q charge is Note that we could have also done the computation directly in terms of a single instanton corresponding to an Euclidean half D3 brane.This leads to an SO(1) theory on the Euclidean brane with one hypermultiplet in the D3−D3 sector, and no vector multiplet.This gives the contribution 1 − 0 = 1 to the charge violation.The mixed D3−D7 sector will give a single half-hypermultiplet contributing 1 2 N c − N f charge violation.These two contributions add up to the result in (4.7).
For ∆Q = 1 we thus expect, as before, a superpotential with a first order pole be generated by point-like instantons.This corresponds to the case 1  2 N c = N f , where indeed there is a superpotential generated with a first order pole [14]. 9We would also expect that for ∆Q = 0 there could be an instanton correction to the moduli space.This is indeed the case [14].

N = 1 duality as T-duality
In this section, as a useful preliminary to the analysis of N = 1 supersymmetric SO and U Sp gauge theories we first review the results of [8] dealing with SU gauge theories.We then turn to the computation of induced charges by orientifold planes and D-branes, and finally derive the dualities of N = 1 supersymmetric SO and U Sp theories from T-dualities of the local models.

Review of the SU case
In this case one starts with K = K3 with N c D7 branes wrapped on it and N f D3 branes appearing as points on it.The N c D7 branes wrapped on K3 induce −N c units of D3 brane charge [23] [24].For later application let us note that, by T-duality, the above statement implies that a D-brane wrapped on K3 gives rise to a codimension-four D-brane charge of (−1).Thus in this configuration the total seven brane charge Q 7 and the total three-brane charge Q 3 are given by If the volume of K3 is small we are in a regime which by T-duality is equivalent to a large volume K3.Note that perturbatively the volume of K3 is related to the bare coupling of the D7 gauge group SU (N c ) by V (K3) = 1/g 2 .Thus, when we enter the small volume regime we are entering a large coupling region of the original theory.Once we dualize we are back to a weak coupling description.This, however, exchanges the D3 and D7 charges.
In particular now we have Solving for N ′ c and N ′ f we find This means we now have a weak coupling description of SU (N f − N c ) with N f flavors in the fundamental of this group.This is still an N = 2 theory.To obtain an N = 1 theory we assume K3 has an extra Z 2 symmetry which inverts the sign of the holomorphic 2-form.We can now mod out the K3 by this Z 2 symmetry to obtain a space S = K3/Z 2 which has h 2,0 = 0. Note that this Z 2 acts only on the middle cohomologies of K3.This does not interfere with the D7 and D3 brane charges as they come from zero and four dimensional cycles of K3.So we expect the same duality to continue to hold by the time we get to N = 1.This is very much in the same spirit as the flow from N = 2 microscopic/macroscopic theories to the dual pairs of N = 1 theories [15].Moreover, the meson field of the dual magnetic theory can naturally arise if the N f D3 branes in the dual theory are forced to be on top of each other.

Computation of induced charges for orientifold planes and D-branes
We now wish to repeat the same count of D-brane charge for the SO and U Sp theories, and see if they are in accord with the expectations.Again we start with K = K3 and assume that modding K3 by the extra Z 2 will not modify the D-brane charge count.This would then be in the same spirit as the flow from the N = 2 microscopic/macroscopic theories to the dual N = 1 theories [16].We will assume that the D3 branes are separate.In the case of SO D7 branes this implies that we have U Sp(2) N f .But the dynamics of these gauge factors will be infrared trivial if N c > 4, which we will be assuming.Therefore, these factors will not affect the field theory dualities.A similar comment applies to the case of U Sp D7 branes.
The main complication for the SO and U Sp cases compared to the SU case is that, in addition to the D7 branes, the orientifold planes can also induce D-brane charges.The D7 and O7 induce 3-brane charges through an interaction of the form B 4 ∧ tr(R ∧ R) where B 4 is the gauge field coupling to D3 charges.This arises for the D7 case from a disk diagram with the boundary on the D7 branes.For the O7 this term arises from the sphere with a single crosscap (RP 2 ).That these are generated follows from the anomaly cancellation considerations for type I strings (which by T-duality is related to the above interactions).In addition, there is the familiar D7 charge induced by the O7.Let us study these contributions in detail.D3-brane charge induced by curved D7-branes Each physical D7 brane will contribute some Q 3 charge.This is because the worldvolume of the seven brane is curved and the curvature of K3 is responsible for generating an effective D3 brane charge, as in the SU case.As mentioned above the string diagram computation involves a disk worldsheet.Being an orientable diagram it is also present with exactly the same value in the nonorientable case we are now dealing with.Thus, as before, each D7 brane will contribute a irrespective of whether we are dealing with SO or U Sp seven branes.Note that for N c /2 D7 branes we will end up getting −N c /2 units of D3 charge.
D7-brane charge of O7 planes The contribution of the O7 plane to the seven brane charge Q 7 is easily determined once we note that O9 plane carries (−16) units of D9 charge (the explanation of SO(32) gauge symmetry as orientifold of type IIB in 10 dimensions [25]).
Once we compactify on a two-torus T 2 and do T-duality we obtain four O7 planes with the same total charge as before, namely (−16) units of D7 charge.Thus each O7 plane carries (−4) units of seven brane charge Q 7 .Note that since the contribution of the O7 plane to D7 charge comes from a diagram with a single crosscap (RP 2 ) and going from SO to U Sp involves the change in sign for diagrams with odd number of crosscaps, the O7 plane will carry +4 units of seven brane charge Q 7 for the U Sp case.Thus we write where the top sign refers to SO D7 branes and the lower sign refers to U Sp D7 branes.
D3-brane charge of curved O7 planes We now wish to compute the D3 charge induced by an O7 wrapped around K3.As noted above, this comes from an RP 2 diagram producing an interaction of the form B 4 ∧ tr(R ∧ R).To get the normalization, it is convenient to proceed as follows.Let us first compute the D5 charge of an O9 wrapped around K3.In the ten dimensional type I theory compactified on K3 the 16 D9 branes and the single O9 plane both contribute to the five brane charge Q 5 .As is familiar [18], upon K3 compactification in order to cancel the D5 charge induced by O9 and D9 we require the addition of 24 five-branes.This means that the induced charges satisfy It follows from our earlier comments that Q 5 (D9) = −1, and therefore from (5.6) we find that Q 5 (O9) = −8.
To find the D3 brane charge of O7 planes wrapped on K3 we use the same trick as before, namely compactify on T 2 and dualize.In this process we obtain four O7 planes wrapped on K3, and therefore each O7 plane wrapped on K3 will induce −8/4 = −2 units of D3 brane charge.Note that this is for the case of SO gauge groups.When we deal with the U Sp groups, as noted earlier, the contribution of the diagrams with a single crosscap change.This implies that for the U Sp case the induced D3 brane charge is +2 units.All in all we find where the top sign refers to SO and the lower sign refers to U Sp.
Total three brane and seven brane charges In the local models we have been considering we have focused on a single orientifold O7 plane and have placed a total of N c /2 physical D7 branes on top of it giving rise to the gauge group SO(N c ) or U Sp(N c ).It follows that the total amount Q 7 of D7 brane charge of this local configuration is given by the number of branes plus the orientifold contribution (5.5) where the top and lower signs in the above refer to SO(N c ) and U Sp(N c ) respectively.The total three brane charge Q 3 arises from the N f physical D3 branes, the D7 brane contribution indicated in (5.4) multiplied by the number of seven branes 1  2 N c , and the O7-plane contribution indicated in (5.7) (5.9)

T-Duality transformations for the SO and U Sp local models
We are now ready to see the effect of T-duality on the above charge assignments.First of all we need to argue that T-duality brings the K3 orientifold back to itself.This would not be the case if we were dealing with T 4 where T -duality would have turned the O7 plane into O3 planes.To see that it is reasonable for K3 note that, if we realize K3 as T 4 /Z 2 the orbifold group is generated by the elements {1, Ω, I, IΩ} where Ω refers to the orientifold action and I refers to the Z 2 inversion of T 4 .Under T-duality, I goes to itself whereas Ω and IΩ get interchanged, so we end up again with a K3 orientifold.This argument leads us to believe that T-duality will bring the K3 orientifold back to itself even away from the orbifold limit of K3 (which we need to assume in order to avoid the zero size instantons concentrated at the fixed points of the orbifold, as discussed in [26]).At any rate we shall assume that for a smooth K3 orientifold the T-dual is still a K3 orientifold.Now let us concentrate on the total D-brane charges in our local models.These were given in eqns.(5.8) and (5.9), which we reproduce for convenience here (5.10) Since T-duality exchanges the D3 and D7 charges, in the dual model, the seven brane charge Q ′ 7 and the three brane charge Q ′ 3 are given by where N ′ c and N ′ f denote respectively the number of D7 branes and D3 branes in the dual model.Solving for N ′ c and N ′ f we find (5.12) This implies that under a T-duality transformation the gauge groups in the local models get exchanged as each theory with 2N f flavors in the fundamental representation of the corresponding group.These are the familiar N = 1 dualities for the SO and U Sp groups with matter in the fundamental [13][14].The additional fundamental meson fields appearing in the magnetic side may arise in the local models (as in [8]) if in the T-dual theory the N f D3 branes are forced to be near each other.In this case, the superpotential term involving the fundamental meson field on the magnetic side will also arise as expected.
We would like to thank M. Bershadsky, P. Cho, K. Intriligator, A. Johansen, T. Pantev and V. Sadov for valuable discussions.
The research of CV is supported in part by NSF grant PHY-92-18167.The research of BZ is supported by D.O.E.Cooperative grant DE-FC02-94ER40818, and a grant from the John Simon Guggenheim Foundation.