4.1 Supersymmetric bosonic configurations and kappa symmetry

To know whether any given on-shell bosonic brane configuration is supersymmetric, and if so, how many supersymmetries are preserved, one must develop some tools analogous to the ones for bosonic supergravity configurations. I will review these first.

Consider any supergravity theory having bosonic (ℬ ) and fermionic (β„± ) degrees of freedom. It is consistent with the equations of motion to set β„± = 0. The question of whether the configuration ℬ preserves supersymmetry reduces to the study of whether there exists any supersymmetry transformation πœ– preserving the bosonic nature of the on-shell configuration, i.e., δβ„± | = 0 β„±=0, without transforming ℬ, i.e., δℬ|β„±=0 = 0. Since the structure of the local supersymmetry transformations in supergravity is

δ ℬ ∝ β„± , δβ„± = 𝒫 (ℬ )πœ– , (202 )
these conditions reduce to 𝒫 (ℬ )πœ– = 0. In general, the Clifford valued operator 𝒫(ℬ ) is not higher than first order in derivatives, but it can also be purely algebraic. Solutions to this equation involve
  1. Differential constraints on the subset of bosonic configurations ℬ. Given the first-order nature of the operator 𝒫 (ℬ), these are simpler than the second-order equations of motion and help to reduce the complexity of the latter.
  2. Differential and algebraic constraints on πœ–. These reduce the infinite dimensional character of the original arbitrary supersymmetry transformation parameter πœ– to a finite dimensional subset, i.e., πœ– = f ℬ(xm)πœ–∞, where the function f ℬ(xm) is uniquely specified by the bosonic background ℬ and the constant spinor πœ–∞ typically satisfies a set of conditions 𝒫iπœ–∞ = 0, where 𝒫i are projectors satisfying 2 𝒫i = 𝒫i and tr𝒫i = 0. These πœ– are the Killing spinors of the bosonic background ℬ. They can depend on the spacetime point, but they are no longer arbitrary. Thus, they are understood as global parameters.

This argument is general and any condition derived from it is necessary. Thus, one is instructed to analyse the condition 𝒫 (ℬ) πœ– = 0 before solving the equations of motion. As a particular example, and to make contact with the discussions in Section 3.1.1, consider 𝒩 = 1 d = 11 supergravity. The only fermionic degrees of freedom are the gravitino components M Ψa = E aΨM. Their supersymmetry transformation is [466Jump To The Next Citation Point]

( 1 ) 1 ( ) δΨa = ∂a + --ωabcΓ bc πœ– − ---- Γ abcde − 8δabΓ cde Rbcdeπœ– . (203 ) 4 288
Solving the supersymmetry preserving condition δΨa = 0 in the M2-brane and M5-brane backgrounds determines the Killing spinors of these solutions to be [466, 467]
M2 − brane πœ– = U− 1βˆ•6πœ–∞ with Γ 012πœ–∞ = ±πœ–∞ , (204 ) − 1βˆ•12 M5 − brane πœ– = U πœ–∞ with Γ 012345πœ–∞ = ± πœ–∞ . (205 )
A similar answer is found for all D-branes in 𝒩 = 2 d = 10 type IIA/B supergravities.

The same question for brane effective actions is treated in a conceptually analogous way. The subspace of bosonic configurations ℬ defined by πœƒ = 0 is compatible with the brane equations of motion. Preservation of supersymmetry requires sπœƒ|ℬ = 0. The total transformation sπœƒ is given by

sπœƒ = δ πœƒ + πœ– + Δ πœƒ + ξμ∂ πœƒ , (206 ) κ μ
where δκπœƒ and ξμ∂μ πœƒ stand for the kappa symmetry and world volume diffeomorphism infinitesimal transformations and Δ πœƒ for any global symmetries different from supersymmetry, which is generated by the Killing spinors πœ–. When restricted to the subspace ℬ of bosonic configurations,
δκπœƒ|ℬ = (πŸ™ + Γ κ|ℬ) κ, (207 ) Δ πœƒ|ℬ = 0 , (208 )
one is left with
sπœƒ | = (1 + Γ |) κ + πœ–. (209 ) ℬ κℬ
This is because Δ πœƒ describes linearly realised symmetries. Thus, kappa symmetry and supersymmetry transformations do generically not leave the subspace ℬ invariant.

We are interested in deriving a general condition for any bosonic configuration to preserve supersymmetry. Since not all fermionic fields πœƒ are physical, working on the subspace πœƒ = 0 is not precise enough for our purposes. We must work in the subspace of field configurations being both physical and bosonic [85Jump To The Next Citation Point].This forces us to work at the intersection of πœƒ = 0 and some kappa symmetry gauge fixing condition. Because of this, I find it convenient to break the general argument into two steps.

  1. Invariance under kappa symmetry. Consider the kappa-symmetry gauge-fixing condition π’«πœƒ = 0, where 𝒫 stands for any field independent projector. This allows us to decompose the original fermions according to
    πœƒ = 𝒫 πœƒ + (πŸ™ − 𝒫)πœƒ . (210)
    To preserve the kappa gauge slice in the subspace ℬ requires
    s𝒫 πœƒ|ℬ = 𝒫 (πŸ™ + Γ κ|ℬ)κ + 𝒫 πœ– = 0. (211)
    This determines the necessary compensating kappa symmetry transformation κ(πœ–) as a function of the background Killing spinors.
  2. Invariance under supersymmetry. Once the set of dynamical fermions (πŸ™ − 𝒫)πœƒ is properly defined, we ask for the set of global supersymmetry transformations preserving them
    s(πŸ™ − 𝒫 )πœƒ|ℬ = 0. (212)
    This is equivalent to
    (πŸ™ + Γ κ|ℬ)κ(πœ–) + πœ– = 0 (213)
    once Eq. (211View Equation) is taken into account. Projecting this equation into the (πŸ™ − Γ κ|ℬ) subspace gives condition
    Γ | πœ– = πœ–. (214) κ ℬ
    No further information can be gained by projecting to the orthogonal subspace (πŸ™ + Γ κ|ℬ).

I will refer to Eq. (214View Equation) as the kappa symmetry preserving condition. It was first derived in [85]. This is the universal necessary condition that any bosonic on-shell brane configuration {Ο•i} must satisfy to preserve some supersymmetry.

Table 5: Set of kappa symmetry matrices Γ κ evaluated in the bosonic subspace of configurations ℬ.


Bosonic kappa symmetry matrix


Γ κ|ℬ = 3!√−1detπ’’πœ–μνργμνρ


√---vμγμ----[ √− det𝒒 μ1μ2 &tidle; ν Γ κ|ℬ = − det(𝒒+ &tidle;H) 2 γ H μ1μ2 + γ νt

] − 1-πœ–μ1...μ5νvν γμ ...μ 5! 1 5

IIA Dp-branes

-----1-----∑ l+1 β„± Γ κ|ℬ = √ −-det(𝒒+-β„±) l=0γ2l+1Γβ™― ∧ e

IIB Dp-branes

Γ | = √----1-----∑ γ τliτ ∧ eβ„± κ ℬ − det(𝒒+ β„±) l=0 2l 3 2

In Table 5, I evaluate all kappa symmetry matrices Γ κ in the subspace of bosonic configurations ℬ for future reference. This matrix encodes information

  1. on the background, both explicitly through the induced world volume Clifford valued matrices γ = E aΓ = ∂ XmE aΓ μ μ a μ m a and the pullback of spacetime fields, such as 𝒒, β„± or &tidle; H, but also implicitly through the background Killing spinors πœ– solving the supergravity constraints 𝒫 (πœ–) = 0, which also depend on the remaining background gauge potentials,
  2. on the brane configuration {Ο•i}, including scalar fields Xm (σ ) and gauge fields, either V1 or V2, depending on the brane under consideration.

Just as in supergravity, any solution to Eq. (214View Equation) involves two sets of conditions, one on the space of configurations {Ο•i} and one on the amount of supersymmetries. More precisely,

  1. a set of constraints among dynamical fields and their derivatives, i i fj(Ο• ,∂Ο• ) = 0,
  2. a set of supersymmetry projection conditions, 𝒫 ′iπœ–∞ = 0, with 𝒫′i being projectors, reducing the dimensionality of the vector space spanned by the original πœ–∞.

The first set will turn out to be BPS equations, whereas the second will determine the amount of supersymmetry preserved by the combined background and probe system.

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