协变变分及其应用
Covariant variation and its applications
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中文总结 AI 辅助
该研究定义了张量场的协变变分算子，推导其性质并与科斯曼导数结合得到总协变变分，还将其异常应用于四维零超曲面的电磁螺旋度通量及任意维度的p-形式。
中文摘要 AI 辅助
我们通过将张量场的李导数与度规变分相结合，定义了张量场的协变变分。该算子保持度规、缩并和霍奇对偶性，但因异常导致其对易子不闭合。我们推导了其代数与几何性质，并将其与科斯曼导数（Kosmann derivative）进行比较。将协变变分与科斯曼导数相结合，可得到具有时空和洛伦兹结构的场的总协变变分，所有这些场均属于度规李导数。此外，我们引入了包含仿射联络、李导数和协变变分的扩展算子族。从协变变分沿超旋转的异常出发，四维零超曲面上会出现电磁螺旋度通量。我们还将协变变分及其异常应用于任意时空维度的张量场，尤其关注d=2p+2维度中的p-形式。
英文摘要
We define a covariant variation of tensor fields by combining its Lie derivative with the metric variation. This operator preserves the metric, contractions, and Hodge duality, but its commutator is not closed due to an anomaly. We derive its algebraic and geometric properties, and compare it with the Kosmann derivative. Combining the covariant variation with Kosmann derivative gives total covariant variation for the fields with both spacetime and Lorentz structure, all of which belong to the metric Lie derivative. Moreover, we introduce families of extended operators which contain the affine connection, Lie derivative, and covariant variation. From the anomaly of the covariant variation along the superrotation, an electromagnetic helicity flux appears at null hypersurfaces in four dimensions. We also apply the covariant variation and its anomaly to tensor fields in arbitrary spacetime dimensions, and especially focus on the $p$-forms in $d=2p+2$ dimensions.
