3.3 Consistency checks

The purpose of this section is to check the consistency of the kinematic structures governing classical bosonic brane effective actions with string dualities [312, 495]. At the level of supergravity, these dualities are responsible for the existence of a non-trivial web of relations among their classical Lagrangians. Here, I describe the realisation of some of these dualities on classical bosonic brane actions. This will allow us to check the consistency of all brane couplings. Alternatively, one can also view the discussions below as independent ways of deriving the latter.

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 3View Image summarises the set of relations between the brane tensions discussed in this review under these symmetries.

View Image

Figure 3: Set of half-BPS branes discussed in this review, their tensions and some of their connections under T-duality and the strongly-coupled limit of type IIA.

M-theory as the strong coupling limit of type IIA:
From the spectrum of 1/2-BPS states in string theory and M-theory, an M2/M5-brane in ℝ1,9 × S1 has a weakly-coupled description in type IIA

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 d = 11 bosonic supergravity fields and the type IIA bosonic ones summarised below [409]

2 − 2ϕ 2 4ϕ 2 ds11 = e 3 ds 10 + e3 (dy + C1) , A3 = C3 + dy ∧ B2 , (54 )
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 S1. In terms of the parameters of the theory, the relation between the M-theory circle R and the 11-dimensional Planck scale ℓp with the type IIA string coupling gs and string length ℓs is
R = gsℓs, ℓ3p = gsℓ3s. (55 )

The same principle should hold for the brane degrees of freedom {ΦA }. 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:

T-duality on closed and open strings:
From the quantisation of open strings satisfying Dirichlet boundary conditions, all D-brane dynamics are described by a massless vector supermultiplet, whose number of scalar fields depends on the number of transverse dimensions to the D-brane. Since D-brane states are mapped among themselves under T-duality [160, 424Jump To The Next Citation Point], one expects the existence of a transformation mapping their classical effective actions under this duality. The question is how such transformation acts on the action. This involves two parts: the transformation of the background and the one of the brane degrees of freedom. Let me focus on the bulk transformation. T-duality is a perturbative string theory duality [241Jump To The Next Citation Point]. It says that type IIA string theory on a circle of radius R and string coupling gs is equivalent to type IIB on a dual circle of radius R ′ and string coupling g′ s related as [121, 122, 240]

α ′ √α-′ R ′ =-- , g′s = gs----, (56 ) R R
when momentum and winding modes are exchanged in both theories. This leaves the free theory spectrum invariant [337], but it has been shown to be an exact perturbative symmetry when including interactions [400, 241]. Despite its stringy nature, there exists a clean field theoretical realisation of this symmetry. The main point is that any field theory on a circle of radius R has a discrete momentum spectrum. Thus, in the limit R → 0, all non-vanishing momentum modes decouple, and one only keeps the original vanishing momentum sector. Notice this is effectively implementing a KK compactification on this circle. This is in contrast with the stringy realisation where in the same limit, the spectrum of winding modes opens up a dual circle of radius ′ R.

Since Type IIA and Type IIB supergravities are field theories, the above field theoretical realisation applies. Thus, the R → 0 compactification limit should give rise to two separate 𝒩 = 2 d = 9 supergravity theories. But it is known [388Jump To The Next Citation Point] that there is just such a unique supergravity theory. In other words, given the type IIA/B field content {φA ∕B} and their KK reduction to d = 9 dimensions, i.e., φA = φA (φ9) and φB = φB (φ9), the uniqueness of 𝒩 = 2 d = 9 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

φA = φA (φB ). (57 )
View Image

Figure 4: Schematic diagram describing the derivation of Buscher’s T-duality rules using type IIA/IIB supergravities.

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

--1- gzz = g′′′ z z ϕ = ϕ′ − 1-log |g′′ ′| 2 zz g′nz′ Bnz = − g′′′ zz′ g = − B-nz′ nz g′z′z′ g′ ′g′′ − B ′ ′B ′ ′ gmn = g′mn − -mz--nz-′---mz--nz- gz′z′ ′ B-′mz′g′nz′ −-B-′nz′g′mz′ Bmn = B mn − g′′′ zz (p+1) ′(p) C′[(mp)1...mp− 1z′g′mp]z′ Cm1...mpz = C m1...mp − p -------′--------- gz′z′ C (mp1)...mp = C ′(mp+..1.m)pz′ − pC[′(mp−.1)..m B′mp ]z′ 1 ′(p−11) p−1 C [m1...mp−2z′B ′mp−1z′g′mp ]z′ − p(p − 1)-----------′------------. (58 ) gz′z′
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 {z, z′} describe both T-dual circles. Notice the dilaton and the g zz transformations do capture the worldsheet relations (56View Equation).

Let me move to the brane transformation. A D(p + 1)-brane wrapping the original circle is mapped under T-duality to a Dp-brane where the dual circle is transverse to the brane [424Jump To The Next Citation Point]. 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 R → 0 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 {φ9 } will be rewritten in terms of the T-dual ten-dimensional fields using the T-duality rules (58View Equation).

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 [483Jump To The Next Citation Point, 252].

3.3.1 M2-branes and their classical reductions

In the following, I discuss the double and direct dimensional reductions of the bosonic M2-brane effective action (40View Equation) to match the bosonic worldsheet string action (6View Equation) and the D2-brane effective action, i.e., the p = 2 version of Eq. (47View Equation). This analysis will also allow us to match/derive the tensions of the different branes.

Connection to the string worldsheet:
Consider the propagation of an M2-brane in an 11-dimensional background of the form (54View Equation). Decompose the set of scalar fields as {XM } = {Xm, Y }, identify one of the world volume directions (ρ) with the KK circle, i.e., partially gauge fix the world volume diffeomorphisms by imposing Y = ρ, and keep the zero modes in the Fourier expansion of all remaining scalar fields {Xm } along the world volume circle, i.e., ∂ Xm = 0 ρ. Under these conditions, which mathematically characterise a double dimensional reduction, the Wess–Zumino coupling becomes

∫ ∫ 3 1 ˆμˆνρ m n (∫ )∫ 𝒜3 = d σ--𝜖 ∂ˆμX ∂ ˆνX Amny = d ρ ℬ2, (59 ) Σ2×S1 Σ2×S1 2 S1 Σ2
where I already used the KK reduction ansatz (54View Equation). Here, ℬ2 stands for the pull-back of the NS-NS two form into the surface Σ2 parameterised by {σ ˆμ}. The DBI action is reduced using the identity satisfied by the induced world-volume metric
( −2ϕ∕3 2ϕ 4ϕ∕3 ) 𝒢μν = e (𝒢 ˆμ4ˆνϕ +∕3 e 𝒞 ˆμ𝒞ˆν) e 4ϕ𝒞∕3ˆμ = ⇒ det𝒢 μν = det𝒢 ˆμˆν . (60 ) e 𝒞 ˆν e
Since the integral over ρ equals the length of the M-theory circle,
∫ ∫ 1 dρ = 2πR = 2πgsls =⇒ Tf = TM2 d ρ = ----′, (61 ) S1 S1 2πα
where I used Eq. (55View Equation), TM2 = 1 ∕(2π)2l3p and absorbed the overall circle length, expressed in terms of type IIA data, in a new energy density scale, matching the fundamental string tension T f defined in Section 2. The same argument applies to the charge density leading to Qf = QM2 2πR. Altogether, the double reduced action reproduces the bosonic effective action (6View Equation) describing the string propagation in a type IIA background. Thus, our classical bosonic M2-brane action is consistent with the relation between half-BPS M2-brane and fundamental strings in the spectrum of M-theory and type IIA.

Connection to the D2-brane:
The direct dimensional reduction of the bosonic M2 brane describes a three-dimensional diffeomorphism invariant theory propagating in 10 dimensions, with eleven scalars as its field content. The latter disagrees with the bosonic field content of a D2-brane, which includes a vector field. Fortunately, a scalar field is Hodge dual, in three dimensions, to a one form. Thus, one expects that by direct dimensional reduction of the bosonic M2-brane action and after world volume dualisation of the scalar field Y along the M-theory circle, one should reproduce the classical D2-brane action [439Jump To The Next Citation Point, 477Jump To The Next Citation Point, 93Jump To The Next Citation Point, 480Jump To The Next Citation Point]. To describe the direct dimensional reduction, consider the Lagrangian [480Jump To The Next Citation Point]

∫ ( ) S = TM2- d3σ v−1 det𝒢 (11)− v + 1𝜖μνρ𝒜 . (62 ) 2 μν 3 μνρ
This is classically equivalent to Eq. (40View Equation) after integrating out the auxiliary scalar density v by solving its algebraic equation of motion. Notice I already focused on the relevant case for later supersymmetric considerations, i.e., QM2 = TM2. The induced world volume fields are
𝒢(11) = e− 23ϕ𝒢 + e43ϕZ Z (63 ) μν μν μ ν 𝒜μνρ = 𝒞μνρ + 3ℬ [μνZ ρ] − 3ℬ[μν𝒞ρ], (64 )
Z ≡ dY + 𝒞 . (65 ) 1
Using the properties of 3 × 3 matrices,
(11) −2ϕ 2ϕ −2ϕ 2 det𝒢μν = e det[𝒢μν + e ZμZ ν] = (det 𝒢μν)[e + |Z |], (66 )
where |Z |2 = ZμZ ν𝒢μν, the action (62View Equation) can be written as
T ∫ ( 1 S = -M2- d3σ v−1e− 2ϕ det𝒢 μν − v + -𝜖μνρ[𝒞μνρ − 3ℬ μν𝒞ρ] 2 3) + v−1(det 𝒢μν)|Z|2 + 𝜖μνρℬ μνZρ . (67 )

The next step is to describe the world volume dualisation and the origin of the U (1) gauge symmetry on the D2 brane effective action [480Jump To The Next Citation Point]. By definition, the identity

d(Z − 𝒞1) ≡ 0 (68 )
holds. Adding the latter to the action through an exact two-form F = dV Lagrange multiplier
1 ∫ − --- F ∧ (Z − 𝒞1), (69 ) 2π
allows one to treat Z as an independent field. For a more thorough discussion on this point and the nature of the U (1 ) gauge symmetry, see [480]. Adding Eq. (69View Equation) to Eq. (67View Equation), one obtains
∫ ( S = TM22 d3σ v−1e− 2ϕ det𝒢 μν − v + 13𝜖μνρ[𝒞μνρ + 3ℱμν𝒞 ρ] −1 2 μνρ + v (det𝒢 μν)|Z | − 𝜖 ℱ μνZρ). (70 )
Notice I already introduced the same gauge invariant quantity introduced in D-brane Lagrangians
ℱμν = F μν − ℬ μν. (71 )
Since Y is now an independent field, it can be eliminated by solving its algebraic equation of motion
Zμ = --v---𝜖μνρℱ . (72 ) 2det 𝒢 μν
Inserting this back into the action and integrating out the auxiliary field &tidle;v = − det(𝒢 )∕v μν by solving its equation of motion, yields
∫ ∘ ----------------- ∫ S = − TD2 d3 σe− ϕ − det(𝒢μν + ℱμν) + TD2 (𝒞3 + ℱ ∧ 𝒞1 ). (73 ) w
This matches the proposed D2-brane effective action, since TM2 = TD2 as a consequence of Eq.s (55View Equation) and (48View Equation).

3.3.2 T-duality covariance

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 (58View Equation) guarantee the covariance of the D-brane effective action functional form [453Jump To The Next Citation Point] 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 as

sXM = ξν∂νXM + ΔXM , (74 ) ν ν s Vμ = ξ ∂νV μ + V ν∂μξ + ∂μc + ΔV μ. (75 )
They involve world volume diffeomorphisms ξν, a U(1) gauge transformation c and global transformations Δ ϕi. Since the background will undergo a T-duality transformation, by assumption, this set of global transformations must include translations along the circle, i.e., ΔZ = 𝜖, m ΔX = ΔV μ = 0, where the original M X scalar fields were split into m {X ,Z }.

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 Z = σp ≡ ρ, identifying one world volume direction with the starting S1 bulk circle and a zero-mode Fourier truncation in the remaining degrees of freedom, m ∂ ρX = ∂ρVμ = 0. Extending this functional truncation to the p-dimensional diffeomorphisms ξ ˆμ, where I split the world volume indices according to {μ } = {ˆμ,ρ } and the space of global transformations, i.e., ∂zΔxM = ∂zΔV μ = 0, 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., i ∂zsϕ |g.f.+trunc= 0, determines

ˆμ μˆ 𝜖′ c(σ ,ρ) = &tidle;c(σ ) + a + 2π-α′ρ (76 )
where ′ a,𝜖 are constants, the latter having mass dimension minus one. The set of transformations in the double dimensional reduction are
&tidle;sXm = ξˆν∂ Xm + &tidle;ΔXm (77 ) ˆν ˆν ˆν &tidle;sVˆμ = ξ ∂ˆνV ˆμ + V ˆν∂ˆμξ + ∂ˆμ&tidle;c + &tidle;ΔV ˆμ (78 ) &tidle;sVρ = ξˆν∂ˆνV ρ + Δ&tidle;V ρ (79 )
where &tidle;ΔV ˆμ = ΔV ˆμ − Vρ∂μˆΔZ, &tidle;ΔV ρ = ΔV ρ + 𝜖′∕2π α′ and &tidle;Δxm satisfies ∂z &tidle;Δxm = 0.

Let me comment on Eq. (79View Equation). Vρ 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 ′ 𝜖 in the original U(1) transformation (76View Equation), describes a global translation along the scalar direction. The interpretation of both observations is that under double-dimensional reduction

(2πα ′) Vρ ≡ Z ′ (80 )
Z ′ becomes the T-dual target space direction along the T-dual circle and 𝜖′ 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 𝒢 and the gauge invariant ℱ in terms of the T-dual background ′ ′ (g , B ) and degrees of freedom ′ ′ ({XM } = {Xm , Z ′}), which will be denoted by primed quantities. This can be achieved by adding and subtracting the relevant pullback quantities. The following identities hold

𝒢 ˆμρ = ∂ ˆμXmgmz (81 ) 𝒢 ρρ = gzz (82 ) 𝒢μˆνˆ= 𝒢 ′ˆμˆν + ∂ ˆμXm ∂νˆXn (gmn − g′mn) − ∂ˆμXm ∂ˆνZ ′g′z′m ′ M ′′ − ∂ˆμZ ∂ˆνX gz′M′ (83 ) ℱ ˆμρ = ∂ ˆμZ′ − ∂ˆμXmBmz (84 ) ′ m n ′ ′ n ′ ℱμˆνˆ= ℱ ˆμˆν − ∂ˆμX ∂ˆνX (Bmn − Bmn ) + ∂ˆμZ ∂ˆνX B z′n + ∂ˆμXm ∂ˆνZ ′B ′mz′ . (85 )
It is a consequence of our previous symmetry discussion that m′ m X = X and ′ Vˆμ = V ˆμ, 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
( ′ ′ det(𝒢μν + ℱ μν) = gzz det 𝒢μˆνˆ+ ℱ ˆμˆν + ∂ˆμXm ∂ˆνXn [(gmn − g′mn ) − (Bmn − B ′mn ) − (gmz − Bmz )(gnz + Bnz )∕gzz] − ∂ Xm ∂ Z ′[(g′ ′ − B ′ ′) − (g − B )∕g ] ˆμ ′ ˆν n m′z ′mz mz mz zz − ∂ˆμZ ∂ˆνX [(gz′n + B nz′) + (gnz + Bnz )∕gzz] ( 1 ) ) − ∂ˆμZ ′∂νˆZ ′ g ′z′z′ −--- . (86 ) gzz
Notice that whenever the bulk T-duality rules (58View Equation) are satisfied, the functional form of the effective action remains covariant, i.e., of the form
∫ ′ ∘----------------- − T ′D(p− 1) dpσ e−ϕ − det(𝒢′ˆμˆν + ℱ ′ˆμˆν). (87 )
This is because all terms in the determinant vanish except for those in the first line. Finally, √ --- e−ϕ gzz equals the T-dual dilaton coupling e−ϕ′ and the original Dp-brane tension TDp becomes the D(p-1)-brane tension in the T-dual theory due to the worldsheet defining properties (56View Equation) after the integration over the world volume circle
∫ TDp d ρ = -----1------2π R = -----1----p-= TD′(p−1). (88 ) (2π)p gslps+1 (2 π)p−1g′sls

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, 453Jump To The Next Citation Point]. First, decompose the WZ Lagrangian density as

+ − ℒW Z = ℒ W Z + ℒW Z ≡ dρ ∧ i∂ρℒW Z + i∂ρ(dρ ∧ ℒW Z) . (89 )
Due to the functional truncation assumed in the double dimensional reduction, the second term vanishes. The D-brane WZ action then becomes
∫ ∫ ℱ − ( − ) Tp Σp+1 ℒW Z = Tp Σp+1 dρ ∧ e ∧ i∂ρC + i∂ρℱ ∧ C (90 )
where − ℱ ≡ i∂ρ(d ρ ∧ ℱ ) and the following conventions are used
1 i∂ρΩ (n) = -------Ω ρμ2...μndσ μ2 ∧ ...dσμn (n − 1)! i∂ρ(Ω(m) ∧ Ω(n)) = i∂ρΩ(m ) ∧ Ω(n) + (− 1)m Ω(m) ∧ i∂ρΩ (n) . (91 )
Using the T-duality transformation properties of the gauge invariant quantity ℱ, derived from our DBI analysis,
( ) ℱ − −→ ℱ ′ − i∂z′B ′ ∧ i∂z′g′ ∕g′z′z′ (92 ) ′ ′ i∂ρℱ −→ − i∂z′g∕gz′z′ (93 )
it was shown in [453] that the functional form of the WZ term is preserved, i.e., T ′ ∫ eℱ ′ ∧ C′ D(p−1) ∂Σ, whenever the condition
′ ′ ′ (− 1)pC ′= i∂ Cp+1 − i∂z′B--∧-i∂z′g-∧ i∂ Cp−1 − i∂z′g- ∧ C− (94 ) p ρ g′z′z′ ρ g′z′z′ p−1
holds (the factor p (− 1) is due to our conventions (91View Equation) and the choice of orientation &tidle;μ1...&tidle;μp μ1...μpρ 𝜖 ≡ 𝜖 and 01...p 𝜖 = 1).

Due to our gauge-fixing condition, Z = ρ, the ± components of the pullbacked world volume forms appearing in Eq. (94View Equation) can be lifted to ± components of the spacetime forms. The condition (94View Equation) is then solved by

( ) p ′ i∂z′g′ ′ i∂zCp+1 = (− 1) C (p) − --′-- ∧ i∂z′Cp (95 ) ( gz′z′ ( ) ) − (p−1) ′ ′ ′ i∂z′g′ ′ C p−1 = (− 1) i∂z′Cp − i∂z′B ∧ C p−2 − g ′′ ′ ∧ i∂z′Cp−2 . (96 ) zz
These are entirely equivalent to the T-duality rules (58View Equation) 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 [208Jump To The Next Citation Point]. 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 C4 appearing in our WZ couplings is not S self-dual, but transforms as

C → C − C ∧ B . (97 ) 4 4 2 2
Using a superindex S to denote an S-dual self-dual 4-form, the latter must be
S 1- C 4 = C4 − 2C2 ∧ B2 . (98 )
Similarly, C6 does not transform as a doublet under S-duality, whereas
CS = C − 1-C ∧ B ∧ B (99 ) 6 6 4 2 2 2
does. It is straightforward to check that Eqs. (95View Equation) and (96View Equation) are equivalent to the ones appearing in [208Jump To The Next Citation Point] using the above redefinitions. Furthermore, one finds
′ CS = C ′ ′ − 6C ′ gm6-]z′ m1...m6 m1...m6z [m1...m5 g′z′z′ ( g′ ′) − 45 C ′[m − C ′z′-[m1z-- B ′m m B ′m m B ′m ]z′ 1 g′z′z′ 2 3 4 5 6 ( g′ ′) − 45C ′[m1m2z ′B ′m3m4 B ′m5m6 − 4B ′m5z′-m6′]z- gz′z′ ′ ′ g′m6 ]z′ − 30C [m1...m4z′Bm5z′--′--- (100 ) gz′z′
, which was not computed in [208Jump To The Next Citation Point].

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 N overlapping parallel D-branes in curved backgrounds, following [395Jump To The Next Citation Point].

3.3.3 M5-brane reduction

The double dimensional reduction of the M5-brane effective action, both in its covariant [417, 8] and non-covariant formulations [420Jump To The Next Citation Point, 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. (47View Equation), but in the dual formulation. The two are related through the world volume dualisation procedure described in [483, 7].

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