论文标题
拉直Frobenius-Schur指示器
Straightening Out the Frobenius-Schur Indicator
论文作者
论文摘要
Frobenius-Schur指示器是一个参数$κ_A= \ pm 1 $ $分配给TQFT中的每个自偶粒子$ a $。如果$κ_A$是负的，那么在$ a $ a $ type $ a $的粒子的世界线中弄平了一个时间表的曲折Zag，可能会导致一个负符号，在这种情况下，与该图相关的振幅并不是不变的变形。这引起了人们对时空变形的简单理论的拓扑不变性的困惑。我们澄清说，鉴于带有负frobenius-schur指标的TQFT，通常使用两种不同的惯例来解释时空图为物理幅度，其中只有一个是同位素不变的。我们阐明了基于Chern-Simons理论的TQFTS在什么意义上，具有负面的Frobenius-Schur指标是同位素不变的，我们解释了Frobenius-Schur指标如何与Chern-Simons理论中的世界界线密切相关。此外，在对图表的无异构性不变的解释中，我们展示了通常可以调用簿记的窍门，以将负标志推向循环的图表值（“环重”），从而使大多数图表的评估都不会从伸直的Zig-Zags和最后一个spepss septemus中添加到均匀的Zig-Zags和Simus。我们解释了可能的条件。然后，我们进一步研究了理论使平面时空图的完全同位素不变性所需的条件，并发现，如果我们成功将Zig-Zags的符号从循环重量推到了循环重量，那么唯一可能的障碍物是由与角落有关的对象给出的，该对象被称为“第三Frobenius-Schurius Schururidainer”。我们最终讨论了这为我们为编织理论提供完整的同位素不变性的程度。
The Frobenius-Schur indicator is a parameter $κ_a=\pm 1$ assigned to each self-dual particle $a$ in a TQFT. If $κ_a$ is negative then straightening out a timelike zig-zag in the worldline of a particle of type $a$ can incur a minus sign and in this case the amplitude associated with the diagram is not invariant under deformation. This has caused some confusion about the topological invariance of even simple theories to space-time deformations. We clarify that, given a TQFT with negative Frobenius-Schur indicators, there are two distinct conventions commonly used to interpret a spacetime diagram as a physical amplitude, only one of which is isotopy invariant. We clarify in what sense TQFTs based on Chern-Simons theory with negative Frobenius-Schur indicators are isotopy invariant, and we explain how the Frobenius-Schur indicator is intimately linked with the need to frame world-lines in Chern-Simons theory. Further, in the non-isotopy-invariant interpretation of the diagram algebra we show how a trick of bookkeeping can usually be invoked to push minus signs onto the diagrammatic value of a loop (the "loop weight"), such that most of the evaluation of a diagram does not incur minus signs from straightening zig-zags, and only at the last step minus signs are added. We explain the conditions required for this to be possible. We then further examine what is required in order for a theory to have full isotopy invariance of planar spacetime diagrams, and discover that, if we have successfully pushed the signs from zig-zags onto the loop weight, the only possible obstruction to this is given by an object related to vertices, known as the "third Frobenius-Schur indicator". We finally discuss the extent to which this gives us full isotopy invariance for braided theories.
