Nonlinear Supersymmetry, Brane-bulk Interactions and Super-Higgs without Gravity

We derive the coupling of a hypermultiplet of N=2 global supersymmetry to the Dirac-Born-Infeld Maxwell theory with linear N=1 and a second nonlinear supersymmetry. At the level of global supersymmetry, this construction corresponds to the interaction with Maxwell brane fields of bulk hypermultiplets, such as the universal dilaton of type IIB strings compactified on a Calabi-Yau manifold. It displays in particular the active role of a four-form field. Constrained N=1 and N=2 superfields and the formulation of the hypermultiplet in its single-tensor version are used to derive the nonlinear realization, allowing a fully off-shell description. Exact results with explicit symmetries and supersymmetries are then obtained. The electric-magnetic dual version of the theory is also derived and the gauge structure of the interaction is exemplified with N=2 nonlinear QED of a charged hypermultiplet. Its Higgs phase describes a novel super-Higgs mechanism without gravity, where the goldstino is combined with half of the hypermultiplet into an N=1 massive vector multiplet.


Introduction
It is notorious that (linear) N = 2 supersymmetry, global or local, forbids a dependence on hypermultiplet scalars of gauge kinetic terms. For instance, in N = 2 supergravity, the scalar manifold is the product of a quaternion-Kähler (Einstein) manifold, for hypermultiplet scalars [1], and a Kähler manifold of a special type for vector multiplet scalars [2]. In global N = 2 supersymmetry, the quaternion-Kähler manifold of hypermultiplet scalars is replaced by a Ricci-flat hyperkähler space [3].
If however (at least) one of the supersymmetries is nonlinearly realized, these restrictions on the action are expected to change. For instance, string theory indicates that the Dirac-Born-Infeld (DBI) Lagrangian describing kinetic terms of brane gauge fields may interact with the dilaton and with its hypermultiplet partners. Moreover, if the dilaton supermultiplet is formulated with one or two antisymmetric tensors, more involved interactions dictated by the gauge symmetries of the theory are certainly allowed. An interesting problem is then to construct an interaction Lagrangian in which, when the second supersymmetry turns nonlinear, both the DBI Lagrangian and its necessary dilaton dependence are simultaneously generated. In other words, if we consider a theory with a broken, nonlinear supersymmetry realized in a goldstino mode, another unbroken linear supersymmetry and a DBI super-Maxwell system coupled to hypermultiplet fields, we certainly expect that the allowed Lagrangians are severely restricted. Analyzing these restrictions is the main motivation of this paper.
In this work, we construct an action invariant under N = 2 global supersymmetry, one of them being nonlinearly realized, involving the Maxwell goldstino multiplet of the nonlinear supersymmetry coupled to a single-tensor N = 2 multiplet [4,5,6], or equivalently to a hypermultiplet with one abelian (shift) isometry. In the absence of this multiplet, the action reduces to the standard super-Maxwell DBI theory, derived in the past from the same symmetry principle [7,8,9]. The coupling of the two multiplets is shown to arise from a N = 2 Chern-Simons (CS) term which, under electric-magnetic duality, amounts to shifting the gauge field strength by the antisymmetric tensor. Moreover, under Poincaré duality of the antisymmetric tensor to a pseudoscalar, the CS coupling becomes a Stückelberg gauging of the pseudoscalar axionic symmetry.
An important property of the single-tensor multiplet is that it admits an off-shell (superspace) formulation, unlike the generic hypermultiplet that can be formulated offshell only at the cost of introducing infinite number of auxiliary fields in the context of harmonic superspace [10]. Thus, our formalism using the single-tensor N = 2 multiplet allows to construct off-shell supersymmetric Lagrangians. By an appropriate change of variables from the N = 2 single-tensor multiplet, one finds an action that couples the goldstino vector multiplet (of the linear supersymmetry) to an N = 2 charged hypermultiplet, describing the low-energy limit of a theory with partial spontaneous supersymmetry breaking from N = 2 to N = 1 [11,12].
The vacuum of this theory exhibits an interesting novel feature: the goldstino is 'absorbed' into a massive vector multiplet of N = 1 linear supersymmetry, leaving a massless N = 1 chiral multiplet associated to flat directions of the scalar potential. The goldstino assembles with one of the two Weyl fermions in the single-tensor multiplet to form a massive Dirac spinor. At one particular point along the flat directions, the vector multiplet becomes massless and the U(1) is restored. This phenomenon is known from D-brane dynamics, where the U(1) world-volume field becomes generically massive due to the CS coupling. A crucial role for the invariance of the action under nonlinear supersymmetry is played by a non-dynamical four-form gauge potential, known again from D-brane dynamics. Hence, a globally supersymmetric combination of Higgs and super-Higgs mechanisms, in the presence of a four-form field, eliminates any massless goldstino fermion related to partial supersymmetry breaking. This interesting new mechanism can be studied in the context of nonlinear N = 2 quantum electrodynamics with one charged hypermultiplet, which after a holomorphic field redefinition and a duality transformation, is equivalent to our setup.
In type IIB superstrings compactified to four dimensions with eight residual supercharges, the dilaton scalar (associated to the string coupling) belongs to a universal hypermultiplet, together with the (Neveu-Schwarz) NS-NS antisymmetric tensor and the (Ramond) R-R scalar and two-form. Its natural basis is therefore a double-tensor supermultiplet, 1 having three perturbative isometries associated to the two axionic shifts of the antisymmetric tensors and an extra shift of the R-R scalar. These isometries form a Heisenberg algebra, which at the string tree-level is enhanced to the quaternion-Kähler and Kähler space SU(2, 1)/SU(2)×U (1). At the level of global N = 2, imposing the Heisenberg algebra of isometries determines a unique hyperkähler manifold of dimension four, depending on a single parameter, in close analogy with the local case of a quaternionic space where the corresponding parameter is associated to the one-loop correction [15]. This manifold is not trivially flat and should describe the rigid limit of the universal hypermultiplet.
The plan of the paper is as follows. In Section 2, we review the construction of the N = 2 simple-tensor and Maxwell supermultiplets in terms of N = 1 superfields and we describe their interaction in a Chern-Simons term, as was earlier partly done in Ref. [9]. In addition we explain how the intricate web of gauge variations in the Stückelberg coupling of the Maxwell and single-tensor supermultiplets leads to the interpretation of one (non-propagating) component of the single-tensor as a four-form field. In Section 3, we reformulate the supermultiplets in chiral N = 2 superspace and then demonstrate how this construction can be used to describe electric-magnetic duality in a manifestly N = 2 covariant way. In Section 4, we first review the construction of the Dirac-Born-Infeld theory from constrained N = 2 superfields describing the goldstino of one non-linear supersymmetry and then extend it to construct its coupling to a single-tensor supermultiplet, engineered by a CS term. We also perform an electromagnetic duality to determine the 'magnetic' version of the theory. WIth the dilaton hypermultiplet of type IIB superstrings in mind, we impose the Heisenberg algebra of perturbative isometries to our theory. In Section 5, we derive the coupling of the Maxwell goldstino multiplet to a charged hypermultiplet and make a detailed analysis of the vacuum structure of N = 2 super-QED with partial supersymmetry breaking. We conclude in Section 6 and two appendices present our conventions and the resolution of a quadratic constraint applied on a N = 2 chiral superfield.

The linear N = Maxwell-dilaton system
Our first objective is to describe, in the context of linear N = 2 supersymmetry, the coupling of the single-tensor multiplet to N = 2 super-Maxwell theory. Since these two supermultiplets admit off-shell realizations, they can be described in superspace without reference to a particular Lagrangian. Gauge transformations of the Maxwell multiplet use a single-tensor multiplet, we then begin with the latter.

The single-tensor multiplet
In global N = 1 supersymmetry, a real antisymmetric tensor field b µν is described by a chiral, spinorial superfield χ α with 8 B + 8 F fields [16] 2 : Chirality of χ α implies linearity of L: DDL = DDL = 0. The linear superfield L is invariant under the supersymmetric gauge transformation 3 of χ α : this is the supersymmetric extension of the invariance of h µνρ under δb µν = 2 ∂ [µ Λ ν] . Considering bosons only, the gauge transformation (2.3) eliminates three of the six components of b µν and the scalar field C ′ . Accordingly, L only depends on the invariant curl h µνρ and on the invariant real scalar C. The linear L describes then 4 B + 4 F fields. Using either χ α or L, we will find two descriptions of the single-tensor multiplet of global N = 2 supersymmetry [4,5,6].
In the gauge-invariant description using L, the N = 2 multiplet is completed with a chiral superfield Φ (8 B +8 F fields in total). The second supersymmetry transformations (with parameter η α ) are where D α and Dα are the usual N = 1 supersymmetry derivatives verifying It is easily verified that the N = 2 supersymmetry algebra closes on L and Φ.
We may try to replace L by χ α with second supersymmetry transformation δ * χ α = − i √ 2 Φ η α , as suggested when comparing Eqs. (2.2) and (2.4). However, with superfields χ α and Φ only, the N = 2 algebra only closes up to a gauge transformation (2.3). This fact, and the unusual number 12 B + 12 F of fields, indicate that (χ α , Φ) is a gaugefixed version of the off-shell N = 2 multiplet. We actually need another chiral N = 1 superfield Y to close the supersymmetry algebra. The second supersymmetry variations are (2.5) One easily verifies that the Y -dependent terms in δ * χ α induce a gauge transformation (2.3). Hence, the linear L and its variation δ * L do not feel Y .
with ∆ ′ real, is then postulated. We will see below that this variation is actually dictated by N = 2 supersymmetry. There exists then a gauge in which Y = 0 but in this gauge the supersymmetry algebra closes on χ α only up to a transformation (2.3). This is analogous to the Wess-Zumino gauge of N = 1 supersymmetry, but in our case, this particular gauge respects N = 1 supersymmetry and gauge symmetry (2.3).
Two remarks should be made at this point. Firstly, the superfield Y will play an important role in the construction of the Dirac-Born-Infeld interaction with non-linear N = 2 supersymmetry. As we will see later on 4 , it includes a four-index antisymmetric tensor field in its highest component. Secondly, a constant (θ-independent) background value Φ breaks the second supersymmetry only, δ * χ α = − i √ 2 Φ η α +. . . It is a natural source of partial supersymmetry breaking in the single-tensor multiplet. Notice that the condition δ * Φ = 0 is equivalent to Dα(Dχ − Dχ) = 0.
An invariant kinetic action for the gauge-invariant single-tensor multiplet involves an arbitrary function solution of the three-dimensional Laplace equation (for the variables L, Φ and Φ) [5]: In the dual hypermultiplet formulation the Laplace equation is replaced by a Monge-Ampère equation. We will often insist on theories with axionic shift symmetry δΦ = ic (c real), dual to a double-tensor theory. In this case, H is a function of L and Φ + Φ so that the general solution of Laplace equation is with an arbitrary analytic function H(V).

The Maxwell multiplet, Fayet-Iliopoulos terms
Take two real vector superfields V 1 and V 2 . Variations provide a representation of N = 2 supersymmetry with 16 B + 16 F fields. We may reduce the supermultiplet by imposing on V 1 and V 2 constraints consistent with the second supersymmetry variations: for instance, the single-tensor multiplet is obtained by requiring V 1 = L and V 2 = Φ + Φ. Another option is to impose a gauge invariance: we may impose that the theory is invariant under 5 where Λ ℓ and Λ c form a single-tensor multiplet,   [17].
The N = 2 gauge-invariant Lagrangian depends on the derivatives of a holomorphic prepotential F (X): (2.14) In the construction of the Maxwell-multiplet in terms of X and W α , one expects a triplet of Fayet-Iliopoulos terms, 5 For clarity, we use the following convention for field variations: δ * refers to the second (N = 2) supersymmetry variations of the superfields and component fields; δ U(1) indicates the Maxwell gauge variations; δ appears for gauge variations of superfields or field components related (by supersymmetry) to δb µν = 2 ∂ [µ Λ ν] . 6 Remember that with this (standard) convention, Wα is minus the complex conjugate of W α . 7 There is a phase choice in the definition of X: a phase rotation of X can be absorbed in a phase choice of η.
with real parameters ξ 1 , ξ 2 and a. They may generate background values of the auxiliary components f X and d 2 of X and V 2 which in general break both supersymmetries: In terms of V 1 and V 2 however, the relation X = 1 2 DDV 1 implies that Im f X is the curl of a three-index antisymmetric tensor (see Subsection 2.4) and that its expectation value is turned into an integration constant of the tensor field equation [18,19]. As a consequence, and the Fayet-Iliopoulos Lagrangian becomes with two real parameters only.
The Maxwell multiplet with superfields (X, W α ) and the single-tensor multiplet (Y, χ α , Φ) have a simple interpretation in terms of chiral superfields on N = 2 superspace. We will use this formalism to construct their interacting Lagrangians in Section 3.

The Chern-Simons interaction
With a Maxwell field F µν = 2 ∂ [µ A ν] (in W α ) and an antisymmetric tensor b µν (in χ α or L), one may expect the presence of a b ∧ F interaction This equality suggests that its N = 2 supersymmetric extension also exists in two forms: either as an integral over chiral superspace of an expression depending on χ α , W α , X, Φ and Y , or as a real expression using L, Φ + Φ, V 1 and V 2 .
In the 'real' formulation, the N = 2 Chern-Simons term is 8 with a real coupling constant g. It is invariant (up to a derivative) under the gauge transformations (2.10) of V 1 and V 2 with L and Φ left inert. Notice that the introduction of Fayet-Iliopoulos terms for V 1 and V 2 corresponds respectively to the shifts Φ + Φ → Φ + Φ − ξ 1 /g and L → L − ξ 2 /g in the Chern-Simons term.
The 'chiral' version uses the spinorial prepotential χ α instead of L. Turning expression (2.18) into a chiral integral and using X = 1 2 DD V 1 leads to which differs from L CS by a derivative. The chiral version of the Chern-Simons term L CS,χ transforms as a derivative under the gauge variation (2.3) of χ α . Its invariance under constant shift symmetry of Im Φ follows from X = 1 2 DD V 1 . It does not depend on Y .
The consistent Lagrangian for the Maxwell-single-tensor system with Chern-Simons interaction is then (2.20) The first two contributions include the kinetic terms and self-interactions of the multiplets while the third describes how they interact. Each of the three terms is separately N = 2 supersymmetric.
Using a N = 1 duality, a linear multiplet can be transformed into a chiral superfield with constant shift symmetry and the opposite transformation of course exists. Hence, performing both transformations, a single-tensor multiplet Lagrangian (L, Φ) with constant shift symmetry of the chiral Φ has a 'double-dual' second version. Suppose that we start with a Lagrangian where Maxwell gauge symmetry acts as a Stückelberg gauging of the single-tensor multiplet: 9 (2.21) The shift symmetry of Im Φ has been gauged and L is invariant under gauge transformations (2.10) combined with 22) and under N = 2 supersymmetry if H verifies Laplace equation (2.7). If we perform a double dualization (L, Φ + Φ) → (Φ +Φ,L), we obtain the dual theorỹ We have then found two classes of couplings of Maxwell theory to the single-tensor multiplet. Firstly, using the supersymmetric extension of the b ∧ F coupling, as in Eqs. (2.20). Secondly, using a Stückelberg gauging (2.21) of the single-tensor kinetic terms. The first version only is directly appropriate to perform an electric-magnetic duality transformation. However, since the second version can always be turned into the first one by a single-tensor -single-tensor duality, electric-magnetic duality of the second version requires this preliminary step: both theories have the same 'magnetic' dual.

The significance of V 1 , X and Y
In the description of the N = 2 Maxwell multiplet in terms of two N = 1 real superfields, V 2 describes as usual the gauge potential A µ , a gaugino λ α and a real auxiliary field d 2 (in Wess-Zumino gauge). We wish to clarify the significance and the field content of the superfields V 1 and X = 1 2 DDV 1 , as well as the related content of the chiral superfield Y used in the description in terms of the spinorial potential χ α of the single-tensor multiplet (Y, χ α , Φ).
The vector superfield V 1 has the N = 2 Maxwell gauge variation δ U (1) V 1 = Λ ℓ , with a real linear parameter superfield Λ ℓ . In analogy with the Wess-Zumino gauge commonly applied to V 2 , there exists then a gauge where This gauge leaves a residual invariance acting on the vector field v 1µ only: This indicates that the vector v µ 1 is actually a three-index antisymmetric tensor, with Maxwell gauge invariance (2.28) By construction, X = 1 2 DDV 1 is gauge invariant. In chiral variables, Hence, while Re f X = d 1 , is the gauge-invariant curl of A µνρ . It follows that the field content (in Wess-Zumino gauge) of V 1 is the second gaugino ψ X , the complex scalar of the Maxwell multiplet x, a real auxiliary field d 1 and the three-form field A µνρ , which corresponds to a single, non-propagating component field. The gauge-invariant chiral X includes the four-form curvature F µνρσ .
At the Lagrangian level, the implication of relations (2.30) is as follows. Suppose that we compare two theories with the same Lagrangian L(u) but either with u = φ, a real scalar, or with u = ∂ µ V µ , as in Eq. (2.30). Since L(φ) does not depend on ∂ µ φ, the scalar φ is auxiliary. The field equations for both theories are The second case allows a supplementary integration constant k related to the possible addition of a 'topological' term proportional to ∂ µ V µ to the Lagrangian [18,19]: In the first case, the same integration constant appears if one considers the following modified theory and field equation: Returning to our super-Maxwell case, the relation is φ = Im f X and the modification of the Lagrangian is then This is the third Fayet-Iliopoulos term, which becomes a 'hidden parameter' [18] when using V 1 instead of X.
Consider finally the single-tensor multiplet (Y, χ α , Φ) and the supersymmetric extension of the antisymmetric-tensor gauge symmetry, as given in Eqs. (2.3) and (2.6): Using expansion (2.29), there is a gauge in which Y reduces simply to and one should identify Im f Y as a four-index antisymmetric tensor field, The antisymmetric tensor C µνρσ describes a single field component which can be gauged away using Λ νρσ . Applying this extended Wess-Zumino gauge to the N = 2 multiplet (Y, χ α , Φ), the fields described by these N = 1 superfields are as given in the following table.
The propagating bosonic fields b µν , C and Φ (four bosonic degrees of freedom) have kinetic terms defined by Lagrangian L ST , Eq. (2.7).

Chiral N = 2 superspace
Many results of the previous section can be reformulated in terms of chiral superfields on N = 2 superspace. We now turn to a discussion of this framework, including an explicitly N = 2 covariant formulation of electric-magnetic duality.

Chiral N = 2 superfields
A chiral superfield on N = 2 superspace can be written as a function of y µ , θ,θ: with y µ = x µ − iθσ µ θ − iθσ µθ and Dα y µ = Dα y µ = 0. Its second supersymmetry variations are with supercharge differential operatorsQ α andQα which we do not need to explicitly write. It includes four N = 1 chiral superfields and 16 B + 16 F component fields and we may use the expansions whereθ and D α are the Grassmann coordinates and the super-derivatives associated with the second supersymmetry. The second supersymmetry variations (3.2) are easily obtained by analogy with the N = 1 chiral supermultiplet: which is obtained by imposing Φ Z = 0 in expansion (3.3), leads to the Maxwell supermultiplet (2.13) [20]. The Bianchi identity D α W α = DαWα is required by δ * Φ Z = 0. The N = 2 Maxwell Lagrangian (2.14) rewrites then as an integral over chiral N = 2 superspace, Considering the unconstrained chiral superfield (3.3) with 16 B + 16 F fields, the reduction to the 8 B + 8 F components of the single-tensor multiplet is done by imposing gauge invariance (2.3) and (2.6). In terms of N = 2 chiral superfields, this gauge symmetry is simply where W is a Maxwell N = 2 superfield parameter (3.5). In terms of N = 1 superfields, this is as in Eqs. (2.3) and (2.6). Hence, a single-tensor superfield Y is a chiral superfield Z with the second expansion (3.3) and with gauge symmetry (3.8).
The chiral version of the Chern-Simons interaction (2.19) can be easily written on N = 2 superspace. Using Y with gauge invariance (3.8) and W to respectively describe the single-tensor and the Maxwell multiplets. Then It is gauge-invariant since for any pair of Maxwell superfields Notice that the lowest component superfield Y of Y does not contribute to the field equations derived from L CS,χ : it only contributes to this Lagrangian with a derivative.
Finally, a second method to obtain an interactive Lagrangian for the Maxwellsingle-tensor system is then obvious. Firstly, a generic N = 2 chiral superfield Z can always be written as Z = W + 2gY.
provides a N = 2 invariant Lagrangian describing 16 B + 16 F (off-shell) interacting fields. There exists a gauge in which W = 0, in which case theory (3.14) describes a massive chiral N = 2 superfield.
Theory (3.14) is actually related to the Chern-Simons Lagrangian (2.20) by electricmagnetic duality, as will be shown below.

Electric-magnetic duality
The description in chiral N = 2 superspace of the Maxwell multiplet allows to derive a N = 2 covariant version of electric-magnetic duality. The Maxwell Lagrangian (2.14) supplemented by the Chern-Simons coupling (2.19) can be written adding Eqs. (3.6) and (3.10). Replace then W by an unconstrained chiral superfield Z (with N = 1 superfieldsẐ,ω α andΦ) and introduce a new Maxwell multiplet W (with N = 1 superfields X and W α ). Using Consider now the Lagrangian Invariance under the gauge transformation of the single-tensor superfield, Eq. (3.8), requires a compensating gauge variation of W, as in Eq. (3.13). Eliminating W leads back to theory (3.15) withẐ = W. This can be seen in two ways. Firstly, the condition i d 2 θ d 2θ WẐ + c.c. = derivative leads toẐ = W, a N = 2 Maxwell superfield, up to a background value. Secondly, using Eqs. (3.16), we see that V 2 imposes the Bianchi identity onω while V 1 cancelsΦ up to an imaginary constant. 10 We will come back to the (important) role of a nonzero background value in the next section. For the moment we disregard it.
On the other hand, we may prefer to eliminateẐ, using its field equation which corresponds to a Legendre transformation exchanging variablesẐ and V. Defining we have The dual (Legendre-transformed) theory is then (3.21) or, expressed in N = 1 superspace, 11 We then conclude that the presence of the Chern-Simons term in the electric theory induces a Stückelberg gauging in the dual magnetic theory.
As explained in Ref. [21], the situation changes when Fayet-Iliopoulos terms (3.7) are present in the electric theory. In the magnetic theory coupled to the single-tensor multiplet, with Lagrangian (3.22), the gauging δ W = 2g W forbids Fayet-Iliopoulos terms for the magnetic Maxwell superfields V 1 and V 2 . Spontaneous supersymmetry breaking by Fayet-Iliopoulos terms in the electric theory finds then a different origin in the magnetic dual.
For our needs, we only consider the Fayet-Iliopoulos term induced by V 1 , i.e. we add to L electric , Eq. (3.15). In turn, this amounts to add to theory (3.17). But, in contrast to expression (3.23), this modification is not invariant under the second supersymmetry: according to the first Eq. (3.4), its δ * variation is not a derivative. 12 To restore N = 2 supersymmetry, we must deform the δ * variation The free, canonically-normalized theory corresponds to F (W) = 1 2 W 2 and F (V) = 1 2 V 2 . 12 It would be a derivative if ω α would be replaced by the Maxwell superfield W α , as in Eq. (3.23). the second term being the usual, undeformed, variations (2.13) and (2.5). Hence, the magnetic theory has a goldstino fermion and linear N = 2 supersymmetry partially breaks to N = 1, as a consequence of the electric Fayet-Iliopoulos term. Concretely, the magnetic theory is now (3.25) One easily checks that N = 2 supersymmetry holds, using the deformed variations (3.24).

Nonlinear N = 2 supersymmetry and the DBI action
In the previous sections, we have developed various aspects of the coupling of a Maxwell multiplet to a single-tensor multiplet in linear N = 2 supersymmetry. With these tools, we can now address our main subject: show how a Dirac-Born-Infeld Lagrangian (DBI) coupled to the single-tensor multiplet arises from non-linearization of the second supersymmetry.
It has been observed that the DBI Lagrangian with nonlinear second supersymmetry can be derived by solving a constraint invariant under N = 2 supersymmetry imposed on the super-Maxwell theory [7,8]. We start with a summary of this result, following mostly Roček and Tseytlin [8], and we then generalize the method to incorporate the fields of the single-tensor multiplet.

The N = 2 super-Maxwell DBI theory
The constraint imposed on the N = 2 Maxwell chiral superfield W is [8] 13 It imposes a relation between the super-Maxwell Lagrangian superfield W 2 and the Fayet-Iliopoulos 'superfield'θθW, Eq. (3.23). The real scale parameter κ has dimension (energy) −2 . In terms of N = 1 superfields, the constraint is equivalent to The third equality leads to which, since W α W β W γ = 0, implies the first two conditions. Solving the third constraint amounts to express X as a function of W W [7] 14 . The DBI theory is then obtained using as Lagrangian the Fayet-Iliopoulos term (3.23) properly normalized: The constraints (4.1) and (4.2) are not invariant under the second linear supersymmetry, with variations δ * . However, one easily verifies that the three constraints (4.2) are invariant under the deformed, nonlinear variation with δ * X unchanged. The deformation preserves the N = 2 supersymmetry algebra. It indicates that the gaugino spinor in W α = −iλ α + . . . transforms inhomogeneously, . ., like a goldstino for the breaking of the second supersymmetry. In other words, at the level of the N = 2 chiral superfield W, The deformed second supersymmetry variations δ * def ormed act on W as the usual variations δ * act on the shifted superfield W − 1 2κθθ . In fact, this superfield transforms like a chiral N = 2 superfield (3.3) with Z = X, ω α = iW α verifying the Bianchi identity and with Φ Z = −i/κ. The latter background value of Φ Z may be viewed as the source of the partial breaking of linear supersymmetry.
Hence, the scale parameter κ introduced in the nonlinear constraint (4.1) appears as the scale parameter of the DBI Lagrangian and also as the order parameter of partial supersymmetry breaking. The Fayet-Iliopoulos term (4.4) has in principle an arbitrary coefficient −ξ 1 /4, as in Eq. (2.17). We have chosen ξ 1 = −κ −1 to canonically normalize gauge kinetic terms.
The DBI Lagrangian is invariant under electric-magnetic duality. 15 In our N = 2 case, the invariance is easily established in the language of N = 2 superspace. We first include the constraint as a field equation of the Lagrangian: The field equation of the N = 2 superfield Λ enforces (4.1). We then introduce two unconstrained N = 2 chiral superfields U and Υ and the modified Lagrangian Since the Lagrange multiplier Υ imposes U = W − 1 2κθθ , the equivalence with (4.6) is manifest. But we may also eliminate W which only appears linearly in the last version of the theory. The result is where W is a Maxwell N = 2 superfield dual to W and ζ an arbitrary real constant. As in Subsection 3.2, N = 2 supersymmetry of the theory with a Fayet-Iliopoulos term requires a nonlinear deformation of the δ * variation of W: W − i 2 1 κ − iζ θθ should be a 'good' N = 2 chiral superfield. Replacing Υ in the Lagrangian and taking ζ = 0 leads to Finally, eliminating U gives the magnetic dual One easily verifies that the resulting theory has the same expression as the initial 'electric' theory (4.4). The Lagrange multiplier Λ −1 imposes constraint (4.1) to −i W, which reduces to Eq. (4.3) applied to −i X. The Lagrangian is then given by the Fayet-Iliopoulos term for this superfield.

Coupling the DBI theory to a single-tensor multiplet: a super-Higgs mechanism without gravity
The N = 2 super-Maxwell DBI theory is given by a Fayet-Iliopoulos term for a Maxwell superfield submitted to the quadratic constraint (4.1), which also provides the source of partial supersymmetry breaking. The second supersymmetry is deformed by the constraint: it is W − 1 2κθθ which transforms as a regular N = 2 chiral superfield. Instead of expression (3.10), we are thus led to consider the following Chern-Simons interaction with the single-tensor multiplet: we see that the Fayet-Iliopoulos term is equivalent to a constant real shift of Φ which, according to variations (2.5), partially breaks supersymmetry. We will choose to expand Φ around Φ = 0 and keep ξ 1 = 0.
Again, the constraint (4.1) imposed by the Lagrange multiplier Λ can be solved to express X as a function of W W : X = X(W W ). The result is [7] where A and B are defined in Appendix B. The DBI Lagrangian coupled to the singletensor multiplet reads then The bosonic Lagrangian depends on a single auxiliary field 16 , d 2 in W α or V 2 : 12) The real scalar field C is the lowest component of the linear superfield L. Contrary to Φ , its background value is allowed by N = 2 supersymmetry. However, a non-zero C would induce a non-zero d 2 which would spontaneously break the residual N = 1 linear supersymmetry. This is visible in the bosonic action which, after elimination of (4.14) First of all, as expected, the theory includes a DBI Lagrangian for the Maxwell field strength F µν , with scale ∼ κ. With the Chern-Simons coupling to the single-tensor multiplet, the DBI term acquires a field-dependent coefficient, It also includes a F ∧ F term which respects the axionic shift symmetry of Im Φ, a b ∧ F coupling induced by (linear) N = 2 supersymmetry and a 'topological' C 4 term induced by the nonlinear deformation. These terms are strongly reminiscent of those found when coupling a D-brane Lagrangian to IIB supergravity. The contribution of the four-form can be eliminated by a gauge choice of the single-tensor symmetry (2.34).
We have however insisted on keeping off-shell (deformed) N = 2 supersymmetry, hence the presence of this term.
The theory also includes a semi-positive scalar potential 17 16 Since X(W W )| θ=0 is a function of fermion bilinears, the auxiliary f Φ does not contribute to the bosonic Lagrangian and χ α does not include any auxiliary field. 17 We only consider 2g Re Φ − ξ 1 > 0, in order to have well-defined positive gauge kinetic terms.
which vanishes only if C is zero. 18 The scalar potential determines then C = 0 but leaves Re Φ arbitrary. Since the vacuum line C = 0 is compatible with linear N = 1 and deformed second supersymmetry. While Φ is clearly massless, C has a mass term The same mass is acquired by the U(1) gauge field coupled to the antisymmetric tensor b µν , and by the goldstino (the U(1) gaugino in W α ) that forms a Dirac spinor with the fermion of the linear multiplet χ α . In other words, the Chern-Simons coupling χW pairs the Maxwell goldstino with the linear multiplet to form a massive vector, while the chiral multiplet Φ remains massless with no superpotential.
At C = Re Φ = 0, gauge kinetic terms are canonically normalized if ξ 1 = −κ −1 . The Maxwell DBI theory (4.4) is of course recovered when the Chern-Simons interaction decouples with g = 0. Notice finally that the kinetic terms L ST of the single-tensor multiplet are given by Eq. (2.7), as with linear N = 2 supersymmetry. Since the nonlinear deformation of the second supersymmetry does not affect δ * L or δ * Φ even if Re Φ = 0, the function H remains completely arbitrary.
The phenomenon described above provides a first instance of a super-Higgs mechanism without gravity: the nonlinear goldstino multiplet is 'absorbed' by the linear multiplet to form a massive vector N = 1 superfield. One may wonder how this can happen without gravity; normally one expects that the goldstino can be absorbed only by the gravitino in local supersymmetry. The reason of this novel mechanism is that the goldstino sits in the same multiplet of the linear supersymmetry as a gauge field which has a Chern-Simons interaction with the tensor multiplet. This will become clearer in Section 5, where we will show by a change of variables that this coupling is equivalent to an ordinary gauge interaction with a charged hypermultiplet, providing non derivative gauge couplings to the goldstino. Actually, this particular super-Higgs mechanism is an explicit realization of a phenomenon known in string theory where the U(1) field of the D-brane world-volume becomes in general massive due to a Chern-Simons interaction with the R-R antisymmetric tensor of a bulk hypermultiplet. 19 We have chosen a description in terms of the single-tensor multiplet because it admits an off-shell formulation well adapted to our problem. Our DBI Lagrangian (4.9), supplemented with kinetic terms L ST , admits however several duality transformations. Firstly, since it only depends on W, we may perform an electric-magnetic duality transformation, as described in Subsection 4.4. Then, for any choice of L ST , we can transform the linear N = 1 superfield L into a chiral Φ ′ . The resulting theory is a hypermultiplet formulation with superfields (Φ, Φ ′ ) and N = 2 supersymmetry realized only on-shell. As already explained in Subsection 2.3, the b∧F interaction is replaced by a Stückelberg gauging of the axionic shift symmetry of the new chiral Φ ′ : the Kähler potential of the hypermultiplet formulation is a function of Φ ′ + Φ ′ − gV 2 . Explicit formulae are given in the next subsection and in Section 5 we will use this mechanism in the case of nonlinear N = 2 QED. Finally, if kinetic terms L ST also respect the shift symmetry of Im Φ, the chiral Φ can be turned into a second linear superfield L ′ , leading to two formulations which are also briefly described below.

Hypermultiplet, double-tensor and single-tensor dual formulations
As already mentioned, using the single-tensor multiplet is justified by the existence of an off-shell N = 2 formulation. The hypermultiplet formulation, with two N = 1 chiral superfields, is however more familiar and the first purpose of this subsection is to translate our results into this formalism. In the DBI theory (4.11), the linear superfield L only appears in These contributions are not invariant under δ * variations: the nonlinear deformation acts on W α and on V 2 . Nevertheless, the linear superfield can be transformed into a new chiral superfield Φ ′ . The resulting 'hypermultiplet formulation' has Lagrangian The Kähler potential is given by the Legendre transformation where U is the solution of In the single-tensor formulation, N = 2 supersymmetry implies that H solves Laplace equation. As a result of the Legendre transformation, the determinant of K is constant and the metric is hyperkähler [5]. It should be noted that the Legendre transformation defines the new auxiliary scalar f Φ ′ of Φ ′ according to Hence, the hypermultiplet formulation has the same number of independent auxiliary fields as the single-tensor version: d 2 and f Φ .
The second supersymmetry variation δ * of Φ ′ is also defined by transformation (4.19): in the hypermultiplet formulation, N = 2 is realized on-shell only, using the Lagrangian function. The nonlinear deformation of variations δ * acts on V 2 . Since The κ-dependent term in the δ * variation of the Kähler potential term in L DBI, hyper. is then the same as the κ-dependent part in g δ * d 2 θ χ α W α + c.c, which is compensated by the variation of the four-form field. This can again be shown using Eqs. (4.18) and (4.19). This hypermultiplet formulation will be used in Section 5, on the example of nonlinear DBI QED with a charged hypermultiplet.
For completeness, let us briefly mention two further formulations of the same DBI theory, using either a double-tensor, or a dual single-tensor N = 2 multiplet. These possibilities appear if Lagrangian (4.11) has a second shift symmetry of Im Φ. This is the case if the single-tensor kinetic Lagrangian has this isometry: We may then transform Φ into a linear superfield L ′ using a N = 1 duality transformation. Keeping L and turning Φ into L ′ leads to a double-tensor formulation with superfields (L, L ′ ). The Lagrangian has the form (4.21) The function G is the Legendre transform of H with respect to its second variable Φ+Φ and the real superfield V 1 (W W ) is defined by the equation It includes the DBI gauge kinetic term in its d 1 component and the Lagrangian depends on the new tensor b ′ µν through the combination 3 ∂ [µ b ′ νρ] −g ω µνρ , where ω µνρ = 3 A [µ F νρ] is the Maxwell Chern-Simons form.
Finally, turning Φ and L into L ′ and Φ ′ , leads to another single-tensor theory with a Stückelberg gauging of both Φ ′ and L ′ , as in theory (2.21). In this case, the Lagrangian is (4.23) While in the double-tensor theory (4.21) the second nonlinear supersymmetry only holds on-shell, it is valid off-shell in theory (4.23). The function H verifies Laplace equation, as required by N = 2 linear supersymmetry. 20 Using the supersymmetric Legendre transformation, one can show that the nonlinear deformation of δ * V 2 , which affects δ * H, is again balanced by the variation of the four-form superfield Y .

The magnetic dual
To perform electric-magnetic duality on theory (4.9), we first replace it with (4.24) Both U and Υ are unconstrained chiral N = 2 superfields. The Lagrange multiplier Υ imposes U = W − 1 2κθθ , which leads again to theory (4.9). The first two terms, which have gauge and N = 2 invariance properties related to the Maxwell character of W are left unchanged. The term quadratic in W has been turned into a linear one using the Lagrange multiplier. Hence, the Maxwell superfield W, which contributes to Lagrangian (4.24) by (4.25) can as well be eliminated: Υ should be such that this contribution is a derivative. In terms of N = 1 chiral superfields, W has components X and W α and since there exists two real superfields V 1 and V 2 such that X = 1 2 DD V 1 and W α = − 1 4 DDD α V 2 , we actually need to eliminate V 1 and V 2 with result In this expression, W is a Maxwell N = 2 superfield, the 'magnetic dual' of the eliminated W. There is a new arbitrary real deformation parameter ζ, allowed by the field equation of V 2 . Notice however that ξ 1 + iζ can be eliminated by a constant complex shift of Φ. Invariance of Υ under the single-tensor gauge variation (3.8) implies that δ W = 2g W = −2gδY and Z ≡ W + 2gY (4.27) is then a gauge-invariant chiral superfield. As already mentioned, any unconstrained chiral N = 2 superfield can be decomposed in this way and our theory may as well be considered as a description of the chiral superfields Z and Y with Lagrangian (4.28) Invariance under the second supersymmetry implies that Z + i 2 (ξ 1 + iζ)θθ transforms as a standard N = 2 chiral superfield and then Eliminating U leads finally to which is the electric-magnetic dual of theory (4.9). 21 The Lagrange multiplier superfield Λ −1 implies now the constraint with Z = X + 2gY , ω α = i W α + 2gχ α and Φ Z = 2gΦ, this constraint corresponds to In this case, and in contrast to the electric case, the constraint leading to the DBI theory is due to the scale Φ Z = 2g Φ : we will actually choose ζ = 0, absorb ξ 1 into Φ Z and consider the constraint Z 2 = 0 with a non-zero background value Φ Z breaking the second supersymmetry. Our magnetic theory is then with constraints the DBI scale arising from Φ Z = φ Z + Φ Z . As in the Maxwell case, the third equation, which also reads implies Zω α = Z 2 = 0 and allows to express Z as a function of ωω and Φ, Z = Z(ωω, Φ), using Φ Z = 2gΦ − ξ 1 . The magnetic theory (4.32) is then simply It is the electric-magnetic dual of expression (4.11). At this point, it is important to recall that ω and Φ are actually N = 1 superfields components of Z = W + 2gY, i.e.
The kinetic terms for the single-tensor multiplet (L, Φ), L = Dχ − Dχ, are included in L ST while Z(ωω, Φ) includes the DBI kinetic terms for the Maxwell N = 1 superfield W α . As in the electric case, the magnetic theory has a contribution proportional to the four-form field included in Y .
The third constraint (4.33) is certainly invariant under the variations (3.4), using Zω α = 0. But with a non-zero background value Φ = φ+ Φ , the spinor ω α transforms nonlinearly, like a goldstino: 22 The solution of the constraint (4.34) is given in Appendix B. The bosonic Lagrangian included in the magnetic theory (4.35) is It depends on a single auxiliary field, the Maxwell real scalar d 2 , with field equation (4.39) Eliminatingd 2 and using Φ Z = 2gΦ − ξ 1 to reintroduce the superfield Φ of the singletensor multiplet and the 'original' Fayet-Iliopoulos term ξ 1 , we finally obtain the magnetic, bosonic Lagrangian

(4.43)
This is to be compared with the electric theory (4.14): (4.44) Hence, the duality from the electric to the magnetic theory corresponds to the transformations which can be also derived from electric-magnetic duality applied on the bosonic DBI theory only.

Double-tensor formulation and connection with the string fields
As mentioned in the introduction, in IIB superstrings compactified to four dimensions with eight residual supercharges, the dilaton belongs to a double-tensor supermultiplet. This representation of N = 2 supersymmetry includes two Majorana spinors, two antisymmetric tensors B µν (NS-NS) and C µν (R-R) with gauge symmetries (4.46) and two (real) scalar fields, the NS-NS dilaton and the R-R scalar, for a total of 4 B + 4 F physical states. In principle, both antisymmetric tensors can be dualized to pseudoscalar fields with axionic shift symmetry, in a version of the effective field theory where the dilaton belongs to a hypermultiplet with four scalars in a quaternion-Kähler 23 manifold possessing three perturbative shift isometries, since the R-R scalar has its own shift symmetry. It is easy to see that only two shift isometries, related to the two antisymmetric tensors, commute, while all three together form the Heisenberg algebra. Indeed, in the double-tensor basis, the R-R field strength is modified [25] due to its anomalous Bianchi identity to 3 Thus, a shift of the R-R scalar C (0) by a constant λ is accompanied by an appropriate transformation of C µν to leave its modified field-strength invariant: It follows that δ gauge , δ ′ gauge and δ H verify the Heisenberg algebra, with a single nonvanishing commutator [δ gauge , δ H ] = δ ′ gauge .
(4.48) 23 For supergravity. The limit of global supersymmetry is a hyperkähler manifold, which is Ricci-flat.
To establish the connection of the general formalism described in the previous subsections with string theory, we would like to identify the double-tensor multiplet with the universal dilaton hypermultiplet and study its coupling to the Maxwell goldstino multiplet of a single D-brane, in the rigid (globally-supersymmetric) limit. To this end, we transform the N = 2 double-tensor into a single-tensor representation by dualizing one of its two N = 1 linear multiplet components L ′ , containing the R-R fields C µν and C (0) , into a chiral basis Φ + Φ. In this basis, the two R-R isometries correspond to constant complex shifts of the N = 1 superfield Φ. Imposing this symmetry to the kinetic function of Eqs. (2.7)-(2.8), one obtains (up to total derivatives, after superspace integration): where α and β are constants. Note that the second term proportional to β can be obtained from the first by shifting L + β/α. For α = 0 however, it corresponds to the free case of quadratic kinetic terms for all fields of the single-tensor multiplet. The coupling to the Maxwell goldstino multiplet is easily obtained using Eqs. (4.12), (4.22) and (2.18). Up to total derivatives, the action is: (4.50) In general, the four-form field is not inert under the variation δ H of Eq. (4.47) [26]. In our single-tensor formalism, δ H L = 0 and δ H Φ = c where c is complex when combined with the axionic shift δ ′ gauge of ImΦ dual to C µν of Eq. (4.46); in addition With this variation, the Lagrangian, including the Chern-Simons interaction, is invariant under the Heisenberg symmetry.
We can now dualize back Φ + Φ to a second linear multiplet L ′ by first replacing it with a real superfield U: where the constant m corresponds to a rescaling of L ′ . Solving for U, delivers the double-tensor Lagrangian (4.54) where as before V 1 = V 1 (W W ) and X = X(W W ) = 1 2 DD V 1 (W W ). It is invariant under variation (4.51) of the four-form superfield combined with δ H L ′ = 2c(αL+β)/m.
After elimination of the Maxwell auxiliary field (choosing m = √ 2)  We expect that this action describes the globally-supersymmetric limit of the effective four-dimensional action of a D-brane coupled to the universal dilaton hypermultiplet of the perturbative type II string. As mentioned previously, its general form in the local case depends also on two constant parameters, upon imposing the perturbative Heisenberg isometries, that correspond to the tree and one-loop contributions [15]. It is tempting to identify these two parameters with α and β of our action. Moreover, by identifying the two antisymmetric tensors b µν and b ′ µν with the respective NS-NS B µν and R-R C µν and the combination C ′ /(αC + β) with the R-R scalar C (0) , as the Heisenberg transformations indicate, one finds that the two actions match up to normalization factors depending on the NS-NS dilaton that should correspond to the scalar C. Finding the precise identifications, which certainly depend on the way one should take the rigid limit that decouples gravity, is an interesting question beyond our present analysis restricted to global supersymmetry.

Nonlinear N = 2 QED
We will now show that the effective theory presented above describing a super-Higgs phenomenon of partial (global) supersymmetry breaking can be identified with the Higgs phase of nonlinear N = 2 QED, up to an appropriate choice of the single-tensor multiplet kinetic terms. We will then analyze its vacuum structure in the generally allowed parameter space.
In linear N = 2 quantum electrodynamics (QED), the Lagrangian couples a hypermultiplet with two chiral superfields (Q 1 , Q 2 ) to the vector multiplet (V 1 , V 2 ) or (X, W α ). The U(1) gauge transformations of the hypermultiplet are linear, and Q 1 and Q 2 have opposite U(1) charges: 57) where L M ax. includes (canonical) gauge kinetic terms and ∆L contains three parameters: The hypermultiplet mass term with coefficient m can be eliminated by a shift of X and ξ 1,2 are the two Fayet-Iliopoulos coefficients. Since There are six real auxiliary fields, f Q 1 , f Q 2 , d 1 and d 2 but only four are actually independent: Since the metric is canonical, det K ij = 1 and trivially hyperkähler. If ξ 1 = ξ 2 = 0, the gauge symmetry is not broken and the hypermultiplet mass m + i X / √ 2 is arbitrary. Any nonzero ξ 1 or ξ 2 induces U(1) symmetry breaking with all fields having the same mass. In any case, N = 2 supersymmetry remains unbroken at the global minimum.
In order to first bring the theory to a form allowing dualization to our single-tensor formulation, we use the holomorphic field redefinition 25 59) 24 We use the same notation for a chiral superfield Φ, Q 1 , Q 2 , . . . and for its lowest complex scalar component field. 25 This field redefinition has constant Jacobian.
with a 2 = 1/ √ 2. The QED Lagrangian becomes Since the Kähler potential is now a function of Φ ′ + Φ ′ , with a Stückelberg gauging of the axionic shift of Φ ′ , the chiral Φ ′ can be dualized to a linear L using a N = 1 Legendre transformation. The result is (5.61) The dual single-tensor QED theory has off-shell N = 2 invariance (the Laplace equation (2.7) is verified) and the two multiplets are now coupled by a N = 2 Chern-Simons interaction (2.19). Notice that the free quadratic kinetic terms of the charged hypermultiplet lead to a highly non-trivial kinetic function in the single-tensor representation. Moreover, there are only four auxiliary fields, f Φ , d 1 and d 2 . The Legendre transformation defines the scalar field C in L as (5.62) and Eqs. (5.59) relate then C and Φ with Q 1 and Q 2 : According to Eq. (4.11), the nonlinear DBI version of N = 2 QED is obtained by replacing in Lagrangian (5.61) X by X(W W ), which includes DBI gauge kinetic terms, by omitting L M ax. which is removed by the third constraint (4.2) and by adding the four-form term i 2κ d 2 θ Y + c.c.: (5.64) Notice that two additional terms appear compared to the action studied in Section 4: an Fayet-Iliopoulos term proportional to ξ 2 and a term linear in Φ which is also invariant under the second (nonlinear) supersymmetry (2.4); they generate, together with ξ 1 the general parameter space of nonlinear QED coupled to a charged hypermultiplet. Without loss of generality, we choose m to be real, while the choice ξ 1 = −1/κ would canonically normalize gauge kinetic terms for a background where Φ vanishes. We may return to chiral superfields (Φ, Φ ′ ) or (Q 1 , Q 2 ) to write the DBI theory as 26 Since X(W W )| θ=0 only depends on fermion fields, the auxiliary fields f 1 and f 2 only contribute to the bosonic Lagrangian by a hypermultiplet mass term to be added to the scalar potential obtained from Eq. (4.16) with the substitutions (since we have chosen g = 1). The complete potential is then 27 66) The analysis is then very simple. The first line vanishes only for The first condition is the usual D-term equation d 2 = 0 for the Maxwell superfield. The second condition is necessary to have a well-defined DBI gauge kinetic term at the minimum. Hence, if m = 0, conditions (5.67), which can always be solved, define the vacuum of the theory. Choosing Q 1 = v and Q 2 = v 2 + ξ 2 , with v real (and arbitrary), we find a massive vector boson which, along with a real scalar and the two Majorana fermions makes a massive N = 1 vector multiplet of mass v 2 + ξ 2 /2. Hence the potentially massless gaugino λ, with its goldstino-like second supersymmetry variation δ * λ α = 26 See Eq. (4.17). 27 The auxiliary d 2 is given in Eq. (4.13). − 1 √ 2κ η α + . . ., has been absorbed in the massive U(1) gauge boson multiplet. This is possible only because the second supersymmetry transformation of the four-form field compensates the gaugino nonlinear variation. The fermion v 2 + ξ 2 ψ Q 1 + v ψ Q 2 is massless and corresponds to the fermion of the chiral superfield Φ in the single-tensor formalism, in agreement with our analysis in Section 4.2 [see below Eq. (4. 16)]. With two real scalars, it belongs to a massless N = 1 chiral multiplet.
If m = 0, a supersymmetric vacuum has Q 1 = Q 2 = 0. It only exists if ξ 2 = 0 and ξ 1 = 0. The second condition is again to have DBI gauge kinetic terms on this vacuum. In this case, the U(1) gauge symmetry is not broken, the goldstino vector multiplet remains massless and the hypermultiplet has mass m. If m = 0, a nonzero Fayet-Iliopoulos coefficient ξ 2 breaks then N = 1 linear supersymmetry. Note that the single-tensor formalism is appropriate for the description of the Higgs phase of nonlinear QED in a manifest N = 1 superfield basis (with respect to the linear supersymmetry), while the charged hypermultiplet representation is obviously convenient for describing the Coulomb phase.
One can finally expand the action (5.65) in powers of κ in order to find the lowest dimensional operators that couple the goldstino multiplet of partial supersymmetry breaking to the N = 2 hypermultiplet. Besides the dimension-four operators corresponding to the gauge factors e ±V 2 , one obtains a dimension-six superpotential interaction ∼ κQ 1 Q 2 W 2 coming from the solution of the nonlinear constraint X = κW 2 + O(κ 3 ); it amounts to a field-dependent correction to the U(1) gauge coupling.

Conclusions
In this work we have studied the interaction of the Maxwell goldstino multiplet of N = 2 nonlinear supersymmetry to a hypermultiplet with at least one isometry. The starting point was to describe the hypermultiplet in terms of a single-tensor multiplet, which admits an off-shell N = 2 formulation, and introduce a coupling using a Chern-Simons interaction. This system describes the coupling of a D-brane to bulk fields of N = 2 compactifications of type II strings, in the rigid limit of decoupled gravity. Using N = 1 and N = 2 dualities, we have also obtained equivalent formulations of the nonlinear Maxwell theory coupled to a matter N = 2 supermultiplet. This web of theories is summarized in the Figure  Specializing to the case of the universal dilaton hypermultiplet, we determined the action completely in the rigid limit, using the Heisenberg symmetry of perturbative string theory, up to an arbitrary constant parameter which, in the quaternion-Kähler case of N = 2 supergravity, corresponds to the string one-loop correction [15]. An interesting open question is to realize this decoupling limit directly from the supergravitycoupled system.
We have shown how the above system applies to the Higgs phase of N = 2 nonlinear QED coupled to a charged hypermultiplet. Allowing a hypermultiplet mass scale and a Fayet-Iliopoulos term in the two-dimensional parameter space, the vacuum structure includes phases with broken and unbroken linear N = 1 supersymmetry and/or U(1) gauge symmetry.
It is interesting to note that in the Higgs phase the goldstino vector multiplet combines with the hypermultiplet to form a N = 1 massive vector and a massless chiral superfield. This novel super-Higgs mechanism is possible without gravity because the hypermultiplet is charged under the U(1) partner of the goldstino. In the N = 1 case, the goldstino multiplet can be gauged only by gravity and is absorbed by the gravitino that acquires a mass.
In principle, it is straightforward to introduce additional hypermultiplets. Obviously only one of them will 'absorb' the goldstino providing mass to the U(1) vector. This action describes also the low-energy limit of spontaneous partial supersymmetry breaking N = 2 → N = 1, when the breaking is 'small' in the matter (hypermultiplet) sector. This is analogous, in the case of a single N = 1 nonlinear supersymmetry, to the effective action of the goldstino coupled to N = 1 multiplets at energies higher than their soft breaking masses. It is then known that this action is obtained by simply identifying the constrained goldstino multiplet with the so-called spurion [23]. One may try to develop the analogy in the N = 2 nonlinear case and derive the structure of possible 'soft' terms associated to the partial N = 2 → N = 1 breaking. As a step further, one could try to integrate out the N = 2 superpartners and obtain the effective action at much lower energies, describing the interactions of the N = 2 goldstino multiplet to N = 1 superfields. This would be directly relevant for constructing brane effective theories involving non-abelian gauge groups and charged matter. It could also be used for studying a supersymmetric extension of the Standard Model in the presence of a second supersymmetry nonlinearly realized due to its breaking at a high scale.

Acknowledgements
We wish to thank L. Alvarez-Gaumé and S. Ferrara for useful conversations. This work has been supported by the Swiss National Science Foundation. The work of J.-P. D. was in part supported by CNRS and the Ecole Polytechnique, Palaiseau. The work of I.A. and P.T. was supported in part by the European Commission under the ERC Advanced Grant 226371 and the contract PITN-GA-2009-237920. I.A. was also supported by the CNRS grant GRC APIC PICS 3747. P. T. is also supported by the "Propondis" Foundation.
Further identities (with identical conventions) can be found in an appendix of Ref. [9].