Flat Dp Boundary State Supercharge Gluing

Let ${\mathcal{S}}_L$ and ${\mathcal{S}}_R$ be the real 16-dimensional left- and right-moving Majorana–Weyl supersymmetry spaces, of opposite chiralities in Type IIA and equal chiralities in Type IIB. For a static flat Dp boundary state, adopt as model data an invertible real-linear map $K_p\in {\mathrm{Iso}}_{\mathbb{R}}({\mathcal{S}}_R,{\mathcal{S}}_L)$, where ${\mathrm{Iso}}_{\mathbb{R}}$ denotes the set of bijective $\mathbb{R}$-linear maps. The unbroken target-supersymmetry pairs are exactly the graph $\{(K_p{ϵ}_R,{ϵ}_R):{ϵ}_R\in {\mathcal{S}}_R\}$, equivalently ${ϵ}_L=K_p{ϵ}_R$. Reversing the Dp orientation replaces $K_p$ by $-K_p$. This is the boundary-state supersymmetry condition, not a consequence of the bosonic endpoint conditions alone; no matrix-to-map identification is used in this adopted abstract formulation.

Polchinski, §2

Polchinski, Chaudhuri, and Johnson, Lecture III §§3.1–3.3