亚纯开弦顶点代数与来自Courant代数胚的扭曲模
Meromorphic open-string vertex algebras and twisted modules from Courant algebroids
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中文总结 AI 辅助
本文从带广义度量和联络的传递Courant代数胚构造亚纯开弦顶点代数及其扭曲模，实现Dirac算子和广义Hodge Laplacian的顶点算子表示，并推广到精确情形下的微分形式。
中文摘要 AI 辅助
我们从光滑流形上装备广义度量和选定广义Levi-Civita联络的传递Courant代数胚E出发，构造了一个1/2Z分次的亚纯开弦顶点代数的层。利用传递锚定丛的完整性原理，我们将这些代数实现为由E的玻色-费米仿射化构造的张量代数丛的平行截面。对于具有适当Clifford模丛数据的偶秩E，我们构造了典范扭曲模的层，其中包含权零的典范加权旋量丛。平行张量的结合代数通过E-联络的迭代作用在旋量截面上，该联络结合了协变微分和Clifford乘法，混合项包含了玻色联络对费米张量的作用。由逆Courant配对和逆广义度量构造的显著态给出顶点算子分量，它们作用于嵌入的旋量截面，分别作为典范Dirac生成算子、其形式伴随算子以及它们的反对易子（即广义Hodge Laplacian）。在具有三形式H和外代数Clifford模的精确Courant情形下，这些算子分别化为d-H∧、其伴随算子以及扭曲Hodge Laplacian，当H=0时约化为普通de Rham算子和Hodge算子。这将在研究二维超对称非线性sigma模型的纲领内，将几何微分算子的顶点算子实现从函数推广到加权旋量和微分形式。
英文摘要
We construct a sheaf of $\frac{1}{2}\mathbb{Z}$-graded meromorphic open-string vertex algebras from a transitive Courant algebroid $E$ equipped with a generalized metric and a chosen generalized Levi-Civita connection on a smooth manifold. Using a holonomy principle for transitive anchored bundles, we realize these algebras as parallel sections of a tensor algebra bundle built from bosonic-fermionic affinizations of $E$. For even-rank $E$ with suitable Clifford module bundle data, we construct a sheaf of canonically twisted modules containing the canonical weighted spinor bundle in weight zero. The associative algebra of parallel tensors acts on spinor sections through iterates of an $E$-connection combining covariant differentiation and Clifford multiplication, with mixed terms incorporating the action of the bosonic connection on fermionic tensors. Distinguished states built from the inverse Courant pairing and inverse generalized metric give vertex-operator components, acting on embedded spinor sections as the canonical Dirac generating operator, its formal adjoint, and their anticommutator, the generalized Hodge Laplacian. In the exact Courant case with three-form $H$ and the exterior algebra Clifford module, the operators specialize to $d-H\wedge$, its adjoint, and the twisted Hodge Laplacian, reducing to the ordinary de Rham and Hodge operators when $H=0$. This extends the vertex-operator realization of geometric differential operators from functions to weighted spinors and differential forms, within the program of studying two-dimensional supersymmetric nonlinear sigma models.
发表机构
- Rutgers University（罗格斯大学）
- Sun Yat-Sen University（中山大学）
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