\section{The Category \dpg} \label{sec:DoPeGDO}

\subsection{Motivation, conventions, generating functions}

This section may seem like an awful way to start a topology paper --- it's
all about formula-based technicalities. Here are its redeeming features
(beyond its usefulness for the later parts of the paper): \begin{itemize}
\item Did you know that quadratic forms (aka ``Gaussians'') form a
category in a natural way?  (Theorem~\ref{thm:GDO}).  \item Did you know
that Feynman diagrams arise in pure algebra in a completely natural way?
\end{itemize}

\begin{motivation} \label{mot:PBW}
The ``PBW Principle'' says that many algebras $U$ are isomorphic, as vector spaces, to
polynomial rings (hence as algebras they are ``polynomial rings with funny multiplications''). Many times
one needs to understand maps between algebras. Primarily, the algebra's own structure: the multiplication
map $m\colon U\otimes U\to U$, perhaps a co-multiplication $\Delta\colon U\to U\otimes U$, and more. Sometimes
one may care about specific special elements in $U$ or some tensor power thereof; say, $R\in U\otimes
U=\Hom(U^{\otimes\emptyset}\to U^{\otimes 2})$. So we need to understand the category of maps between
algebras and their tensor powers, and hence, by PBW, the category of maps between polynomial rings. This
category is way too big --- one can encode an infinite amount of information into a map between polynomial
rings (no matter the base fields) --- and so no finite computer can fully store a general such map. Hence we
develop a theory of ``maps between polynomial rings that can be described using finite formulas (of a
certain kind)'' and we are lucky that the maps we care about later in this paper can indeed be described by
formulas of that kind. Those maps/formulas are ``{\bf Do}cile {\bf Pe}rturbed
{\bf G}aussian differential operators'', and they make a category, \dpg, which is the main
object of study for this section.
\end{motivation}

\begin{convention} Throughout this paper we will use lower case Latin letters such as $z$, $y$, $b$, $a$,
$x$, and $t$ to denote the generators of polynomial rings. Each such generator comes with a dual (whose
purpose will be explained shortly), and the dual will always be denoted by the corresponding Greek letter:
$z^\ast=\zeta$, $y^\ast=\eta$, $b^\ast=\beta$, $a^\ast=\alpha$, $x^\ast=\xi$, and $t^\ast=\tau$. If $C$ is a
finite set, we will denote by $z_C=\{z_c\}_{c\in C}$ the set of variables denoted by the letter $z$ with an
index $c\in C$; likewise there's $y_C$, $x_C$, etc. We will regard $z_C$ sometimes as a set and sometimes as
a column vector, as appropriate. We extend duality to indexed variables:
$z^\ast_C=\zeta_C=\{z^\ast_c=\zeta_c\}_{c\in C}$. We will sometimes treat $\zeta_C$ (or $\eta_C$, etc) as a
row vector.
\end{convention}

Next, we establish a bijection
\begin{equation} \label{eq:calG}
  \calG\colon\Hom(\bbQ[z_A]\to\bbQ[z_B])\to\bbQ[z_B]\llbracket\zeta_a\rrbracket
\end{equation}
between linear maps from polynomials in variables $z_A$ to polynomials
in variables $z_B$ ($A$ and $B$ are finite sets) and a certain class of
power series in the output variables and the duals of the input variables
(more precisely, power series in the Greek variables corresponding to
the inputs, with coefficients that are polynomials in the Latin variables
corresponding to the outputs).

\begin{definition} Let $A$ and $B$ be finite sets and let
$L\colon\bbQ[z_A]\to\bbQ[z_B]$ be linear. Let $\calL=\calG(L)$, the
generating function of $L$, be defined as follows:
\begin{equation} \label{eq:calG1}
  \calL = \calG(L)
  \coloneqq \sum_{n\in\bbN^A}\frac{\zeta_A^n}{n!}L(z_A^n)
  \in \bbQ[z_B]\llbracket\zeta_a\rrbracket.
\end{equation}
Here $\bbN$ denotes the non-negative integers,
$n=(n_a)_{a\in A}$ is a multi-index, $\zeta_A^n\coloneqq\prod_{a\in A}\zeta_a^{n_a}$ and likewise
$z_A^n\coloneqq\prod_{a\in A}z_a^{n_a}$, and $n!\coloneqq\prod_{a\in A}n_a!$. Extending $L$ without changing
its name to an operator $L\colon\bbQ[z_A]\llbracket\zeta_a\rrbracket\to\bbQ[z_B]\llbracket\zeta_a\rrbracket$
by treating the $\zeta_A$'s as scalars, and recalling the definition of the exponential function, we find
that~\eqref{eq:calG1} can also be written as
\[ \calL = \calG(L) = L\left(\bbe^{\zeta_A\cdot z_A}\right), \]
where $\zeta_A\cdot z_A\coloneqq\sum_{a\in A}\zeta_az_a$.
\end{definition}

\begin{proposition} $\calG\colon\Hom(\bbQ[z_A]\to\bbQ[z_B])\to\bbQ[z_B]\llbracket\zeta_a\rrbracket$ is a
bijection. If $\calL\in\bbQ[z_B]\llbracket\zeta_a\rrbracket$ and $p\in\bbQ[z_A]$ then 
\[ \calG^{-1}(\calL)(p)
  = \left.p(\partial_{\zeta_a})\calL(\zeta_a,z_b)\right|_{\zeta_a=0}
  = \left.\calL(\partial_{z_a},z_b)p(z_a)\right|_{z_a=0}
\]
\qed
\end{proposition}

\begin{example} \label{exa:GenFun} Consider $L_i\colon\bbQ[z]\to\bbQ[z]$ for $i=1,2,3,4$,
where $L_1(p)=p$ is the identity, $L_2(p)=p(z+1)$ is the shift,
$L_3(p)=p'$ is differentiation, and $L_4(p)=\int_0^zp$ is definite
integration. Then
\[ \calG(L_1)=\bbe^{\zeta z},
  \qquad \calG(L_2)=\bbe^{\zeta(z+1)},
  \qquad \calG(L_3)=\zeta\bbe^{\zeta z},
  \qquad\text{and}\qquad \calG(L_4)=(\bbe^{\zeta z}-1)/\zeta.
\]
A few further examples of generating functions, closer in spirit to the ones we care for the most in this paper,
are in Section~\ref{ssec:realistic}, right below.
\end{example}

Linear maps between polynomial rings can be composed, and it is useful to know how their corresponding
generating functions compose\footnotemark:

\footnotetext{Below and throughout we use ``$\act$'' for left-to-right composition: $L\act M=M\circ L$.}

\begin{proposition} \label{prop:LMcomposition} Let $A$, $B$, and $C$ be finite
sets, and let $L\in\Hom(\bbQ[z_A]\to\bbQ[z_B])$ and
$M\in\Hom(\bbQ[z_B]\to\bbQ[z_C])$. Then, with $b$ standing for all
elements of $B$,
\begin{equation} \label{eq:LMcomposition}
  \calG(L\act M)
  = \left(\calG(L)|_{z_b\to\partial_{\zeta_b}}\calG(M)\right)_{\zeta_b=0}
  = \left(\calG(M)|_{\zeta_b\to\partial_{z_b}}\calG(L)\right)_{z_b=0}.
\end{equation}
\qed
\end{proposition}

Said differently, $\calG$ is an isomorphism of categories from the
category of polynomial rings in finitely many generators to the category
$\frakG$ whose objects are finite sets with morphisms $\mor_\frakG(A\to
B)=\bbQ[z_B]\llbracket\zeta_A\rrbracket$ and compositions
\begin{equation} \label{eq:frakGComposition}
  \calL\act\calM = \left(\calL|_{z_b\to\partial_{\zeta_b}}\calM\right)_{\zeta_b=0}
  = \left(\calM|_{\zeta_b\to\partial_{z_b}}\calL\right)_{z_b=0},
\end{equation}
where $\calL\in\mor_\frakG(A\to B)$ and $\calM\in\mor_\frakG(B\to C)$.

\begin{comment} \label{com:contraction}
We call the operation in~\eqref{eq:frakGComposition} ``contraction of the variable pairs
$(\zeta_b,z_b)$ for $b\in B$''.
\end{comment}

\begin{comment} \label{com:DoubleContraction}
There is an easily-provable third version for the composition
formula~\eqref{eq:frakGComposition}, which treats $\calL$ and $\calM$
and Greek and Latin letters more symmetrically:
\begin{equation} \label{eq:DoubleContraction}
  \calL\act\calM =
    \left.\bbe^{\sum\partial_{z_b}\partial_{\zeta_b}}(\calL\cdot\calM)\right|_{z_b=\zeta_b=0},
\end{equation}
where the indices $b$ run through the set $B$. Here
$\calL\cdot\calM$ stands for the ordinary product
of power series $\bbQ[z_B]\llbracket\zeta_A\rrbracket
\otimes \bbQ[z_C]\llbracket\zeta_B\rrbracket \to \bbQ[z_{A\cup
B}]\llbracket\zeta_{B\cup C}\rrbracket$.\footnote{Strictly speaking this
is valid only if there are no name clashes, namely if $A\cap B=B\cap
C=\emptyset$. That's a non-issue --- if needed the labels in $B$ can be
temporarily renamed before the formula is applied.}
\end{comment}

\begin{comment} \label{com:integration}
If you are familiar with formal Gaussian integration, especially as
it is used in physics and especially in perturbation theory where one
allows themselve to pretend that integrals always converge
(e.g.~\cite{Polyak:Feynman}), then there is another easily verified form
for the composition formula (see
also~\cite{Abdesselam:FeynmanDiagramsInAlgebraicCombinatorics}):
\begin{equation} \label{eq:integration}
  \calL\act\calM
  = \left.\bbe^{\sum\partial_{z_b}\partial_{\zeta_b}}(\calL\cdot\calM)\right|_{z_b=\zeta_b=0}
  \propto \int\bbe^{-\sum_bz_b\zeta_b}(\calL\cdot\calM)\prod_{b\in B}dz_bd\zeta_b.
\end{equation}

Much of this paper can be re-written in terms of the above formula
and Gaussian integration, yet we prefer to use this fact only
for inspiration\footnotemark.  There is simply nothing to gain:
everything one can do with integration we can also do directly
with~\eqref{eq:DoubleContraction}, a bit more simply. Yet there
is a lesson to learn from~\eqref{eq:integration}: compositions
may have simple formulas (and indeed they do) if $\calL$ and
$\calM$ are themselves Gaussians or perturbed Gaussians, for
then the integral in~\eqref{eq:integration} would be Gaussian
or perturbed Gaussian, and these are known to be computable.
\endpar{\ref{com:integration}}
\end{comment}

\footnotetext{The constant of proportionality in Equation~\eqref{eq:integration}
has some $2\pi$ factors in it.  We don't really want dreadful transcendental
numbers in an algebra paper.}

\begin{discussion} \label{disc:Qh}
Later in this paper we will also want to consider power series in
the mold of $\bbe^{z}\in\bbQ\llbracket z\rrbracket$ or $(1-z)^{-1}$.
The generating function formalism does not extend to power series
in the most naive way: the space $\Hom\left(\bbQ\llbracket
z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket\right)$ is {\em not}
isomorphic to some space of ``generating functions'' such as
$\bbQ\llbracket\zeta_A,z_B\rrbracket$. Indeed, $\bbQ\llbracket
z_A\rrbracket$ is of uncountable dimension over $\bbQ$, and
$\Hom\left(\bbQ\llbracket z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket\right)$
is quite wild\footnotemark. One standard way to get around this is to introduce a
``small'' parameter $\hbar$ and insist that it be present in power series,
as in $\bbe^{\hbar z}$ and $(1-\hbar z)^{-1}$. But first, a discussion and a
convention.

\footnotetext{\label{foot:ContHom}\
One may be tempted to restrict attention in $\Hom\left(\bbQ\llbracket
z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket\right)$ to {\em
continuous} homomorphisms (relative to the power series topology;
see e.g.~\cite[Chapter XVI]{Kassel:QuantumGroups}). That's wrong in
our context --- many of the homomorphisms we care about are simply not
continuous relative to the power series topology. See an example in
Footnote~\ref{foot:ncont}.}

In analysis the identity $(1-\hbar z)^{-1} = \sum \hbar^nz^n$ holds true even if $|z|$ isn't small, provided
$\hbar$ is small enough\footnote{How small? $|h|$ must be smaller than $|z|^{-1}$, so $\hbar$ must be
determined {\em after} $z$.}. In algebra, if we want to enrich $\bbQ[z]$ so as to allow such
identities\footnote{And yet without making $z$ small, that is, without switching to $Q\llbracket z\rrbracket$,
which our formalism can't handle.} we need to do two things:
\begin{itemize}
\item Tensor multiply $\bbQ[z]$ with $\bbQ[\hbar]$ to get $\bbQ[z,\hbar]$, so as to allow coefficient
depending on~$\hbar$.
\item Complete relative to the $\hbar$-adic topology so as to get $\bbQ[z]\llbracket\hbar\rrbracket$, where
series like $\sum \hbar^nz^n$ make sense.
\end{itemize}
{\red MORE. ??? Add somewhere a comment that exponentials make sense in both
$\bbQ\llbracket z\rrbracket$ and $\bbQ[z]\llbracket\hbar\rrbracket$,
yet $\Hom(\bbQ\llbracket z\rrbracket\to\bbQ\llbracket
z\rrbracket)$ is of uncountable dimension while
$\Hom_{\bbQ\llbracket\hbar\rrbracket}(\bbQ[z]\llbracket\hbar\rrbracket\to\bbQ[z]\llbracket\hbar\rrbracket)$
is countable.}

{\red MORE. This whole discussion is still murky. Does God really care about $\hbar$?}
\endpar{\ref{disc:Qh}}
\end{discussion}

\begin{convention}[and subtle point] \label{conv:Qh}
We slightly abuse notation and use $\Qh$ as a symbol for both steps:
\[ \Qh[x,y,z] \coloneqq \bbQ[x,y,z]\llbracket\hbar\rrbracket. \]
Note that $\Qh$ is not a ring but a name for an operator: tensor with $\bbQ[\hbar]$ and complete relative to
the $\hbar$-adic topology. In particular, $\Qh$ isn't $\bbQ\llbracket\hbar\rrbracket$ and $\Qh[z]$ isn't
$\bbQ\llbracket\hbar\rrbracket[z]$. Indeed, $\bbe^{\hbar z}$ and $(1-\hbar z)^{-1}$ are both members of
$\Qh[z]$ but not of $\bbQ\llbracket\hbar\rrbracket[z]$.

Yet we further abuse notation, and when $\Qh$ is on its own, we will regard it as the ring
$\bbQ\llbracket\hbar\rrbracket$. So ``$\omega\in\Qh$'' means that $\omega$ is a power series in $\hbar$
with rational coefficients.

With all this said, in much of this paper one can read $\Qh$ to simply
mean ``$\bbQ$, also with a small parameter $\hbar$'', with only a minor
disloyalty to precision. \endpar{\ref{conv:Qh}}
\end{convention}

Everything said so far work over $\Qh$ as well as over $\bbQ$. The same bijection as in~\eqref{eq:calG},
\[ \calG\colon\Hom_\Qh(\Qh[z_A]\to\Qh[z_B])\to\Qh[z_B]\llbracket\zeta_a\rrbracket, \]
with the same definition~\eqref{eq:calG1} and the same composition law~\eqref{eq:LMcomposition}.

{\red MORE. A continuity clause is missing.}

\begin{convention} \label{conv:GreekCompletion}
We also automatically complete spaces relative to the Greek letters
$\alpha$, $\beta$, $\pi$, $\tau$, $\eta$, $\xi$, and $\zeta$, and also when
they come with subscripts. So if $a$ and $b$ are elements of some algebra
$U$ then $\bbe^{\alpha_1a+\beta_2b}$ always makes sense, and should be
regarded as an element of $U\llbracket\alpha_1,\beta_2\rrbracket$.
\end{convention}

\draftcut \subsection{Some Real Life Examples} \label{ssec:realistic}
Let us briefly meet a few generating functions of the type that we will
care about the most in this paper. But first,

\begin{convention} \label{conv:id} Throughout this paper we often put labels on tensor
factors in a tensor product instead of ordering them; hence we
often write $U^{\otimes A}$, where $U$ is a vector space and $A$
is a finite set, instead of $U^{\otimes n}$, where $n$ is a natural
number\footnotemark. If $U$ has a prescribed unit $1\in U$ and if $z\in
U$ and $i\in A$, we write $z_i$ for ``$z$ placed in tensor factor $i$
(with $1$ in all other tensor factors)''. Thus for example, we often write $z_1+z_2$ for $z\otimes 1+1\otimes z$.
If $\psi\colon U^{\otimes
A}\to U^{\otimes B}$ is a map, we often emphasize its domain and range
by writing ``$\psi^A_B$''.
\end{convention}

\footnotetext{These conventions only make sense in strict monoidal categories. They are consistent with the
``identity'' world view as opposed to the ``geography'' view; see~\cite{Talk:Toronto-1912}.}

We start with some examples from the realm of commutative polynomials. Here $U=\bbQ[z]$ denotes the ring of
commutative polynomials in a variable $z$.

\begin{example} \label{exa:m}
Let $m\colon U\otimes U\to U$ be the
multiplication of polynomials. With the language of Convention~\ref{conv:id}, we choose labels $i,j,k$ and write
$m^{ij}_k\colon\bbQ[z_i,z_j]\simeq U_i\otimes U_j\to
U_k\simeq\bbQ[z_k]$. But now $m^{ij}_k$ is given by $z_i,z_j\mapsto z_k$,
and so \[ \calG(m^{ij}_k) = m^{ij}_k(\bbe^{\zeta_iz_i+\zeta_jz_j}) =
\bbe^{(\zeta_i+\zeta_j)z_k}. \]
Note that $\calG(m^{ij}_k)$ is a Gaussian --- the exponential of a quadratic expression.
\end{example}

\begin{example} Similarly, there is a coproduct $\Delta\colon U\to U\otimes U$, better written
as $\Delta^i_{jk}$, given by $z_i\mapsto z_j+z_k$. We have
\[ \calG(\Delta^i_{jk}) = \Delta(\bbe^{\zeta_i z_i}) = \bbe^{\zeta_i(z_j+z_k)}. \]
Again, this is a Gaussian expression.
\end{example}

\begin{example} \label{exa:sigma}
A bit silly but nevertheless useful is the relabelling map $\sigma^i_j\colon U_i\to U_j$, which is
merely the identity map $U\to U$, albeit with a change-of-label for the unique tensor factor that appears. We have
\[ \calG(\sigma^i_j) = \sigma^i_j(\bbe^{\zeta_iz_i}) = \bbe^{\zeta_iz_j}. \qquad\text{(A Gaussian!)} \]
\end{example}

\begin{example} There is an inner product $P\colon U\otimes U\to\bbQ$ given by $\langle z^n,z^m\rangle=\delta_{nm}n!$,
and by a quick computation we have
\[ \calG(P^{ij}) = \bbe^{\zeta_i\zeta_j}. \qquad\text{(A Gaussian!)} \]
Note that there are no Latin letters in the above expression, because it is the generating function of a morphism
whose target space is a polynomial ring on $0$ variables.
\end{example}

\begin{example} On a finite dimensional vector space $V$ an inner product
$P$ would have an inverse $R\in V\otimes V$ such that $R_{ij}\act
P^{jk}=\sigma^k_i$ (with $\sigma$ like in Example~\ref{exa:sigma}). We
cannot have that here because $U$ is infinite-dimensional. We come close
with $R_{ij}=\bbe^{\hbar z_iz_j}\in\Qh U^{\otimes\{i,j\}}$, which satisfies
$R_{ij}\act P^{jk} = \hbar^{\deg}\sigma^k_i$, where $\hbar^{\deg}$ is the operator defined by
$\hbar^{\deg}(z^n)=\hbar^nz^n$. We then have
\[ \calG(R_{ij}) = \bbe^{\hbar z_iz_j}. \qquad\text{(A Gaussian!)} \]
Note that there are no Greek letters in the above expression, because it is the generating function of a morphism
whose domain space is a polynomial ring on $0$ variables.
\end{example}

Our next few examples are minimally-non-commutative having to do first with the Heisenberg algebra $\bbH$ and then with
the unique non-commutative 2D Lie algebra $\fraka$. But first,

\begin{convention} In support of the PBW principle
(Motivation~\ref{mot:PBW}) we will often consider both commutative and
non-commutative algebras generated by the same set of generators. In such
cases we will use ordinary {\it italics} for the generators regarded
within commutative algebras, yet {\bf boldface} letters for the same
generators regarded within non-commutative algebras. The following
definition is an example.
\end{convention}

\begin{definition} Let $\bbH$ denote the Heisenberg algebra, the free
associative algebra with generators $\bp$ and $\bx$ modulo the ``canonical commutation relation''
$[\bp,\bx]=1$. The ``$\bp$ before $\bx$'' PBW ordering map (or ``normal ordering'',
as physicists would call it) $\bbO\colon\bbQ[p,x]\to\bbH$ defined by
$p^mx^n\mapsto\bp^m\bx^n$ is a vector space isomorphism of a (commutative)
polynomial algebra with the (non-commutative) algebra $\bbH$.
\end{definition}

\begin{example} \label{exa:Heis}
Let $hm$ be the multiplication map of $\bbH$, turned into a map between
polynomial rings by using $\bbO$ to identify $\bbH$ with $\bbQ[p,x]$;
namely, let $hm^{ij}_k$ be the composition
\[ \begin{CD} \bbQ[p_i,x_i,p_j,x_j] @>\bbO_i\otimes\bbO_j>> \bbH_i\otimes\bbH_j
  @>m^{ij}_k>> \bbH_k @>\bbO_k^{-1}>> \bbQ[p_k,x_k], \end{CD}
\]
where $m\colon\bbH\otimes\bbH\to\bbH$ is the (non-commutative) multiplication map of $\bbH$.
Then
\begin{equation} \label{eq:Ghm}
  \calG(hm^{ij}_k) = \bbe^{-\xi_i\pi_j+(\pi_i+\pi_j)p_k+(\xi_i+\xi_j)x_k}, \qquad \text{(a Gaussian!),}
\end{equation}
for indeed, using the Weyl form of the canonical commutation relation,
\begin{equation} \label{eq:Weyl}
  \bbe^{\xi\bx}\bbe^{\pi\bp} = \bbe^{-\xi\pi}\bbe^{\pi\bp}\bbe^{\xi\bx}
  \qquad\text{(in $\bbH\llbracket\pi,\xi\rrbracket$; see Convention~\ref{conv:GreekCompletion})},
\end{equation}
we have
\begin{multline*}
  \calG(hm^{ij}_k)
  = \bbe^{\pi_ip_i+\xi_ix_i+\pi_jp_j+\xi_jx_j} \act \bbO_i\otimes\bbO_j \act m^{ij}_k \act \bbO_k^{-1} 
  = \bbe^{\pi_i\bp_i}\bbe^{\xi_i\bx_i}\bbe^{\pi_j\bp_j}\bbe^{\xi_j\bx_j} \act m^{ij}_k \act \bbO_k^{-1} \\
  = \bbe^{\pi_i\bp_k}\bbe^{\xi_i\bx_k}\bbe^{\pi_j\bp_k}\bbe^{\xi_j\bx_k} \act \bbO_k^{-1}
  = \bbe^{-\xi_i\pi_j}\bbe^{(\pi_i+\pi_j)\bp_k}\bbe^{(\xi_i+\xi_j)\bx_k} \act \bbO_k^{-1}
  = \bbe^{-\xi_i\pi_j+(\pi_i+\pi_j)p_k+(\xi_i+\xi_j)x_k}.
\end{multline*}
Note that as in Example~\ref{exa:m}, $\frakg(m^{ij}_k)=\bbe^{(\pi_i+\pi_j)p_k+(\xi_i+\xi_j)x_k}$, so the only
``contribution'' of the non-commutativity of $hm$ is the term $-\xi_i\pi_j$ in~\eqref{eq:Ghm}.
\end{example}

Our last example for this section is split between a definition, a proposition, two proofs, and a discussion.

\begin{definition} \label{def:am}
Let $\eps$ be a parameter, let $\fraka_\eps$ be the 2D Lie algebra with generators $\ba$ and
$\bx$ and relation $[\ba,\bx]=\eps\bx$, and let $\bbA_\eps=\calU(\fraka_\eps)$ be the universal enveloping
algebra of $\fraka_\eps$. Let $\bbO\colon\bbQ[a,x]\to\bbA_\eps$ be the ``$\ba$ before $\bx$'' ordering map given
by $a^mx^n\mapsto\ba^m\bx^n$ (by PBW, it is a vector space isomorphism). Let $am^{ij}_{\eps;k}$ be the
composition
\[ \begin{CD} \bbQ[a_i,x_i,a_j,x_j] @>\bbO_i\otimes\bbO_j>> \bbA_{\eps;i}\otimes\bbA_{\eps;j}
  @>m^{ij}_{\eps;k}>> \bbA_{\eps;k} @>\bbO_k^{-1}>> \bbQ[a_k,x_k], \end{CD}
\]
where $m^{ij}_{\eps;k}$ is the multiplication map of $\bbA_\eps$.
\end{definition}

\begin{proposition} \label{prop:am}
\[
  \calG(am^{ij}_{\eps;k}) = \exp\left(
    (\alpha_i+\alpha_j)a_k+(\bbe^{-\eps\alpha_j}\xi_i+\xi_j)x_k
  \right)
  \qquad
  \left(\parbox{1.45in}{\rm\centering nearly Gaussian, see Discussion~\ref{disc:gam}}\right)
\]
\end{proposition}

\begin{proof}[Proof 1] We first need a Weyl-style exponentiated
relation (cf.\ \eqref{eq:Weyl}). Start with $\bx\ba = (\ba-\eps)\bx$,\footnotemark\
iterate to get $\bx\ba^n = (\ba-\eps)^n\bx$, sum over $n$ with coefficients
$\frac{\alpha^n}{n!}$ to get $\bx\bbe^{\alpha\ba} = \bbe^{\alpha(\ba-\eps)}\bx =
\bbe^{\alpha\ba}\bbe^{-\eps\alpha}\bx$, iterate again to get $\bx^n\bbe^{\alpha\ba} =
\bbe^{\alpha\ba}(\bbe^{-\eps\alpha})^n\bx^n$, and sum again with coefficients $\frac{\xi^n}{n!}$ to get the
exponentiated relation $\bbe^{\xi\bx}\bbe^{\alpha\ba} = \bbe^{\alpha\ba}\bbe^{\bbe^{-\eps\alpha}\xi\bx}$.

\footnotetext{\label{foot:ncont}\
In continuation of Footnote~\ref{foot:ContHom}: We have
just shown that $am(x\otimes a^n) = (a-\eps)^nx = (-\eps)^nx+\text{higher
powers}$. But $x\otimes a^n\rightarrow 0$ while $(-\eps)^nx\nrightarrow 0$, so $am$
is not continuous.}

Now proceed as in Example~\ref{exa:Heis}:
\begin{multline} \label{eq:GamMain}
  \calG(am^{ij}_{\eps;k})
  = \bbe^{\alpha_ia_i+\xi_ix_i+\alpha_ja_j+\xi_jx_j} \act \bbO_i\otimes\bbO_j \act m^{ij}_{\eps;k} \act \bbO_k^{-1}
  = \bbe^{\alpha_i\ba_i}\bbe^{\xi_i\bx_i}\bbe^{\alpha_j\ba_j}\bbe^{\xi_j\bx_j} \act m^{ij}_{\eps;k} \act \bbO_k^{-1} \\
  = \bbe^{\alpha_i\ba_k}\bbe^{\xi_i\bx_k}\bbe^{\alpha_j\ba_k}\bbe^{\xi_j\bx_k} \act \bbO_k^{-1}
  \overset{\text{\footnotesize(key)}}{=}
     \bbe^{(\alpha_i+\alpha_j)\ba_k}\bbe^{(\bbe^{-\eps\alpha}\xi_i+\xi_j)\bx_k} \act \bbO_k^{-1}
  = \bbe^{(\alpha_i+\alpha_j)a_k+(\bbe^{-\eps\alpha}\xi_i+\xi_j)x_k}.
\end{multline}
\end{proof}

\noindent{\em Proof 2.} We reprove the
key equality of Equation~\eqref{eq:GamMain},
$\bbe^{\alpha_i\ba}\bbe^{\xi_i\bx}\bbe^{\alpha_j\ba}\bbe^{\xi_j\bx} =
\bbe^{(\alpha_i+\alpha_j)\ba}\bbe^{(\bbe^{-\eps\alpha}\xi_i+\xi_j)\bx}$.
Let $\rho$ be the 2-dimensional representation of $\fraka_\eps$
given by $\rho(\ba)=\begin{pmatrix}\eps&0\\0&0\end{pmatrix}$
and $\rho(\bx)=\begin{pmatrix}0&1\\0&0\end{pmatrix}$. The
represenattion $\rho$ is faithful on $\fraka_\eps$ so it extends
to a faithful representation of group-like elements in (the
Greek-letter completion of) $\bbA_\eps$\footnote{That's an algebraic version of the fact that faithful
representations of a Lie algebra are also faithful on a neighborhood of the identity element of the corresponding
Lie group.}, so it is enough to prove that
$\bbe^{\alpha_i\rho(\ba)}\bbe^{\xi_i\rho(\bx)}\bbe^{\alpha_j\rho(\ba)}\bbe^{\xi_j\rho(\bx)}
= \bbe^{(\alpha_i+\alpha_j)\rho(\ba)}\bbe^{(\bbe^{-\eps\alpha}\xi_i+\xi_j)\rho(\bx)}$.
This we do by brute force matrix exponentiation (see also \cite[Gam.nb]{Self}):

\input{Gam.tex}
\vskip -7mm \ \qed

\begin{discussion} \label{disc:gam}
The exponent of $\calG(am^{ij}_{\eps;k})$ in itself has an exponential
term in it ($\bbe^{-\eps\alpha_j}$) hence $\calG(am^{ij}_{\eps;k})$ is
not a Gaussian in $\{\alpha_i,\alpha_j,\xi_i,\xi_j,a_k,x_k\}$, and hence
some of the techniques that we introduce in later sections, to compose
Gaussian generating functions, appear to break. We have two ways around
this, we need both of them below, and in fact, one reason we included
Proposition~\ref{prop:am} is to forewarn that these two ways are needed:

\begin{enumerate}

\item If $\eps$ is considered as ``small'' we can expand relative to $\eps$ and find
\[ \calG(am^{ij}_{\eps;k}) = \exp\left(
    (\alpha_i+\alpha_j)a_k+(\xi_i+\xi_j)x_k + \sum_{m\geq 1}\frac{(-\eps)^m\alpha_j^m}{m!}\xi_i x_k.
  \right)
\]
This is a prime example of a ``perturbed Gaussian'', and the lesson to
take is that we will need to look beyond Gaussians and at perturbation
theory.

\item Even if $\eps=1$, $\calG(am^{ij}_{k})$ is a Gaussian
in the variables $\{\xi_i,\xi_j,x_k\}$ if the variables
$\{\alpha_i,\alpha_j,a_k\}$ are held fixed, so contractions involving
$\xi$'s and $x$'s create no problems. As for contractions of
$\alpha$'a and $a$'s, $\calG(am^{ij}_{k})$ is {\em nearly} Gaussian
in $\{\alpha_i,\alpha_j,a_k\}$ for fixed $\{\xi_i,\xi_j,x_k\}$:
the offending term is the term $\bbe^{-\alpha_j}\xi_ix_k$. That
term is a manageable perturbation. It is a bit hard to summarize
what ``manageable'' means beyond saying ``whatever is subject to the
techniques of Section~\ref{ssec:PDE}''. Yet in short, the manageability
here stems from the fact that the quadratic term $(\alpha_i+\alpha_j)a_k$
is ``bipartite'', involving only $\alpha a$ terms but no $\alpha\alpha$'s
or $aa$'s, while the perturbation term $\bbe^{-\alpha_j}\xi_ix_k$ involves
only variables from one side of the partition: only the $\alpha$'s.

The lesson to take is that sometimes we will need to use Gaussian
techniques twice, relative to to different sets of variables, while
holding the variables from the other set fixed.

\end{enumerate}
\end{discussion}

\draftcut \subsection{Gaussian Differential Operators} \label{ssec:GDO}

In the examples we care about (see Motivation~\ref{mot:PBW} and
Section~\ref{ssec:realistic}) the generating functions turn out
to be perturbed Gaussians, whose perturbations are in some sense
``docile''\footnotemark. Hence we seek to define a category \dpg\ of
docile perturbed Gaussian generating functions, with ``differential
operator'' compositions as in Proposition~\ref{prop"LMcomposition}. We
start with the unperturbed version, \gdo:

\footnotetext{Or perhaps, we care about those examples precisely because their generating functions are
docile perturbed Gaussians.}

\begin{definition} \label{def:GDO}
\gdo\ is the category with
objects finite sets and, if $A$ and $B$ are finite, with $\mor(A\to B)$
the set of ``Gaussians in $\zeta_A\cup z_B$'':
\[ \mor(A\to B) = \left\{\omega\bbe^Q\right\}, \]
where $\omega\in\Qh$ is a scalar and where $Q$ is a
``small'' quadratic expression in $\zeta_A\cup z_B$ with
coefficients in $\Qh$. To define ``small'' and the composition law, we
decompose quadratics in $\zeta_A\cup z_B$ into a Greek-Latin part $E$,
and Greek-Greek part $F$, and a Latin-Latin part $G$:
\[ Q = \sum_{i\in A,j\in B}E_{ij}\zeta_iz_j
  + \frac12\sum_{i,j\in A}F_{ij}\zeta_i\zeta_j
  + \frac12\sum_{i,j\in B}G_{ij}z_iz_j.
\]
With this, ``small'' means that $G$ must be a multiple of
$\hbar$.\footnote{The formulas below make sense either if the $G$
terms are always small or if the $F$ terms are always small. Mostly,
in the applications $G$ will be small and so we made the condition ``$G$
is small'' be a part of the definition of \gdo. Rarely we will encounter
cases where $F$ is small but $G$ isn't. See Discussion~\ref{}.} Also,
we define the composition of $\omega_1\bbe^{Q_1}\in\mor(A\to B)$
and $\omega_2\bbe^{Q_2}$ to be $\omega\bbe^Q$, with
\begin{equation} \label{eq:gdocompositions} \begin{aligned}
  E &= E_1(I-F_2G_1)^{-1}E_2, &
  F &= F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T, \\
  G &= G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2, &
  \omega &= \omega_1\omega_2\det(I-F_2G_1)^{-1/2},
\end{aligned} \end{equation}
where $(E,F,G)$ and $(E_i,F_i,G_i)$ are the Greco-Roman decompositions of
$Q$ and of $Q_i$ as above.  Finally, the identity morphism in $\mor(A\to
A)$ is declared to be $\bbe^{\zeta_A\cdot z_A}$.\endpar{\ref{def:GDO}}
\end{definition}

\begin{comment} \label{com:QGood}
The formulas in Definition~\ref{def:GDO} may appear
unfriendly. But appearances are deceiving. Note that the rank of the
space of quadratics in a certain set of variables is in itself quadratic
in the number of variables, and quadratics grow very slowly relative to
exponentials. Hence the storage and time requirements to store and compute with elements of \gdo\ are much milder
than those for many other computations in quantum algebra, which tend to be exponential.
\end{comment}

\begin{theorem} \label{thm:GDO}
(i) \gdo\ is indeed a category (the composition law is associative, the identity morphisms are identity morphisms).
\newline (ii) The explicit composition law of~\eqref{eq:gdocompositions} agrees with the ``differential
operator'' one of~\eqref{eq:LMcomposition}.
\end{theorem}

\begin{proof} Part (i) can be verified by explicit matrix computations. It
can also be implemented and tested, and seeing that we are committed
to computability, we do that in Appendix~\ref{app:GDOCompositions}. Finally,
part (i) follows from part (ii) and the fact that the composition law
of~\eqref{eq:LMcomposition} is obviously associative. Hence we concentrate
on proving (ii). We do it in two ways: pictorial, right below, for those who
are familiar with diagrammatic algebra, and pure algebraic, on page~\pageref{pf:GDO:algebraic}.
\end{proof}

\parpic[r]{\input{figs/GDOMorphism.pdf_t}}
\noindent{\it Pictorial proof of Theorem~\ref{thm:GDO}, (ii).} This proof assumes familiarity with the kind of
diagrammatics that occurs with Feynman diagrams in quantum field theory and/or with exponentials of connected
diagrams as they occur in, say,~\cite{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}. Pictorially, we view
morphisms in $\mor_\gdo(A\to B)$ as in the picture on the right: we put the Greek input variables corresponding
to $A$ on the left, the Latin output variable corresponding to $B$ on the right, we indicate the scalar
coefficient $\omega$ at the top, and we use the bulk of the picture to indicate $Q$ and its Greco-Roman
decomposition, with an obvious ``Greek facing'' placement of $F$, ``Latin facing'' placement of $G$, and
``across the divide'' placement of $E$. Note that $Q$ is exponentiated and that exponentials are ``reservoirs
of multiple copies'' $\bbe^x=1+x+xx/2+xxx/6+\ldots$. We emphasize this by drawing $E$, $F$, and $G$ as
having multiple shadows.

\Needspace{23mm} % 22mm is not enough.
\parpic[r]{\input{figs/GDOComposition.pdf_t}}
With this language, a composition as in~\eqref{eq:LMcomposition} of
a pair of morphisms as on the right is interpreted as ``sum over all
possible contractions of Latin-side ends in $\bbe^{Q_1}$ with Greek-side
ends in $\bbe^{Q_2}$ (provided their labels, which are elements of $B$,
agree)''. Thus to figure out, say, the $E$ part of the output, we need to
figure out all the ways to travel from $A$ to $C$ across the composition
of $\bbe^{Q_1}$ and $\bbe^{Q_2}$ by carrying out such contractions.

\Needspace{31mm} % 31mm is not enough.
\parpic[r]{\input{figs/E.pdf_t}}
The most obvious way to travel across is the direct route: contract $E_1$
with $E_2$. This contributes a term proportional to $E_1E_2$ to the
output $E$. Another possibility is to travel along $E_1$, then $F_2$, then $G_1$, then $E_2$, producing a term
proportional to $E_1F_2G_1E_2$. Another possibility is to take the $F_2G_1$ detour twice, producing a term 
proportional to $E_1(F_2G_1)^2E_2$. In general, and with proper accounting of the combinatorial factors (it
turns out that all proportionality factors are $1$), we get 
\[ E = \sum_{r=0}^\infty E_1(F_2G_1)^rE_2 = E_1(I-F_2G_1)^{-1}E_2, \]
where the last equality was obtained by summing a geometric series, and where convergence is assured by the
``smallness'' condition on $G$ in Definition~\ref{def:GDO}.

Similar reasonings justify the formulas for $F$ and for $G$.

\Needspace{25mm}
\parpic[r]{\input{figs/FGCycles.pdf_t}}
Yet there is one further contribution to $\bbe^{Q_1}\act\bbe^{Q_2}$,
coming from closed $F_2G_1$ cycles as on the right (but of an arbitrary
length $r$). This contribution is a scalar that modifies $\omega_1\omega_2$,
and it is
$ \exp\left(\sum_{r=1}^\infty\frac{1}{2r}\tr(F_2G_1)^r\right)
  = \exp(-\frac12\tr\log(1-F_2G_1))
  = \det(1-F_2G_1)^{-1/2},
$
justifying the last part of Equation~\eqref{eq:gdocompositions}. Note that
in the last formula we used the familiar quantum field theory dictum to
``divide each diagram by the order of its symmetry group'' to get the
$1/2r$ factor, and that throughout the proof we regarded only connected
diagrams and exponentiated the result, as per the dictum ``the logarithm
of the partition function is generated by connected diagrams''.
\endpar{pictorial}

{\red MORE. Add a section about piggyback Gaussians.}

\draftcut \subsection{A Baby \dpg\ and the Statement of the main \dpg\ Theorem} \label{ssec:baby}

In this section we introduce a ``baby'' version of \dpg, in which the most interesting features of the
``mature'' versions are present, yet some inconveniences regarding weights are censored.

\begin{definition} \label{def:DoPeGDO}
Let $\Omega$ be some ring of ``scalars'' and let
$\eps$ be a formal parameter. Like \gdo, let $\dpg_b$ be the category with
objects finite sets and, if $A$ and $B$ are finite, with $\mor(A\to B)$
the set of ``docile perturbed Gaussians in $\zeta_A\cup z_B$'':
\[ \mor(A\to B) = \left\{\omega\bbe^{Q+P}\right\}, \]
where $\omega$ and $Q$ are $\eps$-independent and otherwise as in Definition~\ref{def:GDO}, and where $P$ is
a power series in $\eps$ of the form $P=\sum_{k\geq 1}P^{(k)}\eps^k$ and where each $P^{(k)}$ is a polynomial
in $\zeta_A\cup z_B$ satisfying the ``docility condition'':
\[ \deg P^{(k)}\leq 2k+2. \]
The composition law of $\dpg_b$ is ``be compatible
with~\eqref{eq:LMcomposition}'' (so this definition becomes complete
only following the discussion of Feynman diagrams below, or in
Section~\ref{ssec:PDE}). \endpar{\ref{def:DoPeGDO}}
\end{definition}

\begin{comment} If we mod out by $\eps^{k_0+1}$ for some $k_0\geq 0$, or
in other words, restrict our attention to $\dpg_b$ ``up to $\eps^{k_0}$'',
then the rank of the space of docile polynomials is polynomial in the
number of variables (cf.\ Comment~\ref{com:QGood}). Hence storing and
manipulating docile polynomials has a chance of being computationally
cheap; later we will see that this is indeed the case.
\end{comment}

We now seek to understand compositions. With the same diagrammatic language as before, we seek to determine
$\omega$, $Q=(E,F,G)$ and $P$, so that the following would hold, where composition is ``all possible
contractions'':
\begin{equation} \label{eq:DoPeGDOCompositions}
  \begin{array}{c} \input{figs/DoPeGDOCompositions.pdf_t} \end{array}
\end{equation}
Looking only at the $\eps$-independent part, it is clear that the composition law for $\omega$ and for $Q$ is
the same as for \gdo~\eqref{eq:gdocompositions} (so \dpg\ is an ``extension'' of \gdo). We just have to find
$P=\sum_{k\geq 1}P^{(k)}\eps^k$ as a function of $Q_{1,2}$ and $P_{1,2}$.

Well, $P^{(k)}$ must get $k$ factors of $\eps$ and it can only get them
from $P_1$ and $P_2$. So $P^{(k)}$ is a sum of diagrams that have at most
$k$ vertices\footnotemark. These vertices can be connected to each other
(including self-connections), or to the outside, either directly, or by
travelling along $E_{1,2}$ lines, or by travelling along $F_2G_1$ or $G_1F_2$ cycles
as before. The latter cycles produce geometric series that sum to either $(I-F_2G_1)^{-1}$ or
$(I-G_1F_2)^{-1}$. We arrive at the following theorem, which we state in a slightly informal manner as a more
rigorous treatment follows in Section~\ref{ssec:PDE}:

\footnotetext{Less than $k$ if a single vertex brings along more
than one factor of $\eps$. Namely, if it comes from $P_{1,2}^{(l\geq 2)}$.}

\Needspace{24mm} % 23mm is not enough
\parpic[r]{\input{figs/FD.pdf_t}}
\begin{theorem}
\picskip{4}
In a composition as in~\eqref{eq:DoPeGDOCompositions} the term $P^{(k)}$ in $P$ is the sum of all
connected Feynman diagrams as on the right, each divided by the order of its automorphism group, and in which the
vertices are determined by $P_1$ and $P_2$ and in which there are five types of propagators (all sampled on
the right):

\begin{enumerate}
\item A $P_1$-to-$P_2$ propagator which equals $(I-F_2G_1)^{-1}$.
\item A $P_1$-to-$P_1$ propagator which equals $(I-F_2G_1)^{-1}F_2$.
\item A $P_2$-to-$P_2$ propagator which equals $G_1(I-G_1F_2)^{-1}$.
\item A Greek-to-$P_2$ propagator which equals $E_1(I-F_2G_1)^{-1}$.
\item A $P_1$-to-Latin propagator which equals $(I-F_2G_1)^{-1}E_2$.
\end{enumerate}

The figure here depicts a contribution to $P^{(4)}$. In general the valencies of vertices may be higher and
self-contractions of two edges coming out of the same vertex are allowed.\qed
\end{theorem}

\parshape 1 0in \textwidth
\begin{proposition}
$\dpg_b$, as defined in Definition~\ref{def:DoPeGDO} and with composition as in the above
theorem, is indeed a category. Namely, with notation as in Equation~\eqref{eq:DoPeGDOCompositions} and with
$P$ as in the theorem, if $P_1$ and $P_2$ are docile then so is $P$.
\end{proposition}

\parshape 1 0in \textwidth
\noindent{\em Proof.} Consider a diagram contributing to $P$ that has $m$ vertices
$v_1,\ldots,v_m$. Each $v_i$ comes from either $P_1$ or $P_2$ and brings
along some power $k_i$ of $\eps$, so the diagram overall contributes a
term $T$ in which the power of $\eps$ is $k=\sum_{i=1}^mk_i$. We need to
show that the degree of $T$ in the Greek and Latin variables satisfies
$\deg T\leq 2k+2$. Indeed, by the docility of $P_1$ and $P_2$ each
$v_i$ contributes at most $2k_i+2$ to that degree. Also, the diagram
is connected\footnote{Da liegt der Hund begraben. Had we used
$\omega\bbe^QP$ instead of $\omega\bbe^{Q+P}$ for the morphisms of \dpg\ 
we'd have had no connectedness here and the docility bound would have
been $\deg P^{(k)}\leq 4k$, leading to slower computations.} so it has
at least $m-1$ edges, and each one contracts to variables, so each one
reduces the overall degree by $2$. So $\deg T\leq\left(\sum_{i=1}^m
2k_i+2\right)-2(m-1)=2k+2$. \qed

\vskip 1mm

{\red MORE. Add a ``formula'' version and a demo.}

The full \dpg\ category needed in this paper is merely a ``garnished''
version of $\dpg_b$, in which every variable has a ``weight'', and some
weight restriction apply. We now turn to its formal definition, which
we give in a slightly informal manner.

\begin{context}
Let $n>0$ be a positive integer, and let us work in some universe of
Latin and Greek variables in which every variable $z$ (or $\zeta$) has a
weight $\wt(z)$ (or $\wt(\zeta)$) with $0\leq\wt(z)\leq n$ (and
$0\leq\wt(\zeta)\leq n$), so that if $z$ and $\zeta$ are dual then
$\wt(z)+\wt(\zeta)=n$. Every monomial in our universe now has a weight,
the sum of the weights of all the variables appearing in it, counted
with multiplicity. The variables $\hbar$ and $\eps$ are special and do
not carry a weight.
\end{context}

\begin{example} \label{exa:MainWeights} In the main context of this paper, that of
Section~\ref{sec:Everything}, we will have $n=2$ and we will have variables $y_i$,
$b_i$, $a_i$, and $x_i$ (where $i$ can run in some sets of
labels), and their duals $\eta_i$, $\beta_i$, $\alpha_i$,
and $\xi_i$, with weights $\wt(y_i,b_i,a_i,x_i)=(1,0,2,1)$ and
$\wt(\eta_i,\beta_i,\alpha_i,\xi_i)=(1,2,0,1)$. In this context,
$\wt(\alpha_3^{62}a_1^8y_{41}^3\hbar^1\eps^7)=62\cdot 0+8\cdot 2+3\cdot 1+1\cdot 0+7\cdot 0=19$.
\end{example}

\begin{definition} A power series $D=\sum D^{(k)}\eps^k$ is called ``docile''
if for every $k$ every monomial appearing in $D^{(k)}$ has weight less than
$n(k+1)$ (with a slight imprecision, this is $\wt(D^{(k)})\leq n(k+1)$). The
same $D$ is called ``$G_n$-docile'' if it is docile and in addition the following
``Condition $G_{n0}$'' holds:
\vskip 2mm
\begin{quote}
{\bf Condition $G_{n0}$.} For any weight-$n$ variable $z$, $\partial_zD^{(0)}$
is affine-linear in the weight-$0$ variables.
\end{quote}
\end{definition}

\begin{comment} Note that if $D$ is docile then $\wt(D^{(0)})\leq n$ so if
also $wt(z)=n$, then $\wt(\partial_zD^{(0)})=0$, so $\partial_zD^{(0)}$ depends only on the weight 0
variables. Condition $G_{n0}$ says that this dependence is particularly simple.
\end{comment}

\begin{example} The generating function of multiplication in the algebra $\bbA_\eps$,
\[ \exp\left( (\alpha_i+\alpha_j)a_k+(\bbe^{-\eps\alpha_j}\xi_i+\xi_j)x_k \right) \]
(see Proposition~\ref{prop:am}), is $G_2$-docile, with variable weights as in Example~\ref{exa:MainWeights}:
$\wt(a,x,\alpha,\xi)=(2,1,0,1)$.
\end{example}

{\red Possible improvement: \dpg\ are things which are $\eps$-weight-docile, $\hbar$-Latin-docile, and have no
Greek-only pairs. Can the last condition also be phrased as a docility condition?}

{\red MORE: State up front a full EDDO/\dpg\ theorem.}

The diagrammatic discussion of this section can be continued and
extended to the full $\dpg_n$ category of Section~\ref{ssec:FullDPG}
but we prefer the more solid grounds of pure algebra as in the next
section, Section~\ref{ssec:PDE}.

\draftcut \subsection{Algebra by means of Partial Differential Equations} \label{ssec:PDE}

Much as we love intuitive graphical reasonings such as in the previous
sections, we also like the more solid grounds of algebra. Hence we repeat
the content of Sections~\ref{ssec:GDO} and~\ref{ssec:baby} in a purely
algebraic language (as it turns out, it is the language of partial
differential equations, though they are only used with power series
and hence we remain in pure algebra).

Recall from Comment~\ref{com:DoubleContraction} that in order to compute compositions of generating functions
we
need to evaluate contractions like
$\left.\bbe^{\sum\partial_{z_b}\partial_{\zeta_b}}(\calL\cdot\calM)\right|_{z_b=\zeta_b=0}$. This inspires the
following slightly more general definition:

\begin{definition} Let $B$ be a finite set, let $F$ be a $B\times B$ matrix, and let $\calE$ be a power
series in variables that include the variable $z_B$. Set the ``partial contraction'' and the ``full
contraction'' of $\calE$ using $F$ to be
\[ [F\colon\calE]_B\coloneqq \bbe^{\frac12\sum_{i,j\in B} F_{ij}\partial_{z_i}\partial_{z_j}}\calE
  \quad\text{and}\quad
  \langle F\colon\calE\rangle_B\coloneqq \left.[F\colon\calE]_B\right|_{z_B\to 0}.
\]
\end{definition}

\begin{note} To ensure convergence one must assume some ``smallness'' condition on either $F$ or
$\calE$. We defer this to a later point.
\end{note}

\begin{note}
In the above definition,
\begin{itemize}
\item $\calE$ replaces the product $\calL\cdot\calM$ of~\eqref{eq:DoubleContraction},
\item we restrict to a single ``type'' of variables $z_B$ instead of the $z_B\cup\zeta_B$
of~\eqref{eq:DoubleContraction} (so $B$ here is ``twice'' the $B$ of~\eqref{eq:DoubleContraction}),
\item instead of a pairing matrix of the form $\begin{pmatrix}0&I\\I&0\end{pmatrix}$ as
in~\eqref{eq:DoubleContraction}, we allow a general matrix $F$.
\end{itemize}
This added generality will become beneficial soon.
\end{note}

\begin{note} The computations of $[F\colon\cdot]_B$ and of $\langle F\colon\cdot\rangle_B$ are equivalent by
``soft'' means: $[F\colon\cdot]_B$ clearly determines $\langle F\colon\cdot\rangle_B$, and we also have
$[F\colon\calE]_B=\left.\left\langle F\colon\left.\calE\right|_{z_b\to
z_b+z'_b}\right\rangle\right|_{z'_b\to z_b}$, where $z'_B$ is a new set of variables indexed by $B$. The
full contraction $\langle F\colon\cdot\rangle_B$ is used in~\eqref{eq:DoubleContraction}, yet the partial
contraction $[F\colon\cdot]_B$ is easier to manipulate as below.
\end{note}

Let $\lambda$ be a formal variable and let $\calZ_\lambda\coloneqq[\lambda F\colon\calE]_B$. Then
$\calZ_\lambda$ (and hence all we care about in this section) is
determined by the following initial value problem, a heat equation:
\begin{equation} \label{eq:heat}
  \calZ_0=\calE \qquad\text{and}\qquad
  \partial_\lambda\calZ_\lambda = \frac12\sum_{i,j\in B}F_{ij}\partial_{z_iz_j}\calZ_\lambda.
\end{equation}
Yet we like to write generating functions as exponentials\footnote{The equations become non-linear, but
as we will see later, their solutions lie in smaller spaces, allowing for more efficient
manipulations.}, and hence the following proposition:

\begin{proposition} \label{prop:synthesis}
With $E=\log\calE$ and $Z_\lambda=\log\calZ_\lambda$ Equation~\eqref{eq:heat}
becomes
\begin{equation} \label{eq:synthesis}
  Z_0=E \qquad\text{and}\qquad
  \partial_\lambda Z_\lambda = \frac12\sum_{i,j\in B}F_{ij}\left(
    \partial_{z_iz_j}Z_\lambda + (\partial_{z_i}Z_\lambda)(\partial_{z_j}Z_\lambda)
  \right).
\end{equation}
\end{proposition}

\begin{proof} Simply substitute $\calZ_\lambda=\bbe^{Z_\lambda}$
into~\eqref{eq:heat} and carry out the differentiations.
\end{proof}

\begin{example} \label{exa:synthesis}
With $B=\{1\}$ a singleton, so we have just one variable
$z=z_1$, with $F$ the $1\times 1$ matrix $(1)$, with $t$ a small
scalar (namely, a commuting extra variable and with all work carried
out over $\bbQ\llbracket t\rrbracket$), and with $\calE=\bbe^{\frac{t}{2}z^2}$,
let us compute
\[ 
  [(1)\colon\calE] = \bbe^{\frac12\partial_z^2}\bbe^{\frac{t}{2}z^2}
  \qquad\text{and}\qquad
  \langle(1)\colon\calE\rangle = [(1)\colon\calE]|_{z=0} =
    \bbe^{\frac12\partial_z^2}\bbe^{\frac{t}{2}z^2}|_{z=0}.
\]
With Proposition~\ref{prop:synthesis} in mind, we set
$Z_\lambda\coloneqq\log[(\lambda)\colon\calE]$ and Equation~\eqref{eq:synthesis} becomes
\[
  Z_0=\frac{t}{2}z^2
  \qquad\text{and}\qquad
  \partial_\lambda Z_\lambda
    = \frac12\left(\partial_z^2Z_\lambda + (\partial_zZ_\lambda)^2 \right).
\]
The differential operator $Z\mapsto\partial_z^2Z + (\partial_zZ)^2$
maps the space of linear combinations of $1$ and of $z^2$ into itself,
and so our solution must be of the form $\frac{f}{2}\cdot
1+\frac{g}{2}z^2$ where $f$ and $g$ are power series in $t$ and
$\lambda$. Substituting this back into the equation, we get the system
\begin{align*}
  f|_{\lambda=0} &= 0, & \partial_\lambda f &= g, \\
  g|_{\lambda=0} &= t, & \partial_\lambda g &= g^2,
\end{align*}
whose solution is $g=\frac{t}{1-t\lambda}$ and $f=\log\frac{1}{1-t\lambda}$. Thus
$Z_\lambda=\log\frac{1}{\sqrt{1-t\lambda}} + \frac{t}{1-t\lambda}\frac{z^2}{2}$, and so
\[ [(1)\colon\calE] = \bbe^{\frac12\partial_z^2}\bbe^{\frac{t}{2}z^2}
  = \bbe^{Z_1} = \frac{1}{\sqrt{1-t}}\exp\left(\frac{t}{1-t}\frac{z^2}{2}\right)
\]
and
\[ \langle(1)\colon\calE\rangle = \bbe^{\frac12\partial_z^2}\bbe^{\frac{t}{2}z^2}|_{z=0}
  = \frac{1}{\sqrt{1-t}}.
\]
\end{example}
\endpar{\ref{exa:synthesis}}

MORE: An exercise about the relationship with integration.

A sometimes-useful alternative to~\eqref{eq:synthesis} is to allow $F$
to be implicitly dependent on $\lambda$ in an arbitrary (differentiable)
manner with the condition $\left.F\right|_{\lambda=0}=0$ and to suppress
the $\lambda$ subscript in $Z_\lambda$. The resulting equation is
\begin{equation} \tag{\ref{eq:synthesis}'}
  \left.Z\right|_{\lambda=0}=E \qquad\text{and}\qquad
  \partial_\lambda Z = \frac12\sum_{i,j\in B}(\partial_\lambda F_{ij})\left(
    \partial_{z_iz_j}Z + (\partial_{z_i}Z)(\partial_{z_j}Z)
  \right).
\end{equation}

We call Equation~\eqref{eq:synthesis} (and its variant
Equation~(\ref{eq:synthesis}')) ``the synthesis equation'',
as it governs how the ``vertices'' in $E$ merge and contract
to synthesize larger and larger connected diagrams, as in the
interpretation below.\footnote{\label{foot:Burger}M.~Pugh told us that
Equation~\eqref{eq:synthesis} is a variant of ``Burger's equation'',
and that it's relationship with the heat equation~\eqref{eq:heat} is a
variant of the ``Cole-Hopf transformation''.}

\Needspace{24mm} % 23mm is not enough.
\parpic[r]{\raisebox{-0mm}{\def\F{$\lambda F$}\input{figs/FDCuts.pdf_t}}}
\begin{interpretation} \label{int:masterequation}
For the initiated, we cannot resist including a Feynman-diagram
interpretation of Equation~\eqref{eq:synthesis}. With $E$ ``the
vertices'' and $F$ ``the contraction tensor'' (roughly, ``the
propagator''), $\calZ_\lambda=\langle\lambda F\colon\bbe^E\rangle$
is the sum of all Feynman diagrams that can be made with
vertices in $E$ and contractions as dictated by $F$, with each
contraction multiplied by an additional factor of $\lambda$. Then
$Z_\lambda=\log\calZ_\lambda$ is the same, except restricting to
connected Feynman diagrams. And then $\partial_\lambda Z_\lambda$
picks out one contraction in $Z_\lambda$. If it is ``separating'',
it contributes an $F$-weighted product of two connected diagrams ---
the term $(\partial_{z_i}Z_\lambda)(\partial_{z_j}Z_\lambda)$. If it not
separating, it can be seen to contribute the $\partial_{z_iz_j}Z_\lambda$
term. See the picture on the right. \endpar{\ref{int:masterequation}}
\end{interpretation}

{\blue
\noindent{\bf Lemma 1.} $\displaystyle \left\langle F\colon\calE\,\bbe^{\sum_{i\in B}y_iz_i}\right\rangle_B
  = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j}
    \left\langle F\colon \left.\calE\right|_{z_B\to z_B+Fy_B}\right\rangle_B$ and
\begin{multline*} \left[F\colon\calE\,\bbe^{\sum_{i\in B}y_iz_i}\right]_B
  = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j+\sum_{i\in B}y_iz_i}
    \left[F\colon \left.\calE\right|_{z_B\to z_B+Fy_B}\right]_B \\
  = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j+\sum_{i\in B}y_iz_i}
    \left(\left.\left[F\colon\calE\right]_B\right|_{z_B\to z_B+Fy_B}\right).
\end{multline*}

\noindent{\bf Lemma 2.} With convergences left to the reader,
\[ \left\langle F\colon\calE\,\bbe^{\frac12\sum_{i,j\in B}G_{ij}z_iz_j}\right\rangle_B
  = \det(1-GF)^{-1/2}\left\langle F(1-GF)^{-1} \colon \calE \right\rangle_B,
\]
and
\begin{multline*} \left[F\colon\calE\,\bbe^{\frac12\sum_{i,j\in B}G_{ij}z_iz_j}\right]_B
  = \det(1-GF)^{-1/2}\bbe^{\frac12\sum_{i,j\in B}(G(I-FG)^{-1})_{ij}z_iz_j} \\
    \cdot\left(\left[F(1-GF)^{-1} \colon \calE \right]_B\right)_{z_B\to(I-FG)^{-1}z_B}.
\end{multline*}

\[ \input{figs/Lemmas.pdf_t} \]
}

{\red MORE.}

\subsection{Full \dpg} \label{ssec:FullDPG}

{\red MORE.}

