A mathematical analysis of the axial anomaly
A mathematical analysis of the axial anomaly
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轴向异常的数学分析
DOI：
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发表时间：
2017
影响因子：
1.2
通讯作者：
E. Rabinovich
中科院分区：
文献类型：
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作者：
E. Rabinovich
正如物理学家所熟知的，欧几里得签名中的无质量自由费米子的轴反常是由相应的狄拉克算符的指数给出的。我们使用巴达林-维尔科维斯基(BV)形式主义以及Costello和Gwilliam的等变量子化方法对这一结果进行了新的数学推导。使用这些方法，我们形式化了两种对轴向反常的传统解释，第一种解释是破坏了量子水平上的电流守恒，第二种解释是阻碍了明确定义的费米子配分函数的存在。此外，在Costello和Gwilliam的形式主义中，异常是在某个阻塞-变形复合体中用上同调类来测量的。我们的主要结果表明，在轴对称的情况下，相关复形准同构于时空流形的De Rham复形，并且反常对应于一个平凡的顶级上同调类，当且仅当相应的Dirac算子的指数为零。
As is well known to physicists, the axial anomaly of the massless free fermion in Euclidean signature is given by the index of the corresponding Dirac operator. We use the Batalin–Vilkovisky (BV) formalism and the methods of equivariant quantization of Costello and Gwilliam to produce a new, mathematical derivation of this result. Using these methods, we formalize two conventional interpretations of the axial anomaly, the first as a violation of current conservation at the quantum level and the second as the obstruction to the existence of a well-defined fermionic partition function. Moreover, in the formalism of Costello and Gwilliam, anomalies are measured by cohomology classes in a certain obstruction–deformation complex. Our main result shows that—in the case of the axial symmetry—the relevant complex is quasi-isomorphic to the complex of de Rham forms of the space–time manifold and that the anomaly corresponds to a top-degree cohomology class which is trivial if and only if the index of the corresponding Dirac operator is zero.