It can then be argued that the moduli space of classical vacua for -dimensional SYM is

Each factor of stands for the position of the D-branes in the -dimensional transverse space, whereas the symmetry group is the residual Weyl symmetry of the gauge group. The latter exchanges D-branes, indicating they should be treated as indistinguishable objects.A remarkable feature of this D-brane description is that a classical geometrical interpretation of D-brane configurations is only available when the matrices are simultaneously diagonalisable. This provides a rather natural venue for non-commutative geometry to appear in D-brane physics at short distances, as first pointed out in [496].

The exploration of further kinematical and dynamical properties of these actions is beyond the scope of
this review. There are excellent reviews on the subject, such as [424, 472, 320], where the connection to
Matrix Theory [48] is also covered. If the reader is interested in understanding how T-duality acts
on non-abelian D-brane effective actions, see [471, 221]. It is also particularly illuminating,
especially for readers not used to the AdS/CFT philosophy, to appreciate that by integrating out
overlapping D-branes at one loop, one is left with an abelian theory describing the
remaining (single) D-brane. The effective dynamics so derived can be reinterpreted as describing
a single D-brane in the background generated by the integrated D-branes, which is
AdS_{5} × S^{5} [365]^{46}.
This is illustrated in Figure 11.

Given the kinematical perspective offered in this review and the relevance of the higher order corrections included in the abelian DBI action, I want to discuss two natural stringy extensions of the SYM description

- Keeping the background fixed, i.e., Minkowski, it is natural to consider the inclusion of higher-order corrections in the effective action, matching the perturbative scattering amplitudes computed in the CFT description of open strings theory, and
- Allowing to vary the background or equivalently, coupling the non-abelian degrees of freedom to curved background geometries. This is towards the direction of achieving a hypothetical covariant formulation of these actions, a natural question to ask given its relevance for the existence of the kappa invariant formulation of abelian D-branes.

In the following, I shall comment on the progress and the important technical and conceptual difficulties regarding the extensions of these non-abelian effective actions.

It is important to stress that, a priori, worldsheet calculations involving an arbitrary number of boundary disk insertions could determine this non-abelian effective action. Since this is technically hard, one can perform other consistency checks. For example, one can compare the D-brane BPS spectrum on tori in the presence of non-trivial magnetic fluxes. This is T-dual to intersecting D-branes, whose spectrum can be independently computed and compared with the fluctuation analysis of the proposed symmetrised non-abelian prescription. It was found in [291, 175, 448] that the proposed prescription was breaking down at order . Further checks at order and were carried over in [103, 346, 345, 347]. The proposal in [346] was confirmed by a first principle five-gluon scattering amplitude at tree level in [387]. The conclusion is that the symmetrised prescription only works up to

These couplings were first found in its form in [266] and in its form in [482]. For further checks on Tseytlin’s proposal using the existence of bound states and BPS equations, see the analysis in [115, 114].

- It must contain a unique trace since this is an effective action derived from string theory disk diagrams involving many graviton insertions in their interior and scalar/vector vertex operators on their boundaries. Since the disk boundary is unique, the trace must be unique.
- It must reduce to N-copies of the particle action when the matrices are diagonal.
- It must yield masses proportional to the geodesic distance for off-diagonal fluctuations.

Having in mind that we required spacetime gauge symmetries to be symmetries of the abelian brane effective actions, it would be natural to include in the above list invariance under target space diffeomorphisms. This was analysed for the effective action kinetic terms in [172]. Instead of discussing this here, I will discuss two non-trivial checks that any such formulation must satisfy.

- to match the Matrix theory linear couplings to closed string backgrounds, and
- to be T-duality covariant, extending the notion I discussed in Section 3.3.2 for single D-branes.

The first was studied in [473, 474] and the second in [395]. Since the results derived from the latter turned out to be consistent with the former, I will focus on the implementation of T-duality covariance for non-abelian D-branes below.

As discussed in Section 3.3, T-duality is implemented by a dimensional reduction. This was already applied for SYM in Eq. (475). Using the same notation introduced there and denoting the world volume direction along which one reduces by , one learns that , where is the T-dual adjoint matrix scalar. Furthermore, covariant derivatives of transverse scalar fields become

Notice this contribution is purely non-abelian and it can typically contribute non-trivially to the potential terms in the effective action. To properly include these non-trivial effects, Myers [395] studied the consequences of requiring T-duality covariance taking as a starting point a properly covariantised version of the multiple D9-brane effective action, having assumed the symmetrised trace prescription described above. Studying T-duality along 9- directions and imposing T-duality covariance of the resulting action, will generate all necessary T-duality compatible commutators, which would have been missed otherwise. This determines the DBI part of the effective action to be [395] with Here indices stand for world volume directions, and indices for transverse directions. To deal with similar commutators arising from the WZ term, one considers [395] where the interior product is responsible for their appearance, for example, as in, Notice one regards as a vector field in the transverse space. In both actions (485) and (487), stands for pullback and it only applies to transverse brane directions since all longitudinal ones are non-physical. Its presence is confirmed by scattering amplitudes calculations [342, 271, 222]. Some remarks are in order.- There exists some non-trivial dependence on the scalars through the arbitrary bosonic closed backgrounds appearing in the action. The latter is defined according to Analogous definitions apply to other background fields.
- There exists a unique trace, because this is an open string effective action that can be derived from worldsheet disk amplitudes. The latter has a unique boundary. Thus, there must be a unique gauge trace [186, 188]. Above, the symmetrised prescription was assumed, not only because one is following Tseytlin and this was his prescription, but also because there are steps in the derivation of T-duality covariance that assumed this property and the scalar field dependence on the background fields (489) is symmetric, by definition.
- The WZ term (487) allows multiple Dp-branes to couple to RR potentials with a form degree greater than the dimension of the world-volume. This is a purely non-abelian effect whose consequences will be discussed below.
- There are different sources for the scalar potential: , its inverse in the first determinant of the DBI and contributions coming from commutators coupling to background field components in the expansion (489).

It was shown in detail in [395], that the bosonic couplings described above were consistent with all the linear couplings of closed string background fields with Matrix Theory degrees of freedom, i.e., multiple D0-branes. These couplings were originally computed in [473] and then extended to Dp-branes in [474] using T-duality once more. We will not review this check here in detail, but as an illustration of the above formalism, present the WZ term for multiple D0-branes that is required to do such matching

Two points are worth emphasising about this matching:- There is no ambiguity of trace in the linear Matrix theory calculations. Myers’ suggestion is to extend this prescription to non-linear couplings.
- Some transverse M5-brane charge couplings are unknown in Matrix theory, but these are absent in the Lagrangian above. This is a prediction of this formulation.

One of the most interesting physical applications of the couplings derived above is the realisation of the dielectric effect in electromagnetism in string theory. As already mentioned above, the non-abelian nature of the degrees of freedom turns on new commutator couplings with closed string fields that can modify the scalar potential. If so, instead of the standard SYM vacua, one may find new potential minima with but . As a toy illustrative example of this phenomenon, consider D0-branes propagating in Minkowski but in a constant background RR four-form field strength

Due to gauge invariance, one expects a coupling of the form Up to total derivatives, this can indeed be derived from the cubic terms in the WZ action above. This coupling modifies the scalar potential to whose extremisation condition becomes The latter allows solutions having lower energy than standard commuting matrices It is reassuring to compare the description above with the one available using the abelian formalism
describing a single brane explained in Section 3. I shall refer to the latter as dual brane description. For the
particular example discussed above, since the D0-branes blow up into spheres due to the electric RR
coupling, one can look for on-shell configurations on the abelian D2-brane effective action in the same
background corresponding to the expanded spherical D0-branes in the non-abelian description. These
configurations exist, reproduce the energy up to corrections and carry no D2-brane
charge [395]. Having reached this point, I am at a position to justify the expansion of pointlike gravitons
into spherical D3-branes, giant gravitons, in the presence of the RR flux supporting AdS_{5} × S^{5}
described in Section 5.9. The non-abelian description would involve non-trivial commutators
in the WZ term giving rise to a fuzzy sphere extremal solution to the scalar potential. The
abelian description reviewed in Section 5.9 corresponds to the dual D3-brane description in
which, by keeping the same background, one searches for on-shell spherical rotating D3-branes
carrying the same charges as a pointlike graviton but no D3-brane charge. For a more thorough
discussion of the comparison between non-abelian solitons and their “dual” abelian descriptions,
see [147, 149, 148, 396].

There exists some body of work constructing classical supersymmetric and kappa invariant actions involving non-abelian gauge fields representing the degrees of freedom of multiple D-branes. This started with actions describing branes of lower co-dimension propagating in lower dimensional spacetimes [461, 462, 190]. It was later extended to multiple D0-branes in an arbitrary number of dimensions, including type IIA, in [411]. Here, both world volume diffeomorphisms and kappa symmetry were assumed to be abelian. It was checked that when the background is super-Poincaré, the proposed action agreed with Matrix Theory [48]. Using the superembedding formalism [460], actions were proposed reproducing the same features in [40, 44, 42, 41, 43], some of them involving a superparticle propagating in arbitrary 11-dimensional backgrounds. Finally, there exists a slightly different approach in which, besides using the superembedding formalism, the world sheet Chan–Paton factors describing multiple D-branes are replaced by boundary fermions. The actions constructed in this way in [303], based on earlier work [304], have similar structure to the ones described in the abelian case, their proof of kappa symmetry invariance is analogous and they reproduce Matrix Theory when the background is super-Poincaré and most of the features highlighted above for the bosonic couplings described by Myers.

- D-brane transverse coordinates being replaced by matrices,
- the existent non-commutative geometry description of D-branes in the presence of a -field in space-time (or a magnetic field strength on the brane) [187, 146, 444].

The general idea behind non-commutative geometry is to replace the space of functions by a non-commutative algebra. In the D-brane context, a natural candidate to consider would be the algebra

As customary in non-commutative geometry, the latter does not yet carry any metric information. Following Connes [145], the construction of a Riemannian structure requires a spectral triple , which, in addition to , also contains a Hilbert space and a self-adjoint operator obeying certain properties. It would be interesting to find triples that describe, in a natural way, metrics relevant for multiple D-branes, incorporating the notion of covariance.Regarding D-branes in the presence of a -field, the main observation is that the structure of an abelian non-commutative gauge theory is similar to that of a non-abelian commutative gauge theory. In both cases, fields no longer commute, and the field strengths are non-linear. Moreover, non-commutative gauge theories can be constructed starting from a non-abelian commutative theory by expanding around suitable backgrounds and taking [443]. This connection suggests it may be possible to relate the gravity coupling of non-commutative gauge theories to the coupling of non-abelian D-brane actions to curved backgrounds (gravity). This was indeed the approach taken in [163] where the stress-tensor of non-commutative gauge theories was derived in this way. In [151], constraints on the kinematical properties of non-abelian D-brane actions due to this connection were studied.

Living Rev. Relativity 15, (2012), 3
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