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Pure-Spinor Euler Divergence Eight

Example

On the patch and with the conventions of Intrinsic Pure-Spinor Volume and Current Cocycle, take E=ρ∂ρ and Ω=ρ7dρ∧d10u. Then div⁡ΩE=8, every infinitesimal Lorentz/Spin generator has zero divergence, and the Euler current has third pole 8α′/(z−w)3. Sixteen unconstrained spinor pairs instead give ordinary Euler divergence 16 and are a different field space.

Facts & Assumptions

Given: The constrained eleven-coordinate chart, its volume and the positive third-pole convention.

[F1]

Intrinsic Pure-Spinor Volume and Current Cocycle proves div⁡ΩE=8, zero divergence for Spin generators, and the current pole formula.

Proof

1.1F1algebra

Directly, div⁡ΩE=ρ−7∂ρ(ρ7ρ)=ρ−7∂ρ(ρ8)=8. A Spin generator has zero divergence by the corresponding calculation in [F1].

2.1F1step 1.1algebra∎

Substituting E into the current formula in [F1] gives 8α′/(z−w)3. In a sixteen-dimensional unconstrained linear spinor space with flat volume, the Euler vector ∑α=116λα∂λα has divergence ∑α=116∂λαλα=16; it does not represent the eleven-dimensional cotangent-quotiented cone system of [F1].

Depends on

Dependency tree · 2 levels

Sources