A Note on Lie-Lorentz Derivatives

TL;DR

Ort\’in defines a Lie-Lorentz derivative $\mathbb{L}_{k}$ that extends the spinor Lie derivative to general Lorentz tensors, with $\mathbb{L}_{k}T = k^{\rho}\nabla_{\rho}T + {\frac{1}{2}}\nabla_{[a}k_{b]}\Gamma_{r}(M^{ab})T$ for pure Lorentz tensors and the appropriate world-index extension; it preserves Clifford multiplication, the covariant derivative, and satisfies the Leibniz rule and the Lie-bracket closure $[\mathbb{L}_{k_{1}},\mathbb{L}_{k_{2}}] = \mathbb{L}_{[k_{1},k_{2}]}$. In supergravity, the Lie-Lorentz derivative provides the bosonic generator of vacuum isometries via $\delta_{P}(k) \equiv -\mathbb{L}_{k}$ with $k^{a} = -i \bar{\varepsilon}_{1}\gamma^{a}\varepsilon_{2}$, so the SUSY commutator on Killing spinor vacua becomes $[\delta_{Q}(\varepsilon_{1}), \delta_{Q}(\varepsilon_{2})] = \delta_{P}(k)$ and leaves the Vielbein invariant. For gauged $N=2$, $d=4$, the commutator includes a gauge transformation, $\delta_{P}(k) + \delta_{e}(\chi)$, reflecting the action on additional fields. A general prescription $[\delta_{Q}(\varepsilon), \delta_{P}(k)] = \delta_{Q}(\mathbb{L}_{k}\varepsilon)$ follows because $\mathbb{L}_{k}$ preserves the supercovariant derivative and maps Killing spinors to Killing spinors, enabling a consistent construction of the vacuum superalgebra. The framework generalizes to higher dimensions and clarifies how isometries act on supergravity fields and their algebras.

Abstract

The definition of ``Lie derivative'' of spinors with respect to Killing vectors is extended to all kinds of Lorentz tensors. This Lie-Lorentz derivative appears naturally in the commutator of two supersymmetry transformations generated by Killing spinors and vanishes for Vielbeins. It can be identified as the generator of the action of isometries on supergravity fields and its use for the calculation of supersymmetry algebras is revised and extended.