\documentclass[11pt,notitlepage]{article}
\usepackage{graphicx, datetime, multicol, qrcode, amsmath, enumitem,
  amssymb, % for \rtimes
  mathtools, % for \coloneqq
  stmaryrd, % for \sslash
  mathbbol, % for \bbe.
  amsthm % for \theoremstyle
}
\usepackage[margin=0.25in,foot=0.2in]{geometry}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}

\def\sheeturl{{\url{http://drorbn.net/21-1350}}}
\def\myurl{http://www.math.toronto.edu/~drorbn}

\def\blue{\color{blue}}
\def\green{\color{green}}
\def\red{\color{red}}

\def\bbe{\mathbb{e}}
\def\bbH{{\mathbb H}}
\def\bbN{{\mathbb N}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calE{{\mathcal E}}
\def\calR{{\mathcal R}}
\def\calT{{\mathcal T}}
\def\calV{{\mathcal V}}
\def\calX{{\mathcal X}}
\def\calY{{\mathcal Y}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\eps{\epsilon}
\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{{\tiny arXiv:}\linebreak[0]{\small #1}}}}
\def\Aut{{\operatorname{Aut}}}
\def\Hom{\operatorname{Hom}}
\def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}}
\def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}}
\def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}}

\pagestyle{empty}

\begin{document}
\parindent 0in

\parbox[b]{1.5in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}:
  \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2122}{2021-22}:
  \href{http://drorbn.net/21-1350}{MAT 1350 Knot Theory}:
}~{\LARGE\bf Perturbed Gaussian Integration}% and Feynman Diagrams
\hfill\parbox[b]{1.4in}{\tiny
  \ldots and Feynman Diagrams
  \newline\null\hfill\sheeturl
  %\newline\null\hfill Source code: \href{http://drorbn.net/21-1350/nb/WG.pdf}{WG}
}~%
\raisebox{0.2em}{
  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/21-1350}
}~\parbox[b]{0.3in}{\tiny\href{http://www.math.toronto.edu/~drorbn/bv.html}{Better\newline vaxxed!}}

\vskip -3mm
\rule{\linewidth}{1pt}

\begin{multicols}{2}

Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah Di dah

\end{multicols}

\parbox[t]{5.4in}{
{\red Gaussian Integration.} $(\lambda_{ij})$ is a symmetric positive definite
matrix and $(\lambda^{ij})$ is its inverse, and $(\lambda_{ijk})$ are the
coefficients of some cubic form. Denote by $(x^i)_{i=1}^n$ the
coordinates of $\bbR^n$, let $(t_i)_{i=1}^n$ be a set of ``dual''
variables, and let $\partial^i$ denote $\frac{\partial}{\partial t_i}$.
Also let $C\coloneqq\frac{(2\pi)^{n/2}}{\det(\lambda_{ij})}$. Then
\begin{multline*}
  \int\limits_{\bbR^n}
    e^{-\frac12\lambda_{ij}x^ix^j+\frac{\epsilon}{6}\lambda_{ijk}x^ix^jx^k}
  = \sum_{m\geq 0}\frac{\epsilon^m}{6^mm!}
    \int\limits_{\bbR^n}
    (\lambda_{ijk}x^ix^jx^k)^m e^{-\frac12\lambda_{ij}x^ix^j} \\
  = \left. \sum_{m\geq 0}\frac{C\epsilon^m}{6^mm!}
      (\lambda_{ijk}\partial^i\partial^j\partial^k)^m
      e^{\frac12\lambda^{\alpha\beta}t_\alpha t_\beta}
    \right|_{t_\alpha=0}
  = \sum_{\substack{m,l\geq 0 \\ 3m=2l }} \frac{C\epsilon^m}{6^mm!2^ll!}
      \left(\lambda_{ijk}\partial^i\partial^j\partial^k\right)^m
      \left(\lambda^{\alpha\beta}t_\alpha t_\beta\right)^l \\
  = \sum_{\substack{m,l\geq 0 \\ 3m=2l }} \frac{C\epsilon^m}{6^mm!2^ll!}
    \left[\begin{array}{c}\input{figs/Pitchforks.pdf_t}\end{array}\right] \\
  = \sum_{\substack{m,l\geq 0 \\ 3m=2l }} \frac{C\epsilon^m}{6^mm!2^ll!}
    \sum_{\substack{m\text{-vertex fully marked} \\ \text{Feynman diagrams }D}}
      \hspace{-18pt}\calE(D)
      \def\Da{$\lambda_{i_1j_1k_1}\lambda_{i_2j_2k_2}
        \lambda^{i_1i_2}\lambda^{j_1j_2}\lambda^{k_1k_2}$}
      \def\Db{$\lambda_{i_1j_1k_1}\lambda_{i_2j_2k_2}
        \lambda^{i_1j_1}\lambda^{k_1k_2}\lambda^{i_2j_2}$}
      \qquad\begin{array}{c}\input{figs/MarkedDiagrams.pdf_t}\end{array} \\
  = C\sum_{\substack{\text{unmarked Feynman} \\ \text{diagrams }D}}
      \frac{\epsilon^{m(D)}\calE(D)}{|\Aut(D)|}.
    \qquad\parbox[t]{3in}{
      {\bf Claim.} The number of pairings that produce a given unmarked
      Feynman diagram $D$ is $\frac{6^mm!2^ll!}{|\Aut(D)|}$.
    }
\end{multline*}
{\bf Proof of the Claim.} The group $G_{m,l}\coloneqq[(S_3)^m\rtimes
S_m]\times[(S_2)^l\rtimes S_l]$ acts on the set of pairings, the action
is transitive on the set of pairings $P$ that produce a given $D$,
and the stabilizer of any given $P$ is $\Aut(D)$. \qed
}

\end{document}


