Observations on the Partial Breaking of $N=2$ Rigid Supersymmetry

We study the partial breaking of $N=2$ rigid supersymmetry for a generic rigid special geometry of $n$ abelian vector multiplets in the presence of Fayet-Iliopoulos terms induced by the Hyper-K\"ahler momentum map. By exhibiting the symplectic structure of the problem we give invariant conditions for the breaking to occur, which rely on a quartic invariant of the Fayet-Iliopoulos charges as well as on a modification of the $N=2$ rigid symmetry algebra by a vector central charge.


Introduction
It is well known that partial breaking of rigid and local supersymmetry can occur [1,2], provided one evades [3][4][5][6][7][8][9][10][11] some no-go theorems [12,13,2] which are satisfied by a certain class of theories. In global supersymmetry, Hughes and Polchinski first pointed out the possibility to realize partial breaking of global supersymmetry [5] and, in four dimensional gauge theories, this was realized for a model of a self-interacting N = 2 vector multiplet, in the presence of N = 2 electric and magnetic Fayet-Iliopoulos terms [8].
This model is closely connected to the Goldstone action of partially broken N = 2 supersymmetry [14] by integrating out the (N = 1) chiral-multiplets components of the N = 2 vector multiplet [15], thus reproducing the supersymmetric Born-Infeld action [16,17]. Multi-field versions which generalize the supersymmetric Born-Infeld theory to an arbitrary number of vector multiplets were then obtained, preserving N = 1 supersymmetry [15,18], or preserving a second non-linearly realized supersymmetry [19,20].
It is the aim of the present note to further elucidate some general conditions for partial supersymmetry breaking to occur which are independent on the particular alignment of the unbroken supersymmetry with respect to the original two supersymmetries, and are also independent of the particular representative of the Fayet-Iliopoulos charge vector which, in our problem, is a triplet * Corresponding author. (1.1) We shall give, in Section 2, the general Ward identities that the N = 2 scalar potential satisfies when Fayet-Iliopoulos symplecticcharge triplets P xM are turned on, explicitly showing that they are modified by a constant traceless matrix C A Furthermore, in Section 2.1, we shall derive in a symplectic covariant manner the modifications of the supersymmetry algebra which, in the framework of N = 2 tensor calculus, was derived in [10].

The rigid Ward identity
It is a well known fact that the Supergravity Ward identity relating the scalar potential V to the shifts of the fermions in the presence of a gauging is a pure trace identity in the R-symmetry SU(N) indices, namely: positive for the spin 1/2-fields and negative for the gravitino. This is true also when the scalar potential is due to the presence of a Fayet-Iliopoulos (FI) term [1,2,4,21].
In the rigid supersymmetric theories with N > 1 the previous statement is violated, since on the right-hand-side of Eq. (2.1) a traceless term can appear related to the presence of electric and magnetic FI terms [5,8,10,15]. In the case of an N = 2 rigid theory, this can be seen either by direct computation of the fermionic shifts of the gauginos or by performing a suitable flat limit of the N = 2 Supergravity parent theory with gravitino and hyperinos constant non-zero shifts.
In the rigid case of a supersymmetric vector-multiplet theory, Eq. (2.1) allows for a traceless constant term C A B in the scalar potential Ward identity, namely (2.1) is modified as follows where λ i A and λ i A ≡ g ik λk A denote the chiral and antichiral projections of the gauginos, respectively. According to the arguments given in [10], the additive constant matrix C B A can be interpreted as a central extension in the supersymmetry algebra, which only affects the commutator of two supersymmetry transformations of the gauge field.
In the N = 2 case the traceless matrix C A B has the following expression In this case, the fermion-shift in the supersymmetry transformation of the chiral gaugino fields can be written, using a symplectic formalism, as where g ik is the rigid Special Kähler metric and the symplectic section U M i is the derivative with respect to the scalar fields z i of the fundamental symplectic section V M (z) of the rigid Special Kaehler Geometry [22,23]: Introducing a triplet of triholomorphic superpotentials W x Eq. (2.6) takes the form Let us now use special coordinates, that is X I = z i ; in this frame we can write A short computation then gives This formula actually coincides with Eq. (23) of [8] and it shows that a non-zero magnetic charge m Ix produces a constant imaginary part of the auxiliary field Y Ix , a necessary condition for partial breaking of supersymmetry. We now use the Special Geometry identity [24]:  (2.14) I ≡ (−Im(F I J )) and R ≡ (Re (F I J )). If we flatten the σ -model coor- (2.15) Finally we may compute the bilinear product in the gaugino shifts (2.16) which coincides with Eq. (2.2), proving our statement. In conclusion, the N = 2 scalar potential of the rigid theory is while, by the identification (2.4), the C A B term can be rewritten as In the general case in which both the F and D-terms are present, we define the following SO(3)-vector: (2.20) Upon diagonalization, the above matrix reads where |ξ | 2 denotes the following quartic invariant I 4 in the FI terms: • If I 4 = 0, N = 2 is spontaneously broken to either N = 1 or N = 0, depending on whether V N=2 > √ I 4 or V N=2 = √ I 4 , respectively. In the latter case one of the eigenvalues vanishes and the corresponding direction in superspace defines the surviving N = 1 supersymmetry. The N = 1 potential is the square of the fermion shifts along the direction of the residual supersymmetry. In the diagonal basis (2.21), this direction is the second one so that: (2.23) In this case, the infra-red dynamics is captured by a Born-Infeld Lagrangian [19]. In the absence of D-terms, ξ 1 = ξ 2 = 0 and |ξ | 2 = |ξ 3 |. Taking for instance m I = (m I , 0, 0) and e I = (e 1 I , e 2 I , 0), we find ξ 1 = ξ 2 = 0 , ξ 3 = 2 m I e 2 I . (2.24) In this case Eq. (2.20) becomes: where we have defined P M = 1 √ 2 C MN P 1 N + iP 2 N . If ξ 3 > 0, the residual N = 1 supersymmetry is along the second direction ( 2 ), while, if ξ 3 < 0, along the first. In the former case the lower diagonal entry of (2.25), along the direction of the preserved supersymmetry, defines the N = 1 potential: (2.26) This is consistent with (2.15) which, in the absence of a D-term, can be written in the form (2.27) Indeed at the N = 1 vacuum where we have used (2.13). From the above expression we conclude that: (2.32) that is the commutator of two supersymmetry transformations on a vector field, in the presence of a magnetic Fayet-Iliopoulos term, amounts to a shift, whose parameter corresponds to a vector central charge in the SU(2) adjoint representation [10,25,26].