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Type Iia Supergravity Killing Spinor Equations

Physical scope

Local massless type IIA supersymmetry criterion in Hamilton's string-frame Clifford convention on a contractible Lorentzian spin patch. The definition requires a bosonic solution of the displayed field equations as well as a nonzero Killing spinor; it asserts no existence, global extension, uniqueness, or classification.

Definition

Let $U$ be a contractible open subset of an oriented and time-oriented smooth ten-dimensional Lorentzian manifold with signature $(- + \dots +)$ . Use the length, field, and string-frame conventions of the adopted massless type IIA action. Supply a principal $S p i n (9, 1)$ -bundle $P \to U$ lifting its oriented, time-oriented orthonormal frame bundle, and the real 32-dimensional Majorana representation $ρ$ of the physical Clifford algebra with generators ${Γ_{a}, Γ_{b}} = 2 η_{a b}$ . The spinor bundle is $S = P \times_{ρ} R^{32}$ , and $\nabla$ is its Levi-Civita spin connection. In an oriented orthonormal frame set $Γ_{11} : = Γ_{0} Γ_{1} \dots Γ_{9}$ and, for a $p$ -form $F$ , define $F \cdot ψ = (1 / p!) F_{a_{1} \dots a_{p}} Γ^{a_{1} \dots a_{p}} ψ$ , with antisymmetrized gamma products and Hamilton’s orientation convention. Clifford multiplication by a vector $X$ is denoted $X \cdot$ .

Take smooth bosonic fields $g_{s}, Φ, B_{2}, C_{1}, C_{3}$ and their local field strengths in the adopted massless action convention. By definition in this item, a local supersymmetric bosonic solution on $U$ is a solution of the source-free bosonic equations stated in the Type IIA Bosonic Field Equations result for which there exists a nonzero smooth $ϵ \in Γ (S)$ satisfying, for every smooth vector field $X$ , $0 = \nabla_{X} ϵ - \frac{1}{4} (i_{X} H_{3}) \cdot Γ_{11} ϵ + \frac{1}{8} e^{Φ} [{\tilde{G}}_{4} \cdot (X \cdot ϵ) + G_{2} \cdot (X \cdot Γ_{11} ϵ)],$ and $0 = [\frac{1}{4} e^{Φ} G_{2} - \frac{1}{3} (d Φ) \cdot Γ_{11} + \frac{1}{12} e^{Φ} {\tilde{G}}_{4} \cdot Γ_{11} + \frac{1}{6} H_{3}] \cdot ϵ .$ The ten-dimensional IIA multiplet has two opposite-chirality Majorana–Weyl supersymmetry parameters; $ϵ$ is their combined 32-real-component Majorana field. This convention does not assert a spin lift on a general non-spin spacetime.

Definition references

M. J. D. Hamilton, §§3.2–3.6 and §6.2, Theorem 6.8

Type Iia Two Derivative Supergravity Action

Type Iia Bosonic Field Equations

Facts & Assumptions

Given: A contractible spin patch with its supplied $S p i n (9, 1)$ lift, smooth fields in the Hamilton massless convention, and a nonzero smooth Majorana spinor candidate.

[A1]

Supergravity Conventions And Normalizations (current batch-56 carrier and current ready Step 1 record inspected; its stated hypotheses match this use): use Lorentzian signature $(- + \dots +)$ , the declared ten-dimensional IIA opposite-chirality Majorana–Weyl pair, and a source-fixed Clifford/Hodge convention.

[A2]

Type Iia Two Derivative Supergravity Action adopts the bosonic fields and local field-strength definitions used in the two displayed spinor operators.

[F1]

Hamilton, §§3.2–3.6: on a supplied oriented, time-oriented Lorentzian spin patch, the associated spinor bundle, spin connection, Clifford multiplication, chirality operator, and real Majorana spinor module are defined in the convention fixed above.

[F2]

Type Iia Bosonic Field Equations: the phrase “bosonic solution” means all its listed source-free Euler–Lagrange equations and potential Bianchi identities hold.

Verification

1.1givenA1F1def. spinor and Clifford definitions

The contraction $i_{X} H_{3}$ is a two-form, so its Clifford action and the action of $\nabla_{X}$ both map sections of $S$ to sections of $S$ . For each $p$ -form $F$ , the defined antisymmetrized Clifford product $F \cdot$ is an endomorphism of $S$ , and $X \cdot$ is also such an endomorphism. Therefore every term of the gravitino equation is a section of $S$ .

1.2givenA2F1F2def. spinor and Clifford definitions

The terms $e^{Φ} G_{2}$ , $(d Φ) \cdot Γ_{11}$ , $e^{Φ} {\tilde{G}}_{4} \cdot Γ_{11}$ , and $H_{3}$ act by Clifford endomorphisms of $S$ ; hence the algebraic equation is a well-defined section of $S$ . The conjunction with [F2] is part of the definition, so a Killing-spinor equation alone is not called an on-shell supersymmetric bosonic solution here.

2.1givenF1F2step 1.1step 1.2∎

Hamilton’s source criterion is precisely the existence of a nonzero spinor satisfying these two systems for a bosonic solution of the IIA field equations. Thus the local criterion is fixed only after the bosonic on-shell hypothesis, spin lift, form conventions, and Clifford convention have all been supplied; it makes no assertion that such a spinor exists.

Depends on

Dependency tree · 2 levels

Sources

M. J. D. Hamilton, The field and Killing spinor equations of M-theory and type IIA/IIB supergravity in coordinate-free notation