论文标题
光谱作用中的循环旋转和一环校正
Cyclic cocycles and one-loop corrections in the spectral action
论文作者
论文摘要
我们对有关频谱作用和单循环量化的循环共体的最新结果进行了可理解的综述。我们表明，当光谱受到仪表电位的扰动时，可以将光谱作用写成一系列的Chern-Simons动作和所有订单的Yang-Mills动作。在奇数中，广义的Chern-Simons表格是针对奇数$（b，b）$ - cocycle集成的，而在均匀的订单中，曲率的功率也与$（b，b）$ - 同伴也是hochschschild Cocycles的cocycles集成在一起。在这两种情况下，Hochschild Cochains均来自泰勒系列的taylor系列扩展tr $（f（d+v））$ $ v =π_d（a）$的功率，但与泰勒的扩展不同，我们以$ a $ $ a $的形式的越来越多地扩展。然后，我们分析了非交通性几何形状中光谱作用的扰动量化，并将其单环的重差异性作为量规理论。我们表明，单循环反对者是相同的Chern-simons-yang-mills形式，因此可以安全地从光谱作用中减去它们。适当的病房身份将扮演至关重要的角色，从而使量子理论在一个循环中的完全光谱进行表述。
We present an intelligible review of recent results concerning cyclic cocycles in the spectral action and one-loop quantization. We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simons actions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integrated against an odd $(b,B)$-cocycle, whereas, in the even orders, powers of the curvature are integrated against $(b,B)$-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action Tr$(f(D+V))$ in powers of $V=π_D(A)$, but unlike the Taylor expansion we expand in increasing order of the forms in $A$. We then analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability as a gauge theory. We show that the one-loop counterterms are of the same Chern-Simons-Yang-Mills form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.
