#  Combinatorial QFT on graphs: first quantization formalism

Ivan Contreras  Department of Mathematics, Amherst College, 31 Quadrangle
Drive, Amherst, MA 01002, USA<EMAIL_ADDRESS>,  Santosh
Kandel  California State University, Sacramento, Department of Mathematics and
Statistics, CA 95819, USA<EMAIL_ADDRESS>,  Pavel Mnev  University of Notre
Dame, Notre Dame, IN 46556, USA  St. Petersburg Department of V. A. Steklov
Institute of Mathematics of the Russian Academy of Sciences, 27 Fontanka, St.
Petersburg, Russia, 191023<EMAIL_ADDRESS>and  Konstantin Wernli  Centre for
Quantum Mathematics, IMADA, University of Southern Denmark, Campusvej 55, 5230
Odense M, Denmark<EMAIL_ADDRESS>
###### Abstract.

We study a combinatorial model of the quantum scalar field with polynomial
potential on a graph. In the first quantization formalism, the value of a
Feynman graph is given by a sum over maps from the Feynman graph to the
spacetime graph (mapping edges to paths). This picture interacts naturally
with Atiyah-Segal-like cutting-gluing of spacetime graphs. In particular, one
has combinatorial counterparts of the known gluing formulae for Green’s
functions and (zeta-regularized) determinants of Laplacians.

The work of K. Wernli was supported by the ERCSyG project, Recursive and Exact
New Quantum Theory (ReNewQuantum) which received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme under grant agreement No. 810573.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation: first quantization formalism
      1. 1.1.1 Version with 1d gravity.
      2. 1.1.2 Heuristics on locality in the first quantization formalism
    2. 1.2 QFT on a graph. A guide to the paper.
    3. 1.3 Acknowledgements
  2. 2 Scalar field theory on a graph
    1. 2.1 Functorial picture
    2. 2.2 Aside: “QFT with corners” (or “extended QFT”) picture
      1. 2.2.1 Source $n$-category
      2. 2.2.2 Target $n$-category
      3. 2.2.3 The QFT functor
  3. 3 Gaussian theory
    1. 3.1 Gaussian theory on a closed graph
      1. 3.1.1 Examples
    2. 3.2 Gaussian theory relative to the boundary
      1. 3.2.1 Dirichlet problem
      2. 3.2.2 Dirichlet-to-Neumann operator
      3. 3.2.3 Partition function and correlators (relative to a boundary subgraph)
      4. 3.2.4 Examples
    3. 3.3 Gluing in Gaussian theory. Gluing of propagators and determinants.
      1. 3.3.1 Cutting a closed graph
      2. 3.3.2 Proof 1 (“functional integral approach”)
      3. 3.3.3 Proof 2 (Schur complement approach)
      4. 3.3.4 Examples
      5. 3.3.5 General cutting/gluing of cobordisms
      6. 3.3.6 Self-gluing and trace formula
    4. 3.4 Comparison to continuum formulation
      1. 3.4.1 Gluing formulae for Green’s functions and determinants
      2. 3.4.2 Example: continuum limit of line and circle graphs
  4. 4 Interacting theory via Feynman diagrams
    1. 4.1 Version relative to a boundary subgraph
    2. 4.2 Cutting/gluing of perturbative partition functions via cutting/gluing of Feynman diagrams
  5. 5 Path sum formulae for the propagator and determinant (Gaussian theory in the first quantization formalism)
    1. 5.1 Quantum mechanics on a graph
    2. 5.2 Path sum formulae on closed graphs
      1. 5.2.1 Paths and h-paths in graphs
      2. 5.2.2 h-path formulae for heat kernel, propagator and determinant
      3. 5.2.3 Resumming h-paths. Path sum formulae for propagator and determinant
      4. 5.2.4 Aside: path sum formulae for the heat kernel and the propagator – “1d gravity” version.
    3. 5.3 Examples
      1. 5.3.1 Circle graph, $N=3$.
      2. 5.3.2 Line graph, $N=3$
    4. 5.4 Relative versions
      1. 5.4.1 h-path formulae for Dirichlet propagator, extension operator, Dirichlet-to-Neumann operator
      2. 5.4.2 Resumming h-paths
      3. 5.4.3 Examples
    5. 5.5 Gluing formulae from path sums
  6. 6 Interacting theory: first quantization formalism
    1. 6.1 Closed graphs
      1. 6.1.1 Edge-to-path maps
      2. 6.1.2 Feynman weights
    2. 6.2 Relative version
    3. 6.3 Cutting and gluing
  7. 7 Conclusion and outlook

##  1\. Introduction

In this paper we study a combinatorial model of the quantum massive scalar
field with polynomial potential on a spacetime given a by a graph $X$. Our
motivation to do so was the study of the first quantization formalism, that we
recall in Section 1.1 below, and in particular its interplay with locality,
i.e. cutting and gluing of the spacetime manifold. At the origin is the
Feynman-Kac formula (4) for the Green’s function of the kinetic operator. In
case the spacetime is a graph, this formula has a combinatorial analog given
by summing over paths with certain weights (see Section 5). These path sums
interact very naturally with cutting and gluing, in a mathematically rigorous
way, see Theorem 3.8 and its proof from path sum formulae 5.5.

A second motivation to study this model was the notion of (extended)
functorial QFTs with source a Riemannian cobordism. Few examples of functorial
QFTs out of Riemannian cobordism categories exist, for instance [15],
[16],[21]. In this paper, we define a graph cobordism category and show that
the combinatorial model defines a functor to the category of Hilbert spaces
(Section 2.1). We also propose an extended cobordism (partial) $n$-category of
graphs and a functor to a target $n$-category of commutative algebras defined
by the combinatorial QFT we are studying (Section 2.2).

Finally, one can use this discrete toy model to approximate the continuum
theory, which in this paper we do only in easy one-dimensional examples (see
Section 3.4). We think that the results derived in this paper will be helpful
to study the interplay between renormalization and locality in higher
dimensions (the two-dimensional case was discussed in detail in [16]).

###  1.1. Motivation: first quantization formalism

We outline the idea of the first quantization picture in QFT in the example of
the interacting scalar field.111We refer the reader to the inspiring
exposition of this idea in [10, Section 3.2].

Consider the scalar field theory on a Riemannian $n$-manifold $M$ perturbed by
a polynomial potential $p(\phi)=\sum_{k\geq 3}\frac{p_{k}}{k!}\phi^{k}$,
defined by the action functional

(1)
$S(\phi)=\int_{M}\left(\frac{1}{2}\phi(\Delta+m^{2})\phi+p(\phi)\right)d^{n}x.$

Here $\phi\in C^{\infty}(M)$ is the field, $\Delta$ is the Laplacian
determined by the metric, $m>0$ is the mass parameter and $d^{n}x$ denotes the
metric volume element.

The partition function is formally given by a (mathematically ill-defined)
functional integral understood perturbatively as a sum over Feynman graphs
$\Gamma$,222In this discussion we will ignore the issue of divergencies and
renormalization.

(2)
$Z_{M}=``\int_{C^{\infty}(M)}\mathcal{D}\phi\;e^{-\frac{1}{\hbar}S(\phi)}"\;=\left({\det}^{\zeta}(\Delta+m^{2})\right)^{-\frac{1}{2}}\cdot\sum_{\Gamma}\Phi_{\Gamma}.$

Here ${\det}^{\zeta}$ is the functional determinant in zeta function
regularization. The weight $\Phi_{\Gamma}$ of a Feynman graph is the product
of Green’s functions $G(x,y)$ of the kinetic operator $\Delta+m^{2}$
associated with the edges of $\Gamma$, integrated over ways to position
vertices of $\Gamma$ at points of $M$ (times the vertex factors, a symmetry
factor and a loop-counting factor):

(3)
$\Phi_{\Gamma}=\frac{\hbar^{|E|-|V|}}{|\mathrm{Aut}(\Gamma)|}\int_{M^{\times
V}}d^{n}x_{1}\cdots d^{n}x_{|V|}\;\prod_{(u,v)\in
E}G(x_{u},x_{v})\cdot\prod_{v\in V}(-p_{\mathrm{val}(v)}).$

Here $V,E$ are the set of vertices and the set of edges of $\Gamma$,
respectively.

Next, one can understand the kinetic operator $\Delta+m^{2}=\colon\widehat{H}$
as a quantum Hamiltonian of an auxiliary quantum mechanical system with
Hilbert space $L^{2}(M)$. Then, one can write the Green’s function $G(x,y)$ as
the evolution operator of this auxiliary system integrated over the time of
evolution:

$G(x,y)=\int_{0}^{\infty}dt\,\langle x|e^{-t\widehat{H}}|y\rangle.$

Replacing the evolution operator (a.k.a. heat kernel) with its Feynman-Kac
path integral representation, one has

(4) $G(x,y)=\int_{0}^{\infty}dt\int_{\gamma\colon[0,t]\rightarrow
M,\;\gamma(0)=x,\,\gamma(t)=y}\mathcal{D}\gamma\;e^{-S^{1q}(\gamma)}.$

Here the inner integral is over paths $\gamma$ on $M$ parameterized by the
interval $[0,t]$, starting at $y$ and ending at $x$; the auxiliary (“first
quantization”) action in the exponent is

(5)
$S^{1q}(\gamma)=\int_{0}^{t}d\tau\left(\frac{\dot{\gamma}^{2}}{4}+m^{2}-\frac{1}{6}R(\gamma)\right),$

where $R$ is the scalar curvature of the metric on $M$;
$\dot{\gamma}^{2}\colon=g_{\gamma(\tau)}(\dot{\gamma},\dot{\gamma})$ is the
square norm of the velocity vector $\dot{\gamma}\in T_{\gamma(\tau)}M$ of the
path w.r.t. the metric $g$ on $M$.333The action (5) can be obtained from the
short-time asymptotics (Seeley-DeWitt expansion) of the heat kernel
$\kappa(x,y;t)=\langle x|e^{-t\widehat{H}}|y\rangle\underset{t\rightarrow
0}{\sim}(4\pi
t)^{-\frac{n}{2}}e^{-\frac{d(x,y)^{2}}{4t}}(1+b_{2}(x,y)t+b_{4}(x,y)t^{2}+\cdots)$,
with $b_{2k}$ smooth functions on $M\times M$ (in particular, on the
diagonal), with $b_{2}(x,x)=-m^{2}+\frac{1}{6}R(x)$; $d(x,y)$ is the geodesic
distance on $M$, see e.g. [26]. One then has
$\displaystyle\kappa(x,y;t)=\lim_{N\rightarrow\infty}\int_{M^{\times(N-1)}}d^{n}x_{N-1}\cdots
d^{n}x_{1}\kappa(x_{N}=x,x_{N-1};\delta t)\cdots\kappa(x_{2},x_{1};\delta
t)\kappa(x_{1},x_{0}=y;\delta
t)=\lim_{N\rightarrow\infty}\int\prod_{j=1}^{N-1}\left((4\pi\delta
t)^{-\frac{n}{2}}dx_{j}\right)\,e^{-\sum_{j=1}^{N}\left(\frac{d(x_{j},x_{j-1})^{2}}{4(\delta
t)^{2}}+(m^{2}-\frac{1}{6}R(x_{j}))\right)\delta t}$. In the r.h.s. one
recognizes the path integral of (4) written as a limit of finite-dimensional
integrals (cf. [12]). We denoted $\delta t=t/N$.

Plugging the integral representation (4) of the Green’s function into (3), one
obtains the following integral formula for the weight of a Feynman graph:

(6)
$\Phi_{\Gamma}=\frac{\hbar^{|E|-|V|}}{|\mathrm{Aut}(\Gamma)|}\int_{0<t_{1},\ldots,t_{|E|}<\infty}dt_{1}\cdots
dt_{|E|}\int_{\boldsymbol{\gamma}\colon\Gamma_{t_{1},\ldots,t_{|E|}}\rightarrow
M}\mathcal{D}\boldsymbol{\gamma}\,e^{-S^{1q}(\boldsymbol{\gamma})}\cdot\\\
\cdot\prod_{v\in V}\left(-p_{\mathrm{val}(v)}\right).$

Here $\Gamma_{t_{1},\ldots,t_{|E|}}$ is the graph $\Gamma$ seen as a metric
graph with $t_{e}$ the length of edge $e$. The outer integral is over metrics
on $\Gamma$, the inner (path) integral is over maps $\boldsymbol{\gamma}$ of
$\Gamma$ to $M$, sending vertices to points of $M$ and edges to paths
connecting those points; $S^{1q}(\boldsymbol{\gamma})$ is understood as a sum
of expressions in the r.h.s. of (5) over edges of $\Gamma$.

We refer to the formula (6), representing the weight of a Feynman graph via an
integral over maps $\Gamma\rightarrow M$ (or, equivalently, as a partition
function of an auxiliary 1d sigma model on the graph $\Gamma$ with target
$M$), as the “first quantization formula.”444As opposed to the functional
integral (2) – the “second quantization formula.”

######  Remark 1.1.

It is known that $\frac{1}{6}R$ appears in the quantum Hamiltonian of the
quantum mechanical system of a free particle on a closed Riemannian manifold
$M$, see for example [1, 27]. Here, the difference is that $\frac{1}{6}R$ is
introduced in the classical action (5) so that $\Delta+m^{2}$ is the quantum
Hamiltonian.

######  Remark 1.2.

One can absorb the determinant factor in the r.h.s. of (2) into the sum over
graphs, if we extend the set of graphs $\Gamma$ to allow them to have circle
connected components (with no vertices), with the rule

(7)
$\Phi_{S^{1}}=-\frac{1}{2}\log{\det}^{\zeta}(\Delta+m^{2})=\frac{1}{2}\int_{0}^{\infty}\frac{dt}{t}\,\operatorname{tr}\left(e^{-t\widehat{H}}\right)\\\
=\frac{1}{2}\int_{0}^{\infty}\frac{dt}{t}\int_{\gamma\colon
S^{1}_{t}\rightarrow M}\mathcal{D}\gamma\,e^{-S^{1q}(\gamma)},$

where the integral in $t$ is understood in zeta-regularized sense;
$S^{1}_{t}=\mathbb{R}/t\mathbb{Z}$ is the circle of perimeter $t$.

####  1.1.1. Version with 1d gravity.

Another way to write the formula (4) is to consider paths $\gamma$
parameterized by the standard interval $I=[0,1]$ (with coordinate $\sigma$)
and introduce an extra field – the metric
$\xi=\underline{\xi}(\sigma)(d\sigma)^{2}$ on $I$:

(8)
$G(x,y)=\int_{(\mathrm{Met}(I)\times\mathrm{Map}(I,M)_{x,y})/\mathrm{Diff(I)}}\mathcal{D}\xi\,\mathcal{D}\gamma\,e^{-\bar{S}^{1q}(\gamma,\xi)}.$

Here $\mathrm{Map}(I,M)_{x,y}$ is the space of paths $\gamma\colon
I\rightarrow M$ from $x$ to $y$; the exponent in the integrand is

(9)
$\bar{S}^{1q}(\gamma,\xi)=\int_{I}\left(\frac{1}{4}(\xi^{-1}\otimes\gamma^{*}g)(d\gamma,d\gamma)+m^{2}-\frac{1}{6}R(\gamma)\right)d\mathrm{vol}_{\xi}\\\
=\int_{0}^{1}\left(\frac{\dot{\gamma}^{2}}{4\underline{\xi}}+m^{2}-\frac{1}{6}R(\gamma)\right)\sqrt{\underline{\xi}}\,d\sigma,$

with $d\mathrm{vol}_{\xi}$ the Riemannian volume form of $I$ induced by $\xi$.
Note that the action (9) is invariant under diffeomorphisms of $I$. One can
gauge-fix this symmetry by requiring that the metric $\xi$ is constant on $I$,
then one is left with integration over the length $t$ of $I$ w.r.t. the
constant metric; this reduces the formula (8) back to (4).

In (8), the Green’s function of the original theory on $M$ is understood in
terms of a 1d sigma-model on $I$ with target $M$ coupled to 1d gravity. For a
Feynman graph, similarly to (6), one has

(10)
$\Phi_{\Gamma}=\hbar^{|E|-|V|}\int_{(\mathrm{Met}(\Gamma)\times\mathrm{Map}(\Gamma,M))/\mathrm{Diff}(\Gamma)}\mathcal{D}\boldsymbol{\xi}\,\mathcal{D}\boldsymbol{\gamma}\,e^{-\bar{S}^{1q}(\boldsymbol{\gamma},\boldsymbol{\xi})}\prod_{v\in
V}\left(-p_{\mathrm{val}(v)}\right)$

– the partition function of 1d sigma model on the Feynman graph $\Gamma$
coupled to 1d gravity on $\Gamma$;
$\bar{S}^{1q}(\boldsymbol{\gamma},\boldsymbol{\xi})$ is understood as a sum of
terms (9) over the edges of $\Gamma$.555If one thinks of the quotient by
$\mathrm{Diff}(\Gamma)$ in (10) as a stack, one can see the symmetry factor
$\frac{1}{|\mathrm{Aut}(\Gamma)|}$ as implicitly contained in the integral. To
see that, one should think of the quotient by $\mathrm{Diff}(\Gamma)$ as first
a quotient by the connected component of the identity map
$\mathrm{Diff}_{0}(\Gamma)$ and then a quotient by the mapping class group
$\pi_{0}\mathrm{Diff}(\Gamma)=\mathrm{Aut}(\Gamma)$. One can interpret the
$1/t$ factor in the r.h.s. of (7) in a similar fashion: $\mathrm{Diff}(S^{1})$
splits as $\mathrm{Diff}(S^{1},\mathrm{pt})\times S^{1}$ –diffeomorphisms
preserving a marked point on $S^{1}$, plus an extra factor $S^{1}$
corresponding to rigid rotations (moving the marked point). It is that extra
$S^{1}$ factor that leads to the factor $1/t=1/\mathrm{vol}(S^{1})$ in the
integration measure over $t$. The factor $1/2$ in (7) comes by the previous
mechanism from the mapping class group $\mathbb{Z}_{2}$ of $S^{1}$
(orientation preserving/reversing diffeomorphisms up to isotopy).

####  1.1.2. Heuristics on locality in the first quantization formalism

Suppose that we have a decomposition $M=M_{1}\cup_{Y}M_{2}$ of $M$ into two
Riemannian manifolds $M_{i}$, with common boundary $Y$. Then locality of
quantum field theory – or, a fictional “Fubini theorem” for the (also
fictional) functional integral – suggests a gluing formula

(11)
$Z_{M}=``\int_{C^{\infty}(Y)}\mathcal{D}\phi_{Y}\;Z_{M_{1}}(\phi_{Y})Z_{M_{2}}(\phi_{Y}),"$

where $Z_{M_{i}}$ is a functional of $C^{\infty}(Y)$, again formally given by
a functional integral understood as a sum over Feynman graphs,666Again, for
the purpose of this motivational section we are not discussing the problem of
divergencies and renormalization. For $n=\dim M=2$, a precise definition of
all involved objects and a proof of the gluing formula (11) can be found in
[16].

(12) $Z_{M_{i}}(\phi_{Y})=``\int_{\begin{subarray}{c}\phi\in C^{\infty}(M)\\\
\phi|_{Y}=\phi_{Y}\end{subarray}}\mathcal{D}\phi\;e^{-\frac{1}{\hbar}S(\phi)}"\\\
=\left({\det}^{\zeta}(\Delta_{M_{i},Y}+m^{2})\right)^{-\frac{1}{2}}\cdot\sum_{\Gamma}\Phi_{\Gamma}(\phi_{Y}),$

where we are putting Dirichlet boundary conditions on the kinetic operator.
Feynman graphs now have bulk and boundary vertices, $V=V^{\mathrm{bulk}}\sqcup
V^{\partial}$, where boundary vertices are required to be univalent. The set
of edges then decomposes as $E=\sqcup_{i=0}^{2}E_{i}$ with where edges in
$E_{i}$ have $i$ endpoints in $V^{\partial}$. The weight of a Feynman graph
then is

(13) $\Phi_{\Gamma}(\phi_{Y})=\\\
=\frac{\hbar^{|E|-|V|}}{|\mathrm{Aut}(\Gamma)|}\int_{M_{i}^{\times
V^{\mathrm{bulk}}}}d^{n}x_{1}\cdots
d^{n}x_{|V^{\mathrm{bulk}}|}\int_{Y^{V^{\partial}}}d^{n-1}y_{1}\cdots
d^{n-1}y_{|V^{\partial}|}\\\ \prod_{v\in
V^{\mathrm{bulk}}}(-p_{\mathrm{val}(v)})\prod_{w\in
V^{\partial}}\phi_{Y}(y_{w})\\\ \prod_{(u,v)\in
E_{0}}G_{M_{i},Y}(x_{u},x_{v})\prod_{(u,v)\in
E_{1}}E_{Y,M_{i}}(x_{u},y_{v})\prod_{(u,v)\in
E_{2}}-\mathrm{DN}_{Y,M_{i}}(y_{u},y_{v}),$

where $G_{M_{i},Y}$ denotes the Green’s function of the operator with
Dirichlet boundary conditions, $E_{Y,M_{i}}(x,y)=\partial_{n_{y}}G(x,y)$ is
the normal derivative of the Green’s function at a boundary point $y\in Y$,
and $\mathrm{DN}_{Y,M_{i}}$ is the Dirichlet-to-Neumann operator associated to
the kinetic operator (see Section 3.4 for details).

Let us sketch an interpretation of the gluing formula for the Green’s function
from the standpoint of the first quantization formalism. Let $x\in M_{1}$,
$y\in M_{2}$ and consider a path $\gamma\colon[0,t]\to M$ with $\gamma(0)=x$
and $\gamma(t)=y$. Then the decomposition $M=M_{1}\cup_{Y}M_{2}$ induces a
decomposition $\gamma=\gamma_{1}*\gamma_{2}*\gamma_{3}$ as follows (“$*$”
means concatenation of paths). Let $t_{0}=0$,
$t_{1}=\mathrm{min}\\{t,\gamma(t)\in Y\\}$ and
$t_{2}=\mathrm{max}\\{t,\gamma(t)\in Y\\}$ and $t_{3}=t$, then
$\gamma_{i}=\gamma|_{[t_{i-1},t_{i}]}$. This gives a decomposition

$P_{M}(x,y)=\bigsqcup_{u,v\in Y}P^{\prime}_{M_{1}}(x,u)\times P_{M}(u,v)\times
P^{\prime}_{M_{2}}(v,y),$

where we have introduced the notation $P_{M}(x,y)$ for the set of all paths
from $x$ to $y$ (of arbitrary length) and $P^{\prime}_{M_{i}}(x,u)$ for the
set of all paths starting at $x\in M_{i}$ and ending at $u\in Y$ and not
intersecting $Y$ in between. See Figure 1.

$x$$y$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$M_{1}$$M_{2}$$Y$ Figure 1.
Decomposing a path $\gamma=\gamma_{1}*\gamma_{2}*\gamma_{3}$.

Paths of a specific length $t$ will be denoted $P^{t}_{M}(x,y)$, or
$(P^{\prime}_{M})^{t}(x,u)$. Assuming a Fubini theorem for the path measure
$\mathcal{D}\gamma$, additivity of the action suggests that we could rewrite
(4) as

(14) $G(x,y)``="\int_{Y\times
Y}d^{n-1}u\,d^{n-1}v\int_{0}^{\infty}dt_{1}\int_{\gamma_{1}\in(P^{\prime}_{M_{1}})^{t_{1}}(x,u)}\mathcal{D}\gamma_{1}\,e^{-S^{\mathrm{1q}}(\gamma_{1})}\\\
\int_{0}^{\infty}dt_{2}\int_{\gamma_{2}\in
P_{M}^{t_{2}}(u,v)}\mathcal{D}\gamma_{2}\,e^{-S^{\mathrm{1q}}(\gamma_{2})}\int_{0}^{\infty}dt_{3}\int_{\gamma_{3}\in(P_{M_{2}})^{t_{3}}(v,y)}\mathcal{D}\gamma_{3}\,e^{-S^{\mathrm{1q}}(\gamma_{1})}.$

Comparing with the gluing formula for the Green’s function777See [16,
Proposition 4.2] and Section 3.4.1 of the present paper for details.

(15) $G(x,y)=\int_{Y\times
Y}d^{n-1}u\,d^{n-1}v\;E_{Y,M_{1}}(x,u)\,\varkappa_{Y,M}(u,v)\,E_{Y,M_{2}}(v,y),$

with $\varkappa_{Y,M}=(\mathrm{DN}_{Y,M_{1}}+\mathrm{DN}_{Y,M_{2}})^{-1}$ the
inverse of the “total” Dirichlet-to-Neumann operator, suggests the following
path integral formulae for the extension operator and $\varkappa$:

(16) $\displaystyle E_{Y,M_{i}}(x,u)$
$\displaystyle=\int_{0}^{\infty}dt\int_{\gamma\in(P^{\prime}_{M_{i}})^{t}(x,u)}\mathcal{D}\gamma\,e^{-S^{\mathrm{1q}}(\gamma)},$
(17) $\displaystyle\varkappa_{Y,M}(u,v)$
$\displaystyle=\int_{0}^{\infty}dt\int_{\gamma\in
P_{M}^{t}(u,v)}\mathcal{D}\gamma\,e^{-S^{\mathrm{1q}}(\gamma)}.$

The results of our paper888Another reason to guess that formula is the fact
that the integral kernel of the Dirichlet-to-Neumann operator is given by a
symmetric normal derivative of the Green’s function
$\mathrm{DN}_{Y,M_{i}}(u,v)=-\partial_{n_{u}}\partial_{n_{v}}G_{M_{i},Y}$ (in
a regularized sense - see [16, Remark 3.4]), and formula (16) for the first
normal derivative of the Green’s function. actually suggest also the following
path integral formula for the Dirichlet-to-Neumann operator:

(18)
$\mathrm{DN}_{Y,M_{i}}(u,v)=\int_{0}^{\infty}dt\int_{\gamma\in(P^{\prime\prime}_{M_{i}})^{t}(u,v)}\mathcal{D}\gamma\,e^{-S^{\mathrm{1q}}(\gamma)}.$

Here $(P^{\prime\prime}_{M_{i}})^{t}(u,v)$ is the set of all paths from
$\gamma\colon[0,t]\to M_{i}$ from $u\in Y$ to $v\in Y$ such that
$\gamma(\tau)\notin Y$ for all $0<\tau<t$.

Assuming these formulae, we have again a “first quantization formula” for
weights of Feynman graphs with boundary vertices

(19)
$\Phi_{\Gamma}(\phi_{Y})=\frac{\hbar^{|E|-|V|}}{|\mathrm{Aut}(\Gamma)|}\int_{0<t_{1},\ldots,t_{|E|}<\infty}dt_{1}\cdots
dt_{|E|}\\\
\int_{\boldsymbol{\gamma}\colon\Gamma_{t_{1},\ldots,t_{|E|}}\rightarrow
M_{i}}\mathcal{D}\boldsymbol{\gamma}\,e^{-S^{1q}(\boldsymbol{\gamma})}\prod_{v\in
V^{\mathrm{bulk}}}\left(-p_{\mathrm{val}(v)}\right)\prod_{v^{\partial}\in
V^{\partial}}\phi_{Y}(\boldsymbol{\gamma}(v^{\partial})).$

Here notation is as in (6), the only additional condition is that
$\boldsymbol{\gamma}$ respects the type of edges in $\Gamma$, that is, for all
$x\in\Gamma_{t_{1},\ldots,t_{|E|}}$ we have $\boldsymbol{\gamma}(x)\in Y$ if
and only if $x\in V^{\partial}$.

###  1.2. QFT on a graph. A guide to the paper.

In this paper we study a toy (“combinatorial” or “lattice”) version of the
scalar field theory (1), where the spacetime manifold $M$ is replaced by a
graph $X$, the scalar field $\phi$ is a function on the vertices of $X$ and
the Laplacian in the kinetic operator is replaced by the graph Laplacian
$\Delta_{X}$. I.e., the model is defined by the action

(20) $S_{X}(\phi)=\sum_{v\in
V_{X}}\frac{1}{2}\phi(v)\left((\Delta_{X}+m^{2}\mathrm{Id})\phi\right)(v)+p(\phi(v)),$

where $V_{X}$ is the set of vertices of $X$ and $p$ is the interaction
potential (a polynomial of $\phi$), as before.

This model has the following properties.

  1. (i)

The “functional integral” over the space of fields is a finite-dimensional
convergent integral (Section 2).

  2. (ii)

The functional integral can be expanded in Feynman graphs, giving an
asymptotic expansion of the nonperturbative partition function in powers of
$\hbar$ (Section 4).

  3. (iii)

Partition functions are compatible with unions of graphs over a subgraph
(“gluing”) – we see this as a graph counterpart of Atiyah-Segal functorial
picture of QFT, with compatibility w.r.t. cutting-gluing $n$-manifolds along
closed $(n-1)$-submanifolds. This functorial property of the graph QFT can be
proven

    1. (a)

directly from the functional integral perspective (by a Fubini theorem
argument) – Section 2.1, or

    2. (b)

at the level of Feynman graphs (Section 4.2).

The proof of functoriality at the level of Feynman graphs relies on the
“gluing formulae” describing the behavior of Green’s functions and
determinants w.r.t. gluing of spacetime graphs (Section 3.3). These formulae
are a combinatorial analog of known gluing formulae for Green’s functions and
zeta-regularized functional determinants on manifolds (Section 3.4).

  4. (iv)

The graph QFT admits a higher-categorical extension which allows cutting-
gluing along higher-codimension “corners”999 “Corners” in the graph setting
are understood à la Čech complex, as multiple overlaps of “bulk” graphs.
(Section 2.2), in the spirit of Baez-Dolan-Lurie extended TQFTs.

  5. (v)

The Green’s function on a graph $X$ can be written as a sum over paths
(Section 5, in particular Table 3), giving an analog of the formula (4);
similarly, the determinant can be written as a sum over closed paths, giving
an analog of (7). This leads to a “first-quantization” representation of
Feynman graphs, as a sum over maps $\Gamma\rightarrow X$, sending vertices of
$\Gamma$ to vertices of $X$ and sending edges of $\Gamma$ to paths on $X$
(connecting the images of the incident vertices) – Section 6. This yields a
graph counterpart of the continuum first quantization formula (6).

  6. (vi)

There are path sum formulae for the combinatorial extension (or “Poisson”)
operators and Dirichlet-to-Neumann operators (Section 5.4, see in particular
Table 4), analogous to the path integral formulae (16) and (18).

  7. (vii)

First quantization perspective gives a visual interpretation of the gluing
formula for Green’s functions and determinants on a graph
$X=X^{\prime}\cup_{Y}X^{\prime\prime}$ in terms of cutting the path into
portions spent in $X^{\prime}$ or in $X^{\prime\prime}$ (Section 5.5), and
likewise an interpretation of the cutting-gluing of Feynman graphs (Section
6.3).

######  Remark 1.3.

A free (Gaussian) version of the combinatorial model we are studying in this
paper was studied in [23]. Our twist on it is the deformation by a polynomial
potential, the path-sum (first quantization) formalism, and the gluing formula
for propagators (the BFK-like gluing formula for determinants was studied in
[23]).

###  1.3. Acknowledgements

We thank Olga Chekeres, Andrey Losev and Donald R. Youmans for inspiring
discussions on the first quantization approach in QFT. I.C., P.M. and K.W.
would like to thank the Galileo Galilei Institute, where part of the work was
completed, for hospitality.

[type=main,title=Notations,nonumberlist,style=myNoHeaderStyle]

##  2\. Scalar field theory on a graph

Let $X$ be a finite graph. Consider the toy field theory on $X$ where fields
are real-valued functions $\phi(v)$ on the set of vertices $V_{X}$, i.e., the
space of fields is the space of 0-cochains on $X$ seen as a 1-dimensional CW
complex,

$F_{X}=C^{0}(X).$

We define the action functional as

(21) $\displaystyle S_{X}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(d\phi,d\phi)+\langle\mu,\frac{m^{2}}{2}\phi^{2}+p(\phi)\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\phi,(\Delta_{X}+m^{2})\phi)+\langle\mu,p(\phi)\rangle$
$\displaystyle=$ $\displaystyle\sum_{e\in
E_{X}}\frac{1}{2}(\phi(v^{1}_{e})-\phi(v^{0}_{e}))^{2}+\sum_{v\in
V_{X}}\left(\frac{m^{2}}{2}\phi(v)^{2}+p(\phi(v))\right).$

Here:

  * •

$d\colon C^{0}(X)\rightarrow C^{1}(X)$ is the cellular coboundary operator (we
assume that $0$ cells carry $+$ orientation and $1$-cells carry some
orientation – the model does not depend on this choice).

  * •

$(,)\colon\mathrm{Sym}^{2}C^{k}(X)\rightarrow\mathbb{R}$ for $k=0,1$ is the
standard metric, in which the cell basis is orthonormal.

  * •

$\langle,\rangle$ is the canonical pairing of chains and cochains; $\mu$ is
the 0-chain given by the sum of all vertices with coefficient $1$.101010The
0-chain $\mu$ is an analog of the volume form on the spacetime manifold in our
model. If we want to consider the field theory on $X$ as a lattice
approximation of a continuum field theory, we would need to scale the metric
$(,)$ and the 0-chain $\mu$ appropriately with the mesh size. Additionally,
one would need to add mesh-dependent counterterms to the action in order to
have finite limits for the partition function and correlators.

  * •

$m>0$ is the fixed “mass” parameter.

  * •

$p(\phi)$ is a fixed polynomial (“potential”),

(22) $p(\phi)=\sum_{k\geq 3}\frac{p_{k}}{k!}\phi^{k}.$

More generally, $p(\phi)$ can be a real analytic function. We will assume that
$\frac{m^{2}}{2}\phi^{2}+p(\phi)$ has a unique absolute minimum at $\phi=0$
and that it grows sufficiently fast111111Namely, we want the integral
$\int_{\mathbb{R}}d\phi\,e^{-\frac{1}{\hbar}(\frac{m^{2}}{2}\phi^{2}+p(\phi))}$
to converge for any $\hbar>0$. at $\phi\rightarrow\pm\infty$, so that the
integral (23) converges measure-theoretically.

  * •

$\Delta_{X}=d^{T}d\colon C^{0}(X)\rightarrow C^{0}(X)$ is the graph Laplace
operator on $0$-cochains; $d^{T}\colon C^{1}(X)\rightarrow C^{0}(X)$ is the
dual (transpose) map to the coboundary operator (in the construction of the
dual, one identifies chains and cochains using the standard metric). The
matrix elements of $\Delta_{X}$ in the cell basis, for $X$ a simple graph
(i.e. without double edges and short loops), are

$(\Delta_{X})_{uv}=\left\\{\begin{array}[]{ll}\mathrm{val}(v)&\mbox{if}\;u=v,\\\
-1&\mbox{if $u\neq v$ and $u$ is connected to $v$ by an edge},\\\
0&\mbox{otherwise},\end{array}\right.$

where $\mathrm{val}(v)$ is the valence of the vertex $v$. More generally, for
$X$ not necessarily simple, one has

$(\Delta_{X})_{uv}=\left\\{\begin{array}[]{ll}\mathrm{val}(v)-2\cdot\\#\\{\mbox{short
loops $v\rightarrow v$}\\}&\mbox{if}\;u=v,\\\ -\\#\\{\mbox{edges $u\rightarrow
v$}\\}&\mbox{if $u\neq v$}.\end{array}\right.$

We will be interested in the partition function

(23) $Z_{X}=\int_{F_{X}}D\phi\;e^{-\frac{1}{\hbar}S_{X}},$

where

$D\phi=\prod_{v\in V_{X}}\frac{d\phi(v)}{\sqrt{2\pi\hbar}}$

is the “functional integral measure” on the space of fields $F_{X}$ (in this
case, just the Lebesgue measure on a finite-dimensional space); $\hbar>0$ is
the parameter of quantization – the “Planck constant.”121212Or one can think
of $\hbar$ as “temperature” if one thinks of (23) as a partition function of
statistical mechanics with $S$ the energy of a state $\phi$. The integral in
the r.h.s. of (23) is absolutely convergent. One can also consider correlation
functions

(24)
$\langle\phi(v_{1})\cdots\phi(v_{n})\rangle=\frac{1}{Z_{X}}\int_{F_{X}}D\phi\;e^{-\frac{1}{\hbar}S_{X}}\phi(v_{1})\cdots\phi(v_{n}).$

######  Remark 2.1.

We stress that in this section we consider the _nonperturbative_ partition
functions/correlators and $\hbar$ is to be understood as an actual positive
number, unlike in the setting of perturbation theory (Section 4) where $\hbar$
becomes a formal parameter.

######  Remark 2.2.

In this paper we use the Euclidean QFT convention for our (toy) functional
integrals, with the integrand $e^{-\frac{1}{\hbar}S}$ instead of
$e^{\frac{i}{\hbar}S}$, in order to have a better measure-theoretic
convergence situation. The first convention leads to absolutely convergent
integrals whereas the second leads to conditionally convergent oscillatory
integrals.

###  2.1. Functorial picture

One can interpret our model in the spirit of Atiyah-Segal functorial picture
of QFT, as a (symmetric monoidal) functor

(25) $\mathrm{GraphCob}\xrightarrow{(\mathcal{H},Z)}\mathrm{Hilb}$

from the spacetime category131313This terminology is taken from [22]. of graph
cobordisms to the category of Hilbert spaces and Hilbert-Schmidt operators.

Here in the source category $\mathrm{GraphCob}$ is as follows:

  * •

The objects are graphs $Y$.

  * •

A morphism from $Y_{\mathrm{in}}$ to $Y_{\mathrm{out}}$ is a graph $X$ which
contains $Y_{\mathrm{in}}$ and $Y_{\mathrm{out}}$ as disjoint subgraphs. We
will write $Y_{\mathrm{in}}\xrightarrow{X}Y_{\mathrm{out}}$ and refer to
$Y_{\mathrm{in}},Y_{\mathrm{out}}$ as “ends” (or “boundaries”) of $X$, or we
will say that $X$ is a “graph cobordism” between $Y_{\mathrm{in}}$ and
$Y_{\mathrm{out}}$.

  * •

The composition is given by unions of graphs with out-end of one cobordism
identified with the in-end of the subsequent one:

(26)
$(Y_{3}\xleftarrow{X^{\prime\prime}}Y_{2})\circ(Y_{2}\xleftarrow{X^{\prime}}Y_{1})=Y_{3}\xleftarrow{X}Y_{1},$

where

$X=X^{\prime}\cup_{Y_{2}}X^{\prime\prime}.$

  * •

The monoidal structure is given by disjoint unions of graphs.

All graphs are assumed to be finite. As defined, $\mathrm{GraphCob}$ does not
have unit morphisms (as usual for spacetime categories in non-topological
QFTs); by abuse of language, we still call it a category.

The target category $\mathrm{Hilb}$ has as its objects Hilbert spaces
$\mathcal{H}$ over $\mathbb{C}$;141414 Alternatively (since we do not put $i$
in the exponent in the functional integral), one can consider Hilbert spaces
over $\mathbb{R}$.  the morphisms are Hilbert-Schmidt operators; the
composition is composition of operators. The monoidal structure is given by
tensor products (of Hilbert spaces and of operators).

The functor (25) is constructed as follows. For an end-graph
$Y\in\mathrm{Ob}(\mathrm{GraphCob})$, the associated vector space is

(27) $\mathcal{H}_{Y}=L^{2}(C^{0}(Y))$

– the space of complex-valued square-integrable functions on the vector space
$C^{0}(Y)=\mathbb{R}^{V_{Y}}$.

For a graph cobordism $Y_{\mathrm{in}}\xrightarrow{X}Y_{\mathrm{out}}$, the
associated operator
$Z_{X}\colon\mathcal{H}_{Y_{\mathrm{in}}}\rightarrow\mathcal{H}_{Y_{\mathrm{out}}}$
is

(28)
$Z_{X}\colon\Psi_{\mathrm{in}}\mapsto\Big{(}\Psi_{\mathrm{out}}\colon\phi_{\mathrm{out}}\mapsto\int_{F_{Y_{\mathrm{in}}}}D\phi_{\mathrm{in}}\;\langle\phi_{\mathrm{out}}|Z_{X}|\phi_{\mathrm{in}}\rangle\Psi_{\mathrm{in}}(\phi_{\mathrm{in}})\Big{)}$

with the integral kernel

(29)
$\langle\phi_{\mathrm{out}}|Z_{X}|\phi_{\mathrm{in}}\rangle\colon=\int_{F_{X}^{\phi_{\mathrm{in}},\phi_{\mathrm{out}}}}[D\phi]^{\phi_{\mathrm{in}},\phi_{\mathrm{out}}}\;e^{-\frac{1}{\hbar}(S_{X}(\phi)-\frac{1}{2}S_{Y_{\mathrm{in}}}(\phi_{\mathrm{in}})-\frac{1}{2}S_{Y_{\mathrm{out}}}(\phi_{\mathrm{out}}))}.$

Here

  * •

$F_{X}^{\phi_{\mathrm{in}},\phi_{\mathrm{out}}}$ is the space of fields on $X$
subject to boundary conditions $\phi_{\mathrm{in}},\phi_{\mathrm{out}}$
imposed on the ends, i.e., it is the fiber of the evaluation-at-the-ends map

$F_{X}\rightarrow F_{Y_{\mathrm{in}}}\times F_{Y_{\mathrm{out}}}$

over the pair $(\phi_{\mathrm{in}},\phi_{\mathrm{out}})$.

  * •

The measure

$[D\phi]^{\phi_{\mathrm{in}},\phi_{\mathrm{out}}}=\prod_{v\in
V_{X}\setminus(V_{Y_{\mathrm{in}}}\sqcup
V_{Y_{\mathrm{out}}})}\frac{d\phi(v)}{\sqrt{2\pi\hbar}}$

stands for the “conditional functional measure” on fields subject to boundary
conditions.

We will also call the expression (29) the partition function on the graph $X$
“relative” to the ends $Y_{\mathrm{in}}$, $Y_{\mathrm{out}}$, or just the
partition function relative to the “boundary” subgraph
$Y=Y_{\mathrm{in}}\sqcup Y_{\mathrm{out}}$, if the distinction between “in”
and “out” is irrelevant. In the latter case we will use the notation
$Z_{X,Y}(\phi_{Y})$, with $\phi_{Y}=(\phi_{\mathrm{in}},\phi_{\mathrm{out}})$.

######  Proposition 2.3.

The assignment (27), (28) is a functor of monoidal categories.

###### Proof.

The main point to check is that composition is mapped to composition. It
follows from Fubini theorem, locality of the integration measure (that it is a
product over vertices of local measures) and additivity of the action:

(30)
$S_{X}(\phi)=S_{X^{\prime}}(\phi|_{X^{\prime}})+S_{X^{\prime\prime}}(\phi|_{X^{\prime\prime}})-S_{Y_{2}}(\phi|_{Y_{2}})$

in the notations of (26). Indeed, it suffices to prove

(31)
$\int_{F_{Y_{2}}}D\phi_{2}\;\langle\phi_{3}|Z_{X^{\prime\prime}}|\phi_{2}\rangle\langle\phi_{2}|Z_{X^{\prime}}|\phi_{1}\rangle\stackrel{{\scriptstyle!}}{{=}}\langle\phi_{3}|Z_{X}|\phi_{1}\rangle$

– again, we are considering the gluing of graph cobordisms as in (26). The
l.h.s. is

$\int_{F_{Y_{2}}}D\phi_{2}\int_{F_{X^{\prime}}^{\phi_{1},\phi_{2}}}[D\phi^{\prime}]^{\phi_{1},\phi_{2}}\int_{F_{X^{\prime\prime}}^{\phi_{2},\phi_{3}}}[D\phi^{\prime\prime}]^{\phi_{2},\phi_{3}}\exp\Big{(}-\frac{1}{\hbar}\Big{(}\big{(}-\frac{1}{2}S_{Y_{1}}(\phi_{1})+\\\
+\underbrace{S_{X^{\prime}}(\phi^{\prime})-\frac{1}{2}S_{Y_{2}}(\phi_{2})\big{)}+\big{(}-\frac{1}{2}S_{Y_{2}}(\phi_{2})+S_{X^{\prime\prime}}(\phi^{\prime\prime})}_{S_{X}(\phi)}-\frac{1}{2}S_{Y_{3}}(\phi_{3})\big{)}\Big{)}\Big{)}\\\
=\int_{F_{X}^{\phi_{1},\phi_{3}}}[D\phi]^{\phi_{1},\phi_{3}}e^{-\frac{1}{\hbar}(S_{X}(\phi)-\frac{1}{2}S_{Y_{1}}(\phi_{1})-\frac{1}{2}S_{Y_{2}}(\phi_{3}))},$

which proves (31). Here we understood that $\phi$ is a field on the glued
cobordism $X$ restricting to $\phi^{\prime},\phi^{\prime\prime}$ on
$X^{\prime},X^{\prime\prime}$, respectively. Compatibility with disjoint
unions is obvious by construction. ∎

######  Remark 2.4.

One can interpret the correlator (24) as the partition function of $X$ seen as
a cobordism $\\{v_{1},\ldots,v_{n}\\}\xrightarrow{X}\varnothing$ applied to
the state
$\phi(v_{1})\otimes\cdots\otimes\phi(v_{n})\in\mathcal{H}_{\\{v_{1},\ldots,v_{n}\\}}$.

######  Remark 2.5.

The combinatorial model we are presenting is intended to be an analog (toy
model) of the continuum QFT, according to the dictionary of Table 1.

combinatorial QFT continuum QFT graph $X$ closed spacetime $n$-manifold $M$;
field $\phi\colon V_{X}\rightarrow\mathbb{R}$ scalar field $\phi\in
C^{\infty}(M)$; action (21) action
$S(\phi)=\int_{M}\frac{1}{2}d\phi\wedge*d\phi+(\frac{m^{2}}{2}\phi^{2}+p(\phi))d\mathrm{vol}$
$=\int_{M}(\frac{1}{2}\phi(\Delta+m^{2})\phi+p(\phi))d\mathrm{vol}$; partition
function (23) functional integral on a closed manifold; graph cobordism
$Y_{\mathrm{in}}\xrightarrow{X}Y_{\mathrm{out}}$ $n$-manifold $M$ with in/out-
boundaries being closed $(n-1)$-manifolds
$\gamma_{\mathrm{in}},\gamma_{\mathrm{out}}$; gluing/cutting of graph
cobordisms gluing/cutting of smooth $n$-cobordisms; matrix element (29)
functional integral with boundary conditions
$\phi_{\mathrm{in}},\phi_{\mathrm{out}}$.

Table 1. Comparison between toy model and continuum QFT.

When we want to emphasize that a graph $X$ is not considered as a cobordism
(or equivalently $X$ is seen as a cobordism
$\varnothing\xrightarrow{X}\varnothing$), we will call $X$ a “closed” graph
(by analogy with closed manifolds).

###  2.2. Aside: “QFT with corners” (or “extended QFT”) picture

Fix any $n\geq 1$. We will describe a (tautological) extension of the
functorial picture above for our graph model as an $n$-extended QFT (with
gluing/cutting along “corners” of codimension up to $n$), in the spirit of
Baez-Dolan-Lurie program [2], [18] of extended topological quantum field
theories.151515 Scalar field theory is not a topological field theory, so,
e.g., we should not expect (and in fact do not have) an analog of the
cobordism hypothesis here: the theory is not recovered from its value on
highest-codimension corner. One has a functor of symmetric monoidal
$n$-categories

(32) $\mathrm{GraphCob}^{n}\xrightarrow{(\mathcal{H},Z)}\mathsf{T}^{n}.$

We proceed to describe its ingredients.

####  2.2.1. Source $n$-category

The source $n$-category $\mathrm{GraphCob}^{n}$ is as follows.

  * •

Objects (a.k.a. $0$-morphisms) are graphs $X^{[0]}$ (the index in brackets is
to emphasize that this is a graph at categorical level $0$).

  * •

A 1-morphism between objects (graphs) $Y_{1}^{[0]},Y_{2}^{[0]}$ is a graph
$X^{[1]}$ together with graph embeddings of $Y_{1}^{[0]},Y_{2}^{[0]}$ into
$X^{[1]}$ with disjoint images.

  * •

For $2\leq k\leq n$, a $k$-morphism between two $(k-1)$-morphisms
$Y_{1}^{[k-1]},Y_{2}^{[k-1]}\in\mathrm{Mor}_{k-1}(A_{1}^{[k-2]},A_{2}^{[k-2]})$
is a graph $X^{[k]}$ equipped with embeddings of $Y_{1,2}^{[k-1]}$ satisfying
the following “maximal disjointness” assumption: in the resulting diagram of
graph embeddings

$\textstyle{X^{[k]}}$$\textstyle{Y_{1}^{[k-1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}^{[k-1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}^{[k-2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}^{[k-2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

the intersection of images of $Y_{1}^{[k-1]},Y_{2}^{[k-1]}$ in $X^{[k]}$ is
the union of images of $A_{1}^{[k-2]}$ and $A_{2}^{[k-2]}$.

######  Example 2.6.

Figure 2 below shows an example of a 2-morphism  in $\mathrm{GraphCob}^{2}$:

Figure 2. An example of a 2-morphism in $\mathrm{GraphCob}^{2}$.

The monoidal structure in $\mathrm{GraphCob}^{n}$ is given by disjoint unions
at each level, and the composition is given by unions (pushouts)
$X_{1}^{[k]}\cup_{Y^{[k-1]}}X_{2}^{[k]}$.

######  Remark 2.7.

In $\mathrm{GraphCob}^{n}$ we only consider “vertical” compositions of
$k$-morphisms, i.e., one can only glue two level $k$ graphs over a level $k-1$
graph, not over any graph at level $k^{\prime}<k$ (otherwise, the composition
would fail the maximal disjointness assumption). One might call this structure
a “partial” $n$-category161616 Or a “Pickwickian $n$-category.” (Cf. “He had
used the word in its Pickwickian sense…He had merely considered him a humbug
in a Pickwickian point of view.” Ch. Dickens, Pickwick Papers.)  (but by abuse
of language we suppress “partial”). On a related point, as in Section 2.1,
there are no unit $k$-morphisms.

####  2.2.2. Target $n$-category

The target $n$-category $\mathsf{T}^{n}$ is as follows.

  * •

Objects are commutative unital algebras over $\mathbb{C}$ (“CUAs”)
$\mathcal{H}^{[0]}$.

  * •

For $1\leq k\leq n-1$, a $k$-morphism between CUAs
$\mathcal{H}_{1}^{[k-1]},\mathcal{H}_{2}^{[k-1]}$ is a CUA $\mathcal{H}^{[k]}$
equipped with injective morphisms of unital algebras of
$\mathcal{H}_{1}^{[k-1]},\mathcal{H}_{2}^{[k-1]}$ into $\mathcal{H}^{[k]}$.

  * •

An $n$-morphism between CUAs $\mathcal{H}_{1,2}^{[n-1]}$ is a linear map
$Z\colon\mathcal{H}_{1}^{[n-1]}\rightarrow\mathcal{H}_{2}^{[n-1]}$. This map
is not required to be an algebra morphism.

The monoidal structure is given by tensor product at all levels. The
composition of $n$-morphisms is the composition of linear maps. The
composition of $k$-morphisms for $k<n$ is given by the balanced tensor product
of algebras over a subalgebra,

$\mathcal{H}_{1}^{[k]}\otimes_{\mathcal{H}^{[k-1]}}\mathcal{H}_{2}^{[k]}.$

####  2.2.3. The QFT functor

The functor (32) is defined as follows.

  * •

For $0\leq k\leq n-1$, a graph
$X^{[k]}\in\mathrm{Mor}_{k}(Y_{1}^{[k-1]},Y_{2}^{[k-1]})$ is mapped to the
commutative unital algebra of functions on $0$-cochains on the graph

$\mathcal{H}^{[k]}(X^{[k]}):=C^{\infty}(C^{0}(X^{[k]})),$

with algebra maps from functions on 0-cochains of
$Y_{1}^{[k-1]},Y_{2}^{[k-1]}$ induced by graph inclusions $i_{1,2}\colon
Y_{1,2}^{[k-1]}\rightarrow X^{[k]}$ :

(33) $(i_{1,2}^{*})^{*}\colon C^{\infty}(C^{0}(Y_{1,2}^{[k-1]}))\rightarrow
C^{\infty}(C^{0}(X^{[k]})).$

Here $i_{1,2}^{*}$ is the restriction of a 0-cochain (field) from $X^{[k]}$ to
$Y_{1,2}^{[k-1]}$, and $(i_{1,2}^{*})^{*}$ is the pullback by this restriction
map.

  * •

A graph
$X^{[n]}\in\mathrm{Mor}_{n}(Y_{\mathrm{in}}^{[n-1]},Y_{\mathrm{out}}^{[n-1]})$
is mapped to the linear map
$Z(X^{[n]})\colon\mathcal{H}^{[n-1]}(Y_{\mathrm{in}}^{[n-1]})\rightarrow\mathcal{H}^{[n-1]}(Y_{\mathrm{out}}^{[n-1]})$
defined by a variant of (28), (29) allowing the in- and out-boundaries to
intersect:

(34)
$Z(X)\colon\Psi_{\mathrm{in}}\mapsto\Big{(}\Psi_{\mathrm{out}}\colon\phi_{\mathrm{out}}\mapsto\\\
\mapsto\int_{F_{X}^{\phi_{\mathrm{out}}}}[D\phi]^{\phi_{\mathrm{out}}}e^{-\frac{1}{\hbar}(S_{X}(\phi)-\frac{1}{2}S_{Y_{\mathrm{out}}}(\phi_{\mathrm{out}})-\frac{1}{2}S_{Y_{\mathrm{in}}}(\phi|_{Y_{\mathrm{in}}}))}\Psi_{\mathrm{in}}(\phi|_{Y_{\mathrm{in}}})\Big{)}.$

Here we suppressed the superscripts $[\cdots]$ for $X,Y$ to lighten the
notation.

It is a straightforward check (by repeating the argument of Proposition 2.3)
that formula (34) is compatible with gluing (vertical composition) of
$n$-morphisms in $\mathrm{GraphCob}^{n}$, see Figure 3.

Figure 3. Composition of 2-morphisms.

######  Remark 2.8.

The category $\mathrm{GraphCob}^{n}$ should be thought of as an
(oversimplified) toy model of Baez-Dolan-Lurie fully extended smooth cobordism
$(\infty,n)$-category, where graphs $X^{[k]}$ model $k$-dimensional smooth
strata.

In this language, if we relabel our graphs by categorical co-level,
$X^{\\{k\\}}\colon=X^{[n-k]}$, we should think of graphs $X^{\\{0\\}}$ as
“bulk,” graphs $X^{\\{1\\}}$ as “boundaries,” graphs $X^{\\{2\\}}$ as
“codimension 2 corners,” etc.

######  Remark 2.9.

We also remark that one can consider a different (simpler) version of the
source category – iterated cospans of graphs w.r.t. graph inclusions, without
any disjointness conditions. While this $n$-category is simpler to define and
admits non-vertical compositions, it has less resemblance to the extended
cobordism category, as here the intersection of strata of codimensions $k$ and
$l$ can have codimension less than $k+l$.

######  Remark 2.10.

Note that the most interesting part of the theory is concentrated in the top
component of the functor (partition functions $Z$) – e.g., the interaction
potential $p(\phi)$ only affects it, not the spaces of states
$\mathcal{H}^{[k]}$. This is why we emphasize (cf. footnote 15) that there is
no analog of the cobordism hypothesis in our model: one cannot recover the
entire functor (in particular, the top component) from its bottom component.

This situation is similar to another example of an extended geometric (non-
topological) QFT – the 2D Yang-Mills theory. In this case, the area form
affects the QFT 2-functor only at the top-dimension stratum (and thus
“obstructs” the cobordism hypothesis), cf. [14].

######  Remark 2.11.

The case $n=1$ of the formalism of this section is slightly
(inconsequentially) different from the non-extended functorial picture of
Section 2.1, with target category $\mathsf{T}^{1}$ instead of $\mathrm{Hilb}$,
with boundaries mapped to algebras of smooth functions on boundary values of
fields rather than Hilbert spaces of $L^{2}$ functions of boundary values of
fields.

In the extended setting, we cannot use $L^{2}$ functions for two reasons: (a)
they don’t form an algebra and (b) the pullback (33) of a function of field
values on codim=2 corner vertices to a codim=1 boundary is generally not
square integrable. (i.e. our QFT functor applied to the inclusion of a corner
graph into the boundary graph does not land in the $L^{2}$ space).

##  3\. Gaussian theory

###  3.1. Gaussian theory on a closed graph

Consider the free case of the model (21), with the interaction $p(\phi)$ set
to zero. The action is quadratic

(35) $S_{X}(\phi)=\frac{1}{2}(\phi,K_{X}\phi),$

where the kinetic operator is

$K_{X}\colon=\Delta_{X}+m^{2}$

– it is a positive self-adjoint operator on $F_{X}$. Let us denote its inverse

$G_{X}\colon=(K_{X})^{-1}$

– the “Green’s function” or “propagator;” we will denote matrix elements of
$G_{X}$ in the basis of vertices by $G_{X}(u,v)$, for $u,v\in V_{X}$.

The partition function (23) for a closed graph $X$ is the Gaussian integral

(36)
$Z_{X}=\int_{F_{X}}D\phi\;e^{-\frac{1}{\hbar}S_{X}(\phi)}=\det(K_{X})^{-\frac{1}{2}}.$

The correlator (24) is given by Wick’s lemma, as a moment of the Gaussian
measure:

$\langle\phi(v_{1})\cdots\phi(v_{2m})\rangle=\hbar^{m}\sum_{\mathrm{partitions}\;\\{1,\ldots,2m\\}=\cup_{i=1}^{m}\\{a_{i},b_{i}\\}}G_{X}(v_{a_{1}},v_{b_{1}})\cdots
G_{X}(v_{a_{m}},v_{b_{m}}).$

####  3.1.1. Examples

######  Example 3.1.

Consider the graph $X$ shown in Figure 4 below:

Figure 4. A line graph on 3 vertices.

The kinetic operator is

$K_{X}=\left(\begin{array}[]{ccc}1+m^{2}&-1&0\\\ -1&2+m^{2}&-1\\\
0&-1&1+m^{2}\end{array}\right).$

Its determinant is:

(37) $\det K_{X}=m^{2}(1+m^{2})(3+m^{2})$

and the inverse is

(38)
$G_{X}=\frac{1}{m^{2}(1+m^{2})(3+m^{2})}\left(\begin{array}[]{ccc}1+3m^{2}+m^{4}&1+m^{2}&1\\\
1+m^{2}&(1+m^{2})^{2}&1+m^{2}\\\ 1&1+m^{2}&1+3m^{2}+m^{4}\end{array}\right).$

######  Example 3.2.

Consider the line graph of length $N$:

Figure 5. A line graph of length $N$.

The kinetic operator is the tridiagonal matrix

$K_{X}=\left(\begin{array}[]{ccccc}1+m^{2}&-1&&&\\\ -1&2+m^{2}&-1&&\\\
&-1&\ddots&\ddots&\\\ &&\ddots&2+m^{2}&-1\\\ &&&-1&1+m^{2}\end{array}\right).$

The matrix elements of its inverse are:171717 One finds this by solving the
finite difference equation $-G(i+1,j)+(2+m^{2})G(i,j)-G(i-1,j)=\delta_{ij}$,
using the ansatz $G(i,j)=A_{+}e^{\beta i}+A_{-}e^{-\beta i}$ for $i\leq j$ and
$G(i,j)=B_{+}e^{\beta i}+B_{-}e^{-\beta i}$ for $i\geq j$, with
$A_{\pm},B_{\pm}$ some coefficients depending on $j$. One imposes single-
valuedness (“continuity”) at $i=j$ and “Neumann boundary conditions”
$G(0,j)=G(1,j)$, $G(N,j)=G(N+1,j)$, which – together with the original
equation at $i=j$ – determines uniquely the solution. One can obtain the
determinant from the propagator using the property $\frac{d}{dm^{2}}\log\det
K_{X}=\operatorname{tr}K_{X}^{-1}\frac{d}{dm^{2}}K_{X}=\sum_{i=1}^{N}G_{X}(i,i)$.

(39)
$G_{X}(i,j)=\frac{\cosh\beta(N-|i-j|)+\cosh\beta(N+1-i-j)}{2\sinh\beta\,\sinh\beta
N},\quad 1\leq i,j\leq N,$

where $\beta$ is related to $m$ by

(40) $\sinh\frac{\beta}{2}=\frac{m}{2}.$

The determinant is:

(41) $\det K_{X}=2\tanh\frac{\beta}{2}\,\sinh\beta N.$

######  Example 3.3.

Consider the circle graph with $N$ vertices shown in Figure 6 below:

Figure 6. A circle graph with $N$ vertices.

The kinetic operator is:

$K_{X}=\left(\begin{array}[]{ccccc}2+m^{2}&-1&&&-1\\\ -1&2+m^{2}&-1&&\\\
&-1&\ddots&\ddots&\\\ &&\ddots&2+m^{2}&-1\\\
-1&&&-1&2+m^{2}\end{array}\right).$

(We are only writing the nonzero entries.) Its inverse is given by

(42)
$G_{X}(i,j)=\frac{\cosh\beta(\frac{N}{2}-|i-j|)}{2\sinh\beta\,\sinh\frac{\beta
N}{2}},\quad 1\leq i,j\leq N.$

Here $\beta$ is as in (40). The determinant is:

(43) $\det K_{X}=4\sinh^{2}\frac{\beta N}{2}.$

For instance, for $N=3$ we obtain

(44) $G_{X}=\frac{1}{m^{2}(m^{2}+3)}\begin{pmatrix}m^{2}+1&1&1\\\
1&m^{2}+1&1\\\ 1&1&m^{2}+1\end{pmatrix}$

and

(45) $\det K_{X}=m^{2}(m^{2}+3)^{2}.$

###  3.2. Gaussian theory relative to the boundary

Consider the Gaussian theory on a graph $X$ with “boundary subgraph” $Y\subset
X$.

####  3.2.1. Dirichlet problem

Consider the following “Dirichlet problem.” For a fixed field configuration on
the boundary $\phi_{Y}\in F_{Y}$, we are looking for a field configuration on
$X$, $\phi\in F_{X}$ such that

(46) $\displaystyle\phi|_{Y}$ $\displaystyle=$ $\displaystyle\phi_{Y},$ (47)
$\displaystyle(K_{X}\phi)(v)$ $\displaystyle=$ $\displaystyle
0\;\;\mathrm{for\;all}\;v\in V_{X}\setminus V_{Y}.$

Equivalently, we are minimizing the action (35) on the fiber
$F_{X}^{\phi_{Y}}$ of the evaluation-on-$Y$ map $F_{X}\rightarrow F_{Y}$ over
$\phi_{Y}$. The solution exists and is unique due to convexity and
nonnegativity of $S_{X}$.

Let us write the inverse of $K_{X}$ as a $2\times 2$ block matrix according to
partition of vertices of $X$ into (1) not belonging to $Y$ (“bulk vertices”)
or (2) belonging to $Y$ (“boundary vertices”):

(48) $(K_{X})^{-1}=\left(\begin{array}[]{c|c}A&B\\\ \hline\cr
C&D\end{array}\right).$

Note that this matrix is symmetric, so $A$ and $D$ are symmetric and
$C=B^{T}$.

Then, we can write the solution of the Dirichlet problem as follows: (47)
implies $K_{X}\phi=\left(\begin{array}[]{c}0\\\ \xi\end{array}\right)$ for
some $\xi\in F_{Y}$. Hence,

$\phi=(K_{X})^{-1}\left(\begin{array}[]{c}0\\\
\xi\end{array}\right)=\left(\begin{array}[]{c}B\xi\\\
D\xi\end{array}\right)\underset{(\ref{Dirichlet
eq1})}{=}\left(\begin{array}[]{c}0\\\ \phi_{Y}\end{array}\right).$

Therefore, $\xi=D^{-1}\phi_{Y}$ and the solution of the Dirichlet problem is

(49) $\phi=\left(\begin{array}[]{c}BD^{-1}\phi_{Y}\\\
\phi_{Y}\end{array}\right).$

####  3.2.2. Dirichlet-to-Neumann operator

Note also that the evaluation of the action $S_{X}$ on the solution of the
Dirichlet problem is

(50)
$S_{X}(\phi)=\frac{1}{2}(\phi,K_{X}\phi)=\frac{1}{2}\Big{(}\left(\begin{array}[]{c}BD^{-1}\phi_{Y}\\\
\phi_{Y}\end{array}\right),\left(\begin{array}[]{c}0\\\
\xi\end{array}\right)\Big{)}\\\
=\frac{1}{2}(\phi_{Y},\xi)=\frac{1}{2}(\phi_{Y},D^{-1}\phi_{Y}).$

The map sending $\phi_{Y}$ to the corresponding $\xi$ (i.e. the kinetic
operator evaluated on the solution of the Dirichlet problem) is a
combinatorial analog of the Dirichlet-to-Neumann operator.181818 Recall that
in the continuum setting, for $X$ a manifold with boundary, the Dirichlet-to-
Neumann operator $\mathrm{DN}\colon C^{\infty}(\partial X)\rightarrow
C^{\infty}(\partial X)$ maps a smooth function $\phi_{\partial}$ to the normal
derivative $\partial_{n}\phi(x)$ on $\partial X$ of the solution $\phi$ of the
Helmholtz equation $(\Delta+m^{2})\phi=0$ subject to Dirichlet boundary
condition $\phi|_{\partial}=\phi_{\partial}$. We will call the operator
$\mathrm{DN}_{Y,X}\colon=D^{-1}\colon F_{Y}\rightarrow F_{Y}$ the
(combinatorial) Dirichlet-to-Neumann operator.191919We put the subscripts in
$\mathrm{DN}_{Y,X}$ to emphasize that we are extending $\phi_{Y}$ into $X$ as
a solution of (47). When we will discuss gluing, the same $Y$ can be a
subgraph of two different graphs $X^{\prime},X^{\prime\prime}$; then it is
important into which graph we are extending $\phi_{Y}$.

Recall (see e.g. [16]) that in the continuum setting, the action of the free
massive scalar field on a manifold with boundary, evaluated on a classical
solution with Dirichlet boundary condition $\phi_{\partial}$ is
$\int_{\partial X}\frac{1}{2}\phi_{\partial}\mathrm{DN}(\phi_{\partial})$.
Comparing with (50) reinforces the idea that is reasonable to call $D^{-1}$
the Dirichlet-to-Neumann operator.

We will denote the operator $BD^{-1}$ appearing in (49) by

(51) $E_{Y,X}=BD^{-1}$

– the “extension” operator (extending $\phi_{Y}$ into the bulk of $X$ as a
solution of the Dirichlet problem).202020 In [23], this operator is called the
_Poisson operator_.

####  3.2.3. Partition function and correlators (relative to a boundary
subgraph)

Let us introduce a notation for the blocks of the matrix $K_{X}$ corresponding
to splitting of the vertices of $X$ into bulk and boundary vertices, similarly
to (48):

(52) $K_{X}=\left(\begin{array}[]{c|c}\widehat{A}=K_{X,Y}&\widehat{B}\\\
\hline\cr\widehat{C}&\widehat{D}\end{array}\right).$

The partition function relative to $Y$ (cf. (29)) is again given by a Gaussian
integral

(53)
$Z_{X,Y}(\phi_{Y})=\int_{F_{X}^{\phi_{Y}}}[D\phi]^{\phi_{Y}}e^{-\frac{1}{\hbar}(S_{X}(\phi)-\frac{1}{2}S_{Y}(\phi_{Y}))}\\\
=\det(K_{X,Y})^{-\frac{1}{2}}e^{-\frac{1}{2\hbar}(\phi_{Y},(\mathrm{DN}_{Y,X}-\frac{1}{2}K_{Y})\phi_{Y})}.$

The normalized correlators (depending on the boundary field $\phi_{Y}$) are as
follows.

  * •

1-point correlator:212121When specifying that a vertex $v$ is in
$V_{X}\setminus V_{Y}$ we will use a shorthand and write $v\in X\setminus Y$.

(54) $\langle\phi(v)\rangle_{\phi_{Y}}=(E_{Y,X}\phi_{Y})(v),\quad v\in
X\setminus Y.$

  * •

Centered $2m$-point correlator:

(55) $\langle\delta\phi(v_{1})\cdots\delta\phi(v_{2m})\rangle_{\phi_{Y}}=\\\
=\hbar^{m}\sum_{\mathrm{partitions}\;\\{1,\ldots,2m\\}=\cup_{i=1}^{m}\\{a_{i},b_{i}\\}}G_{X,Y}(v_{a_{1}},v_{b_{1}})\cdots
G_{X,Y}(v_{a_{m}},v_{b_{m}}),\\\ v_{1},\ldots,v_{2m}\in X\setminus Y.$

Here:

    * –

$\delta\phi(v)\colon=\phi(v)-\langle\phi(v)\rangle_{\phi_{Y}}=\phi(v)-(E_{Y,X}\phi_{Y})(v)$
is the fluctuation of the field w.r.t. its average;

    * –

$G_{X,Y}\colon=(K_{X,Y})^{-1}$ is the “propagator with Dirichlet boundary
condition on $Y$” (or “propagator relative to $Y$”).

  * •

Non-centered correlators follow from (55), e.g.

(56)
$\langle\phi(v_{1})\phi(v_{2})\rangle_{\phi_{Y}}=\hbar\,G_{X,Y}(v_{1},v_{2})+(E_{Y,X}\phi_{Y})(v_{1})\cdot(E_{Y,X}\phi_{Y})(v_{2}).$

######  Remark 3.4.

In our notations, the subscript $X,Y$ (as in $K_{X,Y}$, $G_{X,Y}$, $Z_{X,Y}$)
stands for an object on $X$ relative to $Y$.222222I.e. we think of $(X,Y)$ as
a _pair_ of 1-dimensional CW complexes, where “pair” has the same meaning as
in, e.g., the long exact sequence in cohomology of a pair. On the other hand,
the subscript $Y,X$ (as in $\mathrm{DN}_{Y,X}$, $E_{Y,X}$) refers to an object
related to extending a field on $Y$ to a classical solution in the “bulk” $X$.

####  3.2.4. Examples

######  Example 3.5.

Consider the graph $X$ shown in Figure 7 below, relative the subgraph $Y$
consisting solely of the vertex $2$.

Figure 7. A graph with two vertices relative to one vertex.

The full kinetic operator is

$K_{X}=\left(\begin{array}[]{c|c}1+m^{2}&-1\\\
\hline\cr-1&1+m^{2}\end{array}\right)$

and the relative version is its top left block, $K_{X,Y}=1+m^{2}$. The
relative propagator is $G_{X,Y}=K_{X,Y}^{-1}=\frac{1}{1+m^{2}}$. The inverse
of the full kinetic operator is

$K_{X}^{-1}=\frac{1}{m^{2}(2+m^{2})}\left(\begin{array}[]{c|c}1+m^{2}&1\\\
\hline\cr 1&1+m^{2}\end{array}\right).$

The DN operator is the inverse of the bottom right block:
$\mathrm{DN}_{Y,X}=\frac{m^{2}(2+m^{2})}{1+m^{2}}$ and the extension operator
(51) is $E_{Y,X}=\frac{1}{1+m^{2}}$.

In particular, the relative partition function is

$Z_{X,Y}(\phi_{Y})=(1+m^{2})^{-\frac{1}{2}}\,e^{-\frac{1}{2\hbar}\left(\frac{m^{2}(2+m^{2})}{1+m^{2}}-\frac{m^{2}}{2}\right)\phi_{Y}^{2}}.$

######  Example 3.6.

Consider the line graph of length $N$ relative to the subgraph consisting of
the right endpoint $Y=\\{N\\}$ (Figure 8).

Figure 8. A line graph relative to one endpoint.

The relative propagator is

$G_{X,Y}(i,j)=\frac{\sinh\beta(N-\frac{1}{2}-|i-j|)+\sinh\beta(N+\frac{1}{2}-i-j)}{2\sinh\beta\,\cosh\beta(N-\frac{1}{2})},\quad
1\leq i,j\leq N-1,$

with $\beta$ as in (40). The DN operator is the inverse of the $N-N$ block
(element) of the absolute propagator (39):

$\mathrm{DN}_{Y,X}=\frac{2\sinh\frac{\beta}{2}\,\sinh\beta
N}{\cosh\beta(N-\frac{1}{2})}.$

The extension operator is

$E_{Y,X}(i,N)=\frac{\cosh\beta(i-\frac{1}{2})}{\cosh\beta(N-\frac{1}{2})},\quad
1\leq i\leq N-1$

and the determinant is

(57) $\det K_{X,Y}=\frac{\cosh\beta(N-\frac{1}{2})}{\cosh\frac{\beta}{2}}.$

######  Example 3.7.

Consider again the line graph, but now relative to both left and right
endpoints, see Figure 9 below.

Figure 9. A line graph relative to both endpoints.

Then we have:

(58) $\displaystyle
G_{X,Y}(i,j)=\frac{\cosh\beta(N-1-|i-j|)-\cosh\beta(N+1-i-j)}{2\sinh\beta\,\sinh\beta(N-1)},\;\;2\leq
i,j\leq N-1,$ (61)
$\displaystyle\mathrm{DN}_{Y,X}=\frac{2\sinh\frac{\beta}{2}}{\sinh\beta(N-1)}\left(\begin{array}[]{cc}\cosh\beta(N-\frac{1}{2})&-\cosh\frac{\beta}{2}\\\
-\cosh\frac{\beta}{2}&\cosh\beta(N-\frac{1}{2})\end{array}\right),$ (62)
$\displaystyle
E_{Y,X}(i,1)=\frac{\sinh\beta(N-i)}{\sinh\beta(N-1)},\;\;E_{Y,X}(i,N)=\frac{\sinh\beta(i-1)}{\sinh\beta(N-1)},$
(63) $\displaystyle\det K_{X,Y}=\frac{\sinh\beta(N-1)}{\sinh\beta}.$

###  3.3. Gluing in Gaussian theory. Gluing of propagators and determinants.

####  3.3.1. Cutting a closed graph

Consider a closed graph $X=X^{\prime}\cup_{Y}X^{\prime\prime}$ obtained from
graphs $X^{\prime},X^{\prime\prime}$ by gluing along a common subgraph
$X^{\prime}\supset Y\subset X^{\prime\prime}$.

######  Theorem 3.8.

  1. (a)

The propagator on $X$ is expressed in terms the data (propagators, DN
operators, extension operators) on $X^{\prime},X^{\prime\prime}$ relative to
$Y$ as follows.

     * •

For both vertices $v_{1},v_{2}\in X^{\prime}$:

(64) $G_{X}(v_{1},v_{2})=G_{X^{\prime},Y}(v_{1},v_{2})+\\\
+\sum_{u_{1},u_{2}\in
Y}E_{Y,X^{\prime}}(v_{1},u_{1})\mathrm{DN}^{-1}_{Y,X}(u_{1},u_{2})E_{Y,X^{\prime}}(v_{2},u_{2}).$

For both vertices in $X^{\prime\prime}$, the formula is similar. Here the
total DN operator is

(65)
$\mathrm{DN}_{Y,X}=\mathrm{DN}_{Y,X^{\prime}}+\mathrm{DN}_{Y,X^{\prime\prime}}-K_{Y}.$

Also, we assume by convention that $G_{X^{\prime},Y}(v_{1},v_{2})=0$ if either
of $v_{1},v_{2}$ is in $Y$. We also set $E_{Y,X^{\prime}}(u,v)=\delta_{u,v}$
if $u,v\in Y$.

     * •

For $v_{1}\in X^{\prime}$, $v_{2}\in X^{\prime\prime}$,

(66) $G_{X}(v_{1},v_{2})=\sum_{u_{1},u_{2}\in
Y}E_{Y,X^{\prime}}(v_{1},u_{1})\mathrm{DN}^{-1}_{Y,X}(u_{1},u_{2})E_{Y,X^{\prime\prime}}(v_{2},u_{2})$

and similarly for $v_{1}\in X^{\prime\prime}$, $v_{2}\in X^{\prime}$.

  2. (b)

The determinant of $K_{X}$ is expressed in terms of the data on
$X^{\prime},X^{\prime\prime}$ relative to $Y$ as follows:

(67) $\det
K_{X}=\det(K_{X^{\prime},Y})\det(K_{X^{\prime\prime},Y})\det(\mathrm{DN}_{Y,X}).$

We will give three proofs of these gluing formulae:

  1. (1)

From Fubini theorem for the “functional integral” (QFT/second quantization
approach).

  2. (2)

From inverting a $2\times 2$ block matrix via Schur complement and Schur’s
determinant formula.

  3. (3)

From path counting (first quantization approach) – later, in Section 5.5.

####  3.3.2. Proof 1 (“functional integral approach”)

First, consider the partition function on $X$ relative to $Y$:

(68)
$Z_{X,Y}(\phi_{Y})=\int_{F_{X}^{\phi_{Y}}}[D\phi]^{\phi_{Y}}e^{-\frac{1}{\hbar}(S_{X}(\phi)-\frac{1}{2}S_{Y}(\phi_{Y}))}\underset{(\ref{S
additivity})}{=}\\\
=\int_{F_{X^{\prime}}^{\phi_{Y}}}[D\phi^{\prime}]^{\phi_{Y}}\int_{F_{X^{\prime\prime}}^{\phi_{Y}}}[D\phi^{\prime\prime}]^{\phi_{Y}}e^{-\frac{1}{\hbar}(S_{X^{\prime}}(\phi^{\prime})+S_{X^{\prime\prime}}(\phi^{\prime\prime})-\frac{3}{2}S_{Y}(\phi_{Y}))}\\\
=(\det K_{X^{\prime},Y})^{-\frac{1}{2}}(\det
K_{X^{\prime\prime},Y})^{-\frac{1}{2}}e^{-\frac{1}{2\hbar}(\phi_{Y},(\mathrm{DN}_{Y,X^{\prime}}+\mathrm{DN}_{Y,X^{\prime\prime}}-\frac{3}{2}K_{Y})\phi_{Y})}.$

Comparing the r.h.s. with (53) as functions of $\hbar$, we obtain the formula
(65) for the total DN operator and the relation for determinants

$\det K_{X,Y}=\det K_{X^{\prime},Y}\cdot\det K_{X^{\prime\prime},Y}.$

The partition function on $X$ can be obtained by integrating (68) over the
field on the “gluing interface” $Y$:

$Z_{X}=\int_{F_{X}}D\phi\,e^{-\frac{1}{\hbar}S_{X}(\phi)}=\int_{F_{Y}}D\phi_{Y}\,e^{\frac{1}{2\hbar}S_{Y}(\phi_{Y})}Z_{X,Y}(\phi_{Y})=\\\
=\int_{F_{Y}}D\phi_{Y}(\det
K_{X,Y})^{-\frac{1}{2}}e^{-\frac{1}{2\hbar}(\phi_{Y},\mathrm{DN}_{Y,X}\phi_{Y})}=(\det
K_{X,Y})^{-\frac{1}{2}}(\det\mathrm{DN}_{Y,X})^{-\frac{1}{2}}.$

Comparing the r.h.s. with (36), we obtain the gluing formula for determinants
(67).

Next, we prove the gluing formula for propagators thinking of them as 2-point
correlation functions. We denote by $\ll\cdots\gg$ correlators not normalized
by the partition function. Consider the case $v_{1},v_{2}\in X^{\prime}$. We
have

$\underbrace{\ll\phi(v_{1})\phi(v_{2})\gg}_{\hbar\,G_{X}(v_{1},v_{2})\cdot
Z_{X}}=\int_{F_{X}}D\phi\,e^{-\frac{1}{\hbar}S_{X}(\phi)}\phi(v_{1})\phi(v_{2})=\\\
=\int_{F_{Y}}D\phi_{Y}\underbrace{\int_{F_{X^{\prime}}^{\phi_{Y}}}[D\phi^{\prime}]^{\phi_{Y}}\phi^{\prime}(v_{1})\phi^{\prime}(v_{2})e^{-\frac{1}{\hbar}(S_{X^{\prime}}(\phi^{\prime})-\frac{1}{2}S_{Y}(\phi_{Y}))}}_{\ll\phi(v_{1})\phi(v_{2})\gg_{\phi_{Y}}^{X^{\prime}}\underset{(\ref{Gaussian
non-centered 2-point
corr})}{=}Z_{X^{\prime},Y}(\phi_{Y})\cdot(\hbar\,G_{X^{\prime},Y}(v_{1},v_{2})+(E_{Y,X^{\prime}}\phi_{Y})(v_{1})\cdot(E_{Y,X^{\prime}}\phi_{Y})(v_{2}))}\cdot\\\
\cdot\underbrace{\int_{F_{X^{\prime\prime}}^{\phi_{Y}}}[D\phi^{\prime\prime}]^{\phi_{Y}}e^{-\frac{1}{\hbar}(S_{X^{\prime\prime}}(\phi^{\prime\prime})-\frac{1}{2}S_{Y}(\phi_{Y}))}}_{Z_{X^{\prime\prime},Y}(\phi_{Y})}\\\
=\int_{F_{Y}}D\phi_{Y}(\det K_{X^{\prime},Y})^{-\frac{1}{2}}(\det
K_{X^{\prime\prime},Y})^{-\frac{1}{2}}e^{-\frac{1}{2\hbar}(\phi_{Y},\mathrm{DN}_{Y,X}\phi_{Y})}\cdot\\\
\cdot(\hbar\,G_{X^{\prime},Y}(v_{1},v_{2})+\sum_{u_{1},u_{2}\in
Y}E_{Y,X^{\prime}}(v_{1},u_{1})\phi_{Y}(u_{1})\phi_{Y}(u_{2})E_{Y,X^{\prime}}(v_{2},u_{2}))\\\
=Z_{X}\cdot\hbar(G_{X^{\prime},Y}(v_{1},v_{2})+\sum_{u_{1},u_{2}\in
Y}E_{Y,X^{\prime}}(v_{1},u_{1})\mathrm{DN}^{-1}_{Y,X}(u_{1},u_{2})E_{Y,X^{\prime}}(v_{2},u_{2})).$

This proves the gluing formula (64).

Finally, consider the case $v_{1}\in X^{\prime}$, $v_{2}\in X^{\prime\prime}$.
By a similar computation we find

$\underbrace{\ll\phi(v_{1})\phi(v_{2})\gg}_{\hbar\,G_{X}(v_{1},v_{2})\cdot
Z_{X}}=\int_{F_{Y}}D\phi_{Y}\underbrace{\int_{F_{X^{\prime}}^{\phi_{Y}}}[D\phi^{\prime}]^{\phi_{Y}}\phi^{\prime}(v_{1})e^{-\frac{1}{\hbar}(S_{X^{\prime}}(\phi^{\prime})-\frac{1}{2}S_{Y}(\phi_{Y}))}}_{=\ll\phi(v_{1})\gg_{\phi_{Y}}^{X^{\prime}}\underset{(\ref{Gaussian
1-point
corr})}{=}Z_{X^{\prime},Y}(\phi_{Y})\cdot(E_{Y,X^{\prime}}\phi_{Y})(v_{1})}\cdot\\\
\cdot\underbrace{\int_{F_{X^{\prime\prime}}^{\phi_{Y}}}[D\phi^{\prime\prime}]^{\phi_{Y}}\phi^{\prime\prime}(v_{2})e^{-\frac{1}{\hbar}(S_{X^{\prime\prime}}(\phi^{\prime\prime})-\frac{1}{2}S_{Y}(\phi_{Y}))}}_{=\ll\phi(v_{2})\gg^{X^{\prime\prime}}_{\phi_{Y}}=Z_{X^{\prime\prime},Y}(\phi_{Y})\cdot(E_{Y,X^{\prime\prime}}\phi_{Y})(v_{2})}\\\
=\int_{F_{Y}}D\phi_{Y}(\det K_{X^{\prime},Y})^{-\frac{1}{2}}(\det
K_{X^{\prime\prime},Y})^{-\frac{1}{2}}e^{-\frac{1}{2\hbar}(\phi_{Y},\mathrm{DN}_{Y,X}\phi_{Y})}\cdot\\\
\cdot\sum_{u_{1},u_{2}\in
Y}E_{Y,X^{\prime}}(v_{1},u_{1})\phi_{Y}(u_{1})\phi_{Y}(u_{2})E_{Y,X^{\prime\prime}}(v_{2},u_{2})\\\
=Z_{X}\cdot\hbar\sum_{u_{1},u_{2}\in
Y}E_{Y,X^{\prime}}(v_{1},u_{1})\mathrm{DN}^{-1}_{Y,X}(u_{1},u_{2})E_{Y,X^{\prime\prime}}(v_{2},u_{2}).$

This proves (66).

####  3.3.3. Proof 2 (Schur complement approach)

Let us introduce the notations

(69) $\overline{G}_{X,Y}=\left(\begin{array}[]{c|c}G_{X,Y}&0\\\ \hline\cr
0&0\end{array}\right),\quad\overline{E}_{Y,X}=\left(\begin{array}[]{c}E_{Y,X}\\\
\mathrm{id}\end{array}\right)$

for the extension of the propagator on $X$ relative to $Y$ by zero to vertices
of $Y$ and the extension of the extension operator by identity to vertices of
$Y$ (the blocks correspond to vertices of $X\setminus Y$ and vertices of $Y$,
respectively).232323 Note that one can further refine the block decompositions
(69) according to partitioning of vertices in $X\setminus Y$ into those in
$X^{\prime}\setminus Y$ and those in $X^{\prime\prime}\setminus Y$. Then the
block $G_{X,Y}$ becomes $\left(\begin{array}[]{cc}G_{X^{\prime},Y}&0\\\
0&G_{X^{\prime\prime},Y}\end{array}\right)$ and the block $E_{Y,X}$ becomes
$\left(\begin{array}[]{c}E_{Y,X^{\prime}}\\\
E_{Y,X^{\prime\prime}}\end{array}\right)$.  Using these notations, gluing
formulae (64), (66) for the propagator can be jointly expressed as

(70)
$G_{X}\stackrel{{\scriptstyle!}}{{=}}\overline{G}_{X,Y}+\overline{E}_{Y,X}\mathrm{DN}_{Y,X}^{-1}\overline{E}_{Y,X}^{T}.$

The r.h.s. here is

$\left(\begin{array}[]{c|c}G_{X,Y}+E\,\mathrm{DN}^{-1}E^{T}&E\,\mathrm{DN}^{-1}\\\
\hline\cr\mathrm{DN}^{-1}E^{T}&\mathrm{DN}^{-1}\end{array}\right)=\left(\begin{array}[]{c|c}\widehat{A}^{-1}+BD^{-1}DD^{-1}C&BD^{-1}D\\\
\hline\cr DD^{-1}C&D\end{array}\right)\\\
=\left(\begin{array}[]{c|c}\widehat{A}^{-1}+BD^{-1}C&B\\\ \hline\cr
C&D\end{array}\right).$

Here we are suppressing the subscript $Y,X$ for $E$ and $\mathrm{DN}$;
notations for the blocks are as in (48), (52). So, the only part to check is
that the 1-1 block above is $A$. It is a consequence of the inversion formula
for $2\times 2$ block matrices, which in particular asserts that the 1-1 block
$\widehat{A}$ of the matrix $K_{X}$ inverse to $G_{X}$ is the inverse of the
Schur complement of the 2-2 block in $G_{X}$, i.e.,

$\widehat{A}^{-1}=A-BD^{-1}C.$

This finishes the proof of the gluing formula for propagators (70).

Schur’s formula for a determinant of a block $2\times 2$ matrix applied to
(48) yields

$\displaystyle\det
K_{X}^{-1}=\det\underbrace{D}_{\mathrm{DN}^{-1}}\cdot\det(\underbrace{A-BD^{-1}C}_{\widehat{A}^{-1}=K_{X,Y}^{-1}})$

and thus

$\det K_{X}=\det\mathrm{DN}\cdot\det K_{X,Y}=\det\mathrm{DN}\cdot\det
K_{X^{\prime},Y}\cdot\det K_{X^{\prime\prime},Y}.$

In the last equality we used that $K_{X,Y}$ is block-diagonal, with blocks
corresponding to $X^{\prime}\setminus Y$ and $X^{\prime\prime}\setminus Y$.
This proves the gluing formula for determinants.

####  3.3.4. Examples

######  Example 3.9.

Consider the gluing of two line graphs of length 2,
$X^{\prime},X^{\prime\prime}$ over a common vertex $Y$ into a line graph $X$
of length 3 as pictured in Figure 10 below.

Figure 10. Gluing two line graphs into a longer line graph.

The data of the constituent graphs $X^{\prime},X^{\prime\prime}$ relative to
$Y$ was computed in Example 3.5. We assemble the data on the glued graph $X$
using the gluing formulae of Theorem 3.8. We have

$\underbrace{\mathrm{DN}_{Y,X}}_{=\mathrm{DN}_{Y,X}(2,2)}=\mathrm{DN}_{Y,X^{\prime}}+\mathrm{DN}_{Y,X^{\prime\prime}}-K_{Y}=\\\
=\frac{m^{2}(2+m^{2})}{1+m^{2}}+\frac{m^{2}(2+m^{2})}{1+m^{2}}-m^{2}=\frac{m^{2}(3+m^{2})}{1+m^{2}}.$

For the propagator we have, e.g.,

$G_{X}(1,1)=G_{X^{\prime},Y}(1,1)+E_{Y,X^{\prime}}(1,2)\mathrm{DN}^{-1}_{Y,X}(2,2)E_{Y,X^{\prime}}(1,2)\\\
=\frac{1}{1+m^{2}}+\frac{1}{1+m^{2}}\cdot\frac{1+m^{2}}{m^{2}(3+m^{2})}\cdot\frac{1}{1+m^{2}}$

and

$G_{X}(1,3)=E_{Y,X^{\prime}}(1,2)\mathrm{DN}^{-1}_{Y,X}(2,2)E_{Y,X^{\prime\prime}}(3,2)=\frac{1}{1+m^{2}}\cdot\frac{1+m^{2}}{m^{2}(3+m^{2})}\cdot\frac{1}{1+m^{2}},$

which agrees with the 1-1 entry and 1-3 entry in (38) respectively.

For the gluing of determinants, we have

$\det K_{X^{\prime},Y}\cdot\det
K_{X^{\prime\prime},Y}\cdot\det\mathrm{DN}_{Y,X}=(1+m^{2})\cdot(1+m^{2})\cdot\frac{m^{2}(3+m^{2})}{1+m^{2}},$

which agrees with (37).

######  Example 3.10.

Consider the circle graph $X$ with $N$ vertices presented as a gluing by the
two endpoints of two line graphs $X^{\prime}$, $X^{\prime\prime}$ of lengths
$N^{\prime},N^{\prime\prime}$ respectively, with
$N=N^{\prime}+N^{\prime\prime}-2$, see Figure 11 below.

Figure 11. Gluing a circle from two intervals.

One can then use the gluing formulae of Theorem 3.8 to recover the propagator
and the determinant on the circle graph (cf. Example 3.3) from the data for
line graphs relative to the endpoints (cf. Example 3.7). E.g. for the
determinant, we have

$\underbrace{\frac{\sinh\beta(N^{\prime}-1)}{\sinh\beta}}_{\det
K_{X^{\prime},Y}}\cdot\underbrace{\frac{\sinh\beta(N^{\prime\prime}-1)}{\sinh\beta}}_{\det
K_{X^{\prime\prime},Y}}\cdot\det\Big{(}\mathrm{DN}_{Y,X^{\prime}}+\mathrm{DN}_{Y,X^{\prime\prime}}-\left(\begin{array}[]{cc}m^{2}&0\\\
0&m^{2}\end{array}\right)\Big{)}=\\\ =\underbrace{4\sinh^{2}\frac{\beta
N}{2}}_{\det K_{X}}.$

Here the $2\times 2$ matrices $\mathrm{DN}_{Y,X^{\prime}}$,
$\mathrm{DN}_{Y,X^{\prime\prime}}$ are given by (61), with $N$ replaced by
$N^{\prime},N^{\prime\prime}$, respectively.

####  3.3.5. General cutting/gluing of cobordisms

Consider the gluing of graph cobordisms (26),

$Y_{1}\xrightarrow{X^{\prime}}Y_{2}\xrightarrow{X^{\prime\prime}}Y_{3}\quad=\quad
Y_{1}\xrightarrow{X}Y_{3}.$

Let us introduce the following shorthand notations

  * •

DN operators: $\mathrm{DN}_{ij,A}\colon=\mathrm{DN}_{Y_{i}\sqcup Y_{j},A}$,
with $i,j\in\\{1,2,3\\}$ and $A\in\\{X^{\prime},X^{\prime\prime},X\\}$. Also,
by $(\mathrm{DN}_{ij,A})_{kl}$ we will denote $Y_{k}-Y_{l}$ block in
$\mathrm{DN}_{ij,A}$.

  * •

“Interface” DN operator:
$\mathrm{DN}_{\mathrm{int}}\colon=(\mathrm{DN}_{Y_{1}\sqcup Y_{2}\sqcup
Y_{3},X})_{Y_{2}-Y_{2}\mathrm{-block}}$.

  * •

Extension operators: $E_{ij,A}=E_{Y_{i}\sqcup Y_{j},A}$. We will also denote
its $A-Y_{k}$ block by $(E_{ij,A})_{k}$.

  * •

Propagators: $G_{A,ij}\colon=G_{A,Y_{i}\sqcup Y_{j}}$.

One has the following straightforward generalization of Theorem 3.8 to the
case of possibly nonempty $Y_{1},Y_{3}$.

######  Theorem 3.11.

The data of the Gaussian theory on the glued cobordism
$Y_{1}\xrightarrow{X}Y_{3}$ can be computed from the data of the constituent
cobordisms $Y_{1}\xrightarrow{X^{\prime}}Y_{2}$,
$Y_{2}\xrightarrow{X^{\prime\prime}}Y_{3}$ as follows

  1. (a)

Glued DN operator $\mathrm{DN}_{13,X}$:

(71)
$\left(\begin{array}[]{cc}(\mathrm{DN}_{12,X^{\prime}})_{11}-(\mathrm{DN}_{12,X^{\prime}})_{12}\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{12,X^{\prime}})_{21}&-(\mathrm{DN}_{12,X^{\prime}})_{12}\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{23,X^{\prime\prime}})_{23}\\\
-(\mathrm{DN}_{23,X^{\prime\prime}})_{32}\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{12,X^{\prime}})_{21}&(\mathrm{DN}_{23,X^{\prime\prime}})_{33}-(\mathrm{DN}_{23,X^{\prime\prime}})_{32}\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{23,X^{\prime\prime}})_{23}\end{array}\right).$

The blocks correspond to vertices of $Y_{1}$ and $Y_{3}$. The interface DN
operator here is

(72)
$\mathrm{DN}_{\mathrm{int}}=(\mathrm{DN}_{12,X^{\prime}})_{22}+(\mathrm{DN}_{23,X^{\prime\prime}})_{22}-K_{Y_{2}}.$

  2. (b)

Extension operator $E_{13,X}$:

(73)
$\left(\begin{array}[]{cc}(E_{12,X^{\prime}})_{1}-(E_{12,X^{\prime}})_{2}\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{12,X^{\prime}})_{21}&-(E_{12,X^{\prime}})_{2}\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{23,X^{\prime\prime}})_{23}\\\
-\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{12,X^{\prime}})_{21}&-\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{23,X^{\prime\prime}})_{23}\\\
-(E_{23,X^{\prime\prime}})_{2}\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{12,X^{\prime}})_{21}&(E_{23,X^{\prime\prime}})_{3}-(E_{23,X^{\prime\prime}})_{2}\mathrm{DN}_{\mathrm{int}}^{-1}(\mathrm{DN}_{23,X^{\prime\prime}})_{23}\end{array}\right).$

Here horizontally, the blocks correspond to vertices of $Y_{1}$, $Y_{3}$;
vertically – to vertices of $X^{\prime}\setminus(Y_{1}\sqcup Y_{2})$, $Y_{2}$
and $X^{\prime\prime}\setminus(Y_{2}\sqcup Y_{3})$.

  3. (c)

Determinant:

(74) $\det K_{X,Y_{1}\sqcup Y_{3}}=\det K_{X^{\prime},Y_{1}\sqcup
Y_{2}}\cdot\det K_{X^{\prime\prime},Y_{2}\sqcup
Y_{3}}\cdot\det\mathrm{DN}_{\mathrm{int}}.$

  4. (d)

Propagator:

     * •

For $v_{1},v_{2}\in X^{\prime}$,

(75)
$G_{X,13}(v_{1},v_{2})=G_{X^{\prime},12}(v_{1},v_{2})+\sum_{u_{1},u_{2}\in
Y}E_{12,X^{\prime}}(v_{1},u_{1})\mathrm{DN}_{\mathrm{int}}^{-1}(u_{1},u_{2})E_{12,X^{\prime}}(v_{2},u_{2})$

and similarly for $v_{1},v_{2}\in X^{\prime\prime}$.

     * •

For $v_{1}\in X^{\prime}$, $v_{2}\in X^{\prime\prime}$,

(76) $G_{X,13}(v_{1},v_{2})=\sum_{u_{1},u_{2}\in
Y}E_{12,X^{\prime}}(v_{1},u_{1})\mathrm{DN}_{\mathrm{int}}^{-1}(u_{1},u_{2})E_{23,X^{\prime\prime}}(v_{2},u_{2})$

and similarly for $v_{1}\in X^{\prime\prime}$, $v_{2}\in X^{\prime}$.

####  3.3.6. Self-gluing and trace formula

As another generalization of Theorem 3.8, one can consider the case of a graph
$X$ relative to a subgraph $Y$ that admits a decomposition $Y=Y_{1}\sqcup
Y_{2}$ where $Y_{1}$ and $Y_{2}$ are isomorphic graphs. Then, specifying a
graph isomorphism $f\colon Y_{1}\to Y_{2}$, we can glue $Y_{1}$ to $Y_{2}$
using $f$ to form a new graph $\tilde{X}$ with a distinguished subgraph
$\tilde{Y}$.242424In the setting of theorem 3.8, we have $X=X^{\prime}\sqcup
X^{\prime\prime}$, and there are no edges between $X^{\prime}$ and
$X^{\prime\prime}$. In the following discussion we will suppress $f$ but
remark that in principle the glued graphs $\tilde{X}$ and $\tilde{Y}$ do
depend on $f$. We have $\tilde{Y}\cong Y_{1}\cong Y_{2}$ if and only if there
are no edges between $Y_{1}$ and $Y_{2}$. See Figure 12.

$X$$\leadsto$$Y_{1}$$Y_{2}$$\tilde{Y}$$\tilde{X}$ Figure 12. An example of
self-gluing.

Then one has the following relation between the Dirichlet-to-Neumann operators
of $Y$ relative to $X$ and $\tilde{Y}$ relative to $\tilde{X}$:

######  Proposition 3.12.

Let $\phi\in C^{0}(Y_{1})\simeq C^{0}(Y_{2})\simeq C^{0}(\tilde{Y})$,
then252525Below we are identifying using $f$ to identify $V(Y_{1})$ and
$V(Y_{2})$, and then also $\phi$ and $(f^{-1})^{*}\phi$.

(77)
$\left((\phi,\phi),\left(\mathrm{DN}_{Y,X}-\frac{1}{2}K_{Y}\right)\begin{pmatrix}\phi\\\
\phi\end{pmatrix}\right)=(\phi,\mathrm{DN}_{\tilde{Y},\tilde{X}}\phi)-\left(\phi,\left(\frac{1}{2}K_{\tilde{Y}}-\frac{1}{2}K_{Y_{1}}\right)\phi\right).$

Equivalently,

(78)
$\left((\phi,\phi),\left(\mathrm{DN}_{Y,X}-\frac{1}{2}K_{Y_{1}}-\frac{1}{2}K_{Y_{2}}\right)\begin{pmatrix}\phi\\\
\phi\end{pmatrix}\right)=(\phi,\mathrm{DN}_{\tilde{Y},\tilde{X}}\phi).$

###### Proof.

We have

$S_{X}=S_{X,Y}+S_{Y_{1}}+S_{Y_{2}}+S_{Y_{1},Y_{2}},$

where the first term contains contributions to the action from vertices in
$X\setminus Y$ and edges with at least one vertex in $X\setminus Y$, while the
last term contains just contributions from edges between $Y_{1}$ and $Y_{2}$.
Evaluating on the subspace of fields $F_{X}^{({\phi,\phi})}$ that agree on
$Y_{1}$ and $Y_{2}$, we get

$S_{X}\big{|}_{F_{X}^{({\phi,\phi})}}=S_{X,Y}+2S_{Y_{1}}(\phi)+S_{Y_{1},Y_{2}}(\phi)$

and $S_{X}-\frac{1}{2}S_{Y}=S_{X,Y}+S_{Y_{1}}+\frac{1}{2}S_{Y_{1},Y_{2}}.$ On
the other hand, we have

$\displaystyle
S_{\tilde{X}}-\frac{1}{2}S_{\tilde{Y}}\big{|}_{F_{\tilde{X}}^{\phi}}$
$\displaystyle=S_{\tilde{X},\tilde{Y}}+\frac{1}{2}S_{\tilde{Y}}(\phi)=S_{X,Y}+\frac{1}{2}S_{Y_{1}}(\phi)+\frac{1}{2}S_{Y_{1},Y_{2}}(\phi)$
$\displaystyle=S_{X}-\frac{1}{2}S_{Y}(\phi)-\frac{1}{2}S_{Y_{1}}.$

Therefore,

$Z_{X,Y}((\phi,\phi))=Z_{\tilde{X},\tilde{Y}}(\phi)e^{-\frac{1}{2\hbar}S_{Y_{1}}(\phi)}.$

Noticing that the relative operators agree $K_{X,Y}=K_{\tilde{X},\tilde{Y}}$,
and using (53), we obtain (77). To see (78), notice that the difference
$K_{Y}-K_{Y_{1}}-K_{Y_{2}}=K_{Y_{1},Y_{2}}=K_{\tilde{Y}}-K_{Y_{1}}$, so adding
$\frac{1}{2}K_{Y_{1},Y_{2}}$ to (77) we obtain (78). ∎

######  Corollary 3.13.

We have the following trace formula

(79)
$Z_{\tilde{X}}=\int_{F_{Y_{1}}}[D\phi]\langle\phi|Z_{X,Y_{1},Y_{2}}|\phi\rangle.$

###### Proof.

We have

$\displaystyle\langle\phi|Z_{X,Y_{1},Y_{2}}|\phi\rangle)$ $\displaystyle=\det
K_{X,Y}^{-\frac{1}{2}}e^{-\frac{1}{2\hbar}((\phi,\phi),(\mathrm{DN}_{Y,X}-\frac{1}{2}K_{Y_{1}}-\frac{1}{2}K_{Y_{2}})(\phi,\phi)}$
$\displaystyle=\det
K_{\tilde{X},\tilde{Y}}^{-\frac{1}{2}}e^{-\frac{1}{2\hbar}((\phi,\mathrm{DN}_{\tilde{Y},\tilde{X}})\phi)}.$

Integrating over $\phi$, we obtain the result. ∎

######  Example 3.14 (Gluing a circle graph from a line graph).

For the line graph $L_{3}$ relative to both endpoints,

$DN_{Y,X}=\frac{m^{2}+3}{m^{2}+2}\begin{pmatrix}m^{2}+1&-1\\\
-1&m^{2}+1\end{pmatrix}.$

In this case we have $K_{Y_{1}}=K_{Y_{2}}=m^{2}$ and

$\begin{pmatrix}1&1\end{pmatrix}\frac{m^{2}+3}{m^{2}+2}\begin{pmatrix}m^{2}+1&-1\\\
-1&m^{2}+1\end{pmatrix}\begin{pmatrix}1\\\
1\end{pmatrix}=\frac{2m^{2}(m^{2}+3)}{m^{2}+2},$

which implies

$\frac{2m^{2}(m^{2}+3)}{m^{2}+2}-\underbrace{m^{2}}_{-\frac{1}{2}K_{Y_{1}}-\frac{1}{2}K_{Y_{2}}}=\frac{m^{2}(m^{2}+4)}{m^{2}+2}.$

Here $m^{2}=K_{Y_{1}}$. On the other hand, $\tilde{X}=C_{2}$ is a circle graph
with $\tilde{Y}$ a point, and we have

$K_{X}=\begin{pmatrix}m^{2}+2&-2\\\ -2&m^{2}+2\end{pmatrix},\qquad
G_{X}=\frac{1}{m^{2}(m^{2}+4)}\begin{pmatrix}m^{2}+2&2\\\
2&m^{2}+2\end{pmatrix},$

therefore the corresponding Dirichlet-to-Neumann operator is

$\mathrm{DN}_{\tilde{Y},\tilde{X}}=\frac{m^{2}(m^{2}+4)}{m^{2}+2},$

as predicted by Proposition 3.12. The relative determinant $K_{Y,X}$ is
$m^{2}+2$ so that the trace formula becomes

$\int_{F_{Y_{1}}}[D\phi]\langle\phi|Z_{X,Y_{1},Y_{2}}|\phi\rangle=(m^{2}+2)^{-\frac{1}{2}}\left(\frac{m^{2}(m^{2}+4)}{m^{2}+2}\right)^{-\frac{1}{2}}=(m^{2}(m^{2}+4))^{-\frac{1}{2}}=Z_{C_{2}}.$

Similarly, for the line graph of length $N$ relative to both endpoints, the
Dirichlet-to-Neumann operator is given by (61) and we have

$\begin{pmatrix}1&1\end{pmatrix}\mathrm{DN}_{Y,X}\begin{pmatrix}1\\\
1\end{pmatrix}=\frac{4\sinh\frac{\beta}{2}(\cosh\beta(N-\frac{1}{2})-\cosh\frac{\beta}{2})}{\sinh\beta(N-1)}.$

On the other hand, the Dirichlet-to-Neumann operator of $\tilde{X}=C_{N-1}$
relative to a single vertex is

$\mathrm{DN}_{\tilde{Y},\tilde{X}}=2\sinh\beta\tanh\beta\frac{N-1}{2}.$

Then one can check that

$\frac{4\sinh\frac{\beta}{2}(\cosh\beta(N-\frac{1}{2})-\cosh\frac{\beta}{2})}{\sinh\beta(N-1)}-m^{2}=2\sinh\beta\tanh\beta\frac{N-1}{2}.$

######  Remark 3.15.

There is of course also common generalization of Theorem 3.11 and Proposition
3.12, where we have several boundary components and are allowed sew any two
isomorphic components together, we leave this statement to the imagination of
the reader.

###  3.4. Comparison to continuum formulation

In this subsection, we compare of results of subsections 3.2 and 3.3 to the
continuum counterparts for a free scalar theory on a Riemannian manifold. For
details on the latter, we refer to [16].

Consider the free scalar theory on a closed Riemannian manifold $M$ defined by
the action

$S(\phi)=\int_{M}\frac{1}{2}d\phi\wedge*d\phi+\frac{m^{2}}{2}\phi^{2}d\mathrm{vol}=\int_{M}\frac{1}{2}\phi(\Delta+m^{2})\phi\,d\mathrm{vol},$

where $\phi\in C^{\infty}(M)$ is the scalar field, $m>0$ is the mass, $*$ is
the Hodge star associated with the metric, $d\mathrm{vol}$ is the metric
volume form and $\Delta$ is the metric Laplacian.

The partition function is defined to be

$Z=``{\int
D\phi\,e^{-\frac{1}{\hbar}S(\phi)}\;}"\colon=\left({\det}^{\zeta}(\Delta+m^{2})\right)^{-\frac{1}{2}},$

where ${\det}^{\zeta}$ stands for the functional determinant understood in the
sense of zeta-regularization. Correlators are given by Wick’s lemma in terms
of the Green’s function $G(x,y)\in C^{\infty}(M\times
M\setminus\mathrm{Diag})$ of the operator $\Delta+m^{2}$.

Next, if $M$ is a compact Riemannian manifold with boundary $\partial M$, one
can impose Dirichlet boundary condition $\phi|_{\partial M}=\phi_{\partial}$ –
a fixed function on $\partial M$ (thus, fluctuations of fields are zero on the
boundary). The unique solution of the Dirichlet boundary value problem on $M$,

$(\Delta+m^{2})\phi=0,\quad\phi|_{\partial}=\phi_{\partial},$

can be written as

(80) $\phi(x)=\int_{\partial
M}\partial^{n}_{y}G_{D}(x,y)\,\phi_{\partial}(y)d\mathrm{vol}^{\partial}_{y}.$

Here:

  * •

$d\mathrm{vol}^{\partial}$ is the Riemannian volume form on $\partial M$
(w.r.t. the induced metric from the bulk).

  * •

$G_{D}\in C^{\infty}(M\times M\setminus\mathrm{Diag})$ is the Green’s function
for the operator $\Delta+m^{2}$ with Dirichlet boundary condition.

  * •

$\partial^{n}$ stands for the normal derivative at the boundary. In
particular, for $x\in M$, $y\in\partial M$,

(81) $\partial^{n}_{y}G_{D}(x,y)=\left.\frac{\partial}{\partial
t}\right|_{t=0}G_{D}(x,\widetilde{y}_{t}),$

where $\widetilde{y}_{t}$, $t\geq 0$ is a curve in $M$ starting at
$\widetilde{y}_{0}=y$ with initial velocity being the inward unit normal to
the boundary.

Then on a manifold with boundary one has the partition function

(82)
$Z(\phi_{\partial})=``{\int_{\phi|_{\partial}=\phi_{\partial}}D\phi\,e^{-\frac{1}{\hbar}S(\phi)}\;}"\\\
=\left({\det}^{\zeta}_{D}(\Delta+m^{2})\right)^{-\frac{1}{2}}e^{-\frac{1}{2\hbar}\int_{\partial
M}\phi_{\partial}\mathrm{DN}(\phi_{\partial})d\mathrm{vol}^{\partial}}\\\
=\left({\det}^{\zeta}_{D}(\Delta+m^{2})\right)^{-\frac{1}{2}}e^{\frac{1}{2\hbar}\int_{\partial
M\times\partial
M}\phi_{\partial}(x)\,\partial^{n}_{x}\partial^{n}_{y}G_{D}(x,y)\,\phi_{\partial}(y)d\mathrm{vol}_{x}^{\partial}d\mathrm{vol}_{y}^{\partial}}.$

Here in the determinant in the r.h.s., $\Delta+m^{2}$ is understood as acting
on smooth functions on $M$ vanishing on $\partial M$ (which we indicate by the
subscript $D$ for “Dirichlet boundary condition”); $\mathrm{DN}\colon
C^{\infty}(\partial M)\rightarrow C^{\infty}(\partial M)$ is the Dirichlet-to-
Neumann operator (see footnote 18). The integral kernel of the DN operator is
$-\partial^{n}_{x}\partial^{n}_{y}G_{D}(x,y)$.

The integral in the exponential in the last line of (82) contains a non-
integrable singularity on the diagonal and has to be appropriately
regularized, cf. Remark 3.4 in [16].

Correlators on a manifold with boundary are:

  * •

One-point correlator:

$\langle\phi(x)\rangle_{\phi_{\partial}}=\int_{\partial
M}\partial^{n}_{y}G_{D}(x,y)\phi_{\partial}(y)d\mathrm{vol}^{\partial}_{y}.$

  * •

Centered two-point correlator:

$\langle\delta\phi(x)\delta\phi(y)\rangle_{\phi_{\partial}}=\hbar\,G_{D}(x,y),$

where $\delta\phi(x)\colon=\phi(x)-\langle\phi(x)\rangle_{\phi_{\partial}}$.

  * •

$k$-point centered correlators are given by Wick’s lemma.

When more detailed notations of the manifolds involved is needed, instead of
$G_{D}$ we will write $G_{M,\partial M}$ (and similarly for
${\det}^{\zeta}_{D}$) and instead of $\mathrm{DN}$ we will write
$\mathrm{DN}_{\partial M,M}$.

Continuing the dictionary of Remark 2.5 to free scalar theory on graphs vs.
Riemannian manifolds, we have the following.

Scalar theory on a graph $X$ | Scalar theory on a Riemannian manifold $M$
---|---
relative to subgraph $Y$ | with boundary $\partial M$
propagator $G_{X,Y}$ |  $G_{D}(x,y)$ with $x,y\in M$
extension operator $E_{Y,X}$ |  $\partial^{n}_{y}G_{D}(x,y)$ with $x\in M,\;y\in\partial M$
DN operator $\mathrm{DN}_{Y,X}$ |  $\mathrm{DN}$ or $-\partial^{n}_{x}\partial^{n}_{y}G_{D}(x,y)$ with $x,y\in\partial M$
$\det K_{X,Y}$ | ${\det}^{\zeta}_{D}(\Delta+m^{2})$
Table 2. Comparing the toy model and continuum theory, continued.

Here in the right column we are writing the integral kernels of operators
instead of operators themselves.

####  3.4.1. Gluing formulae for Green’s functions and determinants

Assume that $M$ is a closed Riemannian manifold cut be a closed codimension 1
submanifold $\gamma$ into two pieces, $M^{\prime}$ and $M^{\prime\prime}$.
Then one can recover the Green’s function for the operator $\Delta+m^{2}$ on
$M$ from Green’s functions on $M^{\prime}$ and $M^{\prime\prime}$, both with
Dirichlet condition on $\gamma$, as follows.262626 This result is originally
due to [4], see also [16].

  * •

For $x,y\in M^{\prime}$, one has

(83)
$G_{M}(x,y)=G_{M^{\prime},\gamma}(x,y)+\int_{\gamma}d\mathrm{vol}^{\gamma}_{w}\int_{\gamma}d\mathrm{vol}^{\gamma}_{z}\,\partial^{n}_{w}G_{M^{\prime},\gamma}(x,w)\varkappa(w,z)\partial^{n}_{z}G_{M^{\prime},\gamma}(z,y).$

Here $\varkappa(w,z)$ is the integral kernel of the inverse of the total (or
“interface”) DN operator

(84)
$\mathrm{DN}_{\mathrm{int}}\colon=\mathrm{DN}_{\gamma,M^{\prime}}+\mathrm{DN}_{\gamma,M^{\prime\prime}}.$

For $x,y\in M^{\prime\prime}$, one has a similar formula to (83).

  * •

For $x\in M^{\prime}$, $y\in M^{\prime\prime}$, one has

(85)
$G_{M}(x,y)=\int_{\gamma}d\mathrm{vol}^{\gamma}_{w}\int_{\gamma}d\mathrm{vol}^{\gamma}_{z}\,\partial^{n}_{w}G_{M^{\prime},\gamma}(x,w)\varkappa(w,z)\partial^{n}_{z}G_{M^{\prime\prime},\gamma}(z,y),$

and similarly if $x\in M^{\prime\prime}$, $y\in M^{\prime}$.

In the case $\dim M=2$, the zeta-regularized determinants satisfy a remarkable
Mayer-Vietoris type gluing formula due to Burghelea-Friedlander-Kappeler [3],

(86)
${\det}^{\zeta}_{M}(\Delta+m^{2})={\det}^{\zeta}_{M^{\prime},\gamma}(\Delta+m^{2})\,{\det}^{\zeta}_{M^{\prime\prime},\gamma}(\Delta+m^{2})\,{\det}^{\zeta}_{\gamma}(\mathrm{DN}_{\mathrm{int}}).$

This formula also holds for higher even dimensions provided that the metric
near the cut $\gamma$ is of warped product type (this is a result of Lee
[17]). In odd dimensions, under a similar assumption, the formula is known to
hold up to a multiplicative constant known explicitly in terms of the metric
on the cut.

Note that formulae (83), (85) have the exact same structure as formulae (64),
(66) for gluing of graph propagators.272727 One small remark is that the
continuum formula for the interface DN operator (84) is similar to (65),
except for the $-K_{Y}$ term in the l.h.s. which is specific to the graph
setting and disappears in the continuum limit.  Likewise, the gluing formulae
for determinants in the continuum setting (86) and in graph setting (67) have
the same structure.

One can also allow the manifold $M$ to have extra boundary components disjoint
from the cut, i.e., to consider $M$ as a composition of two cobordisms
$\gamma_{1}\xrightarrow{M^{\prime}}\gamma_{2},\;\gamma_{2}\xrightarrow{M^{\prime\prime}}\gamma_{3}$.
One then has the corresponding gluing formulae which have the same structure
as the formulae of Theorem 3.11. In particular, one has a gluing formula for
continuum DN operators (see [21]) similar to the formula (71) in the graph
setting .

####  3.4.2. Example: continuum limit of line and circle graphs

The action of the continuum theory on an interval $[0,L]$ evaluated on a
smooth field $\phi\in C^{\infty}([0,L])$ can be seen as a limit of Riemann
sums

$S(\phi)=\lim_{N\rightarrow\infty}\sum_{i=2}^{N}\frac{\epsilon_{N}}{2}\left(\frac{\phi(i\epsilon_{N})-\phi((i-1)\epsilon_{N})}{\epsilon_{N}}\right)^{2}+\sum_{i=1}^{N}\epsilon_{N}\frac{m^{2}}{2}\phi(i\epsilon_{N})^{2},$

where in the r.h.s. we denoted $\epsilon_{N}=L/N$. The r.h.s. can be seen as
the action of the graph theory on a line graph with $N=L/\epsilon$ vertices,
where the mass is scaled as $m\mapsto\epsilon m$ and then the kinetic operator
is scaled as $K\mapsto\epsilon^{-1}K$ (and thus the propagator scales as
$G\rightarrow\epsilon G$), where we consider the limit $\epsilon\rightarrow 0$
(we are approximating the interval by a portion of a 1d lattice and taking the
lattice spacing to zero).

Applying the scaling above to the formulae of Example 3.7, we obtain the
following for the propagator (58):

$G_{\mathrm{graph}}(x,y)\underset{\epsilon\rightarrow 0}{\sim}\frac{\cosh
m(L-|x-y|)-\cosh m(L-x-y)}{\sinh mL},$

where we denoted $x=i\epsilon,y=j\epsilon$ – we think of $i,j$ as scaling with
$\epsilon$ so that $x,y$ remain fixed. The r.h.s. above is the Green’s
function for the operator $\Delta+m^{2}$ on an interval $[0,L]$ with Dirichlet
boundary conditions at the endpoints.282828 For the formulae pertaining to the
continuum theory on an interval, see e.g. [16, Appendix A.1].  For the DN
operator (61), we obtain

$\mathrm{DN}_{\mathrm{graph}}\underset{\epsilon\rightarrow
0}{\sim}\frac{m}{\sinh mL}\left(\begin{array}[]{cc}\cosh mL&-1\\\ -1&\cosh
mL\end{array}\right).$

The r.h.s. is the correct DN operator of the continuum theory on the interval.

For the determinant (63), we have

$\det K_{\mathrm{graph}}\underset{\epsilon\rightarrow
0}{\sim}\epsilon^{-N}\frac{\sinh mL}{m}.$

For comparison, the zeta-regularized determinant on the interval is

${\det}^{\zeta}_{D}(\Delta+m^{2})=\frac{2\sinh mL}{m}.$

It differs from the graph result by a scaling factor $\epsilon^{N}$ and an
extra factor $2$ which exhibits a discrepancy between the two regularizations
of the functional determinant – lattice vs. zeta regularization.

######  Remark 3.16.

One can similarly consider the continuum limit for the line graph of Example
3.2, without Dirichlet condition at the endpoints. Its continuum counterpart
is the theory on an interval $[0,L]$ with _Neumann_ boundary conditions at the
endpoints, cf. footnote 17. Likewise, in the continuum limit for line graphs
relative to one endpoint (Example 3.6), one recovers the continuum theory with
Dirichlet condition at one endpoint and Neumann condition at the other.

For example, the zeta-determinant for Neumann condition at both ends is
${\det}_{N-N}^{\zeta}(\Delta+m^{2})=2m\sinh mL$. For Dirichlet condition at
one end and Neumann at the other, one has
${\det}^{\zeta}_{D-N}(\Delta+m^{2})=2\cosh mL$. These formulae are related to
the continuum limit of the discrete counterparts ((41) for Neumann-Neumann and
(57) for Neumann-Dirichlet boundary conditions) in the same way as in the
Dirichlet-Dirichlet case (by scaling with $\epsilon^{N}$ and an extra factor
of 2).

In the same vein, we can consider a circle of length $L$ as a limit of circle
graphs (Example 3.3) with spacing $\epsilon$. Then in the scaling limit, from
(42) we have

$G_{\mathrm{graph}}(x,y)\underset{\epsilon\rightarrow 0}{\sim}\frac{\cosh
m(\frac{L}{2}-|x-y|)}{2m\sinh\frac{mL}{2}},$

where the r.h.s. coincides with the continuum Green’s function on a circle.
For the determinant (43), we have

(87) $\det K_{\mathrm{graph}}\underset{\epsilon\rightarrow
0}{\sim}\epsilon^{-N}4\sinh^{2}\frac{mL}{2}.$

For comparison, the corresponding zeta-regularized functional determinant is

${\det}^{\zeta}(\Delta+m^{2})=4\sinh^{2}\frac{mL}{2},$

which coincides with the r.h.s. of (87) up to the scaling factor
$\epsilon^{N}$.

##  4\. Interacting theory via Feynman diagrams

Consider scalar field theory on a closed graph $X$ defined by the action (21)
– the perturbation of the Gaussian theory by an interaction potential
$p(\phi)$:

$S_{X}(\phi)=\underbrace{\frac{1}{2}(\phi,K_{X}\phi)}_{S^{0}_{X}(\phi)}+\underbrace{\langle\mu,p(\phi)\rangle}_{S^{\mathrm{int}}_{X}(\phi)}.$

The partition function (23) can be computed by perturbation theory – the
Laplace method for the $\hbar\rightarrow 0$ asymptotics of the integral, with
corrections given by Feynman diagrams (see e.g. [11]):

(88)
$Z_{X}=\int_{F_{X}}D\phi\,e^{-\frac{1}{\hbar}(S^{0}_{X}(\phi)+S^{\mathrm{int}}_{X}(\phi))}=\ll
e^{-\frac{1}{\hbar}S_{X}^{\mathrm{int}}(\phi)}\gg^{0}=\\\ =\ll\sum_{n\geq
0}\frac{(-1)^{n}}{\hbar^{n}n!}\sum_{v_{1},\ldots,v_{n}\in
V_{X}}p(\phi(v_{1}))\cdots p(\phi(v_{n}))\gg^{0}\underset{\hbar\rightarrow
0}{\sim}\\\ \underset{\hbar\rightarrow
0}{\sim}\det(K_{X})^{-\frac{1}{2}}\cdot\sum_{\Gamma}\frac{\hbar^{-\chi(\Gamma)}}{|\mathrm{Aut}(\Gamma)|}\Phi_{\Gamma,X}.$

Here:

  * •

$\ll\cdots\gg^{0}$ stands for the non-normalized correlator in the Gaussian
theory.

  * •

The sum in the r.h.s. is over finite (not necessarily connected) graphs
$\Gamma$ (Feynman graphs) with vertices of valence $\geq 3$; $\chi(\Gamma)\leq
0$ is the Euler characteristic of a graph; $|\mathrm{Aut}(\Gamma)|$ – the
order of the automorphism group of the graph.

  * •

The weight of the Feynman graph is292929 We are using sans-serif font to
distinguish vertices of the Feynman graph $\mathsf{u},\mathsf{v}$ from the
vertices of the spacetime graph $u,v$.

(89) $\Phi_{\Gamma,X}=\sum_{f\colon V_{\Gamma}\rightarrow
V_{X}}\prod_{\mathsf{v}\in
V_{\Gamma}}(-p_{\mathrm{val}(\mathsf{v})})\cdot\prod_{(\mathsf{u},\mathsf{v})\in
E_{\Gamma}}G_{X}(f(\mathsf{u}),f(\mathsf{v})),$

where $p_{k}$ are the coefficients of the potential (22); $G_{X}$ is the
Green’s function of the Gaussian theory. The sum over $f$ here – the sum over
$|V_{\Gamma}|$-tuples of vertices of the spacetime graph $X$ – is a graph QFT
analog of the configuration space integral formula for the weight of a Feynman
graph (cf. e.g. [16]).

The expression in the r.h.s. of (88) is a power series in nonnegative powers
in $\hbar$, with finitely many graphs contributing at each order.303030This is
due to the restriction that valencies in $\Gamma$ are $\geq 3$, which is in
turn due to the assumption that $p(\phi)$ contains only terms of degree $\geq
3$. We denote the r.h.s. of (88) by $Z_{X}^{\mathrm{pert}}$ – the perturbative
partition function. It is an asymptotic expansion of the measure-theoretic
integral in the l.h.s. of (88) – the nonperturbative partition function.

###  4.1. Version relative to a boundary subgraph

Let $X$ be graph and $Y\subset X$ a “boundary” subgraph. We have the following
relative version of the perturbative expansion (88):

(90)
$Z_{X,Y}(\phi_{Y})=\int_{F_{X}^{\phi_{Y}}}[D\phi]^{\phi_{Y}}e^{-\frac{1}{\hbar}(S_{X}(\phi)-\frac{1}{2}S_{Y}(\phi_{Y}))}=\\\
=e^{-\frac{1}{2\hbar}S_{Y}^{\mathrm{int}}(\phi_{Y})}\ll
e^{-\frac{1}{\hbar}S^{\mathrm{int}}_{X\setminus Y}(\phi)}\gg^{0}_{\phi_{Y}}\\\
\underset{\hbar\rightarrow 0}{\sim}\det(K_{X,Y})^{-\frac{1}{2}}\cdot
e^{-\frac{1}{2\hbar}((\phi_{Y},(\mathrm{DN}_{Y,X}-\frac{1}{2}K_{Y})\phi_{Y})+S_{Y}^{\mathrm{int}}(\phi_{Y}))}\cdot\sum_{\Gamma}\frac{\hbar^{-\chi(\Gamma)}}{|\mathrm{Aut}(\Gamma)|}\Phi_{\Gamma,(X,Y)}(\phi_{Y}).$

Here the sum is over Feynman graphs $\Gamma$ with vertices split into two
subsets – “bulk” vertices $V^{\mathrm{bulk}}_{\Gamma}$ and “boundary” vertices
$V^{\partial}_{\Gamma}$ – with bulk vertices of valence $\geq 3$ and univalent
boundary vertices. In graphs $\Gamma$ we are not allowing edges connecting two
boundary vertices (while bulk-bulk and bulk-boundary edges are allowed). The
weight of a Feynman graph is a polynomial in the boundary field $\phi_{Y}$:

(91) $\Phi_{\Gamma,(X,Y)}(\phi_{Y})=\\\
=\sum_{f\colon{\tiny\begin{array}[]{c}V_{\Gamma}^{\mathrm{bulk}}\rightarrow
V_{X}\setminus V_{Y}\\\ V_{\Gamma}^{\partial}\rightarrow
V_{Y}\end{array}}}\prod_{\mathsf{v}\in
V_{\Gamma}^{\mathrm{bulk}}}(-p_{\mathrm{val}(\mathsf{v})})\cdot\prod_{\mathsf{u}^{\partial}\in
V_{\Gamma}^{\partial}}\phi(f(\mathsf{u}^{\partial}))\cdot\\\
\cdot\prod_{(\mathsf{u},\mathsf{v})\in E_{\Gamma}^{\mathrm{bulk-
bulk}}}G_{X,Y}(f(\mathsf{u}),f(\mathsf{v}))\cdot\prod_{(\mathsf{u}^{\partial},\mathsf{v})\in
E_{\Gamma}^{\mathrm{bdry-
bulk}}}E_{Y,X}(f(\mathsf{u}^{\partial}),f(\mathsf{v})).$

The sum over $f$ here can be seen as a sum over tuples of bulk and boundary
vertices in $X$. Similarly to (89), it is a graph QFT analog of a
configuration space integral formula for the Feynman diagrams in the
interacting scalar field theory on manifolds with boundary (cf. [16]), where
one is integrating over configurations of $n$ bulk points and $m$ boundary
points on the spacetime manifold.

We will denote the r.h.s. of (90) by $Z^{\mathrm{pert}}_{X,Y}(\phi_{Y})$.

######  Example 4.1.

Figure 13 is an example of a map $f$ contributing to the Feynman weight (91):

Figure 13. A Feynman graph with boundary vertices and a map contributing to
its Feynman weight.

The full Feynman weight of the graph on the left is:

$\Phi_{\Gamma,(X,Y)}(\phi_{Y})=\sum_{a,b\in V_{X}\setminus V_{Y},\;c,d\in
V_{Y}}(p_{3})^{2}G_{X,Y}(a,b)^{2}E_{Y,X}(c,a)E_{Y,X}(d,b)\,\phi(c)\phi(d),$

where we denoted
$a=f(\mathsf{a}),\,b=f(\mathsf{b}),\,c=f(\mathsf{c}),\,d=f(\mathsf{d})$.

######  Remark 4.2.

  1. (i)

By the standard argument, due to multiplicativity of Feynman weights w.r.t.
disjoint unions of Feynman graphs, the sum over graphs $\Gamma$ in (88), (90)
can be written as the exponential of the sum over connected Feynman graphs,
$\sum_{\Gamma}\cdots=e^{\sum_{\Gamma\,\mathrm{connected}}\cdots}$.

  2. (ii)

One can rewrite the r.h.s. of (90) without the DN operator in the exponent in
the prefactor, but instead allowing graphs $\Gamma$ with boundary-boundary
edges. The latter contribute extra factors
$-\mathrm{DN}_{Y,X}(\mathsf{u}^{\partial},\mathsf{v}^{\partial})$ in the
Feynman weight (91).

  3. (iii)

Unlike the case of closed $X$, the sum over $\Gamma$ in the r.h.s. of (90)
generally contributes infinitely many terms to each nonnegative order in
$\hbar$ (for instance, in the order $O(\hbar^{0})$, one has 1-loop graphs
formed by trees connected to a cycle). However, there are finitely many graphs
contributing to a given order in $\hbar$, in any fixed polynomial degree in
$\phi_{Y}$. Moreover, one can introduce a rescaled boundary field $\eta_{Y}$
so that

(92) $\phi_{Y}=\sqrt{\hbar}\,\eta_{Y}.$

Then (90) expressed as a function of $\eta_{Y}$ is a power series in
nonnegative half-integer powers of $\hbar$, with finitely many graphs
contributing at each order.313131The power of $\hbar$ accompanying a graph is
$\hbar^{|E_{\Gamma}|-|V_{\Gamma}^{\mathrm{bulk}}|-\frac{1}{2}|V_{\Gamma}^{\partial}|}$,
i.e., one can think that with this normalization of the boundary field,
boundary vertices contribute $1/2$ instead of $1$ to the Euler characteristic
of a Feynman graph. We also note that the rescaling (92) is rather natural, as
the expected magnitude of fluctuations of $\phi_{Y}$ around zero is
$O(\sqrt{\hbar})$.

###  4.2. Cutting/gluing of perturbative partition functions via
cutting/gluing of Feynman diagrams

As in Section 3.3.1, consider a closed graph
$X=X^{\prime}\cup_{Y}X^{\prime\prime}$ obtained from graphs
$X^{\prime},X^{\prime\prime}$ by gluing along a common subgraph
$X^{\prime}\supset Y\subset X^{\prime\prime}$ (but now we consider the
interacting scalar QFT).

As we know from Proposition 2.3, the nonperturbative partition functions
satisfy the gluing formula

$Z_{X}=\int_{F_{Y}}D\phi_{Y}\,Z_{X^{\prime},Y}(\phi_{Y})Z_{X^{\prime\prime},Y}(\phi_{Y}).$

Replacing both sides with their expansions (asymptotic series) in $\hbar$, we
have the gluing formula for the perturbative partition functions

(93)
$Z^{\mathrm{pert}}_{X}=\int_{F_{Y}}D\phi_{Y}\,Z^{\mathrm{pert}}_{X^{\prime},Y}(\phi_{Y})Z^{\mathrm{pert}}_{X^{\prime\prime},Y}(\phi_{Y}).$

This latter formula admits an independent proof in the language of Feynman
graphs which we will sketch here (adapting the argument of [16]).

Consider “decorations” of Feynman graphs $\Gamma$ for the theory on $X$ by the
following data:

  * •

Each vertex $\mathsf{v}$ of $\Gamma$ is decorated by one of three symbols
$\\{X^{\prime},X^{\prime\prime},Y\\}$, meaning that in the Feynman weight
$f(\mathsf{v})$ is restricted to be in $V_{X^{\prime}}\setminus V_{Y}$, in
$V_{X^{\prime\prime}}\setminus V_{Y}$, or in $V_{Y}$, respectively.

  * •

Each edge $e=(\mathsf{u},\mathsf{v})$ of $\Gamma$ is decorated by either $u$
or $c$ (“uncut” or “cut”), corresponding to the splitting of the Green’s
function on $X$ in Theorem 3.8:

$G_{X}(f(\mathsf{u}),f(\mathsf{v}))=G_{X}^{u}(f(\mathsf{u}),f(\mathsf{v}))+G_{X}^{c}(f(\mathsf{u}),f(\mathsf{v})).$

Here:

    * –

If $\mathsf{u},\mathsf{v}$ are both decorated with $X^{\prime}$, the “uncut”
term is $G_{X}^{u}\colon=G_{X^{\prime},Y}$. Similarly, if
$\mathsf{u},\mathsf{v}$ are both decorated with $X^{\prime\prime}$,
$G_{X}^{u}\colon=G_{X^{\prime\prime},Y}$. For all other decorations of
$\mathsf{u},\mathsf{v}$, $G_{X}^{u}\colon=0$. Because of this, we will impose
a selection rule: $u$-decoration is only allowed for $X^{\prime}-X^{\prime}$
or $X^{\prime\prime}-X^{\prime\prime}$ edges.

    * –

The “cut” term is

$G_{X}^{c}(f(\mathsf{u}),f(\mathsf{v}))\colon=\sum_{w_{1},w_{2}\in
Y}E_{Y,\alpha}(f(\mathsf{u}),w_{1})\mathrm{DN}_{Y,X}^{-1}(w_{1},w_{2})E_{Y,\beta}(f(\mathsf{v}),w_{2}),$

where $\alpha,\beta$ are the decorations of $\mathsf{u},\mathsf{v}$ (and we
understand $E_{Y,Y}$ as identity operator).

Let $\mathrm{Dec}(\Gamma)$ denote the set of all possible decorations of a
Feynman graph $\Gamma$. Theorem 3.8 implies that for any Feynman graph
$\Gamma$ its weight splits into the contributions of its possible decorations:

$\Phi_{\Gamma,X}=\sum_{\Gamma^{\mathrm{dec}}\in\mathrm{Dec}(\Gamma)}\Phi_{\Gamma^{\mathrm{dec}},X},$

where in the summand on the r.h.s., we have restrictions on images of vertices
of $\Gamma$ as prescribed by the decoration, and we only select either cut or
uncut piece of each Green’s function.

Thus, the l.h.s. of (93) can be written as

(94)
$Z_{X}^{\mathrm{pert}}=\det(K_{X})^{-\frac{1}{2}}\cdot\sum_{\Gamma^{\mathrm{dec}}}\frac{\hbar^{-\chi(\Gamma)}}{|\mathrm{Aut}(\Gamma^{\mathrm{dec}})|}\Phi_{\Gamma^{\mathrm{dec}},X}.$

where on the right we are summing over all Feynman graphs with all possible
decorations.

The r.h.s. of (93) is:

(95)
$\ll\det(K_{X^{\prime},Y})^{-\frac{1}{2}}\det(K_{X^{\prime\prime},Y})^{-\frac{1}{2}}e^{-\frac{1}{\hbar}S_{Y}^{\mathrm{int}}(\phi_{Y})}\cdot\\\
\cdot\sum_{\Gamma^{\prime},\Gamma^{\prime\prime}}\frac{\hbar^{-\chi(\Gamma^{\prime}\cup\Gamma^{\prime\prime})}}{|\mathrm{Aut}(\Gamma^{\prime}\cup\Gamma^{\prime\prime})|}\Phi_{\Gamma^{\prime},(X^{\prime},Y)}(\phi_{Y})\Phi_{\Gamma^{\prime\prime},(X^{\prime\prime},Y)}(\phi_{Y})\gg_{Y}=\\\
\det(K_{X})^{-\frac{1}{2}}\cdot\left\langle
e^{-\frac{1}{\hbar}S_{Y}^{\mathrm{int}}(\phi_{Y})}\cdot\sum_{\Gamma^{\prime},\Gamma^{\prime\prime}}\frac{\hbar^{-\chi(\Gamma^{\prime}\cup\Gamma^{\prime\prime})}}{|\mathrm{Aut}(\Gamma^{\prime}\cup\Gamma^{\prime\prime})|}\Phi_{\Gamma^{\prime},(X^{\prime},Y)}(\phi_{Y})\Phi_{\Gamma^{\prime\prime},(X^{\prime\prime},Y)}(\phi_{Y})\right\rangle_{Y},$

where
$\ll\cdots\gg_{Y}\colon=\int_{F^{Y}}D\phi_{Y}\,e^{-\frac{1}{2\hbar}(\phi_{Y},\mathrm{DN}_{Y,X}\phi_{Y})}\cdots$
is the non-normalized Gaussian average w.r.t. the total DN operator;
$\langle\cdots\rangle_{Y}$ is the corresponding normalized average.

The correspondence between (94) and (95) is as follows. Consider a decorated
graph $\Gamma^{\mathrm{dec}}$ and form out of it subgraphs
$\Gamma^{\prime},\Gamma^{\prime\prime}$ in the following way. Let us cut every
cut edge in $\Gamma$ (except $Y-Y$ edges) into two, introducing two new
boundary vertices. Then we collapse every edge between a newly formed vertex
and a $Y$-vertex. $\Gamma^{\prime}$ is the subgraph of $\Gamma$ formed by
vertices decorated by $X^{\prime}$ and uncut edges between them, and those
among the newly formed boundary vertices which are connected to an
$X^{\prime}$-vertex by an edge; $\Gamma^{\prime\prime}$ is formed similarly.

Then the contribution of $\Gamma^{\mathrm{dec}}$ to (94) is equal to the
contribution of a particular Wick pairing for the term in (95) corresponding
to the induced pair of graphs $\Gamma^{\prime},\Gamma^{\prime\prime}$, and
picking a term in the Taylor expansion of
$e^{-\frac{1}{\hbar}S^{\mathrm{int}}_{Y}(\phi_{Y})}$ corresponding to
$Y$-vertices in $\Gamma^{\mathrm{dec}}$. The sum over all decorated Feynman
graphs in (94) recovers the sum over all pairs
$\Gamma^{\prime},\Gamma^{\prime\prime}$ and all Wick contractions in (95).
This shows Feynman-graph-wise the equality of (94) and (95). One can also
check that the combinatorial factors work out similarly to the argument in
[16, Lemma 6.10].

######  Example 4.3.

Figure 14 is an example of a decorated Feynman graph on $X$ (on the left;
vertex decorations $X^{\prime},Y,X^{\prime\prime}$ are according to the labels
in the bottom) and the corresponding contribution to (95) on the right.

Figure 14. An example of cutting a Feynman graph.

Dashed edges on the right denote the Wick pairing for
$\langle\cdots\rangle_{Y}$ and are decorated with $\mathrm{DN}_{Y,X}^{-1}$.
Circle vertices are the boundary vertices of graphs
$\Gamma^{\prime},\Gamma^{\prime\prime}$ or equivalently the vertices formed by
cutting the $c$-edges of $\Gamma^{\mathrm{dec}}$.

##  5\. Path sum formulae for the propagator and determinant (Gaussian theory
in the first quantization formalism)

###  5.1. Quantum mechanics on a graph

Following the logic of Section 1.1, we now want to understand the kinetic
operator $\Delta_{X}+m^{2}$ of the second quantized theory as the Hamiltonian
of an auxiliary quantum mechanical system – a quantum particle on the graph
$X$.323232 This model of quantum mechanics on a graph – as a model for the
interplay between the operator and path integral formalisms – was considered
in [19, 20], see also [5].  The space of states $\mathcal{H}_{X}$ for graph
quantum mechanics on $X$ is $\mathbb{C}^{V}$, i.e. the space of
$\mathbb{C}$-valued functions on $V$. The graph Schrödinger equation333333
Here we are talking about the Wick-rotated Schrödinger equation (i.e.
describing quantum evolution in imaginary time), or equivalently the heat
equation.  on $X$ is

(96) $\frac{\partial}{\partial
t}\ket{\psi(t,v)}(t,v)=-\left(\Delta_{X}+m^{2}\right)\ket{\psi(t,v)},$

where $\ket{\psi(t,v)}$ is a (time-dependent) state, i.e. a vector in
$\mathbb{C}^{V}$. The explicit solutions to (96) are given by

(97)
$\ket{\psi(t_{f})}=e^{-(t_{f}-t_{0})(\Delta_{X}+m^{2})}\ket{\psi(t_{0})}.$

One can explicitly solve equation (97) by summing over certain paths on $X$,
see equations (104), (102), (120) below, in a way reminiscent of Feynman’s
path integral.343434This analogy is discussed in more detail in [19, 20]. This
graph quantum mechanics is the first step of our first quantization approach
to QFT on a graph.

###  5.2. Path sum formulae on closed graphs

####  5.2.1. Paths and h-paths in graphs

We start with some terminology. A _path_ $\gamma$ from a vertex $u$ to a
vertex $v$ of a graph $X$ is a sequence

$\gamma=(u=v_{0},e_{0},v_{1},\ldots,e_{k-1},v_{k}=v)$

where $v_{i}$ are vertices of $X$ and $e_{i}$ is an edge between $v_{i}$ and
$v_{i+1}$.353535 For simplicity of the exposition, we assume that the graph
$X$ has no short loops. The generalization allowing short loops is
straightforward: in the definition of a path and h-path, the edges traversed
$e_{i}$ are not allowed to be short loops (and in the formulae involving the
valence of a vertex, it should be replaced with valence excluding the
contribution of short loops). This is ultimately due to the fact that short
loops do not contribute to the graph Laplacian $\Delta_{X}$. We denote
$V(\gamma)$ the ordered collection $(v_{0},v_{1},\ldots,v_{k})$ of vertices of
$\gamma$. We call $l(\gamma)=k$ the length of the path, and denote
$P_{X}^{k}(u,v)$ the set of paths in $X$ of length $k$ from a vertex $u$ to a
vertex $v$. We denote by $P_{X}(u,v)=\cup_{k=0}^{\infty}P_{X}^{k}(u,v)$ the
set of paths of any length from $u$ to $v$. We also denote by
$P_{X}^{k}=\cup_{u,v\in X}P_{X}^{k}(u,v)$, and
$P_{X}=\cup_{k=0}^{\infty}P_{X}^{k}$ the sets of paths between any two
vertices of $X$.

Below we will also need a variant of this notion that we call _hesitant
paths_. Namely, a _hesitant path_ (or “h-path”) from a vertex $u$ to a vertex
$v$ is a sequence

$\tilde{\gamma}=(u=v_{0},e_{0},v_{1},\ldots,e_{k-1},v_{k}=v),$

but now we allow the possibility that $v_{i+1}=v_{i}$, in which case $e_{i}$
is allowed to be any edge starting at $v_{i}=v_{i+1}$. In this case we say
that $\tilde{\gamma}$ _hesitates_ at step $i$. If $v_{i+1}\neq v_{i}$, then we
say that $\tilde{\gamma}$ jumps at step $i$. As before, we say that such a
path has length $l(\tilde{\gamma})=k$, and we introduce the notion of the
_degree_ of a h-path as363636Diverting slightly from the notation of [9].

(98) $\deg(\tilde{\gamma})=|\\{i|v_{i}\neq v_{i+1},0\leq i\leq
l(\gamma)-1\\}|,$

i.e. the degree is the number of jumps of a h-path. We denote by

$h(\tilde{\gamma})=|\\{i|v_{i}=v_{i+1},0\leq i\leq l(\gamma)-1\\}|$

the number of hesitations of $\tilde{\gamma}$. Obviously
$l(\tilde{\gamma})=\deg(\tilde{\gamma})+h(\tilde{\gamma})$. We denote the set
of h-paths from $u$ to $v$ by $\Pi_{X}(u,v)$, and the set of length $k$
hesitant paths by $\Pi^{k}_{X}(u,v)$. There is an obvious concatenation
operation

(99)
$\begin{array}[]{ccc}\Pi^{k}_{X}(u,v)\times\Pi^{l}_{X}(v,w)&\to&\Pi^{k+l}_{X}(u,w)\\\
(\tilde{\gamma}_{1},\tilde{\gamma}_{2})&\mapsto&\tilde{\gamma}_{1}*\tilde{\gamma}_{2}\end{array}$

Observe that for every h-path $\tilde{\gamma}$ there is a usual (“non-
hesitant”) path $\gamma$ of length $l(\gamma)=\deg(\tilde{\gamma})$ given by
simply forgetting repeated vertices, giving a map
$P\colon\Pi(u,v)\twoheadrightarrow P(u,v)$. See Figure 15.

$A$$B$$C$$D$$E$$e_{0}$$e_{1}$$e_{2}$$e_{3}$

$A$$B$$C$$D$$E$$e_{0}$$f_{0}$$e_{1}$$f_{1}$$e_{2}$$f_{2}$$f_{3}$$e_{3}$$f_{4}$

Figure 15. Left: The path $\gamma=(A,e_{0},B,e_{1},C,e_{2},D,e_{3},E)$ from
$A$ to $E$ with length $l(\gamma)=4$. Right: the h-path
$\tilde{\gamma}=(A,f_{0},A,e_{0},B,f_{1},B,e_{1},C,f_{2},C,f_{3},C,e_{2},D,f_{4},D,e_{3},E)$
with $l(\tilde{\gamma})=9,\deg(\tilde{\gamma})=4,h(\tilde{\gamma})=5$ and
$P(\tilde{\gamma})=\gamma$.

A (hesitant) path is called _closed_ if $v_{k}=v_{0}$, i.e. the first and last
vertex agree. The cyclic group $C_{k}$ acts on closed paths of length $k$ by
shifting the vertices and edges. We call the orbits of this group action
_cycles_ (i.e. closed paths without a preferred start- or end-point), and
denote them by $\Gamma_{X}$ for equivalence classes of h-paths, and $C_{X}$
for equivalence classes of regular paths. A cycle $[\tilde{\gamma}]$ is called
_primitive_ if its representatives have trivial stabilizer under this group
action. Equivalently, this means that there is no $k>1$ and
$\tilde{\gamma}^{\prime}$ such that

$\tilde{\gamma}=\underbrace{\tilde{\gamma}^{\prime}*\tilde{\gamma}^{\prime}*\ldots*\tilde{\gamma}^{\prime}}_{k\text{
times}},$

i.e. the cycle is traversed exactly once. In general, the order of the
stabilizer of $\tilde{\gamma}$ is precisely the number of traverses. We will
denote this number by $t(\tilde{\gamma})$. Obviously, it is well-defined on
cycles.

######  Example 5.1.

In the $N=3$ circle graph, the two closed paths
$\tilde{\gamma}_{1}=(1,(12),2,(23),3,(31),1)$ and
$\tilde{\gamma}_{2}=(2,(23),3,(31),1,(12))$ define the same primitive cycle,
while the closed path $\tilde{\gamma}_{3}=(1,(31),3,(23),2,(12),1)$ defines a
different cycle (since the graph is traversed in a different order). The
closed (hesitant) path $\tilde{\gamma}_{4}=(1,(12),1,(12),1)$ is not
primitive, since $\gamma_{4}=(1,(12),1)*(1,(12),1)$.

####  5.2.2. h-path formulae for heat kernel, propagator and determinant

It is a simple observation that

(100) $p_{X}^{k}(u,v):=|P_{X}^{k}(u,v)|=\langle u|(A_{X})^{k}|v\rangle,$

where $A_{X}$ denotes the adjacency matrix of the graph $X$, $|v\rangle$
denotes the state which is 1 at $v$ and vanishes elsewhere, and

$\langle u|A|v\rangle=A_{uv}$

denotes the $(u,v)$-matrix element of the operator $A$ (in the bra-ket
notation for the quantum mechanics on $X$). We consider the heat operator

(101) $e^{-t\Delta_{X}}\colon C^{0}(X)\to C^{0}(X),$

which is the propagator of the quantum mechanics on the graph $X$ (97).

Suppose that $X$ is regular, i.e. all vertices have the same valence $n$. Then
$\Delta_{X}=n\cdot I-A_{X}$ and (100) implies that the heat kernel $\langle
u|e^{-t\Delta_{X}}|v\rangle$ is given by

(102) $\langle
u|e^{-t\Delta_{X}}|v\rangle=e^{-tn}\sum_{k=0}^{\infty}\frac{t^{k}}{k!}p_{X}^{k}(u,v)=e^{-tn}\sum_{\gamma\in
P_{X}(u,v)}\frac{t^{l(\gamma)}}{l(\gamma)!}.$

One can think of the r.h.s. as a discrete analog of the Feynman path integral
formula where one is integrating over all paths (see [19]).

For a general graph, one can derive a formula for the heat kernel in terms of
h-paths, by using the formula $\Delta=d^{T}d$. Namely, one has (see [9])

(103) $\langle
u|\Delta_{X}^{k}|v\rangle=\sum_{\tilde{\gamma}\in\Pi_{X}^{k}(u,v)}(-1)^{\deg(\tilde{\gamma})}.$

This implies the following formula for the heat kernel:

(104) $\langle
u|e^{-t\Delta_{X}}|v\rangle=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}(-1)^{k}\sum_{\tilde{\gamma}\in\Pi^{k}_{X}(u,v)}(-1)^{\deg(\tilde{\gamma})}=\sum_{\tilde{\gamma}\in\Pi_{X}(u,v)}\frac{t^{l(\tilde{\gamma})}}{l(\tilde{\gamma})!}(-1)^{h(\tilde{\gamma})}.$

Here we have used that
$l(\tilde{\gamma})+\deg(\tilde{\gamma})=h(\tilde{\gamma})\text{ mod }2$.

Then we have the following h-path sum formula for the Green’s function:

######  Lemma 5.2.

The Green’s function $G_{X}$ is given by

(105) $\left\langle
u|G_{X}|v\right\rangle=m^{-2}\sum_{k=0}^{\infty}(m^{-2})^{k}\sum_{\tilde{\gamma}\in\Pi^{k}_{X}}(-1)^{h(\tilde{\gamma})}=m^{-2}\sum_{\tilde{\gamma}\in\Pi_{X}(u,v)}(m^{-2})^{l(\tilde{\gamma})}(-1)^{h(\tilde{\gamma})}.$

###### Proof.

By expanding $m^{2}G_{X}=(m^{-2}K_{X})^{-1}=(1+m^{-2}\Delta_{X})^{-1}$ in
powers of $m^{-2}$ using the geometric series,373737This series converges
absolutely if the operator norm of $m^{-2}\Delta_{X}$ is less that one, or
equivalently $m^{2}>\lambda_{max}(\Delta_{X})$, i.e. $m^{2}$ is larger than
the largest eigenvalues of $\Delta_{X}$. we obtain

(106) $\left\langle
u\left|(1+m^{-2}\Delta_{X})^{-1}\right|v\right\rangle=\sum_{\tilde{\gamma}\in\Pi_{X}(u,v)}(-m^{-2})^{l(\tilde{\gamma})}(-1)^{\deg(\tilde{\gamma})},$

which proves (105). Alternatively, one can prove (105) by integrating the heat
kernel $e^{-tK_{X}}=e^{-tm^{2}}e^{-t\Delta_{X}}$ for $K_{X}$ over the time
parameter $t$ and using the Gamma function identity

$\int_{0}^{\infty}dt\ e^{-tm^{2}}\frac{t^{k}}{k!}=(m^{-2})^{k+1}.$

∎

In equation (105), we see two slightly different ways of interpreting the path
sum formula. In the middle we see that when expanding in powers of $m^{2}$,
the coefficient of $m^{-2(k+1)}$ is given by a signed count of h-path of
length $k$, and that the sign is determined by the number of hesitations. On
the right hand side we interpret the propagator as a weighted sum over all
h-paths, in accordance with the first quantization picture.

We have the following formula for the determinant of the kinetic operator
(normalized by $1/m^{2}$) in terms of closed h-paths or h-cycles:

######  Lemma 5.3.

The determinant of $K_{X}/m^{2}$ is given by

(107) $\displaystyle\log\det\left(\frac{K_{X}}{m^{2}}\right)$
$\displaystyle=-\sum_{v\in X}\sum_{\tilde{\gamma}\in\Pi^{\geq
1}_{X}(v,v)}\frac{(m^{-2})^{l(\tilde{\gamma})}}{l(\tilde{\gamma})}(-1)^{h(\tilde{\gamma})}$
$\displaystyle=-\sum_{[\tilde{\gamma}]\in\Gamma^{\geq
1}_{X}}\frac{(m^{-2})^{l(\tilde{\gamma})}}{{t(\tilde{\gamma})}}(-1)^{h(\tilde{\gamma})}.$

###### Proof.

Expand383838Again, this power series converges absolutely for
$m^{2}>\lambda_{max}(\Delta_{X})$.

$\displaystyle\log\det\left(\frac{K_{X}}{m^{2}}\right)$
$\displaystyle=\mathrm{tr}\log(1+m^{-2}\Delta_{X})=-\sum_{v\in
X}\sum_{k=0}^{\infty}\frac{(-m^{-2})^{k}}{k}\langle
v|\Delta_{X}^{k}|v\rangle,$

which implies (107). ∎

Note that in the expression in the middle of (107), we are summing over
h-paths of length at least 1 with a fixed starting point. To obtain the right
hand side, we sum over orbits of the group action of $C_{k}$ on closed paths
of length $k$, the size of the orbit of $\tilde{\gamma}$ is exactly
$l(\tilde{\gamma})/t(\tilde{\gamma})$.

######  Remark 5.4.

Both h-paths and paths form monoids w.r.t. concatenation, with $P$ a monoid
homomorphism. A map $s$ from a monoid to $\mathbb{R}$ or $\mathbb{C}$ is
called _multiplicative_ if it is a homomorphism of monoids, i.e.

(108)
$s(\tilde{\gamma}_{1}*\tilde{\gamma}_{2})=s(\tilde{\gamma}_{1})s(\tilde{\gamma}_{2}).$

Notice that in the path sum expression for the propagator (105), we are
summing over h-paths $\tilde{\gamma}$ with the weight

(109)
$s(\tilde{\gamma}):=(m^{-2})^{l(\tilde{\gamma})}(-1)^{h(\tilde{\gamma})}.$

Below it will be important that this weight is in fact multiplicative, which
is obvious from the definition.

######  Remark 5.5.

Using multiplicativity of $s$, we can resum over iterates of primitive cycles
to rewrite the right hand side of (107):

$\displaystyle\log\det\left(\frac{K_{X}}{m^{2}}\right)=-\sum_{\begin{subarray}{c}[\tilde{\gamma}]\in\Gamma^{\geq
1}_{X}\\\ \tilde{\gamma}\text{ primitive }\end{subarray}}\sum_{k\geq
1}\frac{s(\tilde{\gamma})^{k}}{k}=\sum_{\begin{subarray}{c}[\tilde{\gamma}]\in\Gamma^{\geq
1}_{X}\\\ \tilde{\gamma}\text{ primitive
}\end{subarray}}\log\left(1-m^{-2l(\tilde{\gamma})}(-1)^{h(\tilde{\gamma})}\right).$

####  5.2.3. Resumming h-paths. Path sum formulae for propagator and
determinant

Summing over the fibers of the map $P\colon\Pi_{X}(u,v)\twoheadrightarrow
P_{X}(u,v)$, we can rewrite (106) as a path sum formula as follows:

######  Lemma 5.6.

If $m^{2}>\mathrm{val}(v)$ for all $v\in X$, we have

(110) $\left\langle
u\left|(1+m^{-2}\Delta_{X})^{-1}\right|v\right\rangle=m^{2}\sum_{\gamma\in
P_{X}(u,v)}\prod_{v_{i}\in V(\gamma)}\frac{1}{m^{2}+\mathrm{val}(v_{i})}.$

###### Proof.

For a path $\gamma\in P^{k}_{X}(u,v)$, the fiber $P^{-1}(\gamma)$ consists of
h-paths $\tilde{\gamma}$ which hesitate an arbitrary number $j_{i}$ of times
at every vertex $v_{i}$ in $V(\gamma)$. For each vertex $v_{i}$, there are
$\mathrm{val}(v_{i})^{j_{i}}$ possibilities for a path to hesitate $j_{i}$
times at $v_{i}$. The length of such a h-path is
$l(\tilde{\gamma})=k+j_{0}+\ldots+j_{k}$ and its degree is
$\deg(\tilde{\gamma})=k$, hence we can rewrite equation (106) as

$\displaystyle\sum_{\tilde{\gamma}\in\Pi_{X}(u,v)}(-m^{-2})^{l(\tilde{\gamma})}(-1)^{\deg(\tilde{\gamma})}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{\gamma\in
P_{X}^{k}(u,v)}\sum_{j_{0},\ldots,j_{k}=0}^{\infty}\mathrm{val}(v_{0})^{j_{0}}\cdot\ldots\cdot\mathrm{val}(v_{k})^{j_{k}}(-m^{-2})^{k+j_{0}+\ldots+j_{k}}(-1)^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{\gamma\in
P_{X}^{k}(u,v)}(m^{-2})^{k}\sum_{j_{0},\ldots,j_{k}}^{\infty}\mathrm{val}(v_{0})^{j_{0}}\cdot\ldots\cdot\mathrm{val}(v_{k})^{j_{k}}(-m^{-2})^{j_{0}+\ldots+j_{k}}$
$\displaystyle=\sum_{\gamma\in P_{X}(u,v)}m^{2}\prod_{v_{i}\in
V(\gamma)}\frac{m^{-2}}{1+m^{-2}\cdot\mathrm{val}(v_{i})}.$

∎

######  Corollary 5.7.

The Green’s function of the kinetic operator has the expression

(111) $\langle u|G_{X}|v\rangle=\sum_{\gamma\in P_{X}(u,v)}\prod_{v_{i}\in
V(\gamma)}\frac{1}{m^{2}+\mathrm{val}(v_{i})}.$

In particular, if $X$ is regular of degree $n$, then

(112) $\langle u|G_{X}|v\rangle=\sum_{\gamma\in
P_{X}(u,v)}\left(\frac{1}{m^{2}+n}\right)^{l(\gamma)+1}=\frac{1}{m^{2}+n}\sum_{k=0}^{\infty}p^{k}_{X}(u,v)(m^{2}+n)^{-k}.$

To derive a path sum formula for the determinant, we use a slightly different
idea, that also provides an alternative proof of the resummed formula for the
propagator. Consider the operator $\Lambda$ which acts on $C^{0}(X)$
diagonally in the vertex basis and sends
$|v\rangle\mapsto(m^{2}+\mathrm{val}(v))|v\rangle$, that is,

$\Lambda=\mathrm{diag}(m^{2}+\mathrm{val}(v_{1}),\ldots,m^{2}+\mathrm{val}(v_{N}))$

in the basis of $C^{0}(X)$ corresponding to an enumeration
$v_{1},\ldots,v_{N}$ of the vertices of $X$. Then, consider the “normalized”
kinetic operator

(113) $\tilde{K}_{X}=\Lambda^{-1}K_{X}=I-\Lambda^{-1}A_{X},$

with $A_{X}$ the adjacency matrix of the graph. Then, we have the simple
generalization of the observation that matrix elements of the $k$-th power of
the adjacency matrix $A_{X}$ count paths of length $k$ (see (100)), namely,
matrix elements of $(\Lambda^{-1}A_{X})^{k}\Lambda^{-1}$ count paths weighted
with

(114) $w(\gamma):=\prod_{v\in V(\gamma)}\frac{1}{m^{2}+\mathrm{val}(v)}.$

Then, we immediately obtain

(115) $\langle u|G_{X}|v\rangle=\langle
u|\tilde{K}_{X}^{-1}\Lambda^{-1}|v\rangle=\sum_{k=0}^{\infty}\left\langle
u\left|(\Lambda^{-1}A_{X})^{k}\Lambda^{-1}\right|v\right\rangle\\\
=\sum_{k=0}^{\infty}\sum_{\gamma\in P^{k}_{X}(u,v)}w(\gamma),$

which is (111).

For the determinant, we have the following statement:

######  Proposition 5.8.

The determinant of the normalized kinetic operator has the expansions

(116) $\displaystyle\log\det\tilde{K}_{X}$ $\displaystyle=-\sum_{v\in
X}\sum_{k=0}^{\infty}\sum_{\gamma\in
P^{k}_{X}(v,v)}\frac{w^{\prime}(\gamma)}{l(\gamma)}$ (117)
$\displaystyle=-\sum_{[\gamma]\in C^{\geq
1}_{X}}\frac{w^{\prime}(\gamma)}{t(\gamma)},$

where for a closed path $\gamma\in P^{k}_{X}(v,v)$,
$w^{\prime}(\gamma)=w(\gamma)\cdot(m^{2}+\mathrm{val}(v)$.393939Note that this
is well-defined on a cycle, we are simply taking the product over all vertices
in the path but without repeating the one corresponding to start-and endpoint.

###### Proof.

To see (116), we simply observe

$\log\det\tilde{K}_{X}=\mathrm{tr}\log(1-\Lambda^{-1}A_{X})=-\mathrm{tr}\sum_{k=0}^{\infty}\frac{(\Lambda^{-1}A_{X})^{k}}{k}=-\sum_{v\in
V}\sum_{k=0}^{\infty}\sum_{\gamma\in
P^{k}_{X}(v,v)}\frac{w^{\prime}(\gamma)}{k}.$

To see the second formula (117), one sums over orbits of the cyclic group
action on closed paths. ∎

In particular, for regular graphs we obtain a formula also derived in [9]:

######  Corollary 5.9.

If $X$ is a regular graph, then

(118) $\log\det\tilde{K}_{X}=-\sum_{k=1}^{\infty}\sum_{[\gamma]\in
C^{k}_{X}}\frac{(m^{2}+n)^{-k}}{t(\gamma)}.$

Another corollary is the following first quantization formula for the
partition function:

######  Theorem 5.10 (First quantization formula for Gaussian theory on closed
graphs).

The partition function of the Gaussian theory on a closed graph can be
expressed by

(119) $\log Z_{X}=\frac{1}{2}\left(\sum_{[\gamma]\in C^{\geq
1}_{X}}\frac{w^{\prime}(\gamma)}{t(\gamma)}-\sum_{v\in
X}\log(m^{2}+\mathrm{val}(v))\right).$

I.e., the logarithm of the partition function is given, up to the
“normalization” term $-\sum_{v\in X}\log(m^{2}+\mathrm{val}(v))$, by summing
over all cycles of length at least 1, dividing by automorphisms coming from
orientation reversing and multiple traversals.

###### Proof.

We have

$\log Z_{X}=-\frac{1}{2}\log\det
K_{X}=-\frac{1}{2}(\log\det\tilde{K}_{X}+\log\det\Lambda),$

from where the theorem follows by Proposition 5.8. ∎

######  Remark 5.11.

Notice that the weight $w(\gamma)$ of the resummed formula (111) is not
multiplicative: if $\gamma_{1}\in P_{X}(u,v)$ and $\gamma_{2}\in P_{X}(v,w)$
then

$\prod_{v_{i}\in\gamma_{1}}\frac{1}{m^{2}+\mathrm{val}(v_{i})}\prod_{v_{i}\in\gamma_{2}}\frac{1}{m^{2}+\mathrm{val}(v_{i})}=\frac{1}{m^{2}+\mathrm{val}(v)}\prod_{v_{i}\in\gamma_{1}*\gamma_{2}}\frac{1}{m^{2}+\mathrm{val}(v_{i})},$

since on the left hand side the vertex $v$ appears twice.

Object h-path sum path sum $\langle u|G_{X}|v\rangle$
$m^{-2}\sum_{\tilde{\gamma}\in\Pi_{X}(u,v)}s(\tilde{\gamma})$ (Eq. (106))
$\sum_{\gamma\in P_{X}(u,v)}w(\gamma)$ (Eq. (111) $\log\det m^{-2}K_{X}$
$-\sum_{[\tilde{\gamma}]\in\Gamma^{\geq
1}_{X}}\frac{s(\tilde{\gamma})}{{t(\tilde{\gamma})}}$ (Eq. (107))
$\log\det\tilde{K}_{X}$ $-\sum_{[\gamma]\in C^{\geq
1}_{X}}\frac{w^{\prime}(\gamma)}{t(\gamma)}$ (Eq. (117))

Table 3. Summary of path sum formulae, closed case.

######  Remark 5.12.

The sum over $k$ in (115), (116) is absolutely convergent for any $m^{2}>0$.
The reason is that the matrix $a=\Lambda^{-1}A_{X}$ has spectral radius
smaller than $1$ for $m^{2}>0$. This in turn follows from Perron-Frobenius
theorem: Since $a$ is a nonnegative matrix, its spectral radius $\rho(a)$ is
equal to its largest eigenvalue (also known as Perron-Frobenius eigenvalue),
which in turn is bounded by the maximum of the row sums of $a$.404040For any
matrix $A$ with entries $a_{ij}$, $\lambda$ an eigenvalue of $A$ and $x$ an
eigenvector for $\lambda$, we have  $|\lambda|=\frac{||\lambda
x||_{\infty}}{||x||_{\infty}}\leq\sup_{||y||_{\infty}=1}||Ay||_{\infty}=\max_{i}\sum_{j}|a_{ij}|.$
Here $||x||_{\infty}=\max_{i}|x_{i}|$ denotes the maximum norm of a vector
$x$. The sum of entries on the $v$-th row of $a$ is
$\frac{\mathrm{val}(v)}{m^{2}+\mathrm{val}(v)}<1$, which implies
$\rho(a)<\max_{v}\frac{\mathrm{val}(v)}{m^{2}+\mathrm{val}(v)}<1$.

In particular, resummation from h-path-sum formula to a path-sum formula
extends the absolute convergence region from $m^{2}>\lambda_{max}(\Delta_{X})$
to $m^{2}>0$.

####  5.2.4. Aside: path sum formulae for the heat kernel and the propagator –
“1d gravity” version.

There is the following generalization of the path sum formula (102) for the
heat kernel for a not necessarily regular graph $X$.

######  Proposition 5.13.

(120) $\langle u|e^{-t\Delta_{X}}|v\rangle=\sum_{\gamma\in
P_{X}(u,v)}W(\gamma;t),$

where the $t$-dependent weight for a path $\gamma$ of length $k$ is given by
an integral over a standard $k$-simplex of size $t$:

(121) $W(\gamma;t)=\int_{\scriptsize\begin{array}[]{c}t_{0},\ldots,t_{k}>0\\\
t_{0}+\cdots+t_{k}=t\end{array}}dt_{1}\cdots
dt_{k}\,e^{-\sum_{i=0}^{k}t_{i}\mathrm{val}(v_{i})},$

where we denoted $v_{0},\ldots,v_{k}$ the vertices along the path.

###### Proof.

To prove this result, note that the Green’s function $G_{X}$ as a function of
$m^{2}$ is the Laplace transform $L$ of the heat kernel $e^{-t\Delta_{X}}$ as
a function of $t$. Thus, one can recover the heat kernel as the inverse
Laplace transform $L^{-1}$ of $G_{X}$. Applying $L^{-1}$ to (111) termwise, we
obtain (120), (121) (note that the product of functions
$\frac{1}{m^{2}+\mathrm{val}(v)}$ is mapped by $L^{-1}$ to the convolution of
functions $L^{-1}(\frac{1}{m^{2}+\mathrm{val}(v)})=e^{-t\mathrm{val}(v)}$). ∎

As a function of $t$, the weight (121) is a certain polynomial in $t$ and
$e^{-t}$ with rational coefficients (depending on the sequence of valences
$\mathrm{val}(v_{i})$). If all valences along $\gamma$ are the same (e.g. if
$X$ is regular), then the integral over the simplex evaluates to
$W(\gamma;t)=\frac{t^{k}}{k!}e^{-t\cdot\mathrm{val}}$ – same as the weight of
a path in (102).

Note also that integrating (121) (multiplied by $e^{-m^{2}t}$) in $t$, we
obtain an integral expression for the weight (114) of a path in the path sum
formula for the Green’s function:

(122) $w(\gamma)=\int_{t_{0},\ldots,t_{k}>0}dt_{0}\cdots
dt_{k}\,e^{-\sum_{i=0}^{k}t_{i}(\mathrm{val}(v_{i})+m^{2})}.$

Here unlike (121) the integral is over $\mathbb{R}_{+}^{k+1}$, not over a
$k$-simplex.

Observe that the resulting formula for the Green’s function

(123) $\langle u|G_{X}|v\rangle=\sum_{\gamma\in
P_{X}(u,v)}\int_{t_{0},\ldots,t_{k}>0}dt_{0}\cdots
dt_{k}\,e^{-\sum_{i=0}^{k}t_{i}(\mathrm{val}(v_{i})+m^{2})}$

bears close resemblance to the first quantization formula (8), where the
proper times $t_{0},\ldots,t_{k}$ should be though of as parametrizing the
worldline metric field $\xi$ (and the path $\gamma$ is the field of the “1d
sigma model”).414141 More explicitly, one can think of the worldline as a
standard interval $[0,1]$ subdivided into $k$ sub-intervals by points
$p_{0}=0<p_{1}<\cdots<p_{k-1}<p_{k}=1$ (we think of $p_{i}$ as moments when
the particle jumps to the next vertex). Then one can think of $\\{t_{i}\\}$ as
moduli of metrics $\xi$ on $[0,1]$ modulo diffeomorphisms of $[0,1]$ relative
to (fixed at) the points $p_{0},p_{1},\ldots,p_{k}$.  We imagine the particle