5.1 Vacuum infinite branes

There exist half-BPS branes in 10- and 11-dimensional Minkowski spacetime. Since their effective actions were discussed in Section 3, we can check their existence and the amount of supersymmetry they preserve, by solving the brane classical equations of motion and the kappa symmetry preserving condition (214View Equation).

First, one works with the bosonic truncation πœƒ = 0. The background, in Cartesian coordinates, involves the metric

2 m n ds = ηmndx dx , m, n = 0,1,...D − 1 (246 )
and all remaining bosonic fields vanish, except for the dilaton, in type IIA/B, which is constant. This supergravity configuration is maximally supersymmetric, i.e., it has Killing spinors spanning a vector space, which is 32-dimensional. In Cartesian coordinates, these are constant spinors πœ– = πœ–∞.

Half-BPS branes should correspond to vacuum configurations in these field theories describing infinite branes breaking the isometry group ISO (1,D − 1) to ISO (1, p) × SO (D − p − 1) and preserving half of the supersymmetries. Geometrically, these configurations are specified by the brane location. This is equivalent to first splitting the scalar fields Xm (σ ) into longitudinal X μ and transverse XI directions, setting the latter to constant values XI = cI (the transverse brane location). Second, one identifies the world volume directions with the longitudinal directions, X μ = σ μ. The latter can also be viewed as fixing the world volume diffeomorphisms to the static gauge. This information can be encoded as an array

p-brane: 1 2 ..p x x x x (247 )

It is easy to check that the above is an on-shell configuration given the structure of the Euler–Lagrange equations and the absence of non-trivial couplings except for the induced world volume metric 𝒒μν, which equals ημν in this case.

To analyse the supersymmetry preserved, one must solve Eq. (214View Equation). Notice that in the static gauge and in the absence of any further excitations, the induced gamma matrices equal

γ = ∂ xmEa Γ = Γ =⇒ γ = Γ , (248 ) μ μ m a μ μ0...μp μ0...μp
where I already used Eam = δam. Thus, Γ κ reduces to a constant Clifford valued matrix standing for the volume of the brane, Γ vol, up to the chirality of the background spinors, which is parameterised by the matrix τ
Γ = Γ τ . (249 ) κ vol
The specific matrices for the branes discussed in this review are summarised in Table 6. Since Γ 2κ = πŸ™ and Tr Γ κ = 0, only half of the vector space spanned by πœ–∞ preserves these bosonic configurations, i.e., all infinite branes preserve half of the supersymmetries. These projectors match the ones derived from bosonic supergravity backgrounds carrying the same charges as these infinite branes.

Table 6: Half-BPS branes and the supersymmetries they preserve.

BPS state



Γ 012πœ– = πœ–


Γ 012345πœ– = πœ–

IIA D2n-brane

n+1 Γ 0...2nΓ β™― πœ– = πœ–

IIB D2n-1-brane

Γ 0...2n− 1τ3niτ2πœ– = πœ–

All these configurations have an energy density equaling the brane tension T since the Hamiltonian constraint is always solved by

β„°2 = T2 det𝒒 = T2. (250 )
From the spacetime superalgebra perspective, these configurations saturate a bound between the energy and the p-form bosonic charge carried by the volume form defined by the brane
β„° = 𝒡 μ1...μp = T πœ–μ1...μp. (251 )
The saturation corresponds to the fact that any excitation above the infinite brane configuration would increase the energy. From the world-volume perspective, the solution is a vacuum, and consequently, it is annihilated by all sixteen world-volume supercharges. These are precisely the ones solving the kappa symmetry preserving condition (214View Equation).
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