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.
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  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 S1. In terms of the parameters of the theory, the relation between the M-theory circle and the 11-dimensional Planck scale with the type IIA string coupling and string length is
The 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:
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  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 S1 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]
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 . 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 . By definition, the identity. Adding Eq. (69) to Eq. (67), one obtains
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  and to review the origin in the interchange between scalar fields and gauge fields on the brane19.
The second question can be addressed by an analysis of the D-brane action bosonic symmetries. These act infinitesimally asmust 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
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 reductionT-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 holdno 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 p-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 circle
Just 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 that the functional form of the WZ term is preserved, i.e., , whenever the condition
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
The expert reader may have noticed that the RR T-duality rules do not coincide with the ones appearing in . 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 the above redefinitions. Furthermore, one finds .
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 .
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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