Explicit Analysis of Kahler Deformations in 4D N=1 Supersymmetric Quiver Theories

Starting from the $\mathcal{N}=2$ SYM$_{4}$ quiver theory living on wrapped $% N_{i}D5$ branes around $S_{i}^{2}$ spheres of deformed ADE fibered Calabi-Yau threefolds (CY3) and considering deformations using \textit{% massive} vector multiplets, we explicitly build a new class of $\mathcal{N}% =1 $ quiver gauge theories. In these models, the quiver gauge group $% \prod_{i}U(N_{i}) $ is spontaneously broken down to $% \prod_{i}SU(N_{i}) $ and Kahler deformations are shown to be given by the real part of the integral $(2,1) $ form of CY3. We also give the superfield correspondence between the $\mathcal{N}=1$ quiver gauge models derived here and those constructed in hep-th/0108120 using complex deformations. Others aspects of these two dual $\mathcal{N}=1$ supersymmetric field theories are discussed.

Recently four dimension N = 1 supersymmetric quiver gauge theories have been subject to an intensive interest [1]- [3]. These theories, which are engineered in different but dual ways, appear as low energy effective field theory of compactification of M-theory on G2 manifolds and type II string compactification on threefolds preserving 1/8 of original supersymmetries [4]- [7]. A remarkable set of such field theoretical system corresponds to those 4D N = 1 supersymmetric quiver gauge theories with gauge group i U (N i ) and which are obtained through deformations of 4D N = 2 i U (N i ) supersymmetric quiver gauge theories living on D5 branes wrapped on ADE fibered Calabi-Yau threefolds (CY3) [8,9].
Two classes ( with and without monodromies) of such 4D N = 1 supersymmetric quiver gauge theories, following from the complex deformation of 4D N = 2 supersymmetric quiver gauge theories, has been constructed in [11,12]. In this paper, we want to derive their mirrors using Kahler deformations rather than complex ones. Note that from the geometric point of view, this kind of dual models follow naturally using algebraic geometry methods and mirror symmetry exchanging Kahler and complex deformations; but from the supersymmetric field theory view the situation is far from obvious and needs a careful treatment. We will show, amongst others, that Kahler deformations in supersymmetric quiver field theories require massive gauge prepotentials; that is a spontaneously broken gauge symmetry i U (N i ) down to i SU (N i ) with all the features that go with this behaviour and too particularly the implementation of a Higgs superpotential and so adding further fundamental matters.
The presentation of this paper is as follows: In section 2, we describe the 4D N = 2 i U (N i ) supersymmetric quiver gauge theories living on D5 branes wrapped on ADE fibered Calabi-Yau threefolds (CY3). We focus our attention on the special example of U (N ) gauge theory engineered on a A 1 fibered CY3 and use a simplest path involving the minimal degrees of freedom. Extension to ADE geometries is straightforward and some of its aspect may be found in [14]. In section 3, we develop the study of the 4D N = 1 i U (N i ) supersymmetric quiver gauge theories following from complex deformations of the N = 2 SYM 4 quiver models. In section 4, we consider the mirror of the previous N = 1 supersymmetric quiver gauge by using Kahler deformations rather than complex ones. In section 5, we give our conclusion. Note that we will work in N = 1 superspace and make use of both real superspace x, θ, θ techniques as well as chiral ones x ± iθσθ, θ, θ . For technical details; see for instance [14,15].
2 4D N = 2 SYM 4 quiver theories: A 1 model The N = 2 supersymmetric A 1 quiver theory in four dimensions involves the following N = 1 degrees of freedom: (i) A U (N ) gauge multiplet V which we take in the WZ gauge as V = −θσ µ θA µ − iθ 2 θλ + iθ 2 θλ + 1 2 θ 2 θ 2 D. This superfield has the special features which will be needed later on. Here Y ∼ I id is the abelian U(1) generator of U (N ) and {T a } refer to the SU (N ) traceless generators. (ii) A chiral multiplet Φ in the adjoint representation of the gauge group U (N ). We will refer to it as adjoint matter and has the two following decompositions. δ m T rΦ m reduces to a polynom in the U(1) superfield Θ. (iii) Four chiral multiplets Q (±,±) with the following U (1) × SU (N ) charges: Q (+,+) ≡ Q + and Q (−,+) ≡ P − are in the representation (±1, N ) and Q (+,−) ≡ P + and Q (−,−) ≡ Q − are in the representation ±1, N . The antichiral superfields are in the complex conjugate of these representations. For convenience, we will work with the normalization of the U(1) charge as [Y, Q ± ] = 2Q ± and [Y, P ± ] = −2P ± . These matter superfields have, in the chiral basis, the following θ-expansions, where q ± , p ± and so on stand for component fields. Note that the chiral composites Q + Q − and P + P − are in the U (N ) adjoint representation and may be expanded as in eqs(2.1,2.2). The same is valid for the hermitian composites Q ± Q * ± and P ± P * ± . Note also that these four Q ± and P ± chiral multiplets form two N = 2 hypermultiplets; one of them encodes the transverse coordinates of D5-branes; it describes their positions in the ten dimensional type IIB string space, and the other is the usual moduli associated with the Kahler deformation of the A 1 singulariy [17].

Action:
The superspace lagrangian density L N =2 (A 1 ) describing the N = 2 dynamics of the previous superfields reads as, where L g (V ) and L ad (Φ) are respectively the gauge covariant lagrangian densities for the U (N ) vector multiplets and adjoint matter superfields. The coupling constants ζ and β are respectively real and complex parameters. They have both a field theoretical and geometric meanings and will play a crucial role in the present study. The supersymmetric scalar potential reads in terms of the auxiliary fields as V = 1 2 T r D 2 + T r (F F * ) + T r f * ± f ± + T r l * ± l ± and the moduli space of its vacuum configuration is given by the following eqs, where we have set The black dot on the ζ axis represents g (I) N =2 and the one on the |β| axis represents g reads as, where ∆t stands for the holomorphic volume (t 1 − t 2 ) of the complex deformation which, by help of eq(2.5), is also equal to β and so eq(2.8) may be rewritten as x 2 + y 2 + z 2 = β 2 .

Mirror N=2 models:
On the supersymmetric field theory side, the ζ and |β| parameters are involved in the N = 2 SYM gauge coupling constant g ≡g N =2 which read, in terms of the type IIB string coupling g s , as, Note that from the above relation, one sees that the N = 2 SYM coupling constant is a real two argument function; g N =2 = g N =2 (ζ, |β|), which we shall naively rewrite as g N =2 (ζ, β). Accordingly, one may think about this gauge coupling constant as describing a flow of N = 2 SYM models interpolating between two extreme models I and II with respective gauge coupling constants g (I) N =2 and g (II) N =2 . The first is, with blown up volume V I and the second involves pure holomorphic volume V II type Weil-Peterson as, Setting ζ = ρ cos ϑ and |β| = ρ sin ϑ; with the spectral parameter ϑ bounded as 0 ≤ ϑ ≤ π 2 , one gets an explicit relation for this N = 2 gauge coupling constant flow g N =2 = g N =2 (ϑ) = gs V (ϑ) . In this view, the theories I and II with respective gauge couplings g N =2 correspond to ϑ = 0 and π 2 , they are mapped to each other under mirror symmetry acting as ϑ → π 4 − ϑ; see figure 1. In A 1 geometric language, the N = 2 gauge models I corresponds to the blowing up of A 1 surface in CY3; but zero holomorphic deformations, S 2 ω (2,0) = 0. The compact part of the A 1 singularity x 2 + y 2 + z 2 = 0 gets a non zero volume as (Re x) 2 + (Re y) 2 + (Re z) 2 = ζ. This positive Kahler parameter ζ is same as in the superfield action eq(2.4). To fix the ideas ζ can be imagined of as corresponding to the derivative of a special Kahker deformation K h, h where h and h are Higgs fields to be specified later on; see eq(4.7). In other words ζ = ∂K F I /∂C where K F I is a linear Kahler deformation as K F I ∼ ζ (C + U ) and where C = υ * H+υH υυ * ; see eqs (5.1-5.4). In the present paper, ζ should be thought of as just the leading case of a non linear Kahler superpotential K H, H so that this Kahler analysis, one may also consider its mirror description using complex deformation of A 1 singularity. In this case the resulting N = 2 gauge model II corresponds exactly to the reverse of previous situation. Here S 2 J (1,1) = 0 but S 2 ω (2,0) = 0. As before the A 1 singularity x 2 + y 2 + z 2 = 0 gets now a holomorphic volume as x 2 + y 2 + z 2 = β 2 where β is as in the super action eq(2.4). Here also this β appears as the derivative of linear complex deformation as W F I ∼ βφ which in general should be thought of as just the leading case of a non linear polynomial superpotential W (φ) so that no complex deformation of this conifold 1 is made here and so it should'nt be confused with geometric transition scenario of [11]. Note also that extension of ζ and β to non linear K ′ and W ′ respectively break N = 2 supersymmetry down to N = 1. From the field theoretical point of view, these two models correspond to choosing the corresponding vevs eqs(2.5-2.6) such that t 1 (φ) = t 2 (φ) and r 1 (c) = r 2 (c) and inversely t 1 (φ) = t 2 (φ) and r 1 (c) = r 2 (c). The two symmetric situations indicate the existence of two mirror N = 1 supersymmetric A 1 quiver gauge theories I and II with gauge couplings g This is equivalent to introducing an extra chiral superpotential W in the adjoint matter superfield and too particularly in the U(1) factor Θ of adjoint matter Φ eq(2.2). In doing so, the lagrangian density In this relation, N = 2 supersymmetry is explicitly broken down to N = 1 due the presence of the non linear super potential W(Θ); but U (N ) gauge invariance is still preserved. The superpotential W (Θ) generating complex deformations has two basic features which in fact are inter-related and play an important role at the quantum level: (i) the holomorphic property ∂W (Θ) ∂Θ * = 0, which permit to benefit from the power of algebraic geometry and (ii) chirality allowing miraculous simplifications. Comparing the above lagrangian density (3.1) with eq(2.4), one learns that complex deformation by the superpotential W (Θ) corresponds to promoting the previous complex FI type linear term with complex coupling constant β, namely β d 2 θ (Θ), to a more general chiral superfunction d 2 θW (Θ). As a consequence W ′ (Θ) is no longer constant as in general it is Θ dependent. It follows then that the constant β of section 2 is now promoted to a U(N ) gauge invariant function P (φ) as, Moreover, putting the relation (3.3) back into the expression of the SYM gauge coupling g, one gets the following running N = 1 gauge coupling constant, Note that N = 2 supersymmetry is recovered at the critical point φ 0 of the superpotential; W ′ (φ 0 ) = 0, and so by expanding around this critical point, one may compute the deviations of the N = 1 gauge coupling from the N = 2 value.
One of the special feature of this expression is that under complex deformation, eq(2.8) becomes showing that the CY3 is indeed a complex deformed A 1 surface fibered on the plane parameterized by the complex variable φ. Furthermore using the relation (3.6) and comparing with eqs(2.7), it is not difficult to see that the superpotential of the adjoint matter considered above is in fact linked to CY3 complex moduli space as follows, where Ω = ω ∧ dτ is a (3, 0)-form on CY3 realized by an A 1 fiber on the complex plane and where one recognizes the FI terms βφ. Such analysis extends straightforwardly to all ADE fibered CY3; with both finite and affine ADE geometries. This aspect and other feature will be exposed in [14].

N = 1 A 1 quiver gauge theory II
Applying mirror symmetry ideas to the above N = 1 A 1 quiver gauge theory I, one expect to be able to build its superfield theoretical dual by starting from the the lagrangian density L N =2 (A 1 ) eq(2.4) and use Kahler deformations as, In superspace, δ Kahler L involve integration over the full superspace measure and reads as, where K is a Kahler superpotential; that is some real superfunction we still have to specify. In what follows, we show that K has much to do with massive gauge superfields.
Massive gauge prepotential: Although natural from geometric point of view due to mirror mirror symmetry exchanging Kahler and complex deformations of CY3 [16], the superfield theoretical formulation of the dual theory II is far from obvious. The point is that in the derivation of N = 1 quiver gauge theories I, the promotion of β to chiral superpotentials W (φ) uses the scalar moduli of adjoint matter Θ. However for the Kahler deformations we are interested in here, one cannot use Θ by deforming the kinetic energy density d 4 θ (Θ * Θ) to, where K adj (Θ * Θ) is a Kahler superpotential for adjoint matter. A field theoretical reason for this is that Θ does not couple to the abelian U(1) gauge prepotential of the U (N ) gauge symmetry. The introduction of Kahler deformations for the Q ± and P ± fundamental matters as, does not solve the problem any more since this leads essentially to quite similar relations to eqs(2.5-

2.7). The adjunction of superpotentials for fundamental matters does not work as well because it
breaks SU (N ) gauge symmetry down to subgroups and this is ruled out by the A 1 fibered CY3 we are considering here. However there is still an issue since a careful analysis for the Kahler analogue of the chiral superpotential of complex deformations of theory I reveals that the difficulty we encounter in theory II is not a technical one. It is linked to the fact that in 4D N = 1 supersymmetric gauge theory II, the N = 1 massless gauge multiplet 1 2 , 1 has no scalar moduli that could play the role of the coordinate of the complex one dimension base of CY3. This is then a serious problem; but fortunately not a basic one since it may be overcome by considering massive N = 1 gauge multiplets U (mass) , which have scalars contrary to massless gauge prepotentials. But how this issue may be implemented in the original N = 2 supersymmetric quiver gauge theory we started with? The answer is by spontaneously breaking the abelian gauge sub-invariance as U (N ) −→ SU (N ). For general ADE geometries, the spontaneous breaking of the quiver gauge symmetry should be as i U (N i ) −→ i SU (N i ). Using this result, one still has to overcome the two following apparent difficulties.
Two more things: (1) From geometric point of view, we know that the variable τ parameterizing the complex one dimension base ( plane) of the CY3 is associated with the complex scalar modulus of the adjoint matter multiplet Φ as shown on 0 2 , 1 2 , In the case of N = 1 massive gauge multiplets U (mass) , one has only one scalar modulus and it is legitimate to ask from where does come the lacking scalar? This is a crucial question since one needs one more scalar to be able to parameterize the two dimension base of CY3. The answer to this question is natural in massive QFT 4 ; the missing scalar degree is in fact hidden in the N = 1 on shell massive gauge representation; it is just the longitudinal degree of freedom of the massive spin one particle A µ .
This a good point in the right direction; but we still need to know how to extract this hidden scalar.
The right answer to this technical difficulty follows from a remarkable feature of N = 1 supersymmetric theory which require complex manifolds [15]. In the language of supersymmetric field theoretical representations, the real scalar c appearing in 0, which one suspects justly to be the right modulus for parameterizing the base of CY3. (2) where 0 2 , 1 2 ⊕ 0 2 , 1 2 are two chiral multiplets. Moreover as the N = 1 massive gauge multiplet 0, 1 2 2 , 1 M may also be decomposed as the sum of a N = 1 massless gauge multiplet and a N = 1 chiral superfield, one then end with the following spectrum: (a) a massless abelian gauge prepotential U and (b) three chiral multiplets H 0,± as shown here below, multiplets should be thought of as Higgs superfields and whose Kahler superpotential, is exactly what we need. (ii) The SU (N ) massless N = 2 vector multiplet which in terms of the N = 1 superfield language we are using here reads as V a and Φ a ; and (iii) finally the two N = 2 hypermultiplets Q ± and P ± describing fundamental matters. From this supersymmetric representation analysis, one learns that dynamics of massive N = 2 vector multiplet may be formulated in N = 1 superspace by starting with a massless vector multiplet U and three chiral ones H 0,± as introduced before. To get a massive gauge superfield, one gives non trivial vevs to H ± ; a fact which is achieved by introducing a superpotential W ext (H + , H 0, , H − ) describing couplings between chiral superfields. Since we are interested by the engineering of N = 1 quiver gauge theory using Kahler deformations, we will not insist on having N = 2 supersymmetric couplings for Higgs superfields. So we restrict the extra superpotential to W ext = W ext (H + , H − ) with the two following requirements: (α) The full scalar potential V of the supersymmertic gauge abelian model namely vanishes in the vacuum ( D U = F 0,± = 0) and (β) at least one of the chiral superfield H ± acquires a vev when minimising V ( ∂V ∂h± = 0). Let take these vevs as, where υ is a complex parameter. A simple candidate for gauge invariant Higgs superpotential fulfilling features (α) and (β) is W ext = mH + H − with mass m linked to ζ and υ; i.e m = m (ζ, υ). With this in mind one can go ahead to work out the Kahler deformation program. In what follows, we describe the main lines and omit details.
The Action for N=1 quiver Theory II: From the above discussion, it follows that the lagrangian density L (II) N =1 (A 1 ) = L N =2 (A 1 ) + δ Kahler L eq(4.1) of the N = 1 supersymmetric quiver model II is given by the following superfunctional, In this relation we have endowed the matter superfield H + with a Kahler potential K H * + e U H + and left H − with a flat geometry. The introduction of a Kahler potential for H − does add nothing new since it is H + that is eaten by the gauge prepotential after symmetry breaking. K H * + e U H + is then crucial in the derivation of N = 1 quiver theories II; it is the mirror of W(Φ) of N = 1 quiver gauge theories I.

More Results
In the lagrangian density L N =2 (A 1 ) eq (2.4)  where Y = H * + e U H + . In this result similarity between Kahler and complex deformation is perfect. It is a consequence of mirror symmetry in this super QFT and may also be rederived from the analysis of the lagrangian density (4.13). The appearance of this composite hermitian superfield Y is not fortuitous; it is just a manifestation of the massive gauge prepotential we have discussed before. Indeed parameterizing H + as, where now H describe quantum fluctuation, we have for Y, But the term υ * H+υH * υυ * + U in the exponential is nothing but the massive gauge prepotential U (mass) of eq(4.5). Eq(5.4) and (5.2) give actually the relation between massive gauge multiplet and Kahler deformations. Moreover the defining eqs for the moduli space of the supersymmetric vacua of Kahler deformations in N = 1 quiver theories II following from (4.13) reads as, Eq (5.5) shows that the blown sphere depends on the coordinate of the base of CY3. Like before, N = 2 supersymmetry is explicitly broken down to N = 1 except at the critical point c 0 of R (c) where it is recovered; but U (N ) gauge invariance is spontaneously broken down to SU (N ). In terms of the quantum fluctuation superfields H and H * eq(5.3), the critical point R ′ (C 0 ) = 0 is translated to, This relation should be thought of as the analogue of ∂W ∂Φ * = 0 in complex deformations. One can also compute the variation of the N = 1 runing gauge coupling g N =1 (c) = g N =1 (ζ, β; c) around the value of the N = 2 one g N =2 (c 0 ) living at the critical point K ′ (c 0 ) = 0. One finds, for a generic point on the N = 2 supersymmetric flow g = g (ϑ), the following dual formula to eq (3.5), Note by the way that one may also work out the mirror of eqs(3.7,3.8). Spliting x, y and z as x = x 1 +ix 2 and so on, one may decompose the complex surface x 2 + y 2 + z 2 = 0 into a compact part x 2 1 + y 2 1 + z 2 1 = 0 and a non compact one. Deformations of compact part as x 2 1 + y 2 1 + (z 1 − ∆r) (z 1 + ∆r) = 0 and substituting ∆r is as in eqs(5.5), one gets the real analogue of eq(3.8) namely Geometrically, this means that R (c) generates Kahler deformations of the CY3 and one can check that R (c) is given by the following, where K (2,1) and K (1,2) are respectively (2, 1) and (1, 2) forms on CY3 and where one recognizes the usual FI term ζc of the N = 1 abelian gauge theories. The correspondence between the two theories is then perfect.

Conclusion
In this paper, we have developed the field theoretic analysis of deformations of 4D N = 2 quiver gauge theories living in D5 branes wrapped on A 1 fibered CY3. Though it looks natural by using algebraic geometry methods and mirror symmetry exchanging complex and Kahler moduli, such study is far from obvious on the field theoretical side. After noting that the gauge coupling constant g N =2 of such a theory is given by a spectral flow with g N =2 (0) and g N =2 π 2 respectively associated with pure Kahler and pure complex deformations in the A 1 fiber, we have considered deformations in the full moduli space of CY3. For complex deformations, geometry implies that we have the two following: (a) If deformations are restricted to the ADE fibers, then N = 2 supersymmetry is preserved, up to a global shift of energy and (b) If they cover the full CY3, then N = 2 supersymmetry is broken down to N = 1. Mirror symmetry implies that similar results are also valid for Kahler deformations. On the superfield theoretical view, this corresponds to adding appropriate superpotential ( complex and Kahler) terms in the original N = 2 SYM 4 . We have studied complex deformations of N = 2 supersymmetric quiver theories by using the method of [8] and given amongst others the field expansion of the N = 1 running gauge coupling constant g N =1 around g N =2 .
We have also developed the explicit analysis for Kahler deformations of N = 2 supersymmetric quiver theories and shown that such real deformations require massive gauge prepotentials U (mass) implying in turn a spontaneously broken U(N ) gauge symmetry down to SU(N ). We have worked out this program explicitly and shown amongst others that Kahler deformations are given by the following. with P (Φ) as in eq (3.3). The analysis we have developed in this paper has the remarkable property of being explicit. It allows superfield realizations of geometric properties of CY3 and offers a powerful method to deal with 4D N = 1 supersymmetric field theories living on wrapped D5. Through this explicit field theoretic study, one also learn that, on the N = 1 supersymmetric field theoretical side, mirror symmetry acts by exchanging the roles of adjoint matters Φ and massive gauge prepotentials U (mass) . On the geometric side, we have shown that Kahler deformations, generated by the real superfield R U (mass) , are given by the real part of the integral of a (2, 1) form on CY3 as shown on eq(5.10).
This analysis may be also extended to incorporate D3 branes by considering affine ADE symmetries.
Details on aspects of this study as well as other issues may be found in [14].