Type IIB Supergravity Killing Spinor Equations

Assume the source-free bosonic solution and convention of Type IIB Two Derivative Supergravity Pseudoaction, Type IIB Five Form Self Duality Constraint, and Type IIB Field Equations with Five Form Self Duality, together with the spinor bundle data of Lorentzian Type IIB Spinor Bundle Data, applied with its metric $g$ identified with the same string-frame metric $g_s$ as the pseudoaction. Thus its full Clifford action, chirality decomposition, and lifted spin connection all use $g_s$; write ${\mathrm{\nabla }}^S:={\mathrm{\nabla }}^{S,g_s}$ and $X^{\mathrm{♭}}:=g_s(X,\cdot )$. Write the two left-handed Majorana–Weyl spinors as a smooth doublet $ϵ=({ϵ}_1,{ϵ}_2)\in \mathrm{\Gamma }(S_{+}\oplus S_{+})$. Let ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, ${\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, ${\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ and set ${\lambda }_1={\sigma }_1$, ${\lambda }_2=i{\sigma }_2$, ${\lambda }_3={\sigma }_3$, acting on the doublet. Define the form-valued matrices

$\tilde{\mathrm{\Omega }}=\frac{1}{8}{e}^{\mathrm{\Phi }}(G_1{\lambda }_2-{\tilde{G}}_3{\lambda }_1+\frac{1}{2}{\tilde{G}}_5{\lambda }_2),\mathrm{\Omega }=e^{\mathrm{\Phi }}(\frac{1}{2}{\tilde{G}}_3{\lambda }_1-G_1{\lambda }_2)\mathrm{.}$

A bosonic solution is called supersymmetric on the patch exactly when there exists a smooth nonzero doublet $ϵ$ such that for every smooth vector field $X$,

${\mathrm{\nabla }}_X^Sϵ+\frac{1}{4}(i_X{H}_3){\lambda }_3\cdot ϵ+\tilde{\mathrm{\Omega }}\cdot (X^{\mathrm{♭}}\cdot ϵ)=0,(d\mathrm{\Phi }+\frac{1}{2}{H}_3{\lambda }_3+\mathrm{\Omega })\cdot ϵ=0.$

Here $i_X$ is contraction, $X^{\mathrm{♭}}=g_s(X,\cdot )$, and Clifford multiplication is extended to form-valued doublet endomorphisms as in Lorentzian Type IIB Spinor Bundle Data. The two-form operator $i_X{H}_3$ preserves chirality. Each component of $\tilde{\mathrm{\Omega }}$ has odd form degree, so its composite with $X^{\mathrm{♭}}$ has even Clifford parity; the gravitino equation therefore takes values in $S_{+}\otimes {\mathbb{R}}^2$. Every term of $d\mathrm{\Phi }+\frac{1}{2}{H}_3{\lambda }_3+\mathrm{\Omega }$ has odd Clifford parity, so the dilatino equation takes values in $S_{-}\otimes {\mathbb{R}}^2$. Both equations use the full supplied Clifford bundle and chirality-preserving connection specified by the spinor-data supplier. This is Hamilton's string-frame convention; the formula is not silently converted to another RR field basis.

Supergravity Conventions And Normalizations

Type IIB Two Derivative Supergravity Pseudoaction

Type IIB Five Form Self Duality Constraint

Type IIB Field Equations with Five Form Self Duality

Lorentzian Type IIB Spinor Bundle Data

Hamilton, §7.2, Theorem 7.9

Hackett-Jones–Smith, §2, equations (1)–(2)

The KSE transcription has been visually checked against Hamilton's printed pp. 44–45. The prefactor in $\tilde{\mathrm{\Omega }}$ is $+1\mathrm{/}8$, as on Hamilton's printed p. 44 and in the component expansion following Proposition 7.10; the earlier track-design minus sign is a transcription error. The auxiliary Hackett-Jones–Smith paper uses a democratic RR field basis and supports the string-frame structure and Clifford convention; its coefficients are not silently transferred to the displayed undoubled Hamilton basis.