I was looking at papers about the SYK model page 33 (equation 112), in which they write

$$\int\mathscr{D}\Sigma\,\mathscr{D}G~e^{-\frac{N}{2}\int\limits_{\left[0,\beta\right]^2} d\tau_1d\tau_2\Sigma\left(G-\frac{1}{N}\sum\limits_{I=1}^N\psi_i\left(\tau_1\right)\psi_i\left(\tau_2\right)\right)}=1.\tag{112}$$

This would follow from $\int d\sigma\,dg\ e^{-\sigma g}=1.$

Wolfram alpha confirms this as a Cauchy principle value.

How would this be proved?


OP asks about heuristic manipulations of a functional integral in Ref. 1. This question (v2) seems to belong more to Phys.SE than Math.SE. Functional integration is a huge topic. In this answer, we will only discuss the underlying ordinary integral.

  1. The underlying integral $$\int_{\mathbb{R}} \mathrm{d}\sigma~ e^{-\sigma g}, \qquad g~\in~\mathbb{R}, $$ in eq. (112) (using the so-called Euclidean formulation cf. footnote 36) is not well-defined.

  2. Presumably it should be defined via a Wick rotation $$\int_{\mathbb{R}} \mathrm{d}\sigma~ e^{\color{red}{i}\sigma g}~=~2\pi \delta(g), \qquad g~\in~\mathbb{R}, $$ (using the so-called Minkowskian formulation).


  1. G. Sarosi, AdS2 holography and the SYK model, arXiv:1711.08482.
  • $\begingroup$ But $\sigma$ is a field. Why is there a factor of $i$ under Wick rotation? $\endgroup$ – Chetan Vuppulury Jun 30 '18 at 12:26
  • $\begingroup$ I updated the answer. $\endgroup$ – Qmechanic Jun 30 '18 at 12:55

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