非线性西格玛模型量子化的再认识
Quantization of the Nonlinear Sigma Model Revisited
摘要：
我们从严格的数学角度重新审视了二维非线性西格玛模型的微扰量化问题。我们的主要贡献是精确的上同调的问题，消除潜在的异常，可能会出现时，试图保持量子化下的对称性。我们考虑的对称性是双重的：（i）一般目标流形的同构协方差;（ii）当目标流形是齐次空间时，等距的传递群。我们表明，有没有异常的情况下（i）和（ii）也是异常的目标齐性空间上的附加假设下，与Friedan的工作。我们进行了一些明确的计算的$O(N)$模型。最后，我们展示了如何一个合适的概念的重整化群建立的Ricci流的非线性西格玛模型的一个循环重整化群流。
英文摘要：
We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomaly-free under additional assumptions on the target homogeneous space, in agreement with the work of Friedan. We carry out some explicit computations for the $O(N)$-model. Finally, we show how a suitable notion of the renormalization group establishes the Ricci flow as the one loop renormalization group flow of the nonlinear sigma model.
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