3.4 Supersymmetric brane effective actions in Minkowski

In the study of global supersymmetric field theories, one learns the superfield formalism is the most manifest way of writing interacting manifestly-supersymmetric Lagrangians [491Jump To The Next Citation Point]. One extends the manifold ℝ1,3 to a supermanifold through the addition of Grassmann fermionic coordinates 𝜃. Physical fields ϕ(x) become components of superfields Φ (x,𝜃), the natural objects in this mathematical structure defined as finite polynomials in a Taylor-like 𝜃 expansion
Φ(x, 𝜃) = ϕ(x) + 𝜃αϕα(x ) + ...
that includes auxiliary (non-dynamical) components allowing one to close the supersymmetry algebra off-shell. Generic superfields do not transform irreducibly under the super-Poincaré group. Imposing constraints on them, i.e., fi(Φ) = 0, gives rise to the different irreducible supersymmetric representations. For a standard reference on these concepts, see [491Jump To The Next Citation Point].

These considerations also apply to the p + 1 dimensional supermultiplets describing the physical brane degrees of freedom propagating in 1,9 ℝ, since these correspond to supersymmetric field theories in 1,p ℝ. The main difference in the GS formulation of brane effective actions is that it is spacetime itself that must be formulated in a manifestly supersymmetric way. By the same argument used in global supersymmetric theories, one would be required to work in a 10- or 11-dimensional superspace, with standard bosonic coordinates xm and the addition of fermionic ones 𝜃, whose representations will depend on the dimension of the bosonic submanifold. There are two crucial points to appreciate for our purposes

  1. the superspace coordinates {xm, 𝜃} will become the brane dynamical degrees of freedom {Xm (σ ), 𝜃(σ)}, besides any additional gauge fields living on the brane;
  2. the couplings of the latter to the fixed background where the brane propagates must also be described in a manifestly spacetime supersymmetric way. The formulation achieving precisely that is the superspace formulation of supergravity theories [491Jump To The Next Citation Point].

Both these points were already encountered in our review of the GS formulation for the superstring. The same features will hold for all brane effective actions discussed below. After all, both strings and branes are different objects in the same theory. Consequently, any manifestly spacetime supersymmetric and covariant formulation should refer to the same superspace. It is worth emphasising the world volume manifold Σ p+1 with local coordinates σμ remains bosonic in this formulation. This is not what occurs in standard superspace formulations of supersymmetric field theories. There exists a classically equivalent formulation to the GS one, the superembedding formulation that extends both the spacetime and the world volume to supermanifolds. Though I will briefly mention this alternative and powerful formulation in Section 8, I refer readers to [460Jump To The Next Citation Point].

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Figure 5: Kappa symmetry and world volume diffeomorphisms allow one to couple the brane degrees of freedom to the superfield components of supergravity in a manifestly covariant and supersymmetric way. Invariance under kappa symmetry forces the background to be on-shell. The gauge fixing of these symmetries connects the GS formulation with world volume supersymmetry, whose on-shell degrees of freedom match the Goldstone modes of the brane supergravity configurations.

As in global supersymmetric theories, supergravity superspace formulations involve an increase in the number of degrees of freedom describing the spacetime dynamics (to preserve supersymmetry covariance). Its equivalence with the more standard component formalism is achieved through the satisfaction of a set of non-trivial constraints imposed on the supergravity superfields. These guarantee the on-shell nature of the physical superfield components. I refer the reader to a brief but self-contained Appendix A where this superspace formulation is reviewed for 𝒩 = 2 type IIA/B d = 10 and 𝒩 = 1 d = 11 supergravities, including the set of constraints that render them on-shell. These will play a very important role in the self-consistency of the supersymmetric effective actions I am about to construct.

Instead of discussing the supersymmetric coupling to an arbitrary curved background at once, my plan is to review the explicit construction of supersymmetric D-brane and M2-brane actions propagating in Minkowski spacetime, or its superspace extension, super-Poincaré.20 The logic will be analogous to that presented for the superstring. First, I will construct these supersymmetric and kappa invariant actions without using the superspace formulation, i.e., using a more explicit component approach. Afterwards, I will rewrite these actions in superspace variables, pointing in the right direction to achieve a covariant extension of these results to curved backgrounds in Section 3.5.

3.4.1 D-branes in flat superspace

In this section, I am aiming to describe the propagation of D-branes in a fixed Minkowski target space preserving all spacetime supersymmetry and being world volume kappa symmetry invariant. Just as for bosonic open strings, the gauge field dependence was proven to be of the DBI form by explicit open superstring calculations [482Jump To The Next Citation Point, 389, 87Jump To The Next Citation Point].21

Here I follow the strategy in [9Jump To The Next Citation Point]. First, I will construct a supersymmetric invariant DBI action, building on the superspace results reported in Section 2. Second, I will determine the WZ couplings by requiring both supersymmetry and kappa symmetry invariance. Finally, as in our brief review of the GS superstring formulation, I will reinterpret the final action in terms of superspace quantities and their pullback to p + 1 world volume hypersurfaces. This step will identify the correct structure to be generalised to arbitrary curved backgrounds.

Let me first set my conventions. The field content includes a set of p + 1 dimensional world volume scalar fields {ZM (σ)} = {Xm (σ), 𝜃 α(σ)} describing the embedding of the brane into the bulk supermanifold. Fermions depend on the theory under consideration

In either case, one defines ¯ðœƒ = 𝜃tC, in terms of an antisymmetric charge conjugation matrix C satisfying

t −1 t Γm = − C Γ mC , C = − C , (101 )
with Γ m satisfying the standard Clifford algebra {Γ , Γ } = 2η m n mn with mostly plus eigenvalues. I am not introducing a special notation above to refer to the tangent space, given the flat nature of the bulk. This is not accurate but will ease the notation below. I will address this point when reinterpreting our results in terms of a purely superspace formulation.

Let me start the discussion with the DBI piece of the action. This involves couplings to the NS-NS bulk sector, a sector that is also probed by the superstring. Thus, both the supervielbein (Em, E α) and the NS-NS 2-form B2 were already identified to be

m m m m α α E = Π = dX + d¯ðœƒΓ d𝜃 , E = d 𝜃 (102 ) m 1 m B2 = − ¯ðœƒΓ â™¯Γ md 𝜃 (dX + 2-¯ðœƒΓ d𝜃), (103 )
in type IIA, whereas in type IIB one replaces Γ â™¯ by τ3. The DBI action
∫ p+1 ∘ -------------- SDBI = − TDp d σ − det(𝒢 + ℱ ) (104 )
will therefore be invariant under the spacetime supersymmetry transformations
δ𝜖𝜃 = 𝜖, δ𝜖Xm = ¯ðœ–Γ m 𝜃 (105 )
if both, the induced world volume metric 𝒢 and the gauge invariant 2-form, ℱ, are. These are defined by
𝒢μν = Πm Πnηmn, Πm = ∂ μXm − ¯ðœƒΓ m∂μ𝜃 (106 ) μ ′ν μ ℱμν = 2 πα Fμν − ℬ μν, (107 )
where ℬ stands for the pullback of the superspace 2-form B2 into the worldvolume, i.e., M N ℬ μν = ∂μZ ∂ νZ BMN. Since B2 is quasi-invariant under (105View Equation), one chooses
m 1- ¯ m ¯ m δ𝜖V = ¯ðœ–Γ â™¯Γ m𝜃dX + 6(¯ðœ–Γ â™¯Γ mðœƒðœƒΓ d𝜃 + ¯ðœ–Γ mðœƒðœƒΓ â™¯Γ d𝜃 ), (108 )
so that δ𝜖ℱ = 0, guaranteeing the invariance of the action (104View Equation) since the set of 1-forms m Π are supersymmetric invariant.

Let me consider the WZ piece of the action

∫ SWZ = Ωp+1 . (109 )
Since invariance under supersymmetry allows total derivatives, the Lagrangian can be characterised in terms of a (p + 2)-form
Ip+2 = dΩp+1, (110 )
δ𝜖Ip+2 = 0 = ⇒ δ𝜖Ωp+1 = dΛp. (111 )
Thus, mathematically, I(p+2) must be constructed out of supersymmetry invariants m {Π , d𝜃, ℱ}.

The above defines a cohomological problem whose solution is not guaranteed to be kappa invariant. Since our goal is to construct an action invariant under both symmetries, let me first formulate the requirements due to the second invariance. The strategy followed in [9Jump To The Next Citation Point] has two steps:

The question is whether T(νp), γ(p) and I(p+2) exist satisfying all the above requirements. The explicit construction of these objects was given in [9Jump To The Next Citation Point]. Here, I simply summarise their results. The WZ action was found to be

ℱ dℒWZ = − TDpℛe , (116 )
where ℛ is the pullback of the field strength of the RR gauge potential C, as defined in Eq. (521View Equation). Using m âˆ•Π = Π Γ m, this can be written as [293]
2l ℛ = ¯E 𝒞 (∕Π)E, 𝒞 (Π∕) = ∑ (Γ )l+1 Π∕--- (117 ) A A l=0 ♯ (2l)!
in type IIA, whereas in type IIB [329Jump To The Next Citation Point]
¯ ∑ l -Π∕2l+1--- ℛ = − E 𝒮B (Π∕) τ1 E, 𝒮B (∕Π) = (τ3) (2l + 1)! . (118 ) l=0
Two observations are in order:
  1. dℒWZ is indeed manifestly supersymmetric, since it only depends on supersymmetric invariant quantities, but ℒWZ is quasi-invariant. Thus, when computing the algebra closed by the set of conserved charges, one can expect the appearance of non-trivial charges in the right-hand side of the supersymmetry algebra. This is a universal feature of brane effective actions that will be conveniently interpreted in Section 3.6.
  2. This analysis has determined the explicit form of all the RR potentials Cp as superfields in superspace. This was achieved by world volume symmetry considerations, but it is reassuring to check that the expressions found above do satisfy the superspace constraints reported in Appendix A.1. I will geometrically reinterpret the derived action as one describing a Dp-brane propagating in a fixed super-Poincaré target space shortly.

Let me summarise the global and gauge symmetry structure of the full action. The set of gauge symmetries involves world volume diffeomorphisms (ξμ), an abelian U (1) gauge symmetry (c) and kappa symmetry (κ). Their infinitesimal transformations are

m m m μ m ¯ m sX = ℒξX + δκX = ξ ∂μX − δκðœƒΓ ðœƒ, (119 ) s𝜃 α = ℒξ𝜃α + δκ𝜃α = ξ μ∂μ𝜃α + δκ𝜃α, (120 ) ν ν sV μ = ℒξVμ + ∂μc + δκV μ = ξ ∂νVμ + Vν∂ μξ + ∂ μc + δκV μ, (121 )
where δκVμ is given in Eq. (112View Equation) and δκ 𝜃 was determined in [9Jump To The Next Citation Point]
(p) (p) ρ(p) δκ¯ðœƒ = ¯κ (𝟙 − γ ), γ = ∘--------------. (122 ) − det(𝒢 + ℱ )
In type IIA, the matrix (p) ρ stands for the p + 1 world volume form coefficient of ℱ 𝒮A(âˆ•Π )e, where
2l+1 ρ(p) = [𝒮 (âˆ•Π )eℱ] , 𝒮 (âˆ•Π ) = ∑ (Γ )l+1 -∕Π------- (123 ) A p+1 A ♯ (2l + 1)! l=0
, while in type IIB, it is given by
∑ Π∕2l ρ(p− 1) = − [𝒞B(∕Π)eℱ τ1]p, 𝒞B (Π∕) = (τ3)l+1 -----. (124 ) l=0 (2l)!
It was proven in [9Jump To The Next Citation Point] that ρ2 = − det(𝒢 + ℱ)𝟙. This proves γ2 (p) equals the identity, as required in our construction.

The set of global symmetries includes supersymmetry (𝜖), bosonic translations (am ) and Lorentz transformations (ωmn ). The field infinitesimal transformations are

ΔXm = δXm + δXm + δ Xm = ¯ðœ–Γ m𝜃 + am + ωm Xn, (125 ) 𝜖 a ω n Δ ðœƒα = δ𝜃α + δ 𝜃 α = 𝜖α + 1ωmn (Γ ðœƒ)α , (126 ) 𝜖 ω 4 mn ΔV μ = δ𝜖V μ, (127 )
with δ𝜖Vμ given in Eq. (108View Equation) and m mp ω n ≡ ω ηpn.

Geometrical reinterpretation of the effective action:
the supersymmetric action was constructed out of the supersymmetric invariant forms {Πm, d𝜃, ℱ }. These can be reinterpreted as the pullback of 10-dimensional superspace tensors to the p + 1 brane world volume. To see this, it is convenient to introduce the explicit supervielbein components EAM (Z ), defined in Appendix A.1, where the index M stands for curved superspace indices, i.e., M = {m, α }, and the index A for tangent flat superspace indices, i.e., A = {a, α}. In this language, the super-Poincaré supervielbein components equal

a a α- α- α- a (¯ a) α- E m = δm , Eα = δα , Em = 0, Eα = ðœƒΓ α-δα . (128 )
manifest that all Clifford matrices a Γ act in the tangent space, as they should. The components (128View Equation) allow us to rewrite all couplings in the effective action as pullbacks
𝒢 (Z) = ∂ ZM Ea (Z )∂ ZN Eb (Z )η , μν μ M ν N ab ℬ μν(Z) = ∂μZM EAM(Z )∂νZN ECN(Z )BAC (Z ), (129 ) M1 A1 Mp+1 Ap+1 𝒞μ1...μp+1(Z) = ∂μ1Z E M1(Z )...∂μp+1Z E Mp+1(Z)CA1...Ap+1(Z ),
of the background superfields EA M, BAC and CA ...A 1 p+1 to the brane world volume. Furthermore, the kappa symmetry transformations (112View Equation) and (122View Equation) also allow a natural superspace description as
δ ZM Ea = 0, δ ZM Eα- = (𝟙 + Γ )κ (130 ) κ M κ M κ
where the kappa symmetry matrix Γ κ is nicely repackaged
∑k (Γ κ) = ∘------1-------- γ(2l+1)Γ l+1∧ eℱ type IIA p = 2k (131 ) (p+1) − det(𝒢 + ℱ ) l=0 ♯ -------1-------k+∑1 l ℱ (Γ κ)(p+1) = ∘-------------- γ(2l)τ3 ∧ e iτ2 typeIIB p = 2k + 1, (132 ) − det(𝒢 + ℱ ) l=0
in terms of the induced Clifford algebra matrices γμ and the gauge invariant tensor ℱ
γ ≡ dσμγ = d σμ∂ ZM Ea (Z )Γ , (133 ) (1) μ′ μ M a ℱ = 2πα F − ℬ2, (134 )
whereas γ (l) stands for the wedge product of the 1-forms γ (1).

We have constructed an effective action describing the propagation of Dp-branes in 10-dimensional Minkowski spacetime being invariant under p + 1 dimensional diffeomorphisms, 10-dimensional supersymmetry and kappa symmetry. The final result resembles the bosonic action (47View Equation) in that it is written in terms of pullbacks of the components of the different superfields EA (Z) M, BAC (Z ) and CA1...Ap+1(Z ) encoding the non-trivial information about the non-dynamical background where the brane propagates in a manifestly supersymmetric way. These superfields are on-shell supergravity configurations, since they satisfy the set of constraints listed in Appendix A.1. It is this set of features that will allow us to generalise these couplings to arbitrary on-shell superspace backgrounds in Section 3.5, while preserving the same kinematic properties.

3.4.2 M2-brane in flat superspace

Let me consider an M2-brane as an example of an M-brane propagating in d = 11 super-Poincaré. Given the lessons from the superstring and D-brane discussions, my presentation here will be much more economical.

First, let me describe d = 11 super-Poincaré as a solution of eleven-dimensional supergravity using the superspace formulation introduced in Appendix A.2. In the following, all fermions will be 11-dimensional Majorana fermions 𝜃 as corresponds to 𝒩 = 1 d = 11 superspace. Denoting the full set of superspace coordinates as M m α {Z } = {X ,𝜃 } with m = 0,...,10 and α = 1,...,32, the superspace description of 𝒩 = 1 d = 11 super-Poincaré is [165Jump To The Next Citation Point, 144]

a a a α α E = dX + d¯ðœƒΓ 𝜃, E --= d𝜃 -, 1- a b α- β R4 = 2 E ∧ E ∧ d𝜃 ∧ d𝜃--(Γ ab)αβ , 1 R7 = --Ea1 ∧ Ea2 ∧ Ea3 ∧ Ea4 ∧ Ea5 ∧ d𝜃α-∧ d𝜃β (Γ a1a2a3a4a5)αβ. (135 ) 5! ---
It includes the supervielbein EA = {Ea, E α} and the gauge invariant field strengths R4 = dA3 and its Hodge dual R7 = dA6 + 1A3 ∧ R4 2, defined as Eq. (550View Equation) in Appendix A.2.

The full effective action can be written as [91Jump To The Next Citation Point]

∫ 3 ∘ --------- ∫ S = − TM2 d σ − det 𝒢μν + TM2 𝒜3, (136 ) a b A M A 𝒢μν = E μ(X, 𝜃)Eν(X, 𝜃)ηab, E μ ≡ ∂μZ E M (X, 𝜃), 1 μνρ B C D 𝒜3 = 3!𝜀 E μ E ν E ρ ABCD (X, 𝜃).
Notice it depends on the supervielbeins A E M (X, 𝜃) and the three form potential CABC (x, 𝜃) superfields only through their pullbacks to the world volume.

Its symmetry structure is analogous to the one described for D-branes. Indeed, the action (136View Equation) is gauge invariant under world volume diffeomorphisms (ξμ) and kappa symmetry (κ) with infinitesimal transformations given by

sXm = ℒ Xm + δ Xm = ξ μ∂ Xm + δ ¯ðœƒΓ m𝜃 , (137 ) ξ κ μ κ s𝜃α = ℒ ξ𝜃α + δκ𝜃α = ξμ∂μ𝜃α + (1 + Γ κ)κ , (138 ) 1 μνρ a b c Γ κ =--√---------𝜀 EμE νE ρΓ abc. (139 ) 3! − det𝒢
The kappa matrix (139View Equation) satisfies Γ 2κ = 𝟙. Thus, δκ𝜃 is a projector that will allow one to gauge away half of the fermionic degrees of freedom.

The action (136View Equation) is also invariant under global super-Poincaré transformations

1- mn δ𝜃 = 𝜖 + 4ωmn Γ ðœƒ, (140 ) δXm = ¯ðœ–Γ m 𝜃 + am + ωm Xn. (141 ) n
Supersymmetry quasi-invariance can be easily argued for since R4 is manifestly invariant. Thus, its gauge potential pullback variation will be a total derivative
δ𝜖A3 = d[¯ðœ–Δ2 ] (142 )
for some spinor-valued two form Δ2.

It is worth mentioning that just as the bosonic membrane action reproduces the string world-sheet action under double dimensional reduction, the same statement is true for their supersymmetric and kappa invariant formulations [192, 476].

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