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\begin{document}
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{\LARGE{\bf Cheat Sheet Pushforwards}}\hfill
\parbox[b]{2.5in}{\tiny
  \null\hfill\url{http://drorbn.net/AcademicPensieve/2017-09/}
  \newline\null\hfill
   modified \today.
}

\vskip -3mm
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\vspace{-8mm}

\begin{multicols}{2}

\[
  \def\U{$\hat\calU(\frakg^\epsilon)$}
  \def\V{\parbox{0.5in}{$(\tau_1,\eta_1,\alpha_1,\xi_1,$ $\null\ \tau_2,\eta_2,\alpha_2,\xi_2)$}}
  \def\W{\parbox{0.5in}{$(\tau,\eta,\alpha,\xi)$}}
  \import{../Talks/LesDiablerets-1708/}{figs/Pushforward.pdf_t}
\]

{\bf Definition.} $PS\coloneqq$(Power Series).

For a vector space $V$, let $\calD_0(V)$ denote the space of distributions on $V$ whose support is $\{0\}$. Via the Laplace transform $\calD_0(V)$ can be identified with $\calS(V)$; we have $\calL_V\colon\calD_0(V)\to\calS(V)$.

{\bf Challenge.} With $\Phi\colon V\to W$ a PS map near $0$ (so $\Phi\in\mor_{PS}(V\to W)\coloneqq W\otimes\calS^+(V^\ast)$ and with $D_f\in\calD_0(V)$, understand $\Phi_\ast D_f\in\calD_0(W)$.

\end{multicols}

\rule{\textwidth}{1pt}

\begin{multicols}{2}

{\bf Challenge.} With $\Phi=(\phi_j(\alpha_i))$ and $Z=\zeta(\partial_{\alpha_i})$, set $\Phi_\ast Z \coloneqq \left. \bbe^{\sum\partial_{\beta_j}\phi_j(\partial_{a_i})}\zeta(a_i)\right|_{a_i=0}$.
With $(a_i, y_i, x_i, t_i) \coloneqq (\partial_{\alpha_i}, \partial_{\eta_i}, \partial_{\xi_i}, \partial_{\tau_i})$, compute/implement $\Phi_\ast Z$, with
\[ Z = \omega\exp\left(\sum \lambda_{ij}t_ia_j + \sum q_{ij}y_ix_j +\epsilon P_0\right), \]
$\lambda_{ij}\in\bbZ$, $\omega,q_{ij}\in R\coloneqq \bbQ(T_i=\bbe^{t_i})$, $P_0\in R[a_i,y_i,x_i]$, and
\begin{eqnarray*}
  \Phi^\ast(\bar\alpha_i) &=&
    \sum\psi^1_{ij}\alpha_j + \epsilon P_1, \\
  \Phi^\ast(\bar\eta_i) &=&
    \sum\psi^2_{ij}\eta_j + \epsilon P_2, \\
  \Phi^\ast(\bar\xi_i) &=&
    \sum\psi^3_{ij}\xi_j + \epsilon P_3, \\
  \Phi^\ast(\bar\tau_i) &=&
    \sum\psi^4_{ij}\tau_j + \sum\gamma_{ij}\eta_i\xi_j + \epsilon P_4,
\end{eqnarray*}
$\psi^{1,4}_{ij}\in\bbZ$, $\psi^{2,3}\in R$, $P_{1,4}\in\bbQ[x_i,y_i]$, $P_{2,3}\in R[x_i,y_i]$, $\gamma_{ij}\in R$.

{\bf Example.} \href{http://drorbn.net/AcademicPensieve/2017-07/nb/Multi-beta-yax.pdf}{2017-07/Multi-beta-yax.nb}: In $\calU_{\gamma^{-1};\gamma\beta}$ where $q=\bbe^\beta$, $\prod_{i=1}^2 \bbe^{\eta_iy}\bbe^{\alpha_ia}\bbe^{\xi_i x} = \bbe^{\eta y}\bbe^{\alpha a}\bbe^{\xi x}\bbe^{\tau t}$, with
\begin{eqnarray*}
\eta & = &
  \eta_1+\eta_2 e^{-\gamma\alpha_1}-\beta\gamma\eta_2^2\xi_1 e^{-\gamma\alpha_1} + \ldots
  = \eta _1 + \delta\eta _2 e^{\beta -\alpha _1 \gamma } \\
\alpha & = &
  \alpha _1+\alpha _2+2 \beta  \eta _2 \xi _1 + \ldots
  = \alpha _1+\alpha_2 - 2 \left(\beta+\log \delta \right)/\gamma \\
\xi & = &
  \xi _1 e^{-\gamma\alpha _2} + \xi_2 - \beta  \gamma  \eta _2 \xi_1^2 e^{-\gamma\alpha _2} + \ldots
  = \delta\xi _1 e^{\beta -\alpha _2 \gamma }+\xi _2 \\
\tau & = &
  -\eta _2 \xi_1 + \beta\eta_2 \xi _1 \left(\gamma  \eta _2 \xi _1+1\right)/2 + \ldots
  = \left(\beta+\log \delta \right)/(\beta\gamma)
\end{eqnarray*}
and $\delta \coloneqq \left(\left(e^{\beta }-1\right) \gamma  \eta_2 \xi _1+e^{\beta }\right)^{-1} = 1-(1+\gamma\eta_1\xi_1)\beta+\ldots$.

\end{multicols}

\[ \includegraphics[width=\linewidth]{PushforwardsAndFeynmanDiagrams.png} \]

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