Heterotic Leading Order Killing Spinor System

Let $M$ be a smooth oriented ten-dimensional Lorentzian manifold with signature $(-,+,\dots ,+)$ and supplied spin structure, and let $U\subseteq M$ be a nonempty open set on which the bosonic fields below are smooth. Write $S_{+}$ and $S_{-}$ for the real Majorana–Weyl spinor bundles on $M$ and use their restrictions to $U$. Let $P_G\to M$ be the heterotic gauge bundle from the preceding postulate and set $E:=\mathrm{ad}{P}_G=P_G{\times }_{\mathrm{A}\mathrm{d}}\mathfrak{g}$. Its smooth connection $A$ restricts to $U$ with curvature $F_A\in {\mathrm{\Omega }}^2(U,E)$. Take a smooth local bosonic heterotic field configuration $(g,\mathrm{\Phi },B_2,A)$ on $U$, with $B_2\in {\mathrm{\Omega }}^2(U)$ and local $H_3\in {\mathrm{\Omega }}^3(U)$ obeying the first-order source-free Bianchi convention. The smooth fermion fields have types $\psi \in \mathrm{\Gamma }(S_{+}{\mathrm{\mid }}_U\otimes T^{\ast }U)$, $\lambda \in \mathrm{\Gamma }(S_{-}{\mathrm{\mid }}_U)$, and $\chi \in \mathrm{\Gamma }(S_{+}{\mathrm{\mid }}_U\otimes E{\mathrm{\mid }}_U)$; the supersymmetry parameter has positive chirality, with its formal section space and nonvanishing condition specified below. These are supplied bundle data; no spin-structure existence is asserted.

For a smooth vector field $V$ on $U$, define the tangent connections by $({\mathrm{\nabla }}_M^{\pm }V{)}^N=({\mathrm{\nabla }}_M^{\mathrm{L}\mathrm{C}}V{)}^N\pm \frac{1}{2}(H_3{)}_M{}^N{}_P{V}^P$, with the raised index formed using $g{\mathrm{\mid }}_U$; $R_{+}$ is the curvature of ${\mathrm{\nabla }}^{+}$ used in the Bianchi identity. Let ${\mathrm{\nabla }}^{-}$ be the connection induced on every covariant tensor slot and on differential forms. For every smooth 4-form $K\in {\mathrm{\Omega }}^4(U)$, define the pointwise source-convention divergence ${\mathrm{\nabla }}_{-}^{†}K\in {\mathrm{\Omega }}^3(U)$ by $({\mathrm{\nabla }}_{-}^{†}K{)}_{mnp}=-g^{qr}({\mathrm{\nabla }}_q^{-}K{)}_{rmnp}$, contracting the first form index. This pointwise definition also agrees with the source's formal-adjoint convention on compactly supported test forms; no compact-support assumption is imposed on its later argument. Define the 3-form

$P=-\frac{1}{4}{e}^{2\mathrm{\Phi }}{\mathrm{\nabla }}_{-}^{†}(e^{-2\mathrm{\Phi }}d{H}_3),$

Use the Clifford multiplication $c:T^{\ast }U\to \mathrm{End}(S_{+}{\mathrm{\mid }}_U\oplus S_{-}{\mathrm{\mid }}_U)$ determined by $g$ and the supplied spin structure, with $c(\xi )c(\eta )+c(\eta )c(\xi )=2g^{-1}(\xi ,\eta )I$. In a coordinate chart write ${\gamma }^m=c(d{x}^m)$ and ${\gamma }_m=g_{mn}{\gamma }^n$. For a smooth $p$-form $Q$, set $\mathrm{cl}(Q)=(1\mathrm{/}p!)Q_{m_1\dots m_p}{\gamma }^{m_1\dots m_p}$, where ${\gamma }^{m_1\dots m_p}={\gamma }^{[m_1}\cdots {\gamma }^{m_p]}$ has unit-weight antisymmetrization, and set ${\gamma }_{mn}={\gamma }_{[m}{\gamma }_{n]}$. The notation $\mathrm{cl}(Q)$ is the source's slash operation.

Use formal power series in $alph{a}^{\mathrm{\prime }}$ with no negative powers, not convergent families at a numerical value of $alph{a}^{\mathrm{\prime }}$. For a fixed smooth vector bundle $V\to U$, define $\mathrm{\Gamma }(V)[\text{\hspace{0.17em}⁣}[{\alpha }^{\mathrm{\prime }}]\text{\hspace{0.17em}⁣}]:=\{{\sum }_{j\ge 0}{\alpha }^{\mathrm{\prime }j}{s}_j:s_j\in \mathrm{\Gamma }(V)\}$, with addition and multiplication by scalar series defined coefficientwise and by the Cauchy product, respectively. All tensor fields have the corresponding formal section types with smooth coefficients; coefficient units compensate the powers of the length-squared parameter ${\alpha }^{\mathrm{\prime }}$. The metric has a nondegenerate Lorentzian constant term $g_0$, so its inverse is defined recursively as a formal series. Keep the gauge bundle and spin structure fixed, write $A=A_0+{\sum }_{j\ge 1}{\alpha }^{\mathrm{\prime }j}{a}_j$ with $A_0$ a smooth connection and $a_j\in {\mathrm{\Omega }}^1(U,E)$, and supply bundle identifications with the reference spinor bundles of $g_0$ for the formal metric deformation. Using these identifications, interpret Clifford multiplication and the induced connections as formal operators on the fixed reference bundles, satisfying the displayed Clifford relation order-by-order. In particular, require $ϵ={\sum }_{j\ge 0}{\alpha }^{\mathrm{\prime }j}{ϵ}_j\in \mathrm{\Gamma }(S_{+}{\mathrm{\mid }}_U)[\text{\hspace{0.17em}⁣}[{\alpha }^{\mathrm{\prime }}]\text{\hspace{0.17em}⁣}]$ with ${ϵ}_0(x)\mathrm{\ne }0$ for every $x\in U$. A parameter divisible by ${\alpha }^{\mathrm{\prime }}$ is inadmissible. Derivatives act coefficientwise; $e^{\pm 2\mathrm{\Phi }}$ is expanded about its smooth constant term $e^{\pm 2{\mathrm{\Phi }}_0}$, and $O({\alpha }^{\mathrm{\prime }k})$ means that all coefficients of degrees below $k$ vanish. The Bianchi relation gives $d{H}_3=O({\alpha }^{\mathrm{\prime }})$ and ${\mathrm{\nabla }}^{-}={\mathrm{\nabla }}_0^{-}+O({\alpha }^{\mathrm{\prime }})$, so $P=O({\alpha }^{\mathrm{\prime }})$ and the displayed ${\alpha }^{\mathrm{\prime }}P$ corrections contribute at order ${\alpha }^{\mathrm{\prime }2}$. Accordingly, these source transformations are retained through $O({\alpha }^{\mathrm{\prime }2})$, with $O({\alpha }^{\mathrm{\prime }3})$ omitted. Evaluate the variations on a bosonic background, meaning $\psi =\lambda =\chi =0$. Read each equality below modulo $O({\alpha }^{\mathrm{\prime }3})$:

For the adjoint-valued curvature $F_A\in {\mathrm{\Omega }}^2(U,E)$, define the bundle-valued Clifford contraction by

$({\mathrm{cl}}_E(F_A)ϵ)(x)=\frac{1}{2}{\gamma }^{mn}ϵ(x)\otimes (F_A{)}_{mn}(x)\in S_{+,x}\otimes E_x\mathrm{.}$

$\delta {\psi }_k={\mathrm{\nabla }}_k^{\mathrm{L}\mathrm{C}}ϵ-\frac{1}{8}(H_3{)}_k{}^{mn}{\gamma }_{mn}ϵ+\frac{{\alpha }^{\mathrm{\prime }}}{4}{P}_k{}^{mn}{\gamma }_{mn}ϵ,$

$\delta \lambda =(\mathrm{cl}(d\mathrm{\Phi })-\frac{1}{2}\mathrm{cl}(H_3)+\frac{3{\alpha }^{\mathrm{\prime }}}{2}\mathrm{cl}(P))ϵ,\delta \chi ={\mathrm{cl}}_E(F_A)ϵ\mathrm{.}$

A bosonic formal background is locally supersymmetric in this definition exactly when there exists a parameter $ϵ$ of the specified formal section type with nowhere-vanishing ${ϵ}_0$ such that all three displayed variations have zero coefficients in degrees $0$, $1$, and $2$ on $U$. The equations are source-specific heterotic formulation data, not an all-orders system or a theorem that the background exists or solves the bosonic equations.

Jock McOrist and Sebastien Picard, §2.1

Washington Taylor, §2.2

Supergravity Conventions And Normalizations

Heterotic Type I Ten Dimensional N1 Supergravity Action

Heterotic Green Schwarz Bianchi And Anomaly Cancellation

McOrist–Picard's paper studies specialized $SU(3)$ geometry after stating these ten-dimensional equations. No $SU(3)$ consequence is imported here. Vanishing variations alone do not prove the full equations of motion. The Type-I system is not inferred from this heterotic definition by S-duality.

The convention input used here is Lorentzian signature $(-,+,\dots ,+)$, natural units, the named length-squared ${\alpha }^{\mathrm{\prime }}$ convention, a supplied spin structure, and one ten-dimensional $N=1$ Majorana–Weyl supersymmetry parameter. The full current definition of Supergravity Conventions And Normalizations was inspected, matches those conventions, and is now listed in batch 56's A-page manifest. This definition uses those adopted conventions and retains the source-specific transformations through second order in ${\alpha }^{\mathrm{\prime }}$, with third-order terms omitted. The linked action postulate supplies the separate heterotic field branch; the linked Bianchi postulate supplies the first-order $H_3$ convention.