The specific dualities I will be appealing to are the strong coupling limit of type IIA string theory, its relation to M-theory and the action of T-duality on type II string theories and D-branes. Figure 3 summarises the set of relations between the brane tensions discussed in this review under these symmetries.

- either as a long string or a D4-brane, if the M2/M5-brane wraps the M-theory circle, respectively
- or as a D2-brane/NS5 brane, if the M-theory circle is transverse to the M2/M5-brane world volume.

The question to ask is: how do these statements manifest in the classical effective action? The answer is by now well known. They involve a double or a direct dimensional reduction, respectively. The idea is simple. The bosonic effective action describes the coupling of a given brane with a fixed supergravity background. If the latter involves a circle and one is interested in a description of the physics nonsensitive to this dimension, one is entitled to replace the d-dimensional supergravity description by a d-1 one using a Kaluza–Klein (KK) reduction (see [197] for a review on KK compactifications). In the case at hand, this involves using the relation between bosonic supergravity fields and the type IIA bosonic ones summarised below [409]

where the left-hand side 11-dimensional fields are rewritten in terms of type IIA fields. The above reduction involves a low energy limit in which one only keeps the zero mode in a Fourier expansion of all background fields on the bulk SThe same principle should hold for the brane degrees of freedom . The distinction between a double and a direct dimensional reduction comes from the physical choice on whether the brane wraps the internal circle or not:

- If it does, one partially fixes the world volume diffeomorphisms by identifying the bulk circle direction with one of the world volume directions , i.e., , and keeps the zero mode in a Fourier expansion of all the remaining brane fields, i.e., where . This procedure is denoted as a double dimensional reduction [192], since both the bulk and the world volume get their dimensions reduced by one.
- If it does not, there is no need to break the world volume diffeomorphisms and one simply truncates the fields to their bulk zero modes. This procedure is denoted as a direct reduction since the brane dimension remains unchanged while the bulk one gets reduced.

Since Type IIA and Type IIB supergravities are field theories, the above field theoretical realisation
applies. Thus, the compactification limit should give rise to two separate
supergravity theories. But it is known [388] that there is just such a unique supergravity theory. In other
words, given the type IIA/B field content and their KK reduction to dimensions, i.e.,
and , the uniqueness of supergravity guarantees the existence
of a non-trivial map between type IIA and type IIB fields in the subset of backgrounds allowing an S^{1}
compactification

This process is illustrated in the diagram of Figure 4. These are the T-duality rules. When expressed in terms of explicit field components, they become [82, 388]

These correspond to the bosonic truncations of the superfields introduced in Appendix A.1. Prime and unprimed fields correspond to both T-dual theories. The same notation applies to the tensor components where describe both T-dual circles. Notice the dilaton and the transformations do capture the worldsheet relations (56).Let me move to the brane transformation. A D()-brane wrapping the original circle is mapped under T-duality to a Dp-brane where the dual circle is transverse to the brane [424]. It must be the case that one of the gauge field components in the original brane maps into a transverse scalar field describing the dual circle. At the level of the effective action, implementing the limit must involve, first, a partial gauge fixing of the world volume diffeomorphisms, to explicitly make the physical choice that the brane wraps the original circle, and second, keeping the zero modes of all the remaining dynamical degrees of freedom. This is precisely the procedure described as a double dimensional reduction. The two differences in this D-brane discussion will be the presence of a gauge field and the fact that the KK reduced supergravity fields will be rewritten in terms of the T-dual ten-dimensional fields using the T-duality rules (58).

In the following, it will be proven that the classical effective actions described in the previous section are interconnected in a way consistent with our T-duality and strongly-coupled considerations. Our logic is as follows. The M2-brane is linked to our starting worldsheet action through double-dimensional reduction. The former is then used to derive the D2-brane effective by direct dimensional reduction. T-duality covariance extends this result to any non-massive D-brane. Finally, to check the consistency of the PST covariant action for the M5-brane, its double dimensional reduction will be shown to match the D4-brane effective action. This will complete the set of classical checks on the bosonic brane actions discussed so far.

It is worth mentioning that the self-duality of the D3-brane effective action under S-duality could also have been included as a further test. For discussions on this point, see [483, 252].

In the following, I discuss the double and direct dimensional reductions of the bosonic M2-brane effective action (40) to match the bosonic worldsheet string action (6) and the D2-brane effective action, i.e., the version of Eq. (47). This analysis will also allow us to match/derive the tensions of the different branes.

The next step is to describe the world volume dualisation and the origin of the gauge symmetry on the D2 brane effective action [480]. By definition, the identity

holds. Adding the latter to the action through an exact two-form Lagrange multiplier allows one to treat as an independent field. For a more thorough discussion on this point and the nature of the gauge symmetry, see [480]. Adding Eq. (69) to Eq. (67), one obtains Notice I already introduced the same gauge invariant quantity introduced in D-brane Lagrangians Since is now an independent field, it can be eliminated by solving its algebraic equation of motion Inserting this back into the action and integrating out the auxiliary field by solving its equation of motion, yields This matches the proposed D2-brane effective action, since as a consequence of Eq.s (55) and (48).

In this section, I extend the D2-brane’s functional form to any Dp-brane using T-duality covariance. My
goal is to show that the bulk T-duality rules (58) guarantee the covariance of the D-brane effective action
functional form [453] and to review the origin in the interchange between scalar fields and gauge fields on the
brane^{19}.

The second question can be addressed by an analysis of the D-brane action bosonic symmetries. These act infinitesimally as

They involve world volume diffeomorphisms , a gauge transformation and global transformations . Since the background will undergo a T-duality transformation, by assumption, this set of global transformations must include translations along the circle, i.e., , , where the original scalar fields were split into .I argued that the realisation of T-duality on the brane action requires one to study its double-dimensional reduction. The latter involves a partial gauge fixing , identifying one world volume direction with the starting S bulk circle and a zero-mode Fourier truncation in the remaining degrees of freedom, . Extending this functional truncation to the -dimensional diffeomorphisms , where I split the world volume indices according to and the space of global transformations, i.e., , the consistency conditions requiring the infinitesimal transformations to preserve the subspace of field configurations defined by the truncation and the partial gauge fixing, i.e., , determines

where are constants, the latter having mass dimension minus one. The set of transformations in the double dimensional reduction are where , and satisfies .Let me comment on Eq. (79). was a gauge field component in the original action. But in its gauge-fixed functionally-truncated version, it transforms like a world volume scalar. Furthermore, the constant piece in the original transformation (76), describes a global translation along the scalar direction. The interpretation of both observations is that under double-dimensional reduction

becomes the T-dual target space direction along the T-dual circle and describes the corresponding translation isometry. This discussion reproduces the well-known massless open string spectrum when exchanging a Dirichlet boundary condition with a Neumann boundary condition.Having clarified the origin of symmetries in the T-dual picture, let me analyse the functional dependence of the effective action. First, consider the couplings to the NS sector in the DBI action. Rewrite the induced metric and the gauge invariant in terms of the T-dual background and degrees of freedom , which will be denoted by primed quantities. This can be achieved by adding and subtracting the relevant pullback quantities. The following identities hold

It is a consequence of our previous symmetry discussion that and , i.e., there is no change in the description of the dynamical degrees of freedom not involved in the circle directions. The determinant appearing in the DBI action can now be computed to be Notice that whenever the bulk T-duality rules (58) are satisfied, the functional form of the effective action remains covariant, i.e., of the form This is because all terms in the determinant vanish except for those in the first line. Finally, equals the T-dual dilaton coupling and the original Dp-brane tension becomes the D(p-1)-brane tension in the T-dual theory due to the worldsheet defining properties (56) after the integration over the world volume circleJust as covariance of the DBI action is determined by the NS-NS sector, one expects the RR sector to do the same for the WZ action. Here I follow similar techniques to the ones developed in [255, 453]. First, decompose the WZ Lagrangian density as

Due to the functional truncation assumed in the double dimensional reduction, the second term vanishes. The D-brane WZ action then becomes where and the following conventions are used Using the T-duality transformation properties of the gauge invariant quantity , derived from our DBI analysis, it was shown in [453] that the functional form of the WZ term is preserved, i.e., , whenever the condition holds (the factor is due to our conventions (91) and the choice of orientation and ).Due to our gauge-fixing condition, , the components of the pullbacked world volume forms appearing in Eq. (94) can be lifted to components of the spacetime forms. The condition (94) is then solved by

These are entirely equivalent to the T-duality rules (58) but written in an intrinsic way.The expert reader may have noticed that the RR T-duality rules do not coincide with the ones appearing in [208]. The reason behind this is the freedom to redefine the fields in our theory. In particular, there exist different choices for the RR potentials, depending on their transformation properties under S-duality. For example, the 4-form appearing in our WZ couplings is not S self-dual, but transforms as

Using a superindex S to denote an S-dual self-dual 4-form, the latter must be Similarly, does not transform as a doublet under S-duality, whereas does. It is straightforward to check that Eqs. (95) and (96) are equivalent to the ones appearing in [208] using the above redefinitions. Furthermore, one finds , which was not computed in [208].In Section 7.1, I will explore the consequences that can be extracted from the requirement of T-duality covariance for the covariant description of the effective dynamics of overlapping parallel D-branes in curved backgrounds, following [395].

The double dimensional reduction of the M5-brane effective action, both in its covariant [417, 8] and non-covariant formulations [420, 420, 78] was checked to agree with the D4-brane effective action. It is important to stress that the outcome of this reduction may not be in the standard D4-brane action form given in Eq. (47), but in the dual formulation. The two are related through the world volume dualisation procedure described in [483, 7].

Living Rev. Relativity 15, (2012), 3
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