###### Abstract

We discuss a locally supersymmetric formulation for the boundary Fayet-Iliopoulos (FI) terms in 5-dimensional gauge theory on , using the four-form multiplier mechanism to introduce the necessary -odd FI coefficient. The physical consequence of the boundary FI terms is studied within the full supergravity framework. For both models giving a flat and a warped spacetime geometry, the only meaningful deformation of vacuum configuration induced by the FI terms is a kink-type vacuum expectation value of the vector multiplet scalar field which generate a 5D kink mass for charged hypermultiplet. This result for the four-form induced boundary FI terms is consistent with the one derived by the superfield formulation of 5D conformal supergravity.

hep-th/0405100

KAIST-TH 2004/04

Fayet-Iliopoulos Terms in 5D Orbifold Supergravity

Hiroyuki Abe^{*}^{*}* ,
Kiwoon Choi^{†}^{†}† and
Ian-Woo Kim^{‡}^{‡}‡

Department of Physics, Korea Advanced Institute of Science and Technology,

Daejeon 305-701, Korea

## 1 Introduction

Theories with extra dimension can provide an attractive mechanism to generate hierarchical structures in 4-dimensional (4D) physics such as the weak to Planck scale hierarchy [1, 2] and the hierarchical Yukawa couplings of quarks and leptons [3], as well as providing a new mechanism to break grand unified symmetry [4] and/or supersymmetry [5, 6, 7]. In regard to generating the scale and/or Yukawa hierarchies, quasi-localization of gravity [2, 8] and/or matter zero modes [3] in extra dimension is particularly interesting since it can generate exponentially different 4D scales and/or Yukawa couplings even when the fundamental parameters of the higher dimensional theory have similar magnitudes. A simple theoretical framework to implement the idea of quasi-localization would be 5D orbifold field theory on . For instance, in 5D orbifold supergravity (SUGRA), the Randall-Sundrum fine tuning [2] of the bulk and brane cosmological constants which is necessary for the gravity quasi-localization can be naturally obtained by gauging the symmetry with a -odd gauge coupling [9, 10, 11]. In the hypermultiplet compensator formulation of 5D off-shell SUGRA [10], this is equivalent to making the compensator hypermultiplet to have a nonzero -odd gauge coupling to the graviphoton. Also a nonzero 5D kink mass of matter hypermultiplet causing the quasi-localization of matter zero mode can be obtained by making the hypermultiplet to have a similar -odd gauge coupling [12, 13].

It has been noted that globally supersymmetric 5D gauge theory allows Fayet-Iliopoulos (FI) terms localized at the orbifold fixed points [14, 15, 16]. In globally supersymmetric 4D theories, FI term is allowed for generic , and can lead to supersymmetry (SUSY) and/or gauge symmetry breakings. However extending the 4D global SUSY to SUGRA severely limits the possible FI terms. In 4D SUGRA, FI term is allowed only when the associated is either an -symmetry [17] or a pseudo-anomalous endowed with the Green-Schwarz anomaly cancellation mechanism [18]. On the other hand, in 5D orbifold SUGRA, there can be a boundary FI term [15] proportional to

even when is neither an -symmetry nor a pseudo-anomalous symmetry, where is the periodic sign function on the orbifold whose fundamental domain is given by . Such boundary FI terms generate a kink-type vacuum expectation value (VEV) of the scalar component of vector multiplet, thereby giving a kink mass to -charged matter hypermultiplets [15, 16, 19, 20] causing the quasi-localization of matter zero modes. It was also pointed out that such a boundary FI term can be induced at one-loop level even when it is absent at tree level [14, 15, 16, 19].

In order to have a boundary FI term proportional to in 5D orbifold SUGRA, one needs to introduce a -odd coupling . It is in fact non-trivial to introduce a -odd coupling in 5D orbifold SUGRA in a manner consistent with local supersymmetry. One known way is the mechanism of Ref. [21] in which the -odd factor appears as a consequence of the equations of motion of the four-form multiplier field. This procedure does not interfere with local SUSY, thus providing an elegant way to construct an orbifold SUGRA with -odd couplings, starting from a theory only with -even couplings.

In this paper we wish to discuss a locally supersymmetric formulation for the boundary FI terms in 5D gauge theory on . We apply the above mentioned four-form mechanism to the known formulation of 5D off-shell SUGRA [10]. We consider a class of simple models with boundary FI terms, and analyze the ground state solutions to examine whether the supersymmetry is broken or not and also the generation of the hypermultiplet kink mass by the FI terms. In Sec. 2, we derive the SUGRA action of bosonic fields containing a boundary FI term proportional to , starting from the 5D off-shell SUGRA with four-form multiplier.

In Sec. 3, we derive the Killing spinor conditions and energy functional in orbifold SUGRA models with FI terms for generic 4D Poincare-invariant background geometry. We then examine in Sec. 4 the vacuum deformation caused by the boundary FI terms and the resulting physical consequences. We will find that for both models giving a flat and a warped spacetime geometry, the vacuum solution preserves supersymmetry and the only meaningful deformation of vacuum configuration induced by the FI terms is a kink-type vacuum expectation value of the vector multiplet scalar field which generate a 5D kink mass for charged hypermultiplet. We then show that this result for the four-form induced boundary FI terms is consistent with the one derived in [22] using the superfield formulation of 5D conformal supergravity [23] (see also [24]).

## 2 5D orbifold supergravity with boundary FI terms

In this section, we construct the action of 5D orbifold SUGRA containing
the boundary FI terms for a bulk gauge symmetry which is originally
neither
an -symmetry nor a pseudo-anomalous symmetry. We will use the off-shell
formulation of 5D SUGRA on which has been developed by Fujita,
Kugo and Ohashi [10]. In this formulation, gravitational
sector of the model is given by the Weyl multiplet and the central charge
vector multiplet which contain the fünfbein
and the -odd graviphoton , respectively, and a consistent
off-shell formulation is obtained by introducing a compensator hypermultiplet
^{4}^{4}4The theories with a single compensator hypermultiplet
can describe only the quaternionic hyperscalar manifolds where is the number of physical hypermultiplets.
To describe other types of hyperscalar geometry, one needs to introduce
additional compensator hypermultiplets. However the physics of FI terms are
mostly independent of the detailed geometry of the hyperscalar manifold.
We thus limit the discussion to the theories with a single compensator
hypermultiplet.. To discuss FI terms, we consider a minimal model with an
ordinary vector multiplet containing -even
gauge field , and also include a physical hypermultiplet .
As it is a theory on , all physical and nonphysical 5D fields have
the following boundary conditions:

where . It will be straightforward to extend our analysis to models containing arbitrary number of vector multiplets and hypermultiplets with more general boundary conditions.

In order to introduce the -odd coupling for boundary FI terms, we apply the four-form mechanism of Ref. [21] to the 5D off-shell SUGRA [10]. For this purpose, we introduce additional vector multiplet as in Ref. [10], so our model contains three vector multiplets at the starting point:

where () are real scalar components, () are -doublet symplectic Majorana spinors, and are -triplet auxiliary components. Note that is -even for -odd , while and are -odd for -even , and the transformation of -doublet is given by . Throughout this paper, represents the 5D coordinate directions with representing the non-compact 4D coordinate directions, and represents the 5D tangent space directions.

As for the hypermultiplets, we have

where () are quaternionic hyperscalars, are symplectic Majorana hyperinos, and are auxiliary components. In the following, we will use frequently a matrix notation for hyperscalars, e.g.,

where are parity eigenstates. In this matrix notation, the symplectic reality condition and the boundary condition are given by

(1) |

An off-shell formulation for the four-form mechanism to generate -odd couplings in 5D orbifold SUGRA has been developed in Ref. [10]. In this formulation, the four-form multiplier corresponds to the dual of the auxiliary scalar component of a linear multiplet . This dualization is defined under the background of the Weyl multiplet and a vector multiplet , and leads to a three-form field as the dual of the constrained vector component of . Here we choose to be the central charge vector multiplet , and then the superconformal invariant couplings of and include

(2) | |||||

where , is the triplet scalar component of the linear multiplet , is the field strength of the gauge field , and . Once we have the above bulk interactions, -odd coupling constants can be obtained by introducing the following superconformal invariant boundary Lagrangian density:

where for the induced 4D metric on the boundaries. A detailed derivation of the above 4-form Lagrangian densities can be found in Ref. [10].

In the above Lagrangian densities, and play the role of Lagrangian multipliers. By varying , we obtain

whose integrability condition leads to

(3) |

Taking the normalization , one finds

(4) |

where is the periodic sign-function obeying

Then the equations of motion for and give

(5) |

Now using the relations (4) and (5), the redundant vector multiplet can be replaced by the central charge vector multiplet multiplied by the -odd factor . This four-form mechanism provides an elegant way to obtain a locally supersymmetric theory of () involving -odd couplings, starting from a locally supersymmetric theory of () and the four-form multiplier multiplet involving only -even couplings.

The action of vector multiplets () is determined by the norm function which is a homogeneous cubic polynomial of the scalar components . As a minimal model incorporating the boundary FI terms, we consider

Note that this form of does not include any -odd coupling, thus the results of Ref. [10] can be straightforwardly applied for our . Then the auxiliary components appear in the bulk Lagrangian as

(6) |

where

for the charge operators .

In this paper, we limit the discussion to the case that commute with the orbifolding transformation . Then under the condition that the model does not contain any -odd coupling before the four-form multiplet is integrated out, the most general form of the hyperscalar charges consistent with the symplectic reality condition (1) is given by

(7) |

where are real constants. If is not an -symmetry, which is the case that we are focusing here, the compensator hyperscalar is neutral under , i.e.,

When , so is an -symmetry, there can be additional FI terms both in the bulk and boundaries [25]. As we will see, in case that has a nonzero charge , the central charge becomes an -symmetry with a -odd gauge coupling after the four-form multiplet is integrated out. Such model has a ground state geometry being a slice of with the AdS curvature [9, 10], so corresponds to the supersymmetric version of the Randall-Sundrum model.

Our goal is to derive the action of physical fields by systematically integrating out all non-physical degrees of freedom. We already noted that the equations of motion of and result in the relations (4) and (5). Using these relations, we find first of all

and

where

(8) | |||||

for () and

(9) |

We remark that the -term in is same as the one noted in Ref. [15]. Here the new generators for hyperscalars are given by

(10) |

and is the inverse matrix of . Note that although was a bulk action in the original theory, it gives boundary terms after the four-form multiplet is integrated out:

There appear additional boundary terms arising from the bulk kinetic term of which is given by

Using , we find

(11) | |||||

where and

(12) | |||||

If the -odd vanishes on the boundaries as was assumed in [10, 21], all operators in would vanish also. However, as we will see, in the presence of the boundary FI terms, develops a kink-type of fluctuation on the boundaries. In this situation, one can not simply assume that operators involving vanish on the boundaries. The values of depend on how to regulate across the boundary. This would cause a UV sensitive ambiguity in the theory, and makes it difficult to find the correct on-shell SUSY transformation and Killing spinor conditions on the boundaries.

It is in fact a generic phenomenon in orbifold field theory that boundary operators can produce a UV sensitive singular behavior of bulk fields on the boundary. Normally the resulting subtleties correspond to higher order effects in the perturbative expansion in powers of dimensionful coupling constants which is equivalent to an expansion in powers of for the cutoff scale . Then the low energy physics below can be described in a UV insensitive manner by limiting the analysis to an appropriate order in the expansion. In our case, all the boundary terms appear in connection with the dimensionful -odd coupling constants . For instance, leads to the integrable boundary FI terms, while the -odd gauge couplings and give rise to the integrable boundary tensions and hyperscalar mass-squares. It turns out that at leading order in the perturbative expansion in , those boundary terms generate the following kink-type fluctuations

(13) |

for the spacetime metric . Then the precise value of boundary operators involving would depend on how to regulate the singular fixed point and the behavior of across the boundary. As in other cases, we can avoid this problem of UV sensitivity by truncating the theory at an appropriate order in the expansion in powers of .

To make this point more explicit, we parametrize in terms of a physical scalar field under the gauge fixing condition

in the unit with the 5D Planck mass :

Then the very special manifold spanned by has the metric

Obviously, is -odd for -odd . For , has a kink-type fluctuation of near the boundaries. Then, simply counting the powers of in the limit , one easily finds , , , and near the boundaries. With this power counting, the first boundary term in (11) is of the order of , while

In the next section, we will explicitly show that the energy functional including only the boundary operators up to takes the standard Bogomolny-squared form. This implies that the analysis at can be a consistent (approximate) framework to examine the effects of the boundary FI terms in orbifold SUGRA. If the operators of are included, the Killing spinor conditions will receive corrections of . However, the values of severely depend on the way of regulating the singular functions and across the boundaries. It is thus expected that one needs informations on the UV completion of orbifold SUGRA in order to make a complete analysis including .

Upon ignoring the higher dimensional boundary terms of , after integrating out all auxiliary fields other than (), we find the following Lagrangian density of bosonic fields:

(14) | |||||

where

(15) |

Here the matrix valued compensator hyperscalar field can be chosen as

(16) |

which corresponds to one of the gauge fixing conditions in the hypermultiplet compensator formulation of off-shell 5D SUGRA [10].

The above action indeed includes the boundary FI term proportional to

which can be identified as the -component of the vector superfield originating from the 5D vector multiplet . According to the action (14), the on-shell value of is given by

(17) |

where are defined in (8). Then after integrating out the auxiliary components , the full 5D scalar potential is given by

(18) | |||||

With the above results, one easily finds

where the ellipsis stands for field-dependent terms. This shows that a nonzero charge of the compensator hyperscalar, i.e., , gives rise to the correctly tuned bulk and boundary cosmological constants yielding a warped Randall-Sundrum geometry with AdS curvature . After the compensator gauge fixing (16), this charge corresponds to a gauge charge. As a result, the graviphoton becomes a gauge field and its auxiliary component has a bulk FI term which leads to the negative bulk cosmological constant in .

Also from the on-shell expression of the auxiliary components, e.g., the order parameter in Eq. (23), one can see that the hypermultiplet obtains the following 5D mass determined by the () charges and the FI coefficient:

(19) |

where we consider only the hyperscalar mass from the -term potential, , and

Note that this hypermultiplet mass reproduces the result of Ref. [12].

## 3 Killing spinor conditions and energy functional

In this section, we derive the Killing spinor equations and the energy functional in 5D orbifold SUGRA model with boundary FI terms for generic 4D Poincare invariant metric configuration:

(20) |

We will employ the relation frequently in the following. Applying the local SUSY transformations of the gravitino , the gauginos , and the compensator and physical hyperinos , we find [10]

(21) |

where , and are given by (16) and (17), respectively, and we use the convention . Here the local SUSY transformation spinor obeys the -orbifolding condition . From the above local SUSY transformations, one can find that the Killing spinor conditions for 4D Poincare invariant spacetime are given by

(22) |

These Killing spinor conditions can be rewritten as

(23) |

where are given by (8), , and . In the above, corresponds to the gravitino Killing condition, is the gaugino Killing condition, is the physical hyperino Killing condition, and comes from the compensator hyperino Killing condition. In fact, is not independent from other Killing conditions since the compensator hypermultiplet is not a physical degree of freedom. However, here we treat it separately for later convenience. Here we remark that () provide two independent Killing conditions even though there is only one physical gaugino. This is essentially due to the existence of the second term in the relation