7.1 D-branes

The perturbative description of D-branes in terms of opens strings [423] allows one to answer the question regarding the enhancement of massless modes raised above in a firmer basis, at least at weak coupling. Consider the spectrum of open strings in the presence of two parallel Dp-branes separated by a physical distance ℓ. As the latter approaches zero, i.e., it becomes smaller than the string scale, there is indeed an enhancement in the number of massless modes. Its origin is in the sector of open strings stretching between D-branes, which is precisely the one captured by the BIon argument. This enhancement is consistent with an enhancement in the gauge symmetry from U (1) × U(1), corresponding to the two separated D-branes, to U (2), corresponding to the overlapping D-branes. The spectrum of massless excitations is then described by a non-abelian vector supermultiplet in the adjoint representation. To understand how this comes about, consider the set of massless scalar excitations. These are described by (Xi )rs, where i labels the transverse directions to the brane, as in the abelian discussion, and the subindices r,s label the D-branes where the open strings are attached. This is illustrated in Figure 10View Image. Since the latter are oriented, there exist 2 N − N such excitations, which arrange themselves into a matrix Xi = XiaT a, with T a being generators of U (2) in the adjoint representation. The conclusion is valid for any number N of D-branes of world volume dimension p + 1 [496Jump To The Next Citation Point].
View Image

Figure 10: Open strings stretched between multiple branes and their matrix representation.

Super-Yang–Mills action:
The previous discussion identifies the appropriate degrees of freedom to describe the low energy dynamics of multiple D-branes in Minkowski at weak coupling as non-abelian vector supermultiplets. Thus, multiple brane effective actions must correspond to supersymmetric non-abelian gauge field theories in p + 1 dimensions. At lowest order in a derivative expansion, these are precisely super-Yang–Mills (SYM) theories. For simplicity of notation, let me focus on d = 10 U (N ) SYM with classical action

∫ ( ) S = d10σ − 1-Tr F F μν + iTr ψ¯Γ μD ψ (470 ) 4 μν 2 μ
where the field strength
Fμν = ∂μA ν − ∂νA μ − igY M [A μ,A ν] (471 )
is the curvature of a U(N ) hermitian gauge field A μ and ψ is a 16-component Majorana–Weyl spinor of SO (1,9). Both fields, A μ and ψ, are in the adjoint representation of U (N ). The covariant derivative Dμ of ψ is given by
D μψ = ∂ μψ − igYM [A μ,ψ ], (472 )
where gYM is the Yang–Mills coupling constant. This action is also usually written in terms of rescaled fields, by absorbing a factor of g YM in both A μ and ψ, to pull an overall coupling constant dependence in front of the full action
1 ∫ ( ) S = --2-- d10σ − Tr FμνF μν + 2iTr ¯ψΓ μD μψ , (473 ) 4gYM
where D ψ = ∂ ψ − i[A ,ψ ] μ μ μ. The action (470View Equation) is invariant under the supersymmetry transformation
i δA μ = --¯ðœ–Γ μψ, (474 ) 2 δ ψ = − 1F Γ μν𝜖, 4 μν
where 𝜖 is a constant Majorana–Weyl spinor in SO (1,9), giving rise to 16 independent supercharges. Classically, this is a well-defined theory; quantum mechanically, it is anomalous. From the string theory perspective, as explained in Section 3.7, this is just an effective field theory, valid at low energies E √ α′ ≪ 1 and weak coupling g ≪ 1 s.

Dimensional reduction:
The low energy effective action for multiple parallel Dp-branes in Minkowski is SYM in p + 1 dimensions. This theory can be obtained by dimensional reduction of the ten-dimensional super Yang–Mills theory introduced above. Thus, one proceeds as described in Section 3.3: assume all fields are independent of coordinates σp+1, ...,σ9. After dimensional reduction, the 10-dimensional gauge field A μ decomposes into a (p + 1)-dimensional gauge field A α and 9 − p adjoint scalar fields XI = 2π α′ΦI45, describing the transverse fluctuations of the D-branes. The reduced action takes the form

1 ∫ S = --2-- dp+1σ Tr (− FαβF αβ − 2(D αΦI)2 + [ΦI,ΦJ ]2 + fermions). (475 ) 4gYM
The p + 1 dimensional YM coupling g2YM can be fixed by matching the expansion of the square root in the gauge fixed abelian D-brane action in a Minkowski background (104View Equation) and comparing it with Maxwell’s theory in the field normalisation used in Eq. (473View Equation)
1 g √ -- g2YM = ---2-′2---- = √-s-(2π α′)p− 2. (476 ) 4π α TDp α ′
Notice also the appearance of a purely non-abelian interaction term in Eq. (475View Equation), the commutator I J 2 [Φ ,Φ ] that acts as a potential term. Indeed, its contribution is negative definite since [ΦI,ΦJ ]† = [ΦJ ,ΦI] = − [ΦI,ΦJ ]. The classical vacuum corresponds to static configurations minimising the potential. This occurs when both the curvature F αβ and the fermions vanish, and for a set of commuting I Φ matrices, at each point of the p + 1 world volume. In this situation, the fields ΦI can be simultaneously diagonalised, so that one has
( I .. ) || x1 0 0 .|| || 0 xI ... 0 || ΦI = || . 2 . || . (477 ) | 0 .. .. 0 | ( .. I ) . 0 0 xN
The N diagonal elements of the matrix ΦI are interpreted as the positions of N distinct D-branes in the I-th transverse direction [496Jump To The Next Citation Point]. Consider a vacuum describing N − 1 overlapping Dp-branes and a single parallel D-brane separated in a transverse direction Φ. This is equivalent to breaking the symmetry group to U (N − 1) × U (1) by choosing a diagonal matrix for Φ with x0 eigenvalue in the first N − 1 diagonal entries and xN ⁄= x0 in the last diagonal entry. The off-diagonal components δΦ will acquire a mass, through the Higgs mechanism. This can be computed by expanding the classical action around the given vacuum. One obtains that this mass is proportional to the distance |x0 − xN | between the two sets of branes
2 (x0-−-xN-)2 M = 2 πα′ , (478 )
according to the geometrical interpretation given to the eigenvalues characterising the vacuum. In light of the open string interpretation, these off-diagonal components do precisely correspond to the open strings stretching between the different D-branes. The latter allow an alternative description in terms of the BIon configurations described earlier, by replacing the N − 1 Dp-branes by its supergravity approximation, though the latter is only suitable at large distances compared to the string scale.

It can then be argued that the moduli space of classical vacua for (p + 1 )-dimensional SYM is

(ℝ9−p)N --------. (479 ) SN
Each factor of ℝ stands for the position of the N D-branes in the (9 − p)-dimensional transverse space, whereas the symmetry group SN 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 ΦI 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 [48Jump To The Next Citation Point] 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 N − 1 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 N − 1 D-branes, which is AdS5 × S5 [365]46. This is illustrated in Figure 11View Image.

View Image

Figure 11: Integrating out the degrees of freedom at one loop corresponding to N − 1 of the D-branes gives rise to an effective action interpretable as an abelian gauge theory in an AdS throat.

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

  1. 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
  2. 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.

Higher-order corrections:
In the abelian theory, it is well known that the DBI action captures all the higher-order corrections in ′ α to the open string effective action in the absence of field strength derivative terms47 [214]. It was further pointed that such derivative corrections were compatible with a DBI expansion by requiring conformal invariance for the bosonic string in [1] and for the superstring in [87]. In the non-abelian theory, such distinction is ambiguous due to the identity

[D μ, D ν]Fρσ = [Fμν, Fρσ], (480 )
relating commutators with covariant derivatives. It was proposed by Tseytlin [482Jump To The Next Citation Point] that the non-abelian extension of SYM including higher-order α′ corrections be given in terms of the symmetrised prescription. The latter consists of treating all Fμν matrices as commuting. Equivalently, the action is completely symmetric in all monomial factors of F of the form tr(F ...F). This reproduces the F 2 and α ′2F 4 terms of the full non-abelian action, but extends it to higher orders
∘ ----------′--- ℒDBI ∝ Str ημν + 2πα F μν. (481 )
The notation Str defines this notion of symmetrised trace for each of the monomials appearing in the expansion of its arguments. For an excellent review describing the history of these calculations, motivating this prescription and summarising the most relevant properties of this action, see [485].

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 (α′)4F 6. Further checks at order α ′3 and α ′4 were carried over in [103, 346Jump To The Next Citation Point, 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 4 F

1 1 1 ℒ = Str[-F 2μν − -(2πα ′)2(F 4 − -(F2μν)2) + O(α ′4)] (482 ) 4 8 4 = tr[1-F 2 − -1-(2πα ′)2(F F F F + 1-F F F F 4 μν 12 μν ρν μλ ρλ 2 μν ρν ρλ μλ 1 1 ′4 − 4FμνF μνFρλF ρλ − 8-FμνF ρλFμνFρλ) + O (α )]. (483 )
These couplings were first found in its Str form in [266] and in its tr 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].

Coupling to arbitrary curved backgrounds:
The above corrections attempted to include higher-order corrections describing the physics of multiple D-branes in Minkowski. More generally, one is interested in coupling D-branes to arbitrary closed string backgrounds. In such situations, one would like to achieve a covariant formulation. This is non-trivial because as soon as the degrees of freedom become non-abelian, they lose their geometrical interpretation. In the abelian case, I X described the brane location. In the non-abelian case, at most, only their eigenvalues xIi may keep their interpretation as the location of the ith brane in the Ith direction. Given the importance and complexity of the problem, it is important to list a set of properties that one would like such a formulation to satisfy. These are the D-geometry axioms [186Jump To The Next Citation Point]. For the case of D0-branes, these follow.

  1. 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.
  2. It must reduce to N-copies of the particle action when the matrices XI are diagonal.
  3. 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.

The first was studied in [473Jump To The Next Citation Point, 474Jump To The Next Citation Point] and the second in [395Jump To The Next Citation Point]. 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. (475View Equation). Using the same notation introduced there and denoting the world volume direction along which one reduces by ρ, one learns that p F μρ → Dμ Φ, where p Φ is the T-dual adjoint matrix scalar. Furthermore, covariant derivatives of transverse scalar fields ΦI become

D ρΦI = ∂ρΦI + i[A ρ,ΦI] = i[A ρ,ΦI ]. (484 )
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 [395Jump To The Next Citation Point] 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-p 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 [395Jump To The Next Citation Point]
∫ ( ∘ -----------------------------------------------------) p+1 −ϕ −1 IJ I SDBI = − TDp d σ STr e − det(P [E μν + E μI(Q − δ) EJ ν] + λF μν) det(Q J) , (485 )
E μν = gμν + B μν , QI J ≡ δIJ + iλ [ΦI,ΦK ]EKJ , and λ = 2πα′. (486 )
Here μ,ν indices stand for world volume directions, and I,J indices for transverse directions. To deal with similar commutators arising from the WZ term, one considers [395Jump To The Next Citation Point]
∫ ( [ ] ) S = T STr P eiλiΦiΦ(∑ C (n)eB ) eλF , (487 ) WZ Dp
where the interior product i Φ is responsible for their appearance, for example, as in,
1 iΦiΦC2 = ΦJΦI C2IJ = -C2IJ [ΦJ ,ΦI]. (488 ) 2
Notice one regards ΦI as a vector field in the transverse space. In both actions (485View Equation) and (487View Equation), P 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.
  1. There exists some non-trivial dependence on the scalars ΦI through the arbitrary bosonic closed backgrounds appearing in the action. The latter is defined according to
    [ ] gμν = exp λΦi ∂xi g0μν(σa,xi)|xi=0 (489) ∞ n = ∑ λ--Φi1 ⋅⋅⋅Φin (∂ i ⋅⋅⋅∂ i)g0 (σa,xi)| i . n=0 n! x1 xn μν x=0
    Analogous definitions apply to other background fields.
  2. 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 ΦI dependence on the background fields (489View Equation) is symmetric, by definition.
  3. The WZ term (487View Equation) 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.
  4. There are different sources for the scalar potential: det QIJ, its inverse in the first determinant of the DBI and contributions coming from commutators coupling to background field components in the expansion (489View Equation).

It was shown in detail in [395Jump To The Next Citation Point], 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

∫ SWZ = μ0 Tr (P [C1 + iλ iΦiΦ (C3 + C1 ∧ B ) (490 ) λ2 ( 1 ) − --(iΦiΦ )2 C5 + C3 ∧ B + --C1 ∧ B ∧ B 2 2 λ3- 3 ( 1- 1- ) − i 6 (iΦiΦ) C7 + C5 ∧ B + 2C3 ∧ B ∧ B + 6C1 ∧ B ∧ B ∧ B 4 ( ( ) )]) + λ-(iΦiΦ )4 C9 + C7 + 1-C5 ∧ B + 1C3 ∧ B ∧ B + -1-C1 ∧ B ∧ B ∧ B ∧ B 24 2 6 24 ∫ ( λ ) = μ0 dtTr C1t + λC1IDt ΦI + i-(C3tJK [ΦK ,ΦJ ] + λ C3IJK Dt ΦI [ΦK ,ΦJ]) + ... . 2
Two points are worth emphasising about this matching:
  1. There is no ambiguity of trace in the linear Matrix theory calculations. Myers’ suggestion is to extend this prescription to non-linear couplings.
  2. 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 Tr ΦI = 0 but Tr (ΦI )2 ⁄= 0. As a toy illustrative example of this phenomenon, consider N D0-branes propagating in Minkowski but in a constant background RR four-form field strength

{ 4 − 2f𝜀IJK for I,J, K ∈ {1, 2,3} R tIJK = 0 otherwise . (491 )
Due to gauge invariance, one expects a coupling of the form
∫ ( ) iλ2μ0 dt Tr ΦI ΦJΦK R4tIJK (t). (492 ) 3
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
λ2T0 I J 2 i 2 ( I J K) 4 V (Φ ) = − ----Tr ([Φ ,Φ ] ) − -λ μ0Tr Φ Φ Φ R tIJK (t) , (493 ) 4 3
whose extremisation condition becomes
0 = [[ΦI ,ΦJ ],ΦK ] + if𝜀 [ΦJ ,ΦK ]. (494 ) IJK
The latter allows SU (2 ) solutions
I f- I I J K Φ = 2 α with [α ,α ] = 2i𝜀IJK α , (495 )
having lower energy than standard commuting matrices
π2ℓ3f4 VN = − ---s--N (N 2 − 1). (496 ) 6gs

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 VN up to 2 1∕N 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 AdS5 × S5 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].

Kappa symmetry and superembeddings:
The covariant results discussed above did not include fermions. Whenever these were included in the abelian case, a further gauge symmetry was required, kappa symmetry, to keep covariance, manifest supersymmetry and describe the appropriate on-shell degrees of freedom. One suspects something similar may occur in the non-abelian case to reduce the number of fermionic degrees of freedom in a manifestly supersymmetric non-abelian formulation. It is important to stress that at this point world volume diffeomorphisms and kappa symmetry will no longer appear together. In all the discussions in this section, world volume diffeomorphisms are assumed to be fixed, in the sense that the only scalar adjoint matrices already correspond to the transverse directions to the brane. Given the projective nature of kappa symmetry transformations, it may be natural to assume that there should be as many kappa symmetries as fermions. In [79], a perturbative approach to determining such transformation

( ) δ ¯ðœƒA = ¯κB (σ ) 𝟙δBA + Γ BA (σ) , A, B = 1,2, ...N 2 (497 ) κ
was analysed for multiple D-branes in super-Poincaré. The idea was to expand the WZ term in covariant derivatives of the fermions and the gauge field strength F, involving some a priori arbitrary tensors. One then computes its kappa symmetry variation and attempts to identify the DBI term in the action at the same order by satisfying the requirement that the total action variation equals
¯ δκℒ = − δκ𝜃 (1 − Γ )𝒯 , (498 )
order by order. In a sense, one is following the same strategy as in [9], determining the different unknown tensors order by order. Unfortunately, it was later concluded in [76] that such an approach could not work.

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 [460Jump To The Next Citation Point], 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.

Relation to non-commutative geometry:
There are at least two reasons why one may expect non-commutative geometry to be related to the description of multiple D-brane actions:

  1. D-brane transverse coordinates being replaced by matrices,
  2. the existent non-commutative geometry description of D-branes in the presence of a B-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

∞ 𝒜 = C (M ) ⊗ MN (C ). (499 )
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 (𝒜, ℋ, D ), which, in addition to 𝒜, also contains a Hilbert space ℋ and a self-adjoint operator D obeying certain properties. It would be interesting to find triples (𝒜, ℋ, D ) 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 B-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 N → ∞ [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.

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