CLASIFICACIÓN DE DAÑO 93-

geometry

In four dimensional spacetime with the usual Minkowski signature, a hypermul- tiplet is composed of four real scalar fields and two Majorana spinors. As a Majorana spinor can be decomposed into two chiral spinors of opposite chiral- ity, one can describenH hypermultiplets in terms of 4nH real scalar fields qX

(X = 1,· · ·,4nH) and 2nH chiral spinors ζA (A= 1, . . . ,2nH) of positive chi-

rality and 2nH chiral spinors ζA of negative chirality. The spinors (ζA, ζA) of

hypermultiplets are called thehyperiniand the 4nH real scalar fieldsqX are the hyperscalars. The kinetic terms of the scalar fields qX are characterized by a sigma model with target space the manifoldMQ, which is constrained byN = 2 supersymmetry to be a quaternionic manifold [147]3

Lhyper=−1₂gXY∂µqX∂µqY. (6.4.1)

The metricgXY is computed from the vielbeinfXiA as

gXY =fXiAfY iA, (6.4.2)

wherefXiA = (fXiA)∗. We denote by f X iA and f

XiA _{= (f}X

iA)∗ the inverse of the

vielbein as a 4nH×4nH matrix and its complex conjugate, thus

f_YiAf_iAX =δYX. (6.4.3)

It can be shown that the vielbein and its inverse satisfy the following reality conditions:

fiAX =fXjBCBAji, fXiA=εijCABfjBX. (6.4.4)

3 _{For a review of quaternionic geometry see [148, 139, 149]. In particular, we shall use the} conventions of appendix B of [149].

6.4. Hypermultiplets and quaternionic-K¨ahler geometry

In our conventions, the matricesij andCAB read:

εij = iσ2, CAB=εij⊗ st, i, j= 1,2, s, t= 1, . . . , nH, (6.4.5)

and CAB the inverse of CAB. In the previous equation, the indices A, B = 1, . . . ,2nH has been decomposed into A ≡ (i, t), B ≡ (j, s) where

i, j= 1,2 andt, s= 1, . . . , nH.

The reality condition for the veilbeinf_XiAcan be translated into the following property of the 2×2 matricesf_Xt ≡(f_Xt)ij

(f_Xt)∗=σ2f_Xtσ2. (6.4.6) This implies thatft_{can be seen asn}

Hone-forms with quaternion entries4written

in the representation where the quaternionics units are ( 2,−iσx) where x =

1,2,35_{. We can then say that f}t

X is aquaternionic vielbeinas at each point of

the scalar manifoldMQ, the 4nH real scalar fieldsqX can be organized intonH

quaternionsqt:

qt=fXtqX. (6.4.7)

In a quaternionic manifold it is possible to define a triplet Jx _{(x = 1,2,3) of}

complex structures :

(Jx)XY =−ifXiA(σ x₎

ijfjAY (6.4.8)

which satisfy the multiplication table of quaternionic units

JxJy =−δxy 4nH+ε

xyz_Jz_{. (6.4.9)}

Any linear combination of the form ˜J =axJx also defines a complex structure

(J2_{=− ) provided that}

||~a||2_{= (a}

1)2+ (a2)2+ (a3)2= 1. (6.4.10)

It follows that at each point of the manifold there is a sphere of complex structures which are related to each other by SU(2) rotations. Note that the quaternionic vielvein f_XiA contains an index i associated to the SU(2) R-symmetry. Therefore, from the definition of the complex structures (6.4.8), it is easy to see to check that the SU(2) R-symmetry can be identified with the SU(2) that rotates the complex structures.

The complex structures are covariantly constant with respect to an SU(2) connectionω_Xij= iωx

X(σx) j i

∇XJ~≡ ∇LCX J~+ 2~ωX×J~= 0, (6.4.11)

4_{The set of quaternions is defined by}

H={q01 +q1i +q2j +q3k|qi∈R}, with the elements of the basis satisfying i.j = k, together with all cyclic permutations and i2_{= j}2_{= k}2_=−1.

5_{Any 2×2 matrixqsatisfying the conditionq}∗ _=σ2_qσ2 _{can be written as a quaternion}

q=q0 _−iqx_σx_withq0_{, q}x_∈

N = 2 supergravity and effective Fayet-Iliopoulos terms.

where∇LC _{is the Levi-Civita covariant derivative on the quaternionic manifold.}

In quaternionic manifolds the SU(2) curvature

~

RXY ≡2∂[X~ωY]+ 2~ωX×~ωY (6.4.12)

is proportional to the quaternionic structureJ~XY:

RXY = 1 4nH gXYR, R~XY = 1 2ν ~JXY, ν= 1 4nH(nH+ 2) R, (6.4.13) withRXY =RZXZY. Here the constantν is proportional to the gravity coupling

constantν=−κ2_{. Since we work in units in whichκ= 1, that isν=−1.}

6.5

Isometries, gauging and scalar potential

In N = 2 supergravity coupled to vector multiplets and hypermultiplets, the only way to generate a scalar potential is to promote some of the symmetries of the scalar manifold to be local symmetries. This implies a choice of the Killing vectors of the scalar manifolds and a choice of vector fields that will be used as gauge fields in the covariant derivatives.

In this thesis we will only consider abelian gauging of the symmetries ofMQ. The gauged symmetry is defined by the transformation with parametersαΛ_:

δqX =kXΛαΛ, (6.5.1)

where kX

Λ are the Killing vectors that we will gauge with the vector fields AΛµ.

To gauge a symmetry all the derivatives of the hyperscalars have to be extended to covariant derivatives. The gauge field is taken from the vector multiplets:

∇µqX=∂µqX−kXΛAΛµ. (6.5.2)

In order to preserve supersymmetry the action has to be corrected adding the following scalar potential

V= (g_αβ¯kα_Λk ¯ β Σ+ 2gXYk X Λk Y Σ)e K_Z¯Λ_ZΣ_{+ 4 U}ΛΣ_−3eK_Z¯Λ_ZΣ Px ΛP x Σ. (6.5.3) where P_Λij =Px

Λ(iσx)ij and PΛ|ij =PΛx(iσx)ij are the moment maps[148, 139,

149] related to the Killing vectorskX

Λ of the quaternionic-K¨ahler manifold, andkαΛ

are the Killing vectors of the special manifold. We also have used the definition

UΛΣ≡eKgαβ¯DαZΛD_β¯ZΣ. (6.5.4)

As the scalar fields of vector multiplets transform in the adjont representation of the gauge group, the Killing vectorsk_Λα, kα_Λ¯ of the special manifold vanish for Abelian gauging. In particular, the sector proportional tog_αβ¯kα_Λk

¯ β

Σ of the scalar