# Stability for QED in d=3 : an overview

J. Dimock
Dept. of Mathematics
SUNY at Buffalo
Buffalo, NY 14260<EMAIL_ADDRESS>Expanded version of talk given at ICMP
2021, Geneva

###### Abstract

We report on a result on quantum electrodynamics on a three dimensional
Euclidean spacetime. The model is formulated on a toroidal lattice with unit
volume and variable lattice spacing. The result is that the renormalized
partition function is bounded above and below uniformly in the lattice
spacing. This is a first step toward showing that the partition function and
correlation functions have limits as the lattice spacing goes to zero.

##  1 Introduction

The topic is quantum electrodynamics (QED) on a three dimensional Euclidean
spacetime, one short of the physical dimension. It is a quantum field theory
for relativistic electrons and photons. The model is defined initially in
finite volume and with a short distance regularization. A basic problem in the
construction of the model is to remove the regularization and take the
infinite volume limit. In this note we report on some recent progress toward
the first of these problems.

The model is formulated on a 3-dimensional toroidal lattice. These will have
the general form

${\mathbb{T}}^{-N}_{N^{\prime}}=(L^{-N}{\mathbb{Z}}/L^{N^{\prime}}{\mathbb{Z}})^{3}$
(1)

where $L$ is a fixed large positive number. We start with
${\mathbb{T}}^{-N}_{0}$ which has lattice spacing $\epsilon\equiv L^{-N}$ and
unit volume. The theory is defined by the action

$S(e,\mathcal{A},\psi)=\frac{1}{2}\|d\mathcal{A}\|^{2}+<\bar{\psi},(\EuFrak{D}_{e}(\mathcal{A})+\bar{m})\psi>+m^{N}<\bar{\psi},\psi>+\varepsilon^{N}$
(2)

Here $\mathcal{A}(b)$ is an abelian gauge field defined on bonds $b$ (nearest
neighbor pairs) in the lattice and
$d\mathcal{A}(p)=\epsilon^{-1}\sum_{b\in\partial p}A(b)$ is the field strength
defined on plaquettes (squares) in the lattice. The notation is
$\|d\mathcal{A}]|^{2}=\sum_{p}\epsilon^{3}|d\mathcal{A}(p)|^{2}$. The
$\bar{\psi}_{\alpha}(x),\psi_{\alpha}(x)$ are fermi fields indexed by lattice
points and spin indices and
$<\bar{\psi},\psi>=\sum_{x,\alpha}\epsilon^{3}\bar{\psi}_{\alpha}(x)\psi_{\alpha}(x)$.
The fermi fields are anti-commuting generators of a Grassmann algebra. The
$\EuFrak{D}_{e}(\mathcal{A})$ is the Wilson form of the covariant lattice
Dirac operator with charge $e$ defined by

$\EuFrak{D}_{e}(\mathcal{A})=\gamma\cdot\nabla_{e\mathcal{A}}-\frac{1}{2}\epsilon\Delta_{e\mathcal{A}}$
(3)

Here $\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}$ are Dirac matrices,
the forward covariant derivative is

$(\partial_{e\mathcal{A},\mu}\psi)(x)=\epsilon^{-1}\Big{(}e^{i\epsilon
eA(x,x+\epsilon e_{\mu})}\psi(x+\epsilon e_{\mu})-\psi(x)\Big{)},$ (4)

$\nabla_{e\mathcal{A},\mu}$ is a symmetric version of this, and
$\Delta_{e\mathcal{A}}$ is the associated Laplacian. The $\bar{m}$ is a
background mass and $m^{N},\varepsilon^{N}$ are respectively mass and energy
density counterterms which will be chosen to depend on the lattice spacing,
thereby renormalizing these quantities. The model is super-renormalizable and
no renormalization is required for charge or field strength.

A basic item of interest is the partition function which is the exponential of
the action integrated over all fields

$\mathsf{Z}_{N}(e)=\int\exp(-S(e,\mathcal{A},\psi))\ D\psi\ D\mathcal{A}$ (5)

Here $D\mathcal{A}=\prod_{b}d(\mathcal{A}(b))$ and $D\psi$ stands for
Grassmann integral over all fermi fields $\bar{\psi},\psi$, the projection
onto the maximal element. Actually the integral over the gauge fields does not
converge because the integrand is invariant under gauge transformations. One
must fix a gauge, that is pick a representative for each gauge orbit and
integrate over them. Once this choice is made the partition function
$\mathsf{Z}_{N}(e)$ is well-defined. The main result is a bound on the
relative partition function uniform in the lattice spacing:

######  Theorem 1.

(Ultraviolet Stability) Assume $\bar{m}>0$ and $e$ is sufficiently small. Then
one can fix the gauge and choose counterterms $m^{N},\varepsilon^{N}$ so that
for all $N$

$K_{-}\leq\frac{\mathsf{Z}_{N}(e)}{\mathsf{Z}_{N}(0)}\leq K_{+}$

for some constants $K_{\pm}$.

The proof can be found in the series of papers [21],[22],[23] (see also
[18],[19],[20]). It uses a block averaging renormalization group technique due
to Balaban, a rigorous version of earlier ideas of Wilson. The method was
introduced by Balaban [1]-[4] for scalar electrodynamics in d=3 . It was
further developed by Balaban, Imbrie, and Jaffe [14],[15],[24] for the abelian
Higgs model in d=3 and by Balaban [5]-[13] for Yang Mills in d=3,4 . There is
also relevant work on block averaged Dirac Green’s functions by Balaban,
O’Carroll, and Shor [16],[17].

The proof should allow one to add local gauge invariant perturbations to the
action like $<dA,J>$ or $<\bar{\psi}\gamma_{\mu}\psi,j_{\mu}>$. Then the
partition function $\mathsf{Z}_{N}(e)$ becomes a generating function
$\mathsf{Z}_{N}(e,J,j)$, derivatives of $\log\mathsf{Z}_{N}(e,J,j)$ in $J,j$
at $J=0,j=0$ give correlation functions. Uniform bounds on the generating
function and analyticity in $J,j$ would give uniform bounds on correlation
functions. The continuum limit $N\to\infty$ for the partition function,
generating function, and correlation functions should also be feasible. The
method works equally well for any finite volume, not just unit volume. But the
infinite volume limit would pose new difficulties.

In the remainder of the paper we explain some of the ideas that go into the
proof. Before proceeding it is convenient to first reformulate the problem by
scaling up to a unit lattice. If $\Psi_{0},A_{0}$ are fields on the large unit
lattice ${\mathbb{T}}^{0}_{N}$, we define a new action by
$S_{0}(e_{0},\Psi_{0},A_{0})=S(e,\Psi_{0,L^{-N}},A_{0,L^{-N}})$ where
$\Psi_{0,L^{-N}}(x)=L^{N}\Psi_{0}(L^{N}x)$ and
$A_{0,L^{-N}}(b)=L^{N/2}A_{0}(L^{N}b)$ are functions on the original lattice
${\mathbb{T}}^{-N}_{0}$. We find that

$S_{0}(e_{0},A_{0},\Psi_{0})=\frac{1}{2}\|dA_{0}\|^{2}+<\bar{\Psi}_{0},(\EuFrak{D}_{e_{0}}(A_{0})+\bar{m}_{0})\Psi_{0}>+m_{0}<\bar{\Psi}_{0},\Psi_{0}>+\varepsilon_{0}|{\mathbb{T}}^{0}_{N}|$
(6)

Here norms, inner products, and operators are now on the unit lattice, and the
parameters have become tiny

$e_{0}=e_{0}^{N}=L^{-N/2}e\hskip
28.45274pt\bar{m}_{0}=\bar{m}_{0}^{N}=L^{-N}\bar{m}$ (7)

as well as $m_{0}=m_{0}^{N}=L^{-N}\bar{m}^{N}$ and
$\varepsilon_{0}=\varepsilon_{0}^{N}=L^{-3N}\varepsilon^{N}$. Our $N\to\infty$
problem now has become an infinite volume problem with scaled parameters.

##  2 Renormalization group transformations

The difficulty with the problem is that we have an unbounded number of
variables. The renormalization group method consists of integrating out a few
variables at a time, and keeping careful track of the effective actions at
each stage. (The transformations on the actions are not invertible and do not
actually form a group. Nevertheless the terminology ”renormalization group” is
ubiquitous.)

In the block averaging method one integrates out all fields which have a fixed
average over $L$-cubes. Let $y$ be a point in the $L$-lattice
${\mathbb{T}}^{1}_{N}$ and let $B(y)$ be the $L$-cube in the unit lattice
${\mathbb{T}}_{N}^{0}$ centered on $y$. For fermion fields $\Psi_{0}$ the
averaging operator in the presence of the gauge field $A_{0}$ is

$(Q(A_{0})\Psi_{0})(y)=L^{-3}\sum_{x\in
B(y)}e^{ie_{0}A_{0}(\Gamma(y,x))}\Psi_{0}(x)$ (8)

Here $A_{0}(\Gamma)=\sum_{b\in\Gamma}A_{0}(b)$ and $\Gamma(y,x)$ is a path
from $x$ to $y$. We use it to parallel translate to the center of the cube
before averaging, so that gauge covariance is preserved. For the gauge field
let $(y,y+Le_{\mu})$ be a bond in the $L$-lattice, and define the averaged
gauge field on this bond to be

$({\cal Q}A_{0})(y,y+Le_{\mu})=\sum_{x\in
B(y)}L^{-4}A_{0}(\Gamma_{x,x+Le_{\mu}})$ (9)

This also preserves gauge covariance.

With these definitions the renormalization group transformation sends the
initial density $\rho_{0}=e^{-S_{0}}$ to a new density $\rho_{1}$ defined for
fields $A_{1},\Psi_{1}$ defined on the $L$-lattice ${\mathbb{T}}^{1}_{N}$ by

$\begin{split}\tilde{\rho}_{1}(A_{1},\Psi_{1})&=\mathsf{N}_{0}\int
DA_{0}D\Psi_{0}\ \delta(A_{1}-{\cal Q}A_{0})\ \delta_{\mathsf{x}}(A_{0})\\\
&\exp\Big{(}-b/L<\bar{\Psi}_{1}-Q(-A_{0})\bar{\Psi}_{0},\Psi_{1}-Q(A_{0})\Psi_{0}>\Big{)}\rho_{0}(A_{0},\Psi_{0})\\\
\end{split}$ (10)

Here $b>0$ is a constant and $\mathsf{N}_{0}$ normalizes the fermion integral.
We have avoided using a delta function for the fermion fields (which would not
work for Grassmann elements) and instead used a Gaussian factor. We have also
begun to fix the gauge by taking a certain maximal tree in each $L$-cube, and
inserting a delta function $\delta_{\mathsf{x}}(A_{0})$ that sets $A_{0}=0$ on
the bonds in each tree. With this axial gauge, and the fixed averages, the
integral over the remaining fields converges and $\tilde{\rho}_{1}$ is well-
defined.

After the block averaging we scale back down defining for $A_{1},\Psi_{1}$ on
the unit lattice ${\mathbb{T}}^{0}_{N-1}$

$\rho_{1}(A_{1},\Psi_{1})=\tilde{\rho}_{1}(A_{1,L},\Psi_{1,L})$ (11)

where $A_{1,L}(b)=L^{-\frac{1}{2}}A_{1}(b/L)$ and
$\Psi_{1,L}(x)=L^{-1}\Psi_{1}(x/L)$ are fields on ${\mathbb{T}}^{1}_{N}$.

We would like the density $\rho_{1}$ to give the same partition function as
$\rho_{0}$. Formally this is true since it satisfies, up to an overall
constant,

$\int\delta_{\mathsf{x}}(A_{1})\rho_{1}(A_{1},\Psi_{1})DA_{1}D\Psi_{1}=\int\delta_{\mathsf{x}}({\cal
Q}A_{0})\delta_{\mathsf{x}}(A_{0})\rho_{0}(A_{0},\Psi_{0})DA_{0}D\Psi_{0}$
(12)

However these integrals do not converge since we do not as yet have a complete
gauge fixing. Nevertheless one can repeat the transformation generating a
sequence $\rho_{0},\rho_{1},\rho_{2},\dots$. After $k$ steps this yields a
density $\rho_{k}(A_{k},\Psi_{k})$ where the fields are defined on the unit
lattice ${\mathbb{T}}^{0}_{N-k}$ with shrunken volume. We continue to iterate
for $K$ steps with $K$ close to $N$ . Then we have a density $\rho_{K}$ on the
lattice ${\mathbb{T}}^{0}_{N-K}$ with bounded volume. Instead of (12) we get
something like

$\int\delta_{\mathsf{x}}(A_{K})\rho_{K}(A_{K},\Psi_{K})DA_{K}D\Psi_{K}=\int\Big{(}\prod^{K}_{j=0}\delta_{\mathsf{x}}({\cal
Q}^{j}A_{0})\Big{)}\rho_{0}(A_{0},\Psi_{0})DA_{0}D\Psi_{0}$ (13)

The right side of (13) does have a more or less complete gauge fixing [23]. It
is an axial gauge with a hierarchical structure, and it provides the gauge
fixed definition of the partition function $Z_{N}(e)$. Thus the identity (13)
shows that the partition function can be computed in principle from $\rho_{K}$
which has a bounded number of variables. Thus the issue is to keep control
over the sequence $\rho_{0},\rho_{1},\rho_{2},\dots$.

##  3 Bounded gauge fields

In analyzing the flow of the renormalization group it is useful to have bounds
on the gauge field, the fermion field is effectively already bounded. We
discuss how to arrange this in a simple case. The actual implementation is
rather more complicated, but the following gives the basic idea.

In the initial density we have a factor $e^{-\frac{1}{2}\|dA_{0}\|^{2}}$ which
is going to suppress large values of $dA_{0}$. To translate this fact into
hard bounds on $dA_{0}$ we divide the torus into cubes of size $M=L^{m}$
larger than $L$. Then we introduce the characteristic functions for each
$M$-cube $\square$ by

$\chi_{0}(,\square,A_{0})=\chi\Big{(}\mathrm{supp}_{p\in\square}|dA_{0}(p)|\leq
p(e_{0})\Big{)}$ (14)

Here

$p(e_{0})=(-\log e_{0})^{p}$ (15)

for some positive integer $p$. Since $e_{0}$ is tiny $p(e_{0})$ is large, but
not too large in the sense that $e_{0}p(e_{0})$ is still very small. Then with
$\zeta_{0}(\square)=1-\chi_{0}(\square)$ we write

$\begin{split}1=&\prod_{\square}\zeta_{0}(\square)+\chi_{0}(\square)=\sum_{\Omega}\prod_{\square\subset\Omega^{c}}\zeta_{0}(\square)\prod_{\square\subset\Omega}\chi_{0}(\square)\equiv\sum_{\Omega}\zeta_{0}(\Omega^{c})\chi_{0}(\Omega)\\\
\end{split}$ (16)

where $\Omega$ is an arbitrary union of $M$-cubes. The identity could be
inserted under the integral sign in (LABEL:polly) and then the sum taken
outside the integral. For the term $\Omega$ the characteristic function
$\chi_{0}(\Omega)$ enforces that $|dA_{0}(p)|\leq p(e_{0})$ everywhere in
$\Omega$. The characteristic function $\zeta_{0}(\Omega^{c})$ enforces that in
each $M$-cube $\square$ in $\Omega^{c}$ there is at least one plaquette $p$
such that $|dA_{0}(p)|>p(e_{0})$. It follow that for
$\square\subset\Omega^{c}$

$\zeta_{0}(\square)e^{-\frac{1}{4}\|dA_{0}\|^{2}_{\square}}\leq
e^{-\frac{1}{4}p(e_{0})^{2}}$ (17)

The quantity $e^{-\frac{1}{4}p(e_{0})^{2}}$ is in fact a very small number; it
satisfies $e^{-\frac{1}{4}p(e_{0})^{2}}={\cal O}(e_{0}^{n})$ for any positive
integer $n$. The bound (17) gives that

$\zeta_{0}(\Omega^{c}))e^{-\frac{1}{4}\|dA_{0}\|^{2}_{\Omega^{c}}}\leq
e^{-\frac{1}{4}p(e_{0})^{2}|\Omega^{c}|_{M}}$ (18)

where $|\Omega^{c}|_{M}$ is the number of $M$ cubes in $\Omega^{c}$. This is
enough for the convergence of the sum over $\Omega$ for we have

$\sum_{\Omega}e^{-\frac{1}{4}p(e_{0})^{2}|\Omega^{c}|_{M}}=\prod_{\square}(1+e^{-\frac{1}{4}p(e_{0})^{2}})\leq\prod_{\square}\exp(e^{-\frac{1}{4}p(e_{0})^{2}})\leq\exp\Big{(}e^{-\frac{1}{4}p(e_{0})^{2}}|{\mathbb{T}}^{0}_{N}|_{M}\Big{)}$
(19)

The last expression is $\exp({\cal O}(e^{n}))\approx 1$ since
$e^{-\frac{1}{4}p(e_{0})^{2}}={\cal O}(L^{-nN/2}e^{n})$ beats
$|{\mathbb{T}}^{0}_{N}|_{M}=L^{3N}M^{-3}$.

Tiny factors like $e^{-\frac{1}{4}p(e_{0})^{2}}$ also ensure that
contributions from large field regions $\Omega^{c}$ are negligible relative to
the contribution of the small field region $\Omega$. They do not require
renormalization and can be bounded without further processing. The
contribution of a small field region does require careful analysis and
renormalization as we will explain.

##  4 The first step

Starting with $\rho_{0}=e^{-S_{0}}$ defined in (6), the density after the
first step defined in (10) has the form

$\tilde{\rho}_{1}(A_{1},\Psi_{1})=\int DA_{0}\ \delta(A_{1}-{\cal
Q}A_{0})\delta_{\mathsf{x}}(A_{0})\exp\Big{(}-\frac{1}{2}\|dA_{0}\|^{2}\Big{)}\rho_{f}(A_{0},\Psi_{1})$
(20)

where the fermion integral is

$\begin{split}\rho_{f}(A_{0},\Psi_{1})=&\mathsf{N}_{0}\int
D\Psi_{0}\exp\Big{(}-bL^{-1}<\bar{\Psi}_{1}-Q(-A_{0})\bar{\Psi}_{0},\Psi_{1}-Q(A_{0})\Psi_{k}>\\\
&-<\bar{\Psi}_{0},(\EuFrak{D}_{e_{0}}(A_{0})+\bar{m}_{0})\Psi_{0}>-m_{0}<\bar{\Psi}_{0},\Psi_{0}>-\varepsilon_{0}|{\mathbb{T}}^{0}_{N}|\Big{)}\\\
\end{split}$ (21)

Our strategy is to make some large/small field splits, and then in the small
field region write the new density as the exponential of an effective action.
In the following for simplicity we only discuss the most important case where
the small field region is the full torus. Furthermore we do not always
explicitly record the characteristic functions bounding the fields.

First for the gauge field we define $A^{\min,\mathsf{x}}_{0}$ to be minimizer
in $A_{0}$ of the quadratic term $\frac{1}{2}\|dA_{0}\|^{2}$ subject to the
constraints that ${\cal Q}A_{0}=A_{1}$ and that $A_{0}$ is in the axial gauge.
This is a linear function of $A_{1}$ and is written
$A^{\min,\mathsf{x}}_{0}=H^{\mathsf{x}}_{0}A_{1}$. We now substitute
$A_{0}=A^{\min,\mathsf{x}}_{0}+Z_{0}$ and integrate over $Z_{0}$ rather than
$A_{0}$. The free gauge action splits as

$\frac{1}{2}\|dA_{0}\|^{2}=\frac{1}{2}\|dA^{\min,\mathsf{x}}_{0}\|^{2}+\frac{1}{2}\|dZ_{0}\|^{2}$
(22)

The delta functions in (20) become $\delta({\cal
Q}Z_{0})\delta_{\mathsf{x}}(Z_{0})$. We identify a Gaussian measure
$d\mu_{C_{0}}$ with covariance $C_{0}$ and a normalization factor
$\EuFrak{Z}_{0}$ by

$\EuFrak{Z}_{0}\ d\mu_{C_{0}}(Z_{0})=\ \delta({\cal
Q}Z_{0})\delta_{\mathsf{x}}(Z_{0})\exp\Big{(}-\frac{1}{2}\|dZ_{0}\|^{2}\Big{)}DZ_{0}$
(23)

The conditions $QZ_{0}=0$ and $Z_{0}$ axial imply
$\|dZ_{0}\|^{2}\geq\textrm{const}\|Z_{0}\|^{2}$ so this is non-degenerate. The
exact meaning of (23) requires parametrizing the hyplane selected by the delta
function [8], [21].

Now the expression (20) becomes

$\tilde{\rho}_{1}(A_{1},\Psi_{1})=\EuFrak{Z}_{0}\exp(-\frac{1}{2}\|dA^{\min,\mathsf{x}}_{0}\|^{2})\int\rho^{f}_{1}\Big{(}A^{\min,\mathsf{x}}_{0}+Z_{0},\Psi_{1}\Big{)})d\mu_{C_{0}}(Z_{0})$
(24)

The integral over $Z_{0}$ is a fluctuation integral, that is an integral over
the deviations from the minimizer.

Before proceeding we change gauges in the minimizer. Instead of the minimizer
$A^{\min,\mathsf{x}}_{0}$ in axial gauge we consider the minimizer
$A^{\min}_{0}$ in the Landau gauge. It is defined to be the minimizer of
$\frac{1}{2}\|dA_{0}\|^{2}$ subject to the constraints that ${\cal
Q}A_{0}=A_{1}$ and a version of the condition that the divergence of $A_{0}$
vanishes. It has the form $A^{\min}_{0}=H_{0}A_{1}$. These two minimizers are
gauge equivalent: $A^{\min,\mathsf{x}}_{0}=A^{\min}_{0}+d\omega$. Thus in
gauge invariant positions like $\|dA^{\min,\mathsf{x}}_{0}\|^{2}$ and
$\rho^{f}_{1}\Big{(}A^{\min,\mathsf{x}}_{0}+Z_{0},\Psi_{1}\Big{)})$ we can
replace $A^{\min,\mathsf{x}}_{0}$ by $A^{\min}_{0}$. The advantage of the
Landau gauge is that we have better control over the minimizing operator
$H_{0}$. In particular one can establish that it has an exponentially decaying
kernel and that it has good bounds on low order derivatives. The remark about
derivatives is not so important in this first step where we are on a unit
lattice. But after many steps the minimizers will be on increasingly fine
lattices and control on derivatives is important.

The Landau gauge minimizer $A^{\min}_{0}$ is only a function of $A_{1}$ and we
introduce the more suggestive notation
$\tilde{\mathcal{A}}_{1}=A^{\min}_{0}=H_{0}A_{1}$. So now we have

$\tilde{\rho}_{1}(A_{1},\Psi_{1})=\EuFrak{Z}_{0}\exp\Big{(}-\frac{1}{2}\|d\mathcal{A}\|^{2}\Big{)}\int\rho^{f}_{1}\Big{(}\mathcal{A}+Z_{0},\Psi_{1}\Big{)})d\mu_{C_{0}}(Z_{0})\
\Big{|}_{\mathcal{A}=\tilde{\mathcal{A}}_{1}}$ (25)

Note that the fluctuation field stays in axial gauge which is better for
convergence of the integrals.

Next in $\rho^{f}_{1}(\mathcal{A}+Z_{0},\Psi_{1}))$ we have a term
$<\bar{\Psi}_{0},\EuFrak{D}_{e_{0}}(\mathcal{A}+Z_{0})\Psi_{0}>$ and we write

$<\bar{\Psi}_{0},\EuFrak{D}_{e_{0}}(\mathcal{A}+Z_{0})\Psi_{0}>=<\bar{\Psi}_{0},\EuFrak{D}_{e_{0}}(\mathcal{A})\Psi_{0}>+V_{0}(\mathcal{A},Z_{0},\Psi_{0})$
(26)

Then $V_{0}$ depends on $Z_{0}$ through

$e^{ie_{0}(\mathcal{A}+Z_{0})}-e^{ie_{0}\mathcal{A}}=e^{ie_{0}\mathcal{A}}(e^{ie_{0}Z_{0}}-1)$
(27)

Introducing a large/small field split in $Z_{0}$ (thanks to
$d\mu_{C_{0}}(Z_{0})$) one can arrange $|Z_{0}|\leq p(e_{0})$ and so
$|e^{ie_{0}Z_{0}}-1|\leq{\cal O}(e_{0}p(e_{0}))$ with the same bound for the
kernel of $V_{0}$. So $V_{0}$ is very small. Similar remarks apply to the term
$<\bar{\Psi}_{1}-Q(-\mathcal{A}-Z_{0})\bar{\Psi}_{0},\Psi_{1}-Q(\mathcal{A}+Z_{0})\Psi_{k}>$;
subtracting off the expression at $Z_{0}=0$ gives another small term which we
include in the definition of $V_{0}$.

Now we have

$\begin{split}&\rho^{f}_{1}(\mathcal{A}+Z_{0},\Psi_{1})\\\
&=\mathsf{N}_{0}\int
D\Psi_{0}\exp\Big{(}-\tilde{\EuFrak{S}}_{1}(\mathcal{A},\Psi_{1},\Psi_{0})-V_{0}(\mathcal{A},Z_{0},\Psi_{0})-m_{0}<\bar{\Psi}_{0},\Psi_{0}>-\varepsilon_{0}|{\mathbb{T}}^{0}_{N}|\Big{)}\\\
\end{split}$ (28)

Here we have isolated the quadratic form

$\begin{split}&\tilde{\EuFrak{S}}_{1}(\mathcal{A},\Psi_{1},\Psi_{0})\equiv
bL^{-1}<\bar{\Psi}_{1}-Q(-\mathcal{A})\bar{\Psi}_{0},\Psi_{1}-Q(\mathcal{A})\Psi_{0}>+<\bar{\Psi}_{0},(\EuFrak{D}_{e_{0}}(\mathcal{A})+\bar{m}_{0})\Psi_{0}>\\\
\end{split}$ (29)

Let $\Psi^{\textrm{crit}}_{0}(\mathcal{A})$ be the critical point of this form
in $\Psi_{0}$. and similarly define
$\bar{\Psi}^{\textrm{crit}}_{0}(\mathcal{A})$. Explicitly we have

$\Psi^{\textrm{crit}}_{0}(\mathcal{A})=H_{0}(\mathcal{A})\Psi_{1}=bL^{-1}\
\Gamma_{0}(\mathcal{A})Q^{T}(-\mathcal{A})\Psi_{1}$ (30)

where

$\Gamma_{0}(\mathcal{A})=\Big{(}\EuFrak{D}_{e_{0}}(\mathcal{A})+\bar{m}_{0}+bL^{-1}\
Q^{T}(-\mathcal{A})Q(\mathcal{A})\Big{)}^{-1}$ (31)

There is a similar expression for
$\bar{\Psi}^{\textrm{crit}}_{0}(\mathcal{A})$.

We expand around the critical point. With the more suggestive notation
$\tilde{\psi}_{1}(\mathcal{A})=\Psi^{\textrm{crit}}_{0}(\mathcal{A})=H_{0}(\mathcal{A})\Psi_{1}$
we substitute $\Psi_{0}=\tilde{\psi}_{1}(\mathcal{A})+W_{0}$ and integrate
over $W_{0}$ rather than $\Psi_{0}$. The quadratic form splits as

$\tilde{\EuFrak{S}}_{1}(\mathcal{A},\Psi_{1},\tilde{\psi}_{1}(\mathcal{A})+W_{0})=\tilde{\EuFrak{S}}_{1}(\mathcal{A},\Psi_{1},\tilde{\psi}_{1}(\mathcal{A}))+<\bar{W}_{0},\Gamma_{0}(\mathcal{A})^{-1}W_{0}>$
(32)

We also identity a Grassmann Gaussian integral with covariance
$\Gamma_{0}(\mathcal{A})$ and formal measure
$d\mu_{\Gamma_{0}(\mathcal{A})}(W_{0})$ given by

$\EuFrak{Z}_{0}(\mathcal{A})\
d\mu_{\Gamma_{0}(\mathcal{A})}(W_{0})=\exp\Big{(}-<\bar{W}_{0},\Gamma_{0}(\mathcal{A})^{-1}W_{0}>\Big{)}DW_{0}.$
(33)

where $\EuFrak{Z}_{0}(\mathcal{A})=\det(\Gamma_{0}(\mathcal{A})^{-1})$. The
interaction is now
$V_{0}(\mathcal{A},Z_{0},\tilde{\psi}_{1}(\mathcal{A})+W_{0})$. We absorb into
it the difference between the mass counterterm at
$\tilde{\psi}_{1}(\mathcal{A})+W_{0}$ and at $\tilde{\psi}_{1}(\mathcal{A})$,
and call the result
$V_{0}(\mathcal{A},Z_{0},\tilde{\psi}_{1}(\mathcal{A}),W_{0})$. So now (28)
has become

$\begin{split}\rho^{f}_{1}(\tilde{\mathcal{A}}_{1}+Z_{0},\Psi_{1})&=\mathsf{N}_{0}\EuFrak{Z}_{0}(\mathcal{A})\
\exp\Big{(}-\tilde{\EuFrak{S}}_{1}(\mathcal{A},\Psi_{1},\psi)-m_{0}<\bar{\psi},\psi>-\varepsilon_{0}|{\mathbb{T}}^{0}_{N}|\Big{)}\\\
&\int
d\mu_{\Gamma_{0}(\mathcal{A})}(W_{0})\exp\Big{(}V_{0}(\mathcal{A},Z_{0},\psi,W_{0})\Big{)}\Big{|}_{\mathcal{A}=\tilde{\mathcal{A}}_{1},\psi=\tilde{\psi}_{1}(\tilde{\mathcal{A}}_{1})}\\\
\end{split}$ (34)

We add some comments about the operators
$\Gamma_{0}(\mathcal{A}),H_{0}(\mathcal{A})$. Controlling the inverse in the
definition (31) of $\Gamma_{0}(\mathcal{A})$ is difficult unless one has a
bound on the field strength $d\mathcal{A}$. Thanks to the factor
$e^{-\frac{1}{2}\|d\mathcal{A}\|^{2}}$ in (25) we can supply such a bound with
a large/small field split. In addition it is useful to establish that the
kernels of $\Gamma_{0}(\mathcal{A}),H_{0}(\mathcal{A})$ have exponential
decay. The mass $\bar{m}_{0}$ in $\Gamma_{0}(\mathcal{A})$ is too tiny to get
any useful estimate, but the term $bL^{-1}\ Q^{T}(-\mathcal{A})Q(\mathcal{A})$
provides an effective mass of order ${\cal O}(L^{-1})$ to do the job for
$\Gamma_{0}(\mathcal{A})$ and hence for $H_{0}(\mathcal{A})$. [17],[21] The
Landau gauge minimizer $H_{0}$ also has exponential decay. [12], [21]. Since
$V_{0}(\tilde{\mathcal{A}}_{1},Z_{0},\tilde{\psi}_{1}(\tilde{\mathcal{A}}_{1}),W_{0})$
has a local structure, some approximate locality in the fundamental fields
$A_{1},\Psi_{1}$ is preserved.

We insert the fermion integral (34) back into the full density (25) and find

$\begin{split}\tilde{\rho}_{1}(A_{1},\Psi_{1})&=\mathsf{N}_{0}\EuFrak{Z}_{0}\EuFrak{Z}_{0}(\mathcal{A})\
\Xi_{0}\Big{(}\mathcal{A},\psi\Big{)}\\\
&\exp\Big{(}-\frac{1}{2}\|d\mathcal{A}\|^{2}-\tilde{\EuFrak{S}}_{1}(\mathcal{A},\Psi_{1},\psi)-m_{0}<\bar{\psi},\psi>-\varepsilon_{0}|{\mathbb{T}}^{0}_{N}|\Big{)}\Big{|}_{\mathcal{A}=\tilde{\mathcal{A}}_{1},\psi=\tilde{\psi}_{1}(\tilde{\mathcal{A}}_{1})}\\\
\end{split}$ (35)

where

$\Xi_{0}\Big{(}\mathcal{A},\psi\Big{)}=\int
d\mu_{\Gamma_{0}(\mathcal{A})}(W_{0})d\mu_{C_{0}}(Z_{0})\chi_{0}(Z_{0})\exp\Big{(}V_{0}(\mathcal{A},Z_{0},\psi,W_{0})\Big{)}$
(36)

is the complete fluctuation integral. Here we have now made explicit a
characteristic function $\chi_{0}(Z_{0})$ enforcing $|Z_{0}|\leq p(e_{0})$

We need to write the fluctuation integral as the exponential of an expression
with good local properties. First we need norms on elements of the Grassman
algebra. Let $\\{\Psi(x)\\}$ be the generators of a Grassman algebra indexed
by a unit lattice. A general element has the form

$E(\Psi)=\sum_{n=0}^{\infty}\frac{1}{n!}\
\sum_{x_{1},\dots,x_{n}}E_{n}(x_{1},\dots,x_{n})\Psi(x_{1})\cdots\Psi(x_{n})$
(37)

Then for $h>0$ a norm is defined by

$\|E\|_{h}=\sum_{n=0}^{\infty}\frac{h^{n}}{n!}\sum_{x_{1},\dots,x_{n}}|E_{n}(x_{1},\dots,x_{n})|$
(38)

This generalizes to families of generators like our
$\\{\bar{\Psi}_{0,\alpha}(x),\Psi_{0,\alpha}(x)\\}$ as well as fields defined
on lattices with spacing $\epsilon$ in which in which case the sum over
$x_{i}$ would be weighted by $\epsilon^{3}$ approximating integrals.

Next we write the interaction $V_{0}$ in a form that can reproduce itself.
These are polymer expansions. Divide the torus ${\mathbb{T}}^{0}_{N}$ into
cubes $\square$ of width $M=L^{m}$ much larger than $L$. A polymer is defined
to be a connected union of $M$-cubes. A polymer expansion is a function of the
form $f=\sum_{X}f(X)$ where the sum is over polymers $X$ and $f(X)$ only
depends on fields in $X$. We generally assume also that $f(X)$ is
exponentially decaying in the size of $X$ in the sense that
$\|f(X)\|_{h}={\cal O}(e^{-\kappa d_{M}(X)})$ for a sufficiently large
constant $\kappa$. The quantity $d_{M}(X)$ is defined by

$Md_{M}(X)=\textrm{ length of the shortest continuum tree joining the cubes in
}X$ (39)

and the factor $e^{-\kappa d_{M}(X)}$ gives tree decay on the scale $M$. This
is sufficient for the convergence of the sum over $X$.

We express $V_{0}$ as polymer functions by

$V_{0}=\sum_{\square}E_{0}(\square)=\sum_{X}E_{0}(X)$ (40)

where first we break the integrals up into integrals $E_{0}(\square)$ over
$M$-cubes $\square$ and then we define $E_{0}(X)$ to be $E_{0}(\square)$ if
$X=\square$ and zero otherwise. We have $\|E_{0}(\square)\|_{h}\leq{\cal
O}(e_{0}p(e_{0}))$ and then trivially

$\|E_{0}(X,\mathcal{A},Z_{0})\|_{h}\leq{\cal O}(e_{0}p(e_{0}))e^{-\kappa
d_{M}(X)}$ (41)

The evaluation of the fluctuation integral (36) uses a standard technique
known as the cluster expansion. (See for example [18]). This says that there
exist polymer functions $E^{\\#}_{0}(X)$ such that

$\begin{split}&\int
d\mu_{\Gamma_{0}(\mathcal{A})}(W_{0})d\mu_{C_{0}}(Z_{0})\chi_{0}(Z_{0})\exp\Big{(}\sum_{X}E_{0}(X,\mathcal{A},Z_{0},\psi,W_{0})\Big{)}\\\
&=\exp\Big{(}\sum_{X}E^{\\#}_{0}(X,\mathcal{A},\psi\Big{)}\\\ \end{split}$
(42)

With our restrictions on the size of $d\mathcal{A}$ and for some
$\kappa^{\prime}<\kappa$ they also satisfy a bound

$\|E^{\\#}_{0}(X,\mathcal{A})\|_{h}\leq{\cal
O}(e_{0}p(e_{0}))e^{-\kappa^{\prime}d_{M}(X)}$ (43)

The key ingredients in the cluster expansion are that the initial functions
$E_{0}(X)$ are sufficiently small, true by our assumption that $e_{0}p(e_{0})$
is small, and the exponential decay of the covariances
$C_{0},\Gamma_{0}(A_{0})$ of the Gaussian measures. In carrying out the proof
it is useful to use the identities

$\begin{split}\int f(Z_{0})\ d\mu_{C_{0}}(Z_{0})=&\int
f\Big{(}C_{0}^{\frac{1}{2}}Z_{0}\Big{)}d\mu_{I}(Z_{0})\\\ \int
f(\bar{W}_{0},W_{0})\ d\mu_{\Gamma_{0}(\mathcal{A})}(\bar{W}_{0},W_{0})=&\int
f\Big{(}\bar{W}_{0},\Gamma_{0}(\mathcal{A})W_{0}\Big{)}d\mu_{I}(\bar{W}_{0},W_{0})\\\
\end{split}$ (44)

which put the non-locality in the function rather than the measure. This
compromises the polymer expansion since
$E_{0}(X,\mathcal{A},C_{0}^{\frac{1}{2}}Z_{0},\psi,\bar{W}_{0},\Gamma_{0}(\mathcal{A})W_{0})$
now depends on $Z_{0},W_{0}$ outside of $X$. But one can restore it by a
further decomposition using random walk expansions for $C_{0}^{\frac{1}{2}}$
and $\Gamma_{0}(\mathcal{A})$. With a new polymer expansion in the exponent
and the ultra-local measures $d\mu_{I}(Z_{0})$ and
$d\mu_{I}(\bar{W}_{0},W_{0})$ the cluster expansion is just a combinatoric
problem.

With bounds on $d\mathcal{A}$ one can also establish a polymer expansion for
the normalizing factor
$\EuFrak{Z}_{0}(\mathcal{A})=\det\Big{(}\EuFrak{D}_{e_{0}}(\mathcal{A})+\bar{m}_{0}+bL^{-1}\
Q^{T}(-\mathcal{A})Q(\mathcal{A})\Big{)}$ of the form [21]

$\EuFrak{Z}_{0}(\mathcal{A})=\EuFrak{Z}_{0}(0)\exp\Big{(}\sum_{X}E^{\det}_{0}(X,\mathcal{A})\Big{)}$
(45)

and $E^{\det}_{0}(X,\mathcal{A})$ has a bound similar to (43).

Now we have with $E_{0}^{\\#}=\sum_{X}E_{0}^{\\#}(X)$, etc.

$\begin{split}\tilde{\rho}_{1}(A_{1},\Psi_{1})&=N_{0}\EuFrak{Z}_{0}\EuFrak{Z}_{0}(0)\exp\Big{(}-\frac{1}{2}\|d\mathcal{A}\|^{2}-\tilde{\EuFrak{S}}_{1}(\mathcal{A},\Psi_{1},\psi)\\\
&-m_{0}<\bar{\psi},\psi>-\varepsilon_{0}|{\mathbb{T}}^{0}_{N}|+E_{0}^{\\#}(\mathcal{A},\psi)+E_{0}^{\det}(\mathcal{A})\Big{)}\Big{|}_{\mathcal{A}=\tilde{\mathcal{A}}_{1},\psi=\tilde{\psi}_{1}(\tilde{\mathcal{A}}_{1})}\\\
\end{split}$ (46)

Next we scale this expression defining
$\rho_{1}(A_{1},\Psi_{1})=\tilde{\rho}_{1}(A_{1,L},\Psi_{1,L})$ for
$A_{1},\Psi_{1}$ on ${\mathbb{T}}^{0}_{N-1}$. Scaled fields
$\mathcal{A}_{1},\psi_{1}(\mathcal{A}_{1})$ on the $L^{-1}$ lattice
${\mathbb{T}}^{-1}_{N-1}$ are given by $\mathcal{A}_{1,L}=H_{0}A_{1,L}$ and
$[\psi_{1}(\mathcal{A}_{1})]_{L}=H_{0}(\mathcal{A}_{1,L})\Psi_{1,L}$. Thanks
to our choice of scaling factors the $\|d\mathcal{A}\|^{2}$ is invariant and
the quadratic fermion term becomes

$\begin{split}\EuFrak{S}_{1}(\mathcal{A}_{1},\Psi_{1},\psi_{1}(\mathcal{A}_{1}))&\equiv
b<\bar{\Psi}_{1}-Q(-\mathcal{A}_{1})\bar{\psi}_{1}(\mathcal{A}_{1}),\Psi_{1}-Q(\mathcal{A}_{1})\psi_{1}(\mathcal{A}_{1})>\\\
&+<\bar{\psi}_{1}(\mathcal{A}_{1}),\Big{(}\EuFrak{D}_{e_{1}}(\mathcal{A}_{1})+\bar{m}_{1}\Big{)}\psi_{1}(\mathcal{A}_{1})>\\\
\end{split}$ (47)

Parameters change by

$e_{1}=L^{\frac{1}{2}}e_{0}\hskip 28.45274pt\bar{m}_{1}=L\bar{m}_{0}\hskip
28.45274ptm_{1}=Lm_{0}\hskip
28.45274pt\bar{\varepsilon}_{1}=L^{3}\varepsilon_{0}$ (48)

and we have up to a multiplicative constant

$\begin{split}\rho_{1}(A_{1},\Psi_{1})=&\exp\Big{(}-\frac{1}{2}\|d\mathcal{A}_{1}\|^{2}-\EuFrak{S}_{1}(\mathcal{A}_{1},\Psi_{1},\psi_{1}(\mathcal{A}_{1}))\\\
&-m_{1}<\bar{\psi}_{1}(\mathcal{A}_{1}),\psi_{1}(\mathcal{A}_{1})>-\varepsilon_{1}|{\mathbb{T}}^{0}_{N-1}|+E_{1}(\mathcal{A}_{1},\psi_{1}(\mathcal{A}_{1}))\Big{)}\\\
\end{split}$ (49)

The function $E_{1}(\mathcal{A},\psi)$ is defined as follows. It has a polymer
expansion $E_{1}=\sum_{X}E_{1}(X)$ where

$E_{1}(X,\mathcal{A},\psi)=({\cal L}E_{0}^{\\#})(X,\mathcal{A},\psi)+({\cal
L}E_{0}^{\det})(X,\mathcal{A},\psi)$ (50)

Here the linear operator ${\cal L}$ reblocks and scales. It is defined for an
$M$-polymer $X$ by

$({\cal
L}E)(X,\mathcal{A},\psi)=\sum_{Y:\bar{Y}=LX}E(Y,\mathcal{A}_{L},\psi_{L})$
(51)

The sum is over all $M$-polymers $Y$ such that $\bar{Y}=LX$, where $\bar{Y}$
is the union of all $LM$ polymers intersecting $Y$.

The sum in (51) tends to increase norms by a factor $L^{3}$. This is not a
problem in this first step, but is a key issue to be dealt with when we repeat
the procedure. The remedy will involve modifying the simple scaling in (48).

##  5 Iteration

Now we repeat the first step and generate the sequence of densities
$\rho_{1},\rho_{1},\rho_{2},\dots$. With $A_{k},\Psi_{k}$ on
${\mathbb{T}}^{0}_{N-k}$ we pass from $\rho_{k}$ to $\rho_{k+1}$ by block
averaging

$\begin{split}&\tilde{\rho}_{k+1}(A_{k+1},\Psi_{k+1})=\mathsf{N}_{k}\int
DA_{k}D\Psi_{k}\ \delta(A_{k+1}-{\cal Q}A_{k})\ \delta_{\mathsf{x}}(A_{k})\\\
&\exp\Big{(}-bL^{-1}<\bar{\Psi}_{k+1}-Q(-\tilde{\mathcal{A}}_{k+1})\bar{\Psi}_{k},\Psi_{k+1}-Q(\tilde{\mathcal{A}}_{k+1})\Psi_{k}>\Big{)}\rho_{k}(A_{k},\Psi_{k})\\\
\end{split}$ (52)

and then scaling
$\rho_{k+1}(A_{k+1},\Psi_{k+1})=\tilde{\rho}_{k+1}(A_{k+1,L},\Psi_{k+1,L})$.
The field $\tilde{\mathcal{A}}_{k+1}$ is yet to be specified.

We continue to consider the case where the small field region is the whole
torus. If this is the only contribution the claim is that up to a
multiplicative constant

$\begin{split}\rho_{k}(A_{k},\Psi_{k})=&\exp\Big{(}-\frac{1}{2}\|d\mathcal{A}_{k}\|^{2}-\EuFrak{S}_{k}(\mathcal{A}_{k},\Psi_{k},\psi_{k}(\mathcal{A}_{k}))\\\
&-m_{k}<\bar{\psi}_{k}(\mathcal{A}_{k}),\psi_{k}(\mathcal{A}_{k})>-\varepsilon_{k}|{\mathbb{T}}^{0}_{N-k}|+E_{k}(\mathcal{A}_{k},\psi_{k}(\mathcal{A}_{k}))\Big{)}\\\
\end{split}$ (53)

Here $\mathcal{A}_{k}={\cal H}_{k}A_{k}$ is a $k$-step Landau gauge minimizer.
The operator ${\cal H}_{k}$ is generated as the composition of $k$ single step
operators. Similarly the $\psi_{k}(\mathcal{A}_{k})={\cal
H}_{k}(\mathcal{A}_{k})\Psi_{k}$ and ${\cal H}_{k}(\mathcal{A})$ is generated
as the composition of $k$ single step operators. The fields
$\mathcal{A}_{k},\psi_{k}(\mathcal{A}_{k})$ are defined on the finer lattice
${\mathbb{T}}^{-k}_{N-k}$, so we are heading back toward the original problem
on ${\mathbb{T}}^{-N}_{0}$, even though the number of fundamental variables
$A_{k},\Psi_{k}$ is decreasing. The quadratic fermion term is

$\begin{split}\EuFrak{S}_{k}(\mathcal{A}_{k},\Psi_{k},\psi_{k}(\mathcal{A}_{k}))\equiv\
&b_{k}<\bar{\Psi}_{k}-Q_{k}(-\mathcal{A}_{k})\bar{\psi}_{k}(\mathcal{A}_{k}),\Psi_{k}-Q_{k}(\mathcal{A}_{k})\psi_{k}(\mathcal{A}_{k})>\\\
&+<\bar{\psi}_{k}(\mathcal{A}_{k}),\Big{(}\EuFrak{D}_{e_{k}}(\mathcal{A}_{k})+\bar{m}_{k}\Big{)}\psi_{k}(\mathcal{A}_{k})>\\\
\end{split}$ (54)

The scaled coupling constant is $e_{k}=L^{k/2}e_{0}=L^{-(N-k)/2}e$ and
$Q_{k}(\mathcal{A}_{k})$ is a $k$-fold averaging operator. The function
$E_{k}$ has a polymer expansion $E_{k}=\sum_{X}E_{k}(X)$. The
$E_{k}(X,\mathcal{A}_{k},\psi_{k}(\mathcal{A}_{k}))$ are restrictions of
functions $E_{k}(X,\mathcal{A},\psi)$ which are invariant under lattice
symmetries and gauge transformations. But note that the gauge invariance only
holds in positions corresponding to the smeared fields
$\mathcal{A}_{k},\psi_{k}(\mathcal{A}_{k})$, not in the fundamental variables
$A_{k},\Psi_{k}$.

We have discussed why $\rho_{1}$ has the form (53). We now assume $\rho_{k}$
has the form (53) and discuss why $\rho_{k+1}$ has this form. We insert (53)
into (LABEL:polly) and follow the treatment of the first step. However there
are new features. Let $A_{k}^{\min,\mathsf{x}}=H^{\mathsf{x}}_{k}A_{k+1}$ be
the minimizer in $A_{k}$ of the term $\|d\mathcal{A}_{k}\|^{2}$ subject to the
constraints $A_{k+1}=QA_{k}$ and $A_{k}$ axial. Expand the action around the
minimizer by $A_{k}=A_{k}^{\min,\mathsf{x}}+Z_{k}$. Then
$\mathcal{A}_{k}={\cal H}_{k}A_{k}$ becomes ${\cal
H}_{k}A^{\min,\mathsf{x}}_{k}+{\cal H}_{k}Z_{k}$. The ${\cal
H}_{k}A^{\min,\mathsf{x}}_{k}={\cal H}_{k}H^{\mathsf{x}}_{k}A_{k+1}$ is mixed
Landau and axial gauge, but it is gauge equivalent to the all Landau
$\tilde{\mathcal{A}}_{k+1}\equiv{\cal H}_{k}H_{k}A_{k+1}$ which will scale to
$\mathcal{A}_{k+1}={\cal H}_{k+1}A_{k+1}$. We make this change for better
estimates. Also defining ${\cal Z}_{k}={\cal H}_{k}Z_{k}$ we can now write

$\mathcal{A}_{k}=\tilde{\mathcal{A}}_{k+1}+{\cal Z}_{k}$ (55)

and in the action we have the split

$\frac{1}{2}\|d\mathcal{A}_{k}\|^{2}=\frac{1}{2}\|d\tilde{\mathcal{A}}_{k+1}\|^{2}+\frac{1}{2}\|d{\cal
Z}_{k}\|^{2}$ (56)

The translation $\mathcal{A}_{k}=\tilde{\mathcal{A}}_{k+1}+{\cal Z}_{k}$
induces changes in the quadratic fermion action and now also in the existing
polymer functions. As in (26) we separate out a leading term and a fluctuating
term by

$\begin{split}\EuFrak{S}_{k}\Big{(}\tilde{\mathcal{A}}_{k+1}+{\cal
Z}_{k},\Psi_{k},\psi_{k}(\tilde{\mathcal{A}}_{k+1}+{\cal
Z}_{k})\Big{)}=&\EuFrak{S}_{k}\Big{(}\tilde{\mathcal{A}}_{k+1},\Psi_{k},\psi_{k}(\tilde{\mathcal{A}}_{k+1})\Big{)}+[\cdots]\\\
E_{k}\Big{(}\tilde{\mathcal{A}}_{k+1}+{\cal
Z}_{k},\psi_{k}(\tilde{\mathcal{A}}_{k+1}+{\cal
Z}_{k}\Big{)}=&E_{k}\Big{(}\tilde{\mathcal{A}}_{k+1},\psi_{k}(\tilde{\mathcal{A}}_{k+1})\Big{)}+[\dots]\\\
\end{split}$ (57)

The fluctuation terms $[\cdots]$ will be discussed further.

The quadratic fermion term now has the form at
$\mathcal{A}=\tilde{\mathcal{A}}_{k+1}$

$bL^{-1}<\bar{\Psi}_{k+1}-Q(-\mathcal{A})\bar{\Psi}_{k},\Psi_{k+1}-Q(\mathcal{A})\Psi_{k}>+\EuFrak{S}_{k}(\mathcal{A},\Psi_{k},\psi_{k}(\mathcal{A}))$
(58)

Let $\Psi^{\textrm{crit}}_{k}(\mathcal{A})=H_{k}(\mathcal{A})\Psi_{k+1}$ be
the critical point in $\Psi_{k}$ of this expression. We expand around this
field by by $\Psi_{k}=\Psi^{\textrm{crit}}(\mathcal{A})+W_{k}$. The field
$\psi_{k}(\mathcal{A})$ becomes

$\psi_{k}(\mathcal{A})=\tilde{\psi}_{k+1}(\mathcal{A})+{\cal
W}_{k}(\mathcal{A})$ (59)

where $\tilde{\psi}_{k+1}(\mathcal{A})={\cal
H}_{k}(\mathcal{A})H_{k}(\mathcal{A})\Psi_{k+1}$ will scale to
$\psi_{k+1}(\mathcal{A})$ and ${\cal W}_{k}(\mathcal{A})={\cal
H}_{k}(\mathcal{A})W_{k}$. The quadratic fermion term (58) splits as

$\tilde{\EuFrak{S}}_{k+1}(\mathcal{A},\Psi_{k+1},\tilde{\psi}_{k+1}(\mathcal{A}))+<\bar{W}_{k},\Gamma_{k}(\mathcal{A})^{-1}W_{k}>\\\
$ (60)

Here $\Gamma_{k}(\mathcal{A})$ is a certain invertible operator similar to
(31) and

$\begin{split}&\tilde{\EuFrak{S}}_{k+1}(\mathcal{A},\Psi_{k+1},\psi)\\\
&=b_{k}L^{-1}<\bar{\Psi}_{k+1}-Q_{k+1}(-\mathcal{A})\bar{\psi},\Psi_{k+1}-Q_{k+1}(\mathcal{A})\psi>+<\bar{\psi},\Big{(}\EuFrak{D}_{e_{k}}(\mathcal{A})+\bar{m}_{k}\Big{)}\psi>\\\
\end{split}$ (61)

where $Q_{k+1}(\mathcal{A})=Q(\mathcal{A})Q_{k}(\mathcal{A})$. The term
$E_{k}(\mathcal{A},\tilde{\psi}_{k+1}(\mathcal{A}))$ in (57) becomes

$E_{k}\Big{(}\mathcal{A},\tilde{\psi}_{k+1}(\mathcal{A})+{\cal
W}_{k}(\mathcal{A})\Big{)}=E_{k}\Big{(}\mathcal{A},\tilde{\psi}_{k+1}(\mathcal{A})\Big{)}+[\dots]$
(62)

The fluctuation part $[\dots]$ will be discussed further.

Continuing as in the first step we identify Gaussian integrals by

$\begin{split}\EuFrak{Z}_{k}\ d\mu_{C_{k}}(Z_{k})=&\ \delta({\cal
Q}Z_{k})\delta_{\mathsf{x}}(Z_{k})\exp\Big{(}-\frac{1}{2}\|d{\cal
H}_{k}Z_{k}\|^{2}\Big{)}DZ_{k}\\\
\EuFrak{Z}_{k}(\mathcal{A})d\mu_{\Gamma_{k}(\mathcal{A})}(W_{k})=&\exp\Big{(}-<\bar{W}_{k},\Gamma_{k}(\mathcal{A})^{-1}W_{k}>\Big{)}DW_{k}\\\
\end{split}$ (63)

Then we find

$\begin{split}\tilde{\rho}_{k+1}(A_{k+1},\Psi_{k+1})&=\mathsf{N}_{k}\EuFrak{Z}_{k}\EuFrak{Z}_{k}(\mathcal{A})\
\Xi_{k}\Big{(}\mathcal{A},\psi\Big{)}\exp\Big{(}-\frac{1}{2}\|d\mathcal{A}\|^{2}-\tilde{\EuFrak{S}}_{k+1}(\mathcal{A},\Psi_{k+1},\psi)\\\
&-m_{k}<\bar{\psi},\psi>-\varepsilon_{k}|{\mathbb{T}}^{0}_{N-k}|+E_{k}(\mathcal{A},\psi)\Big{)}\Big{|}_{\mathcal{A}=\tilde{\mathcal{A}}_{k+1},\psi=\tilde{\psi}_{k+1}(\tilde{\mathcal{A}}_{k+1})}\\\
\end{split}$ (64)

where the fluctuation integral is

$\Xi_{k}\Big{(}\mathcal{A},\psi\Big{)}=\int
d\mu_{\Gamma_{k}(\mathcal{A})}(W_{k})d\mu_{C_{k}}(Z_{k})\chi_{k}(Z_{k})\exp\Big{(}E_{k}^{\prime}(\mathcal{A},{\cal
Z}_{k},\psi,{\cal W}_{k}(\mathcal{A}))\Big{)}$ (65)

Here again we have implicitly made a large/small field split to introduce a
characteristic function $\chi_{k}(Z_{k})$ enforcing a bound like $|Z_{k}|\leq
p(e_{k})=(-\log e_{k})^{p}$.

The $E^{\prime}_{k}$ are the terms $[\cdots]$ in (57),(62). They inherit an
expansion $E^{\prime}_{k}=\sum_{X}E^{\prime}_{k}(X)$. However the
$E^{\prime}_{k}(X,\mathcal{A},{\cal Z}_{k},\psi,{\cal W}_{k}(\mathcal{A}))$
depend on the fundamental variables $Z_{k},W_{k}$ outside $X$ since ${\cal
Z}_{k}={\cal H}_{k}Z_{k}$ and ${\cal W}_{k}(\mathcal{A})={\cal
H}_{k}(\mathcal{A})W_{k}$ and the kernels of ${\cal H}_{k},{\cal
H}_{k}(\mathcal{A})$ are only exponentially decaying. When we shift to
ultralocal Gaussian measures with unit covariance it gets worse with ${\cal
Z}_{k}={\cal H}_{k}C_{k}^{\frac{1}{2}}Z_{k}$ and ${\cal
W}_{k}(\mathcal{A})={\cal H}_{k}(\mathcal{A})\Gamma_{k}(\mathcal{A})W_{k}$.
The remedy as before is to break ${\cal Z}_{k},{\cal W}_{k}(\mathcal{A})$ and
hence $E^{\prime}_{k}(X)$ into local pieces using random walk expansions, now
for each the operators of ${\cal H}_{k},C_{k}^{\frac{1}{2}},{\cal
H}_{k}(\mathcal{A}),\Gamma_{k}(\mathcal{A})$. This leads to a polymer
expansion with $Z_{k},W_{k}$ strictly localized. Then one can do a cluster
expansion as before. The result is that the fluctuation integral has the form

$\Xi_{k}(\mathcal{A},\psi)=\exp\Big{(}E_{k}^{\\#}(\mathcal{A},\psi)\Big{)}=\exp\Big{(}\sum_{X}E_{k}^{\\#}(X,\mathcal{A},\psi)\Big{)}$
(66)

with good bounds on $E_{k}^{\\#}(X)$.

We also need a polymer expansion for
$\EuFrak{Z}_{k}(\mathcal{A})=\det(\Gamma_{k}(\mathcal{A})^{-1})$, the Gaussian
normalization factor in (63). With restrictions on the size of $d\mathcal{A}$
one can show

$\EuFrak{Z}_{k}(\mathcal{A})=\EuFrak{Z}_{k}(0)\exp\Big{(}E_{k}^{\det}(\mathcal{A})\Big{)}=\EuFrak{Z}_{k}(0)\exp\Big{(}\sum_{X}E_{k}^{\det}(X,\mathcal{A})\Big{)}$
(67)

with good bounds on $E_{k}^{\det}(X)$.

At this point we have the generalization of the $k=0$ case (46)

$\begin{split}&\tilde{\rho}_{k+1}(A_{k+1},\Psi_{k+1})=\mathsf{N}_{k}\EuFrak{Z}_{k}\EuFrak{Z}_{k}(0)\
\exp\Big{(}-\frac{1}{2}\|d\mathcal{A}\|^{2}-\tilde{\EuFrak{S}}_{k+1}(\mathcal{A},\Psi_{k+1},\psi)\\\
&-m_{k}<\bar{\psi},\psi>-\varepsilon_{k}|{\mathbb{T}}^{0}_{N-k}|+E_{k}(\mathcal{A},\psi)+E^{\\#}_{k}(\mathcal{A},\psi)+E_{k}^{\det}(\mathcal{A})\Big{)}\Big{|}_{\mathcal{A}=\tilde{\mathcal{A}}_{k+1},\psi=\tilde{\psi}_{k+1}(\tilde{\mathcal{A}}_{k+1})}\\\
\end{split}$ (68)

Before scaling we make some further adjustments. Our goal is to show that
things do not grow too rapidly as we iterate the procedure. There is a
potential problem with the polymer functions $E_{k}(X)$ which tend to grow
like $L^{3}$ in each step as noted at the end of section 4. However this only
occurs for terms with a few fields, since scaling also supplies a factor
$L^{-\frac{1}{2}}$ for gauge fields, and $L^{-1}$ for fermion fields.
Furthermore we gain powers of $L^{-\frac{1}{2}}$ from
$e_{k}=L^{-\frac{1}{2}}e_{k+1}$. Taking account the symmetries of the
$E_{k}(X)$ the only relevant terms are the energy density (constants) and the
fermion mass ( $\int\bar{\psi}\psi$ ). Boson mass terms
$\int|\mathcal{A}|^{2}$ are forbidden by gauge invariance. Other terms shrink
and are said to be irrelevant; for example four fermion terms shrink like
$L^{-1}$. There are no marginal terms.

Accordingly we can write

$E_{k}(X,\mathcal{A},\psi)=-\varepsilon_{k}^{*}(X)\textrm{Vol}(X)-\int\bar{\psi}[m_{k}^{*}(X)]\psi+({\cal
R}E_{k})(X,\mathcal{A},\psi)$ (69)

and with a suitable choice of $\varepsilon_{k}^{*}(X),m_{k}^{*}(X)$ the
remainder $({\cal R}E_{k})(X)$ shrinks under scaling. Summing over $X$ gives

$\sum_{X}E_{k}(X,\mathcal{A},\psi)=-\varepsilon^{*}_{k}|{\mathbb{T}}^{0}_{N-k}|-m^{*}_{k}\int\bar{\psi}\psi+\sum_{X}({\cal
R}E_{k})(X,\mathcal{A},\psi)$ (70)

where $\varepsilon^{*}_{k}=\sum_{X\supset\square}\varepsilon_{k}^{*}(X)$ and
$m^{*}_{k}=\sum_{X\supset\square}m^{*}_{k}(X)$ for any $M$-cube $\square$.

We make these adjustments in (68). After scaling we find that up to a constant
$\rho_{k+1}$ has the claimed form:

$\begin{split}&\rho_{k+1}(A_{k+1},\Psi_{k+1})=\exp\Big{(}-\frac{1}{2}\|d\mathcal{A}\|^{2}-\EuFrak{S}_{k+1}(\mathcal{A},\Psi_{k+1},\psi)\\\
&\hskip 28.45274pt\hskip 28.45274pt-
m_{k+1}<\bar{\psi},\psi>-\varepsilon_{k+1}|{\mathbb{T}}^{0}_{N-{k+1}}|-E_{k+1}(\mathcal{A},\psi)\Big{)}\Big{|}_{\mathcal{A}=\mathcal{A}_{k+1},\psi=\psi_{k+1}(\mathcal{A}_{k+1})}\\\
\end{split}$ (71)

The coupling constant in $\EuFrak{S}_{k+1}$ is now
$e_{k+1}=L^{\frac{1}{2}}e_{k}$ and we have the new parameters:

$\begin{split}\varepsilon_{k+1}=&L^{3}(\varepsilon_{k}+\varepsilon^{*}_{k})\\\
m_{k+1}=&L(m_{k}+m^{*}_{k})\\\ E_{k+1}=&{\cal L}({\cal
R}E_{k}+E_{k}^{\\#}+E_{k}^{\det})\\\ \end{split}$ (72)

Here $\varepsilon^{*}_{k}=\varepsilon^{*}_{k}(E_{k})$,
$m^{*}_{k}=m^{*}_{k}(E_{k})$, $E_{k}^{\\#}=E_{k}^{\\#}(m_{k},E_{k})$, and
${\cal L}$ is the reblocking and scaling operation (51). The possibility of
dangerous growth is now isolated in $m_{k},\varepsilon_{k}$.

We need to study the flow of these equations, a problem in discrete dynamical
systems. We start with $E_{0}=0$ and the renormalization problem is to choose
initial values $\varepsilon_{0}=\varepsilon^{N}_{0}$ $m_{0}=m^{N}_{0}$ such
that $\varepsilon_{k}$ and $m_{k}$ take specified final values and $E_{k}$
stays small. We stop after $K<N$ steps with $N-K={\cal O}(1)$, and arbitrarily
ask for final values $\varepsilon_{K}=0,m_{K}=0$. (This does not give massless
fermions since the background mass is $\bar{m}_{K}=L^{-(N-K)}\bar{m}>0.)$ The
existence of a unique solution can be formulated as a fixed point problem in a
Banach space of sequences $\\{\varepsilon_{k},m_{k},E_{k}\\}^{K}_{k=0}$
satisfying the boundary conditions. The fixed point exists if $e$ is
sufficiently small. This leads to a result that looks something like the
following [21].

######  Theorem 2.

Let $m>0$, and $e$ be sufficiently small. Then for any $N$ and $N-K={\cal
O}(1)$ there exists a unique sequence
$\\{\varepsilon_{k},m_{k},E_{k}\\}^{K}_{k=0}$ satisying (72) with the initial
condition $E_{0}=0$ and the final condition $\varepsilon_{K}=0,m_{K}=0$. With
$e_{k}=e^{-(N-k))/2}e$ and $h_{k}=e_{k}^{-\frac{1}{4}}$, the solution
satisfies

$|\varepsilon_{k}|\leq e_{k}^{\frac{1}{4}}\hskip 28.45274pt|m_{k}|\leq
e_{k}^{\frac{3}{4}}\hskip 28.45274pt\|E_{k}(X)\|_{h_{k}}\leq
e_{k}^{\frac{1}{4}}e^{-\kappa d_{M}(X)}$ (73)

With this sequence the density $\rho_{k}$ has the form (53) for all $0\leq
k\leq K$.

The fractional exponents here are an artifact of the proof and have no
fundamental significance.

Theorem 2 gives good bounds for the partition function $\mathsf{Z}_{N}(e)$ if
one takes it to be
$\int\delta_{\mathsf{x}}(A_{K})\rho_{K}(A_{K},\Psi_{K})DA_{k}D\Psi_{K}$. But
this density $\rho_{K}$ is only the global small field contribution. For the
true partition function we have to include the contribution from terms which
have some large field regions.

The history of small field regions is given by a non-increasing sequence
$\mathbf{\Omega}=(\Omega_{1},\cdots,\Omega_{k})$ with $\Omega_{j}$ a union of
$L^{-(k-j)}M$ cubes in ${\mathbb{T}}^{-k}_{N-k}$. It is created as follows. In
passing from $k$ to $k+1$ a new small field region $\Omega_{k+1}$ is created.
It is a union of $LM$ cubes (or larger) in ${\mathbb{T}}^{-k}_{N-k}$, and is
defined with bounds depending on $p(e_{k+1})=(-\log e_{k+1})^{p}$ to keep pace
with the running coupling constant $e_{k+1}$. The $p(e_{k})$ are decreasing in
$k$ so the small field constraints are becoming tighter as we proceed. The new
small field regions are introduced only where needed, namely inside the old
small field region: $\Omega_{k+1}\subset\Omega_{k}$. At the end of step $k+1$
the $\Omega_{k+1}$ is scaled down to a union of $M$ cubes in
${\mathbb{T}}^{-k-1}_{N-k-1}$.

The complete expression for $\rho_{k}$ involves a sum over histories
$\mathbf{\Omega}$:

$\rho_{k}(A_{k},\Psi_{k})=\sum_{\mathbf{\Omega}}\rho_{k,\mathbf{\Omega}}(A_{k},\Psi_{k})$
(74)

In the small field region $\Omega_{k}$ for $\rho_{k,\mathbf{\Omega}}$ the
action density is identical with the global small field case we have discussed
at length. The contribution of the large field region $\Omega_{k}^{c}$ is much
smaller is estimated along the lines discussed in section 3.

There are difficulties with carrying out this program. The density
$\rho_{k,\mathbf{\Omega}}(A_{k},\Psi_{k})$ is naturally expressed in terms of
smeared fields $\mathcal{A}_{k,\mathbf{\Omega}},\psi_{k,\mathbf{\Omega}}$
which depend on the history $\mathbf{\Omega}$. These depend on the fundamental
fields $A_{k},\Psi_{k}$ through operators ${\cal H}_{k,\mathbf{\Omega}}$ and
${\cal H}_{k,\mathbf{\Omega}}(\mathcal{A})$. Estimates and local expansions
for these operators are needed and the tool is again random walk expansions.
But because the history $\mathbf{\Omega}$ has a multiscale structure these are
multi-scale random walk expansions. These operators also embody coupling
between large and small field regions, which is abundant and needs to be
controlled. Another area of difficulty is the presence of localized
characteristic functions which can be a source of non-locality when subjected
to expansions around critical points. Finally at each step there is not a
single large/small split as our notation would indicate, but several such.
Things get rather complicated.

These difficulties can be overcome, and when we divide by the free partition
function we get the stability bound of Theorem 1, which we recall is the bound
uniform in $N$

$K_{-}\leq\mathsf{Z}_{N}(e)/\mathsf{Z}_{N}(0)\leq K_{+}$ (75)

## References

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# FourierKAN-GCF: Fourier Kolmogorov-Arnold Network - An Effective and
Efficient Feature Transformation for Graph Collaborative Filtering

Jinfeng Xu  The University of Hong KongHong KongChina<EMAIL_ADDRESS>,
Zheyu Chen  The Hong Kong Polytechnic UniversityHong KongChina
<EMAIL_ADDRESS>,  Jinze Li  The University of Hong KongHong
KongChina<EMAIL_ADDRESS>,  Shuo Yang  The University of Hong
KongHong KongChina<EMAIL_ADDRESS>,  Wei Wang  Shenzhen MSU-BIT
UniversityShenzhenChina<EMAIL_ADDRESS>,  Xiping Hu  Beijing Institute of
TechnologyBeijingChina<EMAIL_ADDRESS>and  Edith C.-H. Ngai  The University
of Hong KongHong KongChina<EMAIL_ADDRESS>
(2024)

###### Abstract.

Graph Collaborative Filtering (GCF) has achieved state-of-the-art performance
for recommendation tasks. However, most GCF structures simplify the feature
transformation and nonlinear operation during message passing in the graph
convolution network (GCN). We revisit these two components and discover that a
part of feature transformation and nonlinear operation during message passing
in GCN can improve the representation of GCF, but increase the difficulty of
training.

In this work, we propose a simple and effective graph-based recommendation
model called FourierKAN-GCF. Specifically, it utilizes a novel Fourier
Kolmogorov-Arnold Network (KAN) to replace the multilayer perceptron (MLP) as
a part of the feature transformation during message passing in GCN, which
improves the representation power of GCF and is easy to train. We further
employ message dropout and node dropout strategies to improve the
representation power and robustness of the model. Extensive experiments on two
public datasets demonstrate the superiority of FourierKAN-GCF over most state-
of-the-art methods. The implementation code is available at
https://github.com/Jinfeng-Xu/FKAN-GCF.

Recommendation, Collaborative Filtering, Graph Neural Network, Kolmogorov-
Arnold Network, Fourier Coefficients

††copyright: acmlicensed††journalyear: 2024††doi: XXXXXXX.XXXXXXX††conference:
Make sure to enter the correct conference title from your rights confirmation
emai; June 03–05, 2018; Woodstock, NY††isbn: 978-1-4503-XXXX-X/18/06††ccs:
Information systems Recommender systems;

##  1\. Introduction

Recommender system has been widely used to alleviate information overload on
the web (Xu et al., 2024). The aim of recommender system is to predict whether
a user will interact with an unobserved item. To this end, collaborative
filtering (CF) addresses it by learning user preferences from similar users.
Many works (He et al., 2020; Wang et al., 2019; Jiang et al., 2023) harnessing
graph-based recommendation to capture latent user preferences from user-item
interactions have gained notable successes. The main reason is that the
interactive data between users and items are naturally graph-structured. For
example, NGCF (Wang et al., 2019) utilizes whole standard GCN in recommender
systems, which retains the feature transformation and nonlinear operation.
However, LightGCN (He et al., 2020) states that both feature transformation
and nonlinear operation are unnecessary for feature extraction in
recommendation field and further proposes a light GCN. Most of the subsequent
graph-based models (Zhou et al., 2023a; Wei et al., 2019; Yu et al., 2023) are
based on LightGCN for further exploration.

However, there are many unfair analyses of NGCF in LightGCN: a) There are two
different feature transformations in NGCF, but LightGCN removed them without
performing separate fine-grained ablation experiments. b) LightGCN also
removed the interaction information representation part of NGCF without giving
reasons. Therefore, we question: ‘Is feature transformation during message
passing in GCN really unnecessary in recommendation?’

We provide an empirical analysis for NGCF and LightGCN in Section 2. Then, we
state that feature transformation in NGCF can enhance the interaction
representation and boost the performance of GCN, but the training process
becomes more complex. In this work, we introduce a simple yet powerful graph-
based recommendation model called FourierKAN-GCF. Particularly, FourierKAN-GCF
incorporates a unique Fourier Kolmogorov-Arnold Network (KAN) in place of the
traditional multilayer perceptron (MLP) within the feature transformation
during message passing in GCNs. This substitution enhances the
representational capabilities and training ease of the GCF. Additionally, we
implement various dropout techniques to further boost the model’s
representational strength and robustness. Extensive experiments on public
datasets demonstrate the superiority of our FourierKAN-GCF over state-of-the-
art methods.

##  2\. Preliminary

###  2.1. NGCF Brief

NGCF retains feature transformation and nonlinear operation during the message
passing in GCN, formally:

(1) $\displaystyle\mathbf{e}_{u}^{(l+1)}$
$\displaystyle=\sigma(\mathbf{W}_{1}\mathbf{e}_{u}^{(l)}+\sum_{i\in\mathcal{N}_{u}}\frac{\mathbf{W}_{1}\mathbf{e}_{i}^{(l)}+\mathbf{W}_{2}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})}{\sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}}),$
$\displaystyle\mathbf{e}_{i}^{(l+1)}$
$\displaystyle=\sigma(\mathbf{W}_{1}\mathbf{e}_{i}^{(l)}+\sum_{u\in\mathcal{N}_{i}}\frac{\mathbf{W}_{1}\mathbf{e}_{u}^{(l)}+\mathbf{W}_{2}(\mathbf{e}_{u}^{(l)}\odot\mathbf{e}_{i}^{(l)})}{\sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}}),$

where $\mathbf{e}_{u}^{(l)}$ and $\mathbf{e}_{i}^{(l)}$ represent the
embedding of user $u$ and item $i$ after $l$ layers message propagation,
respectively. $\mathcal{N}_{u}$ and $\mathcal{N}_{i}$ denote the interacted
item set with $u$ and interacted user set with $i$, respectively.
$\mathbf{W}_{1}$ and $\mathbf{W}_{2}$ are two trainable weight matrices to
perform feature transformation in each layer. $\sigma$ is the nonlinear
activation function. It is worth noting that
$\mathbf{W}_{1}\mathbf{e}_{i}^{(l)}$ and $\mathbf{W}_{1}\mathbf{e}_{u}^{(l)}$
can be regarded as the aggregated representation from neighbors, while
$\mathbf{W}_{2}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})$ and
$\mathbf{W}_{2}(\mathbf{e}_{u}^{(l)}\odot\mathbf{e}_{i}^{(l)})$ can be
regarded as the aggregated representation from interaction information. The
final embeddings are calculated as:

(2)
$\mathbf{\hat{e}}_{u}=\mathbf{e}_{u}^{(1)}||\mathbf{e}_{u}^{(2)}||...||\mathbf{e}_{u}^{(L)},\quad\mathbf{\hat{e}}_{i}=\mathbf{e}_{i}^{(1)}||\mathbf{e}_{i}^{(2)}||...||\mathbf{e}_{0}^{(L)},$

where $||$ is the concatenation operation. NGCF only concatenates layer-$1$ to
layer-$L$ and ignores ego layer-$0$, since ego layer-$0$ has already been
considered in the first term $\mathbf{W}_{1}\mathbf{e}_{u}^{(l)}$ and
$\mathbf{W}_{1}\mathbf{e}_{i}^{(l)}$ of message passing and propagation.

###  2.2. LightGCN Brief

LightGCN provides an analysis of the feature transformation and nonlinear
operation in NGCF. It mainly provides four observations:
1\. Removing the entire feature transformation including both $\mathbf{W}_{1}$
and $\mathbf{W}_{2}$ leads to consistent improvements over NGCF.
2\. Removing only nonlinear operation $\sigma$ will lead to a small
deterioration of performance.
3\. Removing both entire feature transformation and nonlinear operation can
improve performance significantly.
4\. The deterioration of NGCF stems from the training difficulty rather than
over-fitting.
To this end, LightGCN removes feature transformation and nonlinear operation
during the message passing in GCN. Formally, LightGCN leverages the user-item
graph to propagate embeddings as:

(3)
$\mathbf{e}_{u}^{(l+1)}=\sum_{i\in\mathcal{N}_{u}}\frac{\mathbf{e}_{i}^{(l)}}{\sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}},\quad\mathbf{e}_{i}^{(l+1)}=\sum_{u\in\mathcal{N}_{i}}\frac{\mathbf{e}_{u}^{(l)}}{\sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}},$

where $\mathbf{e}_{u}^{(l)}$ and $\mathbf{e}_{i}^{(l)}$ represent the
embedding of user $u$ and item $i$ after $l$ layers message propagation,
respectively. $\mathcal{N}_{u}$ and $\mathcal{N}_{i}$ denote the interacted
item set with $u$ and interacted user set with $i$, respectively. LightGCN
only retains the aggregation of neighbors’ representation but removes the
aggregation of interaction information representation. The final embeddings
are calculated as:

(4)
$\mathbf{\hat{e}}_{u}=\sum_{l=0}^{L}\frac{\mathbf{e}_{u}^{(l)}}{L+1},\quad\mathbf{\hat{e}}_{i}=\sum_{l=0}^{L}\frac{\mathbf{e}_{i}^{(l)}}{L+1},$

where $L$ is the total layer number. LightGCN considers the ego layer-$0$
cause it removes the first term $\mathbf{W}_{1}\mathbf{e}_{u}^{(l)}$ and
$\mathbf{W}_{1}\mathbf{e}_{i}^{(l)}$ in NGCF.

We argue that observation 4 is the main reason why LightGCN performs better
than NGCF in most cases. Observations 1-3 were not sufficient to verify that
the entire feature transformation and nonlinear operation do not contribute to
feature extraction. Besides, we argue that LightGCN verified the feature
transformation not contributing to learning better features by comparing NGCF
with removed both $\mathbf{W}_{1}$ and the whole interaction representation
aggregation part
$\mathbf{W}_{2}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})$, which is not
a fair comparison. We point out that $\mathbf{e}_{u}^{(l)}$ and
$\mathbf{e}_{i}^{(l)}$ naturally contain information about user preferences
and item properties that can be adequately described through multiple feature
dimensions. Therefore, $\mathbf{W}_{1}$ is an unnecessary feature
transformation part. However, we point out that the interaction representation
aggregation part
$\mathbf{W}_{2}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})$ contains
valuable interaction information, which can not be easily extracted by the
heuristic rule. In this case, $\mathbf{W}_{2}$ is a necessary feature
transformation part that contributes to feature extraction. In the next
subsection, we provide empirical evidence for these arguments.

Table 1. Performance comparison of LightGCN, NGCF, and four variants of NGCF
in terms of Recall@20 and NDCG@20.

Dataset | Metrics | LightGCN | NGCF | NGCF-f1 | NGCF-f2 | NGCF-i | NGCF-n
---|---|---|---|---|---|---|---
MOOC | Recall@20 | 0.3307 | 0.3361 | 0.3377$\uparrow$ | 0.3357$\downarrow$ | 0.3301$\downarrow$ | 0.3343$\downarrow$
NDCG@20 | 0.1811 | 0.1894 | 0.1926$\uparrow$ | 0.1897$\uparrow$ | 0.1824$\downarrow$ | 0.1878$\downarrow$
Games | Recall@20 | 0.0447 | 0.0379 | 0.0414$\uparrow$ | 0.0388$\uparrow$ | 0.0362$\downarrow$ | 0.0373$\downarrow$
NDCG@20 | 0.0227 | 0.0196 | 0.0209$\uparrow$ | 0.0192$\downarrow$ | 0.0181$\downarrow$ | 0.0196$-$

###  2.3. Re-Analysis for NGCF

We implement four simplified variants of NGCF:

  * •

NGCF-f1, which removes feature transformation matrix $\mathbf{W}_{1}$.

  * •

NGCF-f2, which removes feature transformation matrix $\mathbf{W}_{2}$.

  * •

NGCF-i, which removes the whole interaction representation aggregation part
$\mathbf{W}_{2}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})$.

  * •

NGCF-n, which removes the nonlinear operation $\sigma$.

We keep all hyper-parameter settings the same as the optimal settings of NGCF.
As Table 1 shows, we conclude the findings that:
1\. For both MOOC and Games datasets, removing the feature transformation
matrix $\mathbf{W}_{1}$ will lead to observed improvements. Therefore, the
feature transformation matrix $\mathbf{W}_{1}$ is unnecessary.
2\. For MOOC dataset, removing the feature transformation matrix
$\mathbf{W}_{2}$ leads to a slight deterioration. However, for Games dataset,
it will lead to a small improvement. We own this to that user-item interaction
in MOOC dataset has higher density than in Games dataset.
3\. Removing the whole interaction representation aggregation part
$\mathbf{W}_{2}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})$ will lead to a
performance degradation on all datasets. It shows that this part can bring
valuable interaction information.
4\. Nonlinear operation only brings a small positive effect.
5\. NGCF outperforms LightGCN on the MOOC dataset verifying that NGCF has
better representation power than LightGCN, but the training is more difficult.
Then, we can draw some conclusions. First, the feature transformation matrix
$\mathbf{W}_{1}$ is unnecessary for NGCF. Besides, the feature transformation
matrix $\mathbf{W}_{2}$ and the whole interaction representation aggregation
part $\mathbf{W}_{2}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})$ is
beneficial for feature extraction. Last, nonlinear operation occupies only a
minor influence.

##  3\. Methodology

In this section, we detail the overall architecture of our FourierKAN-GCF,
NGCF, and LightGCN in Figure 1. Our Fourier Kolmogorov-Arnold Network provides
a powerful interaction representation aggregation part in NGCF to achieve a
significant performance improvement and make it easy to train.

Figure 1. Overview of FourierKAN-GCF, NGCF, and LightGCN

###  3.1. Kolmogorov-Arnold Network (KAN)

Kolmogorov-Arnold Network (Liu et al., 2024) is a promising alternative to
Multi-Layer Perceptron (MLP). MLP is inspired by the universal approximation
theorem (Cybenko, 1989). KAN focuses on the Kolmogorov-Arnold representation
theorem (Kolmogorov, 1961). KAN can be formed as:

(5)
$\operatorname{KAN}=f(\mathbf{x})=\sum_{q=1}^{2n+1}\Phi_{q}(\sum_{p=1}^{n}\phi_{q,p}(\mathbf{x}_{p})),$

where $\phi_{q,p}$ are univariate functions that map each input variable
$\mathbf{x}_{p}$ such $\phi_{q,p}:[0,1]\rightarrow\mathbb{R}$ and
$\phi_{q}:\mathbb{R}\rightarrow\mathbb{R}$. $\phi_{q,p}(\mathbf{x}_{p})$ is
trainable activation function. In the KolmogovArnold theorem, the inner
functions form a KAN layer with $n_{in}$ = $n$ and $n_{out}$ = $2n+1$, and the
outer functions form a KAN layer with $n_{in}$ = $2n+1$ and $n_{out}$ = $n$.
So the Kolmogorov-Arnold representations in this formula are simply
compositions of two KAN layers. A useful trick is that it includes a basis
function $b(\mathbf{x})$ such that the activation function $\phi(\mathbf{x})$
is the sum of the basis function $b(\mathbf{x})$ and the spline function:

(6)
$\phi(\mathbf{x})=\mathbf{w}(b(\mathbf{x})+\operatorname{spline}(\mathbf{x})).$

$\operatorname{spline}(\mathbf{x})$ is parameterized as a linear combination
of B-splines:

(7)
$b(x)=\operatorname{silu}(\mathbf{x})=\frac{x}{1+e^{-\mathbf{x}}},\quad\operatorname{spline}(\mathbf{x})=\sum_{i}c_{i}B_{i}(\mathbf{x}),$

where $c_{i}$ is trainable.

###  3.2. Fourier KAN

However, KAN is more difficult to train than MLP due to the spline function,
which does not satisfy our motivation to provide a more effective and
efficient method to replace
$\mathbf{W}_{2}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})$ part. The
essence of KAN is to form an arbitrary function by superposition of multiple
nonlinear functions. Our goal can be converted into finding the split from a
complex function into multiple relatively simple nonlinear functions.
Naturally, the Fourier Coefficients (Nussbaumer and Nussbaumer, 1982) is an
interesting choice. Therefore, we propose the following equation:

(8)
$\phi_{F}(\mathbf{x})=\sum_{i=1}^{d}\sum_{k=1}^{g}\left(\cos\left(k\mathbf{x}_{i}\right)\cdot
a_{ik}+\sin\left(k\mathbf{x}_{i}\right)\cdot b_{ik}\right),$

where $d$ is the dimension number of features. Fourier coefficients $a_{ik}$
and $b_{ik}$ are trainable. Hyper-parameter $g$ is the gridsize, which plays a
critical role in controlling the number of terms (frequencies) used in the
Fourier series expansion. Specifically, $g$ determines how many different sine
and cosine terms are included in the Fourier Coefficients corresponding to
each input dimension. The Fourier Coefficients has a significant advantage in
computational efficiency and solves the training difficulty caused by the
spline function.

###  3.3. FourierKAN-GCF

The message passing in our FourierKAN-GCF is defined as:

(9) $\displaystyle\mathbf{e}_{u}^{(l+1)}$
$\displaystyle=\sigma(\mathbf{e}_{u}^{(l)}+\sum_{i\in\mathcal{N}_{u}}\frac{\mathbf{e}_{i}^{(l)}+\phi_{F}(\mathbf{e}_{i}^{(l)}\odot\mathbf{e}_{u}^{(l)})}{\sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}}),$
$\displaystyle\mathbf{e}_{i}^{(l+1)}$
$\displaystyle=\sigma(\mathbf{e}_{i}^{(l)}+\sum_{u\in\mathcal{N}_{i}}\frac{\mathbf{e}_{u}^{(l)}+\phi_{F}(\mathbf{e}_{u}^{(l)}\odot\mathbf{e}_{i}^{(l)})}{\sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}}),$

where $\phi_{F}(\cdot)$ is simplified single layer Fourier KAN function. We
remove the unnecessary transform matrix $\mathbf{W}_{1}$ in NGCF and utilize
our Fourier KAN function to replace the transform matrix $\mathbf{W}_{2}$. The
final user embedding and item embedding are calculated as:

(10)
$\mathbf{\hat{e}}_{u}=\mathbf{e}_{u}^{(1)}||\mathbf{e}_{u}^{(2)}||...||\mathbf{e}_{u}^{(L)},\quad\mathbf{\hat{e}}_{i}=\mathbf{e}_{i}^{(1)}||\mathbf{e}_{i}^{(2)}||...||\mathbf{e}_{i}^{(L)},$

where $||$ is the concatenation operation.

###  3.4. Dropout Strategies

To further prevent the over-fitting problem for FourierKAN-GCF, we adopt
message dropout and node dropout strategies. For the message dropout, we
randomly change some message passing being propagated in Eq 9 by zero, with a
probability $1-p_{m}$, where $p_{m}$ is the message dropout ratio. For the
node dropout, we randomly drop $1-p_{n}$ nodes of the matrix, where $p_{n}$ is
the node dropout ratio.

Table 2. Performance comparison of baselines and FourierKAN-GCF in terms of
R@K and N@K. The superscript ∗ indicates the improvement is statistically
significant where the p-value is less than 0.05.

Datasets |  1MOOC |  2Games
---|---|---
Metrics | R@10 | R@20 | R@50 | N@10 | N@20 | N@50 | R@10 | R@20 | R@50 | N@10 | N@20 | N@50
BPR-MF | 0.2401 | 0.3353 | 0.4813 | 0.1571 | 0.1898 | 0.2261 | 0.0210 | 0.0369 | 0.0699 | 0.0135 | 0.0183 | 0.0265
BUIR | 0.2354 | 0.3196 | 0.4801 | 0.1589 | 0.1835 | 0.2224 | 0.0227 | 0.0384 | 0.0749 | 0.0143 | 0.0192 | 0.0282
NGCF | 0.2453 | 0.3361 | 0.4799 | 0.1722 | 0.1894 | 0.2349 | 0.0231 | 0.0379 | 0.0782 | 0.0142 | 0.0196 | 0.0274
LR-GCCF | 0.2466 | 0.3336 | 0.4809 | 0.1678 | 0.1938 | 0.2294 | 0.0253 | 0.0440 | 0.0815 | 0.0171 | 0.0224 | 0.0317
LightGCN | 0.2402 | 0.3307 | 0.4773 | 0.1581 | 0.1811 | 0.2217 | 0.0271 | 0.0447 | 0.0844 | 0.0174 | 0.0227 | 0.0326
UltraGCN | 0.2372 | 0.3194 | 0.4701 | 0.1737 | 0.1962 | 0.2307 | 0.0277 | 0.0459 | 0.0844 | 0.0173 | 0.0230 | 0.0331
IMP-GCN | 0.2170 | 0.2788 | 0.4183 | 0.1531 | 0.1717 | 0.2057 | 0.0273 | 0.0461 | 0.0839 | 0.0171 | 0.0232 | 0.0323
w/o MD | 0.2518 | 0.3523 | 0.4912 | 0.1814 | 0.2116 | 0.2449 | 0.0267 | 0.0452 | 0.0825 | 0.0165 | 0.0221 | 0.0314
w/o ND | 0.2603 | 0.3527 | 0.4839 | 0.1800 | 0.2071 | 0.2439 | 0.0257 | 0.0452 | 0.0809 | 0.0161 | 0.0219 | 0.0309
KAN-GCF | 0.2536 | 0.3417 | 0.4984 | 0.1758 | 0.2024 | 0.2396 | 0.0272 | 0.0451 | 0.0837 | 0.0172 | 0.0229 | 0.0325
FourierKAN-GCF | 0.2595 | 0.3564 | 0.5065 | 0.1866 | 0.2147 | 0.2462 | 0.0287 | 0.0473 | 0.0856 | 0.0176 | 0.0252 | 0.0342

1Dataset can be accessed at http://moocdata.cn. 2Dataset can be accessed at
http://jmcauley.ucsd.edu/data/amazon/links.html.

###  3.5. Model Training

To learn model parameters, we adopt the Bayesian Personalized Ranking (BPR)
(Rendle et al., 2012) loss function as our optimization criterion. The core
objective of BPR is to enhance the divergence in the predictive preference
between positive and negative items within each user-item triplet
$(u,i_{p},i_{n})\in\mathcal{D}$, where $\mathcal{D}$ signifies the collection
of training data, the term positive item $p$ pertains to an item with which
the user $u$ has interacted, whereas the negative item $n$ is selected
randomly from the pool of items without interaction with user $u$.

(11)
$\mathcal{L}=\sum_{(u,i_{p},i_{n})\in\mathcal{D}}-\ln\sigma(\mathbf{\hat{e}}_{u}^{T}\mathbf{\hat{e}}_{i_{p}}-\mathbf{\hat{e}}_{u}^{T}\mathbf{\hat{e}}_{i_{n}})+\lambda\|\mathbf{\Theta}\|^{2},$

where $\lambda$ controls the $L_{2}$ regularization strength, $\sigma$ is the
Sigmoid function, and $\mathbf{\Theta}$ denotes model parameters.

Table 3. Statistics of experimental datasets. Dataset | # Users | # Items | # Interaction | Sparsity
---|---|---|---|---
MOOC | 82535 | 1302 | 458453 | 99.57%
Games | 50677 | 16897 | 454529 | 99.95%

##  4\. Experiment

###  4.1. Datasets and Evaluation Metrics

We conduct our experiments on two real-world datasets: MOOC and Anazon Video
Games (Games for short). Details can be found in Table 3. For a fair
comparison, we follow previous work (Zhou et al., 2023b) to sort all the
observed user-item interactions of each dataset in chronological order based
on the interaction timestamps. Then, we split each dataset with a ratio of
7:1:2 for training, validation, and testing. Regarding evaluation metrics, we
adopt two well-established metrics: Recall@K (R@K) and NDCG@K (N@K). We report
the average metrics of all users in the test dataset among $K$ = 10, $K$ = 20,
and $K$ = 50.

###  4.2. Baselines and Experimental Settings

To verify the effectiveness of FourierKAN-GCF, we select two traditional
matrix factorization models (BPR-MF (Rendle et al., 2012) and BUIR (Lee et
al., 2021)) and five state-of-the-art GCN-based models (NGCF (Wang et al.,
2019), LR-GCCF (Chen et al., 2020), LightGCN (He et al., 2020), UltraGCN (Mao
et al., 2021), and IMP-GCN (Liu et al., 2021)) for comparison. For a fair
comparison, we implement all baselines under a unified framework (Zhou et al.,
2023b), fix the embedding size of both users and items to 64 for all models,
initialize the embedding parameters with the Xavier initialization (Glorot and
Bengio, 2010), and use Adam (Kingma and Ba, 2014) as the optimizer. Besides,
we carefully tune the hyper-parameters of each baseline following their
published papers to find optimal settings. For our FourierKAN-GCF, we tune the
$\lambda$ of $L_{2}$ regularization term in {0, 1e-2, 1e-1, 1, 10}, and tune
the layer number $L$ from 1 to 4. The gridsize $g$ of Fourier KAN is searched
from {1, 2, 4, 8}. We further grid search the message dropout ratio $p_{m}$
and node dropout ratio $p_{n}$ in {0.0, 0.1, 0.2, 0.3}.

###  4.3. Performance Comparison

The results of our experiments are listed in Table 2. We can identify that the
majority of GCN-based methods outperform traditional matrix factorization
methods, which is a common trend. Our FourierKAN-GCF outperforms all baselines
in two datasets across various metrics. We owe our superiority to FourierKAN,
which is easier to train and has greater representation power than MLP. It is
worth noting that FourierKAN-GCF can adjust the training difficulty by
adjusting the grid size $g$.

###  4.4. Ablation Study

Table 2 also demonstrates the significance of dropout strategies. We use w/o
MD and w/o ND to denote without message dropout and without node dropout,
respectively. This ablation study shows that both message dropout and node
dropout play a distinct role in improving model representation power and model
robustness. We use KAN-GCF to denote a variant that utilizes our standard KAN
function to replace transform matrix $\mathbf{W}_{2}$ in NGCF. It shows that
standard KAN is better than MLP but worse than our Fourier KAN.

###  4.5. Sensitivity Analysis

To analyze the hyper-parameter sensitivity of FourierKAN-GCF, we test the
performance of FourierKAN-GCF on two datasets with different hyper-parameters.
The optimal layer number $L$ is 3 on all datasets, shown in Figure
LABEL:fig:l. Besides, Figure LABEL:fig:g demonstrates that for MOOC and Games
datasets, optimal ($p_{m}$, $p_{n}$) pairs are (0.2, 0.2) and (0.1, 0.1),
respectively. As Figure LABEL:fig:dropout illustrated, for the relatively
dense dataset MOOC, $g$ = 4 is the best grid size. For the relatively sparse
dataset Games, $g$ = 2 is the best grid size. This further demonstrates that
FourierKAN can be easily adapted to various datasets.

##  5\. Conclusion

In this paper, we revisit the processes of feature transformation and
nonlinear operations during message passing in the NGCF. We assert that
feature transformation in NGCF can enhance the interaction representation and
boost the performance of GCN but add complexity in training. To reduce the
training difficulty while benefiting from these representational enhancements,
we introduce a novel feature transformation method called FourierKAN. Inspired
by KAN, FourierKAN employs a Fourier Coefficients instead of the Spline
function in standard KAN, which further reduces the difficulty of training
while increasing the representation power. Furthermore, based on FourierKAN,
we introduce a simple and effective graph-based recommendation model
(FourierKAN-GCF) with extra dropout strategies. Extensive experiments on
public datasets demonstrate the superiority of our model over the state-of-
the-art methods.

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${\bar{F}}_{1\cdots n}^{\dagger}=F^{t_{1}\cdots t_{n}}_{1\cdots n}$ (D.37)

###### Proof.

We show it by induction on $n$. Let us assume (D.36) for the case of $n-1$. We
first show

${\bar{F}}_{1,2\cdots n}^{\dagger}={\cal C}_{12\cdots
n}\left(e_{1}^{11}+e_{1}^{22}R_{1,2\cdots n}\right)^{\dagger}{\cal
C}_{12\cdots n}=F_{n\cdots 2,1}^{t_{1}\cdots t_{n}}\,.$

Making use of the induction assumption we show

$\displaystyle F_{12\cdots n}^{\dagger}$ $\displaystyle=$
$\displaystyle(F_{2\cdots n}F_{1,2\cdots n})^{\dagger}=F_{1,2\cdots
n}^{\dagger}F_{2\cdots n}^{\dagger}$ $\displaystyle=$ $\displaystyle{\cal
C}_{1\cdots n}F_{n\cdots 2,1}^{t_{1}\cdots t_{n}}{\cal C}_{1\cdots n}\,{\cal
C}_{1\cdots n}F_{n\cdots 2}^{t_{2}\cdots t_{n}}{\cal C}_{1\cdots n}$
$\displaystyle=$ $\displaystyle{\cal C}_{1\cdots n}F_{n\cdots 21}^{t_{1}\cdots
t_{n}}{\cal C}_{1\cdots n}\,.$

∎

######  Corollary D.2.

The inverse of total $F$ is given as follows.

$F_{1\cdots n}F_{n\cdots 21}^{t_{1}\cdots
t_{n}}=\widehat{\delta}^{-1}_{1\cdots n}$ (D.38)

Equivalently, we have

$F_{1\cdots n}^{-1}=F_{n\cdots 21}^{t_{1}\cdots
t_{n}}\,{\widehat{\delta}}_{1\cdots n}$ (D.39)

###### Proof.

It follows from (D.35) that ${\bar{F}}^{\dagger}=F_{n\cdots 1}^{t_{1}\cdots
t_{n}}\widehat{\delta}_{12\cdots n}$. From (D.37) we have (D.38). ∎

##  Appendix E Lemmas for Diagonalizing the $C$ operator

######  Lemma E.1.

Let $X^{+}$ be the generator of the quantum group $U_{q}(sl_{2})$ and
$X_{i}^{+}$ the matrix representation of $X^{+}$ acting on the $i$th site. We
have

${\widetilde{\Delta}}_{1\cdots
n}(X^{+})=\left(X_{n}^{+}+e_{n}^{11}{\widetilde{A^{+}}}^{\dagger}_{1\cdots
n-1}(\xi_{n}){\widetilde{\Delta}}_{1\cdots
n-1}(X^{+})+e_{n}^{22}{\widetilde{\Delta}}_{1\cdots
n-1}(X^{+}){\widetilde{D^{+}}}_{1\cdots
n-1}(\xi_{n})\right)\widehat{\delta}^{12\cdots n}_{n}\,.$ (E.1)

######  Lemma E.2.

The diagonalized form of $\Delta^{(n-1)}(X^{+})$ is expressed in terms of
local operators $X_{i}^{+}$ as follows.

${\widetilde{\Delta}}_{1\cdots
n}(X^{+})=\sum_{i=1}^{n}X_{i}^{+}\widehat{\delta}^{1\cdots n}_{i}$ (E.2)

######  Lemma E.3.

Operator $C^{+}$ in the $F$-basis is expressed in terms of operator $D^{+}$
and $X_{i}^{+}$ as follows.

${\widetilde{C^{+}}}_{1\cdots
n}(\lambda)=\sum_{i=1}^{n}({\widetilde{D^{+}}}_{1\cdots i-1,i+1\cdots
n}(\lambda)-q^{-1}{\widetilde{D^{+}}}_{1\cdots
n}(\lambda))\,X_{i}^{+}\widehat{\delta}_{i}^{1\cdots n}\,.$ (E.3)

## Acknowledgment

The authors would like to thank Prof. S. Miyashita for encouragement and keen
interest in this work. One of the authors (C.M.) would like to thank Dr. K.
Shigechi for his introduction to the mathematical physics of integrable
models. This work is partially supported by Grant-in-Aid for Scientific
Research (C) No. 20540365.

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