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Flat Dp Brane Half BPS Projection

Physical scope

Static flat Type II Dp boundary sector with zero worldvolume flux and no backreaction; conditionally counts the preserved constant target supersymmetries from the flat KSE and boundary-state gluing, without extending to curved, fluxed, or dynamical branes.

Statement

In a source-free flat Type II background, let $S_{L}, S_{R}$ be the two real 16-dimensional Majorana–Weyl supersymmetry spaces of the appropriate IIA/IIB chiralities. For a static flat Dp boundary state, the preserved constant supersymmetries are exactly the graph ${(K_{p} ϵ_{R}, ϵ_{R}) : ϵ_{R} \in S_{R}}$ of the invertible map $K_{p}$ in Flat Dp Boundary State Supercharge Gluing. Hence they form a 16-dimensional real subspace of the 32-dimensional supersymmetry space. Equivalently, with $Γ_{D p} (ϵ_{L}, ϵ_{R}) = (K_{p} ϵ_{R}, K_{p}^{- 1} ϵ_{L}),$ the preserved parameters obey $Γ_{D p} ϵ = ϵ$ , and $Γ_{D p}^{2} = 1$ . This is the conditional half-BPS projection.

Facts & Assumptions

Given: A source-free flat ten-dimensional Type II bosonic solution with constant dilaton and vanishing field strengths, a static flat Dp boundary condition, and the exact source gluing map and KSE interfaces.

[F1]

Flat Dp Boundary State Supercharge Gluing: the unbroken boundary-state supersymmetries obey $ϵ_{L} = K_{p} ϵ_{R}$ , where $K_{p} : S_{R} \to S_{L}$ is invertible.

[F2]

Dp Brane Open String Boundary Condition: a static flat Dp-brane has Neumann tangent and fixed Dirichlet normal endpoint data; Type IIA uses even $p$ and Type IIB odd $p$ .

[A1]

Type Iia Supergravity Killing Spinor Equations: a local on-shell IIA spinor obeys the displayed gravitino and dilatino equations on a supplied spin patch. In the stated flat solution all flux and dilaton-gradient terms vanish, so the gravitino equation reduces to $\nabla_{X} ϵ = 0$ . Its current Step 1 record is ready for its declared dependencies.

[A2]

Type IIB Supergravity Killing Spinor Equations: a local on-shell IIB doublet obeys the displayed equations in the stated spinor bundle. In the stated flat solution $H_{3} = 0$ , $G_{1} = {\tilde{G}}_{3} = {\tilde{G}}_{5} = 0$ , and $d Φ = 0$ , so $\tilde{Ω} = Ω = 0$ and the gravitino equation reduces to $\nabla_{X} ϵ = 0$ . Its current Step 1 record is escalated because its upstream readiness inputs are missing, so use remains conditional.

Proof

1.1givenF1algebra

On $S_{L} \oplus S_{R}$ , define $Γ_{D p}$ by the displayed block map. Since $K_{p}$ is invertible by [F1], both compositions are defined and $Γ_{D p}^{2} (ϵ_{L}, ϵ_{R}) = (K_{p} K_{p}^{- 1} ϵ_{L}, K_{p}^{- 1} K_{p} ϵ_{R}) = (ϵ_{L}, ϵ_{R})$ . Thus it is an involution.

2.1givenF1step 1.1algebra

The fixed-point equation $Γ_{D p} (ϵ_{L}, ϵ_{R}) = (ϵ_{L}, ϵ_{R})$ is equivalent to the two equations $ϵ_{L} = K_{p} ϵ_{R}$ and $ϵ_{R} = K_{p}^{- 1} ϵ_{L}$ ; either one implies the other. The map $ϵ_{R} \mapsto (K_{p} ϵ_{R}, ϵ_{R})$ is therefore a linear bijection from $S_{R}$ to the fixed-point space.

3.1givenF1A1A2F2step 2.1algebra∎

By [A1] or [A2], according to Type IIA or IIB, the flat on-shell KSE reduce to $\nabla_{X} ϵ = 0$ ; in a Cartesian parallel frame these solutions are the constant elements of each chiral 16-real-dimensional space. The boundary-state condition [F1] retains exactly the graph counted in step 2.1, so the preserved space has real dimension $16$ , half of $\dim_{R} (S_{L} \oplus S_{R}) = 32$ . This conclusion remains conditional on current KSE readiness and adopted boundary-state gluing; it asserts nothing about nonflat or fluxed backgrounds.

Flat Dp Brane Half BPS Projection

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources