Sorry about the dumb title.

I'd like to understand Wick's theorem. More specifically, I have seen it pop up in several different contexts and I am really puzzled by the different statements of it that I have seen. My own background/interest is in moduli of curves, if that helps.

The first version, that is also the only one I have seen more than one time, is in the context of infinite wedge space. Here Wick's theorem is a formula about how to decompose any product of the fermionic operators $\psi_k$ and their adjoints $\psi_k^\ast$ as a sum over normally ordered products. This is for instance how it is explained in Kac-Raina.

A second version is in "Graphs on surfaces" by Lando and Zvonkin as Theorem 3.2.5. Here it is a statement about how to integrate a polynomial against a Gaussian measure on the real line. If $\langle f \rangle$ denotes the integral $\frac{1}{\sqrt{2\pi}}\int_{\mathbb R} f(x) \exp(-x^2/2) dx$, then Wick's theorem states that if *f* is a product $f = f_1 f_2 \cdots f_{2k}$ of linear polynomials, then $\langle f \rangle$ can be written as an explicit sum of products of pairs $\langle f_if_j\rangle$.

Now I can somehow believe that the two theorems above are talking about the same thing or that the second is a special case of the first. But what really got me scratching my head was the following statement from page 2 of Getzler & Kapranov's paper on modular operads (sorry for the lengthy quote):

[...] As a model for this calculation, take the formula for the enumeration of graphs known in mathematical physics as Wick’s Theorem. Consider the asymptotic expansion of the integral $W(\xi,\hbar) = \log \int \exp \frac 1 \hbar \left( x \xi - \frac{x^2}{2} + \sum_{2g-2+n > 0} \frac{a_{g,n}\hbar^g x^n}{n!}\right) \frac{dx}{\sqrt{2\pi\hbar}}$ considered as a power series in $\xi$ and $\hbar$. (The asymptotic expansion is independent of the domain of integration, provided it contains 0.) Let $\Gamma((g,n))$ be the set of isomorphism classes of connected graphs $G$, with a map $g$ from the vertices Vert(G) of $G$ to $\{0,1,2,...\}$ and having exactly $n$ legs numbered from 1 to $n$, such that $g = b_1 + \sum_{v\in \mathrm{Vert}(G)} g(v)$ where $b_1$ is the first Betti number of the graph. If $v$ is a vertex of $G$, denote by $n(g)$ its valence, and let |Aut(G)| be the cardinality of the automorphism group of $G$. Wick’s Theorem states that $W \sim \frac 1 \hbar \left(\frac{\xi^2}{2} + \sum_{2g-2+n>0} \frac{\hbar^g\xi^n}{n!} \sum_{G\in \Gamma((g,n))} \frac{1}{|\mathrm{Aut}(G)|} \prod_{v\in \mathrm{Vert}(G)} a_{g(v),n(v)}\right)$.

I also heard a version of Wick's theorem at a talk of Rahul Pandharipandhe about two months ago, which I will not be able to state correctly here since I can't really make sense of my notes. In that version of Wick's theorem one studied an $n$-fold product of a variety with itself by interpreting it as a configuration space of $n$ "particles" moving on the variety. The goal was to simplify certain complicated products of cohomology classes given by diagonals (= particles coinciding) and Chern classes of the tangent/cotangent bundle at one of the "particles". This was all done pictorially, and one represented the diagonals as a line connecting the two points, which at least shows some connection with the Wick formalism since I think I have at one point seen these lines between particles also in the context of Feynman diagrams.

Can someone give a hint about how these Wick theorems fit together?