We considerN = 2 supergravity coupled tonV vector multiplets [136, 137, 138].
For a modern review, see [139, 140, 141].
Since the gravity multiplet contains a vector field, thegraviphoton, this theory admitsnV + 1 vector fieldsAΛµ where Λ = 0, . . . , nV. ThenV vector multiplets
contain as well nV complex scalar fields zα, α = 1, . . . , nV, parametrizing a
Special K¨ahler manifold. The kinetic terms of the scalar and vector fields are : Lvector = 1 4Im(N)ΛΣF Λ|µνFΣ µν− 1 8e −1Re(N) ΛΣµνρσFµνΛFρσΣ −gαβ¯∂µzα∂µz¯ ¯ β, = 12ImNΛΣFµν+ΛF +Σ|µν−g αβ¯∂µzα∂µz¯ ¯ β, (6.3.1)
6.3. Vector multiplets and special geometry
where Fµν±Λis the self-dual combination1:
Fµν±Λ= 12FµνΛ ∓1
2ieεµνρσF
Λ|ρσ, FΛ
µν=∂µAΛν −∂νAΛµ. (6.3.2)
The couplings of vector multiplets to N = 2 supergravity are characterized by
special geometry. The latter relies heavily on the existence of duality transfor- mations for vector fields in supersymmetric theories. Duality transformations generalize the electric-magnetic duality of Maxwell’s equations without sources. Defining the magnetic duals of the field strengths F+Λ
µν as
Gµν+Λ= 2i ∂L
∂Fµν+Λ
=NΛΣF+Σ|µν, (6.3.3)
the equations of motion and Bianchi identities read
∂µImFµν+Λ = 0, Bianchi Identity,
∂µImGµν+Σ= 0, Equations of motion. (6.3.4)
Then, the duality transformations are linear transformationsS ∈G`(2nV+ 2,R)
of the field strengths and their magnetic duals
˜ F+ ˜ G+ =S F+ G+ , (6.3.5)
that preserve the Bianchi identities and equations of motion for vector fields. The relation between the field strengths and their magnetic duals (6.3.3) determines the transformation rule of the coupling matrixNΛΣ under (6.3.5):
N =⇒(C+DN)(A+BN)−1, (6.3.6) where A, B, C, D are (nV + 1)×(nV + 1) real matrices that characterize the
duality transformation S= A B C D ∈G`(2nV + 2,R). (6.3.7)
If equations (6.3.3) are derived from a Lagrangian L, the matrix NΛΣ should
be symmetric. Asking the symmetry of NΛΣ to be preserved under a fractional
transformation (6.3.6) restricts the duality transformations to be given by sym- plectic matrices S :
S ∈Sp(2nV + 2,R). (6.3.8)
As the coupling matrix NΛΣ transforms under duality transformations, the
scalar fieldszαshould also transform in a specific way. The action of the duality
1In our convention the Levi-Civita tensor satisfiesε 0123= 1.
N = 2 supergravity and effective Fayet-Iliopoulos terms.
transformations on the scalar fields is much more transparent once we introduce asymplectic section which depends on the scalar fields of vector multiplets and transforms linearly under symplectic rotations. Special geometry can be com- pletely defined in terms of this symplectic section.
The symplectic section is given by:
v= ZΛ FΛ , Λ = 0,· · ·nV (6.3.9)
and is endowed with a symplectic scalar product hv|¯vi=−vT 0nV − nV nV 0nV ¯ v. (6.3.10)
Here ZΛ and FΛ are functions of the coordinates zα (α = 1, . . . , nV) of the
scalar fields of the vector multiplets. Recall that the index Λ runs from 0 tonV
wherenV is the number of vector multiplets whereas α= 1, . . . , nV because the
graviphoton that appears in the graviton multiplet is not related to any scalar fields. This is compensated by the freedom to re-scale the symplectic section: the symplectic section is a projective section.
Symplectic rotations act linearly on the symplectic section as: ZΛ FΛ =⇒ S ZΛ FΛ . (6.3.11)
We see that the upper partZΛand the lower partF
Λof the symplectic section
transform respectively as the field strengthsF±Λ
µν and their magnetic dualsG±µνΛ.
This can be understood from the following remark: ZΛ and F
Λ are the fermi-
fermi components of the superspace generalization of electric and magnetic field strenghts ˆFΛ µν and ˆGΛ|µν: ˆ FµνΛ =F Λ µν+ ¯Z Λ¯ ψµiψ j νεji+ZΛεjiψ¯iµψjν, ˆ GΛ|µν =GΛ|µν+ ¯FΛψ¯iµψ j νεji+FΛεjiψ¯iµψjν. (6.3.12)
A special manifold is a K¨ahler manifold in which the K¨ahler potential is not a fundamental quantity but is given by the following symplectic invariant expres- sion:
K=−log (−ihv|¯vi) =−log−i ZΛF¯Λ−FΛZ¯Λ
. (6.3.13)
In special geometry, the kinetic terms of both scalar and vector fields are com- puted from the symplectic section as follow
gαβ¯=∂α∂β¯K(z,¯z) = ihDαv|Dβ¯v¯i, NΛ≡ FΛ D¯α¯F¯Λ ZΣ D¯ ¯ αZ¯Σ −1 . (6.3.14)
6.3. Vector multiplets and special geometry
Here the covariant derivatives are defined by
Dαv=∂αv+ (∂αK)v, Dα¯v¯=∂α¯v¯+ (∂α¯K)¯v. (6.3.15)
In contrast to the metricNΛΣ of the vector fields, the metric of the scalar
fields is a symplectic invariant quantity. When the theory is gauged, the electric- magnetic duality is explicitly broken by the introduction of electric charges. In particular, the scalar potential generated by the gauging is not symplectic invari- ant.
Special geometry and prepotentials
A special geometry is saidto admit a prepotential when the lower component of the symplectic section (the variableFΛ) can be expressed as derivative of a scalar
functionF(Z) depending only on the upper part of the symplectic section (ZΛ):
FΛ=
∂
∂ZΛF(Z). (6.3.16)
F(Z) is restricted to be an homogeneous function of second degree in theZΛfields and is called theprepotential. Interestingly, any symplectic section can be rotated to a section admitting a prepotential [140]. In the presence of a prepotential the coupling matrix NΛΣcan be written in the form2
NΛΣ= ¯FΛΣ+ 2i
Im(FΛ∆) Im(FΣΓ)Z∆ZΓ
Im(F∆Γ)Z∆ZΓ
. (6.3.17)
For phenomenogical reasons the most interesting sections are those related to a prepotential only after a symplectic rotation. For instance, in N = 2 compactifications of string theory it is not possible to describe chiral fermions, and partial supersymmetry breaking to N = 1 is only possible with symplectic sections that do not admit a prepotential [142]. Nevertheless, prepotentials are still interesting since they provide a handy way to classify special mani- folds, and they simplify the study of symplectic invariant properties of the action.
Minimal special geometries correspond to quadratic prepotential defined with a metricηΛΣof signature (1, n): F(Z) =−iZΛηΛΣZΣ, ηΛΣ= 1 01×n 0n×1 − n . (6.3.18)
The corresponding scalar manifold of the vector multiplets is U(1) SU(SU(1,n)n). Al- though unusual, minimal special geometry is not incompatible with string theory. To the best of our knowledge, there is so far only one case in which it
2F
N = 2 supergravity and effective Fayet-Iliopoulos terms.
occurs in string theory [143]. This are theN = 2 vacua coming from theN = 3 flux compactification onT6/Z2studied by Frey and Polchinski [144].
Very special K¨ahler geometriesare characterized by cubic prepotentials
F(Z) = idΛΣΓ
ZΛZΣZΓ
Z0 , (6.3.19)
wheredΛΣΓis a real symmetric tensor. Very special K¨ahler geometries are famil-
iar in string theory where they occur in many different compactifications as for example in toroidal compactifications of the heterotic string with possible Wilson lines [139], in compactifications of type II string theories on Calabi-Yau three- folds, in compactification of typeIIstring theories on orientifolds likeK3×T2/Z2
in the presence ofD3 andD7 branes [145, 146].