We construct a consistent supersymmetric action for brane chiral and vector multiplets in a six-dimensional chiral gauged supergravity. A nonzero brane tension can be accommodated by allowing for a brane-localized Fayet-Iliopoulos term proportional to the brane tension. When the brane chiral multiplet is charged under the bulk, we obtain a nontrivial coupling to the extra component of the gauge field strength and a singular scalar self-interaction term. In this article, we consider the supersymmetrisation of the brane tension action in a way compatible with the bulk SUSY in 6D Salam-Sezgin supergravity. We find that a branelocalised Fayet-Ilioupolos term proportional to each brane tension must be introduced to cancel the SUSY variation of the brane tension term. In particular, brane world models in six dimensions have drawn much attention because the brane tension generates a nonzero deficit angle in extra dimensions without curving the 4D spacetime. This feature has been first pursued in the framework of spontaneous compactification due to gauge fluxes in 6D Einstein-Maxwell theory, but ended up with a fine-tuning condition for the brane tension due to flux quantization or conservation. Furthermore, we still need some symmetry to ensure that a bulk tuning condition is stable against the quantum corrections. In the context of D = 5, N = 2, Yang-Mills Supergavity compactified on orbifold we consider the supersymmetric coupling of matter fields propagating on the brane. To solve the discrepancy between the bulk actions with limited field content and the wide range of brane actions that involve all the possible forms, we have constructed a new formulation of IIA supergravity up to quartic order in fermions. In the process of pursuing this goal we have realized that all descriptions of the effective field theory of Type IIA string theory available in the literature are inefficient for our purpose. The realization of the total system in supergravity is rather obscure.
Brane Multiplets in D = 5 , N = 2 Supergravity
In the on-shell formulation of D = 5 , N = 2 supergravity, compactified on orbifold, we extend the results describing the interaction of the bulk fields with matter which is assumed to be confined on the brane. We consider a five-dimensional Yang-Mills supergravity model. The field content of the model is
The supergravity multiplet consists of the fünfbein, two gravitini and the graviphoton. Moreover, there exist vector multiplets, counting the Yang-Mills fields. The spinor and the scalar fields included in the vector multiplets are presented. The bulk Lagrangian is
Recalling the linearized supersymmetry transformations of the bulk fields
The requirement of N = 1 local supersymmetry invariance on the branes determines the on-shell couplings of these fields to the gravity and gauge multiplets. These can be found following Nöther’s procedure. This procedure is used in the on-shell formulation of local supersymmetry, where the role of the gauge field is played by the gravitino, while the gauge current is the supercurrent. However in the case of supersymmetry besides the modification of the Lagrangian the transformation laws should be also modified accordingly.
The original Lagrangian is
with the brane part including the interactions of the matter fields, localized on the brane, with the projections of the bulk fields, gravity and gauge fields, on the brane.
We must modify the original theory by adding new terms, so that the total Lagrangian is
Using the gravitino transformation law one gets
The above results are easily extended in the case that the original brane action has the structure of a general model
Summarizing, the interactions of a set of chiral multiplets localized on the brane designated by the index, with the bulk gravity and gauge fields are found to be
It then turns that with this interpretation the terms in the action are exactly those encountered in the ordinary D=4 supergravity involving the interactions of the chiral fields among themselves and their interactions with the gravity and gauge multiplets. Exception to it are additional terms where bulk fields, involving the radion multiplet, interact with the multiplets on the brane.
Bulk Branes in Salam-Sezgin Supergravity
We present a new anomaly-free gauged N = 1 supergravity model in six dimensions. We construct a consistent supersymmetric action for brane chiral and vector multiplets in a six-dimensional chiral gauged supergravity. When the brane chiral multiplet is charged under the bulk, we obtain a nontrivial coupling to the extra component of the gauge field strength and a singular scalar self-interaction term. The six-dimensional Salam-Sezgin supergravity consists of gravity coupled to a dilaton field, a Kalb-Ramond (KR) field, along with the SUSY fermionic partners, the gravitino, the dilatino. Moreover, it also contains a bulk vector multiplet that gauges the R-symmetry of six-dimensional supergravity. All the bulk fermions are 6D Weyl. In order to do this analysis, we need the spinor part of the action and in particular the part that is quadratic in fermionic terms. This is given by
The complete bulk Langrangian up to four fermion terms is
The local N = 2 SUSY transformations are
We can show that the action for the Lagrangian is invariant under the above SUSY transformations up to the trilinear fermion terms and the Bianchi identities as follows
Thus, as will be seen later, the SUSY variation of the brane action can be cancelled with the bulk variation by modifying the Bianchi identities.
We consider a nonzero brane tension as well as brane matter multiplets: a brane chiral multiplet, the superfield of which has an charge, and a brane vector multiplet. Then, by employing the Noether method for the local SUSY, we find that the supersymmetric action for the bulk-brane system is composed of the original bulk action with the field strength tensors being replaced by the modified field strenght tensors, respectively, and the brane action as follows
with
We introduce a gravitino mass term on the brane. Then, the brane action is supplemented by the supersymmetric gravitino mass terms as
Let us add to the bulk Lagrangian a term for a brane located at the position, where includes the internal space 2D coordinate. This brane Lagrangian will be given by
The SUSY transformation of the brane action is non-vanishing as follows
The gravitino contains a piece of the gauge field strength as
As a consequence of introducing the localised FI terms, we have seen that the brane tension action is made compatible with the bulk SUSY transformations. The supersymmetric action of the bulk and brane system up to four fermion terms is
All the fermionic SUSY transformations are modified as
The important ingredient of the above modifications is that we have a brane term linear in the brane-localised FI term. In other words, there is a brane coupling to the magnetic flux, which is proportional to the brane tension.
There are some known anomaly-free models including the non-abelian gauge fields in 6D gauged supergravity. In these cases, an abelian flux can be also turned on in the direction of the non-abelian gauge fields. For instance, in the model with hyperino, the U(1) contained in E6 can also develop a nonzero flux, still maintaining the warped solution that was obtained for the Salam-Sezgin supergravity.
Bulk and Brane System in Type IIA Supergravity
We will present here the new dual formulation with action, available for the IIA case only. A proper action will be constructed in this formulation. It is this formulation that we will apply in our construction of the bulk and brane system. We will call this the dual formulation. Note that the standard action for IIA supergravity can be obtained by integrating out the dual potentials in the action.
There are also terms quartic in fermions in the action. It turns out that it is important to take them into account to check the T-duality invariance of partition sum. We collect 4-fermionic terms in D=10 IIA supergravity action which are obtained from circle reduction of the D=11 action
where
We are in the position to read of the values of the sources and constants introduced in the dualization procedure and sum the different contributions to obtain the full action of the bulk and brane system in the Einstein frame
The source current arising from the dualization of the two-form is given by
In the total action the kinetic terms of the axionic scalars contain the gauge covariant derivatives
The bulk action has a set of shift symmetries
which are gauged upon inclusion of a D6-brane, resulting in the covariant derivatives given above.
In the sequel we will assume that this is the case so that the part of the Lagrangian including the kinetic terms of the scalar fields encoded in the total action may be written as
Let us now take a closer look at the scalar potential evaluated for the F-fl at con figuration in which we performed our dimensional reduction. From what we said above it reduces to
and depends on the Kähler structure moduli as well as the dualization constants. First we stress again that the Kähler moduli should be viewed as fixed at the F-flat value consistent with the calibration condition and are thus not to be treated dynamically in this potential.
In the context of D = 5, N = 2, Yang-Mills Supergavity compactified on orbifold we consider the supersymmetric coupling of matter fields propagating on the brane. Working in the onshell scheme we have derived the terms of the brane action which are relevant for studying the mechanisms of supersymmetry and gauge symmetry breaking. This rather complicated task, along with the derivation of additional terms coupling the brane fields to the radion multiplet, and other even combination of odd fields, which complete the Lagrangian. The complete brane action including these terms and the mechanisms of supersymmetry and the gauge symmetry breaking in particular unified models, in which both gravity and gauge forces propagate in the bulk. We have constructed a consistent SUSY action for brane matter multiplets in a 6D chiral gauged supergravity. Introducing brane chiral multiplets charged under the U(1)R, we derived the supersymmetric coupling to the brane by modifying both the gauge field strength and the field strength for the KR field together with the necessary modifications of the fermionic SUSY transformations. We studied in detail the spectrum in the general warped background where codimension-two branes were supporting the necessary conical singularities. The above property for the massless gravitino and its mass suppression with extra operators, should also hold for the other fermionic states of the spectrum. In particular the gauginos which correspond to the directions of isometry of the internal space should have the same feature. This procedure offers an alternative way to obtain light fermions in models with extra dimensions. We have constructed new formulations of Type IIA supergravity. We constructed bulk theories with a unified treatment of all potentials. Due to the doubling of degrees of freedom one had to impose extra duality constraints and thus a proper action was not possible. Using these actions all bulk and brane systems can be described. Apart from being a tool to understand the supersymmetric domain walls we were interested, it can be expected that the new effective theories of D = 10 supergravity will have more general applications in the future.
There are also terms quartic in fermions in the action. It turns out that it is important to take them into account to check the T-duality invariance of partition sum. We collect 4-fermionic terms in D=10 IIA supergravity action which are obtained from circle reduction of the D=11 action
where
We are in the position to read of the values of the sources and constants introduced in the dualization procedure and sum the different contributions to obtain the full action of the bulk and brane system in the Einstein frame
The abbreviations used in the above are
The source current arising from the dualization of the two-form is given by
In the total action the kinetic terms of the axionic scalars contain the gauge covariant derivatives
The bulk action has a set of shift symmetries
which are gauged upon inclusion of a D6-brane, resulting in the covariant derivatives given above.
In the sequel we will assume that this is the case so that the part of the Lagrangian including the kinetic terms of the scalar fields encoded in the total action may be written as
Let us now take a closer look at the scalar potential evaluated for the F-fl at con figuration in which we performed our dimensional reduction. From what we said above it reduces to
In the context of D = 5, N = 2, Yang-Mills Supergavity compactified on orbifold we consider the supersymmetric coupling of matter fields propagating on the brane. Working in the onshell scheme we have derived the terms of the brane action which are relevant for studying the mechanisms of supersymmetry and gauge symmetry breaking. This rather complicated task, along with the derivation of additional terms coupling the brane fields to the radion multiplet, and other even combination of odd fields, which complete the Lagrangian. The complete brane action including these terms and the mechanisms of supersymmetry and the gauge symmetry breaking in particular unified models, in which both gravity and gauge forces propagate in the bulk. We have constructed a consistent SUSY action for brane matter multiplets in a 6D chiral gauged supergravity. Introducing brane chiral multiplets charged under the U(1)R, we derived the supersymmetric coupling to the brane by modifying both the gauge field strength and the field strength for the KR field together with the necessary modifications of the fermionic SUSY transformations. We studied in detail the spectrum in the general warped background where codimension-two branes were supporting the necessary conical singularities. The above property for the massless gravitino and its mass suppression with extra operators, should also hold for the other fermionic states of the spectrum. In particular the gauginos which correspond to the directions of isometry of the internal space should have the same feature. This procedure offers an alternative way to obtain light fermions in models with extra dimensions. We have constructed new formulations of Type IIA supergravity. We constructed bulk theories with a unified treatment of all potentials. Due to the doubling of degrees of freedom one had to impose extra duality constraints and thus a proper action was not possible. Using these actions all bulk and brane systems can be described. Apart from being a tool to understand the supersymmetric domain walls we were interested, it can be expected that the new effective theories of D = 10 supergravity will have more general applications in the future.
No comments:
Post a Comment