3.6 Symmetries: spacetime vs world volume

The main purpose of this section is to discuss the global symmetries of brane effective actions, the algebra they close and to emphasise the interpretation of some of the conserved charges appearing in these algebras before and after gauge fixing of the world volume diffeomorphisms and kappa symmetry.

To prove that background symmetries give rise to brane global symmetries, one must first properly define the notion of superisometry of a supergravity background. This involves a Killing superfield ξ(Z ) satisfying the properties

a b ℒξ(E ⊗s E )ηab = 0 , (160 ) ℒ ξR4 = ℒ ξR7 = 0 , M -theory (161 ) ℒ ξH3 = ℒ ξϕ = ℒ ξRk = 0 . Type IIA/B (162 )
ℒ ξ denotes the Lie derivative with respect to ξ, η is either the d = 11 or d = 10 Minkowski metric on the tangent space, depending on which superspace we are working on and {Rk, H3 } are the different M-theory or type IIA/B field strengths satisfying the generalised Bianchi identities defined in Appendix A. Notice these are the superfield versions of the standard bosonic Killing isometry equations. Invariance of the field strengths allows the corresponding gauge potentials to have non-trivial transformations
1 ℒξA3 = dΔ2 , ℒ ξA6 = d Δ5 − --Δ2 ∧ R4 , M -theory (163 ) 2 ℒξB2 = dλ1, ℒξCp+1 = dωp − dωp −2 ∧ H3 , Type IIA/B (164 )
for some set of superfield forms {Δ2, Δ5, ωi}.

The invariance of brane effective actions under the global transformations

δξZM = ξM (Z ), (165 )
was proven in [94Jump To The Next Citation Point]. The proof can be established by analysing the DBI and WZ terms of the action separately. If the brane has gauge field degrees of freedom, one can always choose its infinitesimal transformation
δV = Z⋆(Δ ), M -theory (166 ) 2 2 δV1 = Z⋆(λ1), Type IIA/B (167 )
where Z⋆ stands for pullback to the world volume, i.e., Z⋆(λ ) = dZM (λ ) 1 1M. This guarantees the invariance of the gauge invariant forms, i.e., ℒξℱ = ℒ ξℋ3 = 0. Furthermore, the transformation of the induced metric
M N a b ℒ ξ𝒢μν = ∂μZ ∂νZ ℒξ(E ME N ηab) , (168 )
vanishes because of Eq. (160View Equation). This establishes the invariance of the DBI action. On the other hand, the WZ action is quasi-invariant by construction due to Eqs. (163View Equation) and (164View Equation). Indeed,
⋆ δℒWZ = Z (dΔ2 ) , M2 -brane ⋆( 1- ) δℒWZ = Z d(Δ5 + 2ℋ3 ∧ Δ2 ) , M5 -brane ℱ ⋆ δℒWZ = ℒ ξ𝒞 ∧ e = Z (dω) . D -branes (169 )

Brane effective actions include the supergravity superisometries ξ(Z) as a subset of their global symmetries. It is important to stress that kappa symmetry invariance is necessary to define a supersymmetric field theory on the brane, but not sufficient. Indeed, any on-shell supergravity background having no Killing spinors, i.e., some superisometry in which fermions are shifted as δ𝜃 = 𝜖, breaks supersymmetry, and consequently, will never support a supersymmetric action on the brane. The derivation discussed above does not exclude the existence of further infinitesimal transformations leaving the effective action invariant. The question of determining the full set of continuous global symmetries of a given classical field theory is a well posed mathematical problem in terms of cohomological methods [50, 51]. Applying these to the bosonic D-string [111] gave rise to the discovery of the existence of an infinite number of global symmetries [113, 112]. These were also proven to exist for the kappa invariant D-string action [110].

3.6.1 Supersymmetry algebras

Since spacetime superisometries generate world-volume global symmetries, Noether’s theorem [406Jump To The Next Citation Point, 407Jump To The Next Citation Point] guarantees a field theory realisation of the spacetime (super)symmetry algebra using Poisson brackets. It is by now well known that such (super)algebras contain more bosonic charges than the ones geometrically realised as (super)isometries. There are several ways of reaching this conclusion:

  1. Grouped theoretically, the anticommutator of two supercharges {Q α, Q β} defines a symmetric matrix belonging to the adjoint representation of some symplectic algebra Sp(N, ℝ), whose order N depends on the spinor representation Q α. One can decompose this representation into irreducible representations of the bosonic spacetime isometry group. This can explicitly be done by using the completeness of the basis of antisymmetrised Clifford algebra gamma matrices as follows
    ∑ {Q α, Q β} = (Γ m1...mpC −1)αβZm1...mp , (170) p
    where the allowed values of p depend on symmetry considerations. The right-hand side defines a set of bosonic charges {Zm1...mp } that typically goes beyond the spacetime bosonic isometries.
  2. Physically, BPS branes in a given spacetime background have masses equal to their charges by virtue of the amount of supersymmetry they preserve. This would not be consistent with the supersymmetry algebra if the latter would not include extra charges, the set {Zm1...mp} introduced above, besides the customary spacetime isometries among which the mass (time translations) always belongs to. Thus, some of the extra charges must correspond to such brane charges. The fact that these charges have non-trivial tensor structure means they are typically not invariant under the spacetime isometry group. This is consistent with the fact that the presence of branes breaks the spacetime isometry group, as I already explicitly discussed in super-Poincaré.
  3. All brane effective actions reviewed above are quasi-invariant under spacetime superisometries, since the WZ term transformation equals a total derivative (169View Equation). Technically, it is a well-known theorem that such total derivatives can induce extra charges in the commutation of conserved charges through Poisson brackets. This is the actual field theory origin of the group theoretically allowed set of charges {Zm1...mp}.

Let me review how these structures emerge in both supergravity and brane effective actions. Consider the most general superPoincaré algebra in 11 dimensions. This is spanned by a Majorana spinor supercharge Q α (α = 1,...,32) satisfying the anti-commutation relations25 [487, 478, 481Jump To The Next Citation Point]

{Q ,Q } = (Γ mC −1) P + 1(Γ mnC −1) Z + 1-(Γ m1...m5C −1) Y . (171 ) α β αβ m 2 αβ mn 5! αβ m1...m5
That this superalgebra is maximal can be argued using the fact that its left-hand side defines a symmetric tensor with 528 independent components. Equivalently, it can be interpreted as belonging to the adjoint representation of the Lie algebra of Sp (32,ℝ). The latter decomposes under its subgroup SO (1,10 ), the spacetime Lorentz isometry group, as
528 → 11 ⊕ 55 ⊕ 462 . (172 )
The irreducible representations appearing in the direct sum do precisely correspond to the bosonic tensor charges appearing in the right-hand side: the 11-momentum Pm, a 2-form charge Zmn, which is 55-dimensional, and a 5-form charge Ym1...m5, which is 462-dimensional.

The above is merely based on group theory considerations that may or may not be realised in a given physical theory. In 11-dimensional supergravity, the extra bosonic charges are realised in terms of electric Ze and magnetic Zm charges, the Page charges [410], that one can construct out of the 3-form potential A3 equation of motion, as reviewed in [467Jump To The Next Citation Point, 466Jump To The Next Citation Point]

∫ -1-- 1- Ze = 4Ω ∂ℳ8 (⋆R4 + 2 A3 ∧ R4) , (173 ) 1 7∫ Zm = --- R4 . (174 ) Ω4 ∂ℳ5
The first integral is over the boundary at infinity of an arbitrary infinite 8-dimensional spacelike manifold ℳ8, with volume Ω7. Given the conserved nature of this charge, it does not depend on the time slice chosen to compute it. But there are still many ways of embedding ℳ8 in the corresponding ten-dimensional spacelike hypersurface ℳ10. Thus, Ze represents a set of charges parameterised by the volume element 2-form describing how ℳ8 is embedded in ℳ10. This precisely matches the 2-form Zmn in Eq. (171View Equation). There is an analogous discussion for Zm, which corresponds to the 5-form charge Ym1...m5. As an example, consider the M2 and M5-brane configurations in Eqs. (20View Equation) and (22View Equation). If one labels the M2-brane tangential directions as 1 and 2, there exists a non-trivial charge Z12 computed from Eq. (173View Equation) by plugging in Eq. (20View Equation) and evaluating the integral over the transverse 7-sphere at infinity. The reader is encouraged to read the lecture notes by Stelle [467Jump To The Next Citation Point] where these issues are discussed very explicitly in a rather general framework including all standard half-BPS branes. For a more geometric construction of these maximal superalgebras in AdS × S backgrounds, see [211] and references therein.

The above is a very brief reminder regarding spacetime superalgebras in supergravity. For a more thorough presentation of these issues, the reader is encouraged to read the lectures notes by Townsend [481Jump To The Next Citation Point], where similar considerations are discussed for both type II and heterotic supergravity theories. Given the importance given to the action of dualities on effective actions, the reader may wonder how these same dualities act on superalgebras. It was shown in [96] that these actions correspond to picking different complex structures of an underlying OSp (1|32) superalgebra.

Consider the perspective offered by the M5-brane effective action propagating in d = 11 superPoincaré. The latter is invariant both under supersymmetry and bulk translations. Thus, through Noether’s theorem, there exist field theory realisations of these charges. Quasi-invariance of the WZ term will be responsible for the generation of extra terms in the calculation of the Poisson bracket of these charges [165Jump To The Next Citation Point]. This was confirmed for the case at hand in [464Jump To The Next Citation Point], where the M5-brane superalgebra was explicitly computed. The supercharges Qα are

∫ Qα = i d5 σ[(π + ¯𝜃Γ mPm )α + i(𝒫i1i2 + 1ℋ ∗0i1i2)(Δ2 )α − i𝜀i1...i5(Δ5 )α ], (175 ) 4 i1i2 i1...i5
where π, Pm and 𝒫ij are the variables canonically conjugate to 𝜃, Xm and Vij. As in any Hamiltonian formalism, world volume indices were split according to σμ = {t, σi}i = 1,...5. Notice that the pullbacks of the forms Δ2 and Δ5 appearing in δℒWZ for the M5-brane in Eq. (169View Equation) do make an explicit appearance in this calculation. The anti-commutator of the M5 brane world volume supercharges equals Eq. (171View Equation) with
∫ δ ℒ Pm = d5σ ------m--, (176 ) ∫ δ(∂tX ) Zmn = − dXm ∧ dXn ∧ dV , (177 ) ℳ5 2 m1...m5 ∫ m1 m5 Y = ℳ dX ∧ ⋅⋅⋅ ∧ dX , (178 ) 5
where all integrals are computed on the 5-dimensional spacelike hypersurface ℳ5 spanned by the M5-brane. Notice the algebra of supercharges depends on the brane dimensionality. Indeed, a single M2-brane has a two dimensional spacelike surface that cannot support the pullback of a spacetime 5-form as a single M5-brane can (see Eq. (178View Equation)). This conclusion could be modified if the degrees of freedom living on the brane would be non-abelian.

Even though my discussion above only applies to the M5-brane in the super-Poincaré background, my conclusions are general given the quasi-invariance of their brane WZ action, a point first emphasised in [165Jump To The Next Citation Point]. The reader is encouraged to read [165Jump To The Next Citation Point, 168] for similar analysis carried for super p-branes, [281] for D-branes in super-Poincaré and general mathematical theorems based on the structure of brane effective actions and [438Jump To The Next Citation Point, 437Jump To The Next Citation Point], for superalgebra calculations in some particular curved backgrounds.

3.6.2 World volume supersymmetry algebras

Once the physical location of the brane is given, the spacetime superisometry group G is typically broken into

G → G0 × G1. (179 )
The first factor G0 corresponds to the world volume symmetry group in (p + 1)-dimensions, i.e., the analogue of the Lorentz group in a supersymmetric field theory in (p + 1)-dimensions, whereas the second factor G1 is interpreted as an internal symmetry group acting on the dynamical fields building (p + 1)-dimensional supermultiplets. The purpose of this subsection is to relate the superalgebras before and after this symmetry breaking process [328].26

The link between both superalgebras is achieved through the gauge fixing of world volume diffeomorphisms and kappa symmetry, the gauge symmetries responsible for the covariance of the original brane action in the GS formalism. Focusing on the scalar content in these theories {Xm, 𝜃}, these transform as

m m m m m sX = k (X ) + ℒ ξX + δκX + δ𝜖X , (180 ) s𝜃 = 𝜖 + δk𝜃 + ℒ ξ𝜃 + (𝟙 + Γ κ)κ + δk𝜃. (181 )
The general Killing superfield was decomposed into a supersymmetry translation denoted by 𝜖 and a bosonic Killing vector fields kM (X ). World volume diffeomorphisms were denoted as ξ. At this stage, the reader should already notice the inhomogeneity of the supersymmetry transformation acting on fermions (the same is true for bosons if the background spacetime has a constant translation as an isometry, as it happens in Minkowski).

Locally, one can always impose the static gauge: X μ = σμ, where one decomposes the scalar fields Xm into world volume directions X μ and transverse directions XI ≡ ΦI. For infinite branes, this choice is valid globally and does describe a vacuum configuration. To diagnose which symmetries act, and how, on the physical degrees of freedom i Φ, one must make sure to work in the subset of symmetry transformations preserving the gauge slice μ μ X = σ. This forces one to act with a compensating world volume diffeomorphism

sX μ|Xμ=σμ = 0 ⇒ ξ μ = − k μ − δκX μ − δ𝜖X μ . (182 )
The latter acts on the physical fields giving rise to the following set of transformations preserving the gauge fixed action
I I μ I sΦ |X μ=σμ = k − k ∂μΦ + ..., (183 ) s𝜃|X μ=σμ = − kμ∂μ𝜃 + ℒk 𝜃 + ... . (184 )
There are two important comments to be made at this point
  1. The physical fields ΦI transform as proper world volume scalars [3]. Indeed, ΦI(σ ) = (Φ ′)I(σ ′) induces the infinitesimal transformation kμ∂μΦI for any kμ(σ ) preserving the p + 1 dimensional world volume. Below, the same property will be checked for fermions.
  2. If the spacetime background allows for any constant I k isometry, it would correspond to an inhomogeneous symmetry transformation for the physical field ΦI. In field theory, the latter would be interpreted as a spontaneous broken symmetry and the corresponding ΦI would be its associated massless Goldstone field. This is precisely matching our previous discussions regarding the identification of the appropriate brane degrees of freedom.

There is a similar discussion regarding the gauge fixing of kappa symmetry and the emergence of a subset of linearly realised supersymmetries on the (p + 1)-dimensional world volume field theory. Given the projector nature of the kappa symmetry transformations, it is natural to assume 𝒫 𝜃 = 0 as a gauge fixing condition, where 𝒫 stands for some projector. Preservation of this gauge slice, determines the kappa symmetry parameter κ as a function of the background Killing spinors 𝜖

s𝜃| = 0 = ⇒ κ = κ(𝜖). (185 ) 𝒫𝜃=0
When analysing the supersymmetry transformations for the remaining dynamical fermions, only certain linear combinations of the original supersymmetries 𝜖 will be linearly realised. The difficulty in identifying the appropriate subset depends on the choice of 𝒫.

Branes in super-Poincaré:
The above discussion can be made explicit in this case. Consider a p + 1 dimensional brane propagating in d dimensional super-Poincaré. For completeness, let me remind the reader of the full set of transformations leaving the brane actions invariant

m m m n m m m sX = a + a nX + ℒ ξX + ¯𝜖Γ 𝜃 + δκX , (186 ) 1- mn s𝜃 = 4 amnΓ 𝜃 + ℒ ξ𝜃 + 𝜖 + δκ𝜃 , (187 )
where I ignored possible world volume gauge fields. Decomposing the set of bosonic scalar fields m X m = 0,1,...d − 1 into world volume directions μ X μ = 0,1,...p and transverse directions XI ≡ ΦI I = p + 1,...d − 1, one can now explicitly solve for the preservation of the static gauge slice X μ = σ μ, which does globally describe the vacuum choice of a p-brane extending in the first p spacelike directions and time. This requires some compensating world volume diffeomorphism
ξμ = − aμ − aμ σν − aμ ΦI − ¯𝜖Γ μ𝜃 − δ X μ, (188 ) ν I κ
inducing the following transformations for the remaining degrees of freedom
sΦI = − aμ∂μΦI − a μνσν∂μΦI − aμJΦJ ∂μΦI + aJ + aIJ ΦJ + aIμσ μ + fermions , (189 ) s𝜃 = − aμνσν∂μ𝜃 + 1aμνΓ μν𝜃 + 1-aIJΓ IJ𝜃. (190 ) 4 4
The subset of linearly realised symmetries is ISO (1,p) × SO (D − (p + 1)). The world volume “Poincaré” group is indeed ISO (1, p), under which ΦI are scalars, whereas 𝜃 are fermions, including the standard spin connection transformation giving them their spinorial nature. SO (D − (p + 1)), the transverse rotational group to the brane is reinterpreted as an internal symmetry, under which I Φ transforms as a vector. The parameters aμI describing the coset SO (1,D − 1)∕(SO (1,p) × SO (D − p − 1)) are generically non-linearly realised, whereas the transverse translations aI act inhomogeneously on the dynamical fields ΦI, identifying the latter as Goldstone massless fields, as corresponds to the spontaneous symmetry breaking of these symmetries due to the presence of the brane in the chosen directions. There is a similar discussion for the 32 spacetime supersymmetries (𝜖). Before gauge fixing all fermions 𝜃 transform inhomogeneously under supersymmetry. After gauge fixing 𝒫 𝜃 = 0, the compensating kappa symmetry transformation κ (𝜖) required to preserve the gauge slice in configuration space will induce an extra supersymmetry transformation for the dynamical fermions, i.e., (1 − 𝒫 )𝜃. On general grounds, there must exist sixteen linear combinations of supersymmetries being linearly realised, whereas the sixteen remaining will be spontaneously broken by the brane. There are many choices for 𝒫𝜃 = 0. In [10], where they analysed this aspect for D-branes in super-Poincaré, they set one of the members of the 𝒩 = 2 fermion pair to zero, leading to fairly simple expressions for the gauge fixed Lagrangian. Another natural choice corresponds to picking the projector describing the preserved supersymmetries by the brane from the spacetime perspective. For instance, the supergravity solution describing M2-branes has 16 Killing spinors satisfying
Γ 𝜖 = ± 𝜖, (191 ) 012
where the ± is correlated with the R4 flux carried by the solution. If one fixes kappa symmetry according to
𝒫 𝜃 = (1 + Γ ⋆)𝜃 = 0, with Γ ⋆ = Γ 3...Γ 9Γ ♯, (192 )
where Γ ♯ stands for the 11-dimensional Clifford algebra matrix, then the physical fermionic degrees of freedom are not only 3-dimensional spinors, but they are chiral spinors from the internal symmetry SO (8) perspective. They actually transform in the s (2, 8-) [91Jump To The Next Citation Point]. Similar considerations would apply for any other brane considered in this review.

Having established the relation between spacetime and world volume symmetries, it is natural to close our discussion by revisiting the superalgebra closed by the linearly realised world volume (super)symmetries, once both diffeomorphisms and kappa symmetry have been fixed. Since spacetime superalgebras included extra bosonic charges due to the quasi-invariance of the brane WZ action, the same will be true for their gauge fixed actions. Thus, these (p + 1)-dimensional world volume superalgebras will include as many extra bosonic charges as allowed by group theory and by the dimensionality of the brane world spaces [81Jump To The Next Citation Point]. Consider the M2-brane discussed above. Supercharges transform in the s (2, 8-) representation of the SO (1, 2) × SO (8) bosonic isometry group. Thus, the most general supersymmetry algebra compatible with these generators, 𝒩 = 8 d = 3, is [81Jump To The Next Citation Point]

I J IJ (IJ) [IJ] (IJ) {Q α, Q β} = δ P (αβ) + Z(αβ) + 𝜀αβZ with (δIJZ = 0) . (193 )
P (αβ) stands for a 3-dimensional one-form, the momentum on the brane; (IJ) Z(αβ) transforms in the + 35 under the R-symmetry group SO(8), or equivalently, as a self-dual 4-form in the transverse space to the brane; Z [IJ] is a world volume scalar, which transforms in the 28 of SO (8), i.e., as a 2-form in the transverse space. The same superalgebra is realised on the non-abelian effective action describing N coincident M2-branes [415] to be reviewed in Section 7.2. Similar structures exist for other infinite branes. For example, the M5-brane gives rise to the d = 6 (2,0) superalgebra [81]
{QIα, QJβ} = ΩIJP [αβ] + Z ((αIJ)β) + Y[[IαJβ]] with (ΩIJY [IJ] = 0). (194 )
Here α, β = 1,...,4 is an index of ∗ SU (4) ≃ Spin(1,5), the natural Lorentz group for spinors in d = 6 dimensions, I,J = 1,...4 is an index of Sp (2) ≃ Spin(5), which is the double cover of the geometrical isometry group SO (5) acting on the transverse space to the M5-brane and ΩIJ is an Sp (2) invariant antisymmetric tensor. Thus, using appropriate isomorphisms, these superalgebras allow a geometrical reinterpretation in terms of brane world volumes and transverse isometry groups becoming R-symmetry groups. The last decomposition is again maximal since P[αβ] stands for 1-form in d = 6 (momentum), Z (IJ) (αβ) transforms as a self-dual 3-form in d = 6 and a 2-form in the transverse space and [IJ] Y[αβ] as a 1-form both in d = 6 and in the transverse space. For an example of a non-trivial world volume superalgebra in a curved background, see [152Jump To The Next Citation Point].

I would like to close this discussion with a remark that is usually not stressed in the literature. By construction, any diffeomorphism and kappa symmetry gauge fixed brane effective action describes an interacting supersymmetric field theory in p + 1 dimensions.27 As such, if there are available superspace techniques in these dimensions involving the relevant brane supermultiplet, the gauge fixed action can always be rewritten in that language. The matching between both formulations generically involves non-trivial field redefinitions. To be more precise, consider the example of 𝒩 = 1 d = 4 supersymmetric abelian gauge theories coupled to matter fields. Their kinetic terms are fully characterised by a Kähler potential. If one considers a D3-brane in a background breaking the appropriate amount of supersymmetry, the expansion of the gauge fixed D3-brane action must match the standard textbook description. The reader can find an example of the kind of non-trivial bosonic field redefinitions that is required in [321]. The matching of fermionic components is expected to be harder.

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