D=4, gauged supergravity in the presence of tensor multiplets

Abstract

Using superspace techniques we construct the general theory describing D = 4, N = 2 supergravity coupled to an arbitrary number of vector and scalar-tensor multiplets. The scalar manifold of the theory is the direct product of a special Kähler and a reduction of a Quaternionic-Kähler manifold. We perform the electric gauging of a subgroup of the isometries of such manifold as well as "magnetic" deformations of the theory discussing the consistency conditions arising in this process. The resulting scalar potential is the sum of a symplectic invariant part (which in some instances can be recast into the standard form of the gauged N = 2 theory) and of a non-invariant part, both giving new deformations. We also show the relation of such theories to flux compactifications of type II string theories.As a next step we perform the gauging of the theory. After dualization, not all of the isometries of the original manifold remain isometries of the final scalar manifold. Moreover, some of these act non-trivially on the tensor fields and therefore cannot become local symmetries without leading to non-linear couplings for the tensor fields. We then discuss which isometries can be made local and therefore "gauged". Always using the superspace formalism we compute the fermion shifts which restore the supersymmetry of the theory and give rise to a potential satisfying the supersymmetry Ward identities.The appearance of tensor fields allows to redefine the gauge field strengths with a shift proportional to the tensor fields F Λ → F Λ + m IΛ B I without breaking supersymmetry, provided we redefine appropriately the fermion transformation laws. Here m IΛ are real constants which can be thought as mass parameters for the tensors. This kind of extension 2 of the theory was first obtained in [13] for six-dimensional supergravity, further extended in [14] and shown in Calabi-Yau compactification of Type II theories in [7].Indeed the gauging we perform after dualization of some of the hypermultiplets scalars is a standard electric gauging, but the appearance of the mass parameters m IΛ in the definition of the new gauge field strengths implies the existence of extended solutions. The shifts of the supersymmetry transformations indeed acquire some extra terms depending on such parameters so that the gravitino's and hyperino's shifts are symplectic invariants. This latter can be interpreted also as a "magnetic" gauging, though its definition is not related to the appearance of magnetic gauge fields. These would lead to the construction of [4] whose consistency is problematic, as explained in [6,7].The scalar potential of the theory follows as usual from the square of the fermionic shifts by using a known Ward identity of N-extended gauged supergravities [15]. Being the square of symplectic invariant quantities, but for a term coming from the gaugino shift when non-Abelian isometries of the Special Kähler manifold are gauged, the potential shows symplectic invariance for Abelian gaugings where such gaugino contribution does not appear. T...

Observations on BI from N = 2 $$ \mathcal{N}=2 $$ supergravity and the general Ward identity

“…It was indeed shown in [15] that partial supersymmetry breaking can be achieved in any symplectic frame (and in particular in one in which the prepotential does exist) using an embedding tensor [16][17][18] with both electric and magnetic components. Consistency of such gaugings requires the introduction of antisymmetric tensor fields dual to scalars [10][11][12][13][14].…”

“…The general form of the gauge-invariant bosonic lagrangian, using the embedding tensor formulation, was given in [12] while specific abelian gaugings were constructed in [10,11]. 1 In this paper, to set the stage for the construction of the gauged model generalizing that of [6], we make a step forward in this direction and give, in a self-contained form, all the relevant identities related to the most general gauging of special Kähler and quaternionic Kähler isometries in a generic N = 2 model.…”

“…More specifically, if we denote by A Λ µ = (A 0 µ , A I µ ), the n + 1 supergravity vector fields, in the new symplectic frame, A 0 µ is consistently identified with the graviphoton while A I µ with the vector fields of the resulting rigid theory. Moreover, denoting by Θ Λ m the components of the embedding tensor which define the gauge generators X Λ in terms of the isometry generators t m of the scalar manifold and by Θ Λ m their magnetic counterparts, consistency of the supergravity gauging requires the following locality condition to be satisfied [10][11][12]:…”

“…non-vanishing magnetic components Θ Λm of the embedding tensor, are not well defined since the corresponding field strengths F Λ µν are not covariantly closed [10][11][12][13][14] …”

“…We will see in section 4 that a natural interpretation of η i can be given in supergravity, as charges associated with the gauging procedure, by performing a different choice of symplectic frame. Postponing this issue to next section, let us consider, for the time being, a gauging of two translational isometries in the hypermultiplet sector involving both electric and magnetic charges [10,11]. This gauging can be described in terms of a (redundant) symplectic vector of gauge generators X M ≡ (X Λ , X Λ ), expressed as linear combinations of the isometry generators t m , m = 1, .…”