3.2 Bosonic actions

After the identification of the relevant degrees of freedom and gauge symmetries governing brane effective actions, I focus on the construction of their bosonic truncations, postponing their supersymmetric extensions to Sections 3.4 and 3.5. The main goal below will be to couple brane degrees of freedom to arbitrary curved backgrounds in a world volume diffeomorphic invariant way.

I shall proceed in order of increasing complexity, starting with the M2-brane effective action, which is purely geometric, continuing with D-branes and their one form gauge potentials and finishing with M5-branes including their self-dual three form field strength13.

Bosonic M2-brane:
In the absence of world volume gauge field excitations, all brane effective actions must satisfy two physical requirements

  1. Geometrically, branes are p + 1 hypersurfaces Σp+1 propagating in a fixed background with metric g mn. Thus, their effective actions should account for their world volumes.
  2. Physically, all branes are electrically charged under some appropriate spacetime p + 1 gauge form Cp+1. Thus, their effective actions should contain a minimal coupling accounting for the brane charges.

Both requirements extend the existent effective action describing either a charged particle (p = 0) or a string (p = 1). Thus, the universal description of the purely scalar field m X brane degrees of freedom must be of the form

∫ ∫ S = − T dp+1σ √−--det𝒒-+ Q π’ž , (35 ) p p Σp+1 p Σp+1 p+1
where Tp and Qp stand for the brane tension and charge density14. The first term computes the brane world volume from the induced metric 𝒒μν
𝒒 = ∂ Xm ∂ Xng (X ), (36 ) μν μ ν mn
whereas the second WZ term π’žp+1 describes the pullback of the target space p + 1 gauge field Cp+1 (X ) under which the brane is charged
---1----μ1...μp+1 m1 mp+1 π’ž(p+1) = (p + 1)!πœ– ∂μ1X ...∂μp+1X Cm1...mp+1 (X ). (37 )
At this stage, one assumes all branes propagate in a background with Lorentzian metric gmn(X ) coupled to other matter fields, such as Cp+1 (X ), whose dynamics are neglected in this approximation. In string theory, these background fields correspond to the bosonic truncation of the supergravity multiplet and their dynamics at low energy is governed by supergravity theories. More precisely, M2 and M5-branes propagate in d = 11 supergravity backgrounds, i.e., m, n = 0,1,...10, and they are electrically charged under the gauge potential A3(X ) and its six-form dual potential A6, respectively (see Appendix A for conventions). D-branes propagate in d = 10 type IIA/B backgrounds and the set {Cp+1 (X )} correspond to the set of RR gauge potentials in these theories, see Eq. (521View Equation).

The relevance of the minimal charge coupling can be understood by considering the full effective action involving both brane and gravitational degrees of freedom (17View Equation). Restricting ourselves to the kinetic term for the target space gauge field, i.e., R = dCp+1, the combined action can be written as

∫ ( 1 ) -R ∧ ⋆R + Qp ˆn ∧ Cp+1 . (38 ) β„³D 2
Here β„³D stands for the D-dimensional spacetime, whereas ˆn is a (D − p − 1)-form whose components are those of an epsilon tensor normal to the brane having a δ-function support on the world volume15. Thus, the bulk equation of motion for the gauge potential Cp+1 acquires a source term whenever a brane exists. Since the brane charge is computed as the integral of ⋆R over any topological (D − p − 2)-sphere surrounding it, one obtains
∫ ∫ ∫ ⋆R = d ⋆ R = Q ˆn = Q , (39 ) ΣD −p− 2 BD−p−1 BD −p−1 p p
where the equation of motion was used in the last step. Thus, minimal WZ couplings do capture the brane physical charge.

Since M2-branes do not involve any gauge field degree of freedom, the above discussion covers all its bosonic degrees of freedom. Thus, one expects its bosonic effective action to be

∫ -------- ∫ S = − T d3σ √ − det 𝒒 + Q π’œ , (40 ) M2 M2 M2 3
in analogy with the bosonic worldsheet string action. If Eq. (40View Equation) is viewed as the bosonic truncation of a supersymmetric M2-brane, then |QM2 | = TM2. Besides its manifest spacetime covariance and its invariance under world-volume diffeomorphisms infinitesimally generated by
δξXm = β„’ ξXm = ξμ∂ μXm, (41 )
this action is also quasi-invariant (invariant up to total derivatives) under the target space gauge transformation δΛA3 = dΛ2 leaving 𝒩 = 1 d = 11 supergravity invariant, as reviewed in Eq. (551View Equation) of Appendix A.2. This is reassuring given that the full string theory effective action (17View Equation) describing both gravity and brane degrees of freedom involves both actions.

Bosonic D-branes:
Due to the perturbative description in terms of open strings [423Jump To The Next Citation Point], D-brane effective actions can, in principle, be determined by explicit calculation of appropriate open string disk amplitudes. Let me first discuss the dependence on gauge fields in these actions. Early bosonic open string calculations in background gauge fields [1Jump To The Next Citation Point], allowed to determine the effective action for the gauge field, with purely Dirichlet boundary conditions [214Jump To The Next Citation Point] or with mixed boundary conditions [354], gave rise to a non-linear generalisation of Maxwell’s electromagnetism originally proposed by Born and Infeld in [108]:

∫ p+1 ∘ ----------------′---- − Σ d σ − det(ημν + 2πα F μν). (42 ) p+1
I will refer to this non-linear action as the Dirac–Born–Infeld (DBI) action. Notice, this is an exceptional situation in string theory in which an infinite sum of different α′ contributions is analytically computable. This effective action ignores any contribution from the derivatives of the field strength F, i.e., ∂μF νρ terms or higher derivative operators. Importantly, it was shown in [1Jump To The Next Citation Point] that the first such corrections, for the bosonic open string, are compatible with the DBI structure. Having identified the non-linear gauge field dependence, one is in a position to include the dependence on the embedding scalar fields Xm (σ) and the coupling with non-trivial background closed string fields. Since in the absence of world-volume gauge-field excitations, D-brane actions should reduce to Eq. (35View Equation), it is natural to infer the right answer should involve
∘ --------------------- − det(𝒒μν + 2π α′Fμν), (43 )
using the general arguments of the preceding paragraphs. Notice, this action does not include any contribution from acceleration and higher derivative operators involving scalar fields, i.e., ∂μνXm terms and/or higher derivative terms.16 This proposal has nice properties under T-duality [24Jump To The Next Citation Point, 77Jump To The Next Citation Point, 16Jump To The Next Citation Point, 75Jump To The Next Citation Point], which I will explore in detail in Section 3.3.2 as a non-trivial check on Eq. (43View Equation). In particular, it will be checked that absence of acceleration terms is compatible with T-duality.

The DBI action is a natural extension of the NG action for branes, but it does not capture all the relevant physics, even in the absence of acceleration terms, since it misses important background couplings, responsible for the WZ terms appearing for strings and M2-branes. Let me stress the two main issues separately:

  1. The functional dependence on the gauge field V 1 in a general closed string background. D-branes are hypersurfaces where open strings can end. Thus, open strings do have endpoints. This means that the WZ term describing such open strings is not invariant under the target space gauge transformation δB2 = dΛ1
    ∫ ∫ ∫ δ Σ2 b = Σ2 d Λ = ∂Σ2 Λ, (44)
    due to the presence of boundaries. These are the D-branes themselves, which see these endpoints as charge point sources. The latter has a minimal coupling of the form ∫ ∂Σ2 V1, whose variation cancels Eq. (44View Equation) if the gauge field transforms as δV1 = dXm (σ )Λm under the bulk gauge transformation. Since D-brane effective actions must be invariant under these target space gauge symmetries, this physical argument determines that all the dependence on the gauge field V1 should be through the gauge invariant combination β„± = 2πα ′dV1 − ℬ.
  2. The coupling to the dilaton. The D-brane effective action is an open string tree level action, i.e., the self-interactions of open strings and their couplings to closed string fields come from conformal field theory disk amplitudes. Thus, the brane tension should include a − 1 gs factor coming from the expectation value of the closed string dilaton − Ο• e. Both these considerations lead us to consider the DBI action
    ∫ ∘ -------------- S = − T dp+1σ e−Ο• − det(𝒒 + β„± ), (45) DBI Dp
    where T Dp stands for the D-brane tension.
  3. The WZ couplings. Dp-branes are charged under the RR potential Cp+1. Thus, their effective actions should include a minimal coupling to the pullback of such form. Such coupling would not be invariant under the target space gauge transformations (525View Equation). To achieve this invariance in a way compatible with the bulk Bianchi identities (523View Equation), the D-brane WZ action must be of the form
    ∫ π’ž ∧ e β„±, (46) Σp+1
    where π’ž stands for the corresponding pullbacks of the target space RR potentials Cr to the world volume, according to the definition given in Eq. (521View Equation). Notice this involves more terms than the mere minimal coupling to the bulk RR potential Cp+1. An important physical consequence of this fact will be that turning on non-trivial gauge fluxes on the brane can induce non-trivial lower-dimensional D-brane charges, extending the argument given above for the minimal coupling [185Jump To The Next Citation Point]. This property will be discussed in more detail in the second part of this review. For a discussion on how to extend these couplings to massive type IIA supergravity, see [255Jump To The Next Citation Point].

Putting together all previous arguments, one concludes the final form of the bosonic D-brane action is:17

∫ ∘ -------------- ∫ S = − T dp+1σ e−Ο• − det(𝒒 + β„±) + Q π’ž ∧ eβ„±. (47 ) Dp Dp Σp+1 Dp Σp+1
If one views this action as the bosonic truncation of a supersymmetric D-brane, the D-brane charge density equals its tension in absolute value, i.e., |QDp | = TDp. The latter can be determined from first principles to be [423Jump To The Next Citation Point, 24Jump To The Next Citation Point]
---1-- ----1---- TDp = g √ α ′(2 π√ α′)p. (48 ) s

Bosonic covariant M5-brane:
The bosonic M5-brane degrees of freedom involve scalar fields and a world volume 2-form with self-dual field strength. The former are expected to be described by similar arguments to the ones presented above. The situation with the latter is more problematic given the tension between Lorentz covariance and the self-duality constraint. This problem has a fairly long history, starting with electromagnetic duality and the Dirac monopole problem in Maxwell theory, see [105] and references therein, and more recently, in connection with the formulation of supergravity theories such as type IIB, with the self-duality of the field strength of the RR 4-form gauge potential. There are several solutions in the literature based on different formalisms:

  1. One natural option is to give-up Lorentz covariance and work with non-manifestly Lorentz invariant actions. This was the approach followed in [420Jump To The Next Citation Point] for the M5-brane, building on previous work [213, 295, 441].
  2. One can introduce an infinite number of auxiliary (non-dynamical) fields to achieve a covariant formulation. This is the approach followed in [384, 502, 375, 177, 66, 98, 99, 100].
  3. One can follow the covariant approach due to Pasti, Sorokin and Tonin (PST-formalism) [416Jump To The Next Citation Point, 418Jump To The Next Citation Point], in which a single auxiliary field is introduced in the action with a non-trivial non-polynomial dependence on it. The resulting action has extra gauge symmetries. These allow one to recover the structure in [420Jump To The Next Citation Point] as a gauge fixed version of the PST formalism.
  4. Another option is to work with a Lagrangian that does not imply the self-duality condition but allows it, leaving the implementation of this condition to the path integral. This is the approach followed by Witten [497], which was extended to include non-linear interactions in [140]. The latter work includes kappa symmetry and a proof that their formalism is equivalent to the PST one.

In this review, I follow the PST formalism. This assigns the following bosonic action to the M5-brane [417Jump To The Next Citation Point]

∫ ( ∘ ----------------- ) S = − T d6σ − det(𝒒 + &tidle;H ) − √ −-det-𝒒---1----∂ a(σ)β„‹ ∗μνδβ„‹ ∂ ρa(σ) M5 M5 μν μν 4∂μa∂μa δ μνρ ∫ ( 1 ) +TM5 π’œ6 + --β„‹3 ∧ π’œ3 . (49 ) 2
As in previous effective actions, all the dependence on the scalar fields Xm is through the bulk fields and their pullbacks to the six-dimensional world volume. As in D-brane physics, all the dependence on the world volume gauge potential V2 is not just simply through its field strength dV2, but through the gauge invariant 3-form
β„‹ = dV − π’œ . (50 ) 3 2 3
The physics behind this is analogous. β„± describes the ability of open strings to end on D-branes, whereas β„‹3 describes the possibility of M2-branes to end on M5-branes [469, 479]18. Its world volume Hodge dual and the tensor &tidle; H μν are then defined as
β„‹ ∗μνρ = -√--1-----πœ€μνρα1α2α3β„‹ α α α , (51 ) 6 − det𝒒 1 2 3 1 ∗ ρ &tidle;H μν = ∘-------β„‹ μνρ∂ a(σ ). (52 ) |(∂a )2|
The latter involves an auxiliary field a(σ) responsible for keeping covariance and implementing the self-duality constraint through the second term in the action (49View Equation). Its auxiliary nature was proven in [418, 416], where it was shown that its equation of motion is not independent from the generalised self-duality condition. The full action also includes a DBI-like term, involving the induced world volume metric 𝒒 = ∂ Xm ∂ Xng (X ) μν μ ν mn, and a WZ term, involving the pullbacks π’œ3 and π’œ6 of the 3-form gauge potential and its Hodge dual in 𝒩 = 1 d = 11 supergravity [11].

Besides being manifestly invariant under six-dimensional world volume diffeomorphisms and ordinary abelian gauge transformations δV2 = dΛ1, the action (49View Equation) is also invariant under the transformation

( ) Λ(σ ) δβ„’ ∂ ρa δa(σ) = Λ (σ ), δV μν = ∘--------(2 ---DBI− (dV2)μνρ∘-------) . (53 ) |(∂a )2| δH&tidle;μν |(∂a )2|
Given the non-dynamical nature of a(σ ), one can always fully remove it from the classical action by gauge fixing the symmetry (53View Equation). It was shown in [417Jump To The Next Citation Point] that for an M5-brane propagating in Minkowski, the non-manifest Lorentz invariant formulation in [420Jump To The Next Citation Point] emerges after gauge fixing (53View Equation). This was achieved by working in the gauge ∂ a(σ) = δ5 μ μ and V = 0 μ5. Since ∂ a μ is a world volume vector, six-dimensional Lorentz transformations do not preserve this gauge slice. One must use a compensating gauge transformation (53View Equation), which also acts on Vμν. The overall gauge fixed action is invariant under the full six-dimensional Lorentz group but in a non-linear non-manifestly Lorentz covariant way as discussed in [420Jump To The Next Citation Point].

As a final remark, notice the charge density Q M5 of the bosonic M5-brane has already been set equal to its tension 5 6 TM5 = 1βˆ•(2π) β„“p.


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