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Let $ f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C},0)$be a germ of a holomorphic function with a critical point at 0 and $\Delta$ is the vanishing cycle which is one of the basis of space $H_{n-1}(f^{-1}(t)\cap B,C)$ where B is a small ball. Functions like $I(t)=\int _{\Delta} \omega $ have an expansion of the form $I(t)=\Sigma a_k t^{\alpha} (\log t)^k$ due to Picard-Lefschetz theory ( $t \rightarrow t e^{2\pi i}$ and use Picard-Lefschetz formula ) so we can think it is the log terms that breaks the triviality of monodromy group and the smooth base change theorem in cohomology.

In quantum field theory,terms like $log \frac {q^2}{m^2}$ also appear, such as in electron vertex functions in QED,where the log terms lead to infrared divergence when m=0.

Question： Do the log terms in QED come form the integral about singularities function like the first paragraph(regard q or m as t in integral)? If so，how to explain KLN theorem which is about the non-existence of infrared divergence?

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    $\begingroup$ Integrals like this in QFT typically come from definite integrals of some variable up to a large cutoff scale. The integrals goes like 1/t, hence the log. The integral over the entire (positive) real axis diverges. So, I'm inclined to say that the reason for getting logs is much simpler than Picard-Lefshetz theory, though I have no deep understanding of the latter. $\endgroup$
    – jwimberley
    Commented Sep 15, 2016 at 20:49
  • $\begingroup$ There is the literature of "analytic continuation of Chern-Simons Theory" etc. But are you asking in greater generality? $\endgroup$
    – AHusain
    Commented Sep 15, 2016 at 20:59
  • $\begingroup$ @jwimberley thank you for your comment. I know a little QFT,but besides those coming form Pauli-Villars regularization or other regularization process,degeneracy in QFT or even QM also gives a $\log \mu$ if we give a small mass $\mu$ to photons ( I saw it in T. D. Lee's QFT book） which I think has nothing to do with large cut off（surely ，they come from small cut off）. Kinoshita,Lee and Nauenberg gave a theorem to avoid infrared divergence which is not hard but interesting. I wonder if it can be gained by other way. $\endgroup$
    – wongdl
    Commented Sep 16, 2016 at 6:08
  • $\begingroup$ @AHusain thank you for the literature. Witten seems to consider using Picard-Lefschetz theory directly to path integral which is infinite-dimensional,but I just want to analytic a 4-d integral in QFT $\endgroup$
    – wongdl
    Commented Sep 16, 2016 at 6:11

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