3.5 Supersymmetric brane effective actions in curved backgrounds

In this section, I extend the supersymmetric and kappa invariant D-brane and M2-brane actions in super-Poincaré to D-branes, M2-branes and M5-branes in arbitrary curved backgrounds. The main goal, besides introducing the formalism itself, is to highlight that the existence of kappa symmetry invariance forces the supergravity background to be on-shell.

In all effective actions under consideration, the set of degrees of freedom includes scalars ZM = {Xm, πœƒα} and it may include some gauge field Vp, whose dependence is always through the gauge invariant combination dVp − ℬp+122. The set of kappa symmetry transformations will universally be given by

δ ZM Ea (X, πœƒ) = 0 , κ M Mα- δκZ E M (X, πœƒ) = (πŸ™ + Γ κ)κ , δκVp = Z ⋆iκBp+1. (143 )
The last transformation is a generalisation of the one encountered in super-Poincaré. Indeed, the kappa variation of the pullback of any Tn n-form satisfies
δκ𝒯n ≡ δκZ ⋆Tn = Z ⋆ℒκTn = Z⋆{d, iκ}Tn, (144 )
where Z⋆ stands for the pullback of Tn to the world volume. The choice in Eq. (143View Equation) guarantees the kappa transformation of dV p removes the total derivative in δ ℬ κ p+1.

The structure of the transformations (143View Equation) is universal, but the details of the kappa symmetry matrix Γ κ depend on the specific theory, as described below. A second universal feature, associated with the projection nature of kappa symmetry transformations, i.e., Γ 2κ = πŸ™, is the correlation between the brane charge density and the sign of Γ κ in Eq. (143View Equation). More specifically, any brane effective action will have the structure

∫ p+1 Sbrane = − Tbrane d σ (β„’DBI − πœ–1β„’W Z ). (145 )
Notice this is equivalent to requiring Tbrane = |Qbrane|, a property that is just reflecting the half-BPS nature of these branes. It can be shown that
δκSbrane ∝ (1 + πœ–1Γ κ)δκπœƒ =⇒ δκ πœƒ = (1 − πœ–1Γ κ)κ. (146 )
The choice of πœ–1 is correlated to the distinction between a brane and an anti-brane. Both are supersymmetric, but preserve complementary supercharges. This ambiguity explains why some of the literature has apparently different conventions, besides the possibility of working with different Clifford algebra realisations23.

3.5.1 M2-branes

The effective action describing a single M2-brane in an arbitrary 11-dimensional background is formally the same as in Eq. (136View Equation)

∫ 3 ∘ --------- ∫ SM2 = − TM2 d σ − det𝒒 μν + TM2 π’œ3, (147 )
with the same definitions for the induced metric 𝒒 and the pull back 3-form π’œ3. The information regarding different 11-dimensional backgrounds is encoded in the different couplings described by the supervielbein EAM(X, πœƒ) and 3-form AABD (X, πœƒ) superfields.

The action (147View Equation) is manifestly 3d-diffeomorphism invariant. It was shown to be kappa invariant under the transformations (143View Equation), without any gauge field, whenever the background superfields satisfy the constraints reviewed in Appendix A.2, i.e., whenever they are on-shell, for a kappa symmetry matrix given by [90Jump To The Next Citation Point]

Γ = --√--1-----πœ€μνρEa(X, πœƒ) Eb(X, πœƒ)Ec (X, πœƒ)Γ , (148 ) κ 3! − det𝒒 μ ν ρ abc
where a m a Eμ (X, πœƒ) = ∂muX Em (X, πœƒ) is the pullback of the curved supervielbein to the world volume.

3.5.2 D-branes

Proceeding in an analogous way for Dp-branes, their effective action in an arbitrary type IIA/B background is

∫ ∘ -------------- ∫ p+1 − Ο• β„± SDp = − TDp d σe − det(𝒒 + β„± ) + TDp π’ž ∧ e , (149 ) ′ A C 1 μ...μ A A β„±μν = 2π α Fμν − E μ Eν BAC , π’žr = --πœ– 1 rEμ11...E μrr CA1...Ar. (150 ) r!
It is understood that a b 𝒒 μν(X,πœƒ ) = E μE νηab and π’ž is defined using the same notation as in Eq. (521View Equation), i.e., as a formal sum of forms, so that the WZ term picks all contributions coming from the wedge product of this sum and the Taylor expansion of eβ„± that saturate the p + 1 world volume dimension. Notice all information on the background spacetime is encoded in the superfields EA (X, πœƒ) M, Ο•(X, πœƒ), B (X, πœƒ) AC and the set of RR potentials {CA1...Ar(X, πœƒ)}.

The action (149View Equation) is p + 1 dimensional diffeomorphic invariant and it was shown to be kappa invariant under the transformations (143View Equation) for V1 in [141Jump To The Next Citation Point, 93Jump To The Next Citation Point] when the kappa symmetry matrix equals

(Γ ) = ∘------1--------∑ γ Γ l+1 ∧ eβ„± type IIA p = 2k, (151 ) κ(p+1) − det(𝒒 + β„± ) l=0 (2l+1) 11 (Γ ) = ∘------1--------∑ γ τ l∧ eβ„± iτ type IIB p = 2k + 1, (152 ) κ(p+1) − det(𝒒 + β„± ) l=0 (2l)3 2
and the background is on-shell, i.e., satisfies the constraints reviewed in Appendix A.1. In the expressions above γ (1) stands for the pullback of the bulk tangent space Clifford matrices
γ(1) = dσ μγμ = dσμEaμ(X, πœƒ)Γ a, (153 )
and γ(r) stands for the wedge product of r of these 1-forms. In [94Jump To The Next Citation Point], readers can find an extension of the results reviewed here when the background includes a mass parameter, i.e., it belongs to massive IIA [434].

3.5.3 M5-branes

The six-dimensional diffeomorphic and kappa symmetry invariant M5-brane [45] is a formal extension of the bosonic one

∫ SM5 = TM5 d6 ξ(β„’0 + β„’WZ ) , √-------- ∘ ----------------- --−--det𝒒-- ∗ μνρ ι β„’0 = − − det(𝒒 μν + H&tidle;μν) + 4(∂a ⋅ ∂a ),(∂μa )(β„‹ ) β„‹ νρι(∂ a) (154 ) β„’ = π’œ + 1-β„‹ ∧ π’œ , (155 ) WZ 6 2 3 3
where all pullbacks refer to superspace. This is kappa invariant under the transformations (143View Equation) for V2, including the extra transformation law
δ a = 0, (156 ) κ
for the auxiliary scalar field introduced in the PST formalism. These transformations are determined by the kappa symmetry matrix
μ [ √ -------- ] Γ = ∘------vμγ--------- γ tν + --−-det𝒒-γ νρH &tidle; − 1-πœ€ μ1...μ5νv γ , (157 ) κ − det(𝒒 + &tidle;H ) ν 2 νρ 5! ν μ1...μ5 μν μν
where γμ = E μaΓ a and the vector fields tμ and vμ are defined by
1 ∂ a tμ = --πœ€μν1ν2ρ1ρ2ιH &tidle;ν1ν2H &tidle;ρ1ρ2vι with vμ ≡ √---μ-----. (158 ) 8 − ∂a ⋅ ∂a

Further comments on kappa symmetry:
κ-symmetry is a fermionic local symmetry for which no gauge field is necessary. Besides its defining projective nature when acting on fermions, i.e., δ πœƒ = (πŸ™ + Γ ) κ κ κ with 2 Γ κ = πŸ™, there are two other distinctive features it satisfies [449Jump To The Next Citation Point]:

  1. the algebra of κ-transformations only closes on-shell,
  2. κ-symmetry is an infinitely reducible symmetry.

The latter statement uses the terminology of Batalin and Vilkovisky [52] and it is a direct consequence of its projective nature, since the existence of the infinite chain of transformations

κ → (1 − Γ )κ ,κ → (1 + Γ )κ ... (159 ) κ 1 1 κ 2
gives rise to an infinite tower of ghosts when attempting to follow the Batalin–Vilkovisky quantisation procedure, which is also suitable to handle the first remark above. Thus, covariant quantisation of kappa invariant actions is a subtle problem. For detailed discussions on problems arising from the regularisation of infinite sums and dealing with Stueckelberg type residual gauge symmetries, readers are referred to [326, 325, 254, 223, 84].

It was later realised, using the Hamiltonian formulation, that kappa symmetry does allow covariant quantisation provided the ground state of the theory is massive [327Jump To The Next Citation Point]. The latter is clearly consistent with the brane interpretation of these actions, by which these vacua capture the half-BPS nature of the (massive) branes themselves24.

For further interesting kinematical and geometrical aspects of kappa symmetry, see [449, 167, 166] and references therein.

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