\chapter{The normal tube and the normal oscillator}\label{s:tube}

This section and the next prove that the transfer operator of the
supersymmetric representation is stable over the $N \asymp W^2$ steps of
the chain, uniformly in $W$, when it is written in coordinates adapted
to its saddle manifold. In this section we set up these coordinates and
study the frozen normal transfer, the normal part of the transfer operator, which is a
supersymmetric harmonic oscillator. In \zcref{s:stab} we compare the
full transfer operator with a product of this oscillator and a heat
semigroup on the saddle manifold. The results are used in three places.
The stability theorem of \zcref{s:stab}, the fixed state of the
normal oscillator (\zcref{p:tube-gap}), and the identification of the
heat generator on the saddle manifold with a multiple of the Casimir
operator in \zcref{l:tube-casimir}, which fixes the time normalization of
one link, are the inputs of the averaging argument of \zcref{s:cons}, and
the endpoint and assembly
arguments of \zcref{s:end,s:assembly} use the coordinates and the gauge of
\zcref{s:tube-gauge}. The transfer-operator approach to one-dimensional
band matrices in the supersymmetric representation goes back to
M.~Shcherbina and T.~Shcherbina (see \cite{shcherbina2019transfer} and
the introduction).

Throughout Part IV we write
\[
 h = W^{-1/2}, \qquad \delta_W = W^{-2} = h^4 ,
\]
as in \zcref{s:hist}. The reader who knows Ornstein--Uhlenbeck
semigroups but not superanalysis may read this section as follows. The
normal variables are Gaussian at scale $h$, and in standardized
coordinates the frozen normal transfer is an Ornstein--Uhlenbeck
operator at a small complex time of order $h^2$, tensored with a finite
fermionic analogue. Every estimate below is an estimate for this
operator and for its perturbations by multipliers, cutoffs, and first-
and second-order differential operators. The fermionic variables enter
only through finite-dimensional linear algebra, which we carry out
before taking any norm. Grassmann variables, Berezin integrals,
supermatrices, $\Str$, and $\Sdet$ are as in \zcref{s:grassmann}, and
\zcref{s:tube-prelim} collects the further superalgebra that we use, from
super vector spaces to invariant Berezin integration on homogeneous
superspaces.

\section{Two scales}\label{s:tube-scales}

We first explain heuristically why there are two scales. Recall from
\zcref{p:rep-exact} that the source ratio is given by powers of the
transfer operator $T_\Gamma$, which multiplies by the
onsite weight $w_\Gamma$ after applying the ambient heat factor
$K_{\tau_{\rm phys}}$. The heat factor has the Gaussian kernel
\begin{equation}\label{e:tube-heat-kernel}
 K_{\tau_{\rm phys}}(R, R') = \exp\Bigl\{-\frac{\Str (R - R')^2}{2h^4}\Bigr\},
\end{equation}
where $R$ and $R'$ are the supermatrix variables of two neighboring sites.
The kernel carries no normalizing constant. Indeed, the Berezin form $\dd R$ of \zcref{d:rep-flat} is
not rescaled on the cycle, its order of the odd coordinates makes the
Gaussian integral \eqref{e:rep-flat-normalization} equal to $+1$ with
$c_k = (2\pi)^{-4k^2}$, and \zcref{s:rep-chain} shows that $K_t1 = 1$
without a constant factor. Recall from
\zcref{s:ov-scales} that the chain stays near the saddle manifold of
\zcref{s:rep-heuristics}, the union of $2k + 1$ orbits of critical points
of the onsite weight, whose balanced component is a copy of the target
superspace $\mathcal M_k$ of \zcref{s:sigma}, and that one link moves the
supermatrix by $h^2 = W^{-1}$. Normal to the saddle manifold, the chain
behaves as a discretized Ornstein--Uhlenbeck process that relaxes over
about $h^{-2}$ links to a size of order $h$, and along it as a random
walk with variance $\delta_W = h^4$ per link, which runs for a time of order
one over the $N \asymp h^{-4}$ links (\zcref{f:tube-scales}). We make
the first statement precise in \zcref{l:tube-quadratic}: in the balanced
signature the exact quadratic normal transfer is the Mehler operator
$\mu^{\mathbf N}$ with $-\log\mu \asymp h^2$. In the other signatures it
also carries the scalar root phase and the zero-point factor computed there.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.15]
 \fill[gray!14] plot[smooth] coordinates {(-0.2,0.55) (2,1.15) (4,1.05) (6,1.55) (8.2,1.35)}
   -- plot[smooth] coordinates {(8.2,0.35) (6,0.55) (4,0.05) (2,0.15) (-0.2,-0.45)} -- cycle;
 \draw[dashed,gray] plot[smooth] coordinates {(-0.2,1.15) (2,1.75) (4,1.65) (6,2.15) (8.2,1.95)};
 \draw[dashed,gray] plot[smooth] coordinates {(-0.2,-1.05) (2,-0.45) (4,-0.55) (6,-0.05) (8.2,-0.25)};
 \draw[thick] plot[smooth] coordinates {(-0.2,0.05) (2,0.65) (4,0.55) (6,1.05) (8.2,0.85)};
 \draw[blue!60!black] plot[smooth,tension=0.4] coordinates {(0.3,0.1) (0.45,0.35) (0.62,0.18) (0.8,0.45)
   (0.98,0.62) (1.15,0.38) (1.33,0.6) (1.5,0.85) (1.7,0.7) (1.88,0.95) (2.05,0.78) (2.25,0.52) (2.45,0.7)
   (2.65,0.9) (2.85,0.66) (3.05,0.83) (3.25,0.62) (3.45,0.42) (3.65,0.6) (3.85,0.8)};
 \foreach \p in {(0.3,0.1),(0.45,0.35),(0.62,0.18),(0.8,0.45),(0.98,0.62),(1.15,0.38),(1.33,0.6),
   (1.5,0.85),(1.7,0.7),(1.88,0.95),(2.05,0.78),(2.25,0.52),(2.45,0.7),(2.65,0.9),(2.85,0.66),(3.05,0.83),
   (3.25,0.62),(3.45,0.42),(3.65,0.6),(3.85,0.8)} \fill[blue!60!black] \p circle (0.03);
 \draw[<->] (6.9,0.97) -- (6.9,1.47) node[midway,right] {\small $h$};
 \draw[<->] (7.6,0.9) -- (7.6,1.98) node[midway,right] {\small $r$};
 \node[below] at (5.3,0.62) {\small saddle manifold};
 \node[above] at (2.1,1.25) {\small normal tube};
 \node[blue!60!black,below] at (2.2,0.0) {\small links of size $h^2$};
\end{tikzpicture}
\caption{The saddle manifold (solid), the region within normal distance
of order $h$ of it where the normal variables of the chain are
concentrated (shaded), and the normal cutoff at a fixed radius $r$
(dashed). Each link moves the supermatrix by $h^2$. The normal
coordinate relaxes after about $h^{-2}$ links, during which the tangent
coordinate moves by about $h$; over the $h^{-4}$ links of the chain the
tangent coordinate moves by a distance of order one.}
\label{f:tube-scales}
\end{figure}

The heuristic suggests the method. We write the transfer operator in
coordinates $(x, U)$, where $x$ is a point of the saddle manifold and
$U$ is a normal displacement, and we restrict $U$ to a ball of fixed
radius $r$ by a cutoff. In the standardized normal coordinate
$Z = U/h$, the frozen normal transfer is, to leading
order, an Ornstein--Uhlenbeck operator at time of order $h^2$, and the
normal relaxation comes from its spectral gap of order $h^2$ off the
vacuum.
Recall from \zcref{s:ov-averaging} that an error of size $h^3$ per step
accumulates to $h^{-1}$ over the chain, so that every error term must be
compared with this gap or with the size $\delta_W$ of the tangent step. We
measure errors relative to the defect operators of a reference
contraction, in the form required by \zcref{l:tools-defect}. The
cutoff and Mehler estimates of \zcref{s:tube-cutoff,s:tube-amplitude}
provide these relative bounds for the frozen normal transfer, and
\zcref{s:tube-gap} proves the Ward identity of \zcref{s:ov-averaging},
the exact preservation of the Gaussian mean on closed vectors, which no
perturbative estimate could give.

\section{Super vector spaces and homogeneous superspaces}
\label{s:tube-prelim}

The tubular coordinates of \zcref{s:tube-geometry} use some superalgebra
beyond that of \zcref{s:grassmann,s:sigma-target}: super vector spaces
with the supertraces and superdeterminants of operators on them, the
stabilizer of a point of the saddle manifold, and the invariant Berezin
volume of the quotient by this stabilizer. We use them to prove the
identity \eqref{e:tube-volume} of \zcref{l:tube-volume}, which is the
supersymmetric analogue of the Weyl integration formula. For a Hermitian
$n \times n$ matrix $H = g\diag(\lambda)g^{-1}$ with $g \in U(n)$, this
formula reads $\dd H = \prod_{i < j}(\lambda_j - \lambda_i)^2\dd\lambda\dd g$
up to a constant factor, where $\dd g$ is the invariant measure on the
orbit of $\diag(\lambda)$. In \eqref{e:tube-volume}, a supermatrix near
the saddle manifold is written as a conjugate $T(R_* + U)T^{-1}$ of a
block-diagonal supermatrix $R_* + U$, the invariant Berezin volume $\nu$
of this section plays the role of the orbit measure $\dd g$, and a
superdeterminant $\varpi$, defined in \eqref{e:tube-half-density}, plays
the role of the Vandermonde factor. In the ordinary case, in which all
entries commute, $\varpi$ is the product of the differences between the
eigenvalues of the two diagonal blocks of $R_* + U$, a cross-Vandermonde
factor. The differences within a block do not appear, since $U$ ranges
over block-diagonal rather than diagonal matrices.
\zcref{r:tube-harmonic-checks} treats this ordinary case. We collect these notions here,
in the concrete form in which we use them. We follow
\cite[Chapter~3]{varadarajan2004supersymmetry} for super linear algebra,
\cite[Section~II.F]{zirnbauer1996riemannian} for symmetric superspaces,
and \cite{alldridge2010invariant} for invariant Berezin integration on
homogeneous superspaces. Recall the supermatrices, their supertrace
$\Str$ and their superdeterminant $\Sdet$ from \zcref{d:grass-supermatrix},
and recall from \zcref{s:sigma-supermatrices,s:sigma-symmetries} that
$U(k, k|2k)$ denotes the pair of the group
$G_0 = \{\diag(g_{\rm B} | g_{\rm F}) : g_{\rm B} \in U(k, k), \ g_{\rm F} \in U(2k)\}$
and the Lie superalgebra $\mathfrak g = \mathfrak{gl}(2k|2k)$ of all
$(2k|2k)$ supermatrices, with the bracket
$[Y, Y'] = YY' - (-1)^{|Y||Y'|}Y'Y$.

\subsection{Super vector spaces}

We define a \emph{super vector space} to be a finite-dimensional complex
vector space $V$ with a decomposition $V = V_0 \oplus V_1$ into an even
part $V_0$ and an odd part $V_1$. If $\dim V_0 = m$ and $\dim V_1 = n$,
we write $\dim V = m|n$, and we call $\operatorname{sdim} V = m - n$ the
\emph{superdimension} of $V$. A homogeneous basis of $V$ is a basis of
$V_0$ followed by a basis of $V_1$, and we write $|i| \in \{0, 1\}$ for
the parity of its $i$th vector. For instance, the columns of size
$(2k|2k)$, with even bosonic and odd fermionic coordinates, form a super
vector space of dimension $2k|2k$, and the $(2k|2k)$ supermatrices are the
matrices of its linear maps in the standard basis. For super vector spaces
$V$ and $W$, the space $\operatorname{Hom}(W, V)$ of linear maps is a super
vector space, whose even part consists of the maps that preserve the
parity and whose odd part of the maps that reverse it. In homogeneous
bases, the matrix unit $E_{i\alpha}$ has the parity $|i| + |\alpha|$, as
in \zcref{s:sigma-integral}. If $\dim V = m|n$ and $\dim W = m'|n'$, then
$\operatorname{Hom}(W, V)$ has dimension $(mm' + nn')|(mn' + nm')$, and
therefore
\begin{equation}\label{e:tube-sdim-hom}
 \operatorname{sdim}\operatorname{Hom}(W, V) = \operatorname{sdim} V\,\operatorname{sdim} W .
\end{equation}
We write $\operatorname{End}(V) = \operatorname{Hom}(V, V)$.

Let $\mathcal R$ be a Grassmann algebra, possibly with auxiliary
generators (\zcref{s:grass-algebra}). An element of $V$ with coefficients
in $\mathcal R$ is a sum $X = \sum_I X_Ie_I$ over a homogeneous basis
$(e_I)$, with coordinates $X_I \in \mathcal R$. It is even if every $X_I$ has
the parity $|I|$. For instance, an even supermatrix over
$\mathcal R$ is an even element of the endomorphisms of the columns, with
its entries as coordinates. We call a map $L$ of the even elements of $V$
with coefficients in $\mathcal R$ into themselves an \emph{even operator}
on $V$ if there exists an even supermatrix $M = (M_{IJ})$ over $\mathcal
R$, indexed by the basis, such that $(LX)_I = \sum_JM_{IJ}X_J$ for all
$X$, and we call $M$ the matrix of $L$. We define
\[
 \Str_V L = \Str M , \qquad \Sdet_V L = \Sdet M ,
\]
the second when $M$ is invertible. These do not depend on the homogeneous
basis. Indeed, a change of homogeneous basis replaces $M$ by
$B^{-1}MB$ for an even invertible supermatrix $B$ with complex entries,
and by \zcref{l:grass-sdet}\ref{i:grass-sdet-trace} and
\ref{i:grass-sdet-mult}, we have $\Str(B^{-1}MB) = \Str M$ and
$\Sdet(B^{-1}MB) = \Sdet M$. For example, $\Sdet_V(c\Id) = c^{\operatorname{sdim} V}$
for every even $c \in \mathcal R$ with nonzero body, and
\zcref{l:grass-sdet}\ref{i:grass-sdet-exp} implies that
$\Sdet_V e^L = e^{\Str_V L}$.

For even supermatrices $A$ on $V$ and $B$ on $W$ with entries in
$\mathcal R$, we let $\lambda_A$ and $\rho_B$ denote the even operators
$X \mapsto AX$ and $X \mapsto XB$ on $\operatorname{Hom}(W, V)$. They
commute, and
\begin{equation}\label{e:tube-tensor-trace}
 \Str_{\operatorname{Hom}(W, V)}(\lambda_A \rho_B) = \Str_V(A)\Str_W(B) .
\end{equation}
Indeed, in homogeneous bases of $V$ and $W$, we have
\[
 (AXB)_{i\alpha} = \sum_{j, \beta}(-1)^{(|j| + |\beta|)(|\beta| + |\alpha|)}A_{ij}B_{\beta\alpha}X_{j\beta} .
\]
The matrix of $\lambda_A\rho_B$ has the diagonal entries
$A_{ii}B_{\alpha\alpha}$, at $(j, \beta) = (i, \alpha)$, and the coordinate
$X_{i\alpha}$ has the parity $|i| + |\alpha|$. Summing the
diagonal entries with the signs $(-1)^{|i| + |\alpha|}$, we obtain
\eqref{e:tube-tensor-trace}. For $W = V$, the identity says that the
operator $Y \mapsto AYA'$ of $\operatorname{End}(V)$ has the supertrace
$\Str(A)\Str(A')$. In particular, the operator
$\operatorname{ad}_A = \lambda_A - \rho_A$ of $\operatorname{End}(V)$ satisfies
\begin{equation}\label{e:tube-ad-trace}
 \Str_{\operatorname{End}(V)}(\operatorname{ad}_A) = \Str(A)\operatorname{sdim} V - \operatorname{sdim} V\Str(A) = 0 .
\end{equation}

\subsection{The stabilizer of a root point}

Fix a signature $p \in \{0, \dots, 2k\}$, and let
$\Sigma = \diag\bigl(\Lambda_{\rm B} \,|\, \diag(\Id_p, -\Id_{2k - p})\bigr)$,
so that $R_* = \kappa_E\Id - \iu s_E\Sigma$ is the root point of signature
$p$ of \zcref{s:rep-heuristics}. We let $V_+$ and $V_-$ denote the
eigenspaces of $\Sigma$ in $\C^{2k|2k}$ for the eigenvalues $1$ and $-1$.
Then $\dim V_+ = k|p$ and $\dim V_- = k|(2k - p)$, so that
\[
 \operatorname{sdim} V_+ = k - p , \qquad
 \operatorname{sdim} V_- = p - k .
\]
We call the signature $p = k$ balanced. In the balanced signature
$\Sigma = \Lambda$, and $V_+$ and $V_-$ are spanned by the retarded and by
the advanced basis vectors of \zcref{s:sigma-supermatrices}. We set
\[
 \mathfrak k = \{Y \in \mathfrak g : \Sigma Y = Y\Sigma\} , \qquad
 \mathfrak m = \{Y \in \mathfrak g : \Sigma Y = -Y\Sigma\} ,
\]
which are the block-diagonal and the block off-diagonal supermatrices
relative to the splitting $V_+ \oplus V_-$, so that
$\mathfrak k = \operatorname{End}(V_+) \oplus \operatorname{End}(V_-)$ and
$\mathfrak m = \operatorname{Hom}(V_-, V_+) \oplus \operatorname{Hom}(V_+, V_-)$,
and for $p = k$ these are the spaces of \eqref{e:sigma-km}. The map
$Y \mapsto \Sigma Y\Sigma$ is an automorphism of order two of $\mathfrak g$,
with the eigenspaces $\mathfrak k$ and $\mathfrak m$ for the eigenvalues
$1$ and $-1$, and therefore
\begin{equation}\label{e:tube-symmetric-pair}
 [\mathfrak k, \mathfrak k] \subset \mathfrak k , \qquad
 [\mathfrak k, \mathfrak m] \subset \mathfrak m , \qquad
 [\mathfrak m, \mathfrak m] \subset \mathfrak k .
\end{equation}
Since the supertrace of a block off-diagonal supermatrix vanishes, the
spaces $\mathfrak k$ and $\mathfrak m$ are orthogonal for the form
$\Str(YY')$, and this form is nondegenerate on each of them. We let
$K_0 = \{g \in G_0 : g\Sigma = \Sigma g\}$, which is isomorphic to
$U(k) \times U(k) \times U(p) \times U(2k - p)$ and connected, and we call
the pair $K = (K_0, \mathfrak k)$ the \emph{stabilizer} of $R_*$ in
$U(k, k|2k)$. For $p = k$, the group $K_0$ is the group of
\zcref{s:sigma-supermatrices}. We let
$\mathfrak m_\R = \mathfrak m_0 \cap \operatorname{Lie}(G_0)$ denote the
real form of the even part of $\mathfrak m$ that is tangent to $G_0$.

The odd elements of $\mathfrak k$ do not change volumes on $\mathfrak m$.
Precisely, let $C = \diag(C_+, C_-)$ be an even supermatrix that commutes
with $\Sigma$, with entries in a Grassmann algebra. Then $\operatorname{ad}_C$
preserves $\mathfrak m$, it acts as $\lambda_{C_+} - \rho_{C_-}$ on
$\operatorname{Hom}(V_-, V_+)$ and as $\lambda_{C_-} - \rho_{C_+}$ on
$\operatorname{Hom}(V_+, V_-)$. By \eqref{e:tube-tensor-trace}, we have
\begin{equation}\label{e:tube-unimodular}
\begin{aligned}
 \Str_{\mathfrak m}(\operatorname{ad}_C)
 &= \bigl(\Str C_+\operatorname{sdim} V_- - \operatorname{sdim} V_+\Str C_-\bigr) \\
 &\qquad + \bigl(\Str C_-\operatorname{sdim} V_+ - \operatorname{sdim} V_-\Str C_+\bigr) = 0 .
\end{aligned}
\end{equation}
By \zcref{l:grass-sdet}\ref{i:grass-sdet-exp}, this implies that
$\Sdet_{\mathfrak m}(e^{\operatorname{ad}_C}) = 1$. Since $K_0$ is
connected, the conjugation $\operatorname{Ad}_k \colon Y \mapsto kYk^{-1}$
also satisfies $\Sdet_{\mathfrak m}\operatorname{Ad}_k = 1$ for all
$k \in K_0$, and therefore for all products $k e^C$ of such elements.

\subsection{The homogeneous superspace and its invariant volume}

We define the superfunctions on the superspace $U(k, k|2k)/K$ as in
\zcref{d:sigma-target}, with $\Lambda$ replaced by $\Sigma$. We fix a
basis $(e_a)$ of the odd part $\mathfrak m_1$ of $\mathfrak m$ that is
orthonormal for the Hilbert--Schmidt inner product
$\langle Y, Y'\rangle = \Tr(Y^*Y')$, as in \zcref{s:sigma-supermatrices}.
The space $\mathfrak m_1$ has dimension $4k^2$ for every signature, and we
set $\Theta = \sum_a\chi_ae_a$. A superfunction is a smooth map $f$ from $G_0$
into the Grassmann algebra of the $\chi_a$ such that
$f(g\mathsf k) = \alpha_{\mathsf k}(f(g))$ for all $\mathsf k \in K_0$,
where $\alpha_{\mathsf k}$ replaces $\Theta$ by $\mathsf k\Theta\mathsf k^{-1}$.
We identify the underlying manifold $G_0/K_0$ with the set of matrices
$g\Sigma g^{-1}$, and we call the matrix of superfunctions
$Q = ge^\Theta\Sigma e^{-\Theta}g^{-1}$ the field. For $p = k$, this is the target
space $\mathcal M_k$ of \zcref{d:sigma-target}. Let $\mathcal O$ be an open
subset of $G_0/K_0$ with coordinates $x$, and let $\mathsf s$ be a section
over $\mathcal O$, that is, a smooth map into $G_0$ with
$\mathsf s(x)\Sigma\mathsf s(x)^{-1} = x$. As in
\zcref{s:sigma-superfunctions}, the superfunctions on $\mathcal O$ are then
the superfunctions of the coordinates $z = (x, \chi)$. We call an even
invertible supermatrix $T$ of superfunctions on $\mathcal O$ with
$T\Sigma T^{-1} = Q$ a \emph{local section} of $U(k, k|2k)$ over
$\mathcal O$. For instance, $T = \mathsf s(x)e^\Theta$ is a local section.
Two local sections $T$ and $T'$ over $\mathcal O$ differ by
$T' = Tk$, where $k$ commutes with $\Sigma$.

The quotient of $\mathrm{GL}(2k|2k)$ by the stabilizer
$\mathrm{GL}(k|p) \times \mathrm{GL}(k|2k - p)$ of $\Sigma$, together with
the real manifold $G_0/K_0$, is a Riemannian symmetric superspace in the
sense of \cite[Definition~2.5]{zirnbauer1996riemannian}, of the type
AIII$|$AIII of Table~2 there, and we call $U(k, k|2k)/K$ a symmetric
superspace. We use this only through the automorphism
$Y \mapsto \Sigma Y\Sigma$, which fixes $K$ and acts as $-\Id$ on
$\mathfrak m$, and through the form $\Str(YY')$, for which $\mathfrak k$
and $\mathfrak m$ are orthogonal and which is positive definite on
$\mathfrak m_\R$.

Let $T$ be a local section over $\mathcal O$. For an even derivation
$\mathcal V$ of the superfunctions on $\mathcal O$, that is, a vector field
in the coordinates $z$ or the product of an odd vector field with an
auxiliary odd parameter as in \zcref{s:sigma-symmetries}, we set
\begin{equation}\label{e:tube-maurer-cartan}
 \mathfrak a(\mathcal V) = T^{-1}(\mathcal VT) ,
\end{equation}
where $\mathcal V$ acts on each entry of $T$, and we write
$\mathfrak a(\mathcal V) = \mathfrak a_{\mathfrak k}(\mathcal V) + \mathfrak a_{\mathfrak m}(\mathcal V)$
for its components in $\mathfrak k$ and $\mathfrak m$. We call
$\mathfrak a$ the \emph{Maurer--Cartan form} of $T$. Its value is an even
element of $\mathfrak g$ with coefficients in the superfunctions. The
Maurer--Cartan form does not change if $T$ is replaced by $gT$ with
$g \in G_0$. If $T$ is replaced by $T' = Tk$, then
\begin{equation}\label{e:tube-gauge-change}
 \mathfrak a'(\mathcal V) = k^{-1}\mathfrak a(\mathcal V)k + k^{-1}\mathcal Vk ,
 \qquad
 \mathfrak a'_{\mathfrak m}(\mathcal V) = k^{-1}\mathfrak a_{\mathfrak m}(\mathcal V)k ,
\end{equation}
since $k^{-1}\mathcal Vk$ commutes with $\Sigma$.

We call a local section $T$ together with a homogeneous basis $(Y_b)$ of
$\mathfrak m$, whose even elements form a basis of $\mathfrak m_\R$, a
\emph{local orbit frame}. In a local orbit frame we let
$\mathcal J_{\mathfrak a}$ denote the even supermatrix, with rows indexed
by the coordinates $z_I$ and columns by the basis $(Y_b)$, such that
\[
 \mathfrak a_{\mathfrak m}(\vartheta^{|I|}\partial_{z_I}) = \vartheta^{|I|}\sum_b(\mathcal J_{\mathfrak a})_{Ib}Y_b ,
\]
where $\vartheta$ is an auxiliary odd parameter and $|I|$ is the parity of
$z_I$. It is the analogue of the Jacobian supermatrix
\eqref{e:grass-jacobian}, with the components of $\mathfrak a_{\mathfrak m}$
in place of the differentials of new coordinates. Its body is invertible:
in the even rows, the body of $\mathfrak a_{\mathfrak m}$ maps the tangent
space of $G_0/K_0$ isomorphically onto $\mathfrak m_\R$, and in the odd
rows, for $T = \mathsf s(x)e^\Theta$, we have
$\mathfrak a_{\mathfrak m}(\vartheta\partial_{\chi_a}) = \vartheta e_a$
up to terms of positive degree in the $\chi$. We define the
invariant Berezin volume $\nu$ by
\begin{equation}\label{e:tube-nu}
 \int f\dd\nu = \int f\,\Sdet(\mathcal J_{\mathfrak a})\dd x\dd\chi
\end{equation}
for superfunctions $f$ whose coefficients have compact support in
$\mathcal O$.

The volume $\nu$ does not depend on the coordinates, among those that
induce a fixed orientation of $G_0/K_0$, which we choose before
\zcref{l:tube-volume}: by the chain rule,
another system of coordinates replaces $\mathcal J_{\mathfrak a}$ by the
product of the Jacobian supermatrix \eqref{e:grass-jacobian} of the
coordinate change with the new $\mathcal J_{\mathfrak a}$, and \zcref{l:grass-sdet}\ref{i:grass-sdet-mult}
and \zcref{t:grass-change}, which applies to changes of coordinates that
preserve orientation, show that \eqref{e:tube-nu} is unchanged. It
does not depend on the local section: by \eqref{e:tube-gauge-change}, a
second section multiplies $\mathcal J_{\mathfrak a}$ by the matrix of
$\operatorname{Ad}_k^{-1}$ on $\mathfrak m$, whose superdeterminant is one by
\eqref{e:tube-unimodular}. The integrals over overlapping coordinate
patches therefore agree, and \eqref{e:tube-nu} defines $\int f\dd\nu$ for
all superfunctions with compactly supported integrable coefficients.
Since $\mathfrak a$ is unchanged under $T \mapsto gT$, the volume $\nu$
is invariant under $G_0$. It is also invariant under the odd symmetries,
by \cite[Theorem~4.13]{alldridge2010invariant}, which we apply below. In
the terminology of \cite{alldridge2010invariant}, $\nu$ is the Berezinian
form whose value in the frame $\mathfrak a_{\mathfrak m}$ is constant. By
\cite[Corollary~4.12]{alldridge2010invariant}, the Berezinian bundle of
$U(k, k|2k)/K$ is the bundle associated with the Berezinian of
$(\mathfrak g/\mathfrak k)^*$, so that an invariant Berezinian form is a
constant element of this Berezinian that is invariant under the
stabilizer. By \cite[Theorem~4.13]{alldridge2010invariant}, applied with
the condition that the Berezinian of $(\mathfrak g/\mathfrak k)^*$ be a
trivial module of the stabilizer, which is
$\Sdet_{\mathfrak m}\operatorname{Ad}_k = 1$ and
$\Str_{\mathfrak m}(\operatorname{ad}_C) = 0$ by
\eqref{e:tube-unimodular}, the form $\nu$ is invariant under the whole of
$U(k, k|2k)$; such a form is unique up to a constant factor (see also
\cite[Remark~4.15]{alldridge2010invariant} for symmetric pairs). In
particular, for $p = k$ the volume $\nu$ is a constant multiple of the
invariant Berezin integral of \zcref{d:sigma-integral}, by the uniqueness
proved in \zcref{s:sigma-integral}.

Near the body point $\Sigma$ we use the exponential chart, in which
$T = e^Y$ with $Y = \sum_by_bY_b \in \mathfrak m$ and the coordinates
$y_b$ (real for even $b$). It is a local section, since the change from
the coordinates $y$ to the coordinates of a section satisfies the
hypotheses of \zcref{l:grass-coordinates}: its body is the exponential map
of $G_0/K_0$ at $\Sigma$, and its linear part in the odd coordinates is
the identity at $\Sigma$. For an even derivation $\mathcal V$, the
differential of the exponential map gives
\[
 \mathfrak a(\mathcal V) = e^{-Y}\mathcal Ve^Y
 = \frac{1 - e^{-\operatorname{ad}_Y}}{\operatorname{ad}_Y}(\mathcal VY)
 = \frac{\sinh\operatorname{ad}_Y}{\operatorname{ad}_Y}(\mathcal VY)
 + \frac{1 - \cosh\operatorname{ad}_Y}{\operatorname{ad}_Y}(\mathcal VY) ,
\]
where the first term lies in $\mathfrak m$ and the second in
$\mathfrak k$, since $\operatorname{ad}_Y$ exchanges $\mathfrak k$ and
$\mathfrak m$ by \eqref{e:tube-symmetric-pair}. It follows that
$\mathcal J_{\mathfrak a}$ is the matrix of the operator
$\sinh(\operatorname{ad}_Y)/\operatorname{ad}_Y$ of $\mathfrak m$, and
\begin{equation}\label{e:tube-exp-density}
 \dd\nu = j(Y)\dd y , \qquad
 j(Y) = \Sdet_{\mathfrak m}\Bigl(\frac{\sinh\operatorname{ad}_Y}{\operatorname{ad}_Y}\Bigr) ,
\end{equation}
which is the formula of \cite[Section~II.F]{zirnbauer1996riemannian}. In
particular $j(0) = 1$, and $j(-Y) = j(Y)$, since $\sinh(u)/u$ is even; in
the language of symmetric spaces, the geodesic symmetry $Y \mapsto -Y$ at
$\Sigma$ preserves $\nu$. We normalize the basis $(Y_b)$ by requiring
that $\dd y$ be the flat Berezin form $\dd Y$ of $\mathfrak m$ defined as
follows. We let $\dd U$ denote the flat Berezin form of $\mathfrak k$,
normalized by $\int e^{-\Str(U^2)/2}\dd U = 1$ over the real form of
$\mathfrak k$ with Hermitian boson--boson block, $\iu$ times a Hermitian
fermion--fermion block, and free odd entries, and we let $\dd Y$ denote
the flat Berezin form of $\mathfrak m$ for which the flat Berezin form
$\dd R$ of \zcref{d:rep-flat} splits as $\dd R = \dd U\dd Y$ along
$\mathfrak g = \mathfrak k \oplus \mathfrak m$. With this choice we say
that $\nu$ is normalized at the root.

\section{Tubular coordinates around the saddle manifold}
\label{s:tube-geometry}

We compute the volume of the tubular coordinates, including every odd
coefficient, prove that the square root of the Berezinian of the
tubular map is harmonic in the normal variables, and derive the
resulting decomposition of the ambient Laplacian.

\subsection{Root spaces and the tubular map}

Fix a signature $p$, and recall from \zcref{s:tube-prelim} the root point
$R_* = \kappa_E\Id - \iu s_E\Sigma$, the eigenspaces $V_\pm$ of $\Sigma$,
and the spaces $\mathfrak k$ and $\mathfrak m$, where $\kappa_E = E/2$ and
$s_E = (1 - E^2/4)^{1/2}$ as in \zcref{e:scaling}. The points $R_*$, for
the $2k + 1$ signatures, are the saddle points of the onsite weight of
\zcref{s:rep} up to conjugation by $\mathrm{GL}(2k|2k)$. All matrices below
are even supermatrices with entries in a Grassmann algebra that contains
all odd coordinates. We call the elements of $\mathfrak k$ normal and those
of $\mathfrak m$ tangent. For a normal $U = \diag(U_+, U_-)$ we set
$S = R_* + U$, with blocks $S_\pm$, and we define the operator $L_S$ on
$\mathcal H_{+-} = \operatorname{Hom}(V_-, V_+)$ and the function
$\varpi$ by
\begin{equation}\label{e:tube-half-density}
 L_S A = S_+ A - A S_- , \qquad \varpi(S) = \Sdet_{\mathcal H_{+-}} L_S .
\end{equation}
Note that a scalar shift of $S$ does not change $L_S$. At the root we
have $L_{R_*} = -2\iu s_E\Id$, and by \eqref{e:tube-sdim-hom} we have
$\operatorname{sdim}\mathcal H_{+-} = (k - p)(p - k)$, so that
$\varpi(R_*) = (-2\iu s_E)^{-(p - k)^2}$. In particular, $\varpi(R_*) = 1$
in the balanced signature.

We fix $r_* \in (0, s_E/8]$ so small that the body of $L_S$ is
invertible, with $\|L_S^{-1}\| \le 1/s_E$, whenever the body of $U$ has
norm at most $2r_*$. Then $\varpi$ and $\varpi^{-1}$ are analytic in the entries of
$U$, with all derivatives bounded in terms of $s_E$, $k$, and $r_*$. For
$0 < r \le r_*$ we call the set of normal $U$ whose body has norm less
than $2r$ the normal tube of radius $r$. All statements of this section
and the next are local to a normal tube with $r \le r_*$. Throughout, the energy $E$ ranges over a compact subset of $(-2, 2)$
that does not depend on the rank $k$, and constants may depend on it.

Recall the stabilizer $K = (K_0, \mathfrak k)$ of $R_*$ from
\zcref{s:tube-prelim}. The symmetric superspace $U(k, k|2k)/K$, with the
underlying manifold $G_0/K_0$, is the real form of the component of the
saddle manifold through $R_*$; in the balanced signature, it is the
balanced component of \zcref{s:rep-heuristics}. Let $T(x)$ be a local
section of $U(k, k|2k)$ over a coordinate patch of $U(k, k|2k)/K$, as in
\zcref{s:tube-prelim}, where $x$ stands for all coordinates of the patch,
even and odd. We call the map
\begin{equation}\label{e:tube-map}
 (x, U) \longmapsto R(x, U) = T(x)(R_* + U) T(x)^{-1}
\end{equation}
the tubular map, and $(x, U)$ the tubular coordinates. We let $\nu$
denote the invariant Berezin volume \eqref{e:tube-nu} on $U(k, k|2k)/K$,
normalized at the root, and $\dd U$ the flat Berezin form of
$\mathfrak k$ of \zcref{s:tube-prelim}. In a second local section $Tk$,
the same points $R$ have the normal coordinate $k^{-1}Uk$. We call a function of
the normal coordinate $U$ alone that is invariant under the conjugations
$U \mapsto kUk^{-1}$, for all even invertible supermatrices $k$ with
Grassmann entries that commute with $\Sigma$, a \emph{normal class
function}. Such functions do not depend on the choice of the local
section, and they are annihilated by the normal charges of
\zcref{s:tube-operator}.

The normal integration in this section and in \zcref{s:stab} runs over a rotated real
form of the normal space, which we now fix. For $\sigma \in \{+, -\}$ let
$r_\sigma = \kappa_E + \sigma\iu s_E$ be the two roots, and set
\begin{equation}\label{e:tube-c}
 m_\sigma = 1 - r_\sigma^2 , \qquad
 c_\sigma = \bigl(W^2m_\sigma + \tfrac14m_\sigma^2\bigr)^{1/2} ,
 \qquad
 \theta_\sigma = \tfrac12\arg c_\sigma ,
\end{equation}
with the branch of positive real part. Then $|\theta_\sigma| \le C|E|$.

\begin{definition}\label{d:tube-cycle}
For each root $\sigma$, let $\mathfrak k_\sigma^\R$ denote the real form of
the normal block of root $\sigma$: Hermitian boson--boson block,
$\iu$ times a Hermitian fermion--fermion block, and free odd entries. We
call $\bigoplus_\sigma e^{-\iu\theta_\sigma}\mathfrak k_\sigma^\R$ the adapted
real normal space. We call the set of points
$T(x)(R_* + U)T(x)^{-1}$ with $x \in G_0/K_0$ and $U$ in the adapted real
normal space whose body has norm less than $2r$ the real tube.
\end{definition}

All operators and Hilbert spaces of this section and \zcref{s:stab} are
defined on the real tube, which is their integration cycle. It differs from the
Sch\"afer--Wegner cycle $\mathcal S$ of \zcref{s:rep}, and the two are
connected in \zcref{s:end,s:assembly}. On the unrotated real normal space
the frozen quadratic kernel, whose coefficient $m_\sigma$ is complex, is a
complex harmonic oscillator whose Hermite eigenfunctions are not
orthogonal; on the rotated space its Gaussian ground state is real, as we
will see in \zcref{l:tube-quadratic}.

Integrals over the real tube depend on two orientations, which we fix
here. First, the volume $\nu$ depends on the
orientation of $G_0/K_0$ induced by the coordinates $x$ in
\eqref{e:tube-nu}, and reversing this orientation changes the sign of
$\nu$. In the balanced signature, $\nu$ is a constant multiple of the
integral of \zcref{d:sigma-integral} by \zcref{s:tube-prelim}, and we
show in \zcref{s:tube-gauge} that the constant is $\pm 1$. We orient
$G_0/K_0$ so that it is $+1$. In an unbalanced signature, either
orientation of $G_0/K_0$ may be used, and \zcref{s:asm-telescope} fixes
one. Second, the flat form $\dd R$ is
holomorphic in the entries of $R$, and its integral over the real tube,
a real cycle in the complex supermatrices, depends on an orientation of
the real tube. We orient the real tube so that the identity
\eqref{e:tube-volume} of the next lemma holds; in its proof, this choice
absorbs a sign $(-1)^{(p - k)^2}$. All later sections use these
orientations. In particular, \zcref{s:tube-gauge} concludes that, in the
balanced signature, $\dd\nu$ is the invariant measure of $\mathcal M_k$
with the orientation of the pairing $\mathcal B_k$, and
\zcref{s:asm-telescope} uses this orientation of the real tube when it
deforms the Sch\"afer--Wegner cycle into the real tube.

\begin{lemma}
Let\label{l:tube-volume} $\mathfrak a$ be the Maurer--Cartan form \eqref{e:tube-maurer-cartan}
of $T$, applied to even derivations $\mathcal V$ of the superfunctions of
the tubular coordinates $(x, U)$. Then
\begin{equation}\label{e:tube-differential}
 T^{-1} (\mathcal V R) T = \mathcal V U + [\mathfrak a_{\mathfrak k}(\mathcal V), U] + [\mathfrak a_{\mathfrak m}(\mathcal V), S] .
\end{equation}
The first two terms are normal and the last is tangent, and the flat
Berezin form $\dd R$ of \zcref{d:rep-flat} satisfies
\begin{equation}\label{e:tube-volume}
 \dd R = \varpi(S)^2 \dd U \dd\nu(x) ,
\end{equation}
including all nilpotent terms, with the orientations fixed above.
\end{lemma}

\begin{proof}
Since $T$ depends only on $x$, the derivation $\mathcal V$ acts on $T$
through its tangent part. Applying $\mathcal V$ to $R = TST^{-1}$ and using
$\mathcal V(T^{-1}) = -T^{-1}(\mathcal VT)T^{-1}$, we have
\[
 T^{-1}(\mathcal VR)T = \mathfrak a(\mathcal V)S + \mathcal VU - S\mathfrak a(\mathcal V)
 = \mathcal VU + [\mathfrak a_{\mathfrak k}(\mathcal V), S] + [\mathfrak a_{\mathfrak m}(\mathcal V), S] .
\]
Since $\mathfrak a_{\mathfrak k}(\mathcal V)$ is block diagonal, it commutes
with $R_*$, and therefore
$[\mathfrak a_{\mathfrak k}(\mathcal V), S] = [\mathfrak a_{\mathfrak k}(\mathcal V), U]$,
and \eqref{e:tube-differential} follows. By \eqref{e:tube-symmetric-pair},
the first two terms lie in $\mathfrak k$ and the last in $\mathfrak m$.

To compute the volume, we use linear coordinates of the ambient
supermatrices adapted to $\mathfrak g = \mathfrak k \oplus \mathfrak m$, the
coordinates of the normal part and those of the tangent part in the basis
$(Y_b)$ of \zcref{s:tube-prelim}, in which $\dd R = \dd U\dd Y$. Let $\mathcal J$ be the Jacobian supermatrix
\eqref{e:grass-jacobian} of the tubular coordinates relative to these
coordinates, whose row indexed by a tubular coordinate $w$ consists of the
coordinates of $\mathcal V_wR$, where
$\mathcal V_w = \vartheta^{|w|}\partial_w$. By \zcref{t:grass-change}, we
have $\dd R = \Sdet(\mathcal J)\dd x\dd U$.

We multiply $\mathcal J$ on the right by the matrix of
$\operatorname{Ad}_T^{-1}$ on $\mathfrak g$. This does not change the
superdeterminant: $T$ is a product of exponentials of even supermatrices
$A$, and $\operatorname{ad}_A$ has supertrace zero on
$\mathfrak g = \operatorname{End}(\C^{2k|2k})$ by \eqref{e:tube-ad-trace}. The rows of the new
matrix $\mathcal J'$ consist of the coordinates of
$T^{-1}(\mathcal V_wR)T$, which are given by \eqref{e:tube-differential}.
For a normal coordinate $w$, we have $\mathfrak a(\mathcal V_w) = 0$, and the
row is the unit vector of $w$ in $\mathfrak k$. For a coordinate $z_I$ of
$x$, we have $\mathcal V_{z_I}U = 0$, and the tangent part of the row is
the row of $\mathcal J_{\mathfrak a}$ multiplied by the matrix of the
operator $Y \mapsto [Y, S]$ of $\mathfrak m$. With the rows ordered as
$(x, U)$ and the columns as $(\mathfrak m, \mathfrak k)$, the matrix
$\mathcal J'$ is therefore block upper triangular, with the diagonal blocks
$\mathcal J_{\mathfrak a} \cdot [Y \mapsto [Y, S]]$ and $\Id$. Then
\zcref{l:grass-sdet}\ref{i:grass-sdet-triangular} and
\ref{i:grass-sdet-mult}, together with \eqref{e:tube-nu}, imply that
\[
 \dd R = \Sdet_{\mathfrak m}\bigl(Y \mapsto [Y, S]\bigr)\dd U\dd\nu(x) .
\]

It remains to compute this superdeterminant. The operator
$Y \mapsto [Y, S]$ preserves the two summands of $\mathfrak m$. It acts
as $-L_S$ on $\mathcal H_{+-} = \operatorname{Hom}(V_-, V_+)$, and on
$\operatorname{Hom}(V_+, V_-)$ as
\[
 B \longmapsto BS_+ - S_- B = (\rho_{S_+} - \lambda_{S_-})B .
\]
Using \eqref{e:tube-tensor-trace} on the two
spaces, we find that for every polynomial $f$ we have
\[
 \Str_{\operatorname{Hom}(V_+, V_-)}f(\rho_{S_+}, \lambda_{S_-})
 = \Str_{\mathcal H_{+-}}f(\lambda_{S_+}, \rho_{S_-}) ,
\]
since the two sides are the same combination of $\Str S_+^i\Str S_-^j$.
Since $\Sdet = \exp\Str\log$, we obtain
\[
\begin{gathered}
 \Sdet(\rho_{S_+} - \lambda_{S_-}) = \Sdet_{\mathcal H_{+-}}(\lambda_{S_+} - \rho_{S_-}) = \varpi(S) , \\
 \Sdet_{\mathcal H_{+-}}(-L_S) = (-1)^{\operatorname{sdim}\mathcal H_{+-}}\varpi(S) = (-1)^{(p - k)^2}\varpi(S) ,
\end{gathered}
\]
and our choice of orientation for the real tube absorbs the sign.
Inserting the two
superdeterminants into the last display, we conclude that
\eqref{e:tube-volume} holds.
\end{proof}

\subsection{The harmonic half-density}

We let $\Delta_{\rm nor} = \Delta_+ + \Delta_-$ denote the flat
Laplacian of the normal variables for the metric $\Str(\dd U_+)^2 +
\Str(\dd U_-)^2$, which is the restriction of the ambient metric of
\eqref{e:tube-heat-kernel}. Precisely, we define $\Delta_+$ to be the
second-order operator with constant coefficients in the entries of
$U_+$ such that
\begin{equation}\label{e:tube-laplace-symbol}
 \Delta_+ e^{\Str(P U_+)} = \Str(P^2) e^{\Str(P U_+)}
\end{equation}
for every even supermatrix $P$ on $V_+$ whose entries lie in an
auxiliary Grassmann algebra, and we define $\Delta_-$ in the same way. For
a function $\phi$ of $U_+$ we let $\nabla_+ \phi$ denote the
supermatrix on $V_+$ such that $\frac{\dd}{\dd t}\phi(U_+ + tX)|_{t = 0}
= \Str(X \nabla_+ \phi)$ for all even $X$, and we set
$|\nabla_+ \phi|^2 = \Str(\nabla_+\phi\, \nabla_+\phi)$. We use the same
notation for $U_-$. Applying \eqref{e:tube-laplace-symbol} to
products of exponentials, we find that
\begin{equation}\label{e:tube-carre}
 \Delta_\pm(\phi\psi) = \phi\,\Delta_\pm\psi + \psi\,\Delta_\pm\phi
 + 2\Str(\nabla_\pm\phi\,\nabla_\pm\psi)
\end{equation}
for an even function $\phi$ and an arbitrary function $\psi$ (with odd
auxiliary parameters in the exponentials when $\psi$ has odd parts), and
therefore
\begin{equation}\label{e:tube-exp-rule}
 e^{-g}\Delta_\pm e^{g} = \Delta_\pm g + |\nabla_\pm g|^2
\end{equation}
for every even function $g$. Moreover, the value of $\Delta_+\phi$ at a point is determined by the
Hessian of $\phi$ there. Precisely, suppose that $\mathcal B$ is a linear map of
$\operatorname{End}(V_+)$ such that
\[
 \frac{\partial^2}{\partial s\partial t}\phi(U_+ + sX + tY)\Big|_{s = t = 0} = \Str(X\, \mathcal B Y)
\]
for all even $X$ and $Y$. Then $\Delta_+\phi = \Str_{\operatorname{End}(V_+)} \mathcal B$,
the supertrace of $\mathcal B$ as an endomorphism of the super vector
space $\operatorname{End}(V_+)$. To see this, note that both sides are linear in the Hessian and that
the Hessians of the exponentials in \eqref{e:tube-laplace-symbol} span
all Hessians. It suffices to check the claim for
$\phi = e^{\Str(PU_+)}$. Then $\mathcal BY = P\Str(PY)\phi$, and
$\Str_{\operatorname{End}(V_+)}\mathcal B = \Str(P^2)\phi$.

Recall the operators $\lambda_A$ and $\rho_B$ of left and right
multiplication, here on $\mathcal H_{+-}$, with even supermatrices $A$ on
$V_+$ and $B$ on $V_-$, and the identity \eqref{e:tube-tensor-trace}
$\Str_{\mathcal H_{+-}}(\lambda_A\rho_B) = \Str(A)\Str(B)$, together with
its form $\Str_{\operatorname{End}(V_+)}(Y \mapsto AYA') = \Str(A)\Str(A')$
for $W = V$.

\begin{lemma}\label{l:tube-harmonic}
On the normal tube of radius $r_*$, we have
\begin{equation}\label{e:tube-cross}
 \Delta_+ \log\varpi = -|\nabla_- \log\varpi|^2 ,
 \qquad
 \Delta_- \log\varpi = -|\nabla_+ \log\varpi|^2 ,
\end{equation}
and consequently
\begin{equation}\label{e:tube-harmonic}
 \Delta_{\rm nor} \varpi = 0 .
\end{equation}
These are identities of all Grassmann coefficients. For every normal
superfunction $f$, with
$\varpi^{-2}\operatorname{div}_{\rm nor}(\varpi^2\nabla_{\rm nor}f)$ defined as
$\Delta_{\rm nor}f + 2\Str(\nabla_{\rm nor}\log\varpi\,\nabla_{\rm nor}f)$,
\begin{equation}\label{e:tube-conjugacy}
 \varpi\bigl[\varpi^{-2}\operatorname{div}_{\rm nor}(\varpi^2\nabla_{\rm nor} f)\bigr]
 = \Delta_{\rm nor}(\varpi f) .
\end{equation}
\end{lemma}

The identity \eqref{e:tube-conjugacy} says that the half-density change
$f \mapsto \varpi f$, which turns the tubular measure $\varpi^2 \dd U$
into the flat measure $\dd U$, conjugates the normal part of the
tubular Laplacian to the flat normal Laplacian with no potential term.
Without harmonicity, a potential $\varpi^{-1}\Delta_{\rm nor}\varpi$
would appear, of order one on the tube.

\begin{proof}
Let $G = L_S^{-1}$, an even invertible operator on $\mathcal H_{+-}$.
We define the even supermatrices $\Gamma_+$ on $V_+$ and $\Gamma_-$ on
$V_-$ by
\begin{equation}\label{e:tube-partial-traces}
 \Str(X\Gamma_+) = \Str_{\mathcal H_{+-}}(G\lambda_X) ,
 \qquad
 \Str(Y\Gamma_-) = \Str_{\mathcal H_{+-}}(G\rho_Y)
\end{equation}
for all even $X$ and $Y$. They exist and are unique, since $\Str$ is a
nondegenerate pairing. Since $\log\Sdet = \Str\log$ and
$\dd L_S = \lambda_{\dd U_+} - \rho_{\dd U_-}$, we have
$\dd\log\varpi = \Str(G\lambda_{\dd U_+}) - \Str(G\rho_{\dd U_-})$, and
therefore
\[
 \nabla_+\log\varpi = \Gamma_+ , \qquad \nabla_-\log\varpi = -\Gamma_- .
\]

We now compute $\Delta_+ \log\varpi$. On the tube, the body of
$\iu L_S$ has spectrum in the half-plane $\{\Re w \ge s_E\}$, since
$\iu L_{R_*} = 2s_E$ and $\|L_U\| \le 4r_* \le s_E/2$.
Using that $\lambda_{S_+}$ and $\rho_{S_-}$ commute, we obtain
\begin{equation}\label{e:tube-resolvent-integral}
 G = \iu \int_0^\infty e^{-\iu t L_S}\dd t
 = \int_0^\infty \lambda_{A(t)}\rho_{B(t)}\dd t ,
 \qquad A(t) = \iu e^{-\iu tS_+} , \quad B(t) = e^{\iu tS_-} ,
\end{equation}
where the integral converges absolutely, since the nilpotent parts
of $S_\pm$ contribute only polynomial factors in $t$.

Differentiating
$\log\varpi$ twice and using $\dd G = -G(\dd L_S)G$, we find that the
Hessian of $\log\varpi$ in $U_+$ is the bilinear form
$(X, Y) \mapsto -\Str_{\mathcal H_{+-}}(G\lambda_X G\lambda_Y)$. Since
$\lambda$ and $\rho$ commute and $\rho_B\rho_{B'} = \rho_{B'B}$,
inserting \eqref{e:tube-resolvent-integral} twice and using
\eqref{e:tube-tensor-trace}, we find that
\[
 \Str_{\mathcal H_{+-}}(G\lambda_X G\lambda_Y)
 = \int_0^\infty\!\!\int_0^\infty \Str\bigl(A(t) X A(t') Y\bigr)\Str\bigl(B(t')B(t)\bigr)\dd t\dd t' .
\]
The Hessian is therefore $\Str(X\mathcal B Y)$, where
\[
 \mathcal B Y = -\iint\Str\bigl(B(t')B(t)\bigr)\, A(t') Y A(t)\dd t\dd t' ,
\]
and the characterization of $\Delta_+$ by the Hessian, together with the
identity $\Str_{\operatorname{End}(V_+)}(Y \mapsto AYA') = \Str(A)\Str(A')$,
implies that
\[
 \Delta_+\log\varpi
 = -\iint \Str A(t)\,\Str A(t')\,\Str\bigl(B(t')B(t)\bigr)\dd t\dd t' .
\]
On the other hand, inserting \eqref{e:tube-resolvent-integral} into
\eqref{e:tube-partial-traces} and using \eqref{e:tube-tensor-trace}, we
have
\[
 \Gamma_- = \int_0^\infty\Str A(t)\, B(t)\dd t ,
\]
and hence
\[
 |\nabla_-\log\varpi|^2 = \Str(\Gamma_-^2)
 = \iint\Str A(t)\,\Str A(t')\,\Str\bigl(B(t)B(t')\bigr)\dd t\dd t' .
\]
Since $\Str(B(t)B(t')) = \Str(B(t')B(t))$, this proves the first
identity in \eqref{e:tube-cross}. The second identity follows in the
same way. The Hessian of $\log\varpi$ in $U_-$ is
\[
 (X, Y) \mapsto -\Str_{\mathcal H_{+-}}(G\rho_X G\rho_Y)
 = -\iint\Str\bigl(A(t)A(t')\bigr)\Str\bigl(XB(t)YB(t')\bigr)\dd t\dd t' ,
\]
and
\[
 \Gamma_+ = \int_0^\infty\Str B(t)\,A(t)\dd t .
\]

Adding the two identities in \eqref{e:tube-cross} and using that the
two blocks are orthogonal, we obtain
$\Delta_{\rm nor}\log\varpi + |\nabla_{\rm nor}\log\varpi|^2 = 0$, which
is \eqref{e:tube-harmonic} by \eqref{e:tube-exp-rule}. Finally, by the
product rule \eqref{e:tube-carre}, we have
\[
 \Delta_{\rm nor}(\varpi f)
 = \varpi\,\Delta_{\rm nor} f
 + 2\varpi\Str(\nabla_{\rm nor}\log\varpi\,\nabla_{\rm nor} f)
 + f\,\Delta_{\rm nor}\varpi ,
\]
where the first two terms on the right add up to
$\varpi[\varpi^{-2}\operatorname{div}_{\rm nor}(\varpi^2\nabla_{\rm nor} f)]$
by the definition in the statement.
Together with \eqref{e:tube-harmonic}, this proves \eqref{e:tube-conjugacy}.
\end{proof}

\begin{remark}\label{r:tube-harmonic-checks}
In the purely even analogue, where $V_+$ and $V_-$ are ordinary vector spaces, the
proof is the classical computation for $\varpi = \prod_{i, a}(x_i - y_a)$
in terms of the eigenvalues $x_i$ of $S_+$ and $y_a$ of $S_-$. The two sides of the first identity in \eqref{e:tube-cross} equal
$-\sum_a(\sum_i (x_i - y_a)^{-1})^2$. The identities \eqref{e:tube-cross} and
\eqref{e:tube-harmonic} were checked numerically, coefficient by coefficient in a finite Grassmann
algebra, at selected complex base points in six signatures with
$\dim V_\pm \le 3$, and the scripts are in the supplementary material
(\texttt{tube\_cross\_identity.py}, \texttt{tube\_super\_check.py} and
\texttt{tube\_ordinary\_check.py}). The second script also confirms the
normalization of the odd part of $\Delta_{\rm nor}$ through the identity
$\Delta_+\Str(U_+^2) = 2(\operatorname{sdim}V_+)^2$, which follows from
\eqref{e:tube-laplace-symbol} and \eqref{e:tube-tensor-trace}.
\end{remark}

\subsection{The decomposition of the ambient Laplacian}
\label{s:tube-operator}

We first define the normal charges. For an even element $\mathfrak e$ of
$\mathfrak k$, possibly with Grassmann coefficients, we let
$\mathcal N_{\mathfrak e}$ denote the first-order differential operator
$f \mapsto \frac{\dd}{\dd t}f(e^{-t\mathfrak e}Ue^{t\mathfrak e})|_{t = 0}$ in
the normal variable, which is the derivative in the direction
$[U, \mathfrak e]$. We call the operators $\mathcal N_{\mathfrak e}$ with
even $\mathfrak e \in \mathfrak k$, and the operators $\mathcal N_{\vartheta\mathfrak e}$
with odd $\mathfrak e \in \mathfrak k$ and an auxiliary odd parameter
$\vartheta$, divided by $\vartheta$, the compact and the odd normal charges.
On the normal oscillator of \zcref{s:tube-oscillator} they are the
operators $\mathfrak j_a$ and $\mathfrak q_a$ of \zcref{s:tube-cutoff}. The
normal charges are divergence-free for the flat normal form $\dd U$, since
the supertrace of $\operatorname{ad}_{\mathfrak e}$ on
$\mathfrak k = \operatorname{End}(V_+) \oplus \operatorname{End}(V_-)$
vanishes by \eqref{e:tube-tensor-trace}. They annihilate every normal class
function, in particular $\varpi$, which is invariant under conjugation by
block-diagonal supermatrices, since conjugating $S$ conjugates $L_S$ by
$\lambda_{k_+}\rho_{k_-^{-1}}$. They also preserve the normal metric
$\Str(\dd U)^2$.

We now define the horizontal covariant fields. Let $T$ be a local section.
For an even element $Y$ of $\mathfrak m$ whose coefficients are
superfunctions of $x$, we let $X_Y$ denote the even derivation of the
superfunctions of $x$ with $\mathfrak a_{\mathfrak m}(X_Y) = Y$, which
exists and is unique since $\mathcal J_{\mathfrak a}$ is invertible, and
we set
\[
 \mathbb D_Y = X_Y + \mathcal N_{\mathfrak a_{\mathfrak k}(X_Y)} ,
\]
an even derivation of the superfunctions of $(x, U)$. By
\eqref{e:tube-differential}, $\mathbb D_Y$ is characterized by
\begin{equation}\label{e:tube-horizontal}
 \mathfrak a_{\mathfrak m}(\mathbb D_Y) = Y , \qquad
 \mathbb D_YU + [\mathfrak a_{\mathfrak k}(\mathbb D_Y), U] = 0 , \qquad
 T^{-1}(\mathbb D_YR)T = [Y, S] .
\end{equation}
We call $\mathbb D_Y$ horizontal because it moves the point $R$ only along
its orbit under conjugation, and covariant because its normal part
compensates the rotation $\mathfrak a_{\mathfrak k}$ of the frame. In a local orbit frame $(T, (Y_b))$, we let
$\mathbb D_b = \mathbb D_{Y_b}$, defined for odd $Y_b$ by
$\mathbb D_{\vartheta Y_b} = \vartheta\mathbb D_b$, and we call the
$\mathbb D_b$ the horizontal covariant fields of the frame. The normal
part of $\mathbb D_b$ is the normal charge of
$\mathfrak e_b = \mathfrak a_{\mathfrak k}(X_{Y_b})$.

Every even derivation $\mathcal V$ of the superfunctions of $(x, U)$
splits into a normal and a horizontal part. Indeed,
$\mathcal V - \mathbb D_{\mathfrak a_{\mathfrak m}(\mathcal V)}$ has
vanishing Maurer--Cartan form, so that it has no tangent part, and by
\eqref{e:tube-horizontal} it is the derivative in $U$ in the direction
$\mathcal VU + [\mathfrak a_{\mathfrak k}(\mathcal V), U]$. For a
superfunction $\phi$ on the tube, we let $\nabla_{\rm nor}\phi \in
\mathfrak k$ denote its gradient in $U$, as in \zcref{s:tube-geometry} for
the two blocks, and we let $\nabla_H\phi \in \mathfrak m$ denote the
horizontal gradient, defined by $\Str(Y\nabla_H\phi) = \mathbb D_Y\phi$
for all even $Y \in \mathfrak m$. Then
\begin{equation}\label{e:tube-derivation-split}
 \mathcal V\phi = \Str\bigl((\mathcal VU + [\mathfrak a_{\mathfrak k}(\mathcal V), U])\nabla_{\rm nor}\phi\bigr)
 + \Str\bigl(\mathfrak a_{\mathfrak m}(\mathcal V)\nabla_H\phi\bigr) .
\end{equation}

We now derive the decomposition of the ambient flat Laplacian
$\Delta_{\rm amb}$ of \eqref{e:tube-heat-kernel}, which is the operator
with constant coefficients in the linear ambient coordinates such that
$\Delta_{\rm amb}e^{\Str(PR)} = \Str(P^2)e^{\Str(PR)}$, as in
\eqref{e:tube-laplace-symbol}. Let $\nabla F$ denote the ambient gradient
of a function $F$ of $R$, defined by
$\frac{\dd}{\dd t}F(R + tX)|_{t = 0} = \Str(X\nabla F)$. For an even
derivation $\mathcal V$, we obtain from \eqref{e:tube-differential} and
the orthogonality of $\mathfrak k$ and $\mathfrak m$ that
\[
 \mathcal VF = \Str\bigl(T^{-1}(\mathcal VR)T \cdot T^{-1}(\nabla F)T\bigr)
 = \Str\bigl((\mathcal VU + [\mathfrak a_{\mathfrak k}(\mathcal V), U])G_{\mathfrak k}\bigr)
 + \Str\bigl(\mathfrak a_{\mathfrak m}(\mathcal V)[S, G_{\mathfrak m}]\bigr) ,
\]
where $G = T^{-1}(\nabla F)T = G_{\mathfrak k} + G_{\mathfrak m}$. The
operator $\operatorname{ad}_S = [S, \cdot \,]$ is invertible on
$\mathfrak m$ on the tube, by the proof of \zcref{l:tube-volume}. Comparing
with \eqref{e:tube-derivation-split} for all $\mathcal V$, we conclude that
\begin{equation}\label{e:tube-gradient-split}
 T^{-1}(\nabla F)T = \nabla_{\rm nor}F + \operatorname{ad}_S^{-1}(\nabla_HF) .
\end{equation}
Applying the definition of $\Delta_{\rm amb}$ to products of
exponentials, as for \eqref{e:tube-carre}, we find that
\[
 \Delta_{\rm amb}(\phi\psi) = \phi\,\Delta_{\rm amb}\psi + \psi\,\Delta_{\rm amb}\phi + 2\mathcal Q_{\rm amb}(\phi, \psi) ,
 \qquad
 \mathcal Q_{\rm amb}(\phi, \psi) = \Str(\nabla\phi\,\nabla\psi) ,
\]
for even $\phi$, and we call $\mathcal Q_{\rm amb}$ the carr\'e du champ of
$\Delta_{\rm amb}$. Inserting \eqref{e:tube-gradient-split}, we find that
\begin{equation}\label{e:tube-carre-split}
 \mathcal Q_{\rm amb}(\phi, \psi)
 = \Str(\nabla_{\rm nor}\phi\,\nabla_{\rm nor}\psi)
 + \Str\bigl(\operatorname{ad}_S^{-1}\nabla_H\phi\,\operatorname{ad}_S^{-1}\nabla_H\psi\bigr)
 = \mathcal Q_{\rm nor}(\phi, \psi) + \mathcal Q_H(\phi, \psi) .
\end{equation}
The second term is the inverse of the horizontal metric
\[
 g_S(Y, Z) = \Str([Y, S][Z, S]) , \qquad Y, Z \in \mathfrak m ,
\]
applied to the horizontal gradients: since $\operatorname{ad}_S$ is skew
for the form $\Str(YZ)$, the metric $g_S$ corresponds to the operator
$-\operatorname{ad}_S^2$ of $\mathfrak m$, and
$\mathcal Q_H(\phi, \psi) = -\Str(\nabla_H\phi\,\operatorname{ad}_S^{-2}\nabla_H\psi)$.
In this sense, the ambient metric is the orthogonal sum of the constant
normal metric and the horizontal metric $g_S$.

Let $\phi$ and $\psi$ be superfunctions on the tube, with $\phi$ even and
with compactly supported coefficients. We integrate by parts in the
linear ambient coordinates, by
\zcref{l:grass-berezin}\ref{i:grass-berezin-odd-ibp} and
\ref{i:grass-berezin-even-ibp}, and we write the result in tubular
coordinates by \eqref{e:tube-volume} and \eqref{e:tube-carre-split}. This
yields
\[
 \int\phi\,\Delta_{\rm amb}\psi\,\varpi^2\dd U\dd\nu
 = -\int\mathcal Q_{\rm nor}(\phi, \psi)\,\varpi^2\dd U\dd\nu
 - \int\mathcal Q_H(\phi, \psi)\,\varpi^2\dd U\dd\nu .
\]
For the first term on the right, we integrate by parts in $U$ and use
the product rule \eqref{e:tube-carre} for $\Delta_{\rm nor}$, and we obtain
\[
\begin{aligned}
 -\int\mathcal Q_{\rm nor}(\phi, \psi)\,\varpi^2\dd U
 &= \int\phi\,\bigl(\Delta_{\rm nor}\psi + 2\Str(\nabla_{\rm nor}\log\varpi\,\nabla_{\rm nor}\psi)\bigr)\varpi^2\dd U \\
 &= \int\phi\,\varpi^{-2}\operatorname{div}_{\rm nor}(\varpi^2\nabla_{\rm nor}\psi)\,\varpi^2\dd U ,
\end{aligned}
\]
with the notation of \zcref{l:tube-harmonic}. Let $L_H(U)$ denote the
operator $\Delta_{\rm amb} - \varpi^{-2}\operatorname{div}_{\rm nor}(\varpi^2\nabla_{\rm nor})$
in tubular coordinates. The last two displays show that
\[
 \int\phi\,L_H(U)\psi\,\varpi^2\dd U\dd\nu = -\int\mathcal Q_H(\phi, \psi)\varpi^2\dd U\dd\nu .
\]
Since $\mathbb D_Y\varpi = 0$, we have
$\mathcal Q_H(\varpi^{-2}\phi, \psi) = \varpi^{-2}\mathcal Q_H(\phi, \psi)$,
and replacing $\phi$ by $\varpi^{-2}\phi$ yields
\begin{equation}\label{e:tube-horizontal-weak}
 \int\phi\,L_H(U)\psi\dd U\dd\nu = -\int\mathcal Q_H(\phi, \psi)\dd U\dd\nu .
\end{equation}
For the same reason, $\mathcal Q_H(\phi, \varpi\psi) = \mathcal Q_H(\varpi\phi, \psi)$,
so that $L_H(U)$ commutes with multiplication by $\varpi$. Together with
\eqref{e:tube-conjugacy}, applied to $\varpi^{-1}\psi$, this implies the
exact identity
\begin{equation}\label{e:tube-decomposition}
 \varpi\,\Delta_{\rm amb}\,\varpi^{-1} = \Delta_{\rm nor} + L_H(U) ,
 \qquad
 L_H(U) = \operatorname{div}_{\nu, \mathbb D}(g_S^{-1}\mathbb D) ,
\end{equation}
for all Grassmann coefficients, where we write
$\operatorname{div}_{\nu, \mathbb D}(g_S^{-1}\mathbb D)$ for the operator
characterized by \eqref{e:tube-horizontal-weak}, the divergence-form
operator of the horizontal metric with respect to $\dd U\dd\nu$. We call
$L_H(U)$ the horizontal operator.

In a local orbit frame, integrating by parts along the fields
$\mathbb D_b$ in \eqref{e:tube-horizontal-weak}, whose normal parts are
divergence-free, shows that $L_H(U)$ is a finite sum of products of at
most two of the fields $\mathbb D_b$, with coefficients that are
superfunctions of $(x, U)$, analytic in $U$. Its ordinary second-order
part has the form $\sum_{b, c}g_S^{bc}X_{Y_b}X_{Y_c}$, summed over the even
$b$ and $c$, where $(g_S^{bc})$ is the inverse of the matrix $(g_S(Y_b, Y_c))$. In
particular, all normal derivatives in $L_H(U)$ come from the normal
charges, and $L_H(U)1 = 0$. The inverse horizontal metric is analytic in
$U$ on the tube. It satisfies $g_S^{-1} - g_{R_*}^{-1} = O(U)$, and its
derivatives of every fixed order are bounded. At $U = 0$, we have
$g_{R_*}(Y, Z) = 4s_E^2\Str(YZ)$.

At $U = 0$, the horizontal operator is a constant multiple of the
Casimir operator of the sigma model. In the balanced signature, $U(k, k|2k)/K$ is the
target space $\mathcal M_k$, and we identify the functions of the tangent
variables $x$ alone with the superfunctions on $\mathcal M_k$ through the
field $Q = T\Lambda T^{-1}$, so that $R(x, 0) = \kappa_E - \iu s_EQ$. For
such a function $\psi$, we write $L_H(0)\psi$ for the restriction of
$L_H(U)\psi$ to $U = 0$.

\begin{lemma}\label{l:tube-casimir}
In the balanced signature, on superfunctions on $\mathcal M_k$,
\begin{equation}\label{e:tube-casimir}
 L_H(0) = -\frac{1}{2s_E^2}\,\mathsf C_k ,
\end{equation}
where $\mathsf C_k$ is the Casimir operator of \zcref{d:sigma-casimir}.
\end{lemma}

In words, at $U = 0$ the horizontal operator is the Laplace--Beltrami
operator of the orbit metric
\[
 g_{R_*}(Y, Z) = 4s_E^2\Str(YZ) = -s_E^2\Str([Y, \Lambda][Z, \Lambda]) ,
\]
that is, of $2s_E^2$ times the metric \eqref{e:sigma-body-metric} on the
vectors $[Y, \Lambda]$ (see the proof of
\zcref{l:sigma-casimir}\ref{i:sigma-casimir-local}), and
$\mathsf C_k$ is minus the Laplace--Beltrami operator of the metric
\eqref{e:sigma-body-metric}, extended to the odd directions as the
supermetric $-\frac12\Str(\dd Q\,\dd Q)$.

\begin{proof}
We first express $L_H(0)$ through its horizontal carr\'e du champ. Let
$\phi$ and $\psi$ be functions of $x$ alone, with $\phi$ even and
compactly supported, and let $\zeta$ be a normal function with compactly
supported coefficients. Then $\mathbb D_Y\phi = X_Y\phi$, while
$\mathbb D_Y\zeta = \mathcal N_{\mathfrak a_{\mathfrak k}(X_Y)}\zeta$ is a
derivative of $\zeta$ along a vector field in $U$ that is divergence-free
and vanishes at $U = 0$. We apply \eqref{e:tube-horizontal-weak} to
$\zeta\phi$ and $\psi$, and we move these normal charges from $\zeta$ to
the coefficients of $\mathcal Q_H$ by parts in $U$; the factors $\phi$ and
$\nabla_H\psi$ do not depend on $U$. The resulting terms vanish at
$U = 0$. Since $\zeta$ is arbitrary, the integral against $\dd\nu$ of
$\phi\,L_H(U)\psi + \mathcal Q_H(\phi, \psi)$ plus these terms vanishes
identically in $U$. At $U = 0$ only the first two terms remain, and we
conclude that the restriction to $U = 0$ is the
operator on $\mathcal M_k$ with the carr\'e du champ $\mathcal Q_H$ at
$U = 0$:
\[
 \int\phi\,L_H(0)\psi\dd\nu = -\int\mathcal Q_H(\phi, \psi)|_{U = 0}\dd\nu
 = -\frac{1}{4s_E^2}\int\Str(\nabla_H\phi\,\nabla_H\psi)\dd\nu .
\]
In the second step we used that $\operatorname{ad}_{R_*}A = -2\iu s_E\Lambda A$
for $A \in \mathfrak m$, so that
$\operatorname{ad}_{R_*}^{-1}A = (\iu/2s_E)\Lambda A$ and, since
$\Lambda A\Lambda = -A$,
\[
 \Str(\operatorname{ad}_{R_*}^{-1}A\,\operatorname{ad}_{R_*}^{-1}B)
 = -\frac{1}{4s_E^2}\Str(\Lambda A\Lambda B) = \frac{1}{4s_E^2}\Str(AB) .
\]

Next, we express $\nabla_H$ through the symmetries $\widehat Y$ of
\zcref{l:sigma-action}. Let $Y \in \mathfrak g$ be homogeneous, with the
parameter $\vartheta_Y$ of \zcref{s:sigma-symmetries}. Applying the even
derivation $\vartheta_Y\widehat Y$ to $Q = T\Lambda T^{-1}$ and using
\eqref{e:sigma-action}, we find
$[(\vartheta_Y\widehat YT)T^{-1} - \vartheta_YY, Q] = 0$. Hence
$\mathfrak a(\vartheta_Y\widehat Y) - T^{-1}\vartheta_YYT$ commutes with
$\Lambda$. Its component in $\mathfrak m$ vanishes, and
$\mathfrak a_{\mathfrak m}(\vartheta_Y\widehat Y) = (T^{-1}\vartheta_YYT)_{\mathfrak m}$.
For an even function $f$ of $x$, \eqref{e:tube-derivation-split} with
$\nabla_{\rm nor}f = 0$ and the orthogonality of $\mathfrak k$ and
$\mathfrak m$ then imply that
\[
 \vartheta_Y\widehat Yf = \Str\bigl(\mathfrak a_{\mathfrak m}(\vartheta_Y\widehat Y)\nabla_Hf\bigr)
 = \Str\bigl(\vartheta_YY\,\widehat\nabla f\bigr) ,
\]
where we write $\widehat\nabla f = T(\nabla_Hf)T^{-1}$.
With $Y = E_{ij}$, this gives $(\widehat\nabla f)_{ji} = (-1)^{|i|}\widehat{E_{ij}}f$, and
therefore
\[
 \Str(\nabla_H\phi\,\nabla_H\psi) = \Str(\widehat\nabla\phi\,\widehat\nabla\psi)
 = \sum_{i, j}(-1)^{|j|}(\widehat{E_{ji}}\phi)(\widehat{E_{ij}}\psi) .
\]

Finally, by \zcref{s:tube-prelim} the volume $\nu$ is a constant multiple
of the integral of \zcref{d:sigma-integral}. Using the integration by
parts formula \eqref{e:sigma-ibp} with the even function $\phi$, and then
the definition \eqref{e:sigma-casimir}, we obtain
\[
\begin{aligned}
 \int\phi\,L_H(0)\psi\dd\nu
 &= -\frac{1}{4s_E^2}\sum_{i, j}(-1)^{|j|}\int(\widehat{E_{ji}}\phi)(\widehat{E_{ij}}\psi)\dd\nu \\
 &= \frac{1}{4s_E^2}\sum_{i, j}(-1)^{|j|}\int\phi\,\widehat{E_{ji}}\widehat{E_{ij}}\psi\dd\nu
 = -\frac{1}{2s_E^2}\int\phi\,\mathsf C_k\psi\dd\nu .
\end{aligned}
\]
Since $\phi$ is arbitrary, this proves \eqref{e:tube-casimir} for even
$\psi$, and the case of odd $\psi$ follows by applying it to
$\vartheta\psi$.
\end{proof}

Together with the time normalization of one link, this identity
determines the generator of the tangent motion of the chain; this is done
in \zcref{s:cons-tangent}.

The tangent Gaussian normalization is fixed by the same geometry. By
the proof of \zcref{l:tube-volume}, the linear map $Y \mapsto [Y, S]$ of
$\mathfrak m$ has Berezinian $\pm \varpi(S)^2$. The normalized Gaussian
integral of $e^{-\Str[Y, S]^2/2}$ over $\mathfrak m$ is therefore
$\varpi(S)^{-2}$ times its value for the flat metric, which cancels the
tubular density. Recall from \eqref{e:tube-exp-density} that the density
$j$ of $\nu$ in the exponential chart is even, with $j(0) = 1$; by the
invariance of $\nu$ under $G_0$, the same holds in the exponential chart
$g_0e^Y$ around every point $g_0\Sigma g_0^{-1}$. Consequently
$j(h^2 Y) = 1 + O(h^4|Y|^2)$ on a chart of fixed size.

\section{The normal oscillator}\label{s:tube-oscillator}

We now realize the normal variables on a Hilbert space, the positive Wick
realization, on which the frozen normal transfer of \zcref{s:tube-gauge}
will be a contraction.


We first consider one normal matrix coordinate, with real variables
$x$, $y$, which we treat as independent standard Gaussian variables, and
Grassmann variables $\alpha$, $\beta$. We let $\E_{\rm odd}$ denote the
Gaussian Berezin functional of the odd variables, the Berezin integral
against $e^{-\alpha\beta}$ normalized by $\E_{\rm odd}1 = 1$. For
$f = f_\emptyset + f_\alpha\alpha + f_\beta\beta + f_{\alpha\beta}\alpha\beta$,
we have
\[
 \E_{\rm odd}(f) = f_\emptyset - f_{\alpha\beta} .
\]
Since $\E_{\rm odd}(\alpha\beta) = -1$, it is not a positive functional,
and we use instead the following positive metric.

\begin{definition}\label{d:tube-wick}
Let $\mathcal H_{\rm odd}$ denote the four-dimensional space spanned by
$1$, $\alpha$, $\beta$, and $\alpha\beta + 1$, with these four vectors
declared orthonormal, and give the functions of $(x, y)$ the Gaussian
$L^2$ norm. We define the normal Hilbert space $\mathcal H_{\rm nor}$ to be
the tensor product of these spaces over the finitely many coordinates of
an orthonormal basis of normal Hermitian matrices and over the two roots,
and the Wick basis to be the product of the Hermite basis in the even
variables and the basis above in the odd variables. We call the inner
product $\langle \cdot, \cdot \rangle$ of $\mathcal H_{\rm nor}$, in which
the Wick basis is orthonormal, the positive Wick metric, and
$\mathcal H_{\rm nor}$ with this metric the positive Wick realization of
the normal variables.
\end{definition}

The degree of the Wick basis defines the normal number operator
$\mathbf N \ge 0$, which counts all excitations, even and odd. We let
$\mathbf N_\sigma$ denote the part of root $\sigma$ and $\mathbf N_F$ the
odd part. The vacuum $\Omega_0 = 1$ is the only Wick monomial of degree
zero.

We let $\ell_0$ denote the tensor product, over the coordinates, of the
Gaussian expectation of the even variables and of the functionals
$\E_{\rm odd}$, and we call it the Gaussian mean. It is the normalized
Gaussian Berezin integral against $e^{-\frac12\sum(x^2 + y^2 + 2\alpha\beta)}$.
We define the \emph{analytic Berezin pairing} of the normal variables by
\begin{equation}\label{e:tube-pairing}
 \mathcal B_{\rm nor}(f, g) = \ell_0(fg) , \qquad f, g \in \mathcal H_{\rm nor} .
\end{equation}
It is bilinear and indefinite, whereas the Wick metric is sesquilinear and
positive. For one coordinate, for instance, $\mathcal B_{\rm nor}(1, 1) = 1$,
$\mathcal B_{\rm nor}(\alpha, \beta) = -1$, and
$\mathcal B_{\rm nor}(\alpha\beta + 1, \alpha\beta + 1) = -1$, while all
three vectors have norm one. In general, $\mathcal B_{\rm nor}$ pairs each
Wick monomial with exactly one Wick monomial of the same degree, with a
sign, since the Hermite functions are real and orthonormal and
$\E_{\rm odd}$ pairs $1$ with $1$, $\alpha$ with $\beta$, and
$\alpha\beta + 1$ with itself. In this way, $\mathcal B_{\rm nor}(f, g)$ becomes the
inner product of $g$ with the vector obtained from $f$ by conjugating its
coefficients in the Wick basis and permuting them with these signs, an
antilinear bijective isometry of $\mathcal H_{\rm nor}$. In particular,
$|\mathcal B_{\rm nor}(f, g)| \le \|f\|\|g\|$.

We use the two forms for different purposes, and we fix the
corresponding words. Let $X$ be a bounded operator on
$\mathcal H_{\rm nor}$, or on a space of which $\mathcal H_{\rm nor}$ is a
tensor factor, with the corresponding pairing and metric. We call the
operator $X^*$ with $\langle X^*f, g\rangle = \langle f, Xg\rangle$ the
(Hilbert) adjoint of $X$, and the operator $X^{\rm tr}$ with
$\mathcal B(X^{\rm tr}f, g) = \mathcal B(f, Xg)$ its (analytic) transpose.
If a linear form on the vectors is written as $\mathcal B(g, \cdot)$, we
call $g$ its dual vector. For an operator on $\mathcal H_{\rm nor}$, the
transpose is the conjugate of the adjoint by the antilinear isometry
above, so that the two have the same norm. The transpose is defined without complex conjugation, so that
it depends analytically on analytic parameters of $X$, while the adjoint
appears in all norm estimates. We let $\mathbf P_0$ denote the
orthogonal projection onto $\C\Omega_0$ and set
$\mathbf P_0^\perp = \Id - \mathbf P_0$. Since every Wick monomial of
positive degree has Gaussian mean zero, we have
$\mathbf P_0f = \ell_0(f)\Omega_0$ for all $f \in \mathcal H_{\rm nor}$. For
an operator $X$ on a Hilbert space of which $\mathcal H_{\rm nor}$ is a
tensor factor, with $\mathbf P_0$ acting on this factor, we call
$\mathbf P_0^\perp X\mathbf P_0^\perp$ its \emph{fast block},
$\mathbf P_0X\mathbf P_0^\perp$ and $\mathbf P_0^\perp X\mathbf P_0$ its
\emph{vacuum row} and \emph{vacuum column}, and $\mathbf P_0X\mathbf P_0$
its \emph{vacuum scalar}.

For the analytic computations it is convenient to also use flat
coordinates. For a single species of $d$ real even coordinates, we call
$L^2(\R^d) \otimes \mathcal H_{\rm odd}$ the flat realization, with
\[
 \mathbf N = \tfrac12(-\Delta + |x|^2 - d) + \mathbf N_F ,
\]
and the two realizations are related by the ground state
$\Omega_B(x) = \pi^{-d/4}e^{-|x|^2/2}$. Since $|\Omega_B|^2$ has variance
$1/2$ and the Wick variables have variance one, the flat coordinate is
the Wick coordinate divided by $\sqrt2$. Mehler's formula
\cite{mehler1866entwicklung} gives the kernel of $e^{-z\mathbf N}$ on the
even variables,
\begin{equation}\label{e:tube-mehler}
 k_z(x, y) = \frac{e^{dz/2}}{(2\pi\sinh z)^{d/2}}
 \exp\Bigl\{-\frac14\coth(z/2)|x - y|^2 - \frac14\tanh(z/2)|x + y|^2\Bigr\} ,
\end{equation}
for $\Re z > 0$. The ground-state transform
$\Omega_B^{-1}e^{-z\mathbf N}\Omega_B$ on the even variables is the
Ornstein--Uhlenbeck semigroup at the complex time $z$. For real $z$ its
kernel is the Gaussian density in $y$ with mean $e^{-z}x$ and covariance
$\frac12(1 - e^{-2z})\Id$, and by analytic continuation the same formula
holds for $\Re z > 0$, with the bilinear square in the exponent.

\section{The adapted transfer operator and the normal gauge}
\label{s:tube-gauge}

We fix the normalization of the transfer operator in tubular
coordinates. Let $w_\Gamma$ be the onsite weight of \zcref{s:rep} with
the source supermatrix $\Gamma$, and let $w_0$ be its value at $\Gamma = 0$.
We set $s_\Gamma = w_\Gamma/w_0$ and call it the source factor. Recall
from \eqref{e:tube-volume} that $\dd R = \varpi^2\dd U\dd\nu$ in tubular
coordinates.

In the balanced signature, the tangent variable $x$ runs over the balanced component, with the points $\kappa_E - \iu s_E Q$ for $Q \in \mathcal M_k$, and by \zcref{s:tube-prelim} the volume $\nu$ agrees with the integral of \zcref{d:sigma-integral} up to a constant factor. The normalization of $\nu$ at the root assigns unit Gaussian mass to the ambient metric, which on the tangent space at the root is a positive multiple of $\Str Y^2$, $Y \in \mathfrak m$. Since $d_k(\Lambda e^{-2Y}) = \Str Y^2 + O(|Y|^4)$, the normalization \eqref{e:sigma-integral-normalization} also gives unit Gaussian mass at the root to a positive multiple of $\Str Y^2$. The Gaussian mass of $e^{-c \Str Y^2}$ does not depend on $c > 0$, since $\mathfrak m$ has superdimension zero, and the constant factor is therefore $\pm 1$. With the orientation of $G_0/K_0$ fixed before \zcref{l:tube-volume}, it is $+1$. Hence $\dd\nu$ is the invariant measure of $\mathcal M_k$ with the orientation of $\mathcal B_k$; \zcref{s:assembly} uses this in the proofs of \zcref{p:asm-adapted,p:asm-jets}.

We let $K^\varpi$ denote the operator
$\varpi K_{\tau_{\rm phys}}\varpi^{-1}$, acting on functions of $(x, U)$
with the flat measure $\dd U\dd\nu$. Its kernel is
\begin{equation}\label{e:tube-raw-kernel}
 K^\varpi(x, U; y, U') = \varpi(U)\varpi(U')K_{\tau_{\rm phys}}\bigl(R(x, U), R(y, U')\bigr) ,
\end{equation}
where we write $\varpi(U) = \varpi(R_* + U)$. The two factors $\varpi$
come from the half-density change. Indeed, multiplying functions by
$\varpi$ turns the pulled-back measure $\varpi(U')^2\dd U'\dd\nu(y)$ into
the flat measure. We let $K^\varpi_{\rm nor}$ denote the
flat normal heat kernel, the operator in the normal variable $U$ with
the kernel $c_{\rm nor}\exp\{-\Str(U - U'')^2/(2h^4)\}$, where $c_{\rm nor}$ is the
constant for which this operator maps the constant function $1$ to itself;
in the balanced signature $c_{\rm nor} = 1$, since the normal space has
superdimension zero. We call it the frozen normal heat kernel of
$K^\varpi$.
Setting $x = y$ in $K^\varpi$ and integrating out the tangent increment
against its normalized Gaussian, we obtain $K^\varpi_{\rm nor}$ up to the
factor $\varpi(U)\varpi(U'')\varpi(S)^{-2} = 1 + O(\delta_W|V|^2)$, with
$S = R_* + (U + U'')/2$ and $V = (U - U'')/h^2$; this factor is treated
in \zcref{s:stab-uncharged}. We compute the spectral decomposition of
$K^\varpi_{\rm nor}$ in \zcref{l:tube-quadratic}.

\begin{definition}\label{d:tube-raw}
We call the realization of the chain on functions of the tubular
coordinates $(x, U)$, with the flat measure $\dd U\dd\nu$, in which one
link acts by $K^\varpi$ and the constant function $1$ is represented by
$\varpi$, the \emph{raw realization}.
\end{definition}

The gauged realization below is obtained from the raw realization by
conjugation with $w_0^{1/2}\gamma_h$, with the Gaussian gauge $\gamma_h$
of \eqref{e:tube-gamma}.

We call the square root $w_0^{1/2}$ of the source-free weight the
half-weight. On the normal tube, we write it as
\begin{equation}\label{e:tube-half-weight}
 w_0^{1/2} = w_{\rm quad}^{1/2}\, w_{\rm anh}^{1/2} ,
\end{equation}
where $w_{\rm quad}^{1/2}$ contains the constant and quadratic Taylor
terms of $\log w_0^{1/2}$ in the normal variables and
$w_{\rm anh}^{1/2}$ is the remaining anharmonic part. We retain the
physical weight $w_0$ in every signature and set
\[
 \zeta_p=w_0(R_*)=
 \left(e^{2\iu\kappa_Es_E}\frac{\kappa_E-\iu s_E}{\kappa_E+\iu s_E}\right)^{k-p},
 \qquad |\zeta_p|=1,\qquad \zeta_k=1.
\]
This follows by evaluating $e^{-\Str(R_*^2)/2}\Sdet(E-R_*)^{-1}$ at the
root of signature $p$. Choose a continuous branch of $\zeta_p^{1/2}$
on the energy interval, with $\zeta_k^{1/2}=1$ in the balanced signature, and fix
$w_0^{1/2}(R_*)=w_{\rm quad}^{1/2}(0)=\zeta_p^{1/2}$ and
$w_{\rm anh}^{1/2}(0)=1$. Thus the two quadratic half-weights retain
one factor $\zeta_p$ per step. The endpoint factors below use the same
physical half-weight, with its inverse at the right endpoint. For
operator norms, we extend $\log w_0^{1/2}$ and
$\log w_{\rm anh}^{1/2}$ from the tube by a fixed smooth invariant cutoff
on a slightly larger tube. The exponentials of the extended functions are
bounded invertible multipliers that agree with the analytic ones on the
tube.

We define the \emph{Gaussian gauge} to be the function
\begin{equation}\label{e:tube-gamma}
 \gamma_h(U) = \exp\Bigl\{-\frac12\sum_\sigma c_\sigma\Str(U_\sigma^2)\Bigr\} ,
\end{equation}
where $U_\sigma$ is the normal block of root $\sigma$ and $c_\sigma$ is the
number of \eqref{e:tube-c}. On the adapted real normal space of
\zcref{d:tube-cycle}, we have $U_\sigma = e^{-\iu\theta_\sigma}X_\sigma$
with $X_\sigma \in \mathfrak k_\sigma^\R$, and
$c_\sigma\Str(U_\sigma^2) = |c_\sigma|\Str(X_\sigma^2)$, since
$e^{-2\iu\theta_\sigma}c_\sigma = |c_\sigma|$. In particular, $\gamma_h$ is real on
the real tube, and its body is a Gaussian whose precision is of order $W$
in each even coordinate. We show in \zcref{l:tube-quadratic} that
conjugation by $\gamma_h$ maps the flat realization of the exact quadratic
normal transfer to the positive Wick realization of
\zcref{s:tube-oscillator}, in the standardized coordinates
$Z_\sigma = (2c_\sigma)^{1/2}U_\sigma$, which are real on the adapted real
normal space. In these coordinates $\gamma_h^2 = e^{-\Str(Z^2)/2}$, the
Gaussian of the Gaussian mean $\ell_0$. We define the analytic Berezin
pairing on the adapted tube by
\begin{equation}\label{e:tube-pairing-ad}
 \mathcal B_{\rm ad}(f, g) = \int f g\,\gamma_h^2\dd U\dd\nu .
\end{equation}
In the balanced signature, $\int\gamma_h^2\dd U = 1$, since the change of
variables to $Z$ has Berezinian one on a space of superdimension zero and
the Gaussian $e^{-\Str(U^2)/2}$ has the integral one for $\dd U$. The normal
part of $\mathcal B_{\rm ad}$ is then the pairing $\mathcal B_{\rm nor}$ of
\eqref{e:tube-pairing}: for normal vectors $u$ and $u'$, with $u'$ even,
and functions $f$ and $f'$ of the tangent variables,
\begin{equation}\label{e:tube-pairing-factor}
 \mathcal B_{\rm ad}(uf, u'f') = \mathcal B_{\rm nor}(u, u')\int ff'\dd\nu .
\end{equation}
In an unbalanced signature, $\int\gamma_h^2\dd U$ is a nonzero constant, a
fixed power of the $c_\sigma$.

We define the adapted transfer operator $A_h$, the exact quadratic
normal transfer $\mathbf M_h^{\rm q}$, the frozen normal transfer
$\mathbf F_h$, and the right and left endpoint factors $g_R$, $g_L$ by
\begin{equation}\label{e:tube-table}
\begin{aligned}
 A_h &= \gamma_h^{-1}w_0^{-1/2}\,T_\Gamma\,w_0^{1/2}\gamma_h
      = s_\Gamma\,\gamma_h^{-1}w_0^{1/2}K^\varpi w_0^{1/2}\gamma_h , \\
 \mathbf M_h^{\rm q} &= \gamma_h^{-1}w_{\rm quad}^{1/2}K^\varpi_{\rm nor}w_{\rm quad}^{1/2}\gamma_h ,
 \qquad
 \mathbf F_h = w_{\rm anh}^{1/2}\,\mathbf M_h^{\rm q}\,w_{\rm anh}^{1/2} , \\
 g_R &= \varpi\,w_0^{-1/2}\gamma_h^{-1} ,
 \qquad
 g_L = \varpi\,w_0^{1/2}\gamma_h^{-1} ,
\end{aligned}
\end{equation}
where $T_\Gamma$ is written in tubular coordinates with the half-density
change, that is, $T_\Gamma = w_\Gamma K^\varpi$. The second expression
for $A_h$ uses $w_\Gamma w_0^{-1/2} = s_\Gamma w_0^{1/2}$. The
source factor $s_\Gamma$ stands on the left. We call the normalization
\eqref{e:tube-table} the symmetric normal gauge.

The two endpoint factors represent the two ends of the chain. In the raw
realization, both ends are the half-density form $\varpi$ of the constant
function $1$, a vector at the right end and the linear form
$F \mapsto \int\varpi F\dd U\dd\nu$ at the left end. In the gauged
realization, vectors $F$ become $\gamma_h^{-1}w_0^{-1/2}F$, so that
$g_R = \gamma_h^{-1}w_0^{-1/2}\varpi$, and
\begin{equation}\label{e:tube-endpoint-duality}
 \int\varpi F\dd U\dd\nu = \mathcal B_{\rm ad}\bigl(g_L, \gamma_h^{-1}w_0^{-1/2}F\bigr)
\end{equation}
for all raw vectors $F$. This identity says that $g_L$ is the dual vector, for the analytic
Berezin pairing, of the linear form that ends the chain, and we call it
the analytic dual of $g_R$. It is not the Hilbert adjoint of $g_R$. The
frozen normal transfer $\mathbf F_h$ is symmetric for the analytic Berezin
pairing, $\mathbf F_h^{\rm tr} = \mathbf F_h$, since
\[
 \gamma_h\mathbf F_h\gamma_h^{-1}
 = w_{\rm anh}^{1/2}w_{\rm quad}^{1/2}K^\varpi_{\rm nor}w_{\rm quad}^{1/2}w_{\rm anh}^{1/2}
\]
and $K^\varpi_{\rm nor}$ has a symmetric kernel. All transposes in Part IV
are transposes for the analytic Berezin pairing, in the sense of
\zcref{s:tube-oscillator}.

\begin{lemma}\label{l:tube-class}
The functions $\varpi^{\pm 1}$, $w_0^{\pm 1/2}$, $w_{\rm quad}^{\pm 1/2}$,
$w_{\rm anh}^{\pm 1/2}$ and $\gamma_h^{\pm 1}$ on the normal tube are normal
class functions. In particular, the endpoint factors $g_R$ and $g_L$ are
normal class functions, and they are annihilated by the normal charges.
\end{lemma}

\begin{proof}
Let $k$ be an even invertible supermatrix with Grassmann entries that
commutes with $\Sigma$. Then $k = \diag(k_+, k_-)$ and
$k(R_* + U)k^{-1} = R_* + kUk^{-1}$. For $\varpi$, conjugation of $S$ by
$k$ conjugates $L_S$ by $\lambda_{k_+}\rho_{k_-^{-1}}$, and
\zcref{l:grass-sdet}\ref{i:grass-sdet-mult} shows that $\varpi$ is
unchanged. The weight $w_0$ is invariant under conjugation by
\zcref{r:rep-symmetry}, and the same holds for $w_0^{1/2}$, whose branch
is fixed at $U = 0$. Since $U \mapsto kUk^{-1}$ is linear, it maps the Taylor part of
each degree of $\log w_0^{1/2}$ at $U = 0$ to the Taylor part of the same
degree. Hence the constant and quadratic terms $\log w_{\rm quad}^{1/2}$ are invariant,
and therefore also $w_{\rm anh}^{1/2} = w_0^{1/2}/w_{\rm quad}^{1/2}$. Finally,
\[
 \Str\bigl((kUk^{-1})_\sigma^2\bigr) = \Str(k_\sigma U_\sigma^2k_\sigma^{-1}) = \Str(U_\sigma^2)
\]
by \zcref{l:grass-sdet}\ref{i:grass-sdet-trace}, so that $\gamma_h$ is
invariant. The products $g_R$ and $g_L$ of \eqref{e:tube-table} are
therefore invariant, and the normal charges, which are derivatives along
conjugations by elements of this kind, annihilate them.
\end{proof}

We keep the name $\mathbf F_h$ for the frozen normal transfer with the
normal cutoffs of \zcref{s:tube-cutoff}, and \zcref{s:tube-powers,s:tube-gap}
study it.

\begin{remark}\label{r:tube-transport}
On the tube, $w_0^{\pm 1/2}$ and $w_{\rm anh}^{\pm 1/2}$ are bounded
analytic multipliers, and the identities \eqref{e:tube-table} are exact
for all odd coefficients and all source jets. Indeed, every normal
determinant in $w_0$ has nonzero body on the tube. Its square root,
its inverse and its finite nilpotent completion are therefore analytic
and bounded there. In the standardized coordinates $U = hZ$, each
derivative of a coefficient carries a factor $h$, and the product rule
gives bounds in the graph norms $\|(\mathbf N + 1)^m f\|$ of the number
operator of \zcref{s:tube-oscillator}, which we use from \zcref{s:tube-cutoff} on. Setting $B_h = \gamma_h^{-1}T_\Gamma\gamma_h$, we have
\[
 A_h = w_0^{-1/2}B_hw_0^{1/2} , \qquad B_h^n = w_0^{1/2}A_h^nw_0^{-1/2} .
\] The condition numbers of the
half-weight therefore appear once in each power. All source and cutoff
multipliers are even functions, and they commute with these scalar
normal multipliers. In the balanced signature the equal even and odd
dimensions cancel the standardizing Berezinians and fix the Gaussian
vacuum pairing at one. In an unbalanced signature the constant
$\varpi(R_*)$ is kept separately until it is suppressed in
\zcref{p:tube-gap}.
\end{remark}

The transfer operator of \zcref{p:rep-exact} applies the onsite weight
after the heat factor. Its analytic transpose ends with a heat factor,
and the identities \eqref{e:tube-table} transport this endpoint without
changing the order of the factors.

\section{The quadratic normal transfer and the supercharge}\label{s:tube-quadratic}

We now compute the exact quadratic normal transfer of \eqref{e:tube-table}
in the positive Wick realization of \zcref{s:tube-oscillator}, and we
introduce the supercharge of the normal variables.

\subsection{The exact quadratic normal transfer}

Recall the roots $r_\sigma$ and the numbers $m_\sigma$, $c_\sigma$ and
$\theta_\sigma$ of \eqref{e:tube-c}, and set
\begin{equation}\label{e:tube-mu}
 A_\sigma = W^2 + \tfrac12 m_\sigma , \qquad
 \mu_\sigma = \frac{W^2}{A_\sigma + c_\sigma} , \qquad
 z_\sigma = -\log\mu_\sigma ,
\end{equation}
with the branches of positive real part, which are complex conjugate
for the two roots. We have $z_\sigma = 2\operatorname{arsinh}(m_\sigma^{1/2}/(2W))$,
so that $\Re z_\sigma \asymp W^{-1} = h^2$ and
$|\Im z_\sigma| \le C|E|\Re z_\sigma$. Recall the standardized
coordinates $Z_\sigma = (2c_\sigma)^{1/2}U_\sigma$ of \zcref{s:tube-gauge},
which scale the even and odd coordinates of each root block by the same
factor. Since $2|c_\sigma|h^2 = 2|m_\sigma|^{1/2}(1 + O(h^4))$
lies between two positive constants, $Z$ is comparable to $U/h$. We write
$U = hZ$ for this relation, absorbing the bounded factors
$(2c_\sigma h^2)^{1/2}$ into constants and cutoff radii, and we call $U$
the unscaled normal coordinate.

We also fix coordinates on a normal block. In the balanced signature, we
write a root block $U_\sigma$ through its even blocks $u_{11}$ and
$u_{22}$ and its odd blocks $\xi_1$ and $\xi_2$, all of size $k \times k$,
and we set
\begin{equation}\label{e:tube-normal-coordinates}
\begin{gathered}
 U_\sigma = \begin{pmatrix} u_{11} & \xi_1 \\ \xi_2 & u_{22}\end{pmatrix} ,
 \qquad
 \Str U_\sigma^2 = \sum_e\bigl(x_e^2 + y_e^2 + 2\alpha_e\beta_e\bigr) , \\
 u_{11} = \sum_ex_ee , \quad u_{22} = \iu\sum_ey_ee , \quad
 \xi_1 = \sum_e\alpha_ee , \quad \xi_2 = \sum_e\beta_ee ,
\end{gathered}
\end{equation}
where $\{e\}$ is a basis of the Hermitian $k \times k$ matrices that is
orthonormal for the form $\Tr(ee')$, the coordinates $x_e$ and $y_e$ are
even, and $\alpha_e$ and $\beta_e$ are odd. The identity for
$\Str U_\sigma^2$ follows from
$\Str U_\sigma^2 = \Tr(u_{11}^2) - \Tr(u_{22}^2) + 2\Tr(\xi_1\xi_2)$. On
the adapted real normal space of \zcref{d:tube-cycle}, the coordinates
$x_e$ and $y_e$ are $e^{-\iu\theta_\sigma}$ times real numbers. In an
unbalanced signature, the two even blocks have different sizes. We then
expand $u_{11}$ and $-\iu u_{22}$ in orthonormal bases of the Hermitian
matrices of their sizes, and we take the pairs of odd coordinates
$\alpha = (\xi_1)_{ij}$ and $\beta = (\xi_2)_{ji}$, so that
$\Str U_\sigma^2$ has the same form.

\begin{lemma}\label{l:tube-quadratic}
In the balanced signature, the exact quadratic normal transfer of
\eqref{e:tube-table} is
\[
 \mathbf M_h^{\rm q} = \mu_+^{\mathbf N_+}\mu_-^{\mathbf N_-}
\]
on the positive Wick realization. In the signature $p \ne k$, it is
$\lambda\,\mu_+^{\mathbf N_+}\mu_-^{\mathbf N_-}$ on the normal space of
that signature, where
\[
 \lambda=\zeta_p(\mu_+\mu_-)^{(p-k)^2/2},
 \qquad |\lambda| = e^{-(p - k)^2\Re z_+}.
\]
We use the same notation $\lambda=1$ in the balanced signature.
\end{lemma}

\begin{proof}
The two constant factors $\zeta_p^{1/2}$ in the quadratic half-weights
contribute $\zeta_p$ once. We first compute the quadratic transfer with
this scalar removed. We treat one even coordinate of root $\sigma$, and write it as
$U = e^{-\iu\theta_\sigma}x$ with $x$ real, as in \zcref{d:tube-cycle}. In
this coordinate, the kernel of one step of the exact quadratic normal
transfer, with the quadratic half-weights, is the symmetric Gaussian kernel
\[
 \frac{We^{-\iu\theta_\sigma}}{\sqrt{2\pi}}\exp\bigl\{-\tfrac12W^2e^{-2\iu\theta_\sigma}(x - y)^2 - \tfrac14m_\sigma e^{-2\iu\theta_\sigma}(x^2 + y^2)\bigr\}
\]
with respect to $\dd y$. We set
\[
 A_\sigma' = e^{-2\iu\theta_\sigma}A_\sigma , \qquad
 c_\sigma' = e^{-2\iu\theta_\sigma}c_\sigma = |c_\sigma| > 0 ,
\]
where the last identity is the definition of $\theta_\sigma$ in \eqref{e:tube-c}. Completing the
square in $y$, we find that
\[
\begin{aligned}
 &\frac{We^{-\iu\theta_\sigma}}{\sqrt{2\pi}}\int_\R e^{-\frac12W^2e^{-2\iu\theta_\sigma}(x - y)^2 - \frac14m_\sigma e^{-2\iu\theta_\sigma}(x^2 + y^2) - \frac12c_\sigma'y^2}\dd y \\
 &\qquad = \Bigl(\frac{W^2}{A_\sigma + c_\sigma}\Bigr)^{1/2}e^{-c_\sigma'x^2/2}
 = \mu_\sigma^{1/2}e^{-|c_\sigma|x^2/2} ,
\end{aligned}
\]
where we used the identity
\[
 \frac{W^4e^{-4\iu\theta_\sigma}}{A_\sigma' + c_\sigma'} = A_\sigma' - c_\sigma' ,
\]
which follows from $A_\sigma^2 - c_\sigma^2 = W^4$.

The same Gaussian integral with an additional linear term $e^{ty}$,
expanded in $t$, shows that the
kernel acts on the $n$th Hermite function of the real Gaussian
$e^{-|c_\sigma|x^2/2}$ by the factor $\mu_\sigma^{n + 1/2}$. These Hermite
functions form an orthonormal basis, so the kernel is diagonal in a real
orthonormal basis, and the corresponding ground state, the Gaussian gauge
$\gamma_h$, is real.

We now add the odd coordinates, which appear in $\Str U_\sigma^2$ in pairs
$(\alpha, \beta)$, through $2\alpha\beta$, as in
\eqref{e:tube-normal-coordinates}. This step is the supersymmetric cancellation
of the zero-point factors.

For one pair, the odd parts of the heat kernel,
of the two quadratic half-weights and of the Gaussian gauge
\eqref{e:tube-gamma} are
\[
 -W^{-2}\exp\{-W^2(\alpha - \alpha')(\beta - \beta')\} , \qquad
 e^{-m_\sigma\alpha\beta/2} , \quad e^{-m_\sigma\alpha'\beta'/2} , \qquad
 e^{-c_\sigma\alpha\beta} ,
\]
where the factor $-W^{-2}$ is the one for which the odd heat kernel maps
$1$ to $1$, with the Berezin integral of \zcref{d:grass-berezin} in
$(\alpha', \beta')$. Collecting the exponents, we find that the odd factor
of $\mathbf M_h^{\rm q}$ acts on functions of $(\alpha', \beta')$ by
\[
 f \longmapsto - W^{-2}e^{(c_\sigma - A_\sigma)\alpha\beta}
 \int\exp\bigl\{-(A_\sigma + c_\sigma)\alpha'\beta' + W^2(\alpha\beta' + \alpha'\beta)\bigr\}f(\alpha', \beta')\dd\alpha'\dd\beta' .
\]
Since the three terms in the last exponent are even and have square zero,
we have
\[
 \exp\bigl\{-(A_\sigma + c_\sigma)\alpha'\beta' + W^2(\alpha\beta' + \alpha'\beta)\bigr\}
 = 1 - (A_\sigma + c_\sigma)\alpha'\beta' + W^2\alpha\beta' + W^2\alpha'\beta - W^4\alpha'\beta'\alpha\beta ,
\]
and we obtain
\begin{equation}\label{e:tube-odd-wick}
\begin{gathered}
 1 \longmapsto W^{-2}(A_\sigma + c_\sigma) = \mu_\sigma^{-1} , \qquad
 \alpha \longmapsto \alpha , \qquad \beta \longmapsto \beta , \\
 \alpha\beta + (2c_\sigma)^{-1} \longmapsto W^{-2}(A_\sigma - c_\sigma)\bigl(\alpha\beta + (2c_\sigma)^{-1}\bigr)
 = \mu_\sigma\bigl(\alpha\beta + (2c_\sigma)^{-1}\bigr) ,
\end{gathered}
\end{equation}
where we used $A_\sigma^2 - c_\sigma^2 = W^4$, which cancels the term
with $\alpha\beta$ in the image of $1$ and gives
$W^{-2}(A_\sigma - c_\sigma) = \mu_\sigma$. The supplementary script
\texttt{tube\_odd\_wick.py} confirms the four images in
\eqref{e:tube-odd-wick} in exact arithmetic, with all their Grassmann
components.

In the standardized odd coordinates
$\tilde\alpha = (2c_\sigma)^{1/2}\alpha$ and
$\tilde\beta = (2c_\sigma)^{1/2}\beta$, these four vectors become $1$,
$\tilde\alpha$, $\tilde\beta$ and $(2c_\sigma)^{-1}(\tilde\alpha\tilde\beta + 1)$,
multiples of the odd Wick basis of \zcref{s:tube-oscillator}.
Hence the odd factor of one pair acts as $\mu_\sigma^{-1}\mu_\sigma^{\mathbf N_F}$
on the Wick realization. This is the Wick identity for the odd
coordinates: each odd coordinate carries the zero-point factor
$\mu_\sigma^{-1/2}$, which cancels the zero-point factor $\mu_\sigma^{1/2}$
of an even coordinate.

Tensoring over an orthonormal basis of normal Hermitian matrices, we obtain
$\mu_\sigma^{\mathbf N_\sigma}$ times the zero-point factor
$\mu_\sigma^{\operatorname{sdim}/2}$, where $\operatorname{sdim}$ is the
superdimension of the normal space of root $\sigma$. In the balanced
signature these superdimensions vanish. In the signature $p$, the normal
blocks $\operatorname{End}(V_+) \cong \mathfrak{gl}(k|p)$ and
$\operatorname{End}(V_-) \cong \mathfrak{gl}(k|2k - p)$ both have the
superdimension $(p - k)^2$, by \eqref{e:tube-sdim-hom}, and the
zero-point factors are $\mu_\sigma^{(p - k)^2/2}$. Their product
$(\mu_+\mu_-)^{(p - k)^2/2}$ has modulus $e^{-(p - k)^2\Re z_+}$,
since $z_- = \bar z_+$. Restoring the scalar $\zeta_p$ gives the stated
$\lambda$ and leaves its modulus unchanged.
\end{proof}

\begin{remark}\label{r:stab-fock}\label{r:tube-two-descriptions}
We use two descriptions of one step of the chain, and only one of them
leads to stability. The differential description \eqref{e:tube-decomposition}
identifies the generator without an expansion around the marginal, and
\zcref{s:cons} turns it into consistency on smooth vectors. The kernel
description, through the midpoint formula of \zcref{s:stab}, bounds the
marginals of the kernels by $C\delta_W$ and thereby verifies, for each step,
the hypothesis of \zcref{l:tools-defect}, which can be iterated over the
$h^{-4}$ steps of the chain on arbitrary inputs.

It is natural to try to prove stability from an energy bound for the
heat generator in the positive Wick metric. \zcref[S]{l:cons-short-heat}
transports this metric through the full Gaussian gauge to a flat positive
coefficient metric and estimates the normal charges and their Hilbert
adjoints there. The ordinary normal Laplacian supplies the gradient
energy needed to absorb the charge terms on a sufficiently small fixed
tube. This gives the bound $\exp(Ct h^{-2})$ at time $t$, which is
$1+O(h^2)$ over one link of duration $h^4/2$, but grows like
$\exp(C h^{-2})$ over a fixed macroscopic time. It therefore does not
give the iterable bound $1+O(h^4)$ per link. Invariance of the Gaussian
Berezin density is not used to assert skew-adjointness of odd charges in
the positive metric.

The obstruction is in the odd normal variables, whose heat has a nilpotent generator, which is
dissipative in no metric. By \eqref{e:tube-odd-wick}, the odd factor of one
step has the eigenvalues $\mu_\sigma^{-1}$, $1$, $1$, and $\mu_\sigma$, and
$|\mu_\sigma|^{-1} = e^{\Re z_\sigma} \ge 1 + ch^2$. The eigenvalue
$\mu_\sigma^{-1}$ is compensated by the even zero-point factor $\mu_\sigma$,
which appears only in the product of the heat with the half-weight. Since a
norm is at least the spectral radius, in every metric of tensor-product
form the odd heat and the odd part of the half-weight cannot both have norm
$1 + O(\delta_W)$. In rank one there are even invariant vectors
$u = \varphi(\Str U_\sigma^2)$, annihilated by the horizontal operator, on
which the real part of the quadratic form of the gauged generator is at
least $ch^{-2}\|u\|^2$; a numerical model in the supplementary material,
\texttt{fock\_dissipativity.py}, finds about $0.2h^{-2}$. For this reason we
integrate the odd normal variables against the complete Mehler factor,
which contains the half-weight, before taking any norm.
\end{remark}

\subsection{The supercharge and localization}

\begin{lemma}\label{l:tube-supercharge}
On the span of the Wick polynomials, let $\mathfrak q_{\rm nor}$ be the sum over
all balanced normal coordinates of the odd differential operators
\[
 \beta(\partial_x - \iu\partial_y) - (x - \iu y)\partial_\alpha .
\]
Then $\mathfrak q_{\rm nor}^2 = 0$ and $\mathfrak q_{\rm nor}\mathfrak q_{\rm nor}^* + \mathfrak q_{\rm nor}^*\mathfrak q_{\rm nor} = 2\mathbf N$,
and $\mathfrak q_{\rm nor}$ preserves the number of each coordinate. Its direct
sum over the number spaces is a closed operator, $\ker\mathfrak q_{\rm nor}$ is
closed, the spectral projections of $\mathbf N$ commute with
$\mathfrak q_{\rm nor}$, and the Wick polynomials in $\ker\mathfrak q_{\rm nor}$ are dense in
$\ker\mathfrak q_{\rm nor}$.
\end{lemma}

The operator $\mathfrak q_{\rm nor}$ is a supercharge of the supersymmetric
harmonic oscillator with number operator $\mathbf N$, in the sense of
these two identities. It is one of the odd normal charges of
\zcref{s:tube-operator}, and therefore every invariant function of $U$
is annihilated by it. To verify this, we write a balanced root block
$U$ through its blocks $u_{11}$, $\xi_1$, $\xi_2$, and $u_{22}$ as in
\eqref{e:tube-normal-coordinates}, and we take
$\mathfrak e = \zeta\begin{pmatrix} 0 & \Id \\ 0 & 0\end{pmatrix} \in \mathfrak k$ with an
odd parameter $\zeta$. For this $\mathfrak e$, we have
\[
 [\mathfrak e, U] = \zeta\begin{pmatrix} \xi_2 & u_{22} - u_{11} \\ 0 & \xi_2\end{pmatrix} ,
\]
so the variations of $u_{11}$, $u_{22}$, $\xi_1$, and $\xi_2$ are $\zeta\xi_2$,
$\zeta\xi_2$, $\zeta(u_{22} - u_{11})$, and $0$. In the coordinates of
\eqref{e:tube-normal-coordinates}, the induced derivation
on functions of $U$ therefore takes the form
\[
 \zeta\sum_e\bigl[\beta_e(\partial_{x_e} - \iu\partial_{y_e}) - (x_e - \iu y_e)\partial_{\alpha_e}\bigr] = \zeta\mathfrak q_{\rm nor} .
\]
The rotation and the standardization of
\zcref{d:tube-cycle} preserve this form, since the field is homogeneous
of degree zero. In the coordinates of the remark after
\zcref{c:grass-normalization}, $\mathfrak q_{\rm nor}$ is the half
$\mathfrak q_+$ of the supercharge $\mathfrak q$ of
\zcref{d:grass-supercharge}; unlike $\mathfrak q$, whose square is a
rotation generator, it squares to zero.

\begin{proof}
For one coordinate, let $a = (\partial_x - \iu\partial_y)/\sqrt2$ and
$b = (\partial_x + \iu\partial_y)/\sqrt2$. Together with their Gaussian
adjoints, they form two independent pairs of bosonic creation and
annihilation operators.
In the odd basis above, set
\[
 f_\alpha = \partial_\alpha , \quad f_\alpha^* = \alpha + \partial_\beta ,
 \qquad
 f_\beta = \partial_\beta , \quad f_\beta^* = \beta - \partial_\alpha .
\]
Their action on the four basis vectors verifies the canonical
anticommutation relations, and that $f_\alpha^*$ and $f_\beta^*$ are the
adjoints of $f_\alpha$ and $f_\beta$ in the positive metric. Since
$x - \iu y = \sqrt2(a + b^*)$ on Gaussian functions, the terms $\beta \cdot \sqrt2a$ and $-\sqrt2a\,\partial_\alpha$
combine into $\sqrt2f_\beta^*a$, and for one coordinate
\[
 \mathfrak q_{\rm nor} = \sqrt2(f_\beta^* a - f_\alpha b^*) .
\]
Each term exchanges one even and one odd excitation of the same
coordinate. From the canonical relations we obtain, for one coordinate,
\[
 \mathfrak q_{\rm nor}^2 = 0 , \qquad
 \mathfrak q_{\rm nor}\mathfrak q_{\rm nor}^* + \mathfrak q_{\rm nor}^*\mathfrak q_{\rm nor}
 = 2(a^*a + b^*b + f_\alpha^*f_\alpha + f_\beta^*f_\beta) ,
\]
and the cross terms between different coordinates cancel by
graded anticommutation. Summing over the coordinates proves the two
identities.

Let $\mathcal H_{\rm nor} = \bigoplus_{n \ge 0}\mathcal E_n$ be the
decomposition into the finite-dimensional eigenspaces of $\mathbf N$,
and let $\mathfrak q_{{\rm nor}, n}$ be the restriction of $\mathfrak q_{\rm nor}$ to
$\mathcal E_n$. We define $\mathfrak q_{\rm nor}$ by
\[
 \mathfrak q_{\rm nor}f = \sum_{n \ge 0}\mathfrak q_{{\rm nor}, n}f_n
 \quad\text{on the domain}\quad
 \Bigl\{f = \sum_{n \ge 0}f_n : \sum_{n \ge 0}\|\mathfrak q_{{\rm nor}, n}f_n\|^2 < \infty\Bigr\} .
\] Componentwise limits show that
this operator is closed, $\ker\mathfrak q_{\rm nor} = \bigoplus_n\ker\mathfrak q_{{\rm nor}, n}$
is closed, and the truncations to finitely many $n$ commute with
$\mathfrak q_{\rm nor}$ and converge strongly to the identity. On the domain of $\mathbf N$, this operator agrees with the differential operator, since the two operators are continuous for the graph norm of $\mathbf N$ and agree on
Wick polynomials.
\end{proof}

\begin{lemma}\label{l:tube-localization}
Let $U$ be a balanced normal supermatrix, and let
$\Str U^2 = \sum_a(x_a^2 + y_a^2 + 2\alpha_a\beta_a)$ in the coordinates
of \eqref{e:tube-normal-coordinates}. Let $f$ be a smooth function of $U$, with
Grassmann coefficients, such that $\mathfrak q_{\rm nor}f = 0$ and such that $f$
and its derivatives are bounded by $Ce^{a|U|^2}$ for some $a < 1/2$; for
instance, $f$ may be a Wick polynomial. Then the normalized Gaussian
Berezin integral of $f$ is the value $f_{\rm body}(0)$ of its body at the
origin. The same holds for a uniformly accretive complex Gaussian
precision, that is, for the Gaussian
$\exp\{-\frac12\sum_a\lambda_a(x_a^2 + y_a^2 + 2\alpha_a\beta_a)\}$ with one
complex number $\lambda_a$ per normal coordinate and $\Re\lambda_a \ge c > 0$,
provided that $a < c/2$;
each term of this exponent is annihilated by $\mathfrak q_{\rm nor}$.
\end{lemma}

\begin{proof}
This is the localization of an integral under a $\mathfrak q_{\rm nor}$-exact
deformation \cite{parisi1979random,schwarz1997supersymmetry}, in the form of \zcref{c:grass-normalization}. We use the coordinates $\zeta_a = x_a + \iu y_a$, $\bar\chi_a = \alpha_a$, and $\chi_a = 2\beta_a$ of the remark after that corollary, in which $\zeta^*\zeta + \chi^*\chi = \Str U^2$ and $\mathfrak q_+ = \mathfrak q_{\rm nor}$. Linear changes of coordinates multiply the Berezin integral by a constant, by \zcref{l:grass-berezin}\ref{i:grass-berezin-linear} for the odd coordinates, so normalized Gaussian integrals are the same in both coordinate systems. We apply \zcref{c:grass-normalization} with $\mathfrak q' = \mathfrak q_+$, $A = \frac12 \Id$, and $\gamma = a$, coefficientwise in the Grassmann coefficients of $f$. Its hypothesis $\gamma < \lambda_A = 1/2$ is our growth bound, and the corollary implies the first claim. For the complex precision, we take the diagonal matrix $A$ with entries $\lambda_a/2$, for which $\lambda_A \ge c/2 > a$.
\end{proof}

In particular, every function of $\Str V^2$, such as the invariant
cutoff of \zcref{s:tube-cutoff}, is annihilated by $\mathfrak q_{\rm nor}$,
including all its nilpotent terms.

\section{Cutoffs and Mehler defects}\label{s:tube-cutoff}

We now prove the estimates for the frozen normal transfer that the comparison of
\zcref{s:stab} needs. They concern the Mehler operator with complex times of order $h^2$, a normal cutoff at a radius $r$ that does not depend on $h$, and the normal charges.

\subsection{Setting}

Let $\mathcal H_{\rm nor} = L^2(\R^d) \otimes \mathcal H_{\rm odd}$ in
the flat realization, with $\mathcal H_{\rm odd}$ a fixed
finite-dimensional fermionic Fock space, and let the coordinates and
odd modes be grouped into finitely many species $\sigma$, with
$\mathbf N = \sum_\sigma\mathbf N_\sigma$. We consider
\[
 \mathbf M = \lambda\prod_\sigma e^{-z_\sigma\mathbf N_\sigma} ,
 \qquad
 c_0h^2 \le \Re z_\sigma \le C_0h^2 ,
 \qquad
 |\Im z_\sigma| \le C_0\Re z_\sigma ,
\]
where $0<|\lambda|\le1$ and $1-|\lambda|^2\le C_0h^2$. We set
$\underline z = \min_\sigma\Re z_\sigma$ and
$D_{\rm nor} = \Id - \mathbf M^*\mathbf M$. We call $\mathbf M$ the Mehler
operator and $D_{\rm nor}$ the normal defect. Since the
$\mathbf N_\sigma$ commute, $\mathbf M$ is a normal contraction,
$\mathbf M^* = \bar\lambda\prod_\sigma e^{-\bar z_\sigma\mathbf N_\sigma}$, and
\[
 D_{\rm nor} = \Id - \mathbf M\mathbf M^*
 = \Id - |\lambda|^2\prod_\sigma e^{-2\Re z_\sigma\mathbf N_\sigma} .
\]
Recall from \zcref{l:tube-quadratic} that the exact quadratic transfer
$\mathbf M_h^{\rm q}$ is of this form, with $z_\sigma$ as in
\eqref{e:tube-mu} and the full scalar $\lambda$, including $\zeta_p$,
of that lemma. In particular, $\mathbf M\Omega_0=\lambda\Omega_0$.
On $\mathbf P_0^\perp$, its defect is comparable to $D_+$ below:
the phase-free defect is bounded below by $ch^2\mathbf P_0^\perp$,
while $1-|\lambda|^2=O(h^2)$ adds at most a fixed multiple of that
defect. The scalar $\lambda$ therefore preserves these comparisons.
The proofs below use only the moduli
$|e^{-z_\sigma n}| = e^{-n\Re z_\sigma}$ and the positive real parts of
the Mehler precisions, and for this reason every statement also holds
for $\mathbf M^*$.
The zero eigenspace of $\mathbf N$ is spanned by $\Omega_0$. We fix
$0 < h \le r \le r_*$. Constants may depend on the dimension, on
$\mathcal H_{\rm odd}$, on $c_0$ and $C_0$, and on the indicated finite
data, but not on $h$, and we indicate dependence on $r$ explicitly. For an
integer $m \ge 0$ we let $\mathcal G^m$ denote the domain of
$(\mathbf N + 1)^m$ with the graph norm $\|(\mathbf N + 1)^mf\|$, and we
say that an operator is bounded in the graph norm of order $m$ if it is
bounded on $\mathcal G^m$.

Let $\eta$ be a smooth function on $[0, \infty)$, with $0 \le \eta \le 1$,
equal to one on $[0, 1]$ and to zero on $[4, \infty)$, and set
$\chi(x) = \eta(|x|^2)$ for $x \in \R^d$. Thus $\chi$ is radial, equals
one on the unit ball, and is supported in the ball of radius two.
We set $\chi_{r/h}(x) = \chi(hx/r) = \eta(h^2|x|^2/r^2)$ and
\[
\begin{gathered}
 \mathbf M^\chi = \chi_{r/h}\mathbf M\chi_{r/h} , \\
 \bar D_R = \Id - (\mathbf M^\chi)^*\mathbf M^\chi , \\
 \bar D_L = \Id - \mathbf M^\chi(\mathbf M^\chi)^* , \\
 D_+ = \Id - e^{-2\underline z(\mathbf N + 1)} .
\end{gathered}
\]
We call $\mathbf M^\chi$ the cutoff Mehler operator; it is a contraction,
since $0 \le \chi_{r/h} \le 1$ and $\|\mathbf M\| \le 1$. We define the invariant cutoff by
\[
 \chi_{\rm inv}(Z) = \eta\Bigl(\frac{h^2\Str(Z^2)}{2r^2}\Bigr) .
\]
Here $x$ is the even flat coordinate: the even coordinates of the
standardized normal supermatrix $Z$ are $\sqrt2 x$, by
\zcref{s:tube-oscillator}. Thus $\frac12\Str(Z^2)$ has body $|x|^2$,
and $\chi_{\rm inv}$ has exactly the body $\chi_{r/h}$. The invariant cutoff differs from
$\chi_{r/h}$ by finitely many nilpotent terms, which we estimate in
\zcref{l:tube-faces}.

\begin{lemma}\label{l:tube-slow}
Let $b$ be a bounded scalar or matrix-valued multiplier with bounded
first derivatives. Then
\[
 \|D_+^{-1/2}bD_+^{1/2}\| + \|D_+^{1/2}bD_+^{-1/2}\|
 \le C(\|b\|_\infty + \|\nabla b\|_\infty) .
\]
If $a$ is a multiplier bounded by one and supported in $|x| \ge cr/h$, then
\[
 \|aD_+^{-1/2}\| + \|D_+^{-1/2}a^*\| \le C_c(1 + r^{-1}) .
\]
Finally, $D_{\rm nor} \le 2\bar D_R$ and $D_{\rm nor} \le 2\bar D_L$ as
quadratic forms.
\end{lemma}

\begin{proof}
The function $u \mapsto (1 - e^{-2u})^{-1}$ is comparable, with absolute
constants, to $1 + u^{-1}$ on $u > 0$. By spectral calculus, it follows
that
\[
 \|D_+^{-1/2}f\|^2 \asymp \|f\|^2 + \underline z^{-1}\|(\mathbf N + 1)^{-1/2}f\|^2 .
\]
The form norm $\|(\mathbf N + 1)^{1/2}f\|$ is equivalent to
$\|\nabla f\| + \||x|f\| + \|f\|$, and the product rule bounds
multiplication by $b$ on the form domain by
$C(\|b\|_\infty + \|\nabla b\|_\infty)$. By duality the same bound
holds on the dual space, with the norm $\|(\mathbf N + 1)^{-1/2}f\|$, and
the first estimate follows from the displayed equivalence.

The form inequality $|x|^2 \le C_d(\mathbf N + 1)$ implies that
$\|a(\mathbf N + 1)^{-1/2}\| \le C_ch/r$. Combining this with
$D_+^{-1/2} \le C(\Id + \underline z^{-1/2}(\mathbf N + 1)^{-1/2})$, which
follows from the scalar inequality above by spectral calculus, we
obtain the second estimate.

For the last assertion we use the identity
\[
 \bar D_R = (1 - \chi_{r/h}^2)
 + \chi_{r/h}\mathbf M^*(1 - \chi_{r/h}^2)\mathbf M\chi_{r/h}
 + \chi_{r/h}D_{\rm nor}\chi_{r/h} ,
\]
and the same identity for $\bar D_L$ with $\mathbf M$ and $\mathbf M^*$
exchanged, which holds since $\mathbf M\mathbf M^* = \mathbf M^*\mathbf M$.
Since $0 \le D_{\rm nor} \le \Id$, we have
\[
 \|D_{\rm nor}^{1/2}f\| \le \|D_{\rm nor}^{1/2}\chi_{r/h}f\| + \|(1 - \chi_{r/h})f\| .
\]
Squaring and using $(1 - \chi_{r/h})^2 \le 1 - \chi_{r/h}^2$, we obtain
$D_{\rm nor} \le 2\bar D_R$, and likewise $D_{\rm nor} \le 2\bar D_L$.
\end{proof}

We call the operators $x_j$ and $\partial_j$ letters. An oscillator
polynomial of degree $s$ is a finite sum of products of at most $s$
letters with fixed finite matrices on $\mathcal H_{\rm odd}$ as
coefficients, and we let $O_s$ denote such a polynomial. All bounds below
are uniform over a fixed bounded set of coefficients.

\begin{lemma}\label{l:tube-polynomial}
For every oscillator polynomial $O_s$ of degree at most $s$,
\[
 \|D_+^{-1/2}O_s\mathbf MD_+^{-1/2}\| \le C_s\underline z^{-\max(1, s/2)} ,
\]
and the same bound holds with the factors in the opposite order.
\end{lemma}

\begin{proof}
In creation and annihilation coordinates, $O_s$ changes the number by
one of finitely many integers, bounded in terms of $s$ and
$\mathcal H_{\rm odd}$, and its norm on the eigenspace of $\mathbf N$
with eigenvalue $n$ is at most $C_s(n + 1)^{s/2}$, since
$\|a_jP_n\| \le \sqrt n$ and $\|a_j^*P_n\| \le \sqrt{n + 1}$ for the
spectral projections $P_n$ of $\mathbf N$. The finitely many shifted
values $n' + 1$ are comparable to $n + 1$, and so are the scalar weights
$1 - e^{-2\underline z(n + 1)}$. On the eigenspace with eigenvalue $n$,
$\mathbf M$ has norm at most $e^{-\underline zn}$. Each component of
fixed number shift is bounded by a constant times
\[
 \sup_{n \ge 0}\frac{(n + 1)^{s/2}e^{-\underline zn}}{1 - e^{-c_s\underline z(n + 1)}} .
\]
For $\underline z(n + 1) \le 1$ this supremum is bounded by
$C_s\underline z^{-1}(n + 1)^{s/2 - 1}$, and for the complementary range
the exponential gives $C_s\underline z^{-s/2}$. Taking the maximum
proves the bound, and taking adjoints proves the opposite order.
\end{proof}

We will also use the following interpolation inequality. If $D$ is a
bounded positive operator with bounded inverse and $X$ is bounded, then
\begin{equation}\label{e:tube-interpolation}
 \|D^{-1/2}XD^{-1/2}\| \le \max\bigl(\|D^{-1}X\|, \|XD^{-1}\|\bigr) .
\end{equation}
Indeed, the function $\xi \mapsto D^{-\xi}XD^{\xi - 1}$ is bounded and
analytic on the strip $0 \le \Re\xi \le 1$, and its norm is at most
$\|XD^{-1}\|$ on $\Re\xi = 0$ and at most $\|D^{-1}X\|$ on $\Re\xi = 1$;
the three-lines theorem gives \eqref{e:tube-interpolation} at
$\xi = 1/2$. It applies to $D = D_+$, since $D_+ \ge 1 - e^{-2\underline z}$.

\subsection{Normal charges and words}

We call an oscillator polynomial $\mathfrak q_a$ of degree at most one
with finite fermionic coefficients an odd charge, and an operator
\[
 \mathfrak j_a = \sum_{i, j}a_{ij}x_i\partial_j + \mathfrak j_{a, F} ,
 \qquad a_{ij} = -a_{ji} ,
\]
with a fixed finite matrix $\mathfrak j_{a, F}$, a compact charge. We
assume that the charges are the normal charges of
\zcref{s:tube-operator}, that is, the actions of the stabilizer
generators. Under this assumption they preserve every $\mathbf N_\sigma$.
Hence they commute with $\mathbf M$ and with $\mathbf P_0$, they annihilate
$\Omega_0$ and its dual, and they commute with the invariant cutoff
$\chi_{\rm inv}$, which is a function of the invariant $\Str(hZ)^2$. We call a
finite product $\mathbf w$ of charges a word, and its number of factors
its degree. Since $\mathfrak j_a$ is quadratic in the letters, its norm on
the eigenspace with eigenvalue $n$ grows like $n$, and the cutoff is
needed to bound its coordinate factor.

We now count, heuristically, the powers of $h$ in a word; the next
lemma makes the count precise. In \zcref{s:stab}, each charge carries the factor
$h^2$, which is the size of the endpoint increment of one link. On the
number levels $n \lesssim h^{-2}$, where the Mehler factor is not small,
an odd charge costs $h^{-1}$, and a compact charge costs
$(r/h)h^{-1} = rh^{-2}$, since the cutoff bounds its coordinate factor
by $r/h$ and its derivative costs $h^{-1}$. A Mehler factor between two
inverse half-defects $D_+^{-1/2}$ absorbs a polynomial of degree $s$ at
the cost $h^{-\max(2, s)}$, by \zcref{l:tube-polynomial}. It thus supplies
the two inverse half-defects, each of size $h^{-1}$, at no extra cost
once $s \ge 2$. For a word with $a$ odd and $b$ compact charges and
$a + b \ge 2$, the count is
\[
 h^{2(a + b)}h^{-a}(rh^{-2})^b = h^ar^b .
\]

\begin{lemma}\label{l:tube-word}
Let $\mathbf w$ be a fixed word with $a$ odd charges and $b$ compact
charges, and let $n = a + b \ge 2$. Then
\[
 \|D_+^{-1/2}h^{2n}\chi_{\rm inv}\,\mathbf w\,\mathbf M\chi_{\rm inv}D_+^{-1/2}\|
 \le C_\mathbf w h^ar^b + C_\mathbf w(r)h ,
\]
and the adjoint satisfies the same bound. On the range of $\mathbf P_0^\perp$, the
same estimate holds with $D_{\rm nor}^{-1/2}\mathbf P_0^\perp$ in place of
$D_+^{-1/2}$. The vacuum row and column vanish:
\begin{equation}\label{e:tube-word-vacuum}
 \mathbf P_0\chi_{\rm inv}\,\mathbf w\,\mathbf M\chi_{\rm inv}
 = \chi_{\rm inv}\,\mathbf w\,\mathbf M\chi_{\rm inv}\mathbf P_0 = 0 .
\end{equation}
For a single odd charge with the factor $h^4$,
$\|D_+^{-1/2}h^4\chi_{\rm inv}\mathfrak q_a\mathbf M\chi_{\rm inv}D_+^{-1/2}\| \le Ch^2$,
and for a single compact charge the corresponding bound is $C(rh + h^2)$.
\end{lemma}

\begin{proof}
We move the coordinate factor of each compact charge to the left, using
$[\partial_i, x_j] = \delta_{ij}$, and we leave the coordinate factors of
the odd charges in place. Every resulting term is of the form
$x^\alpha O_s$ with $|\alpha| = j \le b$ and $s \le a + j$, where the finite
matrix terms of the compact charges only lower these degrees.
Commuting a coordinate past a derivative removes one coordinate and one
derivative. It therefore preserves this relation, and induction on the degree
of the word proves the expansion.

We now define the multiplier $b_\alpha$ by
\[
 \chi_{\rm inv}x^\alpha = (r/h)^j\,b_\alpha .
\] The matrix-valued
multiplier $b_\alpha$ and its first derivatives are bounded, since
$r/h \ge 1$ and each nilpotent term of $\chi_{\rm inv}$ is a multiplier
of the same kind times a power of $(h/r)^2$. Using \zcref{l:tube-slow} at
both ends and \zcref{l:tube-polynomial}, we bound the contribution of
this term by
\[
 C_\mathbf w h^{2n}(r/h)^jh^{-\max(2, a + j)} .
\]
For $j = b$, we have $\max(2, a + j) = n$, since $a + b = n \ge 2$, and
the bound reduces to $C_\mathbf w h^ar^b$. For $j < b$, the bound carries
the power $h^{a + 2(b - j)}$ with $a + 2(b - j) \ge 2$ if $a + j \ge 2$, and
the power $h^{2n - j - 2}$ with $2n - j - 2 \ge 1$ if $a + j < 2$; thus every such term
is at most $C_\mathbf w(r)h$. The opposite order follows from the same
proof, or by taking adjoints. On the range of $\mathbf P_0^\perp$, the scalar weights of
$D_{\rm nor}$ and $D_+$ are comparable, which gives the assertion with
$D_{\rm nor}^{-1/2}\mathbf P_0^\perp$.

Since every charge commutes with $\chi_{\rm inv}$ and $\mathbf M$, we have
$\mathbf w\mathbf M\chi_{\rm inv} = \chi_{\rm inv}\mathbf M\mathbf w$ and
$\chi_{\rm inv}\mathbf w = \mathbf w\chi_{\rm inv}$. Together with
$\mathbf w \mathbf P_0 = \mathbf P_0\mathbf w = 0$, these identities give \eqref{e:tube-word-vacuum}. For a
single odd charge, the same argument with $s = 1$ and $j = 0$ gives
$h^4\underline z^{-1} \le Ch^2$. A compact charge gives a term with
$j = s = 1$ and a bounded finite matrix term, which are bounded by
$h^4(r/h)\underline z^{-1} \le Crh$ and by $Ch^2$.
\end{proof}

The reference $\mathbf M^\chi$ uses the scalar cutoff $\chi_{r/h}$, since
$\chi_{\rm inv}$ is not a contraction in the positive metric. The next
lemma bounds the difference between the two cutoffs relative to the
normal defect.

\begin{lemma}\label{l:tube-faces}
There is a decomposition
\[
\begin{gathered}
 [\chi_{r/h}, \mathbf M] = \mathbf P_0^\perp \mathbf K_h\mathbf P_0^\perp + \mathbf R_h , \\
 \|D_{\rm nor}^{-1/2}\mathbf P_0^\perp \mathbf K_h\mathbf P_0^\perp D_{\rm nor}^{-1/2}\| \le C(r)h^2 , \\
 \|\mathbf R_h\| \le C(r)h^{-L}e^{-cr^2/h^2} ,
\end{gathered}
\]
for some $L$. The same holds for the operators $b_h\mathbf M\chi_{r/h}$
and $\chi_{r/h}\mathbf Mb_h$, where $b_h$ is a fixed matrix-valued
multiplier supported in a fixed enlargement of the transition annulus
of $\chi_{r/h}$ such that $\|b_h\|_\infty \le C(r)h^2$. In particular, it
holds for the nilpotent terms of $\chi_{\rm inv}$.
\end{lemma}

\begin{proof}
For one even coordinate of species $\sigma$, the kernel of
$e^{-z_\sigma\mathbf N_\sigma}$ is \eqref{e:tube-mehler}, and the even
kernel of $\mathbf M$ is the product of these factors. In the sector we
have $\Re\coth(z/2) \ge c/\underline z$, $\Re\tanh(z/2) \ge c\underline z$
and $|z| \le C\underline z$. Completing the squares, we find that the
absolute row and column integrals of the kernel are bounded, and that
with an additional factor $|x - y|$ they are bounded by
$C\underline z^{1/2}$. The finite odd Mehler factor is bounded.

Let $a$ be an annular multiplier, equal to one on a fixed neighborhood
of the transition annulus of $\chi_{r/h}$. If
$\chi_{r/h}(x) \ne \chi_{r/h}(y)$ and $x$ or $y$ lies outside the support
of $a$, then $|x - y| \ge cr/h$. The part of the kernel with
$|x - y| \ge cr/h$ has Schur norm at most $C(r)h^{-L}e^{-cr^2/(h^2\underline z)}$,
by the Gaussian bound. For the remaining part,
$|\chi_{r/h}(x) - \chi_{r/h}(y)| \le C(h/r)|x - y|$. We therefore have
\[
 [\chi_{r/h}, \mathbf M] = aS_ha + \mathbf R'_h ,
 \qquad
 \|S_h\| \le C(h/r)\underline z^{1/2} \le Ch^2/r ,
\]
with a remainder $\mathbf R'_h$ of the stated Gaussian size. Applying the annulus estimate of
\zcref{l:tube-slow} on both sides, we obtain
\[
 \|D_+^{-1/2}aS_haD_+^{-1/2}\| \le C(h^2/r)(1 + r^{-1})^2 .
\]
On the range of $\mathbf P_0^\perp$ the defects $D_{\rm nor}$ and $D_+$ are
comparable, which implies the bound on
$D_{\rm nor}^{-1/2}\mathbf P_0^\perp \mathbf K_h\mathbf P_0^\perp D_{\rm nor}^{-1/2}$. The vacuum row and
column are Gaussian-small, because
$[\chi_{r/h}, \mathbf M]\mathbf P_0 = (\lambda\chi_{r/h} - \mathbf M\chi_{r/h})\mathbf P_0$,
$\mathbf P_0[\chi_{r/h}, \mathbf M] = \mathbf P_0(\chi_{r/h}\mathbf M - \lambda\chi_{r/h})$, and
$(1 - \chi_{r/h})\Omega_0$ is a Gaussian tail. We put them and $\mathbf R'_h$
into $\mathbf R_h$.

For $b_h\mathbf M\chi_{r/h}$, we localize the input of the Mehler kernel
to an enlarged annulus. The discarded part again has separation $cr/h$
and the same Gaussian bound. The retained kernel has operator norm at
most $C(r)h^2$ by Schur's test. Its output and its input are both
supported in annuli, so the annulus estimate of \zcref{l:tube-slow} on
each side gives the bound with inverse half-defects on both sides, as for
the commutator, and the opposite order follows by taking adjoints. For the
invariant cutoff, finite Taylor expansion in the nilpotent part gives
terms $(h/r)^{2j}\eta^{(j)}(h^2|x|^2/r^2)\,F_j$ with $j \ge 1$, fixed
finite matrices $F_j$ and the same annular support. These terms satisfy
the hypothesis, since $(h/r)^{2j} \le C(r)h^2$.
\end{proof}

\section{Mehler kernels with amplitudes}\label{s:tube-amplitude}

The transfer kernel of \zcref{s:stab} inserts into the Mehler kernel
amplitudes that need not preserve the normal number. We treat them
through exact identities for their kernels. For simplicity we take one
species and first remove the constant scalar $\lambda$, so that
$\lambda^{-1}\mathbf M = e^{-z\mathbf N}$. The scalar is restored by
multiplying the amplitude by $\lambda$; all bounds below are unchanged
since $|\lambda|\le1$. For several species the
Gaussian below is replaced by the product of the Gaussians of the
species. In the midpoint and increment variables
\[
 U = \frac{h(x + y)}{2} , \qquad V = \frac{x - y}{h} ,
\]
the even kernel \eqref{e:tube-mehler} is
\[
\begin{gathered}
 k_z(x, y) = \frac{e^{dz/2}}{(2\pi\sinh z)^{d/2}}
 \exp\bigl\{-\tfrac12c_z|V|^2 - t_z|U|^2\bigr\} , \\
 c_z = \tfrac12h^2\coth(z/2) , \quad t_z = h^{-2}\tanh(z/2) .
\end{gathered}
\]
In the sector we have $\Re c_z, \Re t_z \ge c$ and $|c_z| + |t_z| \le C$.
Here $U$ is the unscaled normal midpoint and $V$ the normalized
increment, and both are of order one under the kernel.

For integers $k, L \ge 0$ and numbers $\varkappa_0, \varkappa_1 \ge 0$, we
let $\mathcal A_{k, L}(\varkappa_0, \varkappa_1)$ denote the class of
$C^k$ maps $a \colon \R^d \times \R^d \to \operatorname{End}(\mathcal H_{\rm odd})$
such that for all $|\alpha| + |\beta| \le k$,
\[
 \|\partial_U^\alpha\partial_V^\beta a(U, V)\| \le \varkappa_1\langle U\rangle^L\langle V\rangle^L ,
 \qquad
 \|\partial_V^\beta a(U, V)\| \le \varkappa_0\langle U\rangle^L\langle V\rangle^L ,
\]
where $\langle u\rangle = (1 + |u|^2)^{1/2}$. We call such maps
amplitudes. Since the second bound includes $\beta = 0$, the parameter $\varkappa_0$
bounds the amplitude itself, while derivatives in the midpoint may cost
$\varkappa_1$. For an amplitude $a$ we define the operator
\[
 (\mathbf M[a]f)(x) = \int k_z(x, y)\,a\Bigl(\frac{h(x + y)}{2}, \frac{x - y}{h}\Bigr)f(y)\dd y .
\]
The odd Mehler factor $e^{-z\mathbf N_F}$ and fixed finite odd
coefficients are absorbed into $a$. For instance, $\mathbf M = \mathbf M[a]$
with $a = \lambda e^{-z\mathbf N_F}$. Finite sums of such operators are treated
in the same way.

\begin{lemma}\label{l:tube-amplitude}
Let $a \in \mathcal A_{k, L}(\varkappa_0, \varkappa_1)$, where $k$ is large
enough depending on the orders $m$ and $s$ below. Constants depend on
$k$, $L$, $m$, $s$, the dimension, $\mathcal H_{\rm odd}$, and the sector.
\begin{enumerate}[label=(\roman*)]
\item We have
$\|\mathbf M[a]\| \le C\sup_{U, V}\langle U\rangle^{-L}\langle V\rangle^{-L}\|a(U, V)\| \le C\varkappa_0$,
and $\mathbf M[a]^* = \mathbf M[a^\sharp]$ with $z$ replaced by $\bar z$
and $a^\sharp(U, V) = a(U, -V)^*$.
\item For all $j$, the following identities hold:
\begin{align*}
 [x_j, \mathbf M[a]] &= h\,\mathbf M[V_ja] ,
 &
 [\partial_j, \mathbf M[a]] &= \mathbf M[b_j] ,
 \\
 x_j\mathbf M[a] &= \mathbf M[(h^{-1}U_j + hV_j/2)a] ,
 &
 \partial_j\mathbf M[a] &= \mathbf M[a^{(j)}] ,
\end{align*}
where $b_j = h\partial_{U_j}a - 2ht_zU_ja$ and
\[
 a^{(j)} = h^{-1}(\partial_{V_j}a - c_zV_ja) + h(\tfrac12\partial_{U_j}a - t_zU_ja) ,
\]
and $[\mathbf N_F, \mathbf M[a]] = \mathbf M[[\mathbf N_F, a]]$.
\item There exists $a_0 \in \operatorname{End}(\mathcal H_{\rm odd})$ such that
\[
 \Bigl\|a_0 - \int\mathfrak g_z(V)\,a(0, V)\dd V\Bigr\| \le Ch^2\varkappa_1 ,
 \qquad
 \|(\mathbf N + 1)^m(\mathbf M[a]\Omega_0 - a_0\Omega_0)\| \le Ch(\varkappa_0 + \varkappa_1)
\]
for every $m \ge 0$, where $\mathfrak g_z$ is the normalized complex
Gaussian \eqref{e:tube-vacuum-gaussian} below and $a_0\Omega_0$ is the
even vacuum times $a_0$ applied to the odd vacuum. In particular, if
$a(0, \cdot)$ has vanishing Gaussian moment, then
$\|(\mathbf N + 1)^m\mathbf M[a]\Omega_0\| \le Ch(\varkappa_0 + \varkappa_1)$.
\item For all integers $m$ and oscillator polynomials $O_s$ of degree at
most $s$,
\begin{gather*}
 \|(\mathbf N + 1)^mO_s\mathbf M[a](\mathbf N + 1)^{-m}\| \le Ch^{-s}(\varkappa_0 + h\varkappa_1) ,
 \\
 \|D_+^{-1/2}O_s\mathbf M[a]\| + \|O_s\mathbf M[a]D_+^{-1/2}\| \le Ch^{-\max(1, s)}(\varkappa_0 + h\varkappa_1) ,
 \\
 \|D_+^{-1/2}O_s\mathbf M[a]D_+^{-1/2}\| \le Ch^{-\max(2, s)}(\varkappa_0 + h\varkappa_1) .
\end{gather*}
In particular, the estimate of \zcref{l:tube-word} on the range of
$\mathbf P_0^\perp$ remains true with $\mathbf M$ replaced by $\mathbf M[a]$ and
multiplied by $C(\varkappa_0 + h\varkappa_1)$, and its vacuum row and
column are given by part (iii).
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with (i). The row and column integrals of
$|k_z(x, y)|\langle U\rangle^L\langle V\rangle^L$ are bounded, since the
increment Gaussian has width $h$ in $x - y$, the midpoint Gaussian
$e^{-c|U|^2}$ absorbs $\langle U\rangle^L$, and
$|e^{dz/2}(2\pi\sinh z)^{-d/2}| \le Ch^{-d}$. Schur's test gives the bound.
The kernel of $\mathbf M[a]^*$ is
\[
 \overline{k_z(y, x)}\,a(U, -V)^* = k_{\bar z}(x, y)\,a^\sharp(U, V) ,
\]
where we used that $k_z$ is symmetric and $\overline{k_z} = k_{\bar z}$.

We now prove (ii). The kernels of $x_j\mathbf M[a]$ and $\mathbf M[a]x_j$ are
the kernel of $\mathbf M[a]$ multiplied by $x_j$ and by $y_j$, and
$x_j = h^{-1}U_j + hV_j/2$, $y_j = h^{-1}U_j - hV_j/2$. The kernel of
$\partial_j\mathbf M[a]$ is the $x_j$-derivative of the kernel, and
integration by parts shows that the kernel of $\mathbf M[a]\partial_j$ is
minus its $y_j$-derivative. We have
\[
 \partial_{x_j}k_z = -(h^{-1}c_zV_j + ht_zU_j)\,k_z ,
 \qquad
 (\partial_{x_j} + \partial_{y_j})k_z = -2ht_zU_j\,k_z ,
\]
and $\partial_{x_j} = \frac h2\partial_{U_j} + h^{-1}\partial_{V_j}$,
$\partial_{x_j} + \partial_{y_j} = h\partial_{U_j}$ on functions of $(U, V)$.
The identities follow from these relations. All of them are algebraic in $z$,
so they hold in the complex sector.

Next, we prove (iii). We write $\Omega_0 = \Omega_B \otimes \Omega_F$. By the
description of the ground-state transform after \eqref{e:tube-mehler},
substituting $y = x - hV$, we obtain
\[
 \frac{(\mathbf M[a]\Omega_0)(x)}{\Omega_B(x)} = \beta_a(x)\Omega_F ,
 \qquad
 \beta_a(x) = \int\mathfrak g_z(V - \lambda_zx)\,a\bigl(hx - \tfrac12h^2V, V\bigr)\dd V ,
\]
where
\begin{equation}\label{e:tube-vacuum-gaussian}
 \mathfrak g_z(w) = (\pi s_z)^{-d/2}\exp\{-w \cdot w/s_z\} ,
 \qquad s_z = (1 - e^{-2z})/h^2 ,
 \qquad \lambda_z = (1 - e^{-z})/h = O(h) .
\end{equation}
In the sector $\Re(1/s_z) \ge c$, so
$|\mathfrak g_z(V - \lambda_zx)| \le Ce^{-c|V|^2 + Ch^2|x|^2}$. We set
$a_0 = \beta_a(0)$. The first bound in (iii) follows from
$|U| \le h^2|V|$ at $x = 0$. Each derivative of $\beta_a$ brings out a
factor $\lambda_z$ or $h\partial_U$, which gives, for $|\nu| \ge 1$,
\[
 |\partial_x^\nu\beta_a(x)| \le Ch(\varkappa_0 + \varkappa_1)\langle x\rangle^Le^{Ch^2|x|^2} .
\]
Writing
\[
 \beta_a(x) - \beta_a(0) = \int_0^1x \cdot \nabla\beta_a(tx)\dd t ,
\]
we find the same bound for $\beta_a - \beta_a(0)$, with $L + 1$ in place
of $L$. Since $(\mathbf N + 1)^m$ is a differential operator of order $2m$
with polynomial coefficients and $e^{-|x|^2/2 + Ch^2|x|^2} \le e^{-|x|^2/4}$
for small $h$, the second bound in (iii) follows.

It remains to prove (iv). We first record how the identities of (ii)
change the class. Let
$b \in \mathcal A_{k, L}(\varkappa_0', \varkappa_1')$ and let $\ell$ be a
letter. A commutator gains a factor $h$:
\[
 [\ell, \mathbf M[b]] = \mathbf M[b'] ,
 \qquad
 b' \in \mathcal A_{k - 1, L + 1}\bigl(Ch(\varkappa_0' + \varkappa_1'), Ch\varkappa_1'\bigr) ,
\]
and $[\mathbf N_F, \mathbf M[b]] = \mathbf M[[\mathbf N_F, b]]$ with
$[\mathbf N_F, b] \in \mathcal A_{k, L}(C\varkappa_0', C\varkappa_1')$. A
letter absorbed into the kernel on the left costs a factor $h^{-1}$:
\[
 \ell\,\mathbf M[b] = \mathbf M[b''] ,
 \qquad
 b'' \in \mathcal A_{k - 1, L + 1}\bigl(C(h^{-1}\varkappa_0' + h\varkappa_1'), Ch^{-1}\varkappa_1'\bigr) ,
\]
and the same holds on the right, since $\mathbf M[b]\ell = \ell\mathbf M[b] - [\ell, \mathbf M[b]]$.

We first bound $(\mathbf N + 1)^m\mathbf M[b](\mathbf N + 1)^{-m}$ for an
integer $m \ge 1$. Writing $\operatorname{ad}_X(Y) = [X, Y]$, we have
\[
 (\mathbf N + 1)^m\mathbf M[b](\mathbf N + 1)^{-m}
 = \sum_{i \le m}\binom mi\operatorname{ad}_{\mathbf N + 1}^i(\mathbf M[b])(\mathbf N + 1)^{-i} .
\]
Since $[\mathbf N, x_j] = -\partial_j$ and $[\mathbf N, \partial_j] = -x_j$,
the commutator rule shows by induction that
$\operatorname{ad}_{\mathbf N + 1}^i(\mathbf M[b])$ is a finite sum of terms
$\Lambda_1\mathbf M[b_\Lambda]\Lambda_2$, where $\Lambda_1$ and $\Lambda_2$
are products of letters of total degree at most $i$ and the
coefficients of $b_\Lambda$ satisfy
$\varkappa_0(b_\Lambda) + h\varkappa_1(b_\Lambda) \le C(\varkappa_0' + h\varkappa_1')$.
Moving the letters of $\Lambda_1$ to the right by the commutator rule
preserves this form with $\Lambda_1 = \Id$. Since a product of at most
$2i$ letters is bounded relative to $(\mathbf N + 1)^i$, part (i) implies
that
\[
 \|(\mathbf N + 1)^m\mathbf M[b](\mathbf N + 1)^{-m}\| \le C(\varkappa_0' + h\varkappa_1') .
\]
For negative $m$ we apply this to $\mathbf M[b]^* = \mathbf M[b^\sharp]$
and take adjoints.

Now let $O_s$ be given. Absorbing its letters on the left, we have
$O_s\mathbf M[a] = \mathbf M[a_s]$ with
$\varkappa_0(a_s) \le Ch^{-s}(\varkappa_0 + h^2\varkappa_1)$ and
$\varkappa_1(a_s) \le Ch^{-s}\varkappa_1$. The first bound in (iv) then
follows from the previous paragraph. For the third, we use
\eqref{e:tube-interpolation} with $X = O_s\mathbf M[a]$, the bound
$\|D_+^{-1}Y\| \le C(\|Y\| + h^{-2}\|(\mathbf N + 1)^{-1}Y\|)$, which
holds by spectral calculus, and its mirror image on the right. We have
\[
 \|O_s\mathbf M[a]\| \le Ch^{-s}(\varkappa_0 + h^2\varkappa_1) ,
 \qquad
 \|(\mathbf N + 1)^{-1}O_s\mathbf M[a]\| \le Ch^{-\max(s - 2, 0)}(\varkappa_0 + h^2\varkappa_1) ,
\]
where the second bound uses that $(\mathbf N + 1)^{-1}$ times a product
of two letters is bounded.

We now bound $\|O_s\mathbf M[a](\mathbf N + 1)^{-1}\|$.
We move the letters of
$O_s$ to the right through $\mathbf M[a]$, one at a time, by
$\ell\mathbf M[b] = \mathbf M[b]\ell + [\ell, \mathbf M[b]]$. This writes
$O_s\mathbf M[a]$ as a sum of terms $\mathbf M[b]\Lambda$, where $\Lambda$
is a product of $p \le s$ letters and $b$ has gone through $s - p$
commutators, so that $\varkappa_0(b) + h^2\varkappa_1(b) \le Ch^{s - p}(\varkappa_0 + \varkappa_1)$
if $p < s$, while $b = a$ if $p = s$. In each term $(\mathbf N + 1)^{-1}$
absorbs the last two letters of $\Lambda$, and the other letters are
absorbed into $\mathbf M[b]$ on the right, at the cost $h^{-\max(p - 2, 0)}$.
Since $s - p - \max(p - 2, 0) \ge 1 - \max(s - 2, 0)$ for $p < s$, we
obtain
\[
 \|O_s\mathbf M[a](\mathbf N + 1)^{-1}\| \le Ch^{-\max(s - 2, 0)}(\varkappa_0 + h\varkappa_1) ,
\]
and the third bound in (iv) follows. The second bound follows in the
same way from
\[
 \|D_+^{-1/2}Y\| \le C\bigl(\|Y\| + h^{-1}\|(\mathbf N + 1)^{-1/2}Y\|\bigr) ,
\]
its mirror image, and the boundedness of $(\mathbf N + 1)^{-1/2}$ times
one letter. The last assertion of (iv) follows from the proof
of \zcref{l:tube-word}, with this bound in place of
\zcref{l:tube-polynomial}.
\end{proof}

For a word with interleaved amplitudes, part (ii) gives explicit
formulas for the commutators. For a compact charge, the Gaussian factor
of $k_z$ is invariant under simultaneous rotations of $x$ and $y$, and
we have
\begin{equation}\label{e:tube-compact-commutator}
 \Bigl[\sum_{i, j}a_{ij}x_i\partial_j, \mathbf M[a]\Bigr]
 = \frac h2\sum_{i, j}a_{ij}\bigl(x_i\mathbf M[\partial_{U_j}a] + \mathbf M[\partial_{U_j}a]x_i
 + \partial_j\mathbf M[V_ia] + \mathbf M[V_ia]\partial_j\bigr) .
\end{equation}
To prove this, note that by (ii) the left side equals $\mathbf M[b]$ with
$b = \sum_{i, j}a_{ij}(U_i\partial_{U_j} + V_i\partial_{V_j})a$, since the
terms $t_zU_iU_ja$, $c_zV_iV_ja$, and $\delta_{ij}a$ cancel by
antisymmetry. Also by (ii), we have
\[
 \mathbf M[U_ib'] = \tfrac h2(x_i\mathbf M[b'] + \mathbf M[b']x_i) ,
 \qquad
 \mathbf M[\partial_{V_j}b' - c_zV_jb'] = \tfrac h2(\partial_j\mathbf M[b'] + \mathbf M[b']\partial_j) .
\]
Applying these with $b' = \partial_{U_j}a$ and $b' = V_ia$, and using
antisymmetry once more to remove the terms with $c_z$, we obtain
\eqref{e:tube-compact-commutator}. For an odd charge
$\mathfrak q_a = \sum_j(F_jx_j + G_j\partial_j)$ with finite matrices
$F_j$, $G_j$, part (ii) gives
\begin{equation}\label{e:tube-odd-commutator}
 [\mathfrak q_a, \mathbf M[a]] = \sum_j\bigl(hF_j\mathbf M[V_ja] + \mathbf M[[F_j, a]]x_j
 + G_j\mathbf M[b_j] + \mathbf M[[G_j, a]]\partial_j\bigr) ,
\end{equation}
where $[\cdot, \cdot]$ is the graded commutator. In both cases the
commutator is $h$ times operators of the same kind, with at most one
letter on either side, except for the matrix commutators $[F_j, a]$ and
$[G_j, a]$, which \zcref{l:stab-moment} controls in \zcref{s:stab} for the
amplitudes that occur there.

\subsection{Finite odd Gaussian increments}

The odd increments in these kernels are handled by one finite Gaussian
computation, which we carry out before taking any norm. Multiplying the
Berezin kernel of an operator $X$ of parity $|X|$ by $\theta - \theta'$,
where $\theta$ and $\theta'$ are the copies of an odd coordinate at the
two endpoints, gives the kernel of the graded commutator
$[\theta, X] = \theta X - (-1)^{|X|}X\theta$. A product of normalized
odd increments $(\theta_a - \theta_a')/\kappa_{\rm o}$ corresponds to
an iterated graded commutator divided by a power of $\kappa_{\rm o}$. We will
use $\kappa_{\rm o} = h$ for the normal odd variables, whose Mehler factor
runs for a time of order $h^2$, and $\kappa_{\rm o} = h^2$ for the tangent odd
variables, whose Gaussian is the odd heat over the time $\delta_W$.

\begin{lemma}\label{l:tube-odd-gaussian}
Let $\mathcal H_0$ be a finite-dimensional fermionic Fock space, let
$0 < \kappa_{\rm o} \le 1$, and let $\mathbf M_0 = e^{-\mathfrak h_0}$, where
$\mathfrak h_0$ is an even element of the algebra of canonical
anticommutation relations of $\mathcal H_0$, quadratic in the creation
and annihilation operators, with coefficients bounded by
$C_0\kappa_{\rm o}^2$. Let $\theta_1, \dots, \theta_n$ be pairwise
anticommuting linear forms in these operators, with coefficients bounded
by $C_0$. For a sequence $\mathbf a = (a_1, \dots, a_l)$ of indices we set
\[
 \Delta_{\mathbf a}\mathbf M_0
 = \kappa_{\rm o}^{-l}\operatorname{ad}_{\theta_{a_1}}\cdots\operatorname{ad}_{\theta_{a_l}}\mathbf M_0 ,
\]
with graded commutators. Then $\Delta_{\mathbf a}\mathbf M_0$ is a finite
sum, over the partial pairings $\pi$ of $\{1, \dots, l\}$, of terms
\[
 \pm \prod_{\{b, b'\} \in \pi}c_{a_ba_{b'}} \cdot \prod_{d \notin \cup \pi}\lambda_{a_d} \cdot \mathbf M_0 ,
\]
where the $c_{ab}$ are numbers with $|c_{ab}| \le C$, the $\lambda_a$ are
linear forms with coefficients bounded by $C\kappa_{\rm o}$, the unpaired
factors are ordered as in $\mathbf a$, and $C$ depends only on $C_0$, $n$
and $\dim\mathcal H_0$. Moreover $\|\mathbf M_0 - \Id\| \le C\kappa_{\rm o}^2$. In
particular, $\|\Delta_{\mathbf a}\mathbf M_0\| \le C_l$,
$\|\Delta_{\mathbf a}\mathbf M_0\| \le C_l\kappa_{\rm o}$ if $l$ is odd, and
$\Delta_{\mathbf a}\mathbf M_0 = c_{\mathbf a}\Id + O(\kappa_{\rm o})$ with a
number $c_{\mathbf a}$.
\end{lemma}

Thus a pair of normalized odd increments has a bounded covariance, and
every unpaired normalized increment carries a factor $\kappa_{\rm o}$.

\begin{proof}
We introduce Grassmann sources $\xi_1, \dots, \xi_n$ that anticommute
with the $\theta_a$, and we set
\[
 L_\xi = \sum_a\xi_a\theta_a ,
\]
which is even. Then
\[
 \operatorname{ad}_{L_\xi}X = \sum_a\xi_a[\theta_a, X]
\]
for all $X$. Hence the coefficient of $\xi_{a_1}\cdots\xi_{a_l}$ in
\[
 e^{L_\xi/\kappa_{\rm o}}\mathbf M_0e^{-L_\xi/\kappa_{\rm o}}
 = \sum_{l \ge 0}\frac1{l!}\kappa_{\rm o}^{-l}\operatorname{ad}_{L_\xi}^l\mathbf M_0
\]
is $\pm \Delta_{\mathbf a}\mathbf M_0$, where we used that the iterated
commutators are antisymmetric in their indices, since the $\theta_a$
anticommute.

We now compute this conjugation. Since $\mathfrak h_0$ is quadratic, the
operator
\[
 L_\xi' = \mathbf M_0L_\xi\mathbf M_0^{-1} = e^{-\operatorname{ad}_{\mathfrak h_0}}L_\xi
\]
is again linear in the creation and annihilation operators, and the
coefficients of $L_\xi - L_\xi'$ are $O(\kappa_{\rm o}^2)$. The anticommutator of two linear
forms is a number. Therefore $[L_\xi, L_\xi']$ is a quadratic polynomial
in the sources with numerical coefficients, it is central, and
$[L_\xi, L_\xi'] = [L_\xi, L_\xi' - L_\xi]$ has coefficients
$O(\kappa_{\rm o}^2)$. The Baker--Campbell--Hausdorff formula with a central
commutator gives
\[
 e^{L_\xi/\kappa_{\rm o}}\mathbf M_0e^{-L_\xi/\kappa_{\rm o}}
 = e^{L_\xi/\kappa_{\rm o}}e^{-L_\xi'/\kappa_{\rm o}}\mathbf M_0
 = \exp\Bigl\{\frac{L_\xi - L_\xi'}{\kappa_{\rm o}} - \frac{[L_\xi, L_\xi']}{2\kappa_{\rm o}^2}\Bigr\}\mathbf M_0 .
\]
We write the first term in the exponent as $\sum_a\xi_a\lambda_a$, with
$\lambda_a = O(\kappa_{\rm o})$, and the second as
$\frac12\sum_{a, b}c_{ab}\xi_a\xi_b$, with numbers $c_{ab} = O(1)$.
Expanding the exponential, which is a polynomial in the finitely many
sources, and extracting the coefficient of $\xi_{a_1}\cdots\xi_{a_l}$,
we obtain the sum over partial pairings. Finally,
$\|\mathbf M_0 - \Id\| \le C\kappa_{\rm o}^2$ since $\mathfrak h_0 = O(\kappa_{\rm o}^2)$
on the finite-dimensional space $\mathcal H_0$.
\end{proof}

\section{Graph powers of the frozen normal transfer}\label{s:tube-powers}

\subsection{Four-term bounds}

All perturbation estimates in this section and the next have the same
form. Let $C$ be a contraction on a Hilbert space with the defect
operators $D_R = \Id - C^*C$ and $D_L = \Id - CC^*$, as in
\zcref{l:tools-defect}, and let $\mathbf T$ be a bounded operator. We say that
$\mathbf T$ satisfies a four-term bound relative to $C$, with the parameters
$a, A_1 \ge 0$ and $0 < \delta_0 \le 1$, if for all vectors $f$ and $g$,
\begin{equation}\label{e:tube-four-term}
 |\langle g, (\mathbf T - C)f\rangle|
 \le a\|D_L^{1/2}g\|\|D_R^{1/2}f\|
 + A_1\sqrt{\delta_0}\bigl(\|g\|\|D_R^{1/2}f\| + \|D_L^{1/2}g\|\|f\|\bigr)
 + A_1\delta_0\|g\|\|f\| .
\end{equation}
We call the first term the bilateral term, the second the one-sided
terms, and the third the scalar term.

\begin{remark}\label{r:tube-four-term-reduction}
The bound \eqref{e:tube-four-term}
is the hypothesis \eqref{e:tools-placements} of \zcref{c:tools-placements}
with $\mathbf T$ in place of $T$ and $A_0 = A_1$, and we use that corollary
in this form. Its constant $A$
of \eqref{e:tools-placements-B} is then $B = \max\{1, 2304A_1^2\}$. If
$a \le 1/16$ and $B\delta_0 \le 1$, the corollary shows that $\mathbf T$ satisfies
\eqref{e:tools-defect-B} with $B/2$ in place of $A_1$, that is,
\[
 \mathbf T^*\mathbf T \le (1 + B\delta_0/2)\Id - \tfrac12D_R ,
 \qquad
 \mathbf T\mathbf T^* \le (1 + B\delta_0/2)\Id - \tfrac12D_L ,
\]
and \zcref{l:tools-defect}(b) then gives $\|\mathbf T^n\| \le e^{Bt/4}$ whenever
$n\delta_0 \le t$.
\end{remark}

\subsection{The frozen normal transfer with cutoffs}

In the normal gauge \eqref{e:tube-table}, the frozen normal
transfer with cutoffs is
\[
 \mathbf F_h = w_{\rm anh}^{1/2}\chi_{\rm inv}\mathbf M_h^{\rm q}\chi_{\rm inv}w_{\rm anh}^{1/2} .
\]
We call the left factor $\chi_{\rm inv}$ the \emph{output cutoff} and the right factor the \emph{input cutoff}; the same names are used for the cutoffs of $A_h$ in later sections. We write $\mathbf F_h$ as $(\Id + \mathfrak b_{+, h})\chi_{\rm inv}\mathbf M\chi_{\rm inv}(\Id + \mathfrak b_{-, h})$,
with $\mathbf M = \mathbf M_h^{\rm q}$, where
$\mathfrak b_{\pm, h} = w_{\rm anh}^{1/2} - 1$, extended as in
\zcref{s:tube-gauge}, are smooth matrix-valued multipliers supported in
$|hx| \le 3r$.

The structure of these multipliers comes from the Taylor expansion at
$U = 0$ of the phase $\phi_{\rm anh} = \log w_{\rm anh}^{1/2}$. Recall from
\zcref{s:rep-heuristics} that $w_0 = e^{-\mathcal F}$ with
$\mathcal F(S) = \frac12\Str S^2 + \Str\log(E - S)$, so that
\[
 \log w_0^{1/2}(R_* + U) = \log w_{\rm quad}^{1/2}(U) + \phi_{\rm anh}(U) .
\]
We claim that $\phi_{\rm anh}$ vanishes to third order at $U = 0$, in all normal
coordinates, even and odd. The term of degree zero vanishes because both physical half-weights
$w_0^{1/2}$ and $w_{\rm quad}^{1/2}$ have the same value
$\zeta_p^{1/2}$ at the root. The first variation
of $\mathcal F$ at $S$ in the direction $U$ satisfies
\[
 \frac{\dd}{\dd t}\mathcal F(S + tU)\Big|_{t = 0} = \Str\bigl((S - (E - S)^{-1})U\bigr) ,
\]
which vanishes at $S = R_*$, since
$(E - R_*)^{-1} = R_*$ by \eqref{e:rep-saddle-inverse}; hence the term of
degree one vanishes. By definition, $\log w_{\rm quad}^{1/2}$ contains
the constant term and the term of degree two. Since $w_0$ is even, $\phi_{\rm anh}$ contains only monomials of even
degree in the odd normal coordinates, and we write
$\phi_{\rm anh} = \sum_{j \ge 0}\phi_{{\rm anh}, 2j}$, where $\phi_{{\rm anh}, 2j}$ has degree $2j$ in them.
Each normal coordinate carries a factor $h$ in the standardized coordinates
$U = hZ$. Measuring the odd coefficients in the monomials of the
standardized odd coordinates, as in \zcref{d:grass-norm} with the weight
$h$, we therefore obtain
\begin{equation}\label{e:tube-phase}
 |\phi_{{\rm anh}, 0}| \le C|hx|^3 , \qquad |\phi_{{\rm anh}, 2}| \le Ch^2|hx| , \qquad
 |\phi_{{\rm anh}, 2j}| \le Ch^4 \quad (j \ge 2) ,
\end{equation}
where $hx$ is the unscaled even normal coordinate. In words, the remaining
body phase $\phi_{{\rm anh}, 0}$ starts in degree three, its coefficient of degree two
in the normal odd variables starts with $h^2$ times one body coordinate,
and its terms of higher odd degree are of order $h^4$. Since
$\mathfrak b_{\pm, h} = e^{\phi_{\rm anh}} - 1$ and $\phi_{\rm anh}$ is bounded on the tube,
\zcref{l:grass-norm}\ref{i:grass-norm-sub} gives the same bounds for the
coefficients of $\mathfrak b_{\pm, h}$. The next lemma states the
consequences that we need as hypotheses, and the paragraph after its proof
verifies them.

\begin{lemma}\label{l:tube-powers}
Let $\mathbf F_h = (\Id + \mathfrak b_{+, h})\chi_{\rm inv}\mathbf M\chi_{\rm inv}(\Id + \mathfrak b_{-, h})$,
and suppose that for each integer $m \ge 0$ the multipliers
$\mathfrak b_h \in \{\mathfrak b_{+, h}, \mathfrak b_{-, h}, \mathfrak b_{+, h}^*, \mathfrak b_{-, h}^*\}$
satisfy
\begin{gather*}
 \|(\mathbf N + 1)^m\mathfrak b_h(\mathbf N + 1)^{-m}\| \le Cr^3 + C_m(r)h ,
 \\
 \|(\mathbf N + 1)^m\mathfrak b_h(\mathbf N + 1)^{-m - 1}\| \le Crh^2 + C_m(r)h^3 ,
 \\
 \|(\mathbf N + 1)^m\mathfrak b_h\Omega_0\| \le C_mh^3 ,
 \qquad
 |\langle\Omega_0, \mathfrak b_h\Omega_0\rangle| \le Ch^4 .
\end{gather*}
Fix a finite set of graph orders. Then for all sufficiently small $r$,
and then all sufficiently small $h$, the operators
$\mathbf F_{m, h} = (\mathbf N + 1)^m\mathbf F_h(\mathbf N + 1)^{-m}$ satisfy
\begin{equation}\label{e:tube-graph-defect}
 \|\mathbf F_{m, h}f\|^2 \le (1 + C_m(r)h^4)\|f\|^2 - c_m\|\bar D_R^{1/2}f\|^2 ,
\end{equation}
where $\bar D_R$ is the right defect of the cutoff Mehler operator and
$D_{\rm nor} \le 2\bar D_R$. Consequently
\[
 \|(\mathbf N + 1)^m\mathbf F_h^j(\mathbf N + 1)^{-m}\| \le e^{C_m(r)jh^4}
 \qquad (jh^4 \le T) ,
\]
with a constant depending also on $T$, and
\begin{equation}\label{e:tube-vacuum-residual}
 \|(\mathbf N + 1)^m(\mathbf F_h\Omega_0 - \lambda\Omega_0)\| \le C_mh^3 .
\end{equation}
Moreover, $\mathbf F_{m, h}$ satisfies \eqref{e:tube-four-term} relative to
$\mathbf M^\chi$ with $\delta_0 = h^4$, $a = Cr + C_m(r)h$, and $A_1 = C_m(r)$.
The opposite graph order and the Hilbert adjoint satisfy the
corresponding bounds with the left defect $\bar D_L$.
\end{lemma}

\begin{proof}
We first show that conjugation by $(\mathbf N + 1)^m$ changes the cutoff
only by a small bilateral perturbation. For an integer $m$,
$(\mathbf N + 1)^m\chi_{r/h}(\mathbf N + 1)^{-m} - \chi_{r/h}$ is a finite
sum of the terms $\operatorname{ad}_{\mathbf N}^l(\chi_{r/h})(\mathbf N + 1)^{-l}$
with $1 \le l \le m$. Every such term contains a derivative of
$\chi_{r/h}$, has its output supported in the transition annulus, and
has norm at most $C_mh/r$. This follows from
$[\mathbf N, \chi_{r/h}] = -(\nabla\chi_{r/h}) \cdot \nabla - \frac12\Delta\chi_{r/h}$
and the bound of oscillator polynomials by powers of $\mathbf N + 1$.
The further commutators with $|x|^2$ insert coordinates, which the
factor $(\mathbf N + 1)^{-l}$ pays, and they do not remove the first
factor $h/r$.

After composition with $\mathbf M\chi_{r/h}$, these terms can be
localized on both sides to enlarged transition annuli, up to an error
$C_m(r)h^{-L_m}e^{-c_mr^2/h^2}$. To see this, we use the representation
\[
 (\mathbf N + 1)^{-l} = \frac1{\Gamma(l)}\int_0^\infty t^{l - 1}e^{-t(\mathbf N + 1)}\dd t ,
\]
whose integrand has Mehler kernels at positive times. If an annular
endpoint at distance of order $r/h$ from the origin is separated by
$cr/h$ from the other endpoint, the Mehler Gaussian has exponent at most
$-cr^2/h^2$ after integration over $t$. (For $t \le 1$ we use the
increment exponent, and for $t \ge 1$ the confining exponent
$-c(|x|^2 + |y|^2)$.) Polynomial
factors are absorbed by reducing $c$, and Schur's test gives the
operator bound. Applying \zcref{l:tube-slow} at both annular ends, we
find that the graph-conjugated cutoff differs from the cutoff Mehler
operator by a bilateral perturbation of size $C_m(r)h$ on the range of
$\mathbf P_0^\perp$, plus Gaussian-small vacuum rows. The invariant cutoff
differs from its body by the nilpotent terms of \zcref{l:tube-faces}.
Every graph commutator of such a term keeps its first factor $h^2$.
The corresponding estimate follows from the product rule and the same
localization, and the vacuum rows of these terms are Gaussian-small.

Next, we bound the multipliers. We set
\[
 \mathfrak b_m = (\mathbf N + 1)^m\mathfrak b_h(\mathbf N + 1)^{-m} .
\]
By spectral calculus, we have
$\|D_+^{-1}Y\| \le C(\|Y\| + h^{-2}\|(\mathbf N + 1)^{-1}Y\|)$ and the
mirror bound on the right. Together with the assumed bounds and their
adjoint versions, this implies
\[
 \|\mathfrak b_mD_+^{-1}\| + \|D_+^{-1}\mathfrak b_m\| \le Cr + C_m(r)h ,
 \qquad
 \|D_+^{-1/2}\mathfrak b_mD_+^{-1/2}\| \le Cr + C_m(r)h ,
\]
where the second bound follows from the first and
\eqref{e:tube-interpolation}. The operator
$D_+^{1/2}\mathbf M^\chi D_+^{-1/2}$ is bounded by \zcref{l:tube-slow},
since $\mathbf M$ commutes with $D_+$. Expanding
$\mathfrak b_{+, m}\mathbf M^\chi + \mathbf M^\chi\mathfrak b_{-, m} + \mathfrak b_{+, m}\mathbf M^\chi\mathfrak b_{-, m}$
and adding the cutoff error of the previous paragraph, we find that the
part of $\mathbf F_{m, h} - \mathbf M^\chi$ between two vectors in the range
of $\mathbf P_0^\perp$ satisfies
\[
 |\langle \mathbf P_0^\perp g, (\mathbf F_{m, h} - \mathbf M^\chi)\mathbf P_0^\perp f\rangle|
 \le (Cr + C_m(r)h)\|D_+^{1/2}\mathbf P_0^\perp g\|\|D_+^{1/2}\mathbf P_0^\perp f\| .
\] On the
range of $\mathbf P_0^\perp$, $D_+$ is comparable to $D_{\rm nor}$, and
$D_{\rm nor} \le 2\bar D_L$, $D_{\rm nor} \le 2\bar D_R$ by
\zcref{l:tube-slow}.

The parts $\mathbf P_0(\cdot)\mathbf P_0^\perp$ and $\mathbf P_0^\perp(\cdot)\mathbf P_0$ of
$\mathbf F_{m, h} - \mathbf M^\chi$ have norm
$C_mh^3$ by the assumed vacuum bounds and the Gaussian-small cutoff
rows, and its vacuum scalar $\mathbf P_0(\cdot)\mathbf P_0$ has size $C_mh^4$. Since
$D_{\rm nor} \ge ch^2\mathbf P_0^\perp$, we have
$\|\mathbf P_0^\perp f\| \le Ch^{-1}\|\bar D_R^{1/2}f\|$, and similarly on the
left. Therefore $\mathbf F_{m, h}$ satisfies the four-term bound
\eqref{e:tube-four-term} relative to $\mathbf M^\chi$ with
\[
 \delta_0 = h^4 , \qquad a = Cr + C_m(r)h , \qquad A_1 = C_m(r) ,
\]
up to a Gaussian-small error, which we add to the scalar term. We choose
$r$ and then $h$ so small that $a \le 1/16$ and $Bh^4 \le 1$, with the
constant $B = \max\{1, 2304A_1^2\}$ of \zcref{r:tube-four-term-reduction}.
We then apply \zcref{r:tube-four-term-reduction} with
$\mathbf T = \mathbf F_{m, h}$, $C = \mathbf M^\chi$, $\delta_0 = h^4$, and
$A_1 = C_m(r)$. Since
the right defect of $\mathbf M^\chi$ is $\bar D_R$, its first inequality is
\eqref{e:tube-graph-defect}, with $c_m = 1/2$ and the constant $B/2$ in the
first term, and its power bound with $t = jh^4$ implies the power bound of
the lemma. The
inequality $D_{\rm nor} \le 2\bar D_R$ is \zcref{l:tube-slow}.

For \eqref{e:tube-vacuum-residual}, we expand $\mathbf F_h\Omega_0 - \lambda\Omega_0$
into the two terms with $\mathfrak b_{\pm, h}$ and the term
$\chi_{\rm inv}\mathbf M\chi_{\rm inv}\Omega_0 - \lambda\Omega_0$. The last is a
Gaussian cutoff tail, since $\mathbf M\Omega_0 = \lambda\Omega_0$. By the assumed
bounds, each multiplier term has the size $O_m(h^3)$, and the boundedness of
the other factors in the graph norm of order $m$ preserves this order.
The opposite graph order follows by the same argument for the adjoint,
since
\[
 \bigl((\mathbf N + 1)^m\mathbf F_h^*(\mathbf N + 1)^{-m}\bigr)^* = (\mathbf N + 1)^{-m}\mathbf F_h(\mathbf N + 1)^m .
\]
\end{proof}

\paragraph{Verification of the hypotheses.}
For the analytic half-weight of the onsite weight, the hypotheses of
\zcref{l:tube-powers} reduce to a finite computation at each graph
order. By \eqref{e:tube-phase}, the coefficients satisfy
\[
 |\mathfrak b_h(x)| \le C\bigl(|hx|^3 + h^2|hx| + h^4\bigr) .
\]
On the tube we have $|hx|^3 \le Crh^2|x|^2$, and the bound of oscillator
polynomials by $\mathbf N + 1$ implies the second hypothesis at $m = 0$. In a
commutator with $\mathbf N$, the first derivative of the cubic remainder
carries the factor $h$ times a quadratic unscaled coordinate, its second
derivative the factor $h^2$ times a linear one, and its higher derivatives
carry at least three powers of $h$. The additional factor
$(\mathbf N + 1)^{-1}$ in each graph commutator pays its differential
order. We obtain the leading bound $Crh^2$, and the bound $C_m(r)h^3$ for
all differentiated terms. The commutators with fixed odd matrices have the same
order, since every nonconstant normal odd coefficient of the remainder
starts with $h^2|hx|$. The smooth extension off the tube and the
derivatives of its cutoff satisfy the same bounds with constants that
depend on $r$. Gaussian vacuum moments give $O_m(h^3)$ for the vacuum
column. Since the Gaussian expectation of the term of degree three vanishes by
parity, the vacuum scalar starts with $h^4$.

\section{The normal contraction and the Ward identity}\label{s:tube-gap}

We now prove that the frozen normal transfer contracts the
complement of the Gaussian mean at the rate $h^2$, and that it preserves
the Gaussian mean exactly on the kernel of the supercharge.

Recall from \zcref{s:ov-averaging} why the exact preservation, the Ward
identity, is needed. By \zcref{l:tube-powers}, $\mathbf F_h$ differs from
the Mehler operator by vacuum rows of order $h^3$ and a vacuum scalar of
order $h^4$, so that a perturbative argument places its top eigenvalue
only at $1 + O(h^4)$, an error of order one over the $h^{-4}$ steps of the
chain. The identity $\mathbf P_0\mathbf F_hf = \mathbf P_0f$ for
$f \in \ker\mathfrak q_{\rm nor}$, proved below, removes this error. As for
the transfer operator of \cite{disertori2021density} (see
\zcref{s:intro-sources}), supersymmetry fixes an eigenvalue at one; here
the supercharge acts only on the normal variables. It is used only to
make the closed subspace $\ker\mathfrak q_{\rm nor} \cap \ker\ell_0$
invariant, since the contraction holds on the whole kernel of the Gaussian
mean.

\begin{proposition}\label{p:tube-gap}
Let $\mathbf F_h$ be the frozen normal transfer of
\zcref{s:tube-powers}, including all nilpotent terms of the invariant
cutoff and the analytic normal half-weight. Let $\mathbf P_0$, $\ell_0$ and
$\Omega_0$ be as in \zcref{s:tube-oscillator}, and fix a finite set of
graph orders $m \ge 0$. For a sufficiently small tube and then all
sufficiently small $h$, the following holds in the balanced signature.
The operator $\mathbf F_h$ maps $\ker\mathfrak q_{\rm nor}$ into itself, and
\begin{equation}\label{e:tube-closed-contraction}
\begin{gathered}
 \mathbf P_0\mathbf F_hf = \mathbf P_0f \quad (f \in \ker\mathfrak q_{\rm nor}) , \\
 \mathbf F_h(\ker\mathfrak q_{\rm nor} \cap \ker\ell_0) \subset \ker\mathfrak q_{\rm nor} \cap \ker\ell_0 , \\
 \|\mathbf F_h^jf\| \le e^{-cjh^2}\|f\| \quad (f \in \ker\mathfrak q_{\rm nor} \cap \ker\ell_0) .
\end{gathered}
\end{equation}
For every prescribed graph order $m$,
\begin{equation}\label{e:tube-graph-contraction}
 \|(\mathbf N + 1)^m\mathbf F_h^jf\| \le e^{-c_mjh^2}\|(\mathbf N + 1)^mf\| ,
 \qquad f \in \ker\mathfrak q_{\rm nor} \cap \ker\ell_0 \cap \mathcal G^m ,
\end{equation}
and $\|(\mathbf N + 1)^m\mathbf F_hf\| \le e^{-c_mh^2}\|(\mathbf N + 1)^mf\|$ for
all $f \in \mathcal G^m$ with $\ell_0(f) = 0$. In every signature
$p \ne k$, the exact quadratic transfer is $\lambda\mu_+^{\mathbf N_+}\mu_-^{\mathbf N_-}$ as in
\zcref{l:tube-quadratic}, its defect satisfies
\begin{equation}\label{e:tube-unbalanced-defect}
 \Id - (\mathbf M_h^{\rm q})^*\mathbf M_h^{\rm q} \ge (1 - |\lambda|^2)\Id \ge ch^2\Id
\end{equation}
on the whole normal space of that signature, and
$\|\mathbf F_h^j\| \le Ce^{-cjh^2}$ there. All constants are uniform for
$E$ in a compact subset of $(-2, 2)$, and the tube and the threshold
for $h$ may depend on $k$ and on the chosen graph orders.
\end{proposition}

In \zcref{s:cons}, we obtain from \zcref{p:tube-gap} and Banach's fixed
point theorem the fixed state $\Omega_h$ of $\mathbf F_h$ in $\ker\mathfrak q_{\rm nor}$ with
$\ell_0(\Omega_h) = 1$.

\begin{proof}
We first show that $\mathbf F_h$ preserves $\ker\mathfrak q_{\rm nor}$. Every factor
of $\mathbf F_h$ commutes with $\mathfrak q_{\rm nor}$ on smooth functions with
Gaussian decay. Indeed, the half-weight and the invariant cutoff are
invariant even multipliers, and $\mathfrak q_{\rm nor}$ is an odd derivation that
annihilates invariant functions. The Mehler operator $\mathbf M_h^{\rm q} = \mu_+^{\mathbf N_+}\mu_-^{\mathbf N_-}$
commutes with $\mathfrak q_{\rm nor}$, since $\mathfrak q_{\rm nor}$ preserves the number of
each coordinate by \zcref{l:tube-supercharge}. Now let $g$ be a Wick
polynomial in $\ker\mathfrak q_{\rm nor}$. By the Gaussian kernel and the
graph bounds of \zcref{l:tube-powers}, we have $\mathbf F_hg \in \mathcal G^1$,
and \zcref{l:tube-supercharge} implies that
\[
 \mathfrak q_{\rm nor}\mathbf F_hg = \mathbf F_h\mathfrak q_{\rm nor}g = 0 .
\]
Since $\ker\mathfrak q_{\rm nor}$ is closed, such polynomials are dense in it, and
$\mathbf F_h$ is bounded, $\mathbf F_h$ preserves $\ker\mathfrak q_{\rm nor}$.

Next, we show that $\mathbf F_h$ preserves the Gaussian mean on
$\ker\mathfrak q_{\rm nor}$. Let $f$ be a Wick polynomial in $\ker\mathfrak q_{\rm nor}$. Then
$w_{\rm anh}^{1/2}\chi_{\rm inv}f$ is in $\ker\mathfrak q_{\rm nor}$ and has Gaussian
decay, and \zcref{l:tube-localization} identifies its Gaussian mean with
its value at the origin, which equals $f_{\rm body}(0) = \ell_0(f)$,
since the invariant multipliers equal one at the origin. The Mehler
operator preserves the Gaussian mean, since $\mathbf P_0\mathbf M_h^{\rm q} = \mathbf P_0$.
Applying the same argument to the output multipliers, we obtain
$\mathbf P_0\mathbf F_hf = \mathbf P_0f$. By density and the boundedness of $\mathbf P_0\mathbf F_h$,
this identity extends to $\ker\mathfrak q_{\rm nor}$, and in particular
$\ker\mathfrak q_{\rm nor} \cap \ker\ell_0$ is invariant.

We now prove the contraction, using the graph defect estimate. Recall from \eqref{e:tube-graph-defect} that
\[
 \|\mathbf F_{m, h}f\|^2 \le (1 + C_mh^4)\|f\|^2 - c_m\|\bar D_R^{1/2}f\|^2 ,
\]
and that $D_{\rm nor} \le 2\bar D_R$. Using $\Re z_\sigma \ge ch^2$, we also find that
\[
 D_{\rm nor} = \Id - \mathbf M^*\mathbf M \ge c_0h^2\mathbf P_0^\perp .
\]
Together these give
\begin{equation}\label{e:tube-normal-defect}
 \|\mathbf F_{m, h}f\|^2 \le (1 + C_mh^4 - c_m'h^2)\|f\|^2 \le e^{-2c_mh^2}\|f\|^2 ,
 \qquad \mathbf P_0f = 0 ,
\end{equation}
for small $h$, after renaming $c_m$. Since $\mathbf N$ commutes with $\mathbf P_0$,
the operator $(\mathbf N + 1)^m$ preserves $\ker\ell_0$, and
\eqref{e:tube-normal-defect} implies the one-step bound for all
$f \in \mathcal G^m$ with $\ell_0(f) = 0$. For
$f \in \ker\mathfrak q_{\rm nor} \cap \ker\ell_0 \cap \mathcal G^m$, the iterates
$\mathbf F_h^if$ remain in $\ker\mathfrak q_{\rm nor} \cap \ker\ell_0$ by the
invariance proved above, and in $\mathcal G^m$ since $\mathbf F_{m, h}$ is
bounded. Iterating the one-step bound proves
\eqref{e:tube-graph-contraction}, and the case $m = 0$ completes
\eqref{e:tube-closed-contraction}.

It remains to treat the unbalanced signatures. By
\zcref{l:tube-quadratic}, the scalar of $\mathbf M_h^{\rm q}$ satisfies
$|\lambda| = e^{-(p - k)^2\Re z_+}$. Its Mehler factor is a contraction,
and $(p - k)^2\Re z_+ \ge ch^2$, which proves
\eqref{e:tube-unbalanced-defect}. Thus the normal defect of the physical
quadratic transfer satisfies $D_{\rm nor}\ge ch^2\Id$.
Applying \eqref{e:tube-graph-defect} with $m=0$ and using
$D_{\rm nor}\le2\bar D_R$, we obtain
\[
 \|\mathbf F_h^j\| \le (1 + Ch^4-c'h^2)^{j/2} \le e^{-cjh^2}
\]
for small $h$, as desired.
\end{proof}