环路空间上旋量子的 Connes 融合
Connes fusion of spinors on loop space
AI总结：
本文通过将环路空间上的旋量丛实现为von Neumann代数双模丛，利用Connes融合逐纤维构造了覆盖环路融合的积，建立了string结构、环路融合与Fock空间Connes融合的新关系。
摘要：
string 流形的环路空间支持一个无限维 Fock 空间丛，它是 spin 流形上旋量丛的类比。环路空间上的这个旋量丛在二维 sigma 模型的描述中作为超弦位形空间上的态丛出现。我们构造了该丛上的一个积，它覆盖了环路的融合，即两个环路沿公共线段的合并。为此，我们将其展示为某个 von Neumann 代数丛上的双模丛，并利用 von Neumann 双模的 Connes 融合逐纤维地实现了我们的积。我们的主要技术是建立 string 结构、环路融合与 Fock 空间的 Connes 融合之间的一种新关系。环路空间上旋量丛上的融合积由 Stolz 和 Teichner 提出，作为探索广义上同调理论、函子场论与指标理论之间关系这一计划的一部分。它与超弦的裤子对世界面、相应的光滑函子场论向下扩展到点，以及底层 string 流形上的高维范畴丛即 stringor 丛有关。
英文摘要：
The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of 2-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle covering the fusion of loops, i.e., the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish a novel relation between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as a part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.
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