自由环群的拟周期路和弦2群模型
Quasi-periodic paths and a string 2-group model from the free loop group
摘要：
在本文中，我们解决了一个模型的存在性的问题，作为一个严格的Lie-2-组使用自由循环组$LSpin$（或更一般的$LG$紧凑的简单单连通李群$G$）的弦2-组。Baez-Crans-Stevenson-Schreiber用一个基环群为弦2-群构造了一个模型。这有一个缺点，它不承认一个行动的循环组$S^1$，这是至关重要的，例如在建设一个（假设）$S^1$ -等变指数的（更高）微分算子。本文指出，用$LG$构造弦2-群的严格模型实际上是有障碍的。我们证明了一个无限维流形的光滑路径承认没有李群结构，也没有非平凡的李交叉模类似的BCSS模型使用的自由循环群的通用中心扩展。然后，我们构造了下一个最好的东西，即一个相干模型的字符串2-组使用自由循环组，与所有结构的明确公式。这对于我们在最后简要讨论的弦群的预期表示论尤其重要。
英文摘要：
In this paper we address the question of the existence of a model for the string 2-group as a strict Lie-2-group using the free loop group $LSpin$ (or more generally $LG$ for compact simple simply-connected Lie groups $G$). Baez-Crans-Stevenson-Schreiber constructed a model for the string 2-group using a based loop group. This has the deficiency that it does not admit an action of the circle group $S^1$, which is of crucial importance, for instance in the construction of a (hypothetical) $S^1$-equivariant index of (higher) differential operators. The present paper shows that there are in fact obstructions for constructing a strict model for the string 2-group using $LG$. We show that a certain infinite-dimensional manifold of smooth paths admits no Lie group structure, and that there are no nontrivial Lie crossed modules analogous to the BCSS model using the universal central extension of the free loop group. Afterwards, we construct the next best thing, namely a coherent model for the string 2-group using the free loop group, with explicit formulas for all structure. This is in particular important for the expected representation theory of the string group that we discuss briefly in the end.
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