by Taylor expanding $\exp(-\frac{\lambda}{4!} \int_M \phi(x)^4 dV_M(x))$. This gives

$$
\begin{align*}
(A.6) \stackrel{\text{heu}}{\equiv} \mathbb{E}_{\text{GFF}}^{M}[\phi(x)\phi(y)] &+ \mathbb{E}_{\text{GFF}}^{M}[\phi(x)\phi(y)\left(-\frac{\lambda}{4!}\right) \int \phi(z)^4 dV_M(z)] \\
&+ \mathbb{E}_{\text{GFF}}^{M}[\phi(x)\phi(y)\frac{1}{2}\left(-\frac{\lambda}{4!}\right)^2 \iint \phi(z)^4 \phi(w)^4 dV_M(z)dV_M(w)] + \dots \\
&= \mathbb{E}_{\text{GFF}}^{M}[\phi(x)\phi(y)] - \frac{\lambda}{4!} \underbrace{\int \mathbb{E}_{\text{GFF}}^{M}[\phi(x)\phi(y)\phi(z)^4] dV_M(z)}_{A} \\
&\quad + \frac{1}{2}\left(-\frac{\lambda}{4!}\right)^2 \iint \mathbb{E}_{\text{GFF}}^{M}[\phi(x)\phi(y)\phi(z)^4\phi(w)^4] dV_M(z)dV_M(w) + \dots
\end{align*}
$$

Treating $\phi(x)$, $\phi(y)$, $\phi(z)$ as (jointly Gaussian!) random variables, by (i) of lemma A.2 and the above rules we write

$$
\begin{align*}
A \stackrel{\text{heu}}{\equiv} 3 \int dV_M(z) v \left( \xrightarrow{\text{t}} \xrightarrow{\text{y}} \bigcirc 2 \right) + 12 \int dV_M(z) v \left( \xrightarrow{\text{t}} \bigcirc 2 \xrightarrow{\text{y}} \right) \\
&= 3 \int G(x, y) G(z, z)^2 dV_M(z) + 12 \int G(x, z) G(y, z) G(z, z) dV_M(z),
\end{align*}
$$

where the factors correspond to the number of ways of getting the same contraction starting from 6 legs (4 on z, 1 on x, y each), and G denotes the Green function (of Δ + m²). One can represent higher order terms using these diagrams in a similar manner. More than that, the diagrams also represent actual physical processes. See Peskin and Schroeder [53].

A.2 Sobolev Spaces over Domains

In this paper we make essential use of the usual $L^2$ Sobolev spaces over Riemannian manifolds. First let $(M,g)$ be a closed Riemannian manifold and $s \in \mathbb{R}$. Then the Sobolev space $W^s(M)$ of order $s$ is defined generally as the closure of $C^\infty(M)$ under a norm $\|\cdot\|_{W^s(M)}$, where the norm $\|\cdot\|_{W^s(M)}$ could be defined in various equivalent ways. We refer to Taylor [78] chapter 4 for a general discussion. For us, $s = \pm 1, \pm \frac{1}{2}$. We rely heavily on the following fact.

**Lemma A.3.** Let $\Lambda_{2s}$ be an elliptic strictly positive formally self-adjoint pseudodifferential operator on $M$ with order $2s$. Then the inner product

$$
\langle - , - \rangle_{W^s} \stackrel{\text{def}}{=} \langle - , \Lambda_{2s} - \rangle_{L^2} \qquad (A.7)
$$

induces an equivalent norm for $W^s(M)$.

In particular, the real power $(\Delta_M + m^2)^s$ of the Helmholtz operator (massive Laplacian) $\Delta_M + m^2$ provides such a candidate for $\Lambda_{2s}$. **Convention:** whenever we use the space $W^s(M)$, the inner product (A.7) with $\Lambda_{2s} = (\Delta_M + m^2)^s$ is understood, unless otherwise specified.

**Remark A.1.** Various regimes of functional calculus can be used to define $(\Delta_M + m^2)^s$. One of them is presented in section 2.2.1 which in fact defines *complex* powers. We also mention a smooth functional calculus presented in Sogge [75] theorem 4.3.1.

Next we discuss important subspaces of $W^s(M)$. Let $A \subset M$ be a closed set and $U \subset M$ an open set. Define

$$
W_A^s(M) \stackrel{\text{def}}{=} \{u \in W^s(M) \mid \text{supp } u \subset A \text{ as a distribution}\}, \qquad (A.8)
$$

$$
W_U^s(M) \stackrel{\text{def}}{=} \text{closure of } C_c^\infty(U) \text{ inside } W^s(M), \qquad (A.9)
$$

$$
W^s(U) \stackrel{\text{def}}{=} W_{M \setminus U}^s(M)^\perp \subset W^s(M). \qquad (A.10)
$$

These are closed subspaces of $W^s(M)$.
**Remark A.2.** We point out right away that by definition, then,

$$W^s(U) \cong W^s(M)/W_{M\setminus U}^s(M), \qquad (A.11)$$

the latter equipped with the quotient norm, which is a more familiar characterization of $W^s(U)$, see Taylor [78] page 339. Our definition as in (A.10) poses the obvious problem that in general $C_c^\infty(U) \not\subset W^s(U)$, at least for $s \notin \mathbb{Z}_+$. We emphasize therefore that what is important in this definition is not the space $W^s(U)$ per se but the following choice for its inner product:

$$\langle f, h \rangle_{W^s(U)} \stackrel{\text{def}}{=} \langle P_{M\setminus U}^{\perp} f, P_{M\setminus U}^{\perp} h \rangle_{W^s(M)}, \qquad (A.12)$$

for any $f, h \in W^s(M)$, in particular for $f, h \in C_c^\infty(U)$, which produces a norm equivalent to the quotient norm, where $P_{M\setminus U}^{\perp}: W^s(M) \to W^s(U)$ denotes the orthogonal projection.

**Remark A.3.** Clearly $W_U^s(M) \subset W_{\bar{U}}^s(M)$ by definition. In general the inclusion is strict (certainly if $U \neq (\bar{U})^\circ$!). See Taylor [78] page 339 and section 4.7 for interesting discussions on conditions for $s$ and $U$ for which equality holds. In particular, $W_\Omega^k(M) = W_{\Omega^\circ}^k(M) = W_{\Omega^k}^k(M)$ if $\Omega \subset M$ is a domain with smooth boundary $\partial\Omega$ (a closed Riemannian manifold with one dimension less) and $k \in \mathbb{Z}_+$. In this case, we use these notations interchangeably.

The rest of this appendix could be read along with section 5.1. Let $s = -1$. Although $C_c^\infty(U) \not\subset W^{-1}(U)$, we have

**Lemma A.4.** Let $U \subset M$ be an open set. Then $P_{M\setminus U}^{\perp}(C_c^\infty(U))$ is dense in $W^{-1}(U)$.

*Proof.* We note $\Delta_M + m^2$ is local and therefore $(\Delta + m^2)(C_c^\infty(U)) \subset C_c^\infty(U)$. It follows from lemma 5.1 and our definition of $W_U^1(M)$ that $(\Delta + m^2)(C_c^\infty(U)) \subset W_{M\setminus U}^{-1}(M)^\perp$ and is dense there, proving the result. $\square$

**Remark A.4.** Clearly, the map $P_{M\setminus U}^{\perp}$ is also injective on $C_c^\infty(U)$; together with lemma A.4 this shows $P_{M\setminus U}^{\perp}$ is a good embedding of $C_c^\infty(U)$ in $W^{-1}(U)$. In fact, this is the same as the embedding of $C_c^\infty(U)$ in $D'(U)$, by remark 5.1. Nevertheless, the smaller class $(\Delta + m^2)(C_c^\infty(U))$, as it is already dense in $W^{-1}(U)$, suffices as a class of test functions to define the GFF with Dirichlet condition over a domain (see remark below lemma 2.2). This reflects the fact that the Cameron-Martin pairing $\langle -,- \rangle_{W^1}$ is more natural than $\langle -,- \rangle_{L^2}$ in treating the GFF (see remark 2.3). We have stuck to $\langle -,- \rangle_{L^2}$ only because this is more practical with functional analysis.

**Remark A.5.** For general $s \in \mathbb{R}$, one could also define

$$W_0^s(U) \stackrel{\text{def}}{=} \text{closure of } C_c^\infty(U) \text{ under (A.12)}. \qquad (A.13)$$

Then $W_U^s(M) \subset W_0^s(U)$. But it cannot generally be compared with $W_{\bar{U}}^s(M)$ (to the author's knowledge). See the exercises in Taylor [78] pages 343-344 for more information.

Next we state the duality results for the various spaces. Recall that $\langle -,- \rangle_{L^2(M)}$ denotes both the inner product of $L^2(M)$ and the distributional pairing between $D'(M)$ and $C^\infty(M)$. Below, we extend it to denote also the pairing between dual Sobolev spaces (see (i) of the lemma below).

**Lemma A.5.** Let $M$ be a closed Riemannian manifold, $U \subset M$ an open set, $A \subset M$ a closed set, and $s \in \mathbb{R}$.

(i) $W^{-s}(M)$ is the dual Banach space, denoted $W^s(M)^*$, of $W^s(M)$ under $\langle -,- \rangle_{L^2}$;

(ii) the annihilator of $W_U^s(M)$ under $\langle -,- \rangle_{L^2}$ is $W_{M\setminus U}^{-s}(M)$, that is,

$$W_{M\setminus U}^{-s}(M) = \{u \in W^{-s}(M) \mid \langle u, f \rangle_{L^2} = 0 \text{ for all } f \in W_U^s(M)\}; \qquad (A.14)$$

the annihilator of $W_A^s(M)$ is accordingly $W_{M\setminus A}^{-s}(M)$;

(iii) $W^s(U)^* \cong W_{\bar{U}}^{-s}(M)$, $W_U^s(M)^* \cong W^{-s}(U)$, these spaces being therefore reflexive.

Finally, when $\Omega \subset M$ is a domain with smooth boundary $\partial\Omega$, we define, in view of lemma 5.1, the Dirichlet Green operator $(\Delta_{\Omega,D} + m^2)^{-1} := (\Delta + m^2)^{-1}P_{M\setminus\Omega^\circ}^{\perp}: W^{-1}(\Omega^\circ) \to W_{\Omega^\circ}^1(M)$. Clearly this agrees with the usual definition. In terms of quadratic forms,

**Lemma A.6** ([73] theorem VII.1). Let $\Omega \subset M$ be a domain with smooth boundary $\partial\Omega$. We have

$$\langle f, (\Delta_{\Omega,D} + m^2)^{-1} h \rangle_{L^2} = \langle P_{M\setminus\Omega^\circ}^{\perp} f, P_{M\setminus\Omega^\circ}^{\perp} h \rangle_{W^{-1}}, \qquad (A.15)$$

for $f, h \in C_c^\infty(\Omega^\circ)$.
A.3 Symbol Convergence Lemma and Heat Kernel

Proof of lemma 3.1. By coordinate invariance of the definition of Ψ<sup>m</sup>(M) it suffices to pick x ∈ M and prove the result for a chart around x and χ(x) = 1. Denote the kernel of χE<sub>ε</sub>χ by E<sub>χ,ε</sub> then in this chart we could write

$$
E_{\chi, \epsilon}(x, y) = \tilde{E}_{\chi, \epsilon}(x, h) = \frac{1}{F_{\epsilon}(x)} \tilde{\psi}\left(\frac{h}{\epsilon}\right) \tilde{\chi}(h), \quad (A.16)
$$

where $h = x - y$. Indeed, by definition of our function $\psi$ and freedom of choosing $\chi$ we could further assume
that for small enough $\epsilon$ one has $\tilde{\chi}(h) \equiv 1$ on the support of $\tilde{\psi}(\cdot/\epsilon)$. Thus under this condition

$$
\sigma_{\chi E_{\epsilon} \chi}(x, \xi) = \int_{\mathbb{R}^d} e^{-i\mathbf{h} \cdot \xi} \frac{1}{F_{\epsilon}(x)} \tilde{\psi}\left(\frac{\mathbf{h}}{\epsilon}\right) \mathrm{d}\mathbf{h} = \frac{\epsilon^d}{F_{\epsilon}(x)} \underbrace{F_1(x) \sigma_{\chi E_1 \chi}(x, \epsilon\xi)}_{\text{indep. of } x} \quad (A.17)
$$

Note that $\sigma_{\chi E_1 \chi}(x, \eta)$ is Schwartz in $\eta$ and $\sigma_{\chi E_1 \chi}(x, 0) = 1$. On the other hand clearly $\sigma_{\chi 1 \chi}(x, \xi) \equiv 1$. Thus for some $U' \subset U$ depending only on the chart and $\chi$, one has

$$
\sup_{x \in K \cap U'} \sup_{\xi} \frac{|\sigma_{\chi(E_\epsilon - 1)\chi}(x, \xi)|}{(\xi)^\delta} \le \begin{cases} C \sup_{x \in K \cap U'} \sup_{|\xi| \le R} (\xi)^{-\delta} |\sigma_{\chi E_1 \chi}(x, \epsilon\xi) - 1| \le C_{K,\chi} \sqrt{\epsilon}, \\ C \sup_{x \in K \cap U'} \sup_{|\xi| \ge R} (\xi)^{-\delta/2} |\sigma_{\chi E_1 \chi}(x, \eta)|, \end{cases} \quad (A.18)
$$

with $R = \epsilon^{-1/2}$. Next we deal with derivatives. Note that by (A.17) all the x-derivatives fall on $1/F_{\epsilon}(x)$ and all $\xi$-derivatives fall on $\sigma_{\chi E_1 \chi}(x, \epsilon\xi)$. Indeed, one has $|\partial_{x}^{\alpha}(1/F_{\epsilon}(x))| \le C_{\alpha}\epsilon^{-d}$ (see Dyatlov and Zworski [21] page 28), and so when there are only x-derivatives we obtain the same bounds as (A.18) only with new constants depending on $\alpha$. When there is at least one $\xi$-derivative,

$$
|\partial_{\xi}^{\beta} (\sigma_{\chi E_1 \chi}(x, \epsilon\xi))| = |\epsilon^{\beta}| (\partial_{\xi}^{\beta} \sigma_{\chi E_1 \chi})(x, \epsilon\xi)| \le C_{\beta, K, \chi} \epsilon^{\beta} |\epsilon\xi|^{\delta-|\beta|}, \quad |\beta| \ge 1. \quad (A.19)
$$

Hence, on account of (A.17) again,

$$
\sup_{x \in K \cap U'} \sup_{\xi} \frac{|\partial_x^\alpha \partial_\xi^\beta \sigma_{\chi(E_\epsilon-1)\chi}(x, \xi)|}{(\xi)^{\delta-|\beta|}} \le C \epsilon^d C_\alpha \epsilon^{-d} C_{\beta,K,\chi} \epsilon^{|\beta|} = C_{\alpha,\beta,K,\chi} \epsilon^{|\beta|}. \quad (A.20)
$$

Consequently, all the $S_{1,0}^\delta$ seminorms of $\sigma_{\chi(E_\epsilon-1)\chi}$ goes to zero as $\epsilon \to 0$. We obtain the result. $\square$

In what follows we sum up some properties of the **heat operator** $e^{-t(\Delta+m^2)}$ of the massive Laplacian (Helmholtz operator) and its Schwartz kernel $p_t(x, y)$ called the **heat kernel**.

**Lemma A.7** ([11] theorems 2.30, 2.38 and pages 92-94). **We have**

(i) $p_t(x, y) \in C^\infty((0, \infty) \times M \times M);$

(ii) we have

$$
(\Delta + m^2)^{-1} = \int_0^\infty e^{-t(\Delta+m^2)} \mathrm{d}t. \qquad (A.21)
$$

In particular, the kernel $G_t$ of $e^{-t(\Delta+m^2)}(\Delta+m^2)^{-1} = (\Delta+m^2)^{-1}e^{-t(\Delta+m^2)}$ is

$$
G_t(x, y) = \int_t^\infty p_s(x, y) \mathrm{d}s, \quad (A.22)
$$

for $x, y \in M$.

(iii) Let $\dim M = n$. There are asymptotic expansions

$$
p_t(x, y) \sim \frac{1}{(4\pi t)^{n/2}} e^{-\frac{1}{4t} d(x, y)^2} \sum_{i=0}^{\infty} f_i(x, y) t^i, \quad (A.23)
$$

$$
\mathrm{tr}_{L^2(M)} (e^{-t(\Delta+m^2)}) \sim \frac{1}{(4\pi t)^{n/2}} \sum_{i=0}^{\infty} a_i t^i \quad (A.24)
$$

as $t \to 0+$, for some real numbers $a_i$ and functions $f_i \in C^\infty(M \times M)$, $i=0, 1, 2, \dots$

(iv) For $t$ large and each $\ell \in \mathbb{N}$,

$$
\|p_t(x, y)\|_{C^\ell} \le C_\ell e^{-tm^2/2} \quad (A.25)
$$

for some constant $C_\ell$.
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[80] Taylor, Michael E., *The Dirichlet-to-Neumann Map and Fractal Variants*, lecture notes, available at: [https://mtaylor.web.unc.edu/notes/fractal-analysis/](https://mtaylor.web.unc.edu/notes/fractal-analysis/)