Supersymmetry and the Systematics of Tduality Rotations in TypeII Superstring Theories
Abstract
We describe a systematic method of studying the action of the
Tduality group on spacetime fermions and RR field
strengths and potentials in typeII string theories, based on
spacetime supersymmetry. The formalism is then used to show that the
couplings of nonAbelian Dbrane charges to RR potentials can be
described by an appropriate Clifford multiplication.
Report No: HIP200107/TH, hepth/0103149
1 Introduction:
Let us denote the massless NSNS background fields by and , and the RR nform field strengths by , where is in typeIIA and in typeIIB theory. It is customary to define two sets of RR potentials, which we denote by and . Denoting by and the sums of nforms and (similarly for ), the relations among them can be written as
(1) 
Here, and the two sets of potentials are related by . Also we denote the two gravitinos, the two dilatinos and two supersymmetry transformation parameters by , and , respectively. In typeIIA theory the fermions labelled by have opposite chiralities, while in typeIIB they have the same chirality.
Let denote the spacetime coordinates and consider background field configurations which are independent of the coordinates , but may vary with the remaining coordinates . The Tduality group acts on these backgrounds such that the equations of motion, when restricted to independent fields, remain invariant. The transformation of NSNS fields under this group is well known and can be obtained in a variety of ways. Here we describe a systematic procedure, developed in [1, 2], for obtaining the transformations of RR field strengths and potentials. As a byproduct, one also obtains the transformations of the gravitinos, dilatinos and supersymmetry parameters. In the last section, we describe an application of the results to the coupling of nonAbelian Dbrane charges to RR backgrounds [3].
The transformation of RR potentials under a single Tduality was first obtained in [4] in the effective lowenergy theory. More recently, the case of single Tduality in the presence of supersymmetry has also been considered in [5, 6, 7]. In [8, 9] it was observed that the potentials transform in the spinor representations of the Tduality group, verified directly in [10]. Also, see [11]. However, this cannot be used to obtain the transformation of and under the nontrivial elements of the full Tduality group.
1.1 Supersymmetry Transformations
We obtain the transformations of massless RR and RNS fields under nontrivial transformations, by demanding compatibility between Tduality and spacetime supersymmetry. In typeII theories, this leads to the invariance of the equations of motion.
Not all terms in the supersymmetry transformations are needed for the analysis. To see the basic structure, let us first consider typeII superstring theories in flat space. The two spacetime supersymmetry transformations act independently on the left and right moving sectors of the worldsheet by interchanging Ramond states with NeveuSchwarz states . For example, schematically, we can write
If we know the action of the Tduality group on the NSNS sector, then, demanding compatibility with supersymmetry, the first two equations determine the Tduality transformations of the RNS sector and those of . One can then use the third equation to determine the Tduality transformation of the RR sector.
In the presence of background fields, the supersymmetry transformations are more complicated. However, terms with the lowest powers of fermions still retain the flatspace structure. For example, for the gravitinos , corresponding to the above three equations we have [12, 13, 1]
(2)  
(3)  
(4) 
Here, are the torsionful spinconnections and are the 10dimensional Lorentz frame indices. is a bispinor constructed out the RR field strengths as
(5) 
The dots “” in equations (2)(4) represent terms containing cubic powers of the spinors. We do not write these out explicitly but emphasize that their contribution will be accounted for in the final results. The strategy now is to first work out the transformation of under nontrivial elements of the duality group. Then from equations (2) and (3) we obtain the corresponding transformation of and . In turn, these can be used in (4) to find the transformation of . Let us first review the transformation of the metric under the duality group.
1.2 Tduality of the Metric
Not all elements of the Tduality group transform the background fields nontrivially. In fact, transformations of the coordinates , as well as constant shifts in are subgroups of the Tduality group. These are the trivial elements. The rest act nontrivially and are symmetries of the equations of motion only when the fields do not depend on the coordinates . Let us parameterize the subgroup of by matrices and . All nontrivial elements fall in this subgroup and are parameterized by and with . The case is equivalent to a coordinate transformation and belongs to the subgroup of , already classified as trivial. We will be interested in the transformation of fields under the subgroup parameterized by the dimensional matrices . In order to write the transformation of the 10dimensional fields in a compact way, it is convenient to enlarge these to 10dimensional matrices and simply by adding the identity matrix for the extra dimensions. By definition, these satisfy . Here is the flat metric with respect to which the orthogonal groups are defined. , similarly for
The transformation of the metric under the subgroup can be written in two equivalent ways [14] ,
(6) 
where the matrices are given by
(7)  
(8) 
From the structure of and one can see that
(9) 
with similar equations for . We do not write down the transformations of other NSNS fields.
1.3 The Local Lorentz Twist
To couple fermions to gravity one introduces vielbeins through . The observation that the Tduality transformation of the metric can be written in two equivalent ways means that there are two possible vielbeins in the dual theory,
(10) 
They are related by a Lorentz transformation ,
(11) 
In string theory, the vielbeins and the associated local Lorentz frame can originate in either the right, or the left moving sector of the worldsheet theory. Depending on this origin, the vielbeins transform to either or . Alternatively, as implied by (11), we can choose a single set of vielbeins, say, , for both the right and left moving sectors, but assume that the associated local Lorentz frames are now twisted with respect to each other by an amount . Below, we show that this local Lorentz twist, induced by the action of the Tduality group, can be undone by absorbing it into the Ramond sector. This, in turn, dictates the transformation of the spacetime spinors and RR fields.
2 Tduality and Supersymmetry
Now we have the ingredients needed to analyse the effect of Tduality on the supersymmetry variations (2)(4). First we need the transformation of the spinconnections . In the dual theory, we denote these by . The subscript indicates that they are defined with respect to . One can show that [1, 2]
(12)  
(13) 
can now be easily obtained from by noting that and are related by a Lorentz transformation (11). Let denote the spinor representation of defined by
(14) 
Then, using (9)(14), one sees that the duality transformations of the spinorial covariant derivatives, are given by
(15) 
(16) 
2.1 Transformation of Spinors
Using the transformation of the spinorial covariant derivatives above, in the supersymmetry variations (2) and (3), one obtains the Tduality action on the supersymmetry parameters as
(17) 
As for , it seems that their transformation is determined only up to terms cubic in the spinors. There are two sources of 3spinor corrections: i) equations (2) and (3) contain 3spinor terms that we have not taken into account, ii) the Tduality transformations and differ by 3spinor terms since and are both quadratic in the spinors. It turns out that contributions of type i and ii cancel each other and the Tduality action on the gravitinos is given by
(18)  
(19) 
The absence of 3spinor corrections can be confirmed by verifying that they are not allowed by the supersymmetry variations of the NSNS fields. This means that Tduality does not mix terms with different spacetime fermion numbers.
A similar analysis of the supersymmetry variations of the dilatinos gives
2.2 The RR Field Strength Bispinor
In the transformed theory, the supersymmetry variation (4) becomes
(20) 
Then, using (17) and (19) along with the dilaton transformation, , one obtains
(21) 
where now,
(22) 
with . To find the transformation of the component field strengths , as well as those of the associated RR potentials and , we need the explicit form of the spinor representation of the Lorentz twist .
3 Transformations of RR Fields
The Tduality group has elements with determinant and . A representative of the latter class is a single Tduality with respect to a coordinate, say . This is a transformation which corresponds to the choice , . It interchanges typeIIA and typeIIB theories [15]. The elements with determinant , of course, form the group which acts within typeIIA or typeIIB theory. We consider these two cases separately.
3.1 The Single Tduality Case
For a single Tduality, say, along , the spinor representation is given by [1]
(23) 
Here, is an arbitrary sign that we choose to be in going from IIA to IIB and vice versa, so that the transformation squares to . Using this in (21), one obtains
(24) 
Relation (1) between the and then gives
(25) 
as first obtained in [4]. The transformation of can be obtained using , as
(26) 
Now we make three simple observations that will greatly simplify things in the next subsection:

Under a single Tduality, and transform in the same way, up to a sign. The sign difference disappears when considering even number of single Tdualities.

The transformation of is independent of and . Hence, it transforms in the same way as would in the flatspace , .

Being forms, , and transform in the same way under transformations of the coordinates , in particular, under its subgroup.
3.2 The Case
The spinor representation of the Lorentz twist induced by an transformation of the backgrounds is given by [2]
(27) 
Here, stand for the blocks of spanned by the index , and stands for an exponentiallike expansion with the products of all matrices antisymmetrized. The appearing above is given by
(28) 
Substituting (27) in (21) one obtains
(29)  
where, are the binomial expansion coefficients.
3.3 Transformation of Potentials
Unlike the case of a single Tduality, the transformation of and cannot be easily worked out by using equation (1). However, these can be obtained by a simple construction combined with the three observations made in subsection 3.1.
Any nontrivial transformation can, in principle, be constructed as a combination of single Tduality transformations and appropriately chosen coordinate rotations. To see this, let denote a unit vector in dimensions. Any rotation, say , can be decomposed as a product of reflections about planes perpendicular to properly chosen axes , i.e., . Equivalently, it can be written as a product of reflections about planes perpendicular to the coordinate axes , and properly chosen rotations that rotate the coordinate axes into the reflection axes . Also setting to a product of the rotations , we get
(30) 
Then the nontrivial transformation implemented by and corresponds to a sequence of single Tduality transformations with , , intertwined with coordinate rotations .
This construction implies that the nontrivial transformations of in (29) can, at least in principle, be constructed by applying a succession of single Tdualities (24) and coordinate rotations, using the decomposition (30).
As emphasized in subsection 3.1, and transform in the same way under a single Tduality (the sign difference is immaterial for even number of Tdualities). They also transform in the same way under coordinate rotations. Therefore, if we construct the action of an element on , by using its decomposition in terms of single Tdualities and rotations, then we will end up with the same equation as for , i.e.,(29), but now with replaced by . In terms a bispinor , defined analogously to (5), the transformation is
(31) 
Needless to say, this formula is also valid for nontrivial transformations.
3.4 Transformation of Potentials
As observed in subsection 3.1, a single Tduality acts on in the same way as it would act on in a flat space with and . Furthermore, both potentials transform in the same way under coordinate rotations. The decomposition (30) then implies that nontrivial transformations of are given by the same formula as that for with and set to .
This result can also be written in a compact form similar to equation (31). For this, let us define a set of Gamma matrices by
(32) 
The hat indicates that is an auxiliary flat metric while the actual spacetime metric is still . Now we construct the spinor representation of the Lorentz twist in this auxiliary flat space. This is obtained from the expression (27) for by replacing by and setting . Furthermore, we can combine the into an bispinor ,
(33) 
Then the nontrivial transformations of can be written as
(34) 
4 An Application: Dbrane Couplings by Clifford Multiplication
The Abelian theory on a single Dbrane couples to the background RR potentials by exterior multiplication , where is the pullback of to the worldvolume and is the Abelian gauge field strength. In the nonAbelian case the interaction also involves the scalar products of the nonAbelian scalars and the RR potentials. Furthermore, the factors of appearing in the pullback of in the static gauge are replaced by the gauge covariant derivatives [16, 17]. These new interactions are written in the static gauge and moreover the replacement of the pullback by a nonAbelian covariant derivative obscures the geometric description of the Dbrane as an embedded surface.
The construction in the last section allows us to go beyond the static gauge and write a covariant and geometric expression that includes the new couplings [3]. This is achieved by replacing the exterior product by a Clifford multiplication defined with respect to the flat metric as in (32). To see this, note that the degree of a Dbrane volume form increases/decreases under a single Tduality transverse/parallel to the brane. Thus, they behave very much like the in (26) and it makes sense to construct a volume bispinor similar to (33),
(35)  
Moreover, the worldvolume gauge fields and transverse scalars can be combined into a form such that
(36) 
Here, and span the tangent and normal bundles to the worldvolume ( is an orthonormal frame index). Let denote the field strength of and . We can also define a bispinor for the generalized Dbrane charges,
(37) 
It is now easy to write a covariant expression for the Dpbrane couplings to all RR potentials in terms of a Clifford product,
(38) 
where is the symmetrized gauge trace and is a trace over the spinor index. The integral over restricts the expression to the Dpbrane worldvolume.
The matrix multiplication and tracing can be carried out and leads to the component form for the generalized WZ action, including the new interactions. In the static gauge, (, it reduces to the known form in [16, 17]. The generalized pullbacks discussed above appear as part of the restriction of to the Dbrane which, using (36), gives
(39)  
The appearance of the normal bundle connection here is a reflection of the covariance of the formalism. However, it does not yet contain the contribution from curved backgrounds for which it may be necessary to consider the gravitational couplings of the brane.
It is a pleasure to thank the organizers of the International Conference on “Supersymmetry and Quantum Field Theory” (Kharkov, July 2529, 2000) for the invitation and hospitality.
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