\chapter{The sigma model on a wire}\label{s:sigma}

This section defines the continuum object behind our main results, the partition function $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ of \zcref{d:partition}, and proves the general facts about it that the rest of the paper uses. The partition function is the Berezin integral of a heat semigroup on a symmetric superspace $\mathcal M_k$, the target space of the supersymmetric sigma model, and its generator is $X$ times an invariant second-order operator, the Casimir operator, plus a potential that carries the sources. We explain heuristically in \zcref{s:sigma-heuristic} why this object describes the chain of \zcref{s:rep}, and we make it precise in \zcref{d:sigma-partition}. In \zcref{p:sigma-regular}, whose proof is in \zcref{s:regularity}, we construct the semigroup on weighted $L^2$ spaces and show that the partition function is well defined. In \zcref{l:sigma-localization} and \zcref{c:sigma-normalization}, we prove a localization identity of Parisi--Sourlas type and deduce that the partition function equals one without sources. The next section, \zcref{s:sigma-radial}, reduces the case of rank one to two scalar diffusions on the domain $D$ of \zcref{d:kernels}, the radial and the singlet diffusion, develops the calculus of rank-one pairings that Part III uses, and shows in \zcref{l:sigma-jets} how the singlet state $\Psi^{X, z}_t$ and the kernels $K_X$ and $I_X$ arise from the partition function.

Part III uses only these results and those of \zcref{s:sigma-radial}, together with \zcref{t:interface}, and the opening of Part II lists what Part IV takes from this section. From \zcref{s:sigma-target}, the reader of Part III needs only the definitions of the field $Q$ and of the invariant integral, the formula \eqref{e:sigma-casimir} for the Casimir operator $\mathsf C_k$, the three properties of $\mathsf C_k$ in \zcref{l:sigma-casimir}, and the integration by parts formula \eqref{e:sigma-ibp}. The algebraic facts behind these definitions are contained in proofs and remarks, which this reader may skip, as well as \zcref{s:regularity}. After \zcref{d:sigma-partition} and \zcref{l:sigma-chains}, such a reader can go on to the statements of \zcref{s:sigma-localization} and then to \zcref{s:sigma-radial}. At rank one, in \zcref{s:sigma-radial}, everything is explicit. The underlying manifold of $\mathcal M_1$ is a hyperbolic plane times a sphere, and on superfunctions that are invariant under the stabilizer of the base point, $\mathsf C_1$ is an explicit second-order operator on the domain $D$ (\zcref{l:sigma-radial-part}). We use the Grassmann calculus of \zcref{s:grassmann} throughout, and we refer to \zcref{s:overview} for the role of the sigma model in the proof.

\section{Why this target space}\label{s:sigma-heuristic}

We first explain heuristically why the limits of the source ratios $T_N^{(k)}$ should be described by a heat semigroup on a superspace of supermatrices $Q$ with $Q^2 = \Id$. The overview explains the same mechanism in \zcref{s:ov-scales}, and the precise statement is \zcref{t:interface}, which is proved in Part IV. The reduction of band matrices to a one-dimensional supersymmetric sigma model, with the scaling parameter $N/W^2$, is due to Fyodorov and Mirlin \cite{fyodorov1991scaling}; see also \cite[(5.39)--(5.41)]{mirlin2000statistics}, \cite[Sections~2.5.4 and~4.3]{mirlin2000supersymmetry}, and \cite{efetov1997supersymmetry}.

By \zcref{p:rep-exact}, the ratio $T_N^{(k)}$ is a ratio of integrals over chains $S_1, \dots, S_N$ of $(2k | 2k)$ supermatrices. The source-free onsite weight is stationary where $S = (E - S)^{-1}$. Writing $S = \kappa_E - \iu s_E Q$, this equation becomes $s_E^2 (\Id - Q^2) = 0$, so that the critical points are the supermatrices $\kappa_E - \iu s_E Q$ with $Q^2 = \Id$, as in \eqref{e:rep-saddle-inverse}. The chain stays near the balanced component of the saddle manifold of \zcref{s:rep-heuristics}, on which $Q$ lies in the orbit of $\Lambda$ under conjugation, and its tangent variables move along this component. The target space $\mathcal M_k$ is this orbit, with its odd directions, and the balanced component of \zcref{s:rep-heuristics} is the set of supermatrices $\kappa_E - \iu s_E Q$ with $Q \in \mathcal M_k$. To first order, each link moves the tangent variables by a heat semigroup at time $(4 s_E^2 W^2)^{-1}$, since the link factor $e^{-W^2 \Str(S_{i + 1} - S_i)^2/2}$ is a Gaussian in the distance between neighbors, the heat kernel $k_{\tau_{\rm phys}}$ of \zcref{d:rep-chain}. The $N$ links therefore run this semigroup for the time $X_N(E)$ of \eqref{e:ov-time}. Since $N s_E/a_N = \pi$, the source factors of the $N$ sites combine to $\exp(-\pi \int_0^1 \Str(Q(t) \Gamma) \dd t)$, where $Q(t)$ is the tangent variable at the site $t N$.

The odd directions of $\mathcal M_k$ matter for the following reason. By \zcref{r:rep-symmetry}, the onsite weight and the links are invariant under conjugation by all even invertible supermatrices, including those with odd entries. On $\mathcal M_k$, the infinitesimal odd symmetries give the localization identity \zcref{l:sigma-localization}, by which the integral of a superfunction annihilated by an odd symmetry is its value at the base point $\Lambda$. This identity implies the normalization $\mathcal Z^{(k)}_{X, \eps}(0, \eps \1) = 1$, which matches $T_N^{(k)}(0, \eps \1) = 1$, and at rank one it produces the corner terms from which the kernels of \zcref{d:kernels} arise (\zcref{s:sigma-radial}).

A sigma model is a field theory whose field takes values in a symmetric space, here $\mathcal M_k$, and the wire is the interval $[0, 1]$ of scaled positions $t = i/N$ along the chain. Formally, the partition function of \zcref{d:sigma-partition} is a path integral over fields $Q \colon [0, 1] \to \mathcal M_k$, whose transfer operator over a length $t$ of the wire is $e^{-t \mathsf G_{X, \Gamma}}$, with the generator $\mathsf G_{X, \Gamma} = X \mathsf C_k + \pi \Str(Q \Gamma)$ of \eqref{e:sigma-generator}. This picture omits two difficulties, which occupy most of Part IV. The tangent and normal variables of the chain are coupled, and the superfunctions involved have no positivity, so that a Feynman--Kac formula on $\mathcal M_k$ is not available. Further, the chain variables lie on a contour rather than on the saddle manifold, and the weights are complex away from it.

\section{The target space}\label{s:sigma-target}

Throughout this section, $k \ge 1$ is fixed. We first describe the superspace $\mathcal M_k = U(k, k | 2k)/(U(k | k) \times U(k | k))$ through its underlying manifold and its odd coordinates, and we then define the field $Q$, the symmetries, the invariant Berezin integral, and the Casimir operator. We give concrete definitions in global coordinates, and we cite the literature for the facts from the theory of homogeneous supermanifolds that we use. Physical background on these spaces can be found in \cite{efetov1997supersymmetry} and \cite{zirnbauer1996riemannian}.

\subsection{Supermatrices}\label{s:sigma-supermatrices}

We use the supermatrices of \zcref{d:grass-supermatrix} of size $(2k | 2k)$. We index their rows and columns by pairs $(a, \sigma)$, where $a \in [2k]$ is a \emph{flavor} and $\sigma \in \{\mathrm B, \mathrm F\}$ is a \emph{grading}. As in \zcref{s:rep}, the flavors $a \le k$ are retarded and the flavors $a > k$ advanced, and we write $|(a, \sigma)| = 0$ if $\sigma = \mathrm B$ and $|(a, \sigma)| = 1$ if $\sigma = \mathrm F$. For a supermatrix $M$, we let $M^{\mathrm{BB}}$, $M^{\mathrm{BF}}$, $M^{\mathrm{FB}}$, and $M^{\mathrm{FF}}$ denote its four blocks of size $2k \times 2k$, and we let $M_{\mathrm{RR}}$, $M_{\mathrm{RA}}$, $M_{\mathrm{AR}}$, and $M_{\mathrm{AA}}$ denote its four flavor blocks, which are supermatrices of size $(k | k)$. For instance, $M^{\mathrm{FF}}_{\mathrm{RA}}$ is the $k \times k$ matrix of the entries with retarded row flavor, advanced column flavor, and grading $\mathrm F$. Recall the supermatrix $\Lambda = \diag(\Lambda_{\mathrm B} | \Lambda_{\mathrm F})$ of \zcref{s:rep}, with $\Lambda_{\mathrm B} = \Lambda_{\mathrm F} = \diag(\Id_k, -\Id_k)$. We let $\Id_{\mathrm R}$ and $\Id_{\mathrm A}$ denote the diagonal supermatrices with entry one at the retarded, respectively advanced, flavors in each grading, and zero elsewhere. Then $\Lambda = \Id_{\mathrm R} - \Id_{\mathrm A}$.

We let $\mathfrak g$ denote the space of all $(2k | 2k)$ supermatrices with complex entries. We let $\mathfrak g_0$ denote the subspace of supermatrices with vanishing blocks $M^{\mathrm{BF}}$ and $M^{\mathrm{FB}}$, and $\mathfrak g_1$ the subspace with vanishing blocks $M^{\mathrm{BB}}$ and $M^{\mathrm{FF}}$, and we call the elements of $\mathfrak g_0$ even and those of $\mathfrak g_1$ odd. For homogeneous $Y, Y' \in \mathfrak g$ of parities $|Y|$ and $|Y'|$, we define the \emph{bracket} $[Y, Y'] = Y Y' - (-1)^{|Y| |Y'|} Y' Y$, and we extend it bilinearly. We call $\mathfrak g$, with this bracket, the Lie superalgebra $\mathfrak{gl}(2k | 2k)$. All facts about it that we use are proved below. In this section and in \zcref{s:cons-tangent}, the letters $Y$ and $Y'$ denote elements of $\mathfrak g$; the local sources $Y_i$ of \zcref{s:rep} do not occur there. We set
\begin{equation}\label{e:sigma-km}
\mathfrak k = \{ Y \in \mathfrak g : \Lambda Y = Y \Lambda \}, \qquad \mathfrak m = \{ Y \in \mathfrak g : \Lambda Y = - Y \Lambda \},
\end{equation}
and $\mathfrak k_j = \mathfrak k \cap \mathfrak g_j$, $\mathfrak m_j = \mathfrak m \cap \mathfrak g_j$ for $j \in \{0, 1\}$. Thus $\mathfrak k$ consists of the supermatrices whose flavor blocks $Y_{\mathrm{RA}}$ and $Y_{\mathrm{AR}}$ vanish, $\mathfrak m$ of those whose blocks $Y_{\mathrm{RR}}$ and $Y_{\mathrm{AA}}$ vanish, and $\mathfrak g = \mathfrak k \oplus \mathfrak m$. The space $\mathfrak m_1$, of dimension $4k^2$, consists of the odd supermatrices supported on the flavor blocks $\mathrm{RA}$ and $\mathrm{AR}$. We equip $\mathfrak m_1$ with the Hilbert--Schmidt inner product $\langle Y, Y' \rangle = \Tr(Y^* Y')$. For a complex matrix $M$, we write $\|M\|_{\mathrm{op}}$ for its operator norm and $\|M\|_{\mathrm{HS}}$ for its Hilbert--Schmidt norm.

For example, at rank one we order the four indices as $(\mathrm{RB}, \mathrm{RF}, \mathrm{AB}, \mathrm{AF})$, as in \zcref{s:sigma-radial}. Then $\Lambda = \diag(1, 1, -1, -1)$, and the elements of $\mathfrak k$ and $\mathfrak m$ are the matrices
\[
\begin{pmatrix} a_1 & \alpha_1 & 0 & 0 \\ \beta_1 & b_1 & 0 & 0 \\ 0 & 0 & a_2 & \alpha_2 \\ 0 & 0 & \beta_2 & b_2 \end{pmatrix} \in \mathfrak k, \qquad
\begin{pmatrix} 0 & 0 & a_3 & \alpha_3 \\ 0 & 0 & \beta_3 & b_3 \\ a_4 & \alpha_4 & 0 & 0 \\ \beta_4 & b_4 & 0 & 0 \end{pmatrix} \in \mathfrak m ,
\]
with complex entries, where the $a_j$ and $b_j$ occupy the even positions and the $\alpha_j$ and $\beta_j$ the odd ones. Thus $\mathfrak m_0$ and $\mathfrak m_1$ are both four-dimensional, and an element of $\mathfrak m_1$ couples the retarded bosonic index with the advanced fermionic one, and the retarded fermionic index with the advanced bosonic one.

We let $U(k, k)$ denote the group of complex $2k \times 2k$ matrices $g$ with $g^* \Lambda_{\mathrm B} g = \Lambda_{\mathrm B}$, and we set
\[
G_0 = \big\{ \diag(g_{\mathrm B} | g_{\mathrm F}) : g_{\mathrm B} \in U(k, k), \ g_{\mathrm F} \in U(2k) \big\}, \qquad K_0 = \{ g \in G_0 : g \Lambda = \Lambda g \}.
\]
The group $K_0$ consists of the elements of $G_0$ whose flavor blocks $\mathrm{RA}$ and $\mathrm{AR}$ vanish. It is isomorphic to $U(k)^4$, its elements are unitary, and conjugation by $\mathsf k \in K_0$ preserves $\mathfrak m_1$ and its inner product.

\subsection{The underlying manifold}\label{s:sigma-body}

We define $\mathcal M_k^0$ to be the set of supermatrices $g \Lambda g^{-1}$ with $g \in G_0$, and we call it the \emph{underlying manifold} of $\mathcal M_k$. The map $g K_0 \mapsto g \Lambda g^{-1}$ is a bijection from $G_0/K_0$ onto $\mathcal M_k^0$, through which $\mathcal M_k^0$ is a smooth manifold of dimension $4k^2$. A point $Q \in \mathcal M_k^0$ is block diagonal, $Q = \diag(Q^{\mathrm{BB}} | Q^{\mathrm{FF}})$, and $\mathcal M_k^0 = H_k \times S_k$, where
\begin{align*}
H_k &= \{ g \Lambda_{\mathrm B} g^{-1} : g \in U(k, k) \} \cong U(k, k)/(U(k) \times U(k)), \\
S_k &= \{ u \Lambda_{\mathrm F} u^{-1} : u \in U(2k) \} \cong U(2k)/(U(k) \times U(k)).
\end{align*}
We equip $\mathcal M_k^0$ with the Riemannian metric whose square norm of a tangent vector $\dot Q$ at $Q$ is
\begin{equation}\label{e:sigma-body-metric}
- \tfrac12 \Str(\dot Q^2) = - \tfrac12 \Tr\big( (\dot Q^{\mathrm{BB}})^2 \big) + \tfrac12 \Tr\big( (\dot Q^{\mathrm{FF}})^2 \big) ,
\end{equation}
and we let $\dd \mathrm{vol}$ denote its volume measure and $\Delta_{\mathcal M_k^0}$ its Laplace--Beltrami operator, which is nonpositive. The metric is invariant under the action $Q \mapsto g Q g^{-1}$ of $G_0$. The space $H_k$ is the Riemannian symmetric space of the noncompact group $U(k, k)$, and $S_k$ is the Grassmannian of $k$-dimensional subspaces of $\C^{2k}$. To see that \eqref{e:sigma-body-metric} is positive definite, it suffices by invariance to consider $Q = \Lambda$. There the tangent vectors have the form $\dot Q = [Y, \Lambda] = -2 \Lambda Y$, where $Y \in \mathfrak m_0$ satisfies $Y^{\mathrm{BB}}_{\mathrm{AR}} = (Y^{\mathrm{BB}}_{\mathrm{RA}})^*$ and $Y^{\mathrm{FF}}_{\mathrm{AR}} = - (Y^{\mathrm{FF}}_{\mathrm{RA}})^*$, and for these
\[
- \tfrac12 \Str(\dot Q^2) = 4 \Tr\big( Y^{\mathrm{BB}}_{\mathrm{RA}} (Y^{\mathrm{BB}}_{\mathrm{RA}})^* \big) + 4 \Tr\big( Y^{\mathrm{FF}}_{\mathrm{RA}} (Y^{\mathrm{FF}}_{\mathrm{RA}})^* \big) .
\]

At rank one, $H_1$ is the set of matrices $\big( \begin{smallmatrix} B & \sqrt{B^2 - 1} \, e^{\iu \theta_{\mathrm B}} \\ - \sqrt{B^2 - 1} \, e^{-\iu \theta_{\mathrm B}} & -B \end{smallmatrix} \big)$ with $B \ge 1$, and $S_1$ is the set of matrices $\big( \begin{smallmatrix} F & \iu \sqrt{1 - F^2} \, e^{\iu \theta_{\mathrm F}} \\ -\iu \sqrt{1 - F^2} \, e^{-\iu \theta_{\mathrm F}} & -F \end{smallmatrix} \big)$ with $F \in [-1, 1]$. In the coordinates $B = \cosh r$ and $F = \cos \vartheta$, the metric \eqref{e:sigma-body-metric} takes the form $\dd r^2 + \sinh^2 r \dd \theta_{\mathrm B}^2$ on $H_1$ and $\dd \vartheta^2 + \sin^2 \vartheta \dd \theta_{\mathrm F}^2$ on $S_1$. Thus $H_1$ is the hyperbolic plane of curvature $-1$ in the hyperboloid model, $S_1$ is the unit sphere, and the coordinates $B$ and $F$ are those of \zcref{d:kernels}. In these coordinates, $\dd \mathrm{vol} = \dd B \dd \theta_{\mathrm B} \dd F \dd \theta_{\mathrm F}$, and on functions of $(B, F)$ the operator $\Delta_{\mathcal M_1^0}$ acts as $\Delta_D$.

We define the \emph{weight} $v$ on $\mathcal M_k^0$ by
\begin{equation}\label{e:sigma-weight}
v(Q) = 1 + \tfrac12 \Tr(\Lambda_{\mathrm B} Q^{\mathrm{BB}}) .
\end{equation}
At rank one, $v = 1 + B$. The following lemma collects the properties of $v$ that we use.

\begin{lemma}\label{l:sigma-body}
\begin{enumerate}[label=(\alph*)]
\item\label{i:sigma-body-trace} For all $Q \in \mathcal M_k^0$, we have $\frac12 \Tr(\Lambda_{\mathrm B} Q^{\mathrm{BB}}) \ge k$, with equality if and only if $Q^{\mathrm{BB}} = \Lambda_{\mathrm B}$, and $\frac12 \Tr(\Lambda_{\mathrm F} Q^{\mathrm{FF}}) \le k$, with equality if and only if $Q^{\mathrm{FF}} = \Lambda_{\mathrm F}$. Moreover, $\|Q^{\mathrm{BB}}\|_{\mathrm{HS}} \le 2 v(Q)$ and $\|Q^{\mathrm{FF}}\|_{\mathrm{op}} = 1$.
\item\label{i:sigma-body-derivatives} For all integers $m \ge 1$, there exists $C_m$ such that $|\nabla^m v| \le C_m v$ on $\mathcal M_k^0$, where $\nabla$ is the Levi-Civita connection.
\item\label{i:sigma-body-volume} There exist $C$ and $q_1$ such that $\mathrm{vol}(\{ v \le R \}) \le C R^{q_1}$ for all $R \ge 1$.
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with \ref{i:sigma-body-trace}. Let $Q^{\mathrm{BB}} = g \Lambda_{\mathrm B} g^{-1}$ with $g \in U(k, k)$, and write $g$ in flavor blocks $a$, $b$, $c$, $e$. Since $g^{-1} = \Lambda_{\mathrm B} g^* \Lambda_{\mathrm B}$, we have
\[
\Lambda_{\mathrm B} Q^{\mathrm{BB}} = \Lambda_{\mathrm B} g g^* \Lambda_{\mathrm B} \ge 0, \qquad \Tr(\Lambda_{\mathrm B} Q^{\mathrm{BB}}) = \Tr(g g^*) .
\]
The relation $g^* \Lambda_{\mathrm B} g = \Lambda_{\mathrm B}$ implies that $a^* a - c^* c = \Id_k$ and $e^* e - b^* b = \Id_k$, and therefore
\[
\Tr(g g^*) = 2k + 2 \Tr(b^* b) + 2 \Tr(c^* c) \ge 2k .
\]
Equality holds if and only if $b = c = 0$, or equivalently if $g$ commutes with $\Lambda_{\mathrm B}$. Since $\Lambda_{\mathrm B} Q^{\mathrm{BB}}$ is positive semidefinite, we have
\[
\|Q^{\mathrm{BB}}\|_{\mathrm{HS}} = \|\Lambda_{\mathrm B} Q^{\mathrm{BB}}\|_{\mathrm{HS}} \le \Tr(\Lambda_{\mathrm B} Q^{\mathrm{BB}}) = 2(v - 1) .
\]
The matrix $Q^{\mathrm{FF}} = u \Lambda_{\mathrm F} u^*$ is Hermitian and unitary, and $\Tr(\Lambda_{\mathrm F} Q^{\mathrm{FF}}) \le 2k$, with equality if and only if the unitary matrix $\Lambda_{\mathrm F} u \Lambda_{\mathrm F} u^*$ is the identity.

We now prove \ref{i:sigma-body-derivatives}. Fix $g' = \diag(g'_{\mathrm B} | g'_{\mathrm F}) \in G_0$, and consider the isometry $\iota_{g'}(Q) = g' Q g'^{-1}$. Then
\[
v \circ \iota_{g'}(Q) = 1 + \tfrac12 \Tr(A_{g'} Q^{\mathrm{BB}}), \qquad A_{g'} = (g'_{\mathrm B})^{-1} \Lambda_{\mathrm B} g'_{\mathrm B} = \Lambda_{\mathrm B} (g'_{\mathrm B})^* g'_{\mathrm B} .
\]
The function $v \circ \iota_{g'}$ is the restriction to $\mathcal M_k^0$ of an affine function of the entries of $Q$. Its covariant derivatives of order $m$ at $\Lambda$ are therefore bounded by
\[
C_m \|A_{g'}\|_{\mathrm{HS}} \le C_m \Tr((g'_{\mathrm B})^* g'_{\mathrm B}) = 2 C_m \big( v(g' \Lambda g'^{-1}) - 1 \big) ,
\]
with $C_m$ independent of $g'$. Since $\iota_{g'}$ is an isometry mapping $\Lambda$ to $g' \Lambda g'^{-1}$, and $G_0$ acts transitively, we obtain \ref{i:sigma-body-derivatives}.

Finally, we prove \ref{i:sigma-body-volume}. It suffices to consider $H_k$, since $S_k$ is compact. Every point of $H_k$ has the form $Q^{\mathrm{BB}} = e^{Y} \Lambda_{\mathrm B} e^{-Y}$ with $Y = \big( \begin{smallmatrix} 0 & w \\ w^* & 0 \end{smallmatrix} \big)$, since $H_k$ is a symmetric space of noncompact type and its exponential map at $\Lambda_{\mathrm B}$ is onto. Let $\sigma_1, \dots, \sigma_k$ be the singular values of $w$.

The curve $s \mapsto e^{sY} \Lambda_{\mathrm B} e^{-sY}$, $0 \le s \le 1$, is a geodesic, whose length $r$ is the distance from $\Lambda_{\mathrm B}$ to $Q^{\mathrm{BB}}$. Using
\[
\Lambda_{\mathrm B} Q^{\mathrm{BB}} = \Lambda_{\mathrm B} e^{2Y} \Lambda_{\mathrm B} = e^{-2Y} ,
\]
we find
\[
r = 2 \Big( \sum_{j = 1}^k \sigma_j^2 \Big)^{1/2}, \qquad v = 1 + \sum_{j = 1}^k \cosh(2 \sigma_j) \ge \cosh(r / \sqrt k) .
\]
Thus $\{ v \le R \}$ is contained in the product of $S_k$ with the ball of radius $\sqrt k \log(2R)$ about $\Lambda_{\mathrm B}$ in $H_k$. Since $H_k$ is homogeneous, its sectional curvature is bounded below, and the Bishop--Gromov inequality bounds the volume of a ball of radius $r$ by $C e^{C r}$. Together with the inclusion above, this implies \ref{i:sigma-body-volume}, since $e^{C \sqrt k \log(2R)} = (2R)^{C \sqrt k}$.
\end{proof}

\subsection{Superfunctions and the field \texorpdfstring{$Q$}{Q}}\label{s:sigma-superfunctions}

Let $\Grass{4k^2}$ denote the Grassmann algebra with odd generators $\chi_1, \dots, \chi_{4k^2}$ (\zcref{d:grass-algebra}). We fix an orthonormal basis $e_1, \dots, e_{4k^2}$ of $\mathfrak m_1$, and we define the \emph{generic odd tangent supermatrix} to be $\Theta = \sum_a \chi_a e_a$. Its nonzero entries lie in the blocks $\mathrm{BF}$ and $\mathrm{FB}$ and are linear combinations of the $\chi_a$. In particular, $\Theta$ is an even supermatrix in the sense of \zcref{d:grass-supermatrix}. For $\mathsf k \in K_0$, we let $\alpha_\mathsf k$ denote the automorphism of the algebra $\Grass{4k^2}$ with $\sum_a \alpha_\mathsf k(\chi_a) e_a = \mathsf k \Theta \mathsf k^{-1}$, which replaces $\Theta$ by $\mathsf k \Theta \mathsf k^{-1}$.

\begin{definition}\label{d:sigma-target}
A \emph{superfunction} on $\mathcal M_k$ is a smooth map $f \colon G_0 \to \Grass{4k^2}$ such that $f(g \mathsf k) = \alpha_\mathsf k(f(g))$ for all $g \in G_0$ and $\mathsf k \in K_0$. We write $f(g; \Theta)$ for $f(g)$ to indicate the dependence on the odd variables. The \emph{field} $Q$ is the matrix of superfunctions
\begin{equation}\label{e:sigma-Q}
Q(g; \Theta) = g \, e^{\Theta} \Lambda e^{-\Theta} g^{-1} = g \Lambda e^{-2 \Theta} g^{-1} .
\end{equation}
\end{definition}

The second form of \eqref{e:sigma-Q} uses $\Theta \Lambda = - \Lambda \Theta$. Since $\mathsf k \Lambda \mathsf k^{-1} = \Lambda$, we have
\[
Q(g \mathsf k; \Theta) = g \Lambda e^{-2 \mathsf k \Theta \mathsf k^{-1}} g^{-1} = \alpha_\mathsf k \big( Q(g; \Theta) \big) ,
\]
and $Q$ is therefore a matrix of superfunctions. Its entries have the parities of their positions, so $Q$ is an even supermatrix. It satisfies
\begin{equation}\label{e:sigma-Q-relations}
Q^2 = \Id, \qquad \Str Q = 0 .
\end{equation}
For the second identity, note that $\Str Q = \Str(\Lambda e^{-2\Theta})$ and that $\Str(\Lambda \Theta^m) = 0$ for all $m \ge 1$. Indeed, for odd $m$ the supermatrix $\Lambda \Theta^m$ has vanishing diagonal blocks, and for even $m \ge 2$ the cyclicity of the supertrace (\zcref{l:grass-sdet}\ref{i:grass-sdet-trace}) and $\Theta \Lambda = - \Lambda \Theta$ give $\Str(\Lambda \Theta^m) = \Str(\Theta \Lambda \Theta^{m - 1}) = - \Str(\Lambda \Theta^m)$.

\zcref[S]{d:sigma-target} is the global description of $\mathcal M_k$ that we use, and we call $(g, \Theta)$ the \emph{regular coordinates}. We call $g \Lambda g^{-1} \in \mathcal M_k^0$ the body point of $(g; \Theta)$. The body $f_\emptyset$ of a superfunction is then a function on $\mathcal M_k^0$, and $Q_\emptyset(g) = g \Lambda g^{-1}$. Let $\mathcal O \subset \mathcal M_k^0$ be open. We call a smooth map $\mathsf s \colon \mathcal O \to G_0$ with $\mathsf s(x) \Lambda \mathsf s(x)^{-1} = x$ for all $x \in \mathcal O$ a \emph{section} over $\mathcal O$. For a section $\mathsf s$, the map $f \mapsto f \circ \mathsf s$ identifies the superfunctions on $\mathcal M_k$, restricted to $\mathcal O$, with the superfunctions on the superdomain $\mathcal O^{4k^2 | 4k^2}$ of \zcref{d:grass-superfunction}, with odd coordinates $\chi_a$. A second section has the form $\mathsf s \mathsf k$ for a smooth map $\mathsf k \colon \mathcal O \to K_0$, and the two identifications differ by the unitary change of odd coordinates $\Theta \mapsto \mathsf k \Theta \mathsf k^{-1}$.

The regular coordinates identify the superfunctions with the sections of a vector bundle over $\mathcal M_k^0$ with fiber $\Grass{4k^2}$, and all analytic estimates below are based on this structure. The word regular distinguishes them from the polar coordinates of \zcref{s:sigma-radial}, which are singular at a point. The regular coordinates are nonsingular everywhere, and integrals in them require only integration by parts, without boundary terms (compare \zcref{s:grass-boundary}).

We use the conventions of \zcref{s:grassmann} for these superfunctions. Coefficients are written to the left of the monomials in the $\chi_a$, and auxiliary odd generators, such as the parameters $\vartheta_Y$ of \zcref{s:sigma-symmetries}, are treated as coefficients (\zcref{s:grass-algebra}). If $a_1, \dots, a_m$ are even superfunctions and $\phi$ is a smooth function on a neighborhood of the range of their bodies, then the superfunction $\phi(a_1, \dots, a_m)$ is defined pointwise by \zcref{d:grass-even-function}; this defines, for instance, $e^{-c d_k}$ and $\psi(d_1/r)$ below, and compositions of such functions are computed by \zcref{l:grass-calculus}\ref{i:grass-calculus-composition}. Since the number $4k^2$ of the generators $\chi_a$ is even, the Berezin integral in the $\chi_a$ commutes with Berezin integrals in other generators (\zcref{l:grass-berezin}\ref{i:grass-berezin-fubini}).

\begin{remark}\label{r:sigma-superfunction-local}
Every superfunction on $\mathcal M_k$ is, near each point of $\mathcal M_k^0$, a smooth function of the entries of $Q$, in the sense of \zcref{d:grass-even-function}. Indeed, near a point of $\mathcal M_k^0$, the bodies of suitable even entries of $Q$ form a coordinate system of $\mathcal M_k^0$, since $\mathcal M_k^0$ is a submanifold of a space of matrices. By \eqref{e:sigma-Q}, the terms of degree one of the odd entries of $Q$ form the matrix $-2 g \Lambda \Theta g^{-1}$, which determines $\Theta$, and their other terms have degree at least three in the $\chi_a$. Solving for $\Theta$ degree by degree, we write the $\chi_a$ as polynomials in the odd entries of $Q$ whose coefficients are smooth functions of the body point. The souls of the even entries are polynomials in the $\chi_a$, so the bodies of the even entries are also smooth functions of the entries of $Q$, and inserting these expressions into $f = \sum_I f_I \chi^I$ proves the claim. For intuition, a superfunction on $\mathcal M_k$ may therefore be thought of as a function of a supermatrix $Q$ with $Q^2 = \Id$, as in the physics literature, and a derivation of the algebra of superfunctions is determined by its values on the entries of $Q$.
\end{remark}

\subsection{Symmetries}\label{s:sigma-symmetries}

We use the name $U(k, k | 2k)$ only for the pair consisting of the group $G_0$ and the Lie superalgebra $\mathfrak g$. The group $G_0$ acts on $\mathcal M_k^0$ by $Q \mapsto g' Q g'^{-1}$, and on superfunctions by left translation of $g$ in the regular coordinates. In the odd directions there are no group elements with complex entries, and the symmetries are described infinitesimally, by derivations of the algebra of superfunctions. For each $Y \in \mathfrak g$, we now construct the derivation $\widehat Y$ that moves $Q$ by its commutator with $Y$. For even $Y$, it is the derivative of the action of $e^{sY}$, and for odd $Y$, it is the odd analogue, written with an odd parameter. In the language of supermanifolds, the $\widehat Y$ are the fundamental vector fields of the action of the supergroup on the homogeneous superspace $\mathcal M_k$; for textbook accounts of supergroups and their actions, see \cite{berezin1987introduction,varadarajan2004supersymmetry}. We construct them directly, in the regular coordinates, and prove the properties that we use. Let $Y \in \mathfrak g$ be homogeneous. We let $\vartheta_Y$ be a real parameter if $Y$ is even, and we work to first order in it. We let $\vartheta_Y$ be an auxiliary odd generator if $Y$ is odd, in the sense of \zcref{s:grass-berezin}. Then $\vartheta_Y Y$ is an even supermatrix and $\vartheta_Y^2 = 0$.

\begin{lemma}\label{l:sigma-action}
\begin{enumerate}[label=(\alph*)]
\item\label{i:sigma-action-exist} For every homogeneous $Y \in \mathfrak g$, there exists a unique derivation $\widehat Y$ of the algebra of superfunctions on $\mathcal M_k$, of the parity of $Y$, such that
\begin{equation}\label{e:sigma-action}
\vartheta_Y \widehat Y Q = [\vartheta_Y Y, Q] .
\end{equation}
\item\label{i:sigma-action-group} For $g' \in G_0$, let $(L_{g'} f)(g; \Theta) = f(g'^{-1} g; \Theta)$. Then $L_{g'} \widehat Y L_{g'}^{-1} = \widehat{g' Y g'^{-1}}$. If $Y$ is even, then $\widehat Y Q$ is the matrix $[Y, Q]$, and if moreover $Y$ lies in the Lie algebra of $G_0$, then $(\widehat Y f)(g; \Theta) = \frac{\dd}{\dd s} f(e^{s Y} g; \Theta)$ at $s = 0$.
\item\label{i:sigma-action-degree} Let $g \in G_0$ and $s \ge 0$, and let $f$ be a superfunction whose coefficients of degree less than $s$ in the $\chi_a$ vanish on a neighborhood of $g$. If $Y = \sum_a y_a e_a \in \mathfrak m_1$, then the terms of degree at most $s$ of
\[
\big( \widehat{g Y g^{-1}} f \big)(g; \Theta) - \sum_a y_a \frac{\partial f}{\partial \chi_a}(g; \Theta)
\]
vanish. If $Y \in \mathfrak k_1$, then the terms of degree at most $s$ of $(\widehat{g Y g^{-1}} f)(g; \Theta)$ vanish.
\item\label{i:sigma-action-bracket} For homogeneous $Y, Y' \in \mathfrak g$, we have
\[
[\widehat Y, \widehat{Y'}] = - (-1)^{|Y| |Y'|} \, \widehat{[Y, Y']} ,
\]
where $[D, D'] = D D' - (-1)^{|D| |D'|} D' D$ for operators $D$ and $D'$ of parities $|D|$ and $|D'|$.
\end{enumerate}
\end{lemma}

Here and below, $\widehat Y Q$ denotes the matrix obtained by applying $\widehat Y$ to each entry of $Q$. A derivation of parity one is an odd derivation, which means that $\widehat Y(f f') = (\widehat Y f) f' + (-1)^{|f|} f \widehat Y f'$ for homogeneous $f$ of parity $|f|$. By the uniqueness in \ref{i:sigma-action-exist}, $\widehat Y$ depends linearly on $Y$, and we extend the definition to all $Y \in \mathfrak g$ by linearity. In words, \ref{i:sigma-action-degree} says that, at a given point, the odd fields $\widehat{g Y g^{-1}}$ with $Y \in \mathfrak m_1$ act to leading order as the derivatives in the odd coordinates, while those with $Y \in \mathfrak k_1$ raise the degree.

\begin{proof}
We first construct $\widehat Y$. Fix $(g; \Theta)$, write $\vartheta = \vartheta_Y$, and adjoin $\vartheta$ to the coefficient algebra, working modulo $\vartheta^2$ when it is even. Set $Z = e^{-\Theta} (\vartheta g^{-1} Y g) e^{\Theta}$, which is an even supermatrix. We look for $A \in \mathfrak m_0$, $\Theta_1 \in \mathfrak m_1$, and $C \in \mathfrak k$, with coefficients in the original algebra $\Grass{4k^2}$, such that
\begin{equation}\label{e:sigma-action-decomposition}
e^{-\Theta} (\vartheta A) e^{\Theta} + \frac{1 - e^{-\mathrm{ad}_\Theta}}{\mathrm{ad}_\Theta} (\vartheta \Theta_1) + \vartheta C = Z ,
\end{equation}
where $\mathrm{ad}_\Theta(Y') = \Theta Y' - Y' \Theta$, and the fraction is defined by its power series, which terminates since $\Theta$ is nilpotent. We solve for the variations $U = \vartheta A$, $V = \vartheta \Theta_1$, and $K = \vartheta C$, all of which are even supermatrices. On the decomposition $\mathfrak g = \mathfrak m_0 \oplus \mathfrak m_1 \oplus \mathfrak k$ with coefficients in the extended algebra, the left side is the identity plus a linear map that raises the degree in the original generators $\chi_a$. Its inverse is a finite Neumann series. It preserves the ideal generated by $\vartheta$, and gives a unique solution $(U, V, K)$ linear in $\vartheta$. Extracting $\vartheta$ on the left defines $A$, $\Theta_1$, and $C$ uniquely in the original algebra. Their coefficients in the $\chi_a$ are linear functions of $g^{-1}Yg$, with operator norms bounded in terms of $k$ and the fixed basis. The parameter stays inside the matrix products in \eqref{e:sigma-action-decomposition}; for odd $Y$, this accounts for $e^{-\Theta}\vartheta = \vartheta e^{\Theta}$.

By the formula for the differential of the exponential map, \eqref{e:sigma-action-decomposition} says that
\[
e^{\vartheta Y} g e^{\Theta} = g (\Id + \vartheta A) e^{\Theta + \vartheta \Theta_1} (\Id + \vartheta C)
\]
to first order in $\vartheta$. We define $\widehat Y f$ by
\begin{equation}\label{e:sigma-action-def}
f \big( g (\Id + \vartheta A); \Theta + \vartheta \Theta_1 \big) = f(g; \Theta) + \vartheta (\widehat Y f)(g; \Theta) ,
\end{equation}
where the left side is defined by Taylor expansion in the direction of the left-invariant vector field $A$ on $G_0$ (\zcref{d:grass-even-function}). Here $A$ has coefficients in $\mathfrak m_0 \otimes \Grass{4k^2}$, and $\mathfrak m_0$ is not contained in the Lie algebra of $G_0$. Since $\mathfrak g_0$ is the complexification of the Lie algebra of $G_0$, the left-invariant derivative $\frac{\dd}{\dd s} f(g e^{s A})$, defined for $A$ in this Lie algebra, extends complex-linearly to $A \in \mathfrak g_0$, and the infinitesimal form of the equivariance under $K_0$ extends in the same way to $\mathfrak k_0$; this is the meaning of the Taylor expansion here and of the equivariance used below. The first-order substitution is multiplicative, so extracting $\vartheta$ on the left gives the graded Leibniz rule for $\widehat Y$, a derivation of the parity of $Y$. It maps superfunctions to superfunctions, since replacing $(g, \Theta)$ by $(g \mathsf k, \mathsf k^{-1} \Theta \mathsf k)$ conjugates the variations $(U, V, K)$ by $\mathsf k^{-1}$, by uniqueness of the decomposition, and hence conjugates $(A, \Theta_1, C)$ in the same way. Since $\Id + \vartheta C$ commutes with $\Lambda$ to first order, we have
\[
Q(g (\Id + \vartheta A); \Theta + \vartheta \Theta_1) = e^{\vartheta Y} Q(g; \Theta) e^{-\vartheta Y} ,
\]
which is \eqref{e:sigma-action}.

For even $Y$, let $A_0 + C_0$ be the decomposition of $g^{-1} Y g \in \mathfrak g_0$ with $A_0 \in \mathfrak m_0$ and $C_0 \in \mathfrak k_0$. To first order in $s$, we have $(\Id + s C_0) e^{\Theta} = e^{\Theta + s [C_0, \Theta]} (\Id + s C_0)$. The solution of \eqref{e:sigma-action-decomposition} is therefore $A = A_0$, $\Theta_1 = [C_0, \Theta]$, and $C = C_0$, and the equivariance of $f$ under $K_0$ implies that $\widehat Y f(g; \Theta) = \frac{\dd}{\dd s} f(e^{s Y} g; \Theta)$ at $s = 0$. The identity $L_{g'} \widehat Y L_{g'}^{-1} = \widehat{g' Y g'^{-1}}$ follows from $e^{\vartheta_Y Y} g'^{-1} g = g'^{-1} e^{\vartheta_Y g' Y g'^{-1}} g$.

It remains to prove uniqueness. It suffices to show that a derivation $\mathcal D$ with $\mathcal D Q = 0$ vanishes. By \zcref{r:sigma-superfunction-local}, every superfunction $f$ is locally a smooth function of the entries of $Q$, and the chain rule of \zcref{l:grass-calculus}\ref{i:grass-calculus-chain} implies that $\mathcal D f = 0$.

We now prove \ref{i:sigma-action-degree}. By \ref{i:sigma-action-group}, we have $(\widehat{g Y g^{-1}} f)(g; \Theta) = (\widehat Y L_g^{-1} f)(\Id; \Theta)$, and $L_g^{-1} f = f(g \, \cdot \, ; \cdot)$ has the same vanishing coefficients near $\Id$. It suffices to consider $g = \Id$. We solve \eqref{e:sigma-action-decomposition} degree by degree in the $\chi_a$, retaining $\vartheta$ in every matrix product. Since $\vartheta A$ and $\vartheta \Theta_1$ are even and $\vartheta$ is odd, the coefficients of $A$ are odd and those of $\Theta_1$ are even. The equation in degree zero is $\vartheta(\Theta_1^{(0)} + C^{(0)}) = \vartheta Y$, hence $\Theta_1^{(0)} + C^{(0)} = Y$, where the superscript denotes the part of degree zero in the original generators. For $Y \in \mathfrak m_1$, we obtain $\Theta_1 = Y + O(\chi^2)$, and for $Y \in \mathfrak k_1$, we obtain $\Theta_1 = O(\chi^2)$, where $O(\chi^j)$ denotes terms of degree at least $j$. In both cases $A = O(\chi)$.

By \eqref{e:sigma-action-def}, $(\widehat Y f)(\Id; \Theta)$ is the sum of the left-invariant derivative of $f$ in the direction $A$ and of the derivative $\sum_a (\Theta_1)_a \partial f/\partial \chi_a$ in the odd direction $\Theta_1 = \sum_a (\Theta_1)_a e_a$. The first term, and the part $O(\chi^2)$ of $\Theta_1$, contribute only terms of degree at least $s + 1$, and the part $Y$ of $\Theta_1$ contributes $\sum_a y_a \partial f/\partial \chi_a$. This proves \ref{i:sigma-action-degree}. The supplementary script \texttt{polar/check\_regular\_action.py} verifies the factorization, its action on $Q$, the changes of section, and these degree bounds at rank one for all sixteen matrix units.

Finally, we prove \ref{i:sigma-action-bracket}. Let $\vartheta = \vartheta_Y$ and $\vartheta' = \vartheta_{Y'}$ be independent parameters, and set $D = \vartheta \widehat Y$ and $D' = \vartheta' \widehat{Y'}$. These are even derivations with $D Q = [\vartheta Y, Q]$ and $D' Q = [\vartheta' Y', Q]$, and they commute with multiplication by constants. We obtain
\[
(D D' - D' D) Q = [\vartheta' Y', [\vartheta Y, Q]] - [\vartheta Y, [\vartheta' Y', Q]] = - \big[ [\vartheta Y, \vartheta' Y'], Q \big] = - \big[ \vartheta \vartheta' [Y, Y'], Q \big] ,
\]
where we used the Jacobi identity for commutators of the even supermatrices $\vartheta Y$, $\vartheta' Y'$, and $Q$, and the identity
\[
\vartheta Y \vartheta' Y' - \vartheta' Y' \vartheta Y = \vartheta \vartheta' \big( Y Y' - (-1)^{|Y| |Y'|} Y' Y \big) .
\]
The derivation $D D' - D' D + \vartheta \vartheta' \widehat{[Y, Y']}$ therefore vanishes on $Q$, and the uniqueness in \ref{i:sigma-action-exist} shows that
\[
D D' - D' D = - \vartheta \vartheta' \, \widehat{[Y, Y']} .
\]
On the other hand, moving the parameters to the left, we find
\[
D D' - D' D = (-1)^{|Y| |Y'|} \vartheta \vartheta' \, [\widehat Y, \widehat{Y'}] .
\]
Comparing the coefficients of $\vartheta \vartheta'$ proves \ref{i:sigma-action-bracket}.
\end{proof}

We call a superfunction $f$ \emph{invariant} under a subspace $\mathfrak h \subset \mathfrak g$ if $\widehat Y f = 0$ for all homogeneous $Y \in \mathfrak h$. The function
\begin{equation}\label{e:sigma-dk}
d_k(Q) = \tfrac12 \Str(\Lambda Q)
\end{equation}
is invariant under $\mathfrak k$. Indeed, for $Y \in \mathfrak k$, the cyclicity of the supertrace gives
\[
\vartheta_Y \widehat Y d_k = \tfrac12 \Str(\Lambda [\vartheta_Y Y, Q]) = \tfrac12 \Str([\Lambda, \vartheta_Y Y] Q) = 0 .
\]
We have $(d_k)_\emptyset = \frac12 \Tr(\Lambda_{\mathrm B} Q^{\mathrm{BB}}) - \frac12 \Tr(\Lambda_{\mathrm F} Q^{\mathrm{FF}})$ on $\mathcal M_k^0$. By \zcref{l:sigma-body}\ref{i:sigma-body-trace}, it is nonnegative and vanishes only at $Q = \Lambda$, and
\begin{equation}\label{e:sigma-dk-body}
v - 1 - k \le (d_k)_\emptyset \le v - 1 + k .
\end{equation}

The following lemma shows that a superfunction that is invariant under $\mathfrak g$ is constant. We use it in the uniqueness of the invariant integral below.

\begin{lemma}\label{l:sigma-equivariant}
Let $f$ be a superfunction on $\mathcal M_k$ that is invariant under $\mathfrak g$. Then $f$ is constant. This also holds for a superfunction $f$ over a connected open set $\mathcal O \subset \mathcal M_k^0$, that is, for a smooth map from the preimage of $\mathcal O$ in $G_0$ to $\Grass{4k^2}$ with the equivariance of \zcref{d:sigma-target}.
\end{lemma}

\begin{proof}
Let $\tilde{\mathcal O} \subset G_0$ be the preimage of $\mathcal O$, where $\mathcal O = \mathcal M_k^0$ in the first statement. It is connected, since $\mathcal O$ and $K_0$ are connected. For $Y$ in the Lie algebra of $G_0$, \zcref{l:sigma-action}\ref{i:sigma-action-group} implies that $\frac{\dd}{\dd s} f(e^{sY} g; \Theta) = 0$ whenever $e^{s Y} g \in \tilde{\mathcal O}$. The vectors $Y g$ span the tangent space of $G_0$ at $g$, and therefore $f(\cdot \, ; \Theta)$ is constant on $\tilde{\mathcal O}$. Let $f_n$ denote the part of $f$ of degree $n$ in the $\chi_a$. Suppose that $f$ has a nonzero part of positive degree, and let $n \ge 1$ be the smallest such degree. The superfunction $f - f_0$ is invariant under $\mathfrak g$, since $f_0$ is constant, and its coefficients of degree less than $n$ vanish.

Fix $g_0 \in \tilde{\mathcal O}$ and $Y = \sum_a y_a e_a \in \mathfrak m_1$. Since $\widehat{g_0 Y g_0^{-1}} (f - f_0) = 0$, \zcref{l:sigma-action}\ref{i:sigma-action-degree}, applied to $f - f_0$ at $g_0$ with $s = n$, shows that the terms of degree at most $n$ of $\sum_a y_a \, \partial (f - f_0)/\partial \chi_a (g_0; \Theta)$ vanish. Its terms of degree $n - 1$ give
\[
\sum_a y_a \, \frac{\partial f_n}{\partial \chi_a}(g_0; \Theta) = 0 \qquad \text{for all } y \in \C^{4k^2} .
\]
An element of $\Grass{4k^2}$ of degree $n \ge 1$ whose derivatives in all the $\chi_a$ vanish is zero, and therefore $f_n(g_0; \cdot) = 0$. Since $f_n$ does not depend on $g$, this contradicts the choice of $n$.
\end{proof}

\subsection{The invariant integral and the Casimir operator}\label{s:sigma-integral}

This subsection defines the invariant Berezin integral and the Casimir operator $\mathsf C_k$ of the sigma model. In \zcref{l:tube-casimir}, Part IV identifies $\mathsf C_k$, up to a constant factor, with the heat generator of the chain restricted to the saddle manifold.

By \zcref{l:sigma-action}, the action of $G_0$ only moves the body point in regular coordinates. A Berezin integral on $\mathcal M_k$ that is invariant under $G_0$ therefore has the form
\begin{equation}\label{e:sigma-integral}
\int f = \int_{\mathcal M_k^0} \Big( \int \rho(\Theta) f(g; \Theta) \dd \chi \Big) \dd \mathrm{vol}(g \Lambda g^{-1}) ,
\end{equation}
where $\int \cdot \dd \chi$ is the Berezin integral of \zcref{d:grass-berezin} in the generators $\chi_a$, and $\rho \in \Grass{4k^2}$ is even and invariant under the automorphisms $\alpha_\mathsf k$. The inner integral is a function of $g \Lambda g^{-1}$, by \zcref{l:grass-berezin}\ref{i:grass-berezin-linear}, since $\alpha_\mathsf k$ is induced by a linear map of $\mathfrak m_1$ of determinant one. (Conjugation by $\mathsf k = \diag(\mathsf k_{\mathrm{RB}}, \mathsf k_{\mathrm{AB}} | \mathsf k_{\mathrm{RF}}, \mathsf k_{\mathrm{AF}})$ acts on the four blocks of $\mathfrak m_1$ with determinants $\det(\mathsf k_{\mathrm{RB}})^k \det(\mathsf k_{\mathrm{AF}})^{-k}$, $\det(\mathsf k_{\mathrm{RF}})^k \det(\mathsf k_{\mathrm{AB}})^{-k}$, and their inverses.) As in \zcref{d:grass-berezin}, a superfunction $f$ is called integrable if $|f|_{1; 1} \in L^1(\dd \mathrm{vol})$, where $|f|_{1; 1}$ denotes the $\ell^1$ norm of the coefficients of $f$, one of the coefficient norms of \zcref{s:sigma-weighted}. For such $f$, the integral \eqref{e:sigma-integral} converges absolutely.

\begin{definition}\label{d:sigma-integral}
We define the \emph{invariant Berezin integral} $f \mapsto \int f$ to be the functional \eqref{e:sigma-integral} with the even element $\rho$ chosen such that $\int \widehat Y f = 0$ for all homogeneous $Y \in \mathfrak g$ and all compactly supported superfunctions $f$, and such that
\begin{equation}\label{e:sigma-integral-normalization}
\int e^{-d_k} = 1 .
\end{equation}
We define the \emph{Berezin pairing} $\mathcal B_k(f, f') = \int f f'$ for superfunctions $f$ and $f'$ such that $f f'$ is integrable.
\end{definition}

There exists an even element $\rho$ with nonzero constant body such that $\int \widehat Y f = 0$ for all homogeneous $Y$ and all compactly supported $f$. In the terminology of \cite{alldridge2010invariant}, this says that the homogeneous superspace $\mathcal M_k$ is unimodular. It follows from \cite[Corollary~4.12 and Theorem~4.13]{alldridge2010invariant}, which together show that such an element exists if and only if the Berezinian of $(\mathfrak g/\mathfrak k)^*$ is a trivial module of the stabilizer; see also \cite{zirnbauer1996riemannian} for the Riemannian symmetric superspaces of random matrix theory. For the invariant integral, this means that the stabilizer of $\Lambda$, which acts on the Berezin densities on the tangent space $\mathfrak g/\mathfrak k \cong \mathfrak m$ (the elements of the Berezinian of $(\mathfrak g/\mathfrak k)^*$) through the Berezinian of its action on $\mathfrak m$, leaves these densities unchanged, so that a density at $\Lambda$ is transported consistently by the action of $U(k, k | 2k)$ and defines an invariant integral of the form \eqref{e:sigma-integral}. Since the stabilizer of $\Lambda$ is connected, this criterion reduces to the condition that, for every $Y \in \mathfrak k$, the linear map $Y' \mapsto [Y, Y']$ of $\mathfrak m$ has supertrace zero, where the supertrace of a parity-preserving linear map of $\mathfrak m$ is its trace on $\mathfrak m_0$ minus its trace on $\mathfrak m_1$. For a diagonal $Y \in \mathfrak k_0$ with diagonal entries $y_i$, this map multiplies the entry $Y'_{ij}$ of $Y' \in \mathfrak m$ by $y_i - y_j$, and its supertrace is $\sum_{i, j} (-1)^{|i| + |j|} (y_i - y_j)$, summed over the pairs of indices in different retarded and advanced sectors. This sum vanishes, since each sector has $k$ even and $k$ odd indices. The off-diagonal entries of $Y \in \mathfrak k_0$ do not contribute to the trace, and for odd $Y$ the map exchanges $\mathfrak m_0$ and $\mathfrak m_1$ and has supertrace zero.

The element $\rho$ is unique up to a constant factor. Indeed, if $\rho'$ is a second one, let $\sigma = \rho' \rho^{-1}$, regarded as a superfunction that is constant in $g$. For compactly supported $f$, the invariance of both integrals gives
\[
\int (\widehat Y \sigma) f = \int \widehat Y (\sigma f) - \int \sigma \, \widehat Y f = 0 ,
\]
where $\int \sigma \widehat Y f$ is the integral of $\widehat Y f$ with the density $\rho'$. Hence $\widehat Y \sigma = 0$ for all $Y$, and $\sigma$ is constant by \zcref{l:sigma-equivariant}. The normalization \eqref{e:sigma-integral-normalization} is possible, since $c \mapsto \int e^{-c d_k}$ is constant on $(0, \infty)$ and tends to a nonzero multiple of the body of $\rho$ as $c \to \infty$; we prove the two statements in \zcref{s:sigma-localization}. We emphasize that $\mathcal B_k$ is bilinear, not sesquilinear, and has no positivity property. The weighted $L^2$ norms below are Hilbert norms, unrelated to $\mathcal B_k$.

Since $\widehat Y$ is a derivation and $\int \widehat Y(f f') = 0$, we have the integration by parts formula
\begin{equation}\label{e:sigma-ibp}
\int (\widehat Y f) f' = - (-1)^{|Y| |f|} \int f \, \widehat Y f'
\end{equation}
for homogeneous $Y$ and $f$ and smooth $f'$, provided at least one of $f$ and $f'$ has compact support.

We now define the Casimir operator. For $i, j$ in the index set of \zcref{s:sigma-supermatrices}, we let $E_{i j} \in \mathfrak g$ denote the supermatrix with entry one at $(i, j)$ and zero elsewhere, which is homogeneous of parity $|i| + |j|$.

\begin{definition}\label{d:sigma-casimir}
We define the \emph{Casimir operator} to be
\begin{equation}\label{e:sigma-casimir}
\mathsf C_k = - \frac12 \sum_{i, j} (-1)^{|j|} \, \widehat{E_{j i}} \, \widehat{E_{i j}} .
\end{equation}
\end{definition}

The definition comes from the quadratic Casimir element of $\mathfrak{gl}(2k | 2k)$, and we now explain the connection. By \zcref{l:sigma-action}\ref{i:sigma-action-bracket}, the operators $\phi(Y) = - \widehat Y$ for even $Y$ and $\phi(Y) = \iu \widehat Y$ for odd $Y$ satisfy
\begin{equation}\label{e:sigma-rep}
[\phi(Y), \phi(Y')] = \phi([Y, Y'])
\end{equation}
for homogeneous $Y$ and $Y'$, which one checks separately in the three cases of two even, one even and one odd, and two odd elements. In other words, $\phi$ is a representation of the Lie superalgebra $\mathfrak g$ by operators on superfunctions. For the bilinear form $\Str(Y Y')$, the basis $((-1)^{|i|} E_{j i})$ is dual to $(E_{i j})$, and the quadratic Casimir element of $\mathfrak{gl}(2k | 2k)$ is defined as the sum $\sum_{i, j} (-1)^{|i|} E_{j i} E_{i j}$ \cite{berezin1987introduction}. Its image under $\phi$ is a multiple of $\mathsf C_k$. Indeed, $\phi(E_{j i}) \phi(E_{i j})$ equals $\widehat{E_{j i}} \widehat{E_{i j}}$ if $E_{i j}$ is even and $- \widehat{E_{j i}} \widehat{E_{i j}}$ if $E_{i j}$ is odd, while $(-1)^{|i|} = (-1)^{|j|}$ in the first case and $(-1)^{|i|} = -(-1)^{|j|}$ in the second, and therefore
\begin{equation}\label{e:sigma-casimir-rep}
\sum_{i, j} (-1)^{|i|} \phi(E_{j i}) \phi(E_{i j}) = \sum_{i, j} (-1)^{|j|} \widehat{E_{j i}} \widehat{E_{i j}} = - 2 \mathsf C_k .
\end{equation}
We will need the following properties of $\mathsf C_k$.

\begin{lemma}\label{l:sigma-casimir}
\begin{enumerate}[label=(\alph*)]
\item\label{i:sigma-casimir-commute} For all homogeneous $Y \in \mathfrak g$, we have $\mathsf C_k \widehat Y = \widehat Y \mathsf C_k$, and $L_{g'} \mathsf C_k = \mathsf C_k L_{g'}$ for all $g' \in G_0$. Moreover, $\mathsf C_k 1 = 0$.
\item\label{i:sigma-casimir-symmetric} For all smooth superfunctions $f$ and $f'$, at least one of which has compact support, we have $\int (\mathsf C_k f) f' = \int f \, \mathsf C_k f'$.
\item\label{i:sigma-casimir-local} Let $\mathsf s$ be a section over a coordinate patch $\mathcal O \subset \mathcal M_k^0$. In the coordinates $(x, \chi)$ given by $\mathsf s$, the operator $\mathsf C_k$ is a second-order differential operator of the form
\begin{equation}\label{e:sigma-casimir-local}
\mathsf C_k = - \Delta_{\mathcal M_k^0} + \sum_{\mu, \nu} n^{\mu \nu} \partial_{x_\mu} \partial_{x_\nu} + \sum_\mu b^\mu \partial_{x_\mu} + c ,
\end{equation}
where each $n^{\mu \nu}$ is the multiplication by a smooth superfunction whose coefficients have degree at least two in the $\chi_a$, and $b^\mu$ and $c$ are linear maps of $\Grass{4k^2}$ (they may contain $\chi_a$ and $\partial_{\chi_a}$) with smooth coefficients in $x$. Moreover, $\mathsf C_k$ is a finite sum of products of two vector fields in the coordinates $(x, \chi)$. In particular, it is a finite sum of products of at most two of the derivatives $\partial_{x_\mu}$ and $\partial_{\chi_a}$, with smooth superfunctions as coefficients on the left.
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with \ref{i:sigma-casimir-commute}. Its first assertion is the statement that the Casimir element is central \cite{berezin1987introduction}; for Lie superalgebras in general, see also \cite{varadarajan2004supersymmetry}. We give the short direct proof. We write $c_{i j} = \phi(E_{i j})$, we recall that $E_{i j}$ has the parity $|E_{i j}| = |i| + |j|$ modulo two, and we set
\[
\Omega = \sum_{i, j} (-1)^{|i|} c_{j i} c_{i j} ,
\]
so that $\Omega = -2 \mathsf C_k$ by \eqref{e:sigma-casimir-rep}. The brackets of the $E_{i j}$, and \eqref{e:sigma-rep}, give
\begin{align*}
[E_{m n}, E_{j i}] &= \delta_{n j} E_{m i} - (-1)^{|E_{m n}| |E_{j i}|} \delta_{i m} E_{j n} , \\
[c_{m n}, c_{j i}] &= \delta_{n j} c_{m i} - (-1)^{|E_{m n}| |E_{j i}|} \delta_{i m} c_{j n} .
\end{align*}

Using $[c, c' c''] = [c, c'] c'' + (-1)^{|c| |c'|} c' [c, c'']$ and $|E_{j i}| = |E_{i j}|$, we find
\begin{align*}
[c_{m n}, \Omega] &= \sum_{i, j} (-1)^{|i|} \Big( [c_{m n}, c_{j i}] \, c_{i j} + (-1)^{|E_{m n}| |E_{i j}|} c_{j i} \, [c_{m n}, c_{i j}] \Big) \\
&= \sum_i (-1)^{|i|} c_{m i} c_{i n} - \sum_j (-1)^{|m| + |E_{m n}| |E_{j m}|} c_{j n} c_{m j} \\
&\qquad + \sum_j (-1)^{|n| + |E_{m n}| |E_{j n}|} c_{j n} c_{m j} - \sum_i (-1)^{|i|} c_{m i} c_{i n} .
\end{align*}
The first and the last sums cancel. In the two middle sums, the difference of the exponents is $(|n| - |m|)(1 + |E_{m n}|)$, which is even since $|E_{m n}| = 1$ whenever $|n| \ne |m|$, and these sums cancel as well. Therefore $\Omega$ supercommutes with every $c_{m n}$. Since $\Omega$ is even and $c_{m n}$ is a nonzero multiple of $\widehat{E_{m n}}$, we obtain $\mathsf C_k \widehat Y = \widehat Y \mathsf C_k$ for all homogeneous $Y$. The second assertion follows from $L_{g'} \widehat Y L_{g'}^{-1} = \widehat{g' Y g'^{-1}}$ and the invariance of the form $\Str(Y Y')$ under conjugation by $g'$, which maps dual bases to dual bases. The identity $\mathsf C_k 1 = 0$ holds since $\widehat Y 1 = 0$.

We now prove \ref{i:sigma-casimir-symmetric}. By linearity, we may assume that $f$ is homogeneous. Let $a = |i| + |j|$. Differentiation does not enlarge support, so the compactly supported factor remains compactly supported in each of the two applications of \eqref{e:sigma-ibp}. We find
\[
\int (\widehat{E_{j i}} \widehat{E_{i j}} f) f' = - (-1)^{a (|f| + a)} \int (\widehat{E_{i j}} f) \, \widehat{E_{j i}} f' = (-1)^a \int f \, \widehat{E_{i j}} \widehat{E_{j i}} f' .
\]
We multiply by $(-1)^{|j|}$, sum over $i$ and $j$, and exchange the names of $i$ and $j$. Since $(-1)^{|j| + a} = (-1)^{|i|}$, we obtain \ref{i:sigma-casimir-symmetric}.

It remains to prove \ref{i:sigma-casimir-local}. We split the sum \eqref{e:sigma-casimir} into the terms with $E_{i j}$ even and those with $E_{i j}$ odd. Let $Y$ be in the Lie algebra of $G_0$. In the coordinates of $\mathsf s$, we have $e^{sY} \mathsf s(x) = \mathsf s(x_s) \mathsf k_s$ with $x_s = e^{sY} x e^{-sY}$ and $\mathsf k_s \in K_0$. By \zcref{l:sigma-action}\ref{i:sigma-action-group} and the equivariance of superfunctions, $\widehat Y$ is therefore the sum of the vector field $V_Y$ of the action of $e^{sY}$ on $\mathcal M_k^0$ and of the infinitesimal rotation of the odd coordinates by $\mathsf k_s$, which has the form $\sum c_{a b}(x) \chi_a \partial_{\chi_b}$ and contains no derivative in $x$. By complex linearity, the same holds for all $Y \in \mathfrak g_0$, with $V_Y$ complex linear in $Y$.

We now compute the part of the even terms of second order in $x$. It is the same in all charts obtained from one chart at $\Lambda$ by the action of an element of $G_0$, which we call translated charts in \zcref{s:sigma-weighted}, by \ref{i:sigma-casimir-commute}. It therefore suffices to compute it at the point $\Lambda$. Let $\mathfrak m_{\R}$ denote the real subspace of $\mathfrak m_0$ described in \zcref{s:sigma-body}, on which the form $\Str(Y Y')$ is real and positive definite, and let $Y_1, \dots, Y_{4k^2}$ be an orthonormal basis of $\mathfrak m_{\R}$ for this form. Let $Z_1, \dots, Z_m$ be a basis of $\mathfrak k_0$, and let $Z^1, \dots, Z^m$ be its dual basis for $\Str(Y Y')$. The decomposition $\mathfrak g_0 = \mathfrak m_0 \oplus \mathfrak k_0$ is orthogonal for this form, by the cyclicity of the supertrace and $\Lambda Y = - Y \Lambda$ for $Y \in \mathfrak m_0$. The element $\sum (-1)^{|i|} E_{j i} \otimes E_{i j}$ of $\mathfrak g_0 \otimes \mathfrak g_0$, summed over the even $E_{i j}$, is the element dual to the form $\Str(Y Y')$ on $\mathfrak g_0$. It can be computed with any basis of $\mathfrak g_0$ and its dual basis, and therefore
\[
\sum_{i, j \colon |i| = |j|} (-1)^{|j|} \, \widehat{E_{j i}} \, \widehat{E_{i j}} = \sum_{\mu} \widehat{Y_\mu} \widehat{Y_\mu} + \sum_\nu \widehat{Z^\nu} \widehat{Z_\nu} .
\]

The part of second order in $x$ of a product of two of the operators $\widehat Y$ at $\Lambda$ is the symmetrized product of the two tangent vectors $V_Y(\Lambda)$. For $Z \in \mathfrak k_0$, the vector $V_Z(\Lambda)$ vanishes, since $e^{sZ}$ commutes with $\Lambda$. For $Y \in \mathfrak m_{\R}$, we have $V_Y(\Lambda) = [Y, \Lambda]$, and
\[
- \tfrac12 \Str\big( [Y, \Lambda] [Y', \Lambda] \big) = 2 \Str(Y Y'), \qquad Y, Y' \in \mathfrak m_{\R} ,
\]
so that the vectors $2^{-1/2} V_{Y_\mu}(\Lambda)$ form an orthonormal basis of $T_\Lambda \mathcal M_k^0$ for the metric \eqref{e:sigma-body-metric}. The second-order part at $\Lambda$ of the right side of the last display is therefore twice that of $\Delta_{\mathcal M_k^0}$, and the even terms of \eqref{e:sigma-casimir}, which are $-\frac12$ times this display, have the second-order part of $-\Delta_{\mathcal M_k^0}$. This is the classical fact that the Casimir operator acts on a Riemannian symmetric space as a multiple of the Laplace--Beltrami operator (compare \cite{helgason1984groups}). The remaining parts of the even terms have order at most one in $x$.

For odd $Y$, the construction of \zcref{l:sigma-action} shows that $\widehat Y$ is the sum of the left-invariant vector field in the direction $A$ and of a vector field in the odd coordinates. Here $A$ has coefficients of odd degree, since $\vartheta_Y A$ is even. A product of two such fields contains second derivatives in $x$ only with coefficients of degree at least two, which completes the proof of \eqref{e:sigma-casimir-local}.

It remains to prove the last assertion of \ref{i:sigma-casimir-local}. We showed above that, in the coordinates of $\mathsf s$, every $\widehat Y$ is a vector field in $(x, \chi)$ with smooth superfunctions as coefficients. For even $Y$, the field is the sum of $V_Y$ and a rotation of the odd coordinates. For odd $Y$, it is the sum of the left-invariant field in the direction $A$ and a vector field in the odd coordinates, and the left-invariant field is the sum of a vector field in $x$ and a rotation of the odd coordinates, by the argument for even $Y$ with right translation by $e^{s A}$ in place of left translation by $e^{s Y}$. Hence $\mathsf C_k$ is a finite sum of products of two such vector fields. For vector fields $V$ and $V'$ in the coordinates $z = (x, \chi)$, the Leibniz rule gives
\[
 V V' = \sum_{I, J} \pm v^I v'^J \partial_{z_I} \partial_{z_J} + \sum_{I, J} v^I (\partial_{z_I} v'^J) \partial_{z_J} ,
 \qquad
 V = \sum_I v^I \partial_{z_I} , \quad V' = \sum_J v'^J \partial_{z_J} ,
\]
which is of the stated form. In particular, the coefficient of $\partial_{x_\mu} \partial_{x_\nu}$ in $\mathsf C_k$ is the multiplication by a superfunction. Together with the previous paragraph, this proves the assertion on $n^{\mu \nu}$.
\end{proof}

\begin{remark}\label{r:sigma-casimir-rank-one}
At rank one, we show in \zcref{l:sigma-radial-part} that on superfunctions invariant under $\mathfrak k$, which are functions of $(B, F)$ in polar coordinates, $\mathsf C_1$ acts as $-d \Delta_D d^{-1}$. The term $- \frac{2(B^2 - 1)}{d} \partial_B + \frac{2(1 - F^2)}{d} \partial_F$ of $d \Delta_D d^{-1}$, which is absent from the Laplace--Beltrami operator of the underlying manifold, comes from the odd directions. (This term is the reason why the singlet diffusion of \zcref{r:intro-singlet} differs from the diffusion with generator $X \Delta_D$ of \zcref{d:kernels}.)
\end{remark}

\begin{remark}\label{r:sigma-laplace}
The normalization of $\mathsf C_k$ is fixed by its body part $-\Delta_{\mathcal M_k^0}$ (\zcref{l:sigma-casimir}\ref{i:sigma-casimir-local}), and in this sense $\mathsf C_k$ is minus the invariant Laplace operator of $\mathcal M_k$. This also holds with the odd directions included. We show in \zcref{l:tube-casimir}, where Part IV uses it, that $\mathsf C_k$ is minus the Laplace--Beltrami operator of the invariant supermetric $-\frac12 \Str(\dd Q \, \dd Q)$ (compare \cite{zirnbauer1996riemannian}), in the weak sense in which the quadratic form of $\mathsf C_k$ for the invariant Berezin integral is given by the inverse of this supermetric. The chain of \zcref{s:rep} produces $\mathsf C_k$ with this normalization: one link of the chain acts on the tangent variables as the heat semigroup of $-\mathsf C_k$ at time $(4 s_E^2 W^2)^{-1}$, as explained in \zcref{s:sigma-heuristic}, and \zcref{l:asm-time} proves this.
\end{remark}

We will need two computations with $\mathsf C_k$, which determine its radial part at rank one in \zcref{l:sigma-radial-part}. For even superfunctions $u$ and $v$, we set
\[
\mathcal Q_{\mathsf C_k}(u, v) = \mathsf C_k(u v) - (\mathsf C_k u) v - u \, \mathsf C_k v .
\]

\begin{lemma}\label{l:sigma-casimir-values}
\begin{enumerate}[label=(\alph*)]
\item\label{i:sigma-carre} For even superfunctions $u$ and $v$, we have
\begin{equation}\label{e:sigma-carre}
\mathcal Q_{\mathsf C_k}(u, v) = - \sum_{i, j} (-1)^{|j|} \big( \widehat{E_{j i}} u \big) \big( \widehat{E_{i j}} v \big) ,
\end{equation}
and $\mathcal Q_{\mathsf C_k}(u, v)_\emptyset = -2 \nabla u_\emptyset \cdot \nabla v_\emptyset$ for the metric \eqref{e:sigma-body-metric}.
\item\label{i:sigma-casimir-Q} We have $\mathsf C_k Q = 0$ entrywise, and
\begin{equation}\label{e:sigma-casimir-two-values}
\mathsf C_k d_k = 0, \qquad \mathsf C_k \Str \big( (\Lambda Q)^2 \big) = - 8 d_k^2 .
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with \ref{i:sigma-carre}. Applying the Leibniz rule twice to \eqref{e:sigma-casimir}, and exchanging $i$ and $j$ in one of the two mixed sums, which is possible since $(-1)^{|i| + |E_{i j}|} = (-1)^{|j|}$, we obtain \eqref{e:sigma-carre}. For odd $E_{i j}$, both factors in \eqref{e:sigma-carre} are odd and their product has no body. For even $E_{i j}$, the field $\widehat{E_{i j}}$ is the sum of the vector field $V_{E_{i j}}$ on $\mathcal M_k^0$ and a rotation of the odd coordinates, as shown in the proof of \zcref{l:sigma-casimir}\ref{i:sigma-casimir-local}, so the body of $\widehat{E_{i j}} u$ is $V_{E_{i j}} u_\emptyset$. The computation of the second-order part in that proof then shows that $\mathcal Q_{\mathsf C_k}(u, v)_\emptyset = -2 \nabla u_\emptyset \cdot \nabla v_\emptyset$.

We now prove \ref{i:sigma-casimir-Q}. Fix $i$ and $j$, let $\vartheta$ and $\vartheta'$ be auxiliary parameters of the parity $a = |E_{i j}|$, and set $D = \vartheta \widehat{E_{i j}}$ and $D' = \vartheta' \widehat{E_{j i}}$. These are even derivations, which commute with multiplication by $\vartheta$ and $\vartheta'$, and by \eqref{e:sigma-action}, we have $D Q = [P, Q]$ and $D' Q = [P', Q]$ with the even supermatrices $P = \vartheta E_{i j}$ and $P' = \vartheta' E_{j i}$. Since $D' D = (-1)^a \vartheta' \vartheta \, \widehat{E_{j i}} \widehat{E_{i j}}$ and $(-1)^{|j| + a} = (-1)^{|i|}$, the definition \eqref{e:sigma-casimir} becomes
\[
- 2 \mathsf C_k f = \sum_{i, j} (-1)^{|i|} \, [D' D f]_{\vartheta' \vartheta} ,
\]
where $[\, \cdot \,]_{\vartheta' \vartheta}$ denotes the coefficient of $\vartheta' \vartheta$.

We have $D' D Q = [P, [P', Q]]$, and multiplying out with $P P' = \vartheta \vartheta' E_{i i}$, $P Q P' = \vartheta \vartheta' Q_{j j} E_{i i}$, $P' Q P = \vartheta' \vartheta Q_{i i} E_{j j}$, and $P' P = \vartheta' \vartheta E_{j j}$, we find
\[
[D' D Q]_{\vartheta' \vartheta} = (-1)^a \big( E_{i i} Q - Q_{j j} E_{i i} \big) - Q_{i i} E_{j j} + Q E_{j j} .
\]
We sum over $i$ and $j$ with the sign $(-1)^{|i|}$. Each of the four terms is then a multiple of $\sum_i (-1)^{|i|} = 0$ or of $\Str Q = 0$, by \eqref{e:sigma-Q-relations}, so that $\mathsf C_k Q = 0$ entrywise. Since $d_k$ is a linear combination of entries of $Q$, this implies the first identity in \eqref{e:sigma-casimir-two-values}.

For the second identity, the Leibniz rule gives
\[
D' D \Str(\Lambda Q \Lambda Q) = 2 \Str \big( \Lambda (D' D Q) \Lambda Q \big) + 2 \Str \big( \Lambda [P, Q] \Lambda [P', Q] \big) ,
\]
where we used the cyclicity of the supertrace for even supermatrices. After the signed sum, the first term gives $-4 \Str(\Lambda (\mathsf C_k Q) \Lambda Q) = 0$. For even supermatrices $M$ and $N$, a direct computation gives $\Str(M P N P') = (-1)^{|j|} \vartheta' \vartheta \, M_{i i} N_{j j}$. Expanding the commutators and using cyclicity, we obtain, with $\Lambda = \diag(\lambda_1, \dots, \lambda_{4k})$,
\[
\big[ \Str \big( \Lambda [P, Q] \Lambda [P', Q] \big) \big]_{\vartheta' \vartheta} = (-1)^{|j|} \Big( 2 \lambda_i \lambda_j Q_{i i} Q_{j j} - \lambda_i (Q \Lambda Q)_{j j} - \lambda_j (Q \Lambda Q)_{i i} \Big) .
\]
The signed sum of the first term is $2 (\Str \Lambda Q)^2 = 8 d_k^2$, and those of the other two terms are multiples of $\Str \Lambda = 0$. Therefore $-2 \mathsf C_k \Str((\Lambda Q)^2) = 2 \cdot 8 d_k^2$, which is the second identity in \eqref{e:sigma-casimir-two-values}.
\end{proof}

\subsection{Coefficient norms and weighted spaces}\label{s:sigma-weighted}\label{s:sigma-norms}

We now define the norms in which we measure the coefficients of superfunctions, here and in Part IV, and the weighted spaces on which the semigroup of \zcref{p:sigma-regular} acts. Part III does not use these norms, except through the spaces $L^2_p$ and the multipliers defined below, and its reader may skip \zcref{d:grass-norm,l:grass-norm}. The norms are defined for the Grassmann algebra $\Grass{r}$ of \zcref{d:grass-algebra} with any number $r$ of generators $\xi_1, \dots, \xi_r$, and they measure the odd coordinates on a scale that may differ from one. Let $\omega = (\omega_1, \dots, \omega_r) \in (0, \infty)^r$ be a weight vector, and set $\omega^I = \prod_{a \in I} \omega_a$.

\begin{definition}\label{d:grass-norm}
For $f \in \Grass{r}$, we define
\[
|f|_{1; \omega} = \sum_{I \subset [r]} \omega^I |f_I|, \qquad |f|_{2; \omega} = \Big( \sum_{I \subset [r]} \omega^{2I} |f_I|^2 \Big)^{1/2} .
\]
We write $|f|_1$ and $|f|_2$ when $\omega_a = 1$ for all $a$, and $|f|_{1; \omega}$ and $|f|_{2; \omega}$ with a number $\omega > 0$ when $\omega_a = \omega$ for all $a$. For a matrix $M = \sum_I M_I \xi^I$ with entries in $\Grass{r}$, where the $M_I$ are complex matrices, we define $\|M\|_{1; \omega} = \sum_I \omega^I \|M_I\|_{\mathrm{op}}$, where $\| \cdot \|_{\mathrm{op}}$ is the operator norm. For superfunctions on a superdomain (\zcref{d:grass-superfunction}), these norms are taken pointwise.
\end{definition}

Equivalently, $|f|_{1; \omega}$ and $|f|_{2; \omega}$ are the $\ell^1$ and $\ell^2$ norms of the coefficients of $f$ in the normalized monomials $\omega^{-I} \xi^I$. If we write $\xi_a = \omega_a \hat \xi_a$, then $|f|_{1; \omega}$ is the norm $| \cdot |_1$ in the monomials $\hat \xi^I$. In Part IV, the weight is the number $\omega = h = W^{-1/2}$, and we call $h^{-|I|} \xi^I$ the \emph{$h$-scaled monomials}. A coefficient of degree $s$ then carries the factor $h^s$, and a derivative in an odd coordinate costs a factor $h^{-1}$, as the following lemma makes precise.

\begin{lemma}\label{l:grass-norm}
Let $\omega \in (0, \infty)^r$, let $f, f' \in \Grass{r}$, and let $M$ and $M'$ be matrices with entries in $\Grass{r}$ such that $M M'$ is defined.
\begin{enumerate}[label=(\alph*)]
\item\label{i:grass-norm-sub} We have $|f f'|_{1; \omega} \le |f|_{1; \omega} |f'|_{1; \omega}$, $|f f'|_{2; \omega} \le |f|_{1; \omega} |f'|_{2; \omega}$, and $\|M M'\|_{1; \omega} \le \|M\|_{1; \omega} \|M'\|_{1; \omega}$.
\item\label{i:grass-norm-monomial} For all $I \subset [r]$, multiplication by $\omega^{-I} \xi^I$ is a partial isometry for $| \cdot |_{2; \omega}$, that is, it is isometric on the span of the monomials $\xi^J$ with $J \cap I = \emptyset$, and it annihilates the other monomials.
\item\label{i:grass-norm-derivative} For all $a \in [r]$ and $i \in \{1, 2\}$, we have $|\partial_{\xi_a} f|_{i; \omega} \le \omega_a^{-1} |f|_{i; \omega}$.
\item\label{i:grass-norm-degree} Suppose that $\omega_a \le \omega_* \le 1$ for all $a$, and that $f_I = 0$ for all $I$ with $|I| < s$. Then $|f|_{i; \omega} \le \omega_*^s |f|_i$ for $i \in \{1, 2\}$.
\item\label{i:grass-norm-series} Let $a \in \Grass{r}$ be even with soul $n$, and let $g$ be holomorphic in a neighborhood of $a_\emptyset$. Then
\[
|g(a)|_{1; \omega} \le \sum_{j = 0}^{\lfloor r/2 \rfloor} \frac{|g^{(j)}(a_\emptyset)|}{j!} \, |n|_{1; \omega}^j ,
\]
and in particular $|e^a|_{1; \omega} \le e^{\Re a_\emptyset + |n|_{1; \omega}}$.
\item\label{i:grass-norm-frame} Let $B \in \C^{r \times r}$ be invertible, let $\xi'_a = \sum_b B_{a b} \xi_b$, and let $|f|'_1$ denote the norm $| \cdot |_1$ in the monomials of $\xi'_1, \dots, \xi'_r$. Then $|f|_1 \le \max(1, \|B\|_\infty)^r |f|'_1$, where $\|B\|_\infty = \max_a \sum_b |B_{a b}|$.
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with \ref{i:grass-norm-monomial}. For disjoint $I$ and $J$, we have $\xi^I \xi^J = \pm \xi^{I \cup J}$ and $\omega^{I \cup J} = \omega^I \omega^J$, while $\xi^I \xi^J = 0$ otherwise. These relations imply \ref{i:grass-norm-monomial} and the bound $|\xi^I f'|_{i; \omega} \le \omega^I |f'|_{i; \omega}$ for $i \in \{1, 2\}$. Expanding $f = \sum_I f_I \xi^I$ and using the triangle inequality, we find
\[
|f f'|_{i; \omega} \le \sum_{I} |f_I| \, |\xi^I f'|_{i; \omega} \le |f|_{1; \omega} |f'|_{i; \omega}, \qquad i \in \{1, 2\} ,
\]
and the bound for matrices follows in the same way from $\|M_I M'_J\|_{\mathrm{op}} \le \|M_I\|_{\mathrm{op}} \|M'_J\|_{\mathrm{op}}$. This proves \ref{i:grass-norm-sub}.

Next, for $a \in J$ we have
\[
\partial_{\xi_a} \xi^J = \pm \xi^{J \setminus \{a\}}, \qquad \omega^{J \setminus \{a\}} = \omega_a^{-1} \omega^J ,
\]
while $\partial_{\xi_a} \xi^J = 0$ for $a \notin J$, and \ref{i:grass-norm-derivative} follows. Statement \ref{i:grass-norm-degree} follows from $\omega^I \le \omega_*^{|I|} \le \omega_*^s$ for $|I| \ge s$. By \eqref{e:grass-even-function} and \ref{i:grass-norm-sub}, we have
\[
|g(a)|_{1; \omega} \le \sum_{j = 0}^{\lfloor r/2 \rfloor} \frac{|g^{(j)}(a_\emptyset)|}{j!} \, |n^j|_{1; \omega} \le \sum_{j = 0}^{\lfloor r/2 \rfloor} \frac{|g^{(j)}(a_\emptyset)|}{j!} \, |n|_{1; \omega}^j ,
\]
which is \ref{i:grass-norm-series}, and $|g^{(j)}(a_\emptyset)| = e^{\Re a_\emptyset}$ for $g = \exp$.

It remains to prove \ref{i:grass-norm-frame}. We write $f = \sum_I f'_I \xi'^I$. By \ref{i:grass-norm-sub}, we have
\[
|\xi'^I|_1 \le \prod_{a \in I} |\xi'_a|_1 \le \|B\|_\infty^{|I|} ,
\]
and the claim follows from the triangle inequality.
\end{proof}

We measure superfunctions on $\mathcal M_k$ in these norms with a scalar weight $\omega \in (0, 1]$, which will be chosen in the proof of \zcref{p:sigma-regular}. Recall from \zcref{s:sigma-superfunctions} that two sections differ by a unitary change of odd coordinates. The norm $|f|_{2; \omega}$ is therefore the same in all sections and defines a function on $\mathcal M_k^0$. In the norm $| \cdot |_{2; \omega}$, a coefficient of odd degree $j$ carries the factor $\omega^j$. We call $| \cdot |_{2; \omega}$ the \emph{degree metric} with parameter $\omega$; Part IV also uses it, with the parameters $\omega$ and $\varkappa_k$. For a superfunction $f$ and $p \in \R$, we set
\begin{equation}\label{e:sigma-Lp}
\|f\|_{p}^2 = \int_{\mathcal M_k^0} v^{2p} |f|_{2; \omega}^2 \dd \mathrm{vol} ,
\end{equation}
and we let $L^2_p$ denote the Hilbert space completion of the compactly supported superfunctions in this norm. Its elements are measurable maps $f \colon G_0 \to \Grass{4k^2}$ with the equivariance of \zcref{d:sigma-target} and $\|f\|_p < \infty$. The norms for different $\omega \in (0, 1]$ are equivalent, with constants depending on $\omega$ and $k$. For $q \ge p$, we have $L^2_q \subset L^2_p$, since $v \ge 1$. Writing $f = \sum_I f_I \chi^I$ and $f' = \sum_I f'_I \chi^I$ in the coordinates of a section, we let
\[
\langle f, f' \rangle_p = \int_{\mathcal M_k^0} v^{2p} \sum_I \omega^{2 |I|} f_I \, \overline{f'_I} \dd \mathrm{vol}
\]
denote the inner product of $L^2_p$, in which the complex conjugate is taken coefficientwise. A change of section acts on the coefficients of each degree by a unitary matrix, so that this inner product is the same in all sections. It is unrelated to $\mathcal B_k$.

Derivatives of superfunctions are taken in charts adapted to the action of $G_0$. Let $\mathfrak m_{\R} = \mathfrak m_0 \cap \operatorname{Lie}(G_0)$, with the inner product $2\Str(X X')$ induced by the metric \eqref{e:sigma-body-metric} under $X \mapsto [X, \Lambda]$. We choose a sufficiently small ball about zero in $\mathfrak m_{\R}$, let $\mathcal O_0$ be its image under $X \mapsto e^X \Lambda e^{-X}$, and use the coordinates of $X$ in an orthonormal basis as normal coordinates. The ball and $\mathcal O_0$ are $K_0$-invariant, and we fix the exponential section
\[
\mathsf s_0(e^X \Lambda e^{-X}) = e^X .
\]
For $g' \in G_0$, we call the map $x \mapsto g' x g'^{-1}$ on $\mathcal O_0$, together with the section $\mathsf s_{g'}(y) = g' \, \mathsf s_0(g'^{-1} y g')$ over $g' \mathcal O_0 g'^{-1}$, a \emph{translated chart}. In a translated chart, a superfunction $f$ becomes $f_{g'}(X, \chi) = f(g' e^X; \Theta)$, and we let $|\nabla^j f|_{2; \omega}(g' \Lambda g'^{-1})$ denote the norm $| \cdot |_{2; \omega}$ of the collection of $j$-th partial derivatives in $X$ of its coefficients, at $X = 0$. The exponential section satisfies $\mathsf s_0(\mathsf k x \mathsf k^{-1}) = \mathsf k \mathsf s_0(x) \mathsf k^{-1}$ for $\mathsf k \in K_0$, so equivariance gives
\[
f_{g'\mathsf k}(X, \chi) = \alpha_{\mathsf k}\big( f_{g'}(\operatorname{Ad}_{\mathsf k} X, \chi) \big) .
\]
Thus two translated charts centered at the same point differ by an orthogonal linear change of normal coordinates and a constant unitary change of odd coordinates. Derivatives of order $j$ transform only among derivatives of that order, and their norms differ by at most a factor depending on $j$ and $k$. We say that a superfunction $m$ is a \emph{multiplier of degree $s \ge 0$} if, for every integer $j \ge 0$, there exists $C_j$ such that $|\nabla^j m|_{1; 1} \le C_j v^s$ on $\mathcal M_k^0$. By \zcref{l:grass-norm}\ref{i:grass-norm-sub}, multiplication by a multiplier of degree $s$ maps $L^2_p$ to $L^2_{p - s}$ boundedly, for all $p$.

The operators of the sigma model look the same in all translated charts. By \zcref{l:sigma-casimir}\ref{i:sigma-casimir-commute}, the expression \eqref{e:sigma-casimir-local} of $\mathsf C_k$ in the chart of $g'$ is the same for all $g'$, and the invariant integral \eqref{e:sigma-integral} has the same density $\rho$ in all of them. In this sense, $\mathcal M_k$ has bounded geometry: all coefficients of $\mathsf C_k$ and of the integral, and all their derivatives, are bounded uniformly over the translated charts.

\begin{lemma}\label{l:sigma-multipliers}
\begin{enumerate}[label=(\alph*)]
\item\label{i:sigma-mult-Q} The entries of $Q^{\mathrm{BB}}$ are multipliers of degree one, the entries of $Q^{\mathrm{BF}}$ and $Q^{\mathrm{FB}}$ are multipliers of degree $\frac12$, and the entries of $Q^{\mathrm{FF}}$ are multipliers of degree zero.
\item\label{i:sigma-mult-d} The soul of $d_k$ has coefficients of degree at least two in the $\chi_a$, and $|d_k - (d_k)_\emptyset|_{1; \omega} \le C \omega^2 v$ for all $\omega \in (0, 1]$.
\end{enumerate}
\end{lemma}

\begin{proof}
By the polar decomposition of $U(k, k)$, we may represent the point $g \Lambda g^{-1}$ with $g = \diag(g_{\mathrm B} | g_{\mathrm F})$ and $g_{\mathrm B}$ positive definite. The proof of \zcref{l:sigma-body}\ref{i:sigma-body-trace} then yields
\[
\|g_{\mathrm B}\|_{\mathrm{op}}^2 = \|g_{\mathrm B}^{-1}\|_{\mathrm{op}}^2 \le \Tr(g_{\mathrm B} g_{\mathrm B}^*) = 2(v - 1) ,
\]
and $g_{\mathrm F}$ is unitary. The entries of $\Lambda e^{-2 \Theta}$ are polynomials in the $\chi_a$ with bounded coefficients, and $Q = g \Lambda e^{-2\Theta} g^{-1}$. The blocks $Q^{\mathrm{BB}}$, $Q^{\mathrm{BF}}$, and $Q^{\mathrm{FB}}$, and $Q^{\mathrm{FF}}$ are therefore bounded by $C \|g_{\mathrm B}\|_{\mathrm{op}}^2$, $C \|g_{\mathrm B}\|_{\mathrm{op}}$, and $C$, respectively, and the same bounds for the derivatives follow from the argument of \zcref{l:sigma-body}\ref{i:sigma-body-derivatives}. For \ref{i:sigma-mult-d}, we write
\[
d_k = \tfrac12 \Str(\Lambda g \Lambda e^{-2 \Theta} g^{-1}) .
\]
The terms of odd degree in $\Theta$ vanish, since $g$ is block diagonal in the gradings and odd powers of $\Theta$ have vanishing diagonal blocks, and by \zcref{l:grass-norm}\ref{i:grass-norm-degree} the terms of degree at least two are bounded by $C \omega^2 v$ in $| \cdot |_{1; \omega}$.
\end{proof}

The entries of $Q^{\mathrm{FF}}$, including their souls, are bounded because $Q^{\mathrm{FF}} = g_{\mathrm F} (\Lambda e^{-2\Theta})^{\mathrm{FF}} g_{\mathrm F}^{-1}$ by \eqref{e:sigma-Q}, which involves the compact factor $g_{\mathrm F}$ only. This is why the fermionic sources in the next section are bounded perturbations of the generator.

Finally, we fix cutoffs adapted to the weight $v$. Let $\psi \in C_c^\infty(\R)$ be equal to one on $[-1, 1]$ and to zero outside $[-2, 2]$, and set
\begin{equation}\label{e:sigma-cutoff}
\psi_R = \psi(R^{-1} \log v), \qquad R \ge 1 .
\end{equation}
Let $r_{\mathrm B}$ denote the distance from $\Lambda_{\mathrm B}$ to $Q^{\mathrm{BB}}$ in $H_k$. By the proof of \zcref{l:sigma-body}\ref{i:sigma-body-volume}, we have $v \ge \cosh(r_{\mathrm B} / \sqrt k)$. Thus $\psi_R$ has compact support: its support has bounded $r_{\mathrm B}$, and $S_k$ is compact. Moreover, $|\nabla \psi_R| + |\nabla^2 \psi_R| \le C R^{-1}$ by \zcref{l:sigma-body}\ref{i:sigma-body-derivatives}.

\section{The partition function}\label{s:sigma-partition-sec}

We now define the partition function. The heuristic origin of its generator in the chain of \zcref{s:rep} is explained in \zcref{s:sigma-heuristic}. Throughout this section, $X > 0$ and $\eps > 0$. Recall from \zcref{s:rep} the source supermatrix $\Gamma = \diag(\eps \Lambda_{\mathrm B} | \Gamma_{\mathrm F})$, with
\[
\Gamma_{\mathrm F} = \diag(b_1 - \iu u_1, \dots, b_k - \iu u_k, -b_1 - \iu u_1, \dots, -b_k - \iu u_k) .
\]
In this section we allow $\Gamma_{\mathrm F}$ to be an arbitrary complex diagonal matrix. Using
\[
2 \pi \eps d_k(Q) = \pi \eps \Tr(\Lambda_{\mathrm B} Q^{\mathrm{BB}}) - \pi \eps \Tr(\Lambda_{\mathrm F} Q^{\mathrm{FF}}) ,
\]
we find
\begin{equation}\label{e:sigma-potential}
\pi \Str(Q \Gamma) = 2 \pi \eps \, d_k(Q) - \pi \Tr\big( Q^{\mathrm{FF}} (\Gamma_{\mathrm F} - \eps \Lambda_{\mathrm F}) \big) .
\end{equation}
The first term is the damping. The second term carries the sources, and it vanishes at the base point $(\mathbf u, \mathbf b) = (0, \eps \1)$, where $\Gamma = \Gamma_0 = \eps \Lambda$. We define the \emph{generator} to be the differential operator
\begin{equation}\label{e:sigma-generator}
\mathsf G_{X, \Gamma} = X \mathsf C_k + \pi \Str(Q \Gamma) ,
\end{equation}
acting on superfunctions, where the second term acts by multiplication. For $z \ge 0$, we also set $\mathsf G_{X, z} = X \mathsf C_k + z d_k$, so that $\mathsf G_{X, \Gamma_0} = \mathsf G_{X, 2 \pi \eps}$. When the parameters are fixed, we write $\mathsf G = \mathsf G_{X, \Gamma}$. With the semigroup written as $e^{-t \mathsf G}$, the body part $-\Delta_{\mathcal M_k^0}$ of $\mathsf C_k$ is a nonnegative operator on the underlying manifold, and the damping term $2 \pi \eps d_k$ has nonnegative body.

The following proposition constructs the semigroup $e^{-t \mathsf G}$ on the weighted spaces $L^2_p$ of \zcref{s:sigma-weighted}. The constant function $1$ belongs to $L^2_{-q}$ for large $q$ but not to $L^2_0$, and this is why we need all the weights at once. Recall that $q_1$ is the exponent of \zcref{l:sigma-body}\ref{i:sigma-body-volume}. We call a superfunction $f$ \emph{rapidly decreasing} if it is smooth and $\|\nabla^j f\|_p < \infty$ for all $j \ge 0$ and $p \in \R$, where $\|\nabla^j f\|_p$ is defined by \eqref{e:sigma-Lp} with $|\nabla^j f|_{2; \omega}$ in place of $|f|_{2; \omega}$.

\begin{proposition}\label{p:sigma-regular}
Let $X > 0$, $\eps > 0$, $R_{\mathrm F} > 0$, and $\Gamma = \diag(\eps \Lambda_{\mathrm B} | \Gamma_{\mathrm F})$ with $\|\Gamma_{\mathrm F}\|_{\mathrm{op}} \le R_{\mathrm F}$. There exists $\omega_0 > 0$, depending on $k$, $X$, $\eps$, and $R_{\mathrm F}$, such that the following statements hold for the norms of \eqref{e:sigma-Lp} with $\omega \in (0, \omega_0]$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:sigma-regular-kato} \emph{Realization.} For every $p \in \R$, the operator $\mathsf G$, defined on the compactly supported smooth superfunctions, is closable in $L^2_p$. Its closure $\mathsf G_{(p)}$ is the operator associated by Kato's first representation theorem with a closed sectorial form, shifted by a constant, whose domain is
\[
V_p = \{ f \in L^2_p : \|\nabla f\|_p + \|v^{1/2} f\|_p < \infty \} .
\]
In particular $-\mathsf G_{(p)}$ generates a holomorphic semigroup $e^{-t \mathsf G_{(p)}}$ on $L^2_p$, and the compactly supported smooth superfunctions are a core for $\mathsf G_{(p)}$.
\item\label{i:sigma-regular-weights} \emph{Consistency and smoothing.} For $q \ge p$ and $f \in L^2_q$, we have $e^{-t \mathsf G_{(q)}} f = e^{-t \mathsf G_{(p)}} f$ for all $t \ge 0$; we therefore write $e^{-t \mathsf G}$. For all $p \in \R$, all integers $m \ge 0$, and all $T > 0$, there exists $C$ such that
\[
\|e^{-t \mathsf G}\|_{L^2_p \to L^2_{p + m/2}} \le C t^{-m/2}, \qquad 0 < t \le T .
\]
For $t > 0$ and $f \in L^2_p$, the superfunction $e^{-t \mathsf G} f$ is rapidly decreasing.
\item\label{i:sigma-regular-transpose} \emph{Transpose symmetry.} For all $p \in \R$, the pairing $\mathcal B_k$ is bounded on $L^2_p \times L^2_{-p}$, and
\[
\mathcal B_k(e^{-t \mathsf G} f, f') = \mathcal B_k(f, e^{-t \mathsf G} f'), \qquad f \in L^2_p, \ f' \in L^2_{-p}, \ t \ge 0 .
\]
\item\label{i:sigma-regular-split} \emph{The split pairing.} We have $1 \in L^2_{-q}$ for all $q > q_1/2$, and $e^{-t \mathsf G} 1$ is rapidly decreasing for all $t > 0$. For $0 < a < T$,
\[
\mathcal B_k(e^{-a \mathsf G} 1, e^{-(T - a) \mathsf G} 1) = \int e^{-T \mathsf G} 1 .
\]
In particular, the left side does not depend on $a$.
\item\label{i:sigma-regular-uniform} \emph{Small damping.} Let $z_0 > 0$. For $0 < z \le z_0$, the semigroup $e^{-t \mathsf G_{X, z}}$ is the one of \ref{i:sigma-regular-kato} with $\Gamma = (z/2\pi) \Lambda$. For $z = 0$, the operator $X \mathsf C_k$, defined on the compactly supported smooth superfunctions, is closable in $L^2_p$, and its closure is the operator associated with a closed sectorial form, shifted by a constant, whose domain is
\[
H^1_p = \{ f \in L^2_p : \|\nabla f\|_p < \infty \} ;
\]
we write $e^{-t \mathsf G_{X, 0}}$ for the semigroup that it generates. For every $p \in \R$, there exists $\omega_0 > 0$ such that, for each fixed $\omega \in (0, \omega_0]$, there is a constant $C_\omega$, depending also on $k$, $X$, $p$, and $z_0$, with
\[
\|e^{-t \mathsf G_{X, z}}\|_{L^2_p \to L^2_p} \le e^{C_\omega t}, \qquad 0 \le z \le z_0, \quad t \ge 0 .
\]
Moreover, let $m_1, \dots, m_n$ be multipliers and $T > 0$. Then the coefficients of the superfunction $e^{-s_n \mathsf G_{X, z}} m_n \cdots e^{-s_1 \mathsf G_{X, z}} m_1 1$, and all their derivatives, are bounded on compact subsets of $\mathcal M_k^0$, uniformly in $0 \le z \le z_0$ and $s_1, \dots, s_n \in [0, T]$.
\end{enumerate}
The thresholds $\omega_0$ can be chosen uniformly for $(X, \eps)$ in compact subsets of $(0, \infty)^2$. With an admissible $\omega > 0$ fixed, the constants in \ref{i:sigma-regular-weights} and \ref{i:sigma-regular-uniform} can likewise be chosen uniformly on these compact sets; in \ref{i:sigma-regular-uniform} the bounds remain uniform for $z \in [0, z_0]$. These constants may grow as $\omega \downarrow 0$. The realizations do not depend on $\omega \in (0, \omega_0]$, since the norms for different $\omega$ are equivalent.
\end{proposition}

The statement allows complex fermionic sources, which are needed in \zcref{s:process}. We prove the proposition in \zcref{s:regularity}, and we now define the partition function.

\begin{definition}\label{d:sigma-partition}
Let $X > 0$, $\eps > 0$, $\mathbf u \in \R^k$ with $\sum_j u_j = 0$, and $\mathbf b \in (0, \infty)^k$, and let $\Gamma$ be the source supermatrix of $(\mathbf u, \mathbf b)$. We define the \emph{partition function} of the sigma model by
\begin{equation}\label{e:sigma-partition}
\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) = \mathcal B_k \big( e^{-a \mathsf G_{X, \Gamma}} 1, \, e^{-(1 - a) \mathsf G_{X, \Gamma}} 1 \big), \qquad 0 < a < 1 ,
\end{equation}
where the semigroup is the one of \zcref{p:sigma-regular}.
\end{definition}

By \zcref{p:sigma-regular}\ref{i:sigma-regular-split}, the right side of \eqref{e:sigma-partition} is independent of $a$, and
\[
\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) = \int e^{-\mathsf G_{X, \Gamma}} 1 .
\]
The split form \eqref{e:sigma-partition} is the one that the chain produces in Part IV, where the two ends of the chain are treated separately (see \zcref{s:end}). By \zcref{c:sigma-normalization} below, we have $\mathcal Z^{(k)}_{X, \eps}(0, \eps \1) = 1$.

We now record two further facts about the semigroup, which we use from \zcref{s:sigma-localization} on. The proof of \zcref{p:sigma-regular} also identifies the domains of the realizations. For all $p \in \R$, we have
\begin{equation}\label{e:sigma-domain-p}
D(\mathsf G_{(p)}) = \{ f \in V_p : \mathsf G f \in L^2_p \} ,
\end{equation}
where $\mathsf G f$ is taken in the sense of distributions, and $\mathsf G_{(p)} f = \mathsf G f$ for $f$ in this domain. For an integer $j \ge 1$ and $p \in \R$, we let $D(\mathsf G^j_{(p)})$ denote the set of $f \in L^2_p$ such that $f, \mathsf G f, \dots, \mathsf G^{j - 1} f \in D(\mathsf G_{(p)})$, and we set $D(\mathsf G^0_{(p)}) = L^2_p$.

Let $X > 0$, $z_0 > 0$, and $T > 0$. We call a superfunction of the form
\begin{equation}\label{e:sigma-chain}
f = e^{-s_n \mathsf G_{X, z}} m_n \cdots e^{-s_1 \mathsf G_{X, z}} m_1 1 ,
\end{equation}
where $m_1, \dots, m_n$ are multipliers, $0 \le z \le z_0$, and $s_1, \dots, s_n \in [0, T]$, a \emph{semigroup product}. For $z>0$, \zcref{p:sigma-regular}\ref{i:sigma-regular-weights} implies that $e^{-t \mathsf G} f$ is rapidly decreasing for $t > 0$, but the bounds degenerate as $t \downarrow 0$. Part III evaluates semigroup products, and their components in the polar coordinates of \zcref{s:sigma-radial}, for $s_n$ close to zero (\zcref{s:res-permanent}), and the proofs in \zcref{s:sigma-scalar,s:sigma-singlet} need the regularity of $e^{-t \mathsf G} f$ up to $t = 0$. The following lemma provides these bounds, including at $z=0$. We prove it in \zcref{s:sigma-chains}, with the domains of powers of the generator and interior elliptic estimates.

\begin{lemma}\label{l:sigma-chains}
Let $X > 0$, $z_0 > 0$, and $T > 0$, and fix multipliers $m_1, \dots, m_n$. Let $\mathsf G = \mathsf G_{X, z}$ with $0 \le z \le z_0$, and let $f$ be the semigroup product \eqref{e:sigma-chain}. In \ref{i:sigma-chains-weighted} and \ref{i:sigma-chains-compact}, the constants do not depend on $z \in [0, z_0]$ or on $s_1, \dots, s_n \in [0, T]$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:sigma-chains-weighted} For every integer $j \ge 0$, there exist $p \in \R$ and $C$ such that $f \in D(\mathsf G^j_{(p)})$ and
\begin{equation}\label{e:sigma-chains-weighted}
\sum_{l = 0}^j \|\mathsf G^l f\|_p + \sum_{i = 0}^{2j} \|\nabla^i f\|_p \le C .
\end{equation}
\item\label{i:sigma-chains-compact} For every compact set $K \subset \mathcal M_k^0$ and every $j \ge 0$, the coefficients of $f$ and their derivatives of order at most $j$, in the translated charts, are bounded on $K$ by a constant.
\item\label{i:sigma-chains-time} Fix $z$ and the times $s_1, \dots, s_n$. Then the coefficients of $e^{-t \mathsf G} f$, in the coordinates of a section, are smooth functions of $t \in [0, \infty)$ and of the body point, with derivatives continuous up to $t = 0$, and $u(t) = e^{-t \mathsf G} f$ solves $\partial_t u = - \mathsf G u$. For every $j \ge 0$, there exist $p \in \R$ and $C$ such that $\|\nabla^i e^{-t \mathsf G} f\|_p \le C$ for all $t \in [0, T]$ and $i \le j$.
\item\label{i:sigma-chains-rapid} Statement \ref{i:sigma-chains-time} holds for every rapidly decreasing superfunction $f$ in place of the semigroup product, and then the last bound holds for every $p \in \R$, with $C$ depending on $p$.
\item\label{i:sigma-chains-scalar} Let $A_{X, z} = - X \Delta_{\mathcal M_k^0} + z (d_k)_\emptyset$, acting on complex functions on $\mathcal M_k^0$, and use the norms \eqref{e:sigma-Lp} with $|f|$ in place of $|f|_{2; \omega}$. For $z>0$, the realization and smoothing statements of \zcref{p:sigma-regular}\ref{i:sigma-regular-kato} and \ref{i:sigma-regular-weights} hold for $A_{X,z}$, with the form domain $V_p$ and the positive-time weight gain stated there. At $z=0$, use the closure of $-X\Delta_{\mathcal M_k^0}$ with form domain $H^1_p$, as in \zcref{p:sigma-regular}\ref{i:sigma-regular-uniform}. The small-damping statement \ref{i:sigma-regular-uniform} and statements \ref{i:sigma-chains-weighted}--\ref{i:sigma-chains-rapid} above hold for $A_{X,z}$ throughout $0\le z\le z_0$, with multipliers that are smooth functions $m$ on $\mathcal M_k^0$ such that $|\nabla^j m| \le C_j v^s$ for all $j$.
\end{enumerate}
\end{lemma}

In particular, part \ref{i:sigma-chains-compact} is the last assertion of \zcref{p:sigma-regular}\ref{i:sigma-regular-uniform}, with last time $s_n \in [0, T]$. The proof of \zcref{p:sigma-regular} in \zcref{s:regularity} deduces that assertion from the lemma, whose proof uses only the other parts of the proposition.

\section{Localization and normalization}\label{s:sigma-localization}

We now prove the supersymmetric localization identity on $\mathcal M_k$. The mechanism is that of \zcref{p:grass-localization}. An odd vector field $\mathfrak q$ with zero divergence, and a $\mathfrak q$-exact and $\mathfrak q$-closed function whose body has a unique nondegenerate minimum, allow one to insert a Gaussian factor into the integral of a $\mathfrak q$-closed function, and then to concentrate the integral at the minimum. Historically, this is the localization principle of Parisi and Sourlas \cite{parisi1979random}, and a general form on compact supermanifolds is due to Schwarz and Zaboronsky \cite{schwarz1997supersymmetry}. Since $\mathcal M_k$ is not compact, their theorem does not apply directly. For the noncompact superspace $H^{2|2}$, the corresponding identity is proved in \cite[Appendix~C]{disertori2010quasi}, and for flat superspace in \cite[Theorem~11.4.5]{bauerschmidt2019introduction}; see also \zcref{l:grass-deformation}. We follow these proofs, with the same three steps as in \zcref{p:grass-localization}, and we control the contributions from infinity by the growth bounds of \zcref{s:sigma-weighted}.

Let $q_{\mathrm A} \in \mathfrak k_1$ denote the odd supermatrix with the blocks $(q_{\mathrm A})^{\mathrm{BF}}_{\mathrm{AA}} = (q_{\mathrm A})^{\mathrm{FB}}_{\mathrm{AA}} = \Id_k$ and with all other entries zero, and define $q_{\mathrm R}$ in the same way with the retarded flavors. We set $\mathfrak q_{\mathrm A} = \widehat{q_{\mathrm A}}$ and $\mathfrak q_{\mathrm R} = \widehat{q_{\mathrm R}}$, and we call them the advanced and retarded \emph{supercharges}. We also define the odd superfunctions
\begin{equation}\label{e:sigma-primitive}
\mathfrak o_{\mathrm A} = - \frac12 \sum_{i, j} (-1)^{|i|} (q_{\mathrm A})_{i j} \, Q_{j i}, \qquad \mathfrak o_{\mathrm R} = \frac12 \sum_{i, j} (-1)^{|i|} (q_{\mathrm R})_{i j} \, Q_{j i} .
\end{equation}
For an auxiliary odd generator $\vartheta$, we have $\vartheta \mathfrak o_{\mathrm A} = - \frac12 \Str(\vartheta q_{\mathrm A} Q)$, where $\vartheta q_{\mathrm A}$ is an even supermatrix.

\begin{lemma}\label{l:sigma-supercharge}
We have $q_{\mathrm A}^2 = \Id_{\mathrm A}$ and $[q_{\mathrm A}, \Lambda] = 0$, and
\[
\mathfrak q_{\mathrm A} d_k = 0, \qquad \mathfrak q_{\mathrm A} \mathfrak o_{\mathrm A} = d_k .
\]
The same holds with $\mathrm R$ in place of $\mathrm A$. Moreover, let $\mathfrak k_{\mathrm A} = \{ Y \in \mathfrak k : Y = \Id_{\mathrm A} Y \Id_{\mathrm A} \}$ denote the advanced subalgebra, and $\mathfrak k_{\mathrm R}$ the retarded one. Then a superfunction that is invariant under $\mathfrak k_{\mathrm A}$ satisfies $\mathfrak q_{\mathrm A} f = 0$, and the entries of $Q_{\mathrm{RR}}$ are invariant under $\mathfrak k_{\mathrm A}$.
\end{lemma}

\begin{proof}
The first two identities are immediate from the definition of $q_{\mathrm A}$. Let $\vartheta$ and $\vartheta'$ be auxiliary odd generators. Using \eqref{e:sigma-action} and $[q_{\mathrm A}, \Lambda] = 0$, we have
\[
\vartheta \mathfrak q_{\mathrm A} d_k = \tfrac12 \Str([\Lambda, \vartheta q_{\mathrm A}] Q) = 0 .
\]
Using also the cyclicity of the supertrace for the even supermatrices $\vartheta q_{\mathrm A}$ and $\vartheta' q_{\mathrm A}$, we find
\[
\vartheta \mathfrak q_{\mathrm A} (\vartheta' \mathfrak o_{\mathrm A}) = - \frac12 \Str\big( \vartheta' q_{\mathrm A} [\vartheta q_{\mathrm A}, Q] \big) = - \frac12 \Str\big( [\vartheta' q_{\mathrm A}, \vartheta q_{\mathrm A}] Q \big) = - \vartheta' \vartheta \, \Str(q_{\mathrm A}^2 Q) ,
\]
where we used $\vartheta' q_{\mathrm A} \vartheta q_{\mathrm A} = \vartheta' \vartheta q_{\mathrm A}^2$ and $\vartheta q_{\mathrm A} \vartheta' q_{\mathrm A} = - \vartheta' \vartheta q_{\mathrm A}^2$. Since $\mathfrak q_{\mathrm A}$ is odd, we have
\[
\vartheta \mathfrak q_{\mathrm A} (\vartheta' \mathfrak o_{\mathrm A}) = - \vartheta \vartheta' \mathfrak q_{\mathrm A} \mathfrak o_{\mathrm A} = \vartheta' \vartheta \mathfrak q_{\mathrm A} \mathfrak o_{\mathrm A} ,
\]
and comparing with the previous display, we obtain
\[
\mathfrak q_{\mathrm A} \mathfrak o_{\mathrm A} = - \Str(\Id_{\mathrm A} Q) = - \Str Q_{\mathrm{AA}} .
\]
By \eqref{e:sigma-Q-relations}, we have $\Str Q_{\mathrm{RR}} + \Str Q_{\mathrm{AA}} = \Str Q = 0$, and hence
\[
d_k = \tfrac12 \big( \Str Q_{\mathrm{RR}} - \Str Q_{\mathrm{AA}} \big) = - \Str Q_{\mathrm{AA}} = \Str Q_{\mathrm{RR}} .
\]
This proves the second identity, and the retarded case is the same. Finally, $q_{\mathrm A} \in \mathfrak k_{\mathrm A}$, and for $Y \in \mathfrak k_{\mathrm A}$ the commutator $[Y, Q]$ has vanishing flavor block $\mathrm{RR}$.
\end{proof}

We say that a superfunction $f$ has \emph{polynomially integrable coefficients} if $\int v^K |f|_{1; 1} \dd \mathrm{vol} < \infty$ for all $K \ge 0$. By the Cauchy--Schwarz inequality and $1 \in L^2_{-q}$, every rapidly decreasing superfunction has this property.

\begin{lemma}\label{l:sigma-localization}
Let $A$ be a smooth superfunction on $\mathcal M_k$ with polynomially integrable coefficients, and suppose that $\mathfrak q_{\mathrm A} A = 0$ or $\mathfrak q_{\mathrm R} A = 0$. Then
\[
\int A = A_\emptyset(\Lambda) .
\]
The same holds entrywise for matrices of superfunctions whose entries satisfy these hypotheses.
\end{lemma}

\begin{proof}
We treat $\mathfrak q = \mathfrak q_{\mathrm A}$ and $\mathfrak o = \mathfrak o_{\mathrm A}$; the retarded case is identical. For $c > 0$, we set
\[
I(c) = \int A e^{-c d_k} .
\]
We will show that $I$ is constant, that $I(c) \to \int A$ as $c \downarrow 0$, and that $I(c) \to A_\emptyset(\Lambda)$ as $c \to \infty$. The polynomial integrability of $A$ is used only in the second step; the first and the third use only that $A$ is integrable, because of the factor $e^{-c d_k}$.

We first record two bounds. Recall from \zcref{l:sigma-multipliers} that the entries of $Q$ are multipliers of degree one and that the soul of $d_k$ is bounded by $C v$. Using also $(d_k)_\emptyset \ge v - 1 - k$, from \eqref{e:sigma-dk-body}, we obtain
\begin{equation}\label{e:sigma-gauss-bound}
|e^{-c d_k}|_{1; 1} \le e^{-c (d_k)_\emptyset} \sum_{j \le 2k^2} \frac{(C c v)^j}{j!} \le C (1 + c v)^{2k^2} e^{-c (v - 1 - k)} ,
\end{equation}
where we used that a product of more than $2k^2$ souls of even elements vanishes. Moreover, in the translated charts, the coefficients of $\mathfrak q$ are bounded by $C v^{1/2}$, by the construction in the proof of \zcref{l:sigma-action}, since $\|g^{-1} q_{\mathrm A} g\|_{\mathrm{op}} \le C v^{1/2}$ for the choice of $g$ in the proof of \zcref{l:sigma-multipliers}.

\emph{Step 1.} We show that $I$ is constant. Let $\psi_R$ be the compactly supported cutoff \eqref{e:sigma-cutoff}, and set $w = \mathfrak o A e^{-c d_k}$. By \zcref{l:sigma-supercharge}, \zcref{l:grass-calculus}\ref{i:grass-calculus-chain}, and $\mathfrak q A = 0$, we have $\mathfrak q e^{-c d_k} = 0$ and $\mathfrak q w = d_k A e^{-c d_k}$, as in the proof of \zcref{l:grass-deformation}. The invariance of the integral and the Leibniz rule give
\[
\int \psi_R \, d_k A e^{-c d_k} = \int \psi_R \, \mathfrak q w = - \int (\mathfrak q \psi_R) \, w .
\]
The cutoff $\psi_R$ depends only on the body point. Using \zcref{l:sigma-body}\ref{i:sigma-body-derivatives}, we find that $|\mathfrak q \psi_R|_{1; 1} \le C v R^{-1}$ and that $\mathfrak q \psi_R$ is supported where $v \ge e^R$. The superfunctions $v w$ and $d_k A e^{-c d_k}$ are integrable, locally uniformly in $c > 0$, by \eqref{e:sigma-gauss-bound} and $|\mathfrak o|_{1; 1} \le C v$. Dominated convergence shows that $I$ is differentiable, and letting $R \to \infty$ in the last identity, we obtain
\[
I'(c) = - \int d_k A e^{-c d_k} = - \lim_{R \to \infty} \int \psi_R \, d_k A e^{-c d_k} = 0 .
\]

\emph{Step 2.} We consider the limit $c \downarrow 0$. For $0 < c \le 1$, the bound \eqref{e:sigma-gauss-bound} implies that $|e^{-c d_k}|_{1; 1} \le C v^{2k^2}$, and the coefficients of $e^{-c d_k}$ converge to those of $1$ as $c \downarrow 0$. Since $A$ has polynomially integrable coefficients, dominated convergence yields
\[
\lim_{c \downarrow 0} I(c) = \int A .
\]

\emph{Step 3.} We consider the limit $c \to \infty$. We use the exponential chart at $\Lambda$. Let $\mathcal O$ be a small neighborhood of zero in $\mathfrak m$, and write $Y = Y_0 + \Theta' \in \mathcal O$, with $Y_0$ in the real subspace of $\mathfrak m_0$ described in \zcref{s:sigma-body} and $\Theta' \in \mathfrak m_1$ with coefficients in the odd generators. The superfunctions
\[
Q = e^{Y} \Lambda e^{-Y} = \Lambda e^{-2Y}
\]
define coordinates on a neighborhood of $\Lambda$, by \zcref{l:grass-coordinates}, since at $Y = 0$ this change of coordinates agrees with the regular coordinates to first order. In these coordinates,
\[
d_k = \frac12 \Str(e^{-2Y}) = \sum_{n \ge 1} \frac{2^{2n - 1}}{(2n)!} \Str(Y^{2n}) ,
\]
since odd powers of $Y \in \mathfrak m$ have vanishing flavor blocks $\mathrm{RR}$ and $\mathrm{AA}$, and therefore vanishing supertrace. By Berezin's formula (\zcref{t:grass-change}), the invariant integral has a density $\rho_{\mathcal O}(Y)$ in these coordinates, and its body at $Y = 0$ is a nonzero constant $\rho_{\mathcal O}(0)$, since $\rho$ has nonzero body.

Since $(d_k)_\emptyset = \Str(Y_0^2) + O(|Y_0|^4)$, where $|Y_0|^2 = \Str(Y_0^2)$, we may choose $\mathcal O$ so small that $(d_k)_\emptyset \ge a |Y_0|^2$ on $\mathcal O$ for some $a > 0$. Let $\phi$ be a smooth function on the underlying manifold with compact support in $\mathcal O$ and equal to one near zero, and split
\[
I(c) = \int \phi A e^{-c d_k} + \int (1 - \phi) A e^{-c d_k} .
\]
On the support of $1 - \phi$, the body of $d_k$ is at least $\max(\delta, v - 1 - k)$ for some $\delta > 0$, and \eqref{e:sigma-gauss-bound} and dominated convergence show that the second integral tends to zero.

In the first integral, we change variables by $Y = c^{-1/2} Y'$ in all even and odd coordinates. Since $\mathfrak m$ has as many even as odd dimensions, this change of variables has Berezinian one (\zcref{l:grass-berezin}\ref{i:grass-berezin-linear}), and
\begin{gather*}
\int \phi A e^{-c d_k} = \int_{\mathfrak m} \big( \phi \rho_{\mathcal O} A \big)(c^{-1/2} Y') \, e^{-\Str(Y'^2) - r_c(Y')} \dd Y' , \\
r_c(Y') = \sum_{n \ge 2} \frac{2^{2n - 1}}{(2n)!} \, c^{1 - n} \Str(Y'^{2n}) .
\end{gather*}
As $c \to \infty$, every coefficient of the integrand converges to the corresponding coefficient of $A_\emptyset(\Lambda) \rho_{\mathcal O}(0) e^{-\Str(Y'^2)}$. Indeed, the terms of positive degree in the odd coordinates of $A$, $\rho_{\mathcal O}$, and $\phi$ come with positive powers of $c^{-1/2}$, and the bodies converge to their values at zero. On the support of $\phi(c^{-1/2} \, \cdot)$, the body of the exponent $\Str(Y'^2) + r_c(Y') = c \, d_k(c^{-1/2} Y')$ is at least $a |Y_0'|^2$, and the souls of this exponent and of $(\phi \rho_{\mathcal O} A)(c^{-1/2} Y')$ are bounded uniformly in $c \ge 1$. We obtain the integrable majorant $C e^{-a |Y_0'|^2}$ for the coefficients, and dominated convergence yields
\[
\lim_{c \to \infty} I(c) = A_\emptyset(\Lambda) \, \gamma_k, \qquad \gamma_k = \rho_{\mathcal O}(0) \int_{\mathfrak m} e^{-\Str(Y'^2)} \dd Y' .
\]
The Gaussian integral is nonzero, by \zcref{p:grass-gauss}, since $\Str(Y'^2) = 2 \Str(Y'_{\mathrm{RA}} Y'_{\mathrm{AR}})$ is nondegenerate in the odd coordinates.

We now conclude. Applying Steps 1 and 3 to $A = 1$, which is possible since they use only the integrability of $e^{-c d_k}$ and $d_k e^{-c d_k}$, we find that $\int e^{-c d_k} = \gamma_k$ for all $c > 0$. In particular, the normalization \eqref{e:sigma-integral-normalization} is possible, since $\gamma_k \ne 0$, and it implies that $\gamma_k = 1$, so that
\begin{equation}\label{e:sigma-gauss-mass}
\int e^{-c d_k} = 1, \qquad c > 0 .
\end{equation}
For general $A$, the three steps give
\[
\int A = \lim_{c \downarrow 0} I(c) = \lim_{c \to \infty} I(c) = A_\emptyset(\Lambda) .
\]
The statement for matrices follows by applying the scalar statement to each entry.
\end{proof}

We now prove that the partition function is one at the base point, and that the semigroup preserves the symmetries of the constant function. The point is that the generator commutes with the vector fields of the stabilizer $\mathfrak k$.

\begin{corollary}\label{c:sigma-normalization}
Let $X > 0$ and $z > 0$. Then
\[
\mathcal B_k \big( e^{-a \mathsf G_{X, z}} 1, e^{-(T - a) \mathsf G_{X, z}} 1 \big) = \int e^{-T \mathsf G_{X, z}} 1 = 1, \qquad 0 < a < T .
\]
In particular, $\mathcal Z^{(k)}_{X, \eps}(0, \eps \1) = 1$ for all $\eps > 0$. Moreover, $e^{-t \mathsf G_{X, z}} 1$ is invariant under $\mathfrak k$ for all $t \ge 0$.
\end{corollary}

\begin{proof}
Write $\mathsf G = \mathsf G_{X, z}$, and let $Y \in \mathfrak k$ be homogeneous. Since $d_k$ is invariant under $\mathfrak k$, \zcref{l:sigma-casimir}\ref{i:sigma-casimir-commute} implies that $\widehat Y \mathsf G f = \mathsf G \widehat Y f$ for all smooth $f$. Fix $c > 0$. The superfunctions $\mathsf G^n e^{-c d_k}$, $n \ge 0$, are rapidly decreasing, by \eqref{e:sigma-gauss-bound} and the same bound for the derivatives. Applying \eqref{e:sigma-domain-p} inductively, we find that $e^{-c d_k} \in D(\mathsf G_{(p)}^n)$ for all $p$ and $n$. Let
\[
\Phi(t) = e^{-t \mathsf G} e^{-c d_k}, \qquad \Phi_1(t) = \widehat Y \Phi(t) .
\]

We first show that $\Phi_1 = 0$. The construction of \zcref{l:sigma-action} shows that the operator $\widehat Y$ is of first order, with coefficients bounded by $C v$ in the translated charts. In particular, $\|\widehat Y f\|_{p - 1} \le C \|f\|_{V_p}$. The norm $\|f\|_{V_p}$ is bounded by the graph norm of $\mathsf G_{(p)}$, by \eqref{e:sigma-coercive}. Since $\Phi(0) \in D(\mathsf G_{(p)}^2)$, the map $t \mapsto \Phi(t)$ is continuously differentiable on $[0, \infty)$ into $D(\mathsf G_{(p)})$, with derivative $-\mathsf G \Phi(t)$. It follows that $\Phi_1$ is continuously differentiable into $L^2_{p - 1}$, and
\[
\Phi_1'(t) = - \widehat Y \mathsf G \Phi(t) = - \mathsf G \Phi_1(t) ,
\]
where we used that $\Phi(t)$ is smooth.

The superfunction $\Phi(t)$ is rapidly decreasing for $t > 0$, by \zcref{p:sigma-regular}\ref{i:sigma-regular-weights}, hence so is $\Phi_1(t)$, and $\Phi_1(t) \in D(\mathsf G_{(p - 1)})$ by \eqref{e:sigma-domain-p}. Moreover $\Phi_1(0) = \widehat Y e^{-c d_k} = 0$, since $d_k$ is invariant under $\mathfrak k$. Thus $\Phi_1$ is a strong solution of $\Phi_1' = -\mathsf G_{(p - 1)} \Phi_1$ with zero initial value, and for $0 < s < t$,
\begin{equation}\label{e:sigma-duhamel-zero}
\frac{\dd}{\dd s} e^{-(t - s) \mathsf G} \Phi_1(s) = e^{-(t - s) \mathsf G} \big( \mathsf G \Phi_1(s) + \Phi_1'(s) \big) = 0 .
\end{equation}
Integrating over $s \in (0, t)$, we obtain $\Phi_1(t) = 0$.

Next, we show that $\int \Phi(t)$ is independent of $t$. Since $\Phi(t)$ is rapidly decreasing and $-\mathsf G \Phi(t)$ is its derivative in every $L^2_p$, we have
\[
\frac{\dd}{\dd t} \int \Phi(t) = - X \int \mathsf C_k \Phi(t) - z \int d_k \Phi(t) .
\]
The first integral on the right vanishes. Indeed, \zcref{l:sigma-casimir}\ref{i:sigma-casimir-symmetric} gives
\[
\int \big( \mathsf C_k \Phi(t) \big) \psi_R = \int \Phi(t) \, \mathsf C_k \psi_R, \qquad |\mathsf C_k \psi_R|_{1; 1} \le C R^{-1} ,
\]
where the bound holds since the coefficients of $\mathsf C_k$ are bounded in translated charts and the derivatives of $\psi_R$ of order one and two are bounded by $C R^{-1}$. The right side tends to zero as $R \to \infty$. For the second integral, we note that $d_k \Phi(t)$ is $\mathfrak q_{\mathrm A}$-closed, by the first step with $Y = q_{\mathrm A}$ and \zcref{l:sigma-supercharge}, and that it has polynomially integrable coefficients. Since $(d_k)_\emptyset(\Lambda) = 0$, \zcref{l:sigma-localization} gives
\[
\int d_k \Phi(t) = \big( d_k \Phi(t) \big)_\emptyset(\Lambda) = 0 .
\]
Together with \eqref{e:sigma-gauss-mass}, we conclude that
\[
\int \Phi(t) = \int e^{-c d_k} = 1, \qquad t \ge 0 .
\]

Finally, we let $c \downarrow 0$. Let $q > q_1/2$. Using \eqref{e:sigma-gauss-bound} and dominated convergence, we find that $e^{-c d_k} \to 1$ in $L^2_{-q}$. Using \zcref{p:sigma-regular}\ref{i:sigma-regular-split}, \ref{i:sigma-regular-transpose}, and the boundedness of $\mathcal B_k$ on $L^2_q \times L^2_{-q}$, we obtain
\[
\mathcal B_k \big( e^{-a \mathsf G} 1, e^{-(T - a) \mathsf G} 1 \big) = \mathcal B_k(e^{-T \mathsf G} 1, 1) = \lim_{c \downarrow 0} \mathcal B_k(e^{-T \mathsf G} 1, e^{-c d_k}) = \lim_{c \downarrow 0} \int e^{-T \mathsf G} e^{-c d_k} = 1 .
\]
The case $z = 2 \pi \eps$, $T = 1$ is the identity $\mathcal Z^{(k)}_{X, \eps}(0, \eps \1) = 1$, since $\mathsf G_{X, \Gamma_0} = \mathsf G_{X, 2 \pi \eps}$.

For the last assertion, let $t > 0$. As $c \downarrow 0$, the superfunctions $e^{-t \mathsf G} e^{-c d_k} = e^{-t \mathsf G/2} \big( e^{-t \mathsf G/2} e^{-c d_k} \big)$ converge to $e^{-t \mathsf G} 1$ in the graph norm of $\mathsf G_{(-q)}$, since $e^{-t \mathsf G/2}$ maps $L^2_{-q}$ boundedly into $D(\mathsf G_{(-q)})$. Since $\widehat Y e^{-t \mathsf G} e^{-c d_k} = 0$ and $\|\widehat Y f\|_{-q - 1} \le C \|f\|_{V_{-q}}$, we obtain $\widehat Y e^{-t \mathsf G} 1 = 0$ in the limit. The case $t = 0$ is trivial.
\end{proof}

We will also use localization for matrices of superfunctions. For a matrix $F = (F_{i j})$ of superfunctions indexed by the retarded indices, we say that $F$ is \emph{retarded-covariant} if $\vartheta_Y \widehat Y F = [\vartheta_Y Y_{\mathrm{RR}}, F]$ for all homogeneous $Y \in \mathfrak k$, where $\widehat Y$ acts entrywise and $Y_{\mathrm{RR}}$ is the retarded flavor block of $Y$, and \emph{advanced-invariant} if $\widehat Y F = 0$ for all $Y \in \mathfrak k_{\mathrm A}$. For instance, $Q_{\mathrm{RR}}$ is retarded-covariant and advanced-invariant, by \eqref{e:sigma-action}, and so is $f \Id_{\mathrm R}$ for every superfunction $f$ invariant under $\mathfrak k$. Advanced-covariant and retarded-invariant matrices are defined in the same way, with the roles of the flavors exchanged.

\begin{corollary}\label{c:sigma-ward}
Let $F$ be a retarded-covariant and advanced-invariant matrix of smooth superfunctions. Then there exists $c_\Lambda \in \C$ such that $F_\emptyset(\Lambda) = c_\Lambda \Id_{\mathrm R}$. If moreover $f$ is a smooth superfunction invariant under $\mathfrak k$ such that $f F$ has polynomially integrable coefficients, then
\[
\int f F = f_\emptyset(\Lambda) \, c_\Lambda \, \Id_{\mathrm R} .
\]
The same holds with the roles of the retarded and advanced flavors exchanged.
\end{corollary}

\begin{proof}
The entries of $f F$ are invariant under $\mathfrak k_{\mathrm A}$, hence $\mathfrak q_{\mathrm A}$-closed by \zcref{l:sigma-supercharge}, and \zcref{l:sigma-localization} implies that
\[
\int f F = f_\emptyset(\Lambda) F_\emptyset(\Lambda) .
\]
It remains to identify $F_\emptyset(\Lambda)$.

Let $Y \in \mathfrak k_0$ be supported on the retarded flavors. Since $\Lambda$ is fixed by $e^{sY}$, the body of $\widehat Y F$ vanishes at $\Lambda$, and covariance implies that $[Y_{\mathrm{RR}}, F_\emptyset(\Lambda)] = 0$ for all such $Y$. In other words, $F_\emptyset(\Lambda)$ commutes with all block diagonal matrices of the retarded flavors. Its off-diagonal blocks have odd entries and vanishing body. It follows that $F_\emptyset(\Lambda) = \diag(c_{\mathrm B} \Id_k | c_{\mathrm F} \Id_k)$ for some $c_{\mathrm B}, c_{\mathrm F} \in \C$. Now let $Y \in \mathfrak k_1$ be the supermatrix with a single entry one in the block $Y^{\mathrm{BF}}_{\mathrm{RR}}$. At $g = \Id$, the matrices $A$ and $\Theta_1$ in the construction of \zcref{l:sigma-action} have degree at least one in the $\chi_a$, and the body of $\widehat Y F$ vanishes at $\Lambda$ as well. Taking the body at $\Lambda$ of the $\mathrm{BF}$ block of the covariance relation, we find $\vartheta_Y (c_{\mathrm F} - c_{\mathrm B}) Y^{\mathrm{BF}}_{\mathrm{RR}} = 0$, and we conclude that $c_{\mathrm B} = c_{\mathrm F}$.
\end{proof}