\chapter{The sigma model of rank one}\label{s:sigma-radial} In this section, we treat the sigma model of rank one, in which the objects of \zcref{s:sigma} can be computed explicitly. We show that the semigroup acts on invariant superfunctions as the semigroup of the singlet diffusion, so that its value on the constant function is the singlet state $\Psi^{X, z}_t$, and we prove the properties of the kernels $K_X$, $\widetilde K_X$, and $I_X$ that \zcref{d:kernels} uses. We then develop the calculus of rank-one pairings, which evaluates the pairings of singlet states with entries of $Q$ inserted, and we show how $K_X$ and $I_X$ arise from the derivatives of the partition function in its source, and $\widetilde K_X$ from an insertion at a single site. Part III uses these results throughout, and Part IV does not use them. From \zcref{s:sigma}, we use its definitions at $k = 1$, the symmetries and the Casimir operator of $\mathcal M_1$ through \zcref{l:sigma-action,l:sigma-casimir,l:sigma-casimir-values}, the bounds of \zcref{l:sigma-body,l:sigma-multipliers}, the semigroups of \zcref{p:sigma-regular} with the regularity of semigroup products in \zcref{l:sigma-chains}, and the localization identity \zcref{l:sigma-localization}, together with the supercharges of \zcref{l:sigma-supercharge} and its consequences in \zcref{c:sigma-normalization,c:sigma-ward}. \zcref[S]{l:sigma-entire}, by which the partition function extends to an entire function of the sources, holds at every rank, and \zcref{s:proc-continuation} uses it in this generality. Throughout this section, $k = 1$, and we order the indices as $(\mathrm{RB}, \mathrm{RF}, \mathrm{AB}, \mathrm{AF})$. Recall that the underlying manifold of $\mathcal M_1$ is the product $H_1 \times S_1$ of the hyperbolic plane and the sphere, and that $\mathcal M_1$ has four odd coordinates. We introduce polar coordinates on $\mathcal M_1$, in which the invariant integral and the generator $\mathsf G_{X, z}$ reduce to explicit expressions in the radial coordinates $(B, F) \in D$. The polar coordinates are singular at the point $\Lambda$, which corresponds to the corner $(1, 1)$ of $D$, and the corresponding corner term, a boundary term of the kind described in \zcref{s:grass-boundary}, is computed by localization. We write $p = (B, F)$ for points of $D$, as in \zcref{d:kernels}. We set $d = B - F$, and $D^\circ = (1, \infty) \times (-1, 1)$. As in \zcref{s:intro-results}, we call $(1, 1)$ the corner of $D$. We recall from \zcref{d:kernels} the operator $\Delta_D$ and the diffusion $(\tilde B_t, \tilde F_t)$ with generator $X \Delta_D$, which we call the \emph{radial diffusion}, and the semigroup $e^{t(X \Delta_D - z d)}$, and we define the \emph{singlet operator} \begin{equation}\label{e:sigma-singlet-operator} \Delta^{\mathrm s}_D = d \, \Delta_D \, d^{-1} = \Delta_D - \frac{2(B^2 - 1)}{d} \partial_B + \frac{2(1 - F^2)}{d} \partial_F \end{equation} on $D$ minus the corner. The second form follows from $\Delta_D d^{-1} = 0$, which one checks directly using $\partial_B d = 1$, $\partial_F d = -1$, and $(B^2 - 1) + (1 - F^2) = d (B + F)$. We will use the identity $\Delta^{\mathrm s}_D (d h) = d \, \Delta_D h$ repeatedly. \section{Polar coordinates}\label{s:sigma-polar} For $p \in D$ and $\theta_{\mathrm B}, \theta_{\mathrm F} \in \R / 2\pi \Z$, we let $Q_0 = Q_0(B, F, \theta_{\mathrm B}, \theta_{\mathrm F})$ denote the block diagonal supermatrix with blocks \[ Q_0^{\mathrm{BB}} = \begin{pmatrix} B & \sqrt{B^2 - 1} \, e^{\iu \theta_{\mathrm B}} \\ - \sqrt{B^2 - 1} \, e^{-\iu \theta_{\mathrm B}} & -B \end{pmatrix}, \qquad Q_0^{\mathrm{FF}} = \begin{pmatrix} F & \iu \sqrt{1 - F^2} \, e^{\iu \theta_{\mathrm F}} \\ - \iu \sqrt{1 - F^2} \, e^{-\iu \theta_{\mathrm F}} & -F \end{pmatrix} , \] which is the point of $\mathcal M_1^0$ with coordinates $(B, \theta_{\mathrm B}, F, \theta_{\mathrm F})$ of \zcref{s:sigma-body}. Let $\xi_1, \dots, \xi_4$ be odd generators, and let $\Theta_\xi$ denote the odd supermatrix in $\mathfrak k_1$ with entries $\xi_1$ at $(\mathrm{RB}, \mathrm{RF})$, $\xi_2$ at $(\mathrm{RF}, \mathrm{RB})$, $\xi_3$ at $(\mathrm{AB}, \mathrm{AF})$, and $\xi_4$ at $(\mathrm{AF}, \mathrm{AB})$. We define the \emph{polar coordinates} by \begin{equation}\label{e:sigma-polar} Q = e^{-\Theta_\xi} Q_0(B, F, \theta_{\mathrm B}, \theta_{\mathrm F}) \, e^{\Theta_\xi} , \end{equation} and we set $n_{\mathrm R} = \xi_1 \xi_2$ and $n_{\mathrm A} = \xi_3 \xi_4$. These are the coordinates of Efetov \cite{efetov1997supersymmetry}, written as in \cite[Section~2.4]{mirlin2000supersymmetry}, up to a linear change of the odd variables. \begin{lemma}\label{l:sigma-polar} \begin{enumerate}[label=(\alph*)] \item\label{i:sigma-polar-chart} The map \eqref{e:sigma-polar} identifies the superfunctions on $\mathcal M_1$, restricted to the open set of body points with $B > 1$ and $|F| < 1$, with the superfunctions of $(B, F, \theta_{\mathrm B}, \theta_{\mathrm F})$ in $D^\circ \times (\R/2\pi\Z)^2$ and of $\xi_1, \dots, \xi_4$. The same holds near the edges $\{B = 1\}$ and $\{F = \pm 1\}$ of $D$ minus the corner, after the polar coordinates $(B, \theta_{\mathrm B})$, respectively $(F, \theta_{\mathrm F})$, are replaced by Cartesian coordinates of the plane, respectively of the sphere. \item\label{i:sigma-polar-entries} In polar coordinates, we have $Q^2 = \Id$ and \begin{align*} Q^{\mathrm{BB}}_{\mathrm{RR}} &= B + d \, n_{\mathrm R}, & Q^{\mathrm{FF}}_{\mathrm{RR}} &= F + d \, n_{\mathrm R}, & Q^{\mathrm{BF}}_{\mathrm{RR}} &= d \, \xi_1, & Q^{\mathrm{FB}}_{\mathrm{RR}} &= - d \, \xi_2, \\ Q^{\mathrm{BB}}_{\mathrm{AA}} &= -B - d \, n_{\mathrm A}, & Q^{\mathrm{FF}}_{\mathrm{AA}} &= -F - d \, n_{\mathrm A}, & Q^{\mathrm{BF}}_{\mathrm{AA}} &= - d \, \xi_3, & Q^{\mathrm{FB}}_{\mathrm{AA}} &= d \, \xi_4 . \end{align*} In particular, $d_1 = \frac12 \Str(\Lambda Q) = d$ and $\frac14 \Str((\Lambda Q)^2) = B^2 - F^2$, with no odd terms. \item\label{i:sigma-polar-top} Let $[f]_{\mathrm{top}}$ denote the coefficient of $\xi_1 \xi_2 \xi_3 \xi_4$ in $f$, and $\langle f \rangle_{\mathrm{top}}$ its average over $\theta_{\mathrm B}$ and $\theta_{\mathrm F}$ with respect to $\frac{\dd \theta_{\mathrm B}}{2\pi} \frac{\dd \theta_{\mathrm F}}{2\pi}$. Then \begin{gather*} [Q^{\mathrm{BB}}_{\mathrm{RR}} Q^{\mathrm{BB}}_{\mathrm{AA}}]_{\mathrm{top}} = [Q^{\mathrm{FF}}_{\mathrm{RR}} Q^{\mathrm{FF}}_{\mathrm{AA}}]_{\mathrm{top}} = - d^2, \\ \langle Q^{\mathrm{BB}}_{\mathrm{RA}} Q^{\mathrm{BB}}_{\mathrm{AR}} \rangle_{\mathrm{top}} = \langle Q^{\mathrm{FB}}_{\mathrm{AR}} Q^{\mathrm{BF}}_{\mathrm{RA}} \rangle_{\mathrm{top}} = - (B^2 - F^2) . \end{gather*} \end{enumerate} \end{lemma} \begin{proof} We begin with \ref{i:sigma-polar-entries}. The flavor blocks of $\Theta_\xi$ are $(\Theta_\xi)_{\mathrm{RR}} = \big( \begin{smallmatrix} 0 & \xi_1 \\ \xi_2 & 0 \end{smallmatrix} \big)$ and $(\Theta_\xi)_{\mathrm{AA}} = \big( \begin{smallmatrix} 0 & \xi_3 \\ \xi_4 & 0 \end{smallmatrix} \big)$, and since $(\xi_1 \xi_2)^2 = 0$, \[ e^{\pm \Theta_\xi}_{\mathrm{RR}} = \begin{pmatrix} 1 + \frac12 \xi_1 \xi_2 & \pm \xi_1 \\ \pm \xi_2 & 1 - \frac12 \xi_1 \xi_2 \end{pmatrix} , \] and similarly for $\mathrm{AA}$. Since $\Theta_\xi$ is block diagonal in the flavors, we have \[ Q_{\mathrm{RR}} = e^{-\Theta_\xi}_{\mathrm{RR}} \diag(B, F) \, e^{\Theta_\xi}_{\mathrm{RR}} , \] and multiplying out gives the first line; the second line is the same computation with $\diag(-B, -F)$. For the supertraces, we use $\Str(\Lambda Q) = \Str Q_{\mathrm{RR}} - \Str Q_{\mathrm{AA}}$, the identity \[ \Str((\Lambda Q)^2) = 2 \Str(Q_{\mathrm{RR}}^2) + 2 \Str(Q_{\mathrm{AA}}^2) - 2 \Str(\Id) , \] which follows from $Q^2 = \Id$, and $\Str(Q_{\mathrm{RR}}^2) = \Str(\diag(B, F)^2)$, which follows from cyclicity. Part \ref{i:sigma-polar-top} is obtained by multiplying out the products in the Grassmann algebra with four generators; for the first identity, only the product of the terms $d \, n_{\mathrm R}$ and $-d \, n_{\mathrm A}$ contributes. (We have also checked all these identities by computer algebra.) We now prove \ref{i:sigma-polar-chart}. The body map is the product of the polar coordinates of the plane and of the sphere, which is a diffeomorphism of $D^\circ \times (\R/2\pi\Z)^2$ onto the set $\mathcal O^\circ$ of body points with $B > 1$ and $|F| < 1$. Let $\mathsf s$ be a section over an open set $V \subset \mathcal O^\circ$, with coordinates $(x, \chi)$. By \eqref{e:sigma-polar}, the regular coordinates are superfunctions of the polar coordinates over the preimage of $V$, namely $x = y(B, F, \theta_{\mathrm B}, \theta_{\mathrm F}; \xi)$ and $\chi = \eta(B, F, \theta_{\mathrm B}, \theta_{\mathrm F}; \xi)$, with $y$ even and $\eta$ odd. We apply \zcref{l:grass-coordinates} with $m = r = 4$, with $(B, F, \theta_{\mathrm B}, \theta_{\mathrm F}; \xi_1, \dots, \xi_4)$ as the old coordinates, and with $(y, \eta)$ as the new ones. The body map $y_\emptyset$ is a diffeomorphism, and it remains to show that the body of the matrix $(\partial_{\xi_a} \eta_b)$ is invertible, or equivalently that the linear map from $(\xi_1, \dots, \xi_4)$ to the terms of degree one of the regular odd coordinates $\Theta$ is injective. \zcref[S]{l:grass-coordinates} then shows that $f \mapsto f(y, \eta)$ identifies the superfunctions over $V$ with the superfunctions of the polar coordinates, which is \ref{i:sigma-polar-chart}. Recall from \zcref{r:sigma-superfunction-local} that the part of degree one of the odd entries of $Q$ is the matrix $-2 g \Lambda \Theta g^{-1}$, which is linear in $\Theta$. Suppose that the terms of degree one of $\Theta$ vanish. Then so do those of $Q^{\mathrm{BF}}_{\mathrm{RR}} = d \xi_1$, $Q^{\mathrm{FB}}_{\mathrm{RR}} = - d \xi_2$, $Q^{\mathrm{BF}}_{\mathrm{AA}} = - d \xi_3$, and $Q^{\mathrm{FB}}_{\mathrm{AA}} = d \xi_4$, and $d > 0$ forces $\xi = 0$. The same argument applies near the edges with Cartesian coordinates, since $d > 0$ there except at the corner. \end{proof} We first identify the invariant integral away from the corner. Let $\mathcal O$ denote the set of body points with $d > 0$, that is, $\mathcal M_1^0$ minus the point $\Lambda$, which is connected, and let $\mathcal O^\circ \subset \mathcal O$ denote the set of body points with $B > 1$ and $|F| < 1$, on which the polar coordinates are defined. The following lemma says that, away from the corner, the invariant integral has the density $d^{-2}$ in polar coordinates, up to a constant factor. \begin{lemma}\label{l:sigma-polar-density} There exists a constant $c_1$ such that \begin{equation}\label{e:sigma-radial-compact} \int f = c_1 \int_D \frac{\langle f \rangle_{\mathrm{top}}}{d^2} \dd B \dd F \end{equation} for all smooth superfunctions $f$ on $\mathcal M_1$ whose coefficients have compact support in $\mathcal O$. \end{lemma} \begin{proof} For $f$ as in the statement, we set \[ J(f) = \int_{D^\circ \times (\R/2\pi\Z)^2} \frac{[f]_{\mathrm{top}}}{d^2} \dd B \dd F \frac{\dd \theta_{\mathrm B}}{2\pi} \frac{\dd \theta_{\mathrm F}}{2\pi} = \int_D \frac{\langle f \rangle_{\mathrm{top}}}{d^2} \dd B \dd F , \] where $[f]_{\mathrm{top}}$ is computed in polar coordinates. Near the edges of $D$, we use the Cartesian coordinates of \zcref{l:sigma-polar}\ref{i:sigma-polar-chart}. Since $\dd B \dd \theta_{\mathrm B} = B^{-1} \dd^2 w$ with $w = \sqrt{B^2 - 1} \, e^{\iu \theta_{\mathrm B}}$, and the analogous factor for the sphere is smooth, $J$ is a Berezin integral with a smooth and nonvanishing density in each of these coordinate systems over $\mathcal O$. By \zcref{l:sigma-polar}\ref{i:sigma-polar-chart} and \zcref{t:grass-change}, the same holds for the invariant integral, and hence there is an even superfunction $\sigma$ on $\mathcal O$ with nonvanishing body such that \[ \int f = J(\sigma f) \] for all $f$ with compact support in $\mathcal O$. We show that $\sigma$ is constant. We first show that $J$ is invariant under $\mathfrak k$, that is, $J(\widehat Y f) = 0$ for all homogeneous $Y \in \mathfrak k$ and all $f$ with compact support in $\mathcal O^\circ$. By the uniqueness in \zcref{l:sigma-action}\ref{i:sigma-action-exist}, the even derivation $\vartheta_Y \widehat Y$ maps $f$ to $f(e^{\vartheta_Y Y} Q e^{-\vartheta_Y Y}) - f(Q)$, to first order in $\vartheta_Y$, since both derivations map $Q$ to $[\vartheta_Y Y, Q]$. At rank one, $K_0$ consists of diagonal unitary matrices. Conjugation by an element of $K_0$ commutes with $\Lambda$, shifts the phases $\theta_{\mathrm B}$ and $\theta_{\mathrm F}$ by constants, and multiplies each $\xi_a$ by a phase, with inverse phases for $\xi_1$ and $\xi_2$ and for $\xi_3$ and $\xi_4$. It therefore preserves $\dd \theta_{\mathrm B} \dd \theta_{\mathrm F}$ and the coefficient of $\xi_1 \xi_2 \xi_3 \xi_4$, and $J$ is invariant under $K_0$. Differentiating, we obtain $J(\widehat Y f) = 0$ for $Y$ in the Lie algebra of $K_0$, and for all $Y \in \mathfrak k_0$ by complex linearity. For the odd generator $Y = E_{\mathrm{RB}, \mathrm{RF}}$ of $\mathfrak k_1$ and $\vartheta = \vartheta_Y$, a direct computation with the matrices $e^{\pm \Theta_\xi}_{\mathrm{RR}}$ of the proof of \zcref{l:sigma-polar} gives \[ e^{\vartheta Y} e^{-\Theta_\xi} = e^{-\Theta_{\xi'}} \Big( \big( 1 - \tfrac12 \vartheta \xi_2 \big) \Id_{\mathrm R} + \Id_{\mathrm A} \Big) , \qquad \xi' = (\xi_1 - \vartheta, \xi_2, \xi_3, \xi_4) , \] to first order in $\vartheta$. Conjugation by the last factor multiplies the flavor block $\mathrm{RA}$ of $Q_0$ by $1 - \frac12 \vartheta \xi_2$ and the block $\mathrm{AR}$ by its inverse, which is the shift of both phases by $\frac{\iu}{2} \vartheta \xi_2$. In polar coordinates, we conclude that \[ \vartheta \widehat Y = - \vartheta \, \partial_{\xi_1} + \tfrac{\iu}{2} \vartheta \xi_2 \big( \partial_{\theta_{\mathrm B}} + \partial_{\theta_{\mathrm F}} \big) . \] Neither coefficient depends on the variable that it differentiates, so $J(\vartheta \widehat Y f) = 0$, by Berezin integration by parts in $\xi_1$ (\zcref{l:grass-berezin}\ref{i:grass-berezin-odd-ibp}) and by periodicity in the phases. The other three generators $E_{\mathrm{RF}, \mathrm{RB}}$, $E_{\mathrm{AB}, \mathrm{AF}}$, and $E_{\mathrm{AF}, \mathrm{AB}}$ of $\mathfrak k_1$ are treated in the same way, with $\xi_2$, $\xi_3$, or $\xi_4$ shifted. Next, we show that $\sigma$ is a function of $(B, F)$. Since $\sigma$ is even, the invariance of both integrals gives, for $Y \in \mathfrak k$ and $f$ with compact support in $\mathcal O^\circ$, \[ J \big( (\widehat Y \sigma) f \big) = J \big( \widehat Y (\sigma f) \big) - J(\sigma \, \widehat Y f) = 0 - \int \widehat Y f = 0 , \] and therefore $\widehat Y \sigma = 0$ for all $Y \in \mathfrak k$. Write $\Theta_\xi = \sum_a \xi_a Y_a$ with $Y_a \in \mathfrak k_1$. By \eqref{e:sigma-action} and the chain rule, we have $\frac{\dd}{\dd s} \sigma(e^{-s \Theta_\xi} Q_0 e^{s \Theta_\xi}) = - \sum_a \xi_a (\widehat{Y_a} \sigma)(e^{-s \Theta_\xi} Q_0 e^{s \Theta_\xi}) = 0$. Hence $\sigma$ has no odd terms in polar coordinates, and by the invariance under $K_0$, it does not depend on the phases. Thus $\sigma = \sigma(B, F)$ on $\mathcal O^\circ$. We now show that $\sigma$ is constant, with the invariance of the integral under the even generator $Y_0 = E_{\mathrm{RB}, \mathrm{AB}}$ of $\mathfrak m_0$. By \zcref{l:sigma-polar}\ref{i:sigma-polar-entries}, the superfunctions $d_1$ and $\frac14 \Str((\Lambda Q)^2)$ equal $d$ and $m = B^2 - F^2$ in polar coordinates, without odd terms. Since $\partial(d, m)/\partial(B, F) = 2d \ne 0$ and $(d, m/d)$ determines $(B, F)$, the pair $(d, m)$ is a system of coordinates on $D^\circ$, whose coordinate vector fields are \[ \partial_d = - \frac{F \partial_B + B \partial_F}{d} , \qquad \partial_m = \frac{\partial_B + \partial_F}{2d} . \] In particular, every $\phi \in C^\infty_{\rm c}(D^\circ)$ is a smooth function of $(d, m)$. Substituting $d_1$ and $\frac14 \Str((\Lambda Q)^2)$ for $d$ and $m$, we obtain a superfunction on $\mathcal O^\circ$ that equals $\phi(B, F)$ in polar coordinates, with no odd terms, and we denote it also by $\phi$. We now compute the action of $\widehat{Y_0}$. By \zcref{l:sigma-action}\ref{i:sigma-action-group}, we have $\widehat{Y_0} Q = [Y_0, Q]$. Using $\Str(M Y_0) = M_{\mathrm{AB}, \mathrm{RB}}$ for supermatrices $M$ and the cyclicity of the supertrace, we find \begin{equation}\label{e:sigma-Y0-action} \begin{gathered} \widehat{Y_0} d_1 = Q_{\mathrm{AB}, \mathrm{RB}} , \qquad \widehat{Y_0} \tfrac14 \Str \big( (\Lambda Q)^2 \big) = \tfrac12 [Q, \Lambda Q \Lambda]_{\mathrm{AB}, \mathrm{RB}} , \\ \widehat{Y_0} Q_{\mathrm{RB}, \mathrm{AB}} = Q^{\mathrm{BB}}_{\mathrm{AA}} - Q^{\mathrm{BB}}_{\mathrm{RR}} , \qquad \widehat{Y_0} Q^{\mathrm{FF}}_{\mathrm{RR}} = 0 . \end{gathered} \end{equation} Next, we compute two phase averages. For a superfunction $A$ on $\mathcal O^\circ$, let $\langle A \rangle$ denote the average of its coefficients over the phases. Since $\Lambda$ commutes with $\Theta_\xi$, we have $[Q, \Lambda Q \Lambda] = e^{-\Theta_\xi} [Q_0, \Lambda Q_0 \Lambda] e^{\Theta_\xi}$. Multiplying the $2 \times 2$ blocks, we obtain $[Q_0, \Lambda Q_0 \Lambda]_{\mathrm{AR}} = 4 \diag(B, F) (Q_0)_{\mathrm{AR}}$. We write $Q_{\mathrm{RA}} = e^{-\Theta_\xi}_{\mathrm{RR}} (Q_0)_{\mathrm{RA}} e^{\Theta_\xi}_{\mathrm{AA}}$ and $Q_{\mathrm{AR}} = e^{-\Theta_\xi}_{\mathrm{AA}} (Q_0)_{\mathrm{AR}} e^{\Theta_\xi}_{\mathrm{RR}}$, and we multiply out. The terms with the phases $e^{\pm \iu (\theta_{\mathrm B} - \theta_{\mathrm F})}$ average to zero, and we find \begin{equation}\label{e:sigma-polar-averages} \begin{gathered} \big\langle Q_{\mathrm{AB}, \mathrm{RB}} Q_{\mathrm{RB}, \mathrm{AB}} \big\rangle = - (B^2 - 1)(1 + n_{\mathrm R})(1 + n_{\mathrm A}) - (1 - F^2) \, n_{\mathrm R} n_{\mathrm A} , \\ \big\langle [Q, \Lambda Q \Lambda]_{\mathrm{AB}, \mathrm{RB}} Q_{\mathrm{RB}, \mathrm{AB}} \big\rangle = - 4B (B^2 - 1)(1 + n_{\mathrm R})(1 + n_{\mathrm A}) - 4F (1 - F^2) \, n_{\mathrm R} n_{\mathrm A} . \end{gathered} \end{equation} In both lines, the first term comes from the product $-(B^2 - 1)$ of the $\mathrm{BB}$ entries of $(Q_0)_{\mathrm{RA}}$ and $(Q_0)_{\mathrm{AR}}$, and the second from the product $1 - F^2$ of their $\mathrm{FF}$ entries, with the sign of $\xi_3 \xi_2 \xi_1 \xi_4 = - n_{\mathrm R} n_{\mathrm A}$. Let $\phi \in C^\infty_{\rm c}(D^\circ)$, and set $\ell_1 = \phi \, Q_{\mathrm{RB}, \mathrm{AB}}$ and $\ell_2 = \phi \, Q_{\mathrm{RB}, \mathrm{AB}} Q^{\mathrm{FF}}_{\mathrm{RR}}$. These superfunctions have compact support in $\mathcal O^\circ$, and the invariance of the integral gives $J(\sigma \, \widehat{Y_0} \ell_i) = \int \widehat{Y_0} \ell_i = 0$. We compute $\langle \widehat{Y_0} \ell_i \rangle_{\mathrm{top}}$ with the chain rule, \eqref{e:sigma-Y0-action}, \eqref{e:sigma-polar-averages}, and $Q^{\mathrm{FF}}_{\mathrm{RR}} = F + d \, n_{\mathrm R}$ and $Q^{\mathrm{BB}}_{\mathrm{AA}} - Q^{\mathrm{BB}}_{\mathrm{RR}} = -2B - d (n_{\mathrm R} + n_{\mathrm A})$ from \zcref{l:sigma-polar}\ref{i:sigma-polar-entries}. We obtain $\langle \widehat{Y_0} \ell_i \rangle_{\mathrm{top}} = \alpha_i \partial_d \phi + \beta_i \partial_m \phi + \gamma_i \phi$, where \[ \begin{gathered} \alpha_1 = -(B^2 - F^2) , \qquad \beta_1 = -2 \big( B (B^2 - 1) + F (1 - F^2) \big) , \qquad \gamma_1 = 0 , \\ \alpha_2 = -F (B^2 - F^2) - d (B^2 - 1) , \qquad \beta_2 = -2 \big( B^2 (B^2 - 1) + F^2 (1 - F^2) \big) , \qquad \gamma_2 = -d^2 . \end{gathered} \] For $\ell_2$, the factor $F + d \, n_{\mathrm R}$ multiplies the top coefficient by $F$ and the coefficient of $n_{\mathrm A}$ by $d$, and we used $F + d = B$. We now return to the coordinates $(B, F)$. Inserting the coordinate vector fields $\partial_d$ and $\partial_m$ computed above, we find $\alpha_i \partial_d \phi + \beta_i \partial_m \phi = V^{(i)} \cdot \nabla \phi$ with \[ V^{(1)} = \big( 1 - B^2 , \, 1 - F^2 \big) , \qquad V^{(2)} = \big( B (1 - B^2) , \, F (1 - F^2) \big) , \] and therefore \[ 0 = J \big( \sigma \, \widehat{Y_0} \ell_i \big) = \int_D \frac{\sigma}{d^2} \big( V^{(i)} \cdot \nabla \phi + \gamma_i \phi \big) \dd B \dd F , \qquad i \in \{1, 2\} . \] Finally, we conclude that $\sigma$ is constant. A direct computation with $\partial_B d = 1$ and $\partial_F d = -1$ gives $\nabla \cdot (d^{-2} V^{(1)}) = 0$ and $\nabla \cdot (d^{-2} V^{(2)}) = -1 = d^{-2} \gamma_2$. Integrating by parts, we find $\int_D d^{-2} \phi \, V^{(i)} \cdot \nabla \sigma \dd B \dd F = 0$ for all $\phi$, so that $V^{(i)} \cdot \nabla \sigma = 0$ on $D^\circ$ for $i = 1, 2$. The vectors $V^{(1)}$ and $V^{(2)}$ are linearly independent at every point of $D^\circ$, since $\det(V^{(1)}, V^{(2)}) = (1 - B^2)(1 - F^2)(F - B) \ne 0$ there. Hence $\nabla \sigma = 0$, and $\sigma = c_1$ for a constant $c_1$ on the connected set $\mathcal O^\circ$, and by continuity on $\mathcal O$. This proves \eqref{e:sigma-radial-compact}. \end{proof} At the corner, the odd part of the Jacobian degenerates, since $d = 0$ there, and the polar coordinates are singular at $\Lambda$. The following lemma is the corresponding integration formula. Its corner term is the boundary term of polar coordinates described in \zcref{s:grass-boundary}, and its bulk term is the classical form of the invariant measure in Efetov's coordinates. \begin{lemma}\label{l:sigma-radial-integral} Let $f$ be a rapidly decreasing superfunction on $\mathcal M_1$. Then \begin{equation}\label{e:sigma-radial-integral} \int f = f_\emptyset(\Lambda) + \lim_{r \downarrow 0} \int_{\{p \in D : d > r\}} \frac{\langle f \rangle_{\mathrm{top}}(B, F)}{d^2} \dd B \dd F . \end{equation} The bulk density $d^{-2} \langle f \rangle_{\mathrm{top}}$ is absolutely integrable on $D$, so the limit is the integral over $D$. \end{lemma} \begin{proof} By \zcref{l:sigma-polar-density}, the identity \eqref{e:sigma-radial-compact} holds for all $f$ with compact support in $\mathcal O$, with a constant $c_1$ that we determine at the end of the proof. Now let $f$ be general. Let $\psi \in C^\infty([0, \infty))$ be equal to one on $[0, 1]$ and to zero on $[2, \infty)$, and set $\psi_r = \psi(d_1/r)$ for $r > 0$. In polar coordinates, we have $\psi_r = \psi(d/r)$, by \zcref{l:sigma-polar}\ref{i:sigma-polar-entries}, and $\psi_r$ is invariant under $\mathfrak k$. We write $f = \psi_r f + (1 - \psi_r) f$. The coefficients of the second term are supported where $d \ge r$. We justify extending \eqref{e:sigma-radial-compact} to this term by exhaustion with the cutoffs $\psi_R$ of \eqref{e:sigma-cutoff}. On $\{B \ge 2\}$, choose body representatives $Q_0 = g \Lambda g^{-1}$ whose hyperbolic block and its inverse have entries bounded by $C B^{1/2}$, and whose spherical block is bounded in each of finitely many spherical charts. Then \[ Q = g \, e^{-g^{-1} \Theta_\xi g} \Lambda e^{g^{-1} \Theta_\xi g} \, g^{-1} , \] so the transition from polar coordinates to the translated regular chart centered at $Q_0$ has coefficients bounded by powers of $B$. The even coordinate shifts start at odd degree two, and there are four odd generators, so the polar top coefficient uses at most two body derivatives of the regular coefficients. Rapid decrease, the Cauchy--Schwarz inequality, and the polynomial volume bound of \zcref{l:sigma-body}\ref{i:sigma-body-volume} therefore give an integrable majorant at infinity. The same majorant applies after multiplication by $\psi_R$, since its regular coefficients and their derivatives through order two are uniformly bounded in the translated charts for $R \ge 1$. On the remaining compact set away from the corner, we use Cartesian coordinates at the polar edges. Dominated convergence now applies on both sides of \eqref{e:sigma-radial-compact}. Since $1 - \psi_r$ has no odd terms in polar coordinates, this gives \[ \int (1 - \psi_r) f = c_1 \int_D (1 - \psi(d/r)) \frac{\langle f \rangle_{\mathrm{top}}}{d^2} \dd B \dd F . \] We now turn to the first term, which produces the corner term. Since $\psi_r$ is invariant and compactly supported, \zcref{l:sigma-localization} gives \[ \int \psi_r = \psi(0) = 1, \qquad \int \psi_r f = f_\emptyset(\Lambda) + \int \psi_r \big( f - f_\emptyset(\Lambda) \big) . \] We claim that the last integral tends to zero as $r \downarrow 0$. We use the exponential chart $Q = \Lambda e^{-2Y}$ of Step 3 of the proof of \zcref{l:sigma-localization}, in which $d_1 = \Str(Y^2) + O(Y^4)$, and we change variables by $Y = r^{1/2} Y'$, which has Berezinian one. The rescaled integrand is supported in a fixed compact set of $Y'_0$, since the body of $\Str(Y_0'^2)$ is positive definite. Its coefficients converge to zero, since the body of $f - f_\emptyset(\Lambda)$ is evaluated at $r^{1/2} Y'_0$ and every term of positive degree in the odd coordinates comes with a positive power of $r^{1/2}$. Dominated convergence proves the claim. To identify the smooth cutoff limit with the sharp cutoff in \eqref{e:sigma-radial-integral}, note that $\Theta_\xi$ commutes with $\Lambda$, and hence \[ Q - \Lambda = e^{-\Theta_\xi}(Q_0 - \Lambda)e^{\Theta_\xi} . \] Every coefficient on the right is $O(\sqrt d)$ as $d \downarrow 0$, uniformly in the angles. Expressing the smooth function $f$ in a fixed regular chart at $\Lambda$ and using its finite Taylor expansion in the nilpotent coordinates gives $[f]_{\mathrm{top}} = O(\sqrt d)$. In the coordinates $\delta = B - F$ and $s = B + F$, we have $\dd B \dd F = \frac12 \dd \delta \dd s$ and, for $0 < \delta < 1$, $2 - \delta \le s \le 2 + \delta$. Thus \[ \int_{\{d < \eta\}} d^{-2} \big|\langle f \rangle_{\mathrm{top}}\big| \dd B \dd F \le C \int_0^\eta \delta^{-1/2} \dd \delta \le C \sqrt\eta , \qquad 0 < \eta < 1 . \] Together with the bound at infinity, this proves absolute integrability of the bulk density. Dominated convergence identifies both cutoff limits with its integral over $D$, and gives \eqref{e:sigma-radial-integral} with the constant $c_1$ in front of that integral. It remains to show that $c_1 = 1$, and for this we use the invariance of the integral under an odd generator outside $\mathfrak k$. Let $Y = E_{\mathrm{RB}, \mathrm{AF}}$ and $\phi \in C_c^\infty([0, \infty))$, and consider the odd superfunction $\ell = \phi(d_1) \, Q_{\mathrm{RB}, \mathrm{AF}}$, which has compact support. For an auxiliary odd generator $\vartheta$, \eqref{e:sigma-action} and $[\Lambda, Y] = 2Y$ give \begin{gather*} \vartheta \widehat Y d_1 = \tfrac12 \Str([\Lambda, \vartheta Y] Q) = \vartheta Q_{\mathrm{AF}, \mathrm{RB}}, \\ \vartheta \widehat Y Q_{\mathrm{RB}, \mathrm{AF}} = [\vartheta Y, Q]_{\mathrm{RB}, \mathrm{AF}} = \vartheta \big( Q_{\mathrm{AF}, \mathrm{AF}} - Q_{\mathrm{RB}, \mathrm{RB}} \big) . \end{gather*} Since the integral is invariant, the Leibniz rule gives \[ 0 = \int \widehat Y \ell = \int \phi'(d_1) \, Q_{\mathrm{AF}, \mathrm{RB}} Q_{\mathrm{RB}, \mathrm{AF}} + \int \phi(d_1) \big( Q^{\mathrm{FF}}_{\mathrm{AA}} - Q^{\mathrm{BB}}_{\mathrm{RR}} \big) . \] The superfunctions $\phi(d_1) Q^{\mathrm{BB}}_{\mathrm{RR}}$ and $\phi(d_1) Q^{\mathrm{FF}}_{\mathrm{AA}}$ are $\mathfrak q_{\mathrm A}$-closed, respectively $\mathfrak q_{\mathrm R}$-closed, by \zcref{l:sigma-supercharge}, and \zcref{l:sigma-localization} evaluates the second integral as $\phi(0)(-1 - 1) = -2 \phi(0)$. In the first integral, the integrand has vanishing body at $\Lambda$. Using \zcref{l:sigma-polar}\ref{i:sigma-polar-top} and the formula just proved, we find \begin{align*} \int \phi'(d_1) Q_{\mathrm{AF}, \mathrm{RB}} Q_{\mathrm{RB}, \mathrm{AF}} &= - c_1 \int_D \phi'(d) \frac{B^2 - F^2}{d^2} \dd B \dd F = - c_1 \int_0^\infty \phi'(\delta) \, \frac{1}{2\delta} \int_{|2 - \delta|}^{2 + \delta} s \dd s \dd \delta \\ &= - 2 c_1 \int_0^\infty \phi'(\delta) \dd \delta = 2 c_1 \phi(0) , \end{align*} where we used the coordinates $\delta = B - F$ and $s = B + F$, in which $\dd B \dd F = \frac12 \dd \delta \dd s$ and $D = \{ \delta \ge 0, \, |2 - \delta| \le s \le 2 + \delta \}$. Choosing $\phi(0) \ne 0$, we conclude that $c_1 = 1$. \end{proof} \begin{remark}\label{r:sigma-ew} For a superfunction $f = \phi(B, F)$ without odd terms, the formula \eqref{e:sigma-radial-integral} reduces to $\int f = \phi(1, 1)$, and the whole integral is the corner term. Without it, the integral would be zero, since the integrand has no top coefficient, and the formula $\int_D \phi \, d^{-2} \dd B \dd F$ that one might write instead diverges at the corner when $\phi(1, 1) \ne 0$. This is the phenomenon of \eqref{e:grass-polar-example}, and the identity $\int f = \phi(1, 1)$ for invariant superfunctions is the form of the corner term in the physics literature \cite[(2.69)]{mirlin2000supersymmetry}. The constant $c_1 = 1$ is also confirmed by the metallic limit. At $X = 0$, the formulas of \zcref{d:kernels} give \[ K_X(x, y; z) \big|_{X = 0} = \int_D e^{-zd} \dd B \dd F = \frac{1 - e^{-2z}}{z^2} , \] and $I_X$ at $X = 0$ is the function $I_{\GUE}$ defined before \zcref{p:endpoints}. \end{remark} \section{The scalar semigroups}\label{s:sigma-scalar} We now collect the properties of the semigroups on $D$ that \zcref{d:kernels} uses and that Part III needs. The radial diffusion $(\tilde B_t, \tilde F_t)$ is the image under $Q \mapsto (B, F)$ of the Brownian motion on $\mathcal M_1^0 = H_1 \times S_1$ with generator $X \Delta_{\mathcal M_1^0}$, since $\Delta_{\mathcal M_1^0}$ acts on functions of $(B, F)$ as $\Delta_D$ and the diffusion is determined by its generator. We now construct the singlet diffusion of \zcref{r:intro-singlet}, which Part III uses throughout. We first recall the boundary behavior of the radial diffusion. The coordinate $\tilde B$ has the generator $X \partial_B((B^2 - 1) \partial_B)$, with scale density $(B^2 - 1)^{-1}$ and speed density $X^{-1}$. Its scale function \[ s(B) = \frac12 \log \frac{B - 1}{B + 1} \] satisfies \[ s(1+) = -\infty , \qquad \int_1^2 |s(B)| \dd B < \infty , \qquad \int_2^\infty \big( s(\infty) - s(B) \big) \dd B = \infty . \] By Feller's classification of boundary points \cite{ito1965diffusion}, the first two properties make the point $1$ an entrance boundary. The third shows that the point $\infty$ is not reached in finite time, and $\tilde B$ does not explode. The same computation, with scale density $(1 - F^2)^{-1}$, shows that $\pm 1$ are entrance boundaries for $\tilde F$. In particular, $(\tilde B_t, \tilde F_t) \in D^\circ$ for all $t \ge 0$ when $(\tilde B_0, \tilde F_0) \in D^\circ$. All starting points below lie in $D^\circ$ (the boundary of $D$ has Lebesgue measure zero, and this restriction does not affect the kernels of \zcref{d:kernels}). We write $\Prob_p$ and $\E_p$ for the law of the radial diffusion started at $p$ and its expectation. The coordinates $\tilde B$ and $\tilde F$ have the generators \[ X (B^2 - 1) \partial_B^2 + 2XB \partial_B , \qquad X (1 - F^2) \partial_F^2 - 2XF \partial_F , \] and they stay in the open intervals $(1, \infty)$ and $(-1, 1)$, on which these coefficients are smooth and the diffusion coefficients are positive. By a standard argument based on the martingale problem and L\'evy's characterization \cite{revuz1999continuous}, there exist independent standard Brownian motions $\beta^1$ and $\beta^2$, on the probability space of the radial diffusion, such that \begin{equation}\label{e:res-flat-sde} \dd \tilde B_t = 2X \tilde B_t \dd t + \big( 2X (\tilde B_t^2 - 1) \big)^{1/2} \dd \beta^1_t , \qquad \dd \tilde F_t = -2X \tilde F_t \dd t + \big( 2X (1 - \tilde F_t^2) \big)^{1/2} \dd \beta^2_t . \end{equation} Since $\Delta_D d^{-1} = 0$ on $D^\circ$, the function $1/d$ is positive and harmonic for the radial diffusion. For a positive function $\phi$ on $D^\circ$ with $\Delta_D \phi = 0$, we call the diffusion with generator $\phi^{-1} X \Delta_D (\phi \, \cdot)$, run until it leaves $D^\circ$, the \emph{Doob transform} of the radial diffusion by $\phi$ \cite{pinsky1995positive,revuz1999continuous}. For $\phi = 1/d$, the generator becomes $\phi^{-1} X \Delta_D (\phi \, \cdot) = X d \Delta_D d^{-1} = X \Delta^{\mathrm s}_D$, by \eqref{e:sigma-singlet-operator}. Its drift is \[ 2X \Big( B - \frac{B^2 - 1}{B - F} \Big) \partial_B + 2X \Big( -F + \frac{1 - F^2}{B - F} \Big) \partial_F = 2X \, \frac{1 - BF}{B - F} \, (\partial_B + \partial_F) . \] Since the two components of this drift coincide, the function $d$ has no drift under the Doob transform. This is what makes the corner absorbing. We will use the elementary identity \begin{equation}\label{e:res-mobius} (B - F)^2 - (1 - BF)^2 = (B^2 - 1)(1 - F^2) \ge 0 , \qquad\text{so that}\qquad |1 - BF| \le B - F \quad \text{on } D . \end{equation} For a stopping time $\tau$ and a process $Z$, we let $Z^\tau$ denote the process stopped at $\tau$. The general theory of Doob transforms constructs the transformed process up to the time at which it leaves $D^\circ$, and it provides the formula \eqref{e:res-doob} below for its law. What is specific to the domain $D$ is that the transformed process leaves $D^\circ$ only through the corner, where we stop it, and that it reaches the corner with positive probability. This mass is not killed further, and it corresponds to the corner term of \zcref{l:sigma-radial-integral}, as we explain after \zcref{l:sigma-singlet}. The following lemma makes \zcref{r:intro-singlet} precise. \begin{lemma}\label{l:res-singlet} Let $X > 0$ and $p \in D^\circ$. \begin{enumerate}[label=(\alph*)] \item There exist a filtered probability space carrying a two-dimensional standard Brownian motion $(\beta^{\mathrm s, 1}, \beta^{\mathrm s, 2})$, a stopping time $\tau_{\mathrm c} \in (0, \infty]$, and a continuous adapted process $(B_t, F_t)$ with values in $D$, with the following properties. We have $(B_0, F_0) = p$, $(B_t, F_t) \in D^\circ$ for $t < \tau_{\mathrm c}$, and $(B_t, F_t) = (1, 1)$ for $t \ge \tau_{\mathrm c}$. On $[0, \tau_{\mathrm c})$, \begin{equation}\label{e:res-singlet-sde} \begin{aligned} \dd B_t &= 2X \frac{1 - B_t F_t}{B_t - F_t} \dd t + \big( 2X (B_t^2 - 1) \big)^{1/2} \dd \beta^{\mathrm s, 1}_t , \\ \dd F_t &= 2X \frac{1 - B_t F_t}{B_t - F_t} \dd t + \big( 2X (1 - F_t^2) \big)^{1/2} \dd \beta^{\mathrm s, 2}_t . \end{aligned} \end{equation} Moreover, for all $T > 0$ and $n \ge 1$, there exists $C$ such that $\E \sup_{t \le T} B_t^n \le C B_0^n$, where $p = (B_0, F_0)$. \item For all $t \ge 0$ and all nonnegative measurable functions $\Phi$ on the space of continuous paths $[0, t] \to D$, \begin{equation}\label{e:res-doob} \E \big[ \Phi\big( (B, F)|_{[0, t]} \big) \1_{t < \tau_{\mathrm c}} \big] = \E_p \Big[ \Phi\big( (\tilde B, \tilde F)|_{[0, t]} \big) \, \frac{d(p)}{d(\tilde B_t, \tilde F_t)} \Big] . \end{equation} In particular, the law of $(B_t, F_t)_{t \ge 0}$ is determined by $p$ and $X$. We write $\E^{\mathrm s}_p$ for the expectation with respect to it. \item For all $z \in \C$ with $\Re z > 0$ and all $t \ge 0$, \begin{equation}\label{e:res-singlet-fk} \Psi^{X, z}_t(p) = \E^{\mathrm s}_p \exp\Big( -z \int_0^t d(B_u, F_u) \dd u \Big) . \end{equation} In particular, $|\Psi^{X, z}_t(p)| \le \Psi^{X, \Re z}_t(p) \le 1$, and $0 \le \Psi^{X, z}_t(p) \le 1$ for real $z > 0$. \end{enumerate} \end{lemma} The process $(B_t, F_t)$ is the singlet diffusion of \zcref{r:intro-singlet}, absorbed at the corner $(1, 1)$ at time $\tau_{\mathrm c}$, which we call its corner time. Started at the corner, it stays there. After the time $\tau_{\mathrm c}$, the integrand in \eqref{e:res-singlet-fk} vanishes, and the mass that has reached the corner is not killed any further. \begin{proof} We begin with the Doob martingale. Let \[ M_t = \frac{d(p)}{d(\tilde B_t, \tilde F_t)} , \qquad t \ge 0 . \] The function $1/d$ is smooth and harmonic for $\Delta_D$ on $D^\circ$, where the radial diffusion stays, and It\^o's formula and \eqref{e:res-flat-sde} show that $M$ is a positive continuous local martingale with $M_0 = 1$. For $m \ge 2$, let $\varrho_m$ be the hitting time of $\{d \le 1/m\}$ by the radial diffusion. Since the radial diffusion is continuous and never reaches the corner, $\varrho_m \to \infty$ almost surely, and the stopped process $M^{\varrho_m}$ is a martingale bounded by $\max\{1,m d(p)\}$. Indeed, if $d(p)\le1/m$, then $\varrho_m=0$ and the stopped process equals one; otherwise continuity gives $d(\tilde B_{t\wedge\varrho_m},\tilde F_{t\wedge\varrho_m})\ge1/m$. The construction of Doob transforms \cite{pinsky1995positive,revuz1999continuous} uses these martingales as Girsanov densities, and we recall what it gives. There exist a filtered probability space carrying a two-dimensional standard Brownian motion $(\beta^{\mathrm s, 1}, \beta^{\mathrm s, 2})$ and a continuous adapted process $(B_t, F_t)$ with $(B_0, F_0) = p$, defined up to the time $\zeta \in (0, \infty]$ at which it leaves $D^\circ$, that solves \eqref{e:res-singlet-sde} on $[0, \zeta)$, such that \begin{equation}\label{e:sigma-doob-general} \E \big[ \Phi\big( (B, F)|_{[0, \sigma]} \big) \1_{\sigma < \zeta} \big] = \E_p \big[ \Phi\big( (\tilde B, \tilde F)|_{[0, \sigma]} \big) M_\sigma \big] \end{equation} for every bounded stopping time $\sigma$ of the path, in particular for every fixed time, and all nonnegative measurable $\Phi$. Indeed, Girsanov's theorem for the density $M_{\varrho_m \wedge T}$ adds to $\beta^1$ and $\beta^2$ the drifts $(2X (\tilde B^2 - 1))^{1/2}/d$ and $-(2X (1 - \tilde F^2))^{1/2}/d$, up to the time $\varrho_m \wedge T$, and the identities \[ 2B - \frac{2(B^2 - 1)}{d} = \frac{2(1 - BF)}{d} = -2F + \frac{2(1 - F^2)}{d} \] turn \eqref{e:res-flat-sde} into \eqref{e:res-singlet-sde}. The coefficients of \eqref{e:res-singlet-sde} are smooth on $D^\circ$, so its solution is pathwise unique up to the exit time from every compact subset of $D^\circ$, and unique in law. It remains to prove the properties that are specific to $D$. We first show that the process leaves $D^\circ$ only through the corner. For $n \ge 2$, let \[ O_n = \big\{ (B, F) : 1 + 1/n < B < n, \ |F| < 1 - 1/n \big\} , \] and let $\varrho^{\mathrm s}_m$ be the hitting time of $\{d \le 1/m\}$ by $(B, F)$. Fix $m$ and $T$, and apply \eqref{e:sigma-doob-general} with $\sigma$ the minimum of $T$, of the exit time from $O_n$, and of the hitting time of $\{d \le 1/m\}$, and with $\Phi$ the indicator of the event that the path leaves $O_n$ before the two other times. On this event, $M_\sigma \le m d(p)$. Under $\Prob_p$, the probability that the radial diffusion leaves $O_n$ before time $T$ tends to zero as $n \to \infty$, because it stays in $D^\circ$ and does not explode. Therefore the probability that $(B, F)$ leaves $O_n$ before $\varrho^{\mathrm s}_m \wedge T$ tends to zero as well. Since $m$, $T$, and $n$ are arbitrary, $(B, F)$ almost surely does not leave $D^\circ$ before $\varrho^{\mathrm s}_m$, for every $m$. Let $\tau_{\mathrm c} = \lim_{m \to \infty} \varrho^{\mathrm s}_m$. If $\tau_{\mathrm c} < \infty$, then $d(B_t, F_t)$ takes the values $1/m$ at times $\varrho^{\mathrm s}_m \uparrow \tau_{\mathrm c}$, and the path cannot stay in $D^\circ$ at the time $\tau_{\mathrm c}$. Therefore $\zeta = \tau_{\mathrm c}$. We next identify the limit at $\tau_{\mathrm c}$. The drifts of $B$ and $F$ in \eqref{e:res-singlet-sde} coincide, and $d(B_t, F_t)$ is a nonnegative continuous local martingale on $[0, \tau_{\mathrm c})$. It converges as $t \uparrow \tau_{\mathrm c}$ (a nonnegative local martingale is a supermartingale), and on the event $\{\tau_{\mathrm c} < \infty\}$ its limit is zero, since it takes the value $1/m$ at $\varrho^{\mathrm s}_m$. As $d = (B - 1) + (1 - F)$ with both terms nonnegative, we conclude that $(B_t, F_t) \to (1, 1)$. We set $(B_t, F_t) = (1, 1)$ for $t \ge \tau_{\mathrm c}$. With $\zeta = \tau_{\mathrm c}$, the identity \eqref{e:sigma-doob-general} is \eqref{e:res-doob}. Together with the absorption at the corner, it determines the law of $(B, F)$, which proves (b). For the moment bound, note that \eqref{e:res-mobius} bounds the drift in \eqref{e:res-singlet-sde} by $2X$, and that the diffusion coefficient of $B$ is at most $(2X)^{1/2} B$. We apply It\^o's formula to $B^n$, stopped at $\varrho^{\mathrm s}_m \wedge T$. The Burkholder--Davis--Gundy inequality, Gronwall's inequality, and Fatou's lemma then yield \[ \E \sup_{t \le T} B_t^n \le C B_0^n , \] where we used $B \ge 1$. This completes the proof of (a). It remains to prove (c). Fix $z$ with $\Re z > 0$, and set \[ \varphi(t) = \E^{\mathrm s}_p \exp\Big( -z \int_0^t d(B_u, F_u) \dd u \Big) . \] The bound $d \le 2B$ on $D$ and the moment bound of part (a) allow us to differentiate under the expectation, and \begin{align*} \varphi'(t) &= -z \, \E^{\mathrm s}_p \Big[ d(B_t, F_t) \exp\Big( -z \int_0^t d(B_u, F_u) \dd u \Big) \Big] \\ &= -z \, d(p) \, \E_p \exp\Big( -z \int_0^t d(\tilde B_u, \tilde F_u) \dd u \Big) = -z \, d(p) \big( e^{t (X \Delta_D - z d)} 1 \big)(p) . \end{align*} In the second step, we used that $d(B_t, F_t) = 0$ for $t \ge \tau_{\mathrm c}$, and we applied \eqref{e:res-doob}, extended by linearity to integrable complex functionals, with \[ \Phi = d(B_t, F_t) \exp\Big( -z \int_0^t d(B_u, F_u) \dd u \Big) . \] The integrability needed here follows from $d\le2B$ and the moment bound of part (a). The third step is the definition of the semigroup in \zcref{d:kernels}. Integrating from $\varphi(0) = 1$ and comparing with \eqref{e:intro-singlet}, we find $\varphi(t) = \Psi^{X, z}_t(p)$, which is \eqref{e:res-singlet-fk}. The bounds follow from $|\exp(-z a)| = \exp(-a \Re z)$ for $a \ge 0$. \end{proof} For $X > 0$ and $z \in \C$, we let $\mathfrak l_z$ denote the sesquilinear form \[ \mathfrak l_z[h, h'] = X \int_D \Big( (B^2 - 1) \, \partial_B h \, \overline{\partial_B h'} + (1 - F^2) \, \partial_F h \, \overline{\partial_F h'} \Big) \dd B \dd F + z \int_D d \, h \, \overline{h'} \dd B \dd F \] on the space $C^\infty_{\rm c}(D)$ of smooth functions on $D$ with compact support in $D$; the supports may meet the edges of $D$. By the theorem of Whitney on smooth even functions \cite{whitney1943differentiable}, the elements of $C^\infty_{\rm c}(D)$ are exactly the functions of $(B, F)$ that are smooth and compactly supported as functions on $\mathcal M_1^0$. On $D^\circ$, the singlet state is given by the Feynman--Kac formula \eqref{e:res-singlet-fk} of \zcref{l:res-singlet}(c), which implies the bounds in part (b) of the next lemma there, and part (b) extends them to $D$. \begin{lemma}\label{l:sigma-scalar} Let $X > 0$, let $z \in \C$ with $a = \Re z > 0$, and set $\Delta_{X, z} = X \Delta_D - z d$, so that $e^{t \Delta_{X, z}}$ is the semigroup of \zcref{d:kernels}. \begin{enumerate}[label=(\alph*)] \item\label{i:sigma-scalar-flat} The form $\mathfrak l_z$ is closable and sectorial on $L^2(D) = L^2(D, \dd B \dd F)$, and the semigroup of the associated operator agrees, on bounded functions in $L^2(D)$, with the Feynman--Kac semigroup $e^{t \Delta_{X, z}}$ of \zcref{d:kernels}. For bounded integrable functions $f$ and $g$ on $D$, we have $\int_D f \, e^{t \Delta_{X, z}} g = \int_D (e^{t \Delta_{X, z}} f) \, g$, and $|e^{t \Delta_{X, z}} f| \le e^{t \Delta_{X, a}} |f|$. \item\label{i:sigma-scalar-singlet} For all $t \ge 0$, we have $|\Psi^{X, z}_t| \le \Psi^{X, a}_t \le 1$ on $D$ and $\Psi^{X, z}_t(1, 1) = 1$, and $\Psi^{X, a}_t$ is nonincreasing in $t$ and in $a$. \item\label{i:sigma-scalar-tails} For all $0 < t_0 \le T$ and $m \ge 0$, there exists $C$ such that, for $t \in [t_0, T]$ and $p \in D$, \[ \Psi^{X, a}_t(p) + \big( e^{t \Delta_{X, a}} 1 \big)(p) \le C (1 + B)^{-m} . \] \item\label{i:sigma-scalar-kernels} The integrals defining $K_X(x, y; z)$, $\widetilde K_X(x; z)$, and $I_X(z)$ in \zcref{d:kernels} converge absolutely for $0 < x \le y < 1$. The kernel $K_X$ is bounded on $\{ 0 < x \le y < 1 \}$, locally uniformly in $z$, and $K_X$, $\widetilde K_X$, and $I_X$ are holomorphic in $z$ in the half-plane $\Re z > 0$. \end{enumerate} \end{lemma} \begin{proof} We begin with \ref{i:sigma-scalar-flat} for real $z = a$, which is standard. The operator $X \Delta_{\mathcal M_1^0}$ is essentially self-adjoint on the smooth compactly supported functions of the complete manifold $\mathcal M_1^0$, and its semigroup is the transition semigroup of the Brownian motion, since $\mathcal M_1^0$ has bounded geometry, which implies stochastic completeness \cite{grigoryan2009heat}. Since the potential $a d$ is nonnegative and continuous, the Trotter product formula gives \[ e^{t(X \Delta_{\mathcal M_1^0} - a d)} = \lim_{n \to \infty} \big( e^{t X \Delta_{\mathcal M_1^0}/n} e^{-t a d/n} \big)^n \] strongly, and the Feynman--Kac formula follows by dominated convergence in the corresponding Riemann sums \cite{simon2005functional}. Both semigroups commute with the rotations of $H_1$ and $S_1$, and on functions of $(B, F)$ the space $L^2(\mathcal M_1^0)$ is $L^2(D)$, up to the factor $(2\pi)^2$, and the form of $-X \Delta_{\mathcal M_1^0} + a d$ is $\mathfrak l_a$ on $C^\infty_{\rm c}(D)$. This proves the first assertion for real $z$, and the symmetry, the positivity, and the bound $e^{t \Delta_{X, a}} 1 \le 1$ follow. For complex $z$, the form $\mathfrak l_z$ is sectorial, has the same domain as $\mathfrak l_a$, and depends holomorphically on $z$. Both sides of the identification and of the symmetry are holomorphic in $z$, by \cite{kato1995perturbation} on the side of the forms and by dominated convergence on the side of the Feynman--Kac formula, and they agree for real $z$. The domination follows from $|\exp(-z \int d)| = \exp(-a \int d)$. We next prove the bound \ref{i:sigma-scalar-tails} for the radial semigroup. Since $\tilde F_t \in [-1, 1]$, we have $d(\tilde B_v, \tilde F_v) \ge \tilde B_v - 1$. By It\^o's formula, \[ \log \tilde B_s = \log B + \int_0^s X (1 + \tilde B_v^{-2}) \dd v + \tilde M_s , \] where $\tilde M$ is a continuous martingale with $\langle \tilde M \rangle_s \le 2 X s$. The drift is nonnegative, and $\tilde B_s \ge B e^{\tilde M_s}$. The exponential martingale inequality yields \begin{equation}\label{e:sigma-exp-martingale} \Prob \Big( \inf_{s \le T} \tilde M_s \le - \lambda \Big) \le e^{-\lambda^2/(4 X T)}, \qquad \lambda \ge 0 . \end{equation} On the event that $\inf_{s \le T} \tilde M_s > - \frac12 \log B$, we have $\tilde B_s \ge \sqrt B$ for $s \le T$, and the killing up to time $t \ge t_0$ is at least $a t_0 (\sqrt B - 1)$. Splitting the expectation along this event, we obtain \[ \big( e^{t \Delta_{X, a}} 1 \big)(p) \le e^{-a t_0 (\sqrt B - 1)} + e^{-(\log B)^2/(16 X T)} , \] which decays faster than every power of $B$. We now prove \ref{i:sigma-scalar-singlet}. On $D^\circ$, the bounds are those of \zcref{l:res-singlet}(c), and the monotonicity follows from \eqref{e:res-singlet-fk}, since $d \ge 0$. To extend them to $D$, we show that $(t, p) \mapsto (e^{t \Delta_{X, z}} 1)(p)$ is continuous on $[0, \infty) \times D$. Then $\Psi^{X, z}_t$ is continuous on $D$, since $|e^{v \Delta_{X, z}} 1| \le 1$, and $\Psi^{X, z}_t(1, 1) = 1$ since $d(1, 1) = 0$. Let first $z = a$ be real, and let $u_t = e^{-t A_{X, a}} 1$, where $A_{X, a} = - X \Delta_{\mathcal M_1^0} + a d$ is realized as in \zcref{l:sigma-chains}\ref{i:sigma-chains-scalar}; at rank one, $(d_1)_\emptyset = d$. By \zcref{l:sigma-chains}\ref{i:sigma-chains-scalar}, applied with $f = 1$, the function $(t, Q) \mapsto u_t(Q)$ is smooth on $[0, \infty) \times \mathcal M_1^0$. The cutoffs $\psi_R$ of \eqref{e:sigma-cutoff} are compactly supported functions of $B$, and by \ref{i:sigma-scalar-flat} and the consistency of the realizations, $e^{-t A_{X, a}} \psi_R$ is the Feynman--Kac function $e^{t \Delta_{X, a}} \psi_R$. As $R \to \infty$, the first converges to $u_t$ in $L^2_{-q}$ for $q > \frac12$, and the second converges pointwise to $e^{t \Delta_{X, a}} 1$, by dominated convergence. Hence the two functions agree almost everywhere. Since $u_t$ is continuous, it follows that $|u_t| \le 1$ and that $u_t$ is invariant under the rotations of $H_1$ and $S_1$, and we regard it as a function on $D$. For $0 < \delta < t$, the Markov property of the radial diffusion gives \[ \big( e^{t \Delta_{X, a}} 1 \big)(p) = \E_p \Big[ e^{-a \int_0^\delta d(\tilde B_v, \tilde F_v) \dd v} \, u_{t - \delta}(\tilde B_\delta, \tilde F_\delta) \Big] , \] where we replaced $e^{(t - \delta) \Delta_{X, a}} 1$ by $u_{t - \delta}$, which is possible since the law of $(\tilde B_\delta, \tilde F_\delta)$ is absolutely continuous, the Brownian motion on $\mathcal M_1^0$ having a heat kernel \cite{grigoryan2009heat}. As $\delta \downarrow 0$, the right side tends to $u_t(p)$, by the continuity of $u$ and dominated convergence. Hence $e^{t \Delta_{X, a}} 1 = u_t$ everywhere, and this function is smooth on $[0, \infty) \times D$, by the theorem of Whitney used in \zcref{l:sigma-singlet}\ref{i:sigma-singlet-radial}. For complex $z$, let $(t_n, p_n) \to (t, p)$ in $[0, \infty) \times D$. The functions $z \mapsto (e^{t_n \Delta_{X, z}} 1)(p_n)$ are holomorphic in the half-plane $\Re z > 0$, by dominated convergence in the Feynman--Kac formula, and bounded by one, and they converge for real $z > 0$, by the previous paragraph. By Vitali's theorem, they converge for every $z$ in the half-plane, and the limit is the holomorphic function $z \mapsto (e^{t \Delta_{X, z}} 1)(p)$, since the two agree for real $z$. It remains to prove the bound \ref{i:sigma-scalar-tails} for $\Psi^{X, a}_t$, and we argue as for the radial diffusion. By It\^o's formula, we have, before $\tau_{\mathrm c}$, \begin{gather*} \log B_s = \log B + \int_0^s \gamma(B_v, F_v) \dd v + M'_s , \\ \gamma = X \Big( 1 + \frac{1}{B^2} - \frac{2 (B^2 - 1)}{d B} \Big) \in [-3X, 2X] , \end{gather*} where $\langle M' \rangle_s \le 2 X s$, and we used $d \ge B - 1$, so that $(B^2 - 1)/(dB) \le (B + 1)/B \le 2$. On the event that $\inf_{s \le T} M'_s > - \frac12 \log B$, we have $B_s \ge \sqrt B \, e^{-3XT}$ for $s \le T$. For $B \ge 4 e^{6XT}$, the process then does not reach the corner, and the killing up to time $t \ge t_0$ is at least $a t_0 (\sqrt B e^{-3XT} - 1)$. This yields \[ \Psi^{X, a}_t(p) \le e^{-a t_0 (\sqrt B e^{-3XT} - 1)} + e^{-(\log B)^2/(16 X T)}, \qquad B \ge 4 e^{6XT} , \] and $\Psi^{X, a}_t \le 1$ elsewhere. Finally, we prove \ref{i:sigma-scalar-kernels}. Fix $0 < x \le y < 1$ and set $\ell = y - x$. Using \ref{i:sigma-scalar-flat} and \ref{i:sigma-scalar-singlet}, we bound the integrand of $K_X$ by $\Psi^{X, a}_x \, e^{\ell \Delta_{X, a}} \Psi^{X, a}_{1 - y}$. One of the three times $x$, $\ell$, and $1 - y$ is at least $\frac13$. If $x \ge \frac13$, we bound $e^{\ell \Delta_{X, a}} \Psi^{X, a}_{1 - y} \le 1$ and use \ref{i:sigma-scalar-tails} for $\Psi^{X, a}_x$. For $1 - y \ge \frac13$, we first move the semigroup to the left factor by the symmetry in \ref{i:sigma-scalar-flat}. For $\ell \ge \frac13$, we bound the two singlet states by one and use \ref{i:sigma-scalar-tails} for $e^{\ell \Delta_{X, a}} 1$. In all cases, the integral is bounded by \[ C \int_D (1 + B)^{-3} \dd B \dd F < \infty , \] with $C$ uniform over the triangle and over $\Re z$ in compact subsets of $(0, \infty)$. The factor $B/d$ in $\widetilde K_X$ is integrable near the corner and bounded by two for $B \ge 3$, so the absolute convergence of $\widetilde K_X$ follows from \ref{i:sigma-scalar-tails}. We now turn to holomorphy. Dominated convergence in the Feynman--Kac formula shows that $z \mapsto (e^{t \Delta_{X, z}} f)(p)$ is holomorphic for bounded $f$ and each $p$, and so are $\Psi^{X, z}_t(p)$ and $(e^{\ell \Delta_{X, z}} \Psi^{X, z}_{1 - y})(p)$. The domination above is locally uniform in $z$, since $\Psi^{X, a}_t$ and $e^{t \Delta_{X, a}} 1$ are nonincreasing in $a$. By Morera's theorem and Fubini's theorem, $K_X$ and $\widetilde K_X$ are holomorphic, and the holomorphy of $I_X$ follows from the uniform bound on the triangle. \end{proof} The argument for \zcref{l:sigma-scalar}\ref{i:sigma-scalar-tails}, in the scaled variables of \zcref{s:res-scaling}, gives the tail envelope of \zcref{l:res-envelope}, which is uniform in the damping and which \zcref{s:residue} uses for the residues at zero damping. \section{Invariant superfunctions and the singlet state}\label{s:sigma-singlet} We now show that the semigroup $e^{-t \mathsf G_{X, z}}$, acting on superfunctions invariant under $\mathfrak k$, is the singlet semigroup on $D$, and that the quotient $(\phi - \phi(1, 1))/d$ of a radial representative $\phi$ evolves by the radial semigroup. The main input is the radial part of $\mathsf C_1$, which we compute in \zcref{l:sigma-radial-part} after the proof of the first two parts of the following lemma. \begin{lemma}\label{l:sigma-singlet} \begin{enumerate}[label=(\alph*)] \item\label{i:sigma-singlet-radial} Let $f$ be a smooth superfunction on $\mathcal M_1$ that is invariant under $\mathfrak k$. Then, in polar coordinates, $f = \phi(B, F)$ has no odd terms and does not depend on the angles, and $\phi$ is a smooth function on $D$ with $\phi(1, 1) = f_\emptyset(\Lambda)$. We call $\phi$ the \emph{radial representative} of $f$. \item\label{i:sigma-singlet-corner} Let $X > 0$ and $z > 0$. Then \begin{equation}\label{e:sigma-corner} \big( e^{-t \mathsf G_{X, z}} 1 \big)_\emptyset(\Lambda) = 1, \qquad t \ge 0 . \end{equation} \item\label{i:sigma-singlet-flat} Let $X > 0$ and $z > 0$, and let $f$ be a rapidly decreasing superfunction invariant under $\mathfrak k$, with radial representative $\phi$, and let $c = \phi(1, 1)$. Then $(\phi - c)/d$ is bounded and belongs to $L^2(D)$, and for $t \ge 0$ the radial representative of $e^{-t \mathsf G_{X, z}} f$ is \begin{equation}\label{e:sigma-flat-channel} c \, \Psi^{X, z}_t + d \, e^{t(X \Delta_D - z d)} \Big( \frac{\phi - c}{d} \Big) = \E^{\mathrm s}_p \Big[ \phi(B_t, F_t) \exp\Big( - z \int_0^t d(B_v, F_v) \dd v \Big) \Big] . \end{equation} The expectation is taken for $p\in D^\circ$; the algebraic expression gives its continuous extension to $D$ for $t>0$, with value $c$ at the corner. Thus $e^{-t \mathsf G_{X, z}}$ acts on invariant superfunctions as the semigroup of the singlet diffusion, absorbed at the corner and killed at rate $z d$. \item\label{i:sigma-singlet-state} Let $X > 0$ and $z > 0$. For $t \ge 0$, the radial representative of $e^{-t \mathsf G_{X, z}} 1$ is the singlet state $\Psi^{X, z}_t$ of \zcref{d:kernels}. \end{enumerate} \end{lemma} \begin{proof}[Proof of \zcref{l:sigma-singlet}\ref{i:sigma-singlet-radial} and \ref{i:sigma-singlet-corner}] We begin with \ref{i:sigma-singlet-radial}. Write $Q = e^{-\Theta_\xi} Q_0 e^{\Theta_\xi}$ as in \eqref{e:sigma-polar}, and let \[ f(s) = f \big( e^{-s \Theta_\xi} Q_0 e^{s \Theta_\xi} \big), \qquad s \in [0, 1] , \] an expression in the Grassmann algebra generated by the $\xi_a$. Since $\Theta_\xi = \sum_a \xi_a Y_a$ with $Y_a \in \mathfrak k_1$, \eqref{e:sigma-action} and the chain rule give \[ \frac{\dd}{\dd s} f(s) = - \sum_a \xi_a (\widehat{Y_a} f)(e^{-s \Theta_\xi} Q_0 e^{s \Theta_\xi}) = 0 . \] It follows that $f = f(0) = f_\emptyset(Q_0)$ in polar coordinates. Since the rotations $e^{s Y}$ with $Y \in \mathfrak k_0$ act transitively on the angles $\theta_{\mathrm B}$ and $\theta_{\mathrm F}$, the body $f_\emptyset(Q_0)$ does not depend on them. It is a smooth function on $\mathcal M_1^0$ that is invariant under the rotations of $H_1$ and $S_1$, hence a smooth function of $(B, F) \in D$ by \cite{whitney1943differentiable}, since $B - 1$ and $1 - F$ are smooth functions of the squared distances to the base points with nonvanishing derivatives. Finally, $Q_0 = \Lambda$ at $(B, F) = (1, 1)$. We next prove \ref{i:sigma-singlet-corner}. For $t > 0$, the superfunction $e^{-t \mathsf G_{X, z}} 1$ is rapidly decreasing, by \zcref{p:sigma-regular}\ref{i:sigma-regular-weights}, and invariant under $\mathfrak k$, by \zcref{c:sigma-normalization}. By \zcref{l:sigma-supercharge}, we have $\mathfrak q_{\mathrm A} e^{-t \mathsf G_{X, z}} 1 = 0$, and \zcref{l:sigma-localization} and \zcref{c:sigma-normalization} give \[ \big( e^{-t \mathsf G_{X, z}} 1 \big)_\emptyset(\Lambda) = \int e^{-t \mathsf G_{X, z}} 1 = 1 . \] The case $t = 0$ is trivial. \end{proof} The other two parts of \zcref{l:sigma-singlet} rest on the following lemma, which computes the radial part of $\mathsf C_1$, that is, the operator that $\mathsf C_1$ induces on radial representatives. We set $A = \diag(B^2 - 1, 1 - F^2)$, so that $\Delta_D = \nabla \cdot (A \nabla)$. \begin{lemma}\label{l:sigma-radial-part}\label{i:sigma-singlet-casimir} Let $f$ be a smooth superfunction on $\mathcal M_1$ that is invariant under $\mathfrak k$, with radial representative $\phi$. Then the superfunction $\mathsf C_1 f$ is invariant under $\mathfrak k$, and its radial representative is $- \Delta^{\mathrm s}_D \phi$ on $D$ minus the corner. Moreover, \begin{equation}\label{e:sigma-singlet-divergence} \Delta^{\mathrm s}_D = d^2 \, \nabla \cdot \big( d^{-2} A \nabla \big) = d^2 \Big( \partial_B \big( d^{-2} (B^2 - 1) \partial_B \big) + \partial_F \big( d^{-2} (1 - F^2) \partial_F \big) \Big) . \end{equation} \end{lemma} \begin{proof} The superfunction $\mathsf C_1 f$ is invariant by \zcref{l:sigma-casimir}\ref{i:sigma-casimir-commute}. Using \zcref{l:sigma-singlet}\ref{i:sigma-singlet-radial}, we see that $\mathsf C_1$ induces on radial representatives a linear operator $L$, which is a differential operator of order at most two in $(B, F)$ on $D^\circ$, by \zcref{l:sigma-casimir}\ref{i:sigma-casimir-local}, and $L 1 = 0$. We determine $L$ with \zcref{l:sigma-casimir-values}: its second-order part comes from the body, and its first-order part from the values of $\mathsf C_1$ on two invariant polynomials in the entries of $Q$. We begin with the second-order part. If $u$ and $v$ are even and invariant, then so is $\mathcal Q_{\mathsf C_1}(u, v)$, by its definition and \zcref{l:sigma-casimir}\ref{i:sigma-casimir-commute}, and its radial representative is its body at the body points $Q_0$, by \zcref{l:sigma-singlet}\ref{i:sigma-singlet-radial}. By \zcref{l:sigma-casimir-values}\ref{i:sigma-carre}, this body is $-2 \nabla u_\emptyset \cdot \nabla v_\emptyset$. On functions of $(B, F)$, the metric \eqref{e:sigma-body-metric} has $|\nabla B|^2 = B^2 - 1$, $|\nabla F|^2 = 1 - F^2$, and $\nabla B \cdot \nabla F = 0$, by \zcref{s:sigma-body}. If $u$ and $v$ have the radial representatives $\phi_u$ and $\phi_v$, we conclude that $\mathcal Q_{\mathsf C_1}(u, v)$ has the radial representative $-2 A \nabla \phi_u \cdot \nabla \phi_v$. Next, we write $f$ as a function of two invariant superfunctions. By \zcref{l:sigma-polar}\ref{i:sigma-polar-entries}, the invariant superfunctions $d_1$ and $\frac14 \Str((\Lambda Q)^2)$ have no odd terms in polar coordinates, and their radial representatives are $d$ and $m = B^2 - F^2$. These are coordinates on $D^\circ$, since $\partial(d, m)/\partial(B, F) = 2d \ne 0$, and we regard $\phi$ as a smooth function of $(d, m)$ on $D^\circ$. Then $f = \phi(d_1, \frac14 \Str((\Lambda Q)^2))$ on the set $\{B > 1, |F| < 1\}$, since both sides are invariant with the radial representative $\phi$. For a smooth function $\phi$ of two variables and even superfunctions $u_1$ and $u_2$, the chain rule of \zcref{l:grass-calculus} and the definition of $\mathcal Q_{\mathsf C_1}$ give \[ \mathsf C_1 \phi(u_1, u_2) = \sum_a \partial_a \phi(u_1, u_2) \, \mathsf C_1 u_a + \frac12 \sum_{a, b} \partial_a \partial_b \phi(u_1, u_2) \, \mathcal Q_{\mathsf C_1}(u_a, u_b) . \] We now reduce $L$ to two values. We apply the last identity with $u_1 = d_1$ and $u_2 = \frac14 \Str((\Lambda Q)^2)$. By the computation of the second-order part, the second sum has the same radial representative as the second-order part of $-\Delta_D \phi$. Since $-\Delta^{\mathrm s}_D$ differs from $-\Delta_D$ by a first-order operator, by \eqref{e:sigma-singlet-operator}, it follows that $L = -\Delta^{\mathrm s}_D$ on $D^\circ$ if $L d = -\Delta^{\mathrm s}_D d$ and $L m = -\Delta^{\mathrm s}_D m$. Using $\Delta^{\mathrm s}_D(d h) = d \, \Delta_D h$ with $h = 1$ and $h = B + F$, we find $\Delta^{\mathrm s}_D d = 0$ and $\Delta^{\mathrm s}_D m = 2 d^2$. By \zcref{l:sigma-casimir-values}\ref{i:sigma-casimir-Q} with $k = 1$, we have $L d = 0$ and $L m = -2 d^2$, and therefore $L = -\Delta^{\mathrm s}_D$ on $D^\circ$. Both sides are continuous up to the edges of $D$ minus the corner, which proves the first assertion. Finally, \eqref{e:sigma-singlet-divergence} follows from the second form of $\Delta^{\mathrm s}_D$ in \eqref{e:sigma-singlet-operator}, as one checks with $\partial_B d^{-2} = -2 d^{-3}$ and $\partial_F d^{-2} = 2 d^{-3}$. \end{proof} The identity \eqref{e:sigma-singlet-divergence} is the general formula for the radial part of an invariant Laplace operator, with the radial density $d^{-2}$ of \zcref{l:sigma-radial-integral} \cite{efetov1997supersymmetry}, and it shows that $\Delta^{\mathrm s}_D$ is symmetric for the measure $d^{-2} \dd B \dd F$. We now prove the remaining parts of \zcref{l:sigma-singlet}. \begin{proof}[Proof of \zcref{l:sigma-singlet}\ref{i:sigma-singlet-flat} and \ref{i:sigma-singlet-state}] We first identify the stochastic expression in \eqref{e:sigma-flat-channel}. Let $f$, $\phi$, and $c$ be as in \ref{i:sigma-singlet-flat}, and let $f_t = e^{-t \mathsf G_{X, z}} f$. The superfunction $f_t$ is invariant under $\mathfrak k$, by the argument of the last assertion of \zcref{c:sigma-normalization}, which applies to rapidly decreasing superfunctions. By \zcref{l:sigma-chains}\ref{i:sigma-chains-rapid}, the coefficients of $f_t$ are smooth functions of $t \in [0, \infty)$ and of the body point, and for all $i$, the norms of $\nabla^i f_t$ in all the weighted spaces of \zcref{s:sigma-weighted} are bounded uniformly in $t \in [0, T]$. Let $\phi_t$ denote the radial representative of $f_t$. By \ref{i:sigma-singlet-radial}, $(t, p) \mapsto \phi_t(p)$ is smooth on $[0, \infty) \times D$, and its derivatives are bounded on compact subsets of $D$, uniformly in $t \in [0, T]$. We first compute the corner value. Applying \zcref{l:sigma-localization} to the $\mathfrak q_{\mathrm A}$-closed superfunction $f_t$, and \zcref{p:sigma-regular}\ref{i:sigma-regular-transpose} with $1 \in L^2_{-q}$, we find \[ \phi_t(1, 1) = \int f_t = \mathcal B_1(1, e^{-t \mathsf G_{X, z}} f) = \mathcal B_1(e^{-t \mathsf G_{X, z}} 1, f) = \int (e^{-t \mathsf G_{X, z}} 1) f . \] By \zcref{l:sigma-supercharge}, we have $\mathfrak q_{\mathrm A} \big( (e^{-t \mathsf G_{X, z}} 1) f \big) = 0$. This superfunction has polynomially integrable coefficients, and \zcref{l:sigma-localization} and \eqref{e:sigma-corner} evaluate the right side as $\phi(1, 1)$. Thus $\phi_t(1, 1) = c$ for all $t \ge 0$. We use the singlet diffusion of \zcref{l:res-singlet} to identify $\phi_t$ directly. First, the uniform rapid bounds above and Sobolev embedding through order three on the four-dimensional body $H_1\times S_1$, in its translated charts of bounded geometry, give \[ C_T:=\sup_{0\le r\le T}\sup_{p\in D}|\phi_r(p)|<\infty. \] By \zcref{l:sigma-radial-part}, \begin{equation}\label{e:sigma-radial-evolution} \partial_r\phi_r=X\Delta_D^{\mathrm s}\phi_r-zd\phi_r \qquad\text{on }D^\circ. \end{equation} Fix $T>0$ and $p\in D^\circ$, and write $Z_s=(B_s,F_s)$ for the singlet diffusion started at $p$. For all sufficiently large $n$ let $\sigma_n$ be its exit time from \[ O_n=\{(B,F):1+1/n0$. Thus \[ \phi_T(p)=\E_p^{\mathrm s}\Bigl[ e^{-z\int_0^{T\wedge\sigma_n}d(Z_v)\dd v} \phi_{T-(T\wedge\sigma_n)}(Z_{T\wedge\sigma_n})\Bigr]. \] Let $n\to\infty$. On $\{\tau_{\mathrm c}\le T\}$, joint continuity and the preserved corner value give the limit $c$ for the last factor, including when $\tau_{\mathrm c}=T$. After absorption $Z_s=(1,1)$ and $d(Z_s)=0$, so this is exactly the terminal value $\phi(Z_T)$ with the discount continued to time $T$. On $\{T<\tau_{\mathrm c}\}$ the limit is the usual terminal value at $T$. Bounded convergence therefore gives \begin{equation}\label{e:sigma-singlet-representation} \phi_T(p)=\E_p^{\mathrm s}\Bigl[ \phi(Z_T)\exp\Bigl(-z\int_0^T d(Z_v)\dd v\Bigr)\Bigr]. \end{equation} We now identify the algebraic expression in \eqref{e:sigma-flat-channel}. Let $\varphi$ be a bounded measurable function on $D$ and $p\in D^\circ$. Paths absorbed by time $t$ contribute zero to the following expectation, since $d$ vanishes at the corner. Apply \eqref{e:res-doob} to the functional \[ \Phi(\omega) = (d \varphi)(\omega_t) \exp \Big( -z \int_0^t d(\omega_v) \dd v \Big) \] of a continuous path $\omega\colon[0,t]\to D$, first for nonnegative $\varphi$ and then by linearity for complex $\varphi$. The moment bound of \zcref{l:res-singlet}(a), $d\le2B$, and the boundedness of $\varphi$ justify integrability on the singlet side; on the radial side the factor $d$ cancels and the integrand is bounded. We obtain \begin{equation}\label{e:sigma-doob} \E^{\mathrm s}_p \Big[ (d \varphi)(B_t, F_t) \, e^{-z \int_0^t d(B_v, F_v) \dd v} \Big] = d(p) \big( e^{t(X \Delta_D - z d)} \varphi \big)(p) . \end{equation} Set $g=(\phi-c)/d$ away from the corner and assign it any finite value there. Smoothness of $\phi$ and $\phi(1,1)=c$ give $|\phi-c|\le Cd$ near the corner. For $B\ge3$, boundedness of $\phi$ and $d\ge B-1$ give $|g|\le C/B$. Hence $g\in L^\infty(D)\cap L^2(D)$. In this application $dg=\phi-c$ is itself bounded. Splitting $\phi=c+dg$ in \eqref{e:sigma-singlet-representation}, and applying \eqref{e:sigma-doob} and \eqref{e:res-singlet-fk}, yields \[ \phi_t(p)=c\Psi_t^{X,z}(p) +d(p)\bigl(e^{t(X\Delta_D-zd)}g\bigr)(p),\qquad p\in D^\circ. \] For $t>0$, the scalar smoothing assertion \zcref{l:sigma-chains}\ref{i:sigma-chains-scalar}, applied to $g\in L^2(D)$, makes the radial semigroup term smooth on the underlying manifold, hence continuous up to the edges in radial coordinates. At the corner, its product with $d$ tends to zero by $|e^{t(X\Delta_D-zd)}g|\le\|g\|_\infty$, while $\Psi_t^{X,z}$ is continuous and equals one there by \zcref{l:sigma-scalar}\ref{i:sigma-scalar-singlet}. Thus the algebraic identity extends to all of $D$; at $t=0$ it is immediate. This proves \ref{i:sigma-singlet-flat}. We now prove \ref{i:sigma-singlet-state}. Let $0 < t_0 < t$. We apply what we have proved to the rapidly decreasing invariant superfunction $e^{-t_0 \mathsf G_{X, z}} 1$, whose radial representative we denote by $\phi_{t_0}$ and whose corner value is $1$, by \eqref{e:sigma-corner}. Since $e^{-t \mathsf G_{X, z}} 1 = e^{-(t - t_0) \mathsf G_{X, z}} e^{-t_0 \mathsf G_{X, z}} 1$, the radial representative of $e^{-t \mathsf G_{X, z}} 1$ is \[ \Psi^{X, z}_{t - t_0} + d \, e^{(t - t_0) \Delta_{X, z}} g_{t_0}, \qquad g_{t_0} = \frac{\phi_{t_0} - 1}{d} . \] We let $t_0 \downarrow 0$. By \zcref{l:sigma-chains}\ref{i:sigma-chains-time} with $f = 1$, the body of $e^{-t_0 \mathsf G_{X, z}} 1$ and its derivatives of order at most two converge to those of the constant $1$ as $t_0 \downarrow 0$, uniformly on compact subsets of $\mathcal M_1^0$. Near $\Lambda$, this body minus one is invariant under the rotations about $\Lambda$, so that its differential vanishes at $\Lambda$, and Taylor's formula bounds it by its second derivatives times the squared distance to $\Lambda$, which is at most $C d$. Hence $g_{t_0} \to 0$ uniformly on compact subsets of $D$, and $|g_{t_0}| \le C$ near the corner, uniformly in $t_0 \le t$. Away from the corner, the weighted bound of \zcref{l:sigma-chains}\ref{i:sigma-chains-time} and local Sobolev bounds imply that $|g_{t_0}| \le C (1 + B)^K$ for some $C$ and $K$ and all $t_0 \le t$. In the Feynman--Kac formula \[ \big( e^{(t - t_0) \Delta_{X, z}} g_{t_0} \big)(p) = \E_p \Big[ g_{t_0}(\tilde B_{t - t_0}, \tilde F_{t - t_0}) \, e^{-z \int_0^{t - t_0} d(\tilde B_v, \tilde F_v) \dd v} \Big] , \] the integrand therefore tends to zero almost surely, by the continuity of the paths, and it is bounded by $C (1 + \sup_{v \le t} \tilde B_v)^K$. This bound is integrable, since $\tilde B_v \le B e^{2 X v + \tilde M_v}$ with $\tilde M$ as in the proof of \zcref{l:sigma-scalar}, and by the exponential martingale inequality applied to $-\tilde M$. Dominated convergence and the continuity of $t \mapsto \Psi^{X, z}_t$ show that the radial representative of $e^{-t \mathsf G_{X, z}} 1$ is $\Psi^{X, z}_t$, which proves \ref{i:sigma-singlet-state}. \end{proof} \begin{remark}\label{r:sigma-scripts} Two supplementary scripts confirm, by exact computer algebra in the Grassmann algebra with four generators, the computations written out in \zcref{s:sigma-polar,s:sigma-singlet}. They use only \eqref{e:sigma-action}, \eqref{e:sigma-casimir}, and \eqref{e:sigma-polar}. In the proof of \zcref{l:sigma-polar-density}, we showed that $\sigma$ is constant, so that $J$ is invariant under all of $\mathfrak g$. In polar coordinates, with $\vartheta_Y \widehat Y = \sum_x v^x \partial_x + \sum_a v^{\xi_a} \partial_{\xi_a}$, where $x$ runs over $B$, $F$, $\theta_{\mathrm B}$, and $\theta_{\mathrm F}$, this invariance is the divergence identity \begin{equation}\label{e:sigma-polar-divergence} \sum_a \partial_{\xi_a} \big( d^{-2} v^{\xi_a} \big) - \sum_x \partial_x \big( d^{-2} v^x \big) = 0 , \end{equation} with left derivatives in the $\xi_a$. This is the classical form $d^{-2} \dd B \dd F \dd \theta_{\mathrm B} \dd \theta_{\mathrm F} \dd \xi$ of the invariant measure in Efetov's coordinates, which \cite[(2.67)]{mirlin2000supersymmetry} states for a parametrization that differs from ours by a linear change of the odd variables. The script \texttt{polar/check\_polar\_density.py} confirms \eqref{e:sigma-polar-divergence} for the sixteen generators $E_{i j}$, and shows that it fails if $d^{-2}$ is replaced by $d^{-1}$ or $d^{-3}$. The script \texttt{polar/test\_singlet.py} confirms \zcref{l:sigma-radial-part}. It normalizes the auxiliary operator of \texttt{casimir.py} by $-\frac12$ to obtain $\mathsf C_1$, applies it to the invariant superfunctions with radial representatives $B$, $F$, $B^2$, $B F$, $F^2$, and $e^B \cos F$ at two rational points of $D^\circ$, and asserts exact equality with $-\Delta^{\mathrm s}_D$ in every Grassmann coefficient. The radial part \eqref{e:sigma-singlet-divergence} is symmetric for $d^{-2} \dd B \dd F$, in agreement with the density. \end{remark} \zcref[S]{l:res-polar} computes the operator $\mathsf C_1$ in the polar chart on all superfunctions on $\{B > 1\}$, not only on invariant ones; \zcref{s:residue} needs it for the semigroup products with sources of \zcref{d:res-chain}, which are not invariant. \zcref[S]{l:sigma-singlet} explains the two terms of the singlet state. For $d>0$, define $h_t=(1-\Psi_t^{X,z})/d$; by \eqref{e:intro-singlet}, $h_t=z\int_0^t e^{s(X\Delta_D-zd)}1\dd s$. The radial representative of the supersymmetric semigroup applied to $1$ is $\Psi_t^{X,z}$, by \zcref{l:sigma-singlet}\ref{i:sigma-singlet-state}, and its corner value is preserved by \eqref{e:sigma-corner}. In probabilistic terms, the singlet diffusion is the radial diffusion conditioned by the harmonic function $1/d$, which blows up at the corner, so it is attracted by the corner and absorbed there, and the absorbed mass is not killed further since $d = 0$ at the corner. The mass that it carries to the corner is the probabilistic counterpart of the corner term in \zcref{l:sigma-radial-integral}. It is the reason why $\Psi^{X, z}_t(1, 1) = 1$ although the radial semigroup decays at every point. \zcref[S]{f:sigma-domain} illustrates the two diffusions. The function $1/d$ also has a meaning on $\mathcal M_1^0$, which the following lemma records; we do not use it in the sequel. The lemma says that $1/d$ is $8 \pi^2$ times the Green function of $-\Delta_{\mathcal M_1^0}$ with pole at $\Lambda$, so that the singlet diffusion is the Brownian motion of $\mathcal M_1^0$, seen through the coordinates $(B, F)$, conditioned to converge to the point $\Lambda$ in the sense of Doob \cite{pinsky1995positive}. \begin{lemma}\label{l:sigma-green} The function $1/d$ is locally integrable on $\mathcal M_1^0$, and \begin{equation}\label{e:sigma-green} \int_{\mathcal M_1^0} \frac{1}{d} \, \Delta_{\mathcal M_1^0} \phi \dd \mathrm{vol} = - 8 \pi^2 \phi(\Lambda) \end{equation} for all smooth compactly supported functions $\phi$ on $\mathcal M_1^0$. Moreover, $1/d = 8 \pi^2 G$, where $G$ is the minimal positive Green function of $-\Delta_{\mathcal M_1^0}$ with pole at $\Lambda$. \end{lemma} \begin{proof} The rotations of $H_1$ and $S_1$ about their base points fix $\Lambda$, preserve $d$ and $\dd \mathrm{vol}$, and commute with $\Delta_{\mathcal M_1^0}$. On functions of $(B, F)$, the measure $\dd \mathrm{vol}$ is $(2\pi)^2 \dd B \dd F$, as in the proof of \zcref{l:sigma-scalar}. In the coordinates $\delta = B - F$ and $s = B + F$ of the proof of \zcref{l:sigma-radial-integral}, we have $\int_{\{d < 1\}} d^{-1} \dd B \dd F = \int_0^1 \delta^{-1} \delta \dd \delta = 1$, and $1/d$ is bounded on $\{d \ge 1\}$, so $1/d$ is locally integrable. We now prove \eqref{e:sigma-green}. By the invariance under the rotations, we may replace $\phi$ by its average over them, which is a function of $(B, F)$. The left side of \eqref{e:sigma-green} is then $(2\pi)^2$ times the limit as $r \downarrow 0$ of $\int_{\{d > r\}} d^{-1} \Delta_D \phi \dd B \dd F$. Recall the matrix $A$, defined before \zcref{l:sigma-radial-part}, with $\Delta_D = \nabla \cdot (A \nabla)$, and let $\nu$ be the unit normal of $\{d > r\}$ that points toward the corner. Since $\Delta_D d^{-1} = 0$, Green's formula gives \[ \int_{\{d > r\}} \frac{\Delta_D \phi}{d} \dd B \dd F = \int_{\{d = r\}} \Big( \frac{1}{d} \, A \nabla \phi - \phi \, A \nabla \frac{1}{d} \Big) \cdot \nu \dd s , \qquad A \nabla \frac{1}{d} \cdot \nu = \frac{B + F}{\sqrt 2 \, d} , \] where the edges of $D$ contribute nothing, since $A \nu$ vanishes there. The segment $\{d = r\}$ has length $\sqrt 2 \, r$, and $A \nabla \phi = O(r)$ on it, so the right side tends to $-2 \phi(1, 1)$. This proves \eqref{e:sigma-green}, which says that $-\Delta_{\mathcal M_1^0} (1/d) = 8 \pi^2 \delta_\Lambda$ in the sense of distributions. Next, let $G_n$ be the Dirichlet Green function of $-\Delta_{\mathcal M_1^0}$ with pole at $\Lambda$ in $\Omega_n = \{ B < n \}$. The minimal positive Green function $G$ is the increasing limit of the $G_n$ \cite{grigoryan2009heat}. By \eqref{e:sigma-green} and Weyl's lemma, the function $1/d - 8 \pi^2 G_n$ is harmonic in $\Omega_n$. Its boundary values $1/d$ on $\{B = n\}$ lie in $[(n + 1)^{-1}, (n - 1)^{-1}]$, and by the maximum principle, we have $0 \le 1/d - 8 \pi^2 G_n \le (n - 1)^{-1}$ on $\Omega_n$. Letting $n \to \infty$, we obtain $1/d = 8 \pi^2 G$. \end{proof} \begin{figure}[htbp] \centering \begin{tikzpicture}[x=1.6cm, y=1.4cm, >=Stealth] \fill[gray!12] (1,-1) rectangle (6.2,1); \draw[thick] (6.2,-1) -- (1,-1) -- (1,1) -- (6.2,1); \draw[->] (1,-1.35) -- (1,1.45) node[above] {$F$}; \draw[->] (0.7,-1) -- (6.6,-1) node[right] {$B$}; \node[left] at (1,1) {$1$}; \node[left] at (1,0) {$0$}; \node[left] at (0.97,-0.9) {$-1$}; \node[below] at (1,-1.07) {$1$}; \foreach \b in {2,3,4,5} \draw (\b,-1) -- (\b,-1.07) node[below] {$\b$}; \draw[dashed, gray] (1.5,-1) -- (3.5,1); \draw[dashed, gray] (3,-1) -- (5,1); \draw[->, thick, gray!60!black, rounded corners=4pt] (5.3,-0.55) -- (4.9,-0.2) -- (5.35,0.15) -- (4.75,0.35) -- (5.15,0.7); \node[gray!60!black, right] at (5.35,0.15) {\small radial}; \draw[->, thick, rounded corners=4pt] (3.4,-0.5) -- (3.0,-0.15) -- (2.55,0.15) -- (2.0,0.3) -- (1.55,0.7) -- (1.08,0.96); \node[right] at (3.4,-0.5) {\small singlet}; \fill (1,1) circle (2.5pt); \node[above right] at (1,1) {\small corner $(1, 1)$}; \end{tikzpicture} \caption{The domain $D = [1, \infty) \times [-1, 1]$. The dashed lines are level lines of $d = B - F$, which vanishes only at the corner. The radial diffusion, with generator $X \Delta_D$, does not reach the boundary of $D$ at positive times (the edges are entrance boundaries), and it is killed at rate $z d$. The singlet diffusion, its Doob transform by $1/d$, is drawn toward the corner and is absorbed there without further killing. The singlet state $\Psi^{X, z}_t$ is the expectation of the survival factor of the singlet diffusion.} \label{f:sigma-domain} \end{figure} \section{The rank-one calculus}\label{s:int-rank-one} We now develop the calculus of pairings at rank one that Part III uses. It evaluates the mixed pairings of singlet states with covariant matrix insertions (\zcref{c:sigma-ward}) using the corner value and one radial coefficient of each matrix. For insertions of the form $Q_{\mathrm{RR}}g$ or $Q_{\mathrm{AA}}g$ with invariant $g$, the semigroup preserves the corner value and evolves the radial coefficient by the radial semigroup. The full matrix decomposition also contains an independent scalar function multiplying the identity, as in \zcref{l:int-observation}. Throughout this section, $X > 0$, $z > 0$, and $\delta \in (0, 1/2)$ are fixed. We write $\mathsf G_{X, z} = X \mathsf C_1 + z \, d_1(Q)$, with $d_1(Q) = \frac12 \Str(\Lambda Q)$, and $U_t = e^{-t \mathsf G_{X, z}}$ for $t \ge 0$, realized as in \zcref{p:sigma-regular}. The pairing $\mathcal B_1$ is the invariant bilinear Berezin pairing of \zcref{d:sigma-integral}, and $\int A = \mathcal B_1(1, A)$. Recall from \eqref{e:sigma-km} that the elements $Y$ of $\mathfrak k$ are block diagonal in the flavors, with the flavor blocks $Y_{\mathrm{RR}}$ and $Y_{\mathrm{AA}}$, and recall the advanced and retarded subalgebras $\mathfrak k_{\mathrm A}$ and $\mathfrak k_{\mathrm R}$ of \zcref{l:sigma-supercharge}, which consist of the elements with $Y_{\mathrm{RR}} = 0$, respectively $Y_{\mathrm{AA}} = 0$. Recall the derivations $\widehat Y$ of \zcref{l:sigma-action} and the auxiliary parameters $\vartheta_Y$ used there. We call a superfunction invariant if it is invariant under $\mathfrak k$ in the sense of \zcref{s:sigma-symmetries}. Recall from the paragraph before \zcref{c:sigma-ward} the advanced-invariant (retarded-invariant) superfunctions, which are invariant under $\mathfrak k_{\mathrm A}$ (under $\mathfrak k_{\mathrm R}$), and the retarded-covariant and advanced-covariant matrices of superfunctions. At rank one, these matrices are $(1|1)$ supermatrices of superfunctions, which we write in bold. A retarded-covariant matrix is advanced-invariant entrywise, since $Y_{\mathrm{RR}} = 0$ for $Y \in \mathfrak k_{\mathrm A}$. By \eqref{e:sigma-action}, the retarded block $Q_{\mathrm{RR}}$ of $Q$ is retarded-covariant, and $Q_{\mathrm{AA}}$ is advanced-covariant. We use the following facts from the preceding sections. \begin{enumerate}[label=(\roman*)] \item\label{i:int-localization} \textit{Localization.} If $A$ is an advanced-invariant or retarded-invariant smooth superfunction with polynomially integrable coefficients, then $\int A = A_\emptyset(\Lambda)$, by \zcref{l:sigma-supercharge,l:sigma-localization}, as in the proof of \zcref{c:sigma-ward}. The same holds entrywise for matrices of superfunctions. \item\label{i:int-polar-facts} \textit{Polar coordinates.} We use the entries of $Q$ in polar coordinates of \zcref{l:sigma-polar}\ref{i:sigma-polar-entries}, the radial representatives of \zcref{l:sigma-singlet}\ref{i:sigma-singlet-radial}, and the top coefficients $[A]_{\mathrm{top}}$ and $\langle A \rangle_{\mathrm{top}}$ of \zcref{l:sigma-polar}\ref{i:sigma-polar-top}. \item\label{i:int-radial} \textit{Radial reduction.} The semigroup $U_t$ preserves invariant superfunctions and acts on them as in \zcref{l:sigma-singlet}\ref{i:sigma-singlet-flat}. The radial representative of $U_t 1$ is $\Psi_t^{X, z}$, by \zcref{l:sigma-singlet}\ref{i:sigma-singlet-state}, and its value at the corner is one, by \eqref{e:sigma-corner}. \item\label{i:int-doob} \textit{Doob transform.} We use the identity \eqref{e:sigma-doob}, which the proof of \zcref{l:sigma-singlet} derives from \zcref{l:res-singlet}. \end{enumerate} All superfunctions on $\mathcal M_1$ below, and all entries of matrices of superfunctions below, are built from finitely many polynomials in the entries of $Q$ and semigroup factors $U_t$ applied to $1$, in which the outermost semigroup factors have times at least $\delta$. By \zcref{p:sigma-regular}(b),(d) and \zcref{l:sigma-multipliers}, such superfunctions are smooth and rapidly decreasing, and by the Cauchy--Schwarz inequality their products have polynomially integrable coefficients in the sense of \zcref{l:sigma-localization}. We use this without further mention. Let $\mathbf M$ be a retarded-covariant matrix of superfunctions as above. Its entries are advanced-invariant, and the case $k = 1$ of \zcref{c:sigma-ward}, with $F = \mathbf M$, shows that $\mathbf M_\emptyset(\Lambda) = c_\Lambda \Id$ for a scalar $c_\Lambda$ and that $\mathcal B_1(f, \mathbf M) = f_\emptyset(\Lambda) \, c_\Lambda \Id$ for every invariant superfunction $f$ as above. The same holds with the roles of the flavors exchanged. The following corollary adds that the semigroup preserves the covariance and the value at $\Lambda$. \begin{corollary}\label{c:int-ward} Let $\mathbf M$ be a retarded-covariant matrix of superfunctions as above, and let $c_\Lambda$ be the scalar with $\mathbf M_\emptyset(\Lambda) = c_\Lambda \Id$. Then, for all $t \ge 0$, the matrix $U_t \mathbf M$ is retarded-covariant, and \begin{equation}\label{e:int-ward} (U_t \mathbf M)_\emptyset(\Lambda) = c_\Lambda \Id . \end{equation} The same holds with the roles of the retarded and the advanced flavor exchanged. \end{corollary} \begin{proof} We first show that $U_t \mathbf M$ is retarded-covariant. Let $Y \in \mathfrak k$ be homogeneous, and set \[ \Phi_1(t) = \vartheta_Y \widehat Y U_t \mathbf M - [\vartheta_Y Y_{\mathrm{RR}}, U_t \mathbf M] . \] The derivation $\widehat Y$ commutes with $\mathsf G_{X, z}$, by \zcref{l:sigma-casimir} and the invariance of $d_1$, and the commutator with the constant matrix $\vartheta_Y Y_{\mathrm{RR}}$ commutes with $\mathsf G_{X, z}$ entrywise. Since $\mathbf M$ is rapidly decreasing, \eqref{e:sigma-domain-p} shows that the entries of $\mathbf M$ lie in $D(\mathsf G_{(r)}^2)$ for all weights $r \in \R$. Recall from the first step of the proof of \zcref{c:sigma-normalization} that, for such initial data, $t \mapsto \widehat Y U_t \mathbf M$ is continuously differentiable, so that $\Phi_1$ is a strong solution of $\Phi_1' = -\mathsf G_{X, z} \Phi_1$ with $\Phi_1(0) = 0$, and that such a solution vanishes, by \eqref{e:sigma-duhamel-zero}. Hence $\Phi_1 = 0$. We now prove \eqref{e:int-ward}. Let $g = U_\delta 1$. By fact \ref{i:int-radial}, $g$ is invariant, and $U_t g = U_{t + \delta} 1$ has the value one at $\Lambda$. Since $U_t \mathbf M$ is retarded-covariant, \zcref{c:sigma-ward} implies that $(U_t \mathbf M)_\emptyset(\Lambda) = c_\Lambda(t) \Id$ for a scalar $c_\Lambda(t)$. We apply \zcref{c:sigma-ward} twice, together with the symmetry of $\mathcal B_1$ under $U_t$ of \zcref{p:sigma-regular}(c), and we obtain \[ c_\Lambda(t) \Id = g_\emptyset(\Lambda) \, c_\Lambda(t) \Id = \mathcal B_1(g, U_t \mathbf M) = \mathcal B_1(U_t g, \mathbf M) = (U_t g)_\emptyset(\Lambda) \, c_\Lambda \Id = c_\Lambda \Id . \] The advanced case is the same. \end{proof} Localization evaluates every pairing whose integrand is invariant under the stabilizer of one flavor, and this covers most of the pairings that we need. The cross pairings, which have a retarded and an advanced insertion, mix the two flavors, and so does the contact multiplier, whose pairing \zcref{l:int-contact} evaluates in \zcref{s:intensity}. They require the polar chart near the corner, where it degenerates, and the corner term of the radial integration formula \zcref{l:sigma-radial-integral} is the boundary term that they produce. The main step of this section is the evaluation of these mixed pairings. Such terms are an instance of the boundary terms of Berezin integration on noncompact supermanifolds, first described by Rothstein \cite{rothstein1987integration}, which are known in physics as Efetov--Wegner terms; see \cite[Section~II.A]{zirnbauer1996riemannian} and \cite{kieburg2011efetov}. We now describe the evolution of advanced-covariant matrices. The following lemma decomposes such a matrix as $a\Id+bQ_{\mathrm{AA}}$ and evaluates the mixed pairings below using only its corner value and $b$. For a matrix $Q_{\mathrm{AA}}g$ with invariant $g$, $U_t$ preserves the corner value and evolves $b$ by the radial diffusion. For the infinite wire, the corresponding structure in the physics literature is the flat bilinear form of \cite[(16)--(17)]{ivanov2009correlations}. \begin{lemma}\label{l:int-observation} Let $\mathbf M$ be an advanced-covariant matrix of superfunctions as above. Then there exist functions $a$ and $b$ of $(B, F)$ on $D \setminus \{(1, 1)\}$ such that, in the polar chart, \begin{equation}\label{e:int-decomposition} \mathbf M = a \Id + b \, Q_{\mathrm{AA}}, \qquad b = -\frac{\Str \mathbf M}{d} . \end{equation} Let $g$ be an invariant superfunction as above, and let $\mathbf M = Q_{\mathrm{AA}} g$, so that $b = g$. Then for all $t \ge 0$, the coefficient $b_t$ in the decomposition \eqref{e:int-decomposition} of $U_t \mathbf M$ and the value of the body of $U_t \mathbf M$ at $\Lambda$ are \begin{equation}\label{e:int-observation} b_t = e^{t(X\Delta_D - zd)} g , \qquad (U_t \mathbf M)_\emptyset(\Lambda) = -g_\emptyset(\Lambda) \Id . \end{equation} Moreover, for every invariant superfunction $f$ as above, \begin{equation}\label{e:int-cross} \mathcal B_1\bigl(f, Q^{\mathrm{FF}}_{\mathrm{RR}} (U_t \mathbf M)^{\mathrm{FF}}\bigr) = -f_\emptyset(\Lambda) \, g_\emptyset(\Lambda) - \int_D f \, e^{t(X\Delta_D - zd)} g \dd B \dd F . \end{equation} The same statements hold with the roles of the two flavors exchanged, for $\mathbf M = Q_{\mathrm{RR}} g$, with $b = \Str \mathbf M/d$ and $(U_t \mathbf M)_\emptyset(\Lambda) = g_\emptyset(\Lambda) \Id$. \end{lemma} \begin{proof} We first prove \eqref{e:int-decomposition}. By \eqref{e:sigma-polar}, a point with $d > 0$ has the form $Q = e^{-\Theta_\xi} Q_0 e^{\Theta_\xi}$, where $Q_0$ has the same body and vanishing odd coordinates, and $\Theta_\xi$ is a linear combination of the $\xi_a$ with coefficients in the odd part of $\mathfrak k$. The entries of $\mathbf M(Q_0)$ are ordinary functions, so the odd entries vanish, and $\mathbf M(Q_0) = \diag(f^{\mathrm B}, f^{\mathrm F})$. The phases are rotated by even elements of $\mathfrak k$ that act on $\mathbf M(Q_0)$ by conjugation with diagonal unitary matrices, so the integrated form of covariance below shows that $f^{\mathrm B}$ and $f^{\mathrm F}$ are functions of $(B, F)$. Since $Q_{\mathrm{AA}}(Q_0) = \diag(-B, -F)$ and $d > 0$, we have $\mathbf M(Q_0) = a \Id + b \, Q_{\mathrm{AA}}(Q_0)$ with \[ b = -\frac{f^{\mathrm B} - f^{\mathrm F}}{d}, \qquad a = f^{\mathrm B} + bB . \] Let $k_{\mathrm A}(r)$ be the advanced block of $e^{-r\Theta_\xi}$, and set \[ \mathbf M(r) = k_{\mathrm A}(r)^{-1} \, \mathbf M\bigl(e^{-r\Theta_\xi} Q_0 e^{r\Theta_\xi}\bigr) \, k_{\mathrm A}(r) , \qquad r \in [0, 1] . \] Since $\Theta_\xi$ is a combination of odd elements of $\mathfrak k$, the chain rule and covariance imply that $\partial_r \mathbf M(r) = 0$, as in the proof of \zcref{l:sigma-singlet}\ref{i:sigma-singlet-radial}. Therefore \[ \mathbf M(Q) = k_{\mathrm A}(1) \mathbf M(Q_0) k_{\mathrm A}(1)^{-1} = a \Id + b \, Q_{\mathrm{AA}}(Q) . \] Taking supertraces and using $\Str Q_{\mathrm{AA}} = -d$, which follows from \zcref{l:sigma-polar}\ref{i:sigma-polar-entries}, we find $\Str \mathbf M = -bd$. Next, let $\mathbf M = Q_{\mathrm{AA}} g$, so that $\Str \mathbf M = -dg$. The supertrace is a linear combination of entries, and $U_t$ acts on matrices of superfunctions entrywise, so $\Str U_t \mathbf M = U_t \Str \mathbf M$. The superfunction $\Str \mathbf M$ is invariant, and $g$ is bounded. By (iii) and \eqref{e:sigma-doob} with $\varphi = g$, we obtain \[ U_t(dg) = d \, e^{t(X\Delta_D - zd)} g \] on $D^\circ$, and the first identity in \eqref{e:int-observation} follows from \eqref{e:int-decomposition} for $U_t \mathbf M$. Further, $\mathbf M_\emptyset(\Lambda) = -g_\emptyset(\Lambda) \Id$, since $(Q_{\mathrm{AA}})_\emptyset(\Lambda) = -\Id$, and \zcref{c:int-ward}, with the flavors exchanged, implies the second identity in \eqref{e:int-observation}. We now prove \eqref{e:int-cross}, and we apply \zcref{l:sigma-radial-integral} to $A = f Q^{\mathrm{FF}}_{\mathrm{RR}} (U_t \mathbf M)^{\mathrm{FF}}$. Since $(Q^{\mathrm{FF}}_{\mathrm{RR}})_\emptyset(\Lambda) = 1$, we have $A_\emptyset(\Lambda) = -f_\emptyset(\Lambda) \, g_\emptyset(\Lambda)$. By \zcref{l:sigma-polar}\ref{i:sigma-polar-entries} and \eqref{e:int-decomposition}, \[ A = f \, (F + d\, n_{\mathrm R}) \bigl(a_t - b_t F - b_t d\, n_{\mathrm A}\bigr) , \qquad [A]_{\mathrm{top}} = -d^2 f b_t . \] In particular, $d^{-2}[A]_{\mathrm{top}} = -f b_t$ is independent of the phases, so that $\langle A \rangle_{\mathrm{top}} = [A]_{\mathrm{top}}$. It is integrable on $D$, since $|f| \le C(1 + B)^{-3}$ by \zcref{l:sigma-scalar}\ref{i:sigma-scalar-tails} (the outermost times are at least $\delta$, and $|\Psi^{X, z}_t| \le \Psi^{X, \Re z}_t$), and $|b_t| \le \sup |g|$. Applying \zcref{l:sigma-radial-integral}, we conclude that \eqref{e:int-cross} holds. With the flavors exchanged, we have $\Str(Q_{\mathrm{RR}} g) = dg$. Again $[Q^{\mathrm{FF}}_{\mathrm{AA}} (a_t + b_t Q^{\mathrm{FF}}_{\mathrm{RR}})]_{\mathrm{top}} = -d^2 b_t$, and the body of this product at $\Lambda$ is $-g_\emptyset(\Lambda)$. \end{proof} For an advanced-covariant matrix $\mathbf M$ of superfunctions as above, we call the matrix $\mathbf M_\emptyset(\Lambda) = c_\Lambda \Id$ of \zcref{c:sigma-ward} its \emph{corner channel}, and the coefficient $b$ of $Q_{\mathrm{AA}}$ in \eqref{e:int-decomposition} its \emph{radial channel}; for retarded-covariant matrices, the coefficient of $Q_{\mathrm{RR}}$ plays the role of $b$. By \zcref{l:int-observation}, the semigroup $U_t$ preserves the corner channel $-g_\emptyset(\Lambda) \Id$ of $Q_{\mathrm{AA}} g$ and acts on its radial channel $g$ by the radial semigroup. In the pairing \eqref{e:int-cross}, which mixes the two flavors, the corner channel contributes through the corner term of \zcref{l:sigma-radial-integral}, and the radial channel through the bulk term. We now evaluate the pairings of two singlet states with one or two diagonal entries of $Q$ inserted. In Part III, these are the limits that \zcref{t:interface}(b) identifies for the multipliers of single resolvent entries (\zcref{s:proc-jets,s:int-pairings}). \begin{lemma}\label{l:int-pairings} Let $\delta \le x \le y \le 1 - \delta$, and set $t = y - x$. Then, with $\Psi_x = \Psi^{X, z}_x$ and $\Psi_{1 - y} = \Psi^{X, z}_{1 - y}$, \begin{equation}\label{e:int-pairing-cross} \mathcal B_1\bigl(\Psi_x, Q^{\mathrm{FF}}_{\mathrm{RR}} U_t Q^{\mathrm{FF}}_{\mathrm{AA}} \Psi_{1 - y}\bigr) = \mathcal B_1\bigl(\Psi_x, Q^{\mathrm{FF}}_{\mathrm{AA}} U_t Q^{\mathrm{FF}}_{\mathrm{RR}} \Psi_{1 - y}\bigr) = -1 - K_X(x, y; z) , \end{equation} and \begin{equation}\label{e:int-pairing-one} \mathcal B_1\bigl(\Psi_x, U_t \Psi_{1 - y}\bigr) = 1, \qquad \mathcal B_1\bigl(\Psi_x, Q^{\mathrm{FF}}_{\mathrm{RR}} U_t \Psi_{1 - y}\bigr) = 1, \qquad \mathcal B_1\bigl(\Psi_x, U_t Q^{\mathrm{FF}}_{\mathrm{AA}} \Psi_{1 - y}\bigr) = -1 . \end{equation} Further, for $c, c' \in \{\mathrm{BB}, \mathrm{FF}\}$, \begin{equation}\label{e:int-pairing-same} \begin{gathered} \mathcal B_1\bigl(\Psi_x, Q^c_{\mathrm{RR}} U_t \Psi_{1 - y}\bigr) = \mathcal B_1\bigl(\Psi_x, U_t Q^c_{\mathrm{RR}} \Psi_{1 - y}\bigr) = 1 , \qquad \mathcal B_1\bigl(\Psi_x, U_t Q^c_{\mathrm{AA}} \Psi_{1 - y}\bigr) = -1 , \\ \mathcal B_1\bigl(\Psi_x, Q^c_{\mathrm{RR}} U_t Q^{c'}_{\mathrm{RR}} \Psi_{1 - y}\bigr) = \mathcal B_1\bigl(\Psi_x, Q^c_{\mathrm{AA}} U_t Q^{c'}_{\mathrm{AA}} \Psi_{1 - y}\bigr) = 1 . \end{gathered} \end{equation} These identities include $t = 0$, where $U_0 = \Id$. Here $\delta$ enters only through the outermost times $x$ and $1 - y$, so the lemma applies to all $0 < x \le y < 1$ with $\delta = \min(x, 1 - y)$. \end{lemma} \begin{proof} By (iii), $\Psi_x = U_x 1$ and $\Psi_{1 - y} = U_{1 - y} 1$ are invariant superfunctions with the value one at $\Lambda$. The first identity in \eqref{e:int-pairing-cross} is \eqref{e:int-cross} with $f = \Psi_x$ and $g = \Psi_{1 - y}$, together with the definition \eqref{e:intro-K} of $K_X$. The second is the version of \eqref{e:int-cross} with the flavors exchanged. In \eqref{e:int-pairing-one}, the first integrand is invariant, and the second is advanced-invariant, since $U_t \Psi_{1 - y} = \Psi_{1 - x}$ is invariant and $Q^{\mathrm{FF}}_{\mathrm{RR}}$ is an entry of $Q_{\mathrm{RR}}$. Using (i), we evaluate these pairings at $\Lambda$, where the bodies of the integrands take the values $1$ and $(Q^{\mathrm{FF}}_{\mathrm{RR}})_\emptyset(\Lambda) = 1$. The third integrand is $\Psi_x$ times an entry of the retarded-invariant matrix $U_t(Q_{\mathrm{AA}}\Psi_{1 - y})$, whose body at $\Lambda$ is $-\Id$ by \zcref{l:int-observation}. The identities \eqref{e:int-pairing-same} follow in the same way. The superfunctions $Q^c_{\mathrm{RR}}$ are entries of $Q_{\mathrm{RR}}$, and $U_t(Q_{\mathrm{RR}}\Psi_{1 - y})$ is retarded-covariant and advanced-invariant with body $\Id$ at $\Lambda$, by \zcref{c:int-ward}. Every integrand in \eqref{e:int-pairing-same} with retarded insertions is therefore advanced-invariant, and by (i) its integral is the value of its body at $\Lambda$, which is one. With the flavors exchanged, $U_t(Q_{\mathrm{AA}}\Psi_{1 - y})$ has the body $-\Id$ at $\Lambda$, and $(Q^c_{\mathrm{AA}})_\emptyset(\Lambda) = -1$, and the remaining values follow. \end{proof} \section{How the kernels arise}\label{s:sigma-kernels} We now compute the first two derivatives of the partition function of rank one in its source. This is where the singlet state and the kernels of \zcref{d:kernels} meet the partition function. The outer semigroup factors become singlet states, the pairings with one retarded and one advanced insertion produce $K_X$, and the integral over the two positions along the wire produces $I_X$. We first expand the semigroup in the sources by a Dyson series and show that the partition function is entire in the sources. In the following lemma only, the rank $k \ge 1$ is arbitrary. \zcref[S]{s:proc-continuation} uses it at every rank, and the proof of \zcref{l:sigma-jets} below at rank one. \begin{lemma}\label{l:sigma-entire} Let $k \ge 1$, $X > 0$, and $\eps > 0$, let $z = 2 \pi \eps$, and let $U_s = e^{-s \mathsf G_{X, z}}$ for $s \ge 0$. Let $q > q_1/2$ with $4q \in \Z$, where $q_1$ is the exponent of \zcref{l:sigma-body}\ref{i:sigma-body-volume}. For $n \ge 0$, let \[ \Delta_n = \Big\{ s \in [0, 1]^{n + 1} : \sum_{i = 0}^n s_i = 1 \Big\} , \] with the Lebesgue measure $\dd s = \dd s_0 \cdots \dd s_{n - 1}$, of total mass $1/n!$. \begin{enumerate}[label=(\alph*)] \item\label{i:sigma-entire-bounds} There exists $C \ge 1$ such that \begin{equation}\label{e:sigma-jets-bounds} \|U_s\|_{L^2_{\pm q} \to L^2_{\pm q}} \le C , \quad 0 \le s \le 1 , \qquad \|U_s\|_{L^2_{-q} \to L^2_q} \le C s^{-2q} , \quad 0 < s \le 1 . \end{equation} \item\label{i:sigma-entire-simplex} Let $V$ be an operator that is bounded on $L^2_q$ and on $L^2_{-q}$, and let $\|V\|_{\pm q}$ denote the larger of its two operator norms. Then, for all $n \ge 0$ and $s \in \Delta_n$, \begin{equation}\label{e:sigma-entire-simplex} \big\| U_{s_0} V U_{s_1} V \cdots V U_{s_n} \big\|_{L^2_{-q} \to L^2_q} \le C^{n + 1} (n + 1)^{2q} \|V\|_{\pm q}^n . \end{equation} \item\label{i:sigma-entire-series} For a complex diagonal $2k \times 2k$ matrix $\Gamma_{\mathrm F}$, let $\Gamma = \diag(\eps \Lambda_{\mathrm B} | \Gamma_{\mathrm F})$, and set \begin{equation}\label{e:sigma-entire-series} V_\Gamma = - \pi \Tr \big( Q^{\mathrm{FF}} (\Gamma_{\mathrm F} - \eps \Lambda_{\mathrm F}) \big) , \qquad \mathcal S_\Gamma = \sum_{n = 0}^\infty (-1)^n \int_{\Delta_n} U_{s_0} V_\Gamma U_{s_1} V_\Gamma \cdots V_\Gamma U_{s_n} \dd s . \end{equation} The series converges in $L(L^2_{-q}, L^2_q)$, locally uniformly in the diagonal entries of $\Gamma_{\mathrm F}$, and $\mathcal S_\Gamma = e^{-\mathsf G_{X, \Gamma}}$ on $L^2_{-q}$, with the semigroup of \zcref{p:sigma-regular}. The function $\Gamma_{\mathrm F} \mapsto \int e^{-\mathsf G_{X, \Gamma}} 1 = \mathcal B_k(1, \mathcal S_\Gamma 1)$ is entire in the diagonal entries of $\Gamma_{\mathrm F}$. In particular, the partition function $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ of \zcref{d:sigma-partition}, as a function of the diagonal entries of $\Gamma_{\mathrm F}$, is the restriction of an entire function on $\C^{2k}$. \end{enumerate} \end{lemma} \begin{proof} We begin with \ref{i:sigma-entire-bounds}. By \zcref{p:sigma-regular}\ref{i:sigma-regular-uniform}, the semigroup $U_s$ is the one of \zcref{p:sigma-regular} with $\Gamma = \eps \Lambda$. Applying \zcref{p:sigma-regular}\ref{i:sigma-regular-weights} on $L^2_{\pm q}$ with $m = 0$, and from $L^2_{-q}$ with $m = 4q$, we obtain \ref{i:sigma-entire-bounds}, since $U_0 = \Id$. For \ref{i:sigma-entire-simplex}, let $s \in \Delta_n$. Some $s_i$ is at least $1/(n + 1)$. We bound the factor $U_{s_i}$ from $L^2_{-q}$ to $L^2_q$ by \eqref{e:sigma-jets-bounds}, which gives the factor $C (n + 1)^{2q}$, the factors to its right on $L^2_{-q}$, and the factors to its left on $L^2_q$. This proves \eqref{e:sigma-entire-simplex}. We now prove \ref{i:sigma-entire-series}. By \zcref{l:sigma-multipliers}\ref{i:sigma-mult-Q}, the entries of $Q^{\mathrm{FF}}$ are multipliers of degree zero. Hence there exists $C_q$ such that $\|V_\Gamma\|_{\pm q} \le C_q \|\Gamma_{\mathrm F} - \eps \Lambda_{\mathrm F}\|_{\mathrm{op}}$ for all $\Gamma_{\mathrm F}$. By \eqref{e:sigma-entire-simplex} with $V = V_\Gamma$, and since $\Delta_n$ has the mass $1/n!$, the $n$th term of the series in \eqref{e:sigma-entire-series} has norm at most \[ \frac{C^{n + 1} (n + 1)^{2q}}{n!} \big( C_q \|\Gamma_{\mathrm F} - \eps \Lambda_{\mathrm F}\|_{\mathrm{op}} \big)^n \] in $L(L^2_{-q}, L^2_q)$. The $n$th term is a homogeneous polynomial of degree $n$ in the diagonal entries of $\Gamma_{\mathrm F} - \eps \Lambda_{\mathrm F}$, with values in $L(L^2_{-q}, L^2_q)$. Therefore the series converges locally uniformly, and $\Gamma_{\mathrm F} \mapsto \mathcal S_\Gamma$ is entire. Next, we identify $\mathcal S_\Gamma$ with the semigroup. By \eqref{e:sigma-potential}, we have $\mathsf G_{X, \Gamma} = \mathsf G_{X, z} + V_\Gamma$ on the compactly supported smooth superfunctions. Since $V_\Gamma$ is bounded, the closure of $\mathsf G_{X, \Gamma}$ in $L^2_{-q}$ of \zcref{p:sigma-regular}\ref{i:sigma-regular-kato} is the closure of $\mathsf G_{X, z}$ plus $V_\Gamma$, with the same domain; here we use one norm on $L^2_{-q}$ for both realizations, which is possible since the realizations do not depend on $\omega$ and the norms for different $\omega$ are equivalent. The semigroup of a bounded perturbation of a generator is given by the Dyson series \cite[Chapter~IX]{kato1995perturbation}, which converges in $L(L^2_{-q})$ and is the series in \eqref{e:sigma-entire-series}. Since $L^2_q$ embeds continuously in $L^2_{-q}$, we obtain $e^{-\mathsf G_{X, \Gamma}} = \mathcal S_\Gamma$ on $L^2_{-q}$. Finally, $1 \in L^2_{-q}$ by \zcref{p:sigma-regular}\ref{i:sigma-regular-split}, and $\mathcal B_k$ is bounded on $L^2_{-q} \times L^2_q$ by \zcref{p:sigma-regular}\ref{i:sigma-regular-transpose}. Hence $\int e^{-\mathsf G_{X, \Gamma}} 1 = \mathcal B_k(1, e^{-\mathsf G_{X, \Gamma}} 1) = \mathcal B_k(1, \mathcal S_\Gamma 1)$ is entire in the diagonal entries of $\Gamma_{\mathrm F}$. For the sources $(\mathbf u, \mathbf b)$ of \zcref{d:sigma-partition}, this is $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$, by \zcref{d:sigma-partition} and \zcref{p:sigma-regular}\ref{i:sigma-regular-split}, and the map from $(\mathbf u, \mathbf b)$ to the diagonal entries of $\Gamma_{\mathrm F}$ is injective. \end{proof} The following lemma is the computation in the sigma model that the proof of \zcref{t:process}(c) in \zcref{s:proc-jets} uses. \begin{lemma}\label{l:sigma-jets} Let $X > 0$ and $\eps > 0$, let $z = 2 \pi \eps$ and $U_s = e^{-s \mathsf G_{X, z}}$ for $s \ge 0$, and let \[ \mathcal Z(t) = \mathcal Z^{(1)}_{X, \eps}(0, \eps + t), \quad t > -\eps, \qquad m_{\mathrm F} = \Tr(\Lambda_{\mathrm F} Q^{\mathrm{FF}}) = Q^{\mathrm{FF}}_{\mathrm{RR}} - Q^{\mathrm{FF}}_{\mathrm{AA}} . \] Then $\mathcal Z$ is smooth on $(-\eps, \infty)$, and \begin{equation}\label{e:sigma-jets-dyson} \begin{gathered} \mathcal Z'(0) = \pi \int_0^1 \mathcal B_1 \big( 1, U_s m_{\mathrm F} U_{1 - s} 1 \big) \dd s , \\ \mathcal Z''(0) = 2 \pi^2 \int_{0 < x < y < 1} \mathcal B_1 \big( 1, U_x m_{\mathrm F} U_{y - x} m_{\mathrm F} U_{1 - y} 1 \big) \dd x \dd y . \end{gathered} \end{equation} For $0 < x \le y < 1$, we have \begin{equation}\label{e:sigma-jets-pairings} \mathcal B_1 \big( 1, U_x m_{\mathrm F} U_{1 - x} 1 \big) = 2, \qquad \mathcal B_1 \big( 1, U_x m_{\mathrm F} U_{y - x} m_{\mathrm F} U_{1 - y} 1 \big) = 4 + 2 K_X(x, y; z) . \end{equation} In particular, \begin{equation}\label{e:sigma-jets-values} \mathcal Z'(0) = 2 \pi, \qquad \mathcal Z''(0) = 4 \pi^2 \big( 1 + I_X(2 \pi \eps) \big) . \end{equation} \end{lemma} \begin{proof} We begin with the generator. At rank one, $u_1 = 0$ and $b_1 = \eps + t$, so that $\Gamma_{\mathrm F} = (\eps + t) \Lambda_{\mathrm F}$ and $\Gamma_{\mathrm F} - \eps \Lambda_{\mathrm F} = t \Lambda_{\mathrm F}$. Inserting this into \eqref{e:sigma-potential}, we obtain \begin{equation}\label{e:sigma-jets-generator} \mathsf G_{X, \Gamma(0, \eps + t)} = \mathsf G_{X, z} - \pi t \, m_{\mathrm F} . \end{equation} We now expand the semigroup in $t$. We apply \zcref{l:sigma-entire} with $k = 1$ and $\Gamma_{\mathrm F} = (\eps + t) \Lambda_{\mathrm F}$, for which $V_\Gamma = - \pi t \, m_{\mathrm F}$, by \eqref{e:sigma-jets-generator}, and we let $q$, $C$, and $\Delta_n$ be as there. The series \eqref{e:sigma-entire-series} becomes \begin{equation}\label{e:sigma-jets-series} e^{-\mathsf G_{X, \Gamma(0, \eps + t)}} = \sum_{n = 0}^\infty (\pi t)^n \int_{\Delta_n} U_{s_0} m_{\mathrm F} U_{s_1} m_{\mathrm F} \cdots m_{\mathrm F} U_{s_n} \dd s , \end{equation} with convergence in $L(L^2_{-q}, L^2_q)$, and \zcref{l:sigma-entire}\ref{i:sigma-entire-series} gives \[ \mathcal Z(t) = \int e^{-\mathsf G_{X, \Gamma(0, \eps + t)}} 1 = \mathcal B_1 \big( 1, e^{-\mathsf G_{X, \Gamma(0, \eps + t)}} 1 \big) . \] By \zcref{l:sigma-multipliers}\ref{i:sigma-mult-Q}, the entries of $Q^{\mathrm{FF}}$ are multipliers of degree zero, so that multiplication by $m_{\mathrm F}$ is bounded on $L^2_q$ and on $L^2_{-q}$, and we let $C_q$ bound both norms. Inserting \eqref{e:sigma-jets-series}, and using \zcref{l:sigma-entire}\ref{i:sigma-entire-simplex} with $V = m_{\mathrm F}$ and the boundedness of $\mathcal B_1$ on $L^2_{-q} \times L^2_q$ of \zcref{p:sigma-regular}\ref{i:sigma-regular-transpose}, we find that, for all $t > -\eps$, \begin{equation}\label{e:sigma-jets-power} \begin{gathered} \mathcal Z(t) = \sum_{n = 0}^\infty (\pi t)^n c_n , \qquad c_n = \int_{\Delta_n} \mathcal B_1 \big( 1, U_{s_0} m_{\mathrm F} U_{s_1} \cdots m_{\mathrm F} U_{s_n} 1 \big) \dd s , \\ |c_n| \le \frac{C' C^{n + 1} (n + 1)^{2q} C_q^n}{n!} , \end{gathered} \end{equation} where $C'$ depends only on $q$. The power series converges for all $t$, so that $\mathcal Z$ is smooth on $(-\eps, \infty)$, with $\mathcal Z'(0) = \pi c_1$ and $\mathcal Z''(0) = 2 \pi^2 c_2$. In the coordinates $x = s_0$ and $y = s_0 + s_1$, the simplex $\Delta_2$ becomes $\{ 0 < x < y < 1 \}$, with $\dd s = \dd x \dd y$, and we obtain \eqref{e:sigma-jets-dyson}. We now evaluate the pairings in \eqref{e:sigma-jets-dyson} with the rank-one calculus of \zcref{s:int-rank-one}, applied with the present $z$. Let $0 < x \le y < 1$, and write $\Psi_s = \Psi^{X, z}_s$. By fact \ref{i:int-radial} of \zcref{s:int-rank-one}, the superfunctions $U_x 1$ and $U_{1 - y} 1$ are invariant, with the radial representatives $\Psi_x$ and $\Psi_{1 - y}$, and we denote them by the same letters. Their values at $\Lambda$ are one, by \eqref{e:sigma-corner}. The transpose symmetry of \zcref{p:sigma-regular}\ref{i:sigma-regular-transpose} implies that $\mathcal B_1(1, U_x f) = \mathcal B_1(\Psi_x, f)$. The pairings in \eqref{e:sigma-jets-dyson} are therefore those of \zcref{l:int-pairings}, with $\delta = \min(x, 1 - y)$ and with the time $y - x$ in place of the $t$ of that lemma. For the first pairing, we expand $m_{\mathrm F}$ and use the second and the third identities in \eqref{e:int-pairing-one} with $y = x$. We obtain \[ \mathcal B_1 \big( 1, U_x m_{\mathrm F} U_{1 - x} 1 \big) = \mathcal B_1 \big( \Psi_x, Q^{\mathrm{FF}}_{\mathrm{RR}} \Psi_{1 - x} \big) - \mathcal B_1 \big( \Psi_x, Q^{\mathrm{FF}}_{\mathrm{AA}} \Psi_{1 - x} \big) = 1 - (-1) = 2 . \] For the second pairing, we expand both factors $m_{\mathrm F}$, and we obtain four pairings. The two with insertions of one flavor equal one, by \eqref{e:int-pairing-same} with $c = c' = \mathrm{FF}$, and the two with insertions of both flavors equal $-1 - K_X(x, y; z)$, by \eqref{e:int-pairing-cross}. In the last two, the corner channel $-\Id$ of the advanced-covariant matrix $U_{y - x}(Q_{\mathrm{AA}} \Psi_{1 - y})$, which the semigroup preserves, contributes the corner term $-1$ of the radial integration formula of \zcref{l:sigma-radial-integral}. Its radial channel $e^{(y - x)(X \Delta_D - z d)} \Psi_{1 - y}$, which evolves by the radial semigroup, contributes the bulk term $-K_X(x, y; z)$, by \zcref{l:int-observation}. Since $m_{\mathrm F} U_{y - x} m_{\mathrm F}$ contains the two pairings of one flavor with the sign $+$ and the two others with the sign $-$, we conclude that \[ \mathcal B_1 \big( 1, U_x m_{\mathrm F} U_{y - x} m_{\mathrm F} U_{1 - y} 1 \big) = 1 + 1 + 2 \big( 1 + K_X(x, y; z) \big) = 4 + 2 K_X(x, y; z) , \] which completes the proof of \eqref{e:sigma-jets-pairings}. It remains to integrate. The integral of $K_X$ over $\{ 0 < x < y < 1 \}$ converges absolutely, by \zcref{l:sigma-scalar}\ref{i:sigma-scalar-kernels}, and the definition \eqref{e:intro-I} shows that it equals $I_X(z)$. Inserting \eqref{e:sigma-jets-pairings} into \eqref{e:sigma-jets-dyson}, we find \begin{gather*} \mathcal Z'(0) = \pi \int_0^1 2 \dd s = 2 \pi, \\ \mathcal Z''(0) = 2 \pi^2 \int_{0 < x < y < 1} \big( 4 + 2 K_X(x, y; z) \big) \dd x \dd y = 2 \pi^2 \big( 2 + 2 I_X(z) \big) , \end{gather*} which is \eqref{e:sigma-jets-values}, since $z = 2 \pi \eps$. \end{proof} The first identity in \eqref{e:sigma-jets-values} expresses the unit intensity of the limiting point process, and the second determines its pair correlation through \zcref{r:proc-pair}; they are the two limits in \eqref{e:intro-jets}. The contact kernel $\widetilde K_X$ arises in the same way from the product $Q^{\mathrm{BB}}_{\mathrm{RR}} Q^{\mathrm{BB}}_{\mathrm{AA}} + Q^{\mathrm{BB}}_{\mathrm{RA}} Q^{\mathrm{BB}}_{\mathrm{AR}}$ of entries of $Q$ at a single site. By \zcref{l:sigma-polar}\ref{i:sigma-polar-top}, we have \[ \big\langle Q^{\mathrm{BB}}_{\mathrm{RR}} Q^{\mathrm{BB}}_{\mathrm{AA}} + Q^{\mathrm{BB}}_{\mathrm{RA}} Q^{\mathrm{BB}}_{\mathrm{AR}} \big\rangle_{\mathrm{top}} = -d^2 - d(B + F) = -2Bd , \] and after multiplication by the density $d^{-2}$ of \zcref{l:sigma-radial-integral}, it produces the weight $B/d$ of \eqref{e:intro-Kc}. The corresponding pairing is evaluated in \zcref{l:int-contact}.