$\mathcal{N}=2$ Supersymmetry Partial Breaking and Tadpole Anomaly

We consider the $U(1) ^{n}$ extension of the effective $\mathcal{N}=2$ supersymmetric $U(1) \times U(1) $ model of $arXiv:1204.2141$; and study the explicit relationship between partial breaking of $\mathcal{N}=2$ supersymmetry constraint and D3 brane tadpole anomaly of type IIB string on Calabi-Yau threefolds in presence of H$^{RR}$ and H$^{NS}$ fluxes. We also comment on supersymmetry breaking in the particular $\mathcal{N}=2$ $U(1) $ Maxwell theory; and study its interpretation in connection with the tadpole anomaly with extra localized flux sources.


Introduction
Breaking N = 2 supersymmetric quantum field theories in 4d space time at two different mass scales has been subject of interest for many years [1]- [21]; and refs therein. This scenario is possible in 4d N = 2 supergravity theory; but not with N = 2 global supersymmetry suspected to live at lower energies below Planck scale. If gravity is neglected, superalgebra relations requires that once one of the two global supercharges Q ± α = (Q α ,Q α ) is broken; say Q − α , the second Q + α has to be broken too. However this constraint can be bypassed in the presence of magnetic Fayet-Iliopoulos (FI) terms induced by non perturbative BPS states such as D-branes of type II strings. With non zero magnetic FI couplings, the supercurrent algebra develops a constant term that violates the SU R (2) R-symmetry of the supercharges [6]; offering as a consequence a way to break N = 2 supersymmetry partially via gauginos instead of gravitinos [4,22,23,24,25]. This idea has been approached in past by using non linear realisation of half of the eight supersymmetric charges [26,27,28,29,30,31,32]; but further developed recently in [33,34,35] by using N = 1 superspace QFT 4 method where a simple, but instructive, effective N = 2 supersymmetric abelian U(1) 2 model, with two breaking scales Λ 1 and Λ 2 , has been engineered.
In this paper, we consider the partial breaking of N = 2 supersymmetry in the effective 4d U (1)×U (1) model of [33,34], to which we refer now on as ADJ effective gauge theory; and study explicitly its relationship with D3 brane tadpole anomaly of type IIB string compactified on local Calabi-Yau threefolds (CY3). To deal with brane realisation of the ADJ construction, we first relax the rank of the abelian group symmetry by considering the effective U (1) 2 model as the leading prototype in the family of 4d N = 2 U (1) n gauge models indexed by n ≥ 2; and then think of this set of abelian gauge models in terms of an effective low energy theory following from D3 branes wrapping 3-cycles in type IIB string on local CY3 with a n-dimensional symplectic homology basis of 3-cycles (A a , B a ), a=1,...,n. In this D3 brane realisation of 4d N = 2 U (1) n gauge theory, partial breaking of global N = 2 supersymmetry is induced by H RR = 0. We also study the missing n = 1 term in the sequence of 4d N = 2 U (1) n gauge models with n ≥ 2; this particular model, which corresponds type IIB string on conifold geometry, is anomalous in agreement with known results in literature; this anomaly may be directly learnt from the naive extension of ADJ condition which is given by the singular equation g κ = 0 requiring the vanishing of the gauge coupling constant g = 0 for finite magnetic FI coupling κ. By trying to engineer a 4d N = 2 U (1) model going beyond ADJ constraint by deforming the singularity like ν + g κ = 0, we end with an explicit breaking of N = 2 supersymmetry down to N = 1. The brane interpretation of this deformation in terms of presence of D7 branes or O3 planes is also studied by using a result from [36].
The presentation is as follows: In section 2, we review the basis of the effective 4d N = 2 U (1) 2 model; and derive the N = 2 ADJ constraint equation and its N = 1 deformation. In section 3, we give the main lines of the N = 1 superfield formulation of the 4d N = 2 U (1) n gauge theory describing the gauge dynamics of n N = 2 gauge multiplets coupled to a single tensor multiplet. In section 4, we study the realisation of ADJ model in type IIB string on CY3 with non trivial fluxes of the 3-form field strengths H N S 3 and H RR 3 . In section 5, we give conclusion and make some comments. In section 6 we give two appendices; in the first appendix we collect some useful tools on type IIB string compactification to 4d space time; and in the second we describe the gauge and supersymmetric transformations of single-tensor and Maxwell multiplets in superspace.

N = U (1) n theory and ADJ constraints
In this section, we review the main lines of the model of [33,34]; and study particular aspects of ADJ constraint equation supporting the partial supersymmetry breaking in this theory. Following [33,34,35], the ADJ model is a 4d N = 2 supersymmetric effective gauge theory where gravity is decoupled; but global N = 2 supersymmetry is broken at two different scales. The simplest version of the model realising the two partial breaking is given by the interacting dynamics of a N = 2 single tensor multiplet T (N =2) with two abelian N = 2 gauge supermultiplets V . Besides special Kahler geometry of the gauge background, the model has a Chern-Simons type interaction between T (N =2) and the linear combination g 1 V

ADJ theory in N = 1 superspace
Because of lack of a simple formulation of 4d supersymmetric gauge theories with 8 supercharges in N = 2 superspace, one is limited to use the standard N = 1 superspace method with the price that only half of supersymmetries is manifestly exhibited; the other half is hidden; but can be linearly realised in absence of magnetic FI couplings.

Fibering N = 2 chiral superspace
A way to deal with the 4 hidden supersymmetric charges is to use N = 2 chiral superspace and think about it in terms of fibration of two copies of N = 1 chiral superspaces; a N f iber = 1 ′ chiral superspace, with odd coordinatesθ α , fibered on a N base = 1 chiral superspace base with odd coordinates θ α . Schematically, this fibration may be represented like In this chiral superspace fibration, typical N = 2 chiral superfields have the structure Φ N =2 = Φ(z, θ,θ) with space time coordinate z µ related to the real x µ by two pure imaginary shifts iυ µ + iυ µ , one from N f iber = 1 ′ fiber and the other from the N base = 1 base as shown on the relation z = y − iθσ θ with y = x − iθσθ. Viewed from fiber, Φ N =2 can be expanded in a finite series ofθ as follows with expansion modes given by N base = 1 superfields: Φ N =1 = Φ (y, θ) and similarly for Ψ N =1 α and F N =1 . The extra dots stand for additional terms involving space time derivatives generated by −iθσ µ θ ∂ µ = −iυ µ ∂ µ . The expansion modes in (2.2) describe N = 1 chiral superfields in the base; and are related amongst others by those N f iber = 1 ′ supersymmetric transformations in the fiber; for examplẽ By imposing appropriate constraint relations on Φ N =2 , one obtains the desired N base = 1 superfields to describe supersymmetric matter or radiation with 8 supercharges. In this way of doing, N = 2 supersymmetric gauge multiplet is then approached by using superfield strength W N =2 with expansion as in (2.2); but satisfying moreover D αD α W N =2 + hc = 0. As this constraint involves both the chiral W N =2 and its adjoint conjugate, one ends with aθ-expansion involving both N = 1 chiral X andD 2X as follows where the role of the extra constant coefficient 1 2κ will be discussed later on; it scales as mass 2 seen that W N =2 = [X] = mass 1 and the N base = 1 chiral gauge superfield strength spinor [W α ] = mass 3/2 ; it may be generated by the particular and asymmetric shift of By asymmetric we mean eq(2.5) but without modifying the X superfield in (2.4). This property may be roughly interpreted as giving a non zero VEV to theθ 2 -component field of expansion of (2.4) as breaking thus the N f iber = 1 ′ supersymmetry in fiber. For a stringy interpretation of the coupling constant 1 κ in terms of 3-form flux through non compact 3-cycles in local CY3; see eq (4.30). A quite similar expansion is valid for the N = 2 tensor multiplet T (N =2) which is described as well by a constrained N = 2 chiral superfield [33,34,35], see also appendix §6.2; it is given by with N f iber = 1 ′ supersymmetric transformations in fiber as With these tools at hand, we turn to describe useful features on superfield spectrum of N = 2 supersymmetric ADJ model. For later use, we will give both the N = 2 chiral superfields spectrum and their splitting in terms of N base = 1 superfields.
more on matter sector The matter sector of ADJ model is quite simple; it involves one 4d N = 2 matter multiplet having two dual realisations as given by eqs (2.11). In the realisation we will be using in present study, N = 2 matter is described by a N = 2 chiral superfield T (N =2) with expansion along fiber direction as in (2.7). From the N base = 1 base view, this expansion has four chiral superfields: two bosonic Y , Φ; and a fermionic superfield doublet χ α = (χ 1 , χ 2 ); altogether they capture 16 + 16 off shell degrees of freedom. This number may be reduced down to 8 + 8 by thinking about Y as an exotic auxiliary superfield playing the role of a Lagrange superfield parameter capturing the constraint on partial breaking of second supersymmetry; and about χ α as a superfield prepotential of a hermitian linear multiplet L given by the relation Observe that L is invariant under the change χ ′ α = χ α + i 4D 2 D α Ω with Ω an arbitrary real superfield; this symmetry together with footnote 1 allows to reduce the 16+16 degrees of freedom down to 8 + 8; for details see appendix §6.2; other features can be found in [33,34,35].
For completeness, notice that using a result on hypermultiplet duality on superfield representations of N = 2 matter multiplet [37, 38,39,40,41], we can show that N = 2 superfield T (N =2) has two dual representations in terms of N base = 1 superfields; the (Φ, L) we will be using in this paper; and a second realisation based on two chiral superfields ADJ constraint First notice that in above (2.11), it looks like if we have three N base = 1 superfields to describe N = 2 matter; this is not exact since Y is some how a "spurious superfield" carrying no physical degrees of freedom; it is a topological object exhibiting very special properties as shown by eqs(2.19-2.21); this is our reason behind putting Y aside in eq(2.11); it breaks N = 2 supersymmetry partially and is one of the nice observations in [34]; there it appears as a Lagrange superfield parameter capturing a constraint relation f (g a , κ a ) = 0 of the model, which to fix ideas may be thought of as giving a relationship between the coupling constants g a and the magnetic FI couplings 1/κ a of the ADJ model; see also eq(3.5) given below for explicit details. In the limit 1 κ a → 0 (2.13) the above constraint is trivially solved and then Y has no role to play in this N = 2 supersymmetric limit. Notice also that the sum on integer n in eq(2.12) rules out the particular case n = 1; since the corresponding condition reads as leading to g 1 = 0 for finite 1/κ 1 ; and then no ADJ theory with one U(1) gauge factor [31,32,33,34,35]. By trying to overcome the constraint g 1 κ 1 = 0 by adding an extra term like ν + with ν a real parameter having same scaling mass dimension as ga κa ; one breaks explicitly N = 2 supersymmetry down to N = 1.

N = 1 superfields in U (1) 2 ADJ model
In our superspace description of U (1) × U (1) ADJ model, we will use a particular set of N = 1 superfields; these are the chiral Φ and hermitian L for representing T (N =2) ; and 2 gauge superfields (V 1 , V 2 ), 2 chiral (X 1 , X 2 ) for representing V . Let us comment this system of superfields.
• T (N =2) sector In the N = 1 superfield realisation given by the first relation of eqs(2.11), the dynamics of T (N =2) is described by two basic superfields and an auxiliary one; these are: (i) the chiral superfield Φ with the usual θ-expansion namely a leading scalar component φ; a Weyl fermions ψ α and auxiliary field F φ ; (ii) the standard hermitian linear multiplet L satisfying the superspace constraint relations D 2 L =D 2 L = 0 following from (2.10); this is a particular superfield with θexpansion in component fields as follows involving the propagating real field C and the field strength of the antisymmetric tensor field B ρσ ; but no auxiliary field. The superfields Φ and L are related under fiber N f iber = 1 ′ supersymmetric variations as follows with Ψ standing for L and Φ.
(iii) an extra auxiliary superfield Y capturing information on non linear realisation of the N f iber = 1 ′ hidden supersymmetry; it is not needed for the closure of transformations (2.17); but will be used to approach partial supersymmetry breaking. Properties of this superfield have been explored in [33,34] where, using gauge fixing 1 method, it has been shown to have the following remarkable θ-expansion for details see appendix 6.2. This relation shows that Y encodes data on the constant antisymmetric tensor C µνρσ = 4!ε µνρσ ∆. Because of its special dependence in θ, Y has no physical degrees of freedom and obeys moreover a nilpotency property In the N = 1 superfield realisation given by the second relation of eqs(2.11), the role of Φ and L gets played by the two chiral superfields Q 1 and Q 2 capturing opposite charge under a U (1) gauge symmetry. The duality transformations between the two matter multiplet realisations are given by Legendre transform in superspace [37, 38,39,40,41]; they may be written as follows with Φ as in the first relation of eqs(2.11) and where Φ ′ is another chiral superfield. We will not need this realisation in this paper; but to fix ideas we give some comments on their dynamics in § 3.2; see eq(3.18).
• V (N =2) a gauge sector The gauge sector of ADJ supersymmetric U(1) 2 model involves two N = 2 abelian Maxwell type multiplets described by the hermitian superfields V with superfields strength θ-expansions along fiber direction as in eq(2.2). Following [33,34], the solution of constraint equations lead to the N = 1 superfields spectrum Gauge fixing of this symmetry leads precisely to eq(2.19); for more details see also [33].
where the hermitian V 1 , V 2 are the usual N base = 1 gauge superfield potentials; and where X 1 , X 2 are two chiral superfields. So the gauge symmetry of the model is . The extra κ 1 , κ 2 are constants and are as in (2.4); they may be put in correspondence with the auxiliary chiral superfield Y as it may be viewed by comparing (2.7) with (2.4); that is: The general form of the superspace lagrangian density L gauge of the gauge superfields depending on the prepotential F (X 1 , X 2 ) reads as follows with gauge lagrangian density in N base = 1 superspace given by and Fayet-Iliopoulos part as where the real ξ a and complex (pure imaginary) e a are constants and where the holomorphic F a = ∂F ∂X a and F ab = ∂ 2 F ∂X a ∂X b .

ADJ constraint and N = 1 deformation
Here we use the N = 1 superfield spectrum of ADJ model to study the derivation of the constraint eq(2.12) and its N = 1 deformation (2.15).

Superfield
given by (2.4); and the T N =2 of (2.7). These N = 2 chiral superfields are remarkable; they have the same scaling mass dimension and quite similarθ-expansions which make them to share some general features. Indeed, though physically different objects, the resemblance between theirθ-expansions could serve as a guide to have more insight into the ADJ construction. This formal property has been used in [33,27] to study the interaction between a N = 2 Maxwell multiplet W N =2 and a tensor T N =2 . There, the formal similarity between the two chiral superfields W N =2 and T N =2 ; in particular their scaling mass dimension and θ-expansions, has been used to build the linear combination of these N = 2 chiral superfield to reach the gauge invariant quantity that plays a central role in the N = 2 Dirac-Born-Infeld U max (1) theory; and also in studying electric-magnetic duality in N = 2 chiral superspace in presence of Chern-Simons coupling. In this relation, F M ax µν is the usual field strength of the Maxwell gauge field potential; and B µν the antisymmetric gauge potential appearing in the tensor multiplet. By exhibiting this formal similarity between W N =2 and T N =2 ; one finds that there exist a correspondence between their N = 1 superfields contents; by comparing theθ-expansions (2.4) and (2.7); as well as topological relations reported in appendix B (6.19) and (6.24), one ends with where Y occupies a place in T N =2 that is similar to the place occupied by X inW N =2 . Obviously the superfields in left and right of table (2.30) have different meanings and carry different degrees of freedom; but as far as fibration of N = 2 supersymmetry is concerned; this correspondence may be used as an indication to get more insight into the general form of constraint equation captured by Y .

Deriving ADJ condition
The ADJ constraint equation is obtained by from N = 2 Chern-Simons couplings between the linear combination of the gauge superfield strengths ( a g aW N =2 a ) and the tensor multiplet T N =2 . In N = 2 chiral superspace where N = 2 supersymmetry is manifest, this CS coupling reads in terms ofW N =2 a and T N =2 as follows where n = 2 for the case of U(1) 2 model; but can generally take any value n ≥ 2 as the case of U(1) n models with n gaugeW N =2 a coupled to T N =2 . By using (2.4) and (2.7) and performing integration with respect toθ; one brings above CS coupling to the form where N = 1 supersymmetry in the base of fibration is manifest and the fibered N = 1 ′ one becomes hidden. Because of linear dependence, the superfield equation of Y leads to the constraint n a=1 g a κ a = 0 , n ≥ 2 (2.33) Notice that the deformation of the CS coupling (2.32) by adding the term ν d 2 θY like preserves gauge symmetry as shown by (6.41) of appendix B; but breaks explicitly N = 2 supersymmetry down to N = 1. Under this deformation, the ADJ constraint becomes ν + n a=1 g a κ a = 0 (2.35)

Superspace Lagrangian
The N = 1 superspace expression of the lagrangian density L describing the interacting dynamics of the above N = 2 supersymmetric system W , T (N =2) can be approached in two manners depending on the N = 1 superfield realisation used to represent the T (N =2) single tensor multiplet.

Using
With the realisation of the tensor multiplet T (N =2) in terms of the superfields the chiral Φ, the hermitian L as well as the Lagrange chiral superfield Y carrying the ADJ constraint; and following [34], the superspace lagrangian density of the N = 2 supersymmetric U(1) 2 model describing coupled dynamics of W (N =2) a and T (N =2) reads in N = 1 superspace as follows with L gauge as in eq(2.26) and In what follows, we first make few comments useful for our later analysis; then we study scalar potential of the gauge model.

Properties of (3.2)
From the above expression of the superspace lagrangian density (3.2), we learn a set of special properties on the superfield realisation of matter using single tensor multiplet; in particular the following: First, the mass constant m in (3.2) can be absorbed by shifting the linear combination g a X a ; it will be dropped out in what follows. Second, the superfields Φ and L are gauge invariant and scale as mass 2 ; their coupling to the gauge multiplet is of Chern-Simons type they involve the remarkable linear combinations g 1 V 1 + g 2 V 2 and g 1 X 1 + g 2 X 2 . Moreover, the contribution of the superfield Y in the full superspace lagrangian density (3.2) appears linearly as follows together with the particular linear combination g 1 κ 1 + g 2 κ 2 . So the auxiliary superfield Y in ADJ theory plays the role of a Lagrange superfield capturing the constraint relation showing that the ratio g 1 g 2 of the two gauge couplings is fixed by the ratio κ 1 κ 2 of the magnetic FI coupling constants. Third, the kinetic energy density of Φ and L involves non polynomial expressions, a square root term L 2 + 2ΦΦ and a logarithm one namely L ln L+ L 2 + 2ΦΦ ; this non linearity may be understood as due to the antisymmetric field B µν . Self interactions of (Φ, L) are also non polynomial and are generally characterized by an arbitrary hermitian prepotential H Φ,Φ; L with superspace lagrangian density as [37, 38,39] L (H)

Scalar potential
The scalar potential of the ADJ model (3.1-3.2) has two contributions as follows a contribution V gauge coming from the auxiliary fields F X a and D a of the gauge multiplets; and another contribution V tens coming from the auxiliary field Φ; seen that L has no auxiliary field. The V gauge contribution reads explicitly as where H ab = Im F ab is the metric of the special-Kähler manifold with inverse H ab . For the V tens contribution, we have where G φφ is the analogue of metric H ab for the matter sector. Substituting the various auxiliary fields by their field equations, we obtain the explicit expression of the full scalar potential of the model. For the contribution V gauge , we have with real r a and complex w a as follows Besides FI coupling constants, they depend on the degrees of freedom of the tensor multiplet namely C and φ. The other contribution is given by where we have set So the total scalar scalar potential reads as Observe the two following features: first for ̺ 2 = 0 and w a = 0, the scalar potential V sca has a non zero value due to the non-vanishing ξ and hence N = 2 supersymmetry breaks down. For ̺ 2 = 0 and ξ a = e a = 0, the scalar potential V sca has as well a non zero value proportional to the magnetic FI coupling as shown on the following expression The stationarity condition of the scalar potential with respect to the various fields namely leads, for the case C 2 + 2 |φ| 2 = 0, to the following equation leading to broken supersymmetric phase for the case where F abc = 0.

Using
Using duality transformations (2.22), one can also express the N = 2 tensor multiplet T (N =2) as a hypermultiplet described by two N = 1 chiral superfields Q 1 and Q 2 . In this realisation of T (N =2) , the previous superspace density (3.2) gets mapped to the equivalent expression where the superfields Q 1 and Q 2 carry opposite charges under the U 1 (1) × U 2 (1) gauge symmetry; but Y playing the same role. The scalar potential of this superfield realisation of the the gauge theory is given by It has the same form as (3.14) but now with the dual expressions and The properties of the scalar potential (3.20), including the description of the two scale breakings of N = 2 supersymmetry, have been explicitly studied in [34].

ADJ model and tadpole anomaly
In this section, we study a D3 brane realisation of ADJ theory and the interpretation of partial supersymmetric breaking in terms of 3-forms fluxes through 3-cycles in CY3. This brane realisation has been succinctly presented in the introduction section; here we use results on type IIB string on local CY3s to describe the underlying geometry and the nature of H RR 3 , H N S 3 fluxes behind N = 2 ADJ model. To reach this goal, we first examine the geometric property the linear combinations of abelian gauge superfields; then we study the geometric derivation of the N = 2 ADJ condition and its N = 1 deformation given by (2.35); and after we give the explicit relationship between ADJ condition and D3 tadpole cancellation anomaly in type IIB.

Linear combinations
By an inspection of the superspace density (3.1), one notices that L N =2 U (1) 2 depends on the following superfield linear combinations and These superfield combinations may a priori be extended to any number n of N = 2 gauge multiplets W N =2 a = (X a , V a ) as follows where the g a 's are gauge coupling constants associated with each abelian U a (1) gauge multiplet (X a , V a ). Similar relations can be written down for V ′ , X ′ and ∂F ∂X . However, because of ADJ constraint relation; the generalisation of the condition (3.5) to arbitrary U(1) n gauge symmetry is valid provided n ≥ 2 as in (2.12). The restriction to the particular n = 1 case leads to singular relation g 1 κ 1 = 0 requiring g = 0 for κ = 0. To overcome this difficulty; one may resolve the g 1 κ 1 = 0 singularity by deforming it like ν + g κ = 0; this leads to g = −κν; however remembering the property of eqs(2.34-2.35), one learns that the deformation by ν breaks explicitly N = 2 supersymmetry down to N = 1. To see the meaning of the linear combinations (4.1-4.4) as well as the interpretation of the deformation ν + n a=1 g a κ a = 0 (4.5) we need to go beyond 4d space time by thinking of: • the N = 2 supersymmetric U (1) n gauge model as a part of an effective theory following from type IIB string compactified on a local CY3; and • the constraint relation (4.5) as corresponding to the D3 tapole anomaly [36] 1 where the 3-form gauge field strengthsĤ RR 3 ,Ĥ N S 3 ; and the numbers T 3 and N D3 will be introduced later on.
To be explicit, we study in what follows the derivation of the linear combinations a ξ a V a and a g a V a from type IIB string compactification on a Calabi -Yau threefold Z 3 with Kahler 2-form J 2 and complex holomorphic 3-form Ω 3 . Then, we turn to the derivation of the linear combinations a e a X a and a 1 κa ∂F ∂X a concerning the chiral superfields.

Kahler sector
To derive the two linear combinations involving the gauge multiplet namely the V = a ξ a V a , depending on FI coupling constants ξ a , and the V ′ = a g a V a involving gauge coupling constants g a , it is interesting to start by describing the θ-expansions of these combinations. Focussing on the 4-vector υ a µ field components of the linear sum of gauge superfields V a which expands in θ-series as Obviously for the case χ a = ξ a , the contribution to ADJ model is given by the D-term θ 2θ 2 ( n a=1 ξ a D a ); and the interpretation of ξ a 's may be obtained by computing the field eqs of the auxiliary D a fields. However, we can reach the same result by looking for the derivation of this quantity from superstring compactification. i) FI coupling constants As a first step toward the ξ a V a 's, we use the 4d space time language of 1-form gauge field potentials V a 1 = υ a µ dx µ to rewrite the gauge component field linear combination ξ a υ a µ as follows n a=1 ξ a υ a µ dx µ = n a=1 ξ a V a 1 ≡ V 1 (4.9) So ξ a υ a µ can be also viewed in terms of a linear combination of the 1-form gauge field potentials V a 1 . The next step is to transform above (4.9) into an integral over full dimensions of the Calabi-Yau threefolds; this is achieved by thinking about the 1-form gauge field V a 1 in 4d space time as due to a 4-form gauge potentialĈ 4 of a D3 brane living in 10d space timeĈ ii) gauge coupling constants To derive the linear combination n a=1 g a υ a µ and the expression of the gauge coupling constants g a , we use the 4d Chern-Simons interaction L 4d CS between the gauge potentials υ a µ and the the antisymmetric field strength H νρσ , g a υ a µ H νρσ ε µνρσ (4.14) that we rewrite, by using wedge product V a 1 ∧ H 3 of 4d space time 1-and 3-forms like If we restrict to first contribution and comparing with (4.11), we end with the following expression for the gauge coupling constants that reads also as follows with α ′ the string constant and where Λ 0 is a cut off playing the role of running scale of the well known renormalisation group equation.
To conclude, the linear combinations of the N = 2 gauge multiplets used in ADJ model have the following geometric interpretation in type IIB string on local CY3 n a=1 ξ a V a

Chiral sector
The X a 's are chiral superfields with θ-components given by complex scalar fields X a | θ=0 = X a ; Weyl spinors D α X a | θ=0 = ψ a α and auxiliary fields 1 2 D α D α X a θ=0 = F a . The linear combination e a X a is a chiral superfield a≥1 e a X a = X + θ. In the embedding of ADJ model into 10d type IIB string compactification on CY3, the complex scalars X a and F a describe the expansion modes of the holomorphic 3-form Ω 3 over the 3-form harmonic basis (α a , β a ) of the local CY3 The electric FI e a and magnetic 1 κa coupling constants are obtained by equating the superpotential with the expression of W (X), build out of G 3 = H RR

Deriving ADJ constraint from type IIB string
In ADJ model, the condition (2.12) is intimately related with the real 4-form C 4 with antisymmetric gauge potential field C µνρσ as in (2.19) and directions filling the 4d space time dimensions This 4-form gauge field potential cannot have field strength F 5 = dC 4 in 4d space time; and thus it should be treated as an auxiliary field as done by ADJ theory. Notice that this is a constant field that may be ignored by setting it to zero; but this corresponds to a particular solution since in general it reads like and captures a constraint relation that we study below. Observe that ∆ scales as mass 2 ; and so the scaling mass dimension of C 4 is mass −2 .

Tadpole anomaly
To get more insight into the meaning of the N = 2 ADJ condition a ga κa = 0 and its N = 1 deformation ν + a ga κa = 0, one has to go beyond 4d space time where the gauge potential C 4 is no longer constant are the gauge field strengths of the antisymmetriĉ B N S µν andB RR µν gauge potentials of type IIB strings. The field equation of the 5-form field strengthF 5 of theĈ 4 gauge potential is given by with T 3 = 1 (2π) 3 α ′2 the D3-brane tension, 2κ 2 10 = (2π) 7 α ′4 ; and where ̺ loc 3 -form stands for the D3 charge density due to localized sources including D7-branes or O3 planes and also of mobile D3-branes [36]; see also [46,47]  leads to various terms that can be organised into three block terms where un-hatted fields refer to 4d space time fields. Notice that the two last terms are precisely the ones given by eqs (4.17), they contribute to Chern-Simons couplings in 4d space time.
The remaining term (4.39), involving the integral ofĤ RR 3 ∧Ĥ N S 3 through full CY3; is a topological term describing the flux Φ f lux of the 6-formĤ RR For compact CY3, this flux has no where to go and so has to vanish; but as we will see in a moment it may be compensated by D3-brane charges coming from local sources. So the contribution of this term to 4d space time lagrangian density reads as where we have used (4.32). This term appears linearly in the ADJ lagrangian density; so it captures a constraint equation requiring the flux Φ f lux to vanish as noticed above. This constraint can be then interpreted as nothing but the vanishing condition given by the integral of (4.35) on CY3 for the particular case ̺ loc 3 = 0. For a non zero ̺ loc 3 density of D3-brane charges as in eq(4.36), the tadpole vanishing condition reads as and the contribution to the 4d space time lagrangian density gets modified like The presence of the N D3 flux breaks explicitly N = 2 supersymmetry down to N = 1; N D3 flux plays the same role as the parameter ν in the deformed constraint eq(4.5).

Revisiting N = 2 ADJ condition and its deformation
Here, we give the general expression of the condition of tadpole cancellation in terms of the H RR fluxes. If assuming that the CY3 is compact, then we have From these relations, we learn where the quantised vectors are as (4.49) and where Ω AB is the usual 2n × 2n symplectic matrix Putting back into (4.44), the tadpole condition becomes The N = 2 ADJ constraint (2.12) may be recovered by requiring N D3 = 0; and choosing the vectors p A and q A like 1 κ a = 2πα ′2 q a , g a = 2πp a+n (4.53) the other p a and q a+n are set to zero.

Conclusion and comments
In this paper, we have studied a D3 brane realisation of partial breaking of N = 2 supersymmetry in ADJ model of ref [34]. This is a particular 4d N = 2 supersymmetric U (1)×U (1)  One of the basic constraint equations in this 4d effective N = 2 supersymmetric U (1) n gauge theory, formulated in N = 1 superspace with lagrangian density L = L gauge + L ST + L CS , is the one given by δL δY = 0; the superfield equation of the auxiliary Y whose full contribution comes from the L CS term namely Because of linear dependence of L CS into the superfield Y , the above superfield equation turns into the constraint n a=1 ga κa = 0. This condition relates the gauge coupling constants g a to magnetic FI couplings 1 κa ; and, upon computing energy of the ground state of the model, one also obtains links between partial supersymmetry breaking scales and the g a , κ a constant parameters as done in [34]. In our brane realisation, the ADJ model is represented by D3 branes wrapping 3-cycles in type IIB on local CY3 in presence of non trivial fluxes of the 3-forms gauge field strengths H RR = 0 which, by using the ndimensional symplectic homology basis of 3-cycles (A a , B a ) of the CY3, reads in general like Φ f lux = q A Ω AB p B with p A and q B as in eq(4. 46 -4.48).
By expanding the above relation as Φ f lux = q a p a+n − q a+n p a ; it follows that the condition n a=1 ga κa = 0 is indeed a particular solution of vanishing Φ f lux = 0 given the choice of fluxes as in eq(4.53). This flux choice corresponds to expanding gauge field strengths H 3 on the basis (α a , β a ) of 3-forms, dual to 3-cycles basis, like H N S 3 = q a α a and H RR 3 = p a+n β a . Geometrically, these values correspond to the local Calabi-Yau picture where the 3-cycles B a are taken in the non compact space approximation. In the case n = 1; describing the special N = 2 U (1) gauge model, the CY3 is the T * S 3 conifold with compact A given by the 3-sphere S 3 and non compact B ≃ R 3 ; and the above expression of the Φ f lux flux reduces to The relationship between ADJ constraint and the flux q A Ω AB p B shows that n a=1 ga κa = 0 is nothing but the vanishing condition of the D3 tadpole anomaly in absence of D7 branes or O3 planes; the particular n = 1 case is therefore anomalous. By taking into account D3 charge density due to localized sources D7-branes or O3 planes; and by using a result from the study of [36], the previous total flux conservation condition gets promoted to the following relatioñ with N D3 as in (4.36). In this picture, the tadpole anomaly can be lifted; but with the price of breaking explicitly N = 2 supersymmetry in the underlying effective field theory down to N = 1. In this situation, the ADJ constraint a ga κa = 0 becomes deformed like ν + a ga κa = 0 as shown by eq(4.45).
Acknowledgement 1 This work is supported by URAC 09/CNRST; Saidi thanks ICTP-Trieste, Italie; where a part of this work has been done.

Appendices
Here we give two appendices; in appendix § 6.1, we collect useful tools on type IIB on CY3; and in appendix § 6.2, we study properties shared by supersymmetric and gauge transformations of the N = 2 gauge superfield W N =2 and the N = 2 tensor multiplet T N =2 used in this paper.

Useful tools on type IIB on CY3
Effective N = 2 supersymmetric QFT models in 4d space-time can be embedded in string compactifications; they are constructed in various manners; in particular by compactifying type II strings on CY3s or heterotic string on K3 × T 2 . These compactifications to 4d N = 2 low energy theories are related by duality symmetries. By decoupling massive modes which are of order compactification scale, one can build the structure of the effective N = 2 supersymmetric theory with decoupled gravity. The massless states following from the CY3 compactification can be organised into N = 2 multiplets as follows: (1) vectors V N =2 ; each contains a complex scalar, a vector and two Weyl fermions.
(3) two other kinds of non standard N = 2 multiplets having an antisymmetric tensor B µν ; these are: (a) the tensor multiplet T N =2 having: 3 scalars, B µν and 2 Weyl fermions; and (b) the vector-tensor R N =2 containing: a scalar, a vector, B µν and 2 Weyl fermions.

Type IIB in 10d
The 10d type IIB supergravity multiplet has 128 +128 on shell degrees of freedom with bosonic sector as follows In addition to the metricĜ M N , we have a real axionĈ 0 and the dilationφ = ln g IIB with g IIB the string coupling. We also have p-forms namely the NS-NSB 2 and the RR p-form gauge field potentialsĈ p together with the corresponding gauge invariant field strengthsF This theory has an S-duality symmetry that allows to combine these fields into SL (2, Z) representations; in particular as where τ may be interpreted as the complex structure of 2-torus as in the embedding of type IIB in F-theory. To avoid confusion between p-form of same rank and also their descendent after compactification, we shall use the notationsB 2 =B N S 2 for NS 2-form gauge potential sourced by the elementary string F1; andĈ 2 =B RR 2 for the RR 2-form gauge potential sourced by the solitonic D1 string.
Type IIB on CY3 To descend to 4d space time, we have to factorise the 10d fields as products of parts; one depending on 4d space time and the other on the internal coordinates. This is achieved by decomposing the 2-and 4-forms on a harmonic basis of form of the local CY3 as followsB Here the set ω I , α a , β a ,ω I stand for a real harmonic basis of p-forms generating the cohomology of the CY3 obeying amongst others the following useful relations Notice that the 10d self duality condition ofF 5 = dĈ 4 implies that the 2-form A I 2 = 1 2! A I µν dx µ ∧ dx ν and the scalars ̺ I in eq(6.5) are related as dA I 2 = * d̺ I ; and so only one of them should be kept; for our concern we have kept A I 2 . The same feature holds for the 1-form gauge field potentials V a 1 and U 1a ; in other words the decomposition of C 4 capturing the right number of degrees of freedom is reduced tô Combining altogether, we learn from above decomposition that the 4d space time spectrum that we obtain, after compactification on CY3, the following multiplets where only bosonic fields are reported with V a 1 = V a µ dx µ standing for 1-form gauge potential in 4d space time. The fields of (6.9) are obtained by inverting eqs(6.5) by using properties of the harmonic basis of the homology cycles of the CY3; they are given by In this appendix, we shed light on those features behind formal similarities between Maxwell W N =2 and single tensor T N =2 superfields; this is done by studying their behaviour under the second supersymmetric charge in the fibration (2.1); as well as their link through gauge symmetry.

Realising N = 2 by two N = 1 gauge superfields
There are different ways to deal with off shell representations of N = 2 gauge and tensor multiplets; here we want to focuss on those aspects shared by their formulation in terms of the N = 2 chiral superfields W N =2 and T N =2 introduced in section 2. These superfields carry 8 B + 8 F degrees of freedom; they may be obtained by first considering two N = 1 superfields having 16 B + 16 F degrees of freedom; and then reducing this number down to 8 B + 8 F degrees by imposing appropriate constraints. The study of the reduction for both gauge W N =2 and tensor T N =2 multiplets is interesting in the sense it allows to get more information on : (i) the link between the second supersymmetry and gauge symmetry; and (ii) the feature behind similarities between W N =2 and T N =2 . Explicitly, we proceed as follows: First, we use N = 1 formalism, where the first supersymmetry is manifest; to study a simple realisation of W N =2 and T N =2 engineered from two N = 1 hermitian gauge superfields V 1 and V 2 ; but handled in different manners.
After that, we draw the line on how this construction extends to the N = 2 chiral superspace; this extension is also important for W N =2 and T N =2 because it constitutes the starting point for studying electric-magnetic duality in N = 2 superspace in presence of the Chern-Simons term given by the coupling Actually, this duality gives another facet on the relationship between W N =2 and T N =2 as shown by eq(2.28); it will not be developed here; but for details see [33] and refs therein.
N = 2 supersymmetry in N = 1 superspace In N = 1 superspace, the hermitian gauge superfields V 1 and V 2 describe two representations of N = 1 supersymmetry; and roughly speaking a N = 2 multiplet. Altogether, V 1 and V 2 carry 16 B + 16 F off shell degrees of freedom; half of them coming from V 1 and the other half from V 2 . With these two superfields, the second supersymmetry is generated by the transformations with supersymmetric parameter ǫ = δθ. These 16 B + 16 F degrees can be reduced down to 8 B + 8 F by imposing constraint eqs on V 1 and V 2 ; it happens that this can be done in two different manners; one leading to (X, W α ), the N = 1 superfields representation of W N =2 ; and the other to the (Φ, L) realisation of T N =2 ; or up to a gauge symmetry, to (Y , χ α , Φ) with L = D α χ α +Dαχα and Y as in (2.19).
In this realisation, the N = 1 hermitian superfields V 1 and V 2 are interpreted as gauge superfield potentials obeying the following U(1) gauge transformations They involve two N = 1 superfield gauge parameters; the hermitian Λ l =Λ l and the chiral Λ c solving the superspace conditions Each one of these superparameters reduces by 4 the number of degrees of freedom in the corresponding gauge superfield. The N = 2 Maxwell chiral superfield strength they can be combined into a N = 2 chiral superfield strength as in (2.4) namely which by substituting can be also expressed like Notice that in analogy with the Wess-Zumino gauge commonly used for V 2 , the vector superfield V 1 in (6.16) has also a Wess-Zumino like expansion leading, after substituting in X = 1 2D 2 V 1 , to the following θ-expansion with X standing for the complex scalar of the N = 2 gauge multiplet and ψ α1 for one of the two gauginos; the other gaugino ψ α2 comes from W α . However, the non propagating complex auxiliary field F X is realised here as F X = −d 1 − i∂ µ υ µ 1 with imaginary part Im F X given by the topological quantity can be realised like 1 κ ε µνρσ where 1 κ is precisely the deformation term appearing in the second line of (2.4).

From
First recall that in this paper, we have considered two kinds of realisations of the tensor multiplet T N =2 : (i) a short representation where T N =2 ≡ (L, Φ) having 8 B + 8 F degrees of freedom; and (ii) a long representation T N =2 ≡ (Y , χ α , Φ) involving 16 B +16 F degrees of freedom obeying the symmetry (6.23). The first one is obtained from the second by gauging away half of the degrees of freedom as in eq(6.11). In fact both short and long representation of T N =2 may be imagined as solutions of the following linear constraint where we have used the notation (U 1 , U 2 ) to avoid confusion with the hermitian multiplets (V 1 , V 2 ) used in eqs (6.16). Notice that eqs(6.20) may thought of as the complement of eqs(6.16) in the space parameterized by (V 1 , V 2 ). A first type of solution of these constraint relations is easily identified by remembering that the N = 1 linear multiplet L satisfies also the conditions D 2 L =D 2 L = 0 exactly as U 1 . Moreover because of the chirality propertiesDαΦ = D α Φ = 0 as well as the relations [D α , ∂ µ ] = Dα , ∂ µ = 0; it follows that (U 1 , U 2 ) are nothing but where we have dropped out the spurious superfield Y . The second type of solution of (6.20) is given by T N =2 ≡ (Y , χ α , Φ); it is convenient for the use of the N = 2 chiral superspace and reads, roughly speaking, in terms of the four N = 1 chiral superfields as follows with the gauge symmetry property where Ω and Υ are two N = 1 hermitian gauge superfield parameters. Notice that the implementation of the superfield Y is required by off shell closure of N = 2 supersymmetry as in eq (2.8). Notice also the three following features: (i) first in the gauge where Y ′ = 0, the chiral superfield Y is given by 1 2D 2 Υ; by comparing this expression with the first relation of (6.16), one learns that Y has same form as the chiral superfield X = 1 2D 2 V 1 . Obviously X and Y are different things; the first one carries propagating physical degrees of freedom; while the second is a pure auxiliary superfield with no physical field.
(ii) second alike for the relations (6.18-6.19) satisfied by the X superfield and allowing adjunction of a topological term to the lagrangian density implemented in superspace formulation by theθ 2 2κ deformation in the second line of (2.4), one has as well a quite similar feature for the superfield Y . Indeed, following [33], there is a gauge where Y reads as with C µνρσ a 4-form field with no propagating degree of freedom.
(iii) third if denoting byŵ then the gauge transformations (6.23) read as Putting this change into we find that it transforms like with N = 2 superfield gauge parameter having the same structure as the Maxwell superfield W N =2 . From this view, it follows that a single-tensor superfield T N =2 is a chiral superfield obeying the gauge symmetry (6.27) with superfield parameter as in (6.28).

Comment on gauge symmetry
Here we comment on the gauge symmetry of the field action term S (Y ) = d 4 xL (Y ) with lagrangian density L (Y ) ≡ L given by L = η d 2 θY + hc (6.30) and coupling constant a pure imaginary number η = iν. This field action S (Y ) is an extra term that has been used in this paper to induce a deformation of the ADJ constraint a ga κa = 0 into ν + a ga κa = 0; it lifts the singularity of the particular equation g 1 κ 1 = 0; but breaks N = 2 supersymmetry explicitly. Gauge invariance of S (Y ) can be studied either directly by using the superspace method taking into account that the superfield Y is a chiral multiplet; or more explicitly by working with component fields. In the first way, the gauge transformation of Y can be learnt from (6.23); it reads as Y ′ = Y + δ gauge Y with δ gauge Y = − 1 2D 2 Υ; where Υ is an arbitrary N = 1 real superfield. To shed light on this gauge transformation and its effect on the action S (Y ); we will use the component field language to first build the explicit θ-expansions of Υ andD 2 Υ; then turn back to study gauge symmetry of (6.30).
• component field analysis We begin by describing the θ-expansion of the superfield Y which can be expressed into two manners: (i) either by using the complex chiral superspace coordinate basis Z c = x − iθσθ, θ where Y is expanded as y + √ 2θΨ + θ 2 F ; or (ii) by working in the real superspace coordinate basis Z R = x µ , θ a ,θȧ where the superfield Y has also an explicitθ dependence as shown below 2 Y = y − iθσ µθ ∂ µ y + 1 4 θ 2θ 2 y + √ 2θΨ + i √ 2 θ 2 ∂ µ Ψσ µθ + θ 2 F Y (6.31) From this expansion, we learn L = iν F Y −F Y (6.32) In the gauge transformation δ gauge Y = − 1 2D 2 Υ, the real superfield parameter Υ has a θ-expansion involving as usual several component field parameters which, for later use can be expressed like where "fermionic" stands for those monomials with odd powers in Grassman variables involving fermionic fields. The components γ, d Υ are real scalar components; the complex scalars M,M are related by complex conjugation; and the real 4-vector w µ is parameterised like w µ = 1 3! ε µνρσ A [νρσ] with A [νρσ] a completely antisymmetric real rank 3-tensor. This dual parametrisation of w µ is dictated by an underlying arbitrariness in dealing with the superfield Υ; by shifting the real superparameter Υ by a hermitian superfield Γ like Υ ′ = Υ + Γ, the quantityD 2 Υ remains invariant provided Γ satisfying the constraintsD 2 Γ = D 2 Γ = 0 , Γ † = Γ (6.34) But these conditions are same as the ones defining linear multiplet L introduced in section 2 and which we recall here for comparisonD 2 L = D 2 L = 0 with L † = L; the real superfield Γ has therefore a similar θ-expansion as eq(2.16); and so it can be expressed like Γ = γ + θσ µθ ε µνρσ ∂ ν ω [ρσ] − 1 4 θ 2θ 2 γ + f ermionic (6.35) showing that the term ε µνρσ ∂ ν ω ρσ is nothing but a gauge transformation of w µ = 1 3! ε µνρσ A [νρσ] with gauge parameter given by the rank 2-antisymmetric ω ρσ = −ω σρ . Taking advantage of the arbitrariness (6.34), one can make convenient choices; in particular we can put the real superfield Υ into the following form; see also [33], With this choice; the initial gauge symmetry (6.23) gets now reduced to gauge symmetry of the 4-vector w µ = 1 3! ε µνρσ A [νρσ] namely w µ → w µ + ε µνρσ ∂ ν ω ρσ (6.37) • more on gauge invariance From the gauge fixed expression (6.36), we can make two useful computations namely the explicit expressions of 1 2D 2 Υ and the integral over Grassman variables iν d 2 θ Y + hc.
For the θ-expansion of 1 2D 2 Υ; we find 1 2D has an interpretation in term of rank 4-antisymmetric field strength F [µνρσ] of the 3form potential field A [νρσ] . Under the residual gauge (6.37), the term ∂ µ w µ is therefore manifestly gauge invariant; this property can be explicitly checked on the transformation ∂ µ w µ → ∂ µ w µ + ε µνρσ ∂ µ ∂ ν ω ρσ (6.40) where the extra term ε µνρσ ∂ µ ∂ ν ω ρσ vanishes identically due to a tensor calculus feature. Comparing (6.31) and (6.38), one remarks, that by using a Wess-Zumino-like gauge, Y can be expressed as Y gauged = iθ 2 ∂ µ w µ . Moreover, using this gauge fixed expression, the field action S = d 4 xL with the lagrangian density L as in eq(6.30) reads therefore like, and, because of (6.40), is manifestly gauge invariant δ gauge S = 0.
In the end of this comment, we would like to add that even if thinking of the scalar ∂ µ w µ = ∆ as the dual of a generic rank 4-tensor gauge potential like ∆ = 1 4! ε µνρσ A [µνρσ] where the completely antisymmetric A [µνρσ] obeys the gauge transformation δA [µνρσ] = ∂ [µ Λ νρσ] , the variation of the quantity 1 4! ε µνρσ A [µνρσ] behaves as a divergence term 1 3! ∂ µ (ε µνρσ Λ νρσ ); and, up on ignoring boundary effects, this variation does not contribute at the level of the action S = −2ν d 4 x 1 4! ε µνρσ A [µνρσ] . However, in this way of doing, the field strength F [µνρστ ] ≡ F (5) , associated to the potential field A [µνρσ] ≡ A (4) , would be a rank 5-antisymmetric tensor field which is invariant under the gauge transformation A (4) → A (4) + dΛ (3) ; but because of space time dimension constraint, the rank 5-tensor should be equal to zero, F (5) = dA (4) = 0; this means that the 4-form A (4) is a pure gauge potential without curvature which may be thought of as A (4) = dA (3) ; and then the gauge parameter Λ (3) as just a shift of the origin of A (3) .