# Supersymmetric Gauge Theory on the Graph

So Matsuura<EMAIL_ADDRESS>Hiyoshi Departments of Physics, and
Research and Education Center for Natural Sciences, Keio University, 4-1-1
Hiyoshi, Yokohama, Kanagawa 223-8521, Japan Kazutoshi Ohta
<EMAIL_ADDRESS>Institute of Physics, Meiji Gakuin University,
Yokohama, Kanagawa 244-8539, Japan

Abstract

We consider two-dimensional ${\cal N}{=}(2,2)$ supersymmetric gauge theory on
discretized Riemann surfaces. We find that the discretized theory can be
efficiently described by using graph theory, where the bosonic and fermionic
fields are regarded as vectors on a graph and its dual. We first analyze the
Abelian theory and identify its spectrum in terms of graph theory. In
particular, we show that the fermions have zero modes corresponding to the
topology of the graph, which can be understood as kernels of the incidence
matrices of the graph and the dual graph. In the continuous theory, a scalar
curvature appears as an anomaly in the Ward-Takahashi (WT) identity associated
with a $U(1)$ symmetry. We find that the same anomaly arises as the deficit
angle at each vertex on the graph. By using the localization method, we show
that the path integral on the graph reduces to an integral over a set of the
zero modes. The partition function is then ill-defined unless suitable
operators are inserted. We extend the same argument to the non-Abelian theory
and show that the path integral reduces to multiple integrals of Abelian
theories at the localization fixed points.

##  1 Introduction

Gauge theory has been recognized as the most fundamental theory describing the
interaction of elementary particles. However, in recent years, the importance
of gauge theories has been extended and it has become to be recognized as a
powerful tool for approaching the quantum theory of gravity through the
gauge/gravity duality [1, 2, 3]. In particular, the gauge/gravity duality
predicted by the superstring theory provides a clear dictionary between the
gauge and gravity theories with supersymmetry. Since supersymmetric gauge
theories can give mathematically accurate descriptions for some physical
quantities by using strong constraints based on supersymmetry, they have
traditionally been studied in analytic ways. However, the prediction of the
gauge/gravity duality should also be applied to the dynamical quantities. To
approach quantum gravity through the supersymmetric gauge theories, therefore,
we need to have the means to analyze their dynamics in a non-perturbative way.

One of the most effective non-perturbative approaches to gauge theories is
lattice gauge theory, which regularizes gauge field theory on a finite lattice
and defines the continuous theory as the continuum limit of the discretized
theory. Various attempts have been made to construct the lattice gauge
theories with preserving a part of supersymmetries on the square lattice [4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. The
relations between these models were investigated in [23, 24, 25, 26, 27], and
several numerical computations based on these models have been carried out
[28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. In two
dimensions, the theory is super renormalizable, and numerical calculations can
be performed without maintaining supersymmetry with a small number of fine-
tunings [29]. In [43], two-dimensional ${\cal N}=(2,2)$ supersymmetric $SU(2)$
Yang-Mills theory is numerically investigated by using a lattice theory with a
conventional Wilson fermion. For reviews, see [44, 45, 46, 47, 48, 49].

Among these lattice theories, the models constructed by Sugino (Sugino models)
[4, 5, 6, 7, 8] have the gauge group $SU(N)$, while the gauge group of the
other models is inevitably $U(N)$. As a result, the link variables in the
Sugino models are expressed by compact unitary matrices as in the conventional
lattice gauge theories. In the Sugino models, the action is written by an
exact form of scalar supercharges constructed by topological twisting. Then
these scalar supersymmetries are manifestly preserved even if the
translational symmetry is explicitly broken by the discretization of space-
time. The problem of vacuum degeneracy of lattice gauge field has been solved
without using an admissibility condition [50] and the tree level improvement
has been proposed [51].

In [52], the Sugino model defined on the usual square lattice is extended to a
theory on a discrete space-time where the two-dimensional Riemann surface is
divided by polygons. This model (the generalized Sugino model) has been
subjected to rigorous analysis using the method of localization [53] and
numerical calculations [54].

In this paper, we reconstruct and analyze the generalized Sugino model in two-
dimensional supersymmetric gauge theories by using graph theory. (For an
introduction to graph theory, see e.g. [55].) We regard the vertices and edges
of the polygons on the discretized Riemann surface as a graph. We assume that
the graph is a directed graph with an edge orientation. We also assume that
faces are assigned to the vertices of the dual graph. The directed graph can
introduce a “difference” between adjacent vertices, and the difference
operator expressed by a matrix is called an incidence matrix. Using this
incidence matrix, it is possible to construct a field theory on the discrete
space-time represented by the graph, including supersymmetry.

The generalized Sugino model has been analyzed using the method of
localization owing to supersymmetry in [53], but the use of graph theory makes
it possible to discuss clearer. In particular, we can see the structure of
zero modes, which play an important role in the localization. The zero modes
appear as a kernel of the incidence matrix in the context of graph theory, and
thus we can use linear algebra to understand their property. In addition, the
analysis by graph theory is beneficial in understanding the anomaly as well
since the zero modes are important in understanding the anomaly of the theory
even in the discretized theory. For analysis of field theories and quantum
mechanics on the graph including the supersymmetry from other viewpoints, see
[56, 57, 58]. See also [59] for localization in the quiver gauge theory by
using the technology of graph theory.

The organization of this paper is as follows. In Sec. 2, we briefly review the
construction of ${\cal N}{=}(2,2)$ supersymmetric Yang-Mills theory on the
smooth Riemann surface by using differential forms. The formulation by the
differential forms makes the relation to the graph structure clearer later. We
also derive the currents and the Ward-Takahashi(WT) identities corresponding
to the global symmetry of the theory. In Sec. 3, we prepare some basics of
graph theory for the discretization of the Riemann surface. We also introduce
some useful matrices including the incidence matrix and summarize their
properties in graph theory. In Sec. 4, we formulate a supersymmetric Abelian
gauge theory by using graph theory and discuss the properties of the fermion
zero modes. We derive the chiral anomaly on the graph and show that the
fermion zero modes play an important role. We see that the anomaly in the WT
identity, which appears as the scalar curvature in the continuous theory,
appears as the deficit angle in the theory on the graph. In Sec. 5, we perform
the path integral for the Abelian theory by using the localization method. We
see that there exists a residual integral over the zero modes after
integrating out non-zero modes. The integral over the zero modes makes the
partition function itself ill-defined, but we also discuss a remedy for this
problem by inserting operators including bi-linear terms of the fermions. In
Sec. 6, we generalize the localization arguments to non-Abelian theory. By the
saddle point approximation in the localization method, non-zero modes give a
Vandermonde type measure and the path integral reduces to multiple integrals
of the Abelian theory one. For this non-Abelian theory, we will see the zero
modes play an important role as well as the Abelian theory. Sec. 7 is devoted
to conclusion and discussion. In Appendix, we give some concrete examples of
the graph structure and properties of the incidence matrix and Laplacian. We
also give the convention of Weyl-Cartan bases, which are used for non-Abelian
theory.

##  2 Supersymmetric Gauge Theory on the Riemann Surface

###  2.1 Action and currents in supersymmetric gauge theory

We start with the review of the supersymmetric gauge theory on the smooth
Riemann surface $\Sigma_{h}$ (continuous space-time), which has the genus
(handles) $h$. The theory considering in this paper is essentially obtained by
dimensional reduction from four-dimensional ${\cal N}{=}1$ supersymmetric
gauge theory with four supercharges, namely two-dimensional ${\cal N}{=}(2,2)$
supersymmetric gauge theory. In general, however, when one simply constructs
this theory on a curved manifold, the supersymmetry is completely broken. The
point is that a part of supersymmetry can be restored by introducing a
specific $U(1)$ gauge field as a background in accordance with the spin
connection of the background space-time. Supersymmetric theory obtained by
this procedure becomes naturally a topologically twisted theory on the curved
space-time, and a half of the supersymmetry is recovered in general. Among the
ways of the topological twistings, which depend on how to turn on the
background gauge field, we choose the so-called topological A-model through,
which is the theory that we consider in this paper. (See also [60, 59] for
more detailed construction.)

Because of the topological twisting, not only the bosonic fields but also the
fermion fields have integer spins. Therefore it is convenient to express the
fields in this theory by differential forms: The 0-form scalar fields are
$\Phi$, $\bar{\Phi}$ and $\eta$, the 1-form vector fields are $A\equiv
A_{\mu}dx^{\mu}$ and $\lambda\equiv\lambda_{\mu}dx^{\mu}$, and 2-form fields
are $Y\equiv\frac{1}{2}Y_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ and
$\chi\equiv\frac{1}{2}\chi_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$, where $\Phi$,
$\bar{\Phi}$, $A$ and $Y$ are bosons and $\eta$, $\lambda$ and $\chi$ are
fermions (Grassmann valued).

We write $Q$ for one of the supercharges, which transforms the fields as111The
notation of the $Q$ transformation has been slightly changed from the previous
works [52, 53, 54].

$\begin{array}[]{lcl}Q\Phi=0,&&\\\
Q\bar{\Phi}=2\eta,&&Q\eta=\frac{i}{2}[\Phi,\bar{\Phi}],\\\
QA=\lambda,&&Q\lambda=-d_{A}\Phi,\\\ QY=i[\Phi,\chi],&&Q\chi=Y,\end{array}$
(2.1)

where $d_{A}\Phi\equiv d\Phi+i[A,\Phi]$ is a covariant exterior derivative for
the adjoint scalar field. We can see that the square of $Q$ generates the
gauge transformation with a parameter $\Phi$, which is denoted by
$Q^{2}=\delta_{\Phi}$.

Using this supercharge, we can write the action of theory in $Q$-exact form

$S=-\frac{1}{2g^{2}}Q\int_{\Sigma_{h}}\operatorname{Tr}\bigg{\\{}\frac{i}{2}\eta[\Phi,\bar{\Phi}]\omega+d_{A}\bar{\Phi}\wedge{*}\lambda+\chi{*}(Y-2F)\bigg{\\}},$
(2.2)

where $\omega$ is a volume (Kähler) form on $\Sigma_{h}$, $*$ represents the
Hodge star operation which maps from an $n$-form to a $(2-n)$-form, and

$F\equiv dA+iA\wedge A$ (2.3)

is a field strength, which will give a kinetic term of the gauge field after
integrating out the auxiliary filed $Y$.

More concretely, after applying the $Q$-transformation, the bosonic and
fermionic parts of the action are given by

$\displaystyle S_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2g^{2}}\int_{\Sigma_{h}}\operatorname{Tr}\bigg{\\{}\frac{1}{4}[\Phi,\bar{\Phi}]^{2}\omega+d_{A}\bar{\Phi}\wedge{*}d_{A}\Phi-Y{*}(Y-2F)\bigg{\\}},$
(2.4) $\displaystyle S_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{2g^{2}}\int_{\Sigma_{h}}\operatorname{Tr}\bigg{\\{}i\eta[\Phi,\eta]\omega+2\eta
d_{A}{*}\lambda-i\lambda\wedge{*}[\bar{\Phi},\lambda]+i\chi{*}[\Phi,\chi]-2\chi{*}d_{A}\lambda\bigg{\\}},$
(2.5)

respectively.

For later convenience, we define a vector of fermionic fields in the following
order,

$\Psi\equiv\begin{pmatrix}\eta\\\ \chi\\\ \lambda\end{pmatrix}.$ (2.6)

Then the fermionic part of the action (2.5) reduces to

$S_{F}=\frac{1}{2g^{2}}\operatorname{Tr}\int_{\Sigma_{h}}\Psi^{T}\wedge{*}\left(i{\not{\cal
D}}_{A}+{\cal M}_{\Phi}\right)\Psi,$ (2.7)

where $\not{\cal D}_{A}$ and ${\cal M}_{\Phi}$ are the Dirac operator and the
mass matrix depending on $\Phi$ and $\bar{\Phi}$, respectively;

$\not{\cal D}_{A}\equiv\begin{pmatrix}0&0&id_{A}^{\dagger}\\\ 0&0&id_{A}\\\
-id_{A}&-id_{A}^{\dagger}&0\end{pmatrix},\quad{\cal
M}_{\Phi}\equiv\begin{pmatrix}i[\Phi,\cdot]&0&0\\\ 0&i[\bar{\Phi},\cdot]&0\\\
0&0&-i[\Phi,\cdot]\\\ \end{pmatrix}.$ (2.8)

Here

$d_{A}^{\dagger}\equiv-{*}d_{A}{*}$ (2.9)

is the co-differential operator, which maps from an $n$-form to an
$(n-1)$-form on $\Sigma_{h}$ and $[\Phi,\cdot]$ represents an adjoint action
induced by $\Phi$.

Let us now consider yet another supercharge $\tilde{Q}$. Since there are two
preserved supercharges on the curved Riemann surface, we have the supercharge
$\tilde{Q}$ in addition to $Q$. The supersymmetry transformation for the
vector multiplet is given by

$\begin{array}[]{lcl}\tilde{Q}\Phi=0,&&\\\
\tilde{Q}A={*}\lambda,&&\tilde{Q}\lambda={*}d_{A}\Phi,\\\
\tilde{Q}\bar{\Phi}=2{*}\chi,&&\tilde{Q}\chi=\frac{i}{2}[\Phi,\bar{\Phi}]\omega,\\\
\tilde{Q}Y=-i[\Phi,\eta]\omega,&&\tilde{Q}\eta=-{*}Y.\end{array}$ (2.10)

Roughly speaking, $\tilde{Q}$ swaps the role of 0-form $\eta$ and 2-form
$\chi$ against the action of $Q$. We can also see the square of $\tilde{Q}$
becomes the gauge transformation, namely $\tilde{Q}^{2}=\delta_{\Phi}$ and $Q$
and $\tilde{Q}$ is anti-commuting with each other;

$\\{Q,\tilde{Q}\\}=0.$ (2.11)

Using the transformation of $\tilde{Q}$, we can also write the action in the
$\tilde{Q}$-exact form

$S=-\frac{1}{2g^{2}}\tilde{Q}\int_{\Sigma_{h}}\operatorname{Tr}\Bigg{\\{}\frac{i}{2}\chi[\Phi,\bar{\Phi}]+\lambda\wedge
d_{A}\bar{\Phi}-\eta(Y-2F)\Bigg{\\}}.$ (2.12)

So the action is also invariant under $\tilde{Q}$. This obeys from the fact
that the action is written by

$S=\frac{1}{4g^{2}}[Q,\tilde{Q}]\int_{\Sigma_{h}}\operatorname{Tr}\left\\{\bar{\Phi}F+\eta\chi\right\\}.$
(2.13)

Using also the anti-commuting relation between $Q$ and $\tilde{Q}$, we can
find that the action is written in both $Q$\- and $\tilde{Q}$-exact forms.
Note also that the part acting the supercharges in (2.13) is invariant under
swapping (a rotation of) $\eta$ and $\chi$.

###  2.2 Symmetries and relations among conserved currents

We next consider the global symmetries of this theory and the associated
Noether currents. In general, if a theory is invariant under a certain global
transformation, the infinitesimal transformation of the action can be written
as

$\delta_{\xi}S=\xi\,\boldsymbol{s}S=\int_{\Sigma_{h}}\left(d\xi*\tilde{J}_{s}+\xi
dI_{s}\right),$ (2.14)

where $\boldsymbol{s}$ is the generator of this transformation, $\xi$ is a
position-dependent parameter, and $\tilde{J}_{s}$ and $I_{s}$ are both one-
form. Then the corresponding Noether current is defined by

$J_{s}=\tilde{J}_{s}+*I_{s},$ (2.15)

which is conserved at least classically;

$d^{\dagger}J_{s}=0.$ (2.16)

Note that this current is invariant if we add any $s$-invariant total
derivative term to the action.

Using this prescription, the Noether current corresponding to $Q$ and
$\tilde{Q}$ symmetries are constructed as

$J_{Q}=\frac{1}{g^{2}}\operatorname{Tr}\left\\{d_{A}\Phi\eta+{*}d_{A}\Phi{*}\chi-\frac{i}{2}[\Phi,\bar{\Phi}]\lambda-{*}Y{*}\lambda\right\\},$
(2.17)

and

$J_{\tilde{Q}}=\frac{1}{g^{2}}\operatorname{Tr}\left\\{-{*}d_{A}\Phi\eta+d_{A}\Phi{*}\chi-\frac{i}{2}[\Phi,\bar{\Phi}]{*}\lambda+{*}Y\lambda\right\\},$
(2.18)

respectively.

The theory also possesses two global $U(1)$ symmetries, $U(1)_{A}$ and
$U(1)_{V}$. The $U(1)_{A}$ symmetry transforms the fields as

$\delta_{A}A=0,\quad\delta_{A}\Phi=2i\theta_{A}\Phi,\quad\delta_{A}\bar{\Phi}=-2i\theta_{A}\bar{\Phi},\quad\delta\Psi=i\theta_{A}\gamma_{A}\Psi,$
(2.19)

where

$\gamma_{A}\equiv\begin{pmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{pmatrix}\,,$
(2.20)

and the $U(1)_{V}$ symmetry transforms the fields as

$\delta_{V}A=0,\quad\delta_{V}\Phi=0,\quad\delta_{V}\bar{\Phi}=0,\quad\delta_{V}\Psi=i\theta_{V}\gamma_{V}\Psi,$
(2.21)

where

$\gamma_{V}\equiv-i\begin{pmatrix}0&{*}&0\\\ -*&0&0\\\ 0&0&{*}\end{pmatrix}.$
(2.22)

The corresponding Noether currents are given by

$\displaystyle J_{A}$ $\displaystyle=\frac{i}{g^{2}}{\rm
Tr}\left(-\bar{\Phi}d_{A}\Phi+\Phi
d_{A}\bar{\Phi}+\eta\lambda-*\lambda*\chi\right)\,,$ (2.23) $\displaystyle
J_{V}$ $\displaystyle=\frac{1}{g^{2}}{\rm
Tr}\left(\eta*\lambda+\lambda*\chi\right)\,.$ (2.24)

Note that $\gamma_{A}$ and $\gamma_{V}$ satisfy
$\gamma_{A}^{2}=\gamma_{V}^{2}=1$. We will soon see that the $U(1)_{A}$
symmetry is anomalous quantum mechanically.

We point out that there are important relationships among the supercurrents
$J_{Q}$, $J_{\tilde{Q}}$ and the current $J_{V}$ [36];

$\displaystyle QJ_{V}$ $\displaystyle=$ $\displaystyle-J_{\tilde{Q}},$ (2.25)
$\displaystyle\tilde{Q}J_{V}$ $\displaystyle=$ $\displaystyle J_{Q}.$ (2.26)

This means that the conservation law of the $U(1)_{V}$ symmetry guarantees the
conservation law of the $\tilde{Q}$ symmetry;

$d^{\dagger}J_{V}=0\quad\Rightarrow\quad d^{\dagger}J_{\tilde{Q}}=0,$ (2.27)

if the $Q$ symmetry is preserved.

###  2.3 Ward-Takahashi identities and anomaly

We next consider the WT identities. In the path-integral formalism, the WT
identity is derived from the obvious invariance under a change of variables,

$\int dX\,{\cal O}(x_{0})e^{-S[X]}=\int dX^{\prime}\,{\cal
O}^{\prime}(x_{0})e^{-S[X^{\prime}]}\,,$ (2.28)

where $X$ expresses the fields of the theory, $X^{\prime}$ is transformed
fields of $X$,222We assume that the ranges of the integration by $X$ and
$X^{\prime}$ are identical. and ${\cal O}(x_{0})$ is a local operator at a
position $x_{0}$. Note here that this identity can be applied regardless of
whether the classical action is invariant under the transformation or not. So
if we consider a general transformation with the parameter $\xi(x)$ and the
generator $\boldsymbol{s}$, the infinitesimal transformation of the action $S$
and the operator ${\cal O}$ are given by

$\delta S=\int_{\Sigma_{h}}dx\,\xi(x)K_{s}(x)\,,$ (2.29)

and

$\delta{\cal O}(x_{0})=\xi(x_{0})\,\boldsymbol{s}{\cal O}(x_{0})\,,$ (2.30)

respectively. In addition, we assume that the integration measure transforms
as

$dX^{\prime}=dX\left(1+\int_{\Sigma_{h}}dx\,\xi(x){\cal A}_{s}(x)\right)\,.$
(2.31)

Then, from (2.28), we obtain the identity for the vacuum expectation value
(vev),

$\int_{\Sigma_{h}}dx\,\xi(x)\Bigl{\langle}\boldsymbol{s}{\cal
O}(x_{0})\delta(x-x_{0})-{\cal O}(x_{0})\left(K_{s}(x)-{\cal
A}_{s}(x)\right)\Bigr{\rangle}=0\,,$ (2.32)

where the vev is defined by

$\left\langle{\cal O}(x)\right\rangle\equiv{Z}^{-1}\int dX\,{\cal
O}(x)e^{-S[X]}\,,$ (2.33)

with the partition function $Z=\int dX\,e^{-S[X]}$. In particular, if we
assume the parameter $\xi(y)$ takes the form $\xi(y)=\xi\delta(y-x)$ for a
specific coordinate $x$ and constant parameter $\xi$, we obtain the identity
for the vev of local variables;

$\Bigl{\langle}{\cal O}(x_{0})\left(K_{s}(x)-{\cal
A}_{s}(x)\right)\Bigr{\rangle}=\delta(x-x_{0})\Bigl{\langle}\boldsymbol{s}{\cal
O}(x_{0})\Bigr{\rangle}\,.$ (2.34)

Let us return to the supersymmetric gauge theory that we are considering. As
discussed in the previous subsection, the action of the supersymmetric theory
is invariant under the two supersymmetry transformations generated by $Q$ and
$\tilde{Q}$, and the two $U(1)$ transformations $U(1)_{A}$ and $U(1)_{V}$.
Therefore, the transformation of the action (2.2) is given by

$\delta_{s}S=\int_{\Sigma_{h}}dx\,\xi(x)d^{\dagger}J_{s}(x)\omega\,,$ (2.35)

up to total derivative, where $J_{s}(x)$ is the corresponding Noether current.

Although it is sufficient only to consider the action (2.2) of the continuous
theory, we further add a supersymmetry breaking term to lead to the discussion
in the next section;

$S_{\mu}=\int_{\Sigma_{h}}dx\,{\cal L}_{\mu}(x),$ (2.36)

which is typically assumed to be mass terms. For the total action $S+S_{\mu}$,
$K_{s}(x)$ is given by

$K_{s}(x)=d^{\dagger}J_{s}(x)\omega+\boldsymbol{s}{\cal L}_{\mu}(x).$ (2.37)

Since the integration measure is invariant under the supersymmetry
transformation $Q$ and $\tilde{Q}$, the corresponding WT identities are given
by

$\displaystyle\left\langle{\cal
O}(x_{0})d^{\dagger}J_{Q}(x)\omega\right\rangle$
$\displaystyle=-\bigl{\langle}{\cal O}(x_{0})Q{\cal
L}_{\mu}(x)\bigr{\rangle}+\delta^{2}(x-x_{0})\bigl{\langle}Q{\cal
O}(x_{0})\bigr{\rangle}\,,$ (2.38) $\displaystyle\left\langle{\cal
O}(x_{0})d^{\dagger}J_{\tilde{Q}}(x)\omega\right\rangle$
$\displaystyle=-\bigl{\langle}{\cal O}(x_{0})\tilde{Q}{\cal
L}_{\mu}(x)\bigr{\rangle}+\delta^{2}(x-x_{0})\bigl{\langle}\tilde{Q}{\cal
O}(x_{0})\bigr{\rangle}\,,$ (2.39)

respectively.

For the $U(1)_{A}$ symmetry, we have to be more careful since the integration
measure is not invariant under the $U(1)_{A}$ transformation. Using the so-
called Fujikawa’s method, we see

$dX^{\prime}=dX\exp\left(i\frac{{\rm
dim}G}{4\pi}\int_{\Sigma_{h}}\theta_{A}(x)R(x)\omega\right)\,,$ (2.40)

where $R(x)$ is the scalar curvature of $\Sigma_{h}$. This is the case where
${\cal A}_{s}\neq 0$ and is nothing but the $U(1)$ anomaly. Thus we obtain the
WT identity for the $U(1)_{A}$ symmetries

$\left\langle{\cal O}(x_{0})\left(d^{\dagger}J_{A}(x)+\frac{{\rm
dim}G}{4\pi}R(x)\right)\omega\right\rangle\\\ =-\bigl{\langle}{\cal
O}(x_{0})\boldsymbol{s}_{A}{\cal
L}_{\mu}(x)\bigr{\rangle}+\delta^{2}(x-x_{0})\bigl{\langle}\boldsymbol{s}_{A}{\cal
O}(x_{0})\bigr{\rangle}\,,$ (2.41)

where $\boldsymbol{s}_{A}$ is the generators of $U(1)_{A}$ transformation. In
particular, by integrating (2.41) over $\Sigma_{h}$ with ${\cal O}=1$, we
obtain

$\int_{\Sigma_{h}}\left\langle d^{\dagger}J_{A}\omega\right\rangle=-\frac{{\rm
dim}G}{4\pi}\int_{\Sigma_{h}}R\omega=-{\rm dim}G\,\chi_{h}\,,$ (2.42)

where $\chi_{h}\equiv 2-2h$ is the Euler characteristics of $\Sigma_{h}$.
Therefore we see that no anomaly appears on the torus $T^{2}$ $(h=1)$.

On the other hand, since the integration measure is invariant under the
$U(1)_{V}$ transformation, the $U(1)_{V}$ symmetry is not anomalous and the
corresponding WT identity symmetry is given by

$\left\langle{\cal
O}(x_{0})d^{\dagger}J_{V}(x)\omega\right\rangle=-\bigl{\langle}{\cal
O}(x_{0})\boldsymbol{s}_{V}{\cal
L}_{\mu}(x)\bigr{\rangle}+\delta^{2}(x-x_{0})\bigl{\langle}\boldsymbol{s}_{V}{\cal
O}(x_{0})\bigr{\rangle}\,.$ (2.43)

where $\boldsymbol{s}_{V}$ is the generator of the $U(1)_{V}$ transformation.

##  3 Graph Theory for Discretized Supersymmetric Gauge Theory

In this section, we consider a discretization of the Riemann surface in order
to regularize the supersymmetric gauge theory discussed in the previous
section. Although the way of the discretization is identical to the model
given in [52], we will reconstruct it from the perspective of graph theory.

###  3.1 Discretized Riemann surface as a graph

To regularize the supersymmetric gauge theory considered in the previous
section, we divide the Riemann surface into polygons, that is, the continuous
Riemann surface is approximated by an object consisting of polyhedra glued
together without gaps. We assume that polyhedrons are connected by edges, and
that there are no vertices in the middle of the edges. We call each polyhedron
that constitutes a polygon a face. As a result, a discretized Riemann surface
is labeled by a set of vertices $V=\\{v_{1},\cdots,v_{n_{V}}\\}$, a set of
edges $E=\\{e_{1},\cdots,e_{n_{E}}\\}$, and a set of faces
$F=\\{f_{1},\cdots,f_{n_{F}}\\}$, where $n_{V}$, $n_{F}$ and $n_{F}$ are the
number of the vertices, edges and faces, respectively. For simplicity, we will
consider only Riemann surfaces without boundaries in this paper.

The number of faces that a vertex $v$ shares is equal to the number of edges
one of whose ends is $v$, which we call the degree of the vertex $v$ and
denote by ${\rm deg}(v)$. Similarly, the number of vertices that a face $f$
shares is equal to the number of edges that consist of $f$, which we call the
degree of the face $f$ and denote by ${\rm deg}(f)$. We can assign a direction
to every edge. We thus express the edge $e$ starting from vertex $v_{s}$ and
end to vertex $v_{t}$ as $e=\\{v_{s},v_{t}\\}$. We call $v_{s}$ the source of
$e$ and $v_{t}$ the target of $e$, and also write $v_{s}=s(e)$ and
$v_{t}=t(e)$ for given $e$, respectively (see Fig. 2).

blah

Figure 1: A piece of the directed graph. $s(e)$ and $t(e)$ stand for “source”
and “target” vertices for a given edge $e$, respectively.

Figure 2: Another piece of the directed graph. $f_{+}(e)$ and $f_{-}(e)$ stand
for the faces that contain the common edge $e$ in the same and the opposite
direction for a given edge $e$, respectively.

Since the Riemann surfaces are orientable by definition, we can also consider
orientations for the surface $f$. Here we adopt the right-handed system and
define the direction of the face as the counter-clockwise rotation when we see
the Riemann surface from the outside. Then the surface $f$ can be expressed as
$f=\\{e_{i_{1}},\cdots,e_{i_{\rm deg}(f)}\\}$ as a list of its constituent
edges along the direction of $f$. Since we do not assume the boundary, every
edge is shared by two faces, and these two faces contain a common edge in
opposite directions from the way of the construction. We then write $f_{+}(e)$
for the face that contains the edge $e$ in the same direction and $f_{-}(e)$
for the face that contains it in the opposite direction (See Fig. 2).

The observation is that the discretized Riemann surface constructed in this
way can be naturally interpreted as a pair of a graph $\Gamma$ and its dual
graph $\check{\Gamma}$. A directed graph is defined as a triple
$(V,E,\varphi)$ where $V$ is a set of vertices, $E$ is a set of edges, and
$\varphi$ is a map $V\times V\to E$. By considering the relation between $V$
and $E$ constructed above as a map $\varphi:V\times V\to E$, we can naturally
regard the triple $(V,E,\varphi)$ as a directed graph $\Gamma$. Note that, in
general graphs, it is possible to draw two or more edges between two definite
vertices, or to draw an edge that returns to the same vertex, but the graph we
are considering here has at most one edge between two definite vertices, and
does not allow edges connecting the same vertex.

On the other hand, the relationship between the faces and edges constructed
above can be regarded as a map $\check{\varphi}:F\times F\to E$, thus
$(F,E,\check{\varphi})$ can also be regarded as a graph, which is the dual
graph $\check{\Gamma}$ of $\Gamma$. Therefore, the polygon partition of the
Riemann surface considered above can be regarded as a pair
$(\Gamma,\check{\Gamma})$ of a graph and its dual graph.

###  3.2 Matrices describing a graph

We introduce some useful matrices which describe the structures, and examine
their properties. These matrices are not only used to define the graph
structures, but also make it possible to use linear algebra to treat the
graph. In the following, we denote the $n_{V}$-, $n_{E}$\- and
$n_{F}$-dimensional vector spaces on $V$, $E$, and $F$ as ${\cal V}_{V}$,
${\cal V}_{E}$ and ${\cal V_{F}}$, respectively. We consider only directed
simple graphs in the following.

#### Incidence matrix

The incidence matrix $L$ is a matrix of size $n_{E}\times n_{V}$ whose
elements are given by

${L^{e}}_{v}=\begin{cases}+1&\text{if $t(e)=v$}\\\ -1&\text{if $s(e)=v$}\\\
0&\text{otherwise}\end{cases}\,,$ (3.1)

which gives a linear mapping ${\cal V}_{V}\to{\cal V}_{E}$ and can be
generated uniquely from the mapping $\varphi:V\times V\to E$ in the triple of
the graph. Note that $L$ is essentially the charge matrix of the quiver
theory. The matrix $L$ acts on a vector
${\boldsymbol{x}}=(x^{1},\cdots,x^{n_{V}})^{T}\in{\cal V}_{V}$ as

$L{\boldsymbol{x}}=(x^{t(e_{1})}-x^{s(e_{1})},\ldots,x^{t(e_{n_{E}})}-x^{s(e_{n_{E}})})^{T}\in{\cal
V}_{E}\,,$ (3.2)

which is a generalization of the forward difference in terms of the lattice
gauge theory. By regarding ${\cal V}_{V}$ and ${\cal V}_{E}$ as analogs of the
spaces of 0-forms and 1-forms, $L$ and $L^{T}$ can be seen as analogs of the
exterior derivative $d$ and its adjoint $d^{\dagger}$ in the differential
geometry.

We see that the equation $L{\boldsymbol{x}}=0$ has unique solution
$x^{v_{1}}=\cdots=x^{v_{n_{V}}}=c$, since this equation is equivalent to
$x^{s(e)}=x^{t(e)}$ and all the vertices are assumed to be connected.
Therefore, for a connected graph, the rank of the incidence matrix $L$ is
$n_{V}-1$ and $\ker L=\\{c{\boldsymbol{1}}_{n_{V}}|c\in{\mathbb{C}}\\}$ where
${\boldsymbol{1}}_{n_{V}}=(1,\cdots,1)^{T}$ in general. This is the analog of
the fact that the unique solution of $df=0$ is $f={\rm const.}$ in the
differential geometry. In the following, we denote the normalized zero mode of
$L$ as

$\boldsymbol{v}_{0}\equiv\frac{1}{\sqrt{n_{V}}}{\boldsymbol{1}}_{n_{V}}\,.$
(3.3)

To specify $\ker L^{T}$, we consider a closed loop $C$ made of edges with a
direction. Correspondingly, we define the vector $\boldsymbol{w}_{C}\in{\cal
V}_{E}$ whose elements are given by

$(\boldsymbol{w}_{C})^{e}=\begin{cases}1&\text{if $C$ includes $e$ in the same
direction}\\\ -1&\text{if $C$ includes $e$ in the opposite direction}\\\
0&\text{otherwise}\end{cases}\,,$ (3.4)

which we call the loop vector associated with the loop $C$. The loop vector
$\boldsymbol{w}_{C}$ satisfies $L^{T}\boldsymbol{w}_{C}=0$ since, for a fixed
vertex $v$ in the loop $C$, there are two edges $e_{1}$ and $e_{2}$ in $C$
which have the end on $v$ and the values of the products
${{L^{T}}^{v}}_{e_{1}}(\boldsymbol{w}_{C})^{e_{1}}$ and
${{L^{T}}^{v}}_{e_{2}}(\boldsymbol{w}_{C})^{e_{2}}$ are always opposite and
cancel with each other from the way of the construction. Therefore all the
loop vectors $\boldsymbol{w}_{C}$ are elements of $\ker L^{T}$. Furthermore,
by counting the dimension, we see that $\ker L^{T}$ is generated by linearly
independent loop vectors. First of all, we can construct $n_{F}$ loop vectors
$\boldsymbol{w}_{f}$ associated with each face $f\in F$. They are not linearly
independent but have one dependence $\sum_{f\in F}\boldsymbol{w}_{f}=0$, so we
can construct $n_{F}-1$ linearly independent loop vectors. In addition, there
are $2h$ independent non-contractible cycles on the genus $h$ Riemann surface.
Since the loop vectors $\boldsymbol{w}_{I}$ $(I=1,\cdots,2h)$ corresponding to
the cycles cannot be constructed from $\boldsymbol{w}_{f}$, the $n_{F}+2h-1$
vectors $\\{\boldsymbol{w}_{f},\boldsymbol{w}_{I}\\}$ are linearly
independent. Here recalling that the rank of $L$ is $n_{V}-1$, we find
$\dim\ker L^{T}=n_{E}-n_{V}+1=n_{F}+2h-1$. Therefore, we can conclude that
$\\{\boldsymbol{w}_{f},\boldsymbol{w}_{I}\\}$ could form a basis of $\ker
L^{T}$.

For the later purpose, we also introduce a matrix (unoriented incidence
matrix) of size $n_{E}\times n_{V}$,

$K^{e}_{\ v}\equiv|L^{e}_{\ v}|=\begin{cases}1&\text{if $t(e)=v$ or
$s(e)=v$}\\\ 0&\text{others}\end{cases}\,.$ (3.5)

#### Dual incidence matrix

We can also define the incidence matrix $\check{L}$ for the dual graph
$\check{\Gamma}$, which is a matrix of size $n_{E}\times n_{F}$ whose elements
are given by

${{\check{L}{}}^{e}}_{f}=\begin{cases}+1&\text{if the edge $e$ on the face $f$
is the forward direction}\\\ -1&\text{if the edge $e$ on the face $f$ is the
backward direction}\\\ 0&\text{otherwise}\end{cases}.$ (3.6)

We call this matrix the dual incidence matrix in the following. As well as the
incidence matrix, if we restrict restrict ourselves to the relationship
between $E$ and $F$, $\check{L}$ and $\check{L}^{T}$ correspond to
$d^{\dagger}$ and $d$ in the differential geometry, respectively. By repeating
the same discussion as $L$, we see that $\ker\check{L}$ is the one-dimensional
vector space generated by the constant vector;
$\ker\check{L}=\\{c{\boldsymbol{1}}_{n_{F}}|c\in{\mathbb{C}}\\}$, and
$\ker\check{L}^{T}$ is generated by independent dual loop vectors whose
dimension is $n_{V}+2h-1$. We denote the normalized zero mode of $\check{L}$
by

$\boldsymbol{u}_{0}\equiv\frac{1}{\sqrt{n_{F}}}{\boldsymbol{1}}_{n_{F}}\,,$
(3.7)

as well as the case of the incidence matrix.

We also introduce the unoriented dual incidence matrix by

$\check{K}^{e}_{\ f}\equiv|\check{L}^{e}_{\ f}|=\begin{cases}1&\text{if the
face $f$ includes the edge $e$}\\\ 0&\text{others}\end{cases}\,.$ (3.8)

#### Laplacian matrices

The Laplacian matrix (Kirchhoff matrix) $\Delta_{V}$ acting on the vertex is
an $n_{V}\times n_{V}$ matrix defined by

$\displaystyle({\Delta_{V}})^{v}_{\ v^{\prime}}$
$\displaystyle=L^{T}L=\begin{cases}\deg(v)&\text{if $v=v^{\prime}$}\\\
-1&\text{if $v\neq v^{\prime}$ and $v$ is adjacent to $v^{\prime}$}\\\
0&\text{otherwise}\end{cases}\,,$ (3.9)

which is also known as the Cartan matrix on the Dynkin diagram (graph) for the
Lie algebra. Note that this matrix is called “Laplacian” since it acts on
$\boldsymbol{x}$ like

$\boldsymbol{x}^{T}\Delta_{V}\boldsymbol{x}=\sum_{e\in
E}(x^{t(e)}-x^{s(e)})^{2}\,,$ (3.10)

which is nothing but a second order difference operator for the “field”
$x^{v}$. This also supports the analogy $L\leftrightarrow d$ and
${L}^{T}\leftrightarrow d^{\dagger}$ because the Laplacian acting on a 0-form
$f$ is $\Delta f=d^{\dagger}df$ in the differential geometry.

We also define the face Laplacian matrix $\Delta_{F}$ by

$\displaystyle({\Delta_{F}})^{f}_{\ f^{\prime}}$
$\displaystyle=\check{L}^{T}\check{L}=\begin{cases}\deg(f)&\text{if
$f=f^{\prime}$}\\\ -1&\text{if $f$ and $f^{\prime}$ share the same edge}\\\
0&\text{otherwise}\end{cases}\,,$ (3.11)

and the edge Laplacian by

$\Delta_{E}\equiv LL^{T}+\check{L}\check{L}^{T}\,.$ (3.12)

#### Cohomology of $L$ and $\check{L}$ and the eigenvalues of the Laplacian
matrices

The incidence and dual incidence matrix represent maps from ${\cal V}_{V}$ to
${\cal V}_{E}$ and from ${\cal V}_{F}$ to ${\cal V}_{E}$, respectively.
Furthermore, the incidence and dual incidence matrices are orthogonal with
each other,

$L^{T}\check{L}=\check{L}^{T}L=0,$ (3.13)

which holds for the same reason that the loop vectors belong to the kernel of
$L^{T}$. This is an analog of the nilpotency (exactness) of the differential
$d$ and $d^{\dagger}$. This means that we can construct the following exact
sequences;

$\begin{split}&0\rightarrow{\cal V}_{V}\xrightarrow{L}{\cal
V}_{E}\xrightarrow{\check{L}^{T}}{\cal V}_{F}\rightarrow 0,\\\
&0\leftarrow{\cal V}_{V}\xleftarrow{L^{T}}{\cal
V}_{E}\xleftarrow{\check{L}}{\cal V}_{F}\leftarrow 0.\end{split}$ (3.14)

Because of this nilpotency, we can define the cohomology group $H^{V}=\ker L$,
$H^{E}=\ker\check{L}^{T}/\operatorname{im}L$ and $H^{F}={\cal
V}_{F}/\operatorname{im}\check{L}^{T}$.

Similar to Hodge’s theorem on a compact orientable Riemannian manifold, the
cohomology group is isomorphic to a set of the kernel of the Laplacian
(harmonic forms). As mentioned in the previous subsection, $\dim\ker
L=\dim\ker\check{L}=1$ and thus we see

$\displaystyle\dim\ker\Delta_{V}$ $\displaystyle=\dim H^{V}=1\,,$ (3.15)
$\displaystyle\dim\ker\Delta_{F}$ $\displaystyle=\dim H^{F}=1\,.$ (3.16)

More explicitly, $\ker\Delta_{V}$ and $\ker\Delta_{F}$ are generated by
$\boldsymbol{v}_{0}$ and $\boldsymbol{u}_{0}$, respectively. Similarly, since
$\dim\ker\check{L}^{T}=n_{V}+2h-1$ and $\dim{\rm im}L=n_{V}-1$, we see

$\displaystyle\dim\ker\Delta_{E}$ $\displaystyle=\dim H^{E}=2h\,.$ (3.17)

Note that these results are consistent with the definition of the Euler
characteristic of the graph $\Gamma$,

$\chi_{h}=\dim H^{V}-\dim H^{E}+\dim H^{F}=2-2h.$ (3.18)

In the following argument, the eigenvectors of the Laplacian matrices play
important roles. Since $\operatorname{rank}L=n_{V}-1$ and
$\operatorname{rank}\check{L}=n_{F}-1$, the rank of $\Delta_{V}$ and
$\Delta_{F}$ are $n_{V}-1$ and $n_{F}-1$, respectively. Therefore, in addition
to the normalized zero modes $\boldsymbol{v}_{0}$ and $\boldsymbol{u}_{0}$
given by (3.3) and $\eqref{eq:f0}$, $\Delta_{V}$ and $\Delta_{F}$ have
$n_{V}-1$ and $n_{F}-1$ linearly independent eigenvectors with non-zero
eigenvalues, respectively. We then denote the orthonormal eigenvectors of
$\Delta_{V}$ and $\Delta_{F}$ as $\\{\boldsymbol{v}_{0},\boldsymbol{v}_{i}\\}$
$(i=1,\cdots,n_{V}-1)$ and $\\{\boldsymbol{u}_{0},\boldsymbol{u}_{a}\\}$
$(a=1,\cdots,n_{F}-1)$ with

$\begin{split}&\Delta_{V}\boldsymbol{v}_{0}=0,\quad\Delta_{V}\boldsymbol{v}_{i}=\lambda_{i}\boldsymbol{v}_{i}\quad(\lambda_{i}\neq
0),\\\
&\Delta_{F}\boldsymbol{u}_{0}=0,\quad\Delta_{F}\boldsymbol{u}_{a}=\mu_{a}\boldsymbol{u}_{a}\quad(\mu_{a}\neq
0),\end{split}$ (3.19)

respectively.

$L\boldsymbol{v}_{i}$ and $\check{L}\boldsymbol{u}_{a}$ become simultaneously
eigenvectors of $\Delta_{E}$ with eigenvalues $\lambda_{i}$ and $\mu_{a}$,
respectively, since

$\begin{split}&\Delta_{E}(L\boldsymbol{v}_{i})=L\Delta_{V}\boldsymbol{v}_{i}=\lambda_{i}(L\boldsymbol{v}_{i}),\\\
&\Delta_{E}(\check{L}\boldsymbol{u}_{a})=\check{L}\Delta_{F}\boldsymbol{u}_{a}=\mu_{a}(\check{L}\boldsymbol{u}_{a}),\end{split}$
(3.20)

where we have used the relation (3.13). We then normalize them by

$\boldsymbol{e}_{i}\equiv
L\boldsymbol{v}_{i}/|L\boldsymbol{v}_{i}|\,,\quad\boldsymbol{e}_{a}\equiv\check{L}\boldsymbol{u}_{a}/|\check{L}\boldsymbol{u}_{a}|\,,$
(3.21)

which are orthogonal vectors in ${\cal V}_{E}$. To complete the orthonormal
basis of ${\cal V}_{E}$, we need to add $2h$ independent normalized vectors
$\\{\boldsymbol{e}_{0}^{I}\\}$ $(I=1,\cdots,2h)$, which belong to
$\ker\Delta_{E}$, since $\dim{\cal V}_{E}=n_{E}=(n_{V}-1)+(n_{F}-1)+2h$ and
$\dim\ker\Delta_{E}=2h$. Therefore the orthonormal eigenvectors of
$\Delta_{E}$ are spanned by
$\\{\boldsymbol{e}_{0}^{I},\boldsymbol{e}_{i},\boldsymbol{e}_{a}\\}$, which
satisfy

$\displaystyle\Delta_{E}\boldsymbol{e}_{0}^{I}=0,\quad\Delta_{E}\boldsymbol{e}_{i}=\lambda_{i}\boldsymbol{e_{i}},\quad\Delta_{E}\boldsymbol{e}_{a}=\mu_{a}\boldsymbol{e_{a}}\,.$
(3.22)

From the argument below Eq. (3.4), we find that the zero eigenvectors
$\boldsymbol{e}_{0}^{I}$ correspond to the independent non-contractible cycles
on the graph $\Gamma$. Note also that the non-zero eigenvalues of $\Delta_{E}$
are common with those of $\Delta_{V}$ and $\Delta_{F}$, namely

$\operatorname{Spec}^{\prime}\Delta_{V}\oplus\operatorname{Spec}^{\prime}\Delta_{F}=\operatorname{Spec}^{\prime}\Delta_{E},$
(3.23)

where $\operatorname{Spec}^{\prime}$ stands for a set of the non-zero
eigenvalues (spectrum) of the Laplacian.

##  4 Abelian Gauge Theory on the Graph

In this section, we formulate a supersymmetric Abelian gauge theory on the
discretized Riemann surface by using graph theory. We will see that
restricting the theory to Abelian makes it easier to see the zero mode
structure of the theory, but will also give us important insight into the
relationship between anomalies and zero modes.

###  4.1 Definition of the model

In the following, we will define several vectors on the vertices, edges, and
faces, which are regarded as fields. To define a covariant theory using these
fields, we have to consider a structure of metric. To this end, we introduce
contravariant and covariant vectors, which are expressed as vectors with upper
and lower indices, respectively.

We consider a $n_{V}$-dimensional bosonic vector
$\boldsymbol{\phi}=(\phi^{1},\phi^{2},\ldots,\phi^{n_{V}})^{T}$ and their
complex conjugate
$\bar{\boldsymbol{\phi}}=(\bar{\phi}^{1},\bar{\phi}^{2},\ldots,\bar{\phi}^{n_{V}})^{T}$,
whose elements $\phi^{v}$ and $\bar{\phi}^{v}$ are regarded as complex bosonic
variables (field) living on the vertex $v\in V$. Note that the position of the
indices is important: the elements of $\boldsymbol{\phi}$ and
$\bar{\boldsymbol{\phi}}$ have upper indices which indicates that they are
contravariant vectors. If we take the transpose, they become covariant
vectors, who have lower indices as
$\boldsymbol{\phi}^{T}=(\phi_{1},\phi_{2},\ldots,\phi_{n_{V}})$ and
$\bar{\boldsymbol{\phi}}^{T}=(\bar{\phi}_{1},\bar{\phi}_{2},\ldots,\bar{\phi}_{n_{V}})$.

We also consider a $n_{E}$-dimensional bosonic vector
$\boldsymbol{U}=(U^{1},U^{2},\ldots,U^{n_{E}})^{T}$, which will be gauge
fields, whose elements are assumed to take the values in $U(1)$. In this case,
we can write $\boldsymbol{U}$ by $\exp\left\\{i\boldsymbol{A}\right\\}$, where
$\boldsymbol{A}=(A^{1},A^{2},\ldots,A^{n_{E}})^{T}$ whose elements are real
variables. Furthermore, we introduce an $n_{F}$-dimensional bosonic vector
$\boldsymbol{Y}=(Y^{1},Y^{2},\dots,Y^{n_{F}})^{T}$ whose elements are assumed
to be real. In addition to the bosonic fields, we also consider the fermionic
fields $\boldsymbol{\eta}=(\eta^{1},\eta^{2},$
$\ldots,\eta^{n_{V}})^{T}\in{\cal V}_{V}$,
$\boldsymbol{\lambda}=(\lambda^{1},\lambda^{2},\ldots,\lambda^{n_{E}})^{T}\in{\cal
V}_{E}$, and
$\boldsymbol{\chi}=(\chi^{1},\chi^{2},\ldots,\chi^{n_{F}})^{T}\in{\cal
V}_{F}$.

We define the supersymmetry transformations of these fields by

$\begin{array}[]{lcl}Q\phi^{v}=0,&&\\\
Q\bar{\phi}^{v}=2\eta^{v},&&Q\eta^{v}=0,\\\
QA^{e}=\lambda^{e},&&Q\lambda^{e}=-{L^{e}}_{v}\phi^{v},\\\
QY^{f}=0,&&Q\chi^{f}=Y^{f}.\end{array}$ (4.1)

Using this symmetry, we write the action of the model in the $Q$-exact form as

$S=-\frac{1}{2g^{2}}Q\left\\{\bar{\phi}_{v}{{L^{T}}^{v}}_{e}\lambda^{e}+\chi_{f}(Y^{f}-2\mu(P^{f}))\right\\},$
(4.2)

where $P^{f}$ is the plaquette variables associated with the face $f$ defined
by

$P^{f}\equiv\prod_{e\in
f}(U^{e})^{{{\check{L}^{T}}{}^{f}}_{e}}=\exp\left\\{i{{{\check{L}^{T}}{}}^{f}}_{e}A^{e}\right\\}\,,$
(4.3)

and $\mu(P^{f})$ is a function called the moment map, which becomes the field
strength $F$ in the continuum limit.

There are several candidates on the moment map. The moment map used in the
original Sugino model is

$\mu(P)=\frac{1}{2i}\frac{P-P^{\dagger}}{1-\frac{1}{\epsilon^{2}}||1-P||^{2}}\,,$
(4.4)

where $||\cdot||$ is a norm of a matrix defined by $||A||\equiv\sqrt{{\rm
tr}(AA^{\dagger})}$, which is now simply $||A||=|A|^{2}$ in the Abelian
theory, and $\epsilon$ is a positive constant parameter chosen in the range
$0<\epsilon<2$. If we do not introduce the denominator, the vacuum condition
$\mu(P)=0$ has two solutions $P=\pm 1$ which contain an unphysical vacuum at
$P=-1$. The denominator is necessary to avoid it. See [5] for more detail.

Another candidate is the tangent type function,

$\mu(P)=\frac{1}{2i}\frac{P-P^{\dagger}}{P+P^{\dagger}}\,,$ (4.5)

proposed in [50]. $P=-1$ is on the pole of this function and thus $\mu(P)=0$
has a unique solution at the physical vacuum.

In addition to them, in Abelian theory, we can use logarithmic type moment
map,

$\mu(P)=-i\log(P)\,.$ (4.6)

Although it suffered from the mathematical difficulty of defining the log of a
matrix in non-Abelian theory, there is no ambiguity in Abelian theory by
choosing the branch of the logarithmic function as $-\pi<\arg z\leq\pi$ for
$z\in{\mathbb{C}}$.

After eliminating the auxiliary field $Y^{f}$, the bosonic part of the action
becomes

$\begin{split}S_{B}&=\frac{1}{2g^{2}}\left\\{\bar{\phi}_{v}{{L^{T}}^{v}}_{e}{L^{e}}_{v^{\prime}}\phi^{v^{\prime}}+\mu(P^{f})^{2}\right\\}\\\
&=\frac{1}{2g^{2}}\left\\{\bar{\boldsymbol{\phi}}^{T}\Delta_{V}\boldsymbol{\phi}+|\boldsymbol{\mu}|^{2}\right\\},\end{split}$
(4.7)

where we have used the definition of the graph vertex Laplacian
$\Delta_{V}=L^{T}L$ and defined
${\boldsymbol{\mu}}\equiv(\mu(P^{f_{1}}),\cdots,\mu(P^{f_{n_{F}}}))^{T}$. This
action describes the free complex scalar field decoupling from Maxwell theory
in the continuum limit.

The fermionic part of the action becomes

$\begin{split}S_{F}&=-\frac{1}{g^{2}}\left\\{\eta_{v}{{L^{T}}^{v}}_{e}\lambda^{e}+\chi_{f}\frac{\delta\mu^{f}}{\delta
A^{e}}\lambda^{e}\right\\}\\\
&=-\frac{1}{g^{2}}\left\\{\boldsymbol{\eta}^{T}L^{T}\boldsymbol{\lambda}+\boldsymbol{\chi}^{T}\frac{\delta\boldsymbol{\mu}}{\delta\boldsymbol{A}}\boldsymbol{\lambda}\right\\},\end{split}$
(4.8)

where $\frac{\delta\boldsymbol{\mu}}{\delta\boldsymbol{A}}$ is an $n_{F}\times
n_{E}$ matrix and is proportional to a transpose of the dual incidence matrix
$\check{L}^{T}$. Indeed, for each component, we find

$\begin{split}\frac{\delta\mu(P^{f})}{\delta
A^{e}}&=i\mu^{\prime}(P^{f})P^{f}{\check{L}^{T}}{{}^{f}}_{e},\end{split}$
(4.9)

where $\mu^{\prime}$ stands for a derivative of the function $\mu$ w.r.t. the
argument.

This model is nothing but the Abelian version of the generalized Sugino model
constructed in [52]. Repeating the discussion in [52], we see that this model
is a discretization of the ${\cal N}{=}(2,2)$ supersymmetric Abelian gauge
theory discussed in Sec.2. Because of the discretization, the
$\tilde{Q}$-symmetry and the $U(1)_{V}$ symmetries are explicitly broken while
the $Q$-symmetry and the $U(1)_{A}$ symmetry are still preserved after
discretization at least classically.

###  4.2 Fermion zero modes

Here we will examine the structure of the fermions in this theory. We first
redefine $\chi^{f}$ by

${\rho}^{f}\equiv i\mu^{\prime}(P^{f})P^{f}\chi^{f},$ (4.10)

where we assume that the prefactor $i\mu^{\prime}(P^{f})P^{f}$ is non-
vanishing and non-singular everywhere333 The argument in this subsection
always holds for $\chi^{f}$ itself in the saddle point approximation, where
$\rho^{f}\sim i\chi^{f}$ with $P^{f}\sim 1$. In the localization method, the
zero modes of $\chi^{f}$ play the same role as the zero modes of $\rho^{f}$.
. Then let us combine all fermionic variables into a single
$(n_{V}+n_{F}+n_{E})$-dimensional vector together such that

$\begin{split}{{\boldsymbol{\Psi}}}&=\begin{pmatrix}{{\boldsymbol{\eta}}}\\\
{\boldsymbol{\rho}}\\\ {{\boldsymbol{\lambda}}}\end{pmatrix}\,.\end{split}$
(4.11)

Then the fermionic part of the action (4.8) can be written as

$\begin{split}S_{F}&=\frac{1}{g^{2}}\boldsymbol{\Psi}^{T}i\not{D}\boldsymbol{\Psi},\end{split}$
(4.12)

where $\not{D}$ is a matrix of size $(n_{V}+n_{F}+n_{E})$,

$\not{D}=\begin{pmatrix}0&iD^{T}\\\ -iD&0\end{pmatrix}$ (4.13)

with

$D=\begin{pmatrix}L&\check{L}\end{pmatrix}\,.$ (4.14)

Let us examine the zero modes of the matrix $\not{D}$, which can be obtained
from the zero modes of $D$ and $D^{T}$.

$D$ is a linear mapping from ${\cal V}_{V}\oplus{\cal V}_{F}$ to ${\cal
V}_{E}$ which transforms $(\boldsymbol{\eta},\boldsymbol{\rho})^{T}\in{\cal
V}_{V}\oplus{\cal V}_{F}$ as

$D\begin{pmatrix}\boldsymbol{\eta}\\\
{\boldsymbol{\rho}}\end{pmatrix}=L{\boldsymbol{\eta}}+\check{L}\boldsymbol{\rho}\,.$
(4.15)

Recalling $\dim\ker L=\dim\ker\check{L}=1$, we can immediately see that $D$
has one zero mode in $\boldsymbol{\eta}$ and $\boldsymbol{\rho}$ each444Note
that the equation $L{\boldsymbol{\eta}}+\check{L}\boldsymbol{\rho}=0$ leads
$L\boldsymbol{\eta}=\check{L}\boldsymbol{\rho}=0$ because of the orthogonality
(3.13). proportional to $\boldsymbol{v}_{0}$ (3.3) and $\boldsymbol{u}_{0}$
(3.7), respectively. Therefore we find
$\operatorname{rank}D=n_{V}+n_{F}-2=n_{E}-2h$ where we have used the
definition $\chi_{h}=n_{V}-n_{E}+n_{F}=2-2h$.

On the other hand, $D^{T}$ is a linear mapping from ${\cal V}_{E}$ to ${\cal
V}_{V}\oplus{\cal V}_{F}$ which transforms $\boldsymbol{\lambda}\in{\cal
V}_{E}$ as

$D^{T}\boldsymbol{\lambda}=\begin{pmatrix}L^{T}{\boldsymbol{\lambda}}\\\
\check{L}^{T}\boldsymbol{\lambda}\end{pmatrix}\,.$ (4.16)

Then the zero modes of $D^{T}$ are the common zero modes of $L^{T}$ and
$\check{L}^{T}$, which are the same as the zero modes
$\\{\boldsymbol{e}_{0}^{I}\\}$ of $\Delta_{E}$. This is a direct result of the
isomorphism $\ker{L}^{T}/\operatorname{im}\check{L}\simeq\ker\Delta_{E}$, that
is, $\ker L^{T}$ is spanned by the $n_{F}+2h-1$ linearly independent loop
vectors $\\{\boldsymbol{e}_{0}^{I},\boldsymbol{w}_{a}\\}$ where
$\boldsymbol{w}_{a}$ $(a=1,\cdots,n_{F}-1)$ are linearly independent loop
vectors corresponding to the faces, and $\operatorname{im}\check{L}$ is
spanned by $\\{\boldsymbol{w}_{a}\\}$ by construction. Therefore
$\ker\Delta_{E}$ is spanned by $\\{\boldsymbol{e}_{0}^{I}\\}$.

In summary, $\not{D}$ has one zero mode $\eta_{0}$ in ${\boldsymbol{\eta}}$,
one zero mode $\rho_{0}$ in $\boldsymbol{\rho}$ and $2h$ zero modes
$\lambda_{0}^{I}$ $(I=1,\cdots,2h)$ in ${\boldsymbol{\lambda}}$, which can be
explicitly written as

$\displaystyle\eta_{0}=\boldsymbol{v}_{0}\cdot\boldsymbol{\eta},\quad\rho_{0}=\boldsymbol{u}_{0}\cdot\boldsymbol{\rho},\quad\lambda_{0}^{I}=\boldsymbol{e}_{0}^{I}\cdot\boldsymbol{\lambda}\,.$
(4.17)

Note that the vertex zero mode $\eta_{0}$ and the edge zero modes
$\lambda_{0}^{I}$ are $Q$-invariant while the face zero mode $\rho_{0}$ is
not. The $Q$-invariance of $\eta_{0}$ is trivial from (4.1), and
$\lambda_{0}^{I}$ is $Q$-invariant since the loop vector
$\boldsymbol{e}_{0}^{I}$ belongs to $\ker L^{T}$. However, $Q$ transforms
$\chi_{0}^{\prime}$ as

$\displaystyle Q\rho_{0}$ $\displaystyle=\frac{1}{\sqrt{n_{F}}}\sum_{f\in
F}Q(i\mu^{\prime}(P^{f})P^{f}\chi^{f})$
$\displaystyle=\frac{1}{\sqrt{n_{F}}}\sum_{f\in
F}\left\\{-\left(\mu^{\prime\prime}(P^{f})(P^{f})^{2}+\mu^{\prime}(P^{f})P^{f}\right)\lambda^{e}\check{L}^{e}_{\
f}\chi^{f}+i\mu^{\prime}(P^{f})P^{f}Y^{f}\right\\}\,,$ (4.18)

which does not vanish in general.

###  4.3 Heat kernel and dimensionality

We can examine the analytic behavior of the fermion spectrum by using the hear
kernel. In the continuous theory, the heat kernel is defined by the following
heat equation;

$\left(\frac{\partial}{\partial t}+\not{\cal D}^{2}\right)h(x,y;t)=0,$ (4.19)

which is formally solved as

$h(x,y;t)=e^{-t\not{\cal D}^{2}}\,.$ (4.20)

In particular, the regularized $U(1)_{A}$ current is evaluated by the heat
kernel;

$\int_{\Sigma_{h}}\left\langle
d^{\dagger}J_{A}\omega\right\rangle_{\text{reg}}=-\operatorname{Tr}\gamma_{A}e^{-t\not{\cal
D}^{2}}=-\operatorname{ind}\not{\cal D}\,.$ (4.21)

We can extend the definition of the heat kernel to the Dirac operator on the
graph. The heat kernel on the graph is simply defined by

${h(t)^{i}}_{j}\equiv{\left(e^{-t\not{D}^{2}}\right)^{i}}_{j},$ (4.22)

using the matrix $\not{D}$ defined by (4.13). Correspondingly, we define the
quantity,

$\begin{split}I(t)&\equiv\operatorname{Tr}_{V\oplus F\oplus
E}\gamma_{A}e^{-t\not{D}^{2}}\\\
&=\operatorname{Tr}_{V}e^{-t\Delta_{V}}+\operatorname{Tr}_{F}e^{-t\Delta_{F}}-\operatorname{Tr}_{E}e^{-t\Delta_{E}}\\\
&=\sum_{l=1}^{n_{V}}e^{-t\lambda^{V}_{l}}+\sum_{m=1}^{n_{F}}e^{-t\lambda^{F}_{m}}-\sum_{n=1}^{n_{E}}e^{-t\lambda^{E}_{n}},\end{split}$
(4.23)

where $\lambda^{V}_{l}$, $\lambda^{F}_{m}$ and $\lambda^{E}_{n}$ are
eigenvalues for the Laplacians $\Delta_{V}$, $\Delta_{F}$ and $\Delta_{E}$,
respectively.

In the large $t$ limit, the contribution of the non-zero modes (eigenvalues)
to $I(t)$ disappears and the difference of the number of the zero modes
survives as the index. Moreover, the contributions from the non-zero modes are
canceled with each other even when the value of $t$ is finite because there is
a one-to-one correspondence between the non-zero modes of
$\\{\Delta_{V},\Delta_{F}\\}$ and $\Delta_{E}$ as shown in Sec. 3.2. Therefore
$I(t)$ is independent of $t$, that is, $I(t)$ gives the index of $\not{D}$
which is equal to the Euler characteristic of the graph, as well as in the
continuous theory.

It is instructive to give some concrete examples of the trace of heat kernels.
The simplest example of graph with genus zero is the tetrahedron
($(n_{V},n_{F},n_{E})=(4,4,6)$). The eigenvalues of the Laplacians are given
by

$\begin{split}\operatorname{Spec}\Delta_{V}&=\\{4,4,4,0\\},\\\
\operatorname{Spec}\Delta_{F}&=\\{4,4,4,0\\},\\\
\operatorname{Spec}\Delta_{E}&=\\{4,4,4,4,4,4\\}.\end{split}$ (4.24)

Then we get

$\operatorname{Tr}\gamma_{A}e^{-t\not{D}^{2}}=(3e^{-4t}+1)+(3e^{-4t}+1)-6e^{-4t}=2,$
(4.25)

where the non-zero modes are canceled with each other order by order.

Similarly, for the hexahedron ($(n_{V},n_{F},n_{E})=(8,6,12)$), we obtain

$\begin{split}\operatorname{Spec}\Delta_{V}&=\\{6,4,4,4,2,2,2,0\\},\\\
\operatorname{Spec}\Delta_{F}&=\\{6,6,4,4,4,0\\},\\\
\operatorname{Spec}\Delta_{E}&=\\{6,6,6,4,4,4,4,4,4,2,2,2\\}\,,\end{split}$
(4.26)

then we obtain

$\operatorname{Tr}\gamma_{A}e^{-t\not{D}^{2}}=(e^{-6t}+3e^{-4t}+3e^{-2t}+1)\\\
+(2e^{-6t}+3e^{-4t}+1)-(3e^{-6t}+6e^{-4t}+3e^{-2t})=2.$ (4.27)

For genus 1, the spectrum of the Laplacians of the $3\times 3$ torus is

$\begin{split}\operatorname{Spec}\Delta_{V}&=\\{6,6,6,6,3,3,3,3,0\\}\,,\\\
\operatorname{Spec}\Delta_{F}&=\\{6,6,6,6,3,3,3,3,0\\}\,,\\\
\operatorname{Spec}\Delta_{E}&=\\{6,6,6,6,6,6,6,6,3,3,3,3,3,3,3,3,0,0\\}\,,\end{split}$
(4.28)

and the index becomes

$\begin{split}\operatorname{Tr}\gamma_{A}e^{-t\not{D}^{2}}&=(4e^{-6t}+4e^{-3t}+1)+(4e^{-6t}+4e^{-3t}+1)-(8e^{-6t}+8e^{-3t}+2)\\\
&=0.\end{split}$ (4.29)

as expected.

Furthermore, the behavior of the heat kernel for each Laplacian (not square of
the Dirac operator) also represents the dimensionality of the graph structure.
If we take the continuum limit of the graph discretization (lattice), we
expect that the space-time goes to the smooth Riemann surface. The heat kernel
on the Riemann surface, which satisfies the heat equation

$\left(\frac{\partial}{\partial t}+\Delta_{x}\right)h(x,y;t)=0\,,$ (4.30)

with the Laplacian $\Delta_{x}$, behaves as

$h(x,y;t)=\frac{1}{4\pi t}e^{-|x-y|/2t}+\cdots,$ (4.31)

for small $t$, while the index (4.21) is independent of $t$. In particular,
the trace of the heat kernel behaves as

$\tilde{h}(t)\equiv\int_{\Sigma_{h}}dx\,h(x,x;t)=\frac{{\rm
Vol}(\Sigma_{h})}{4\pi t}+\cdots\,,$ (4.32)

and, in the large $t$ limit, the trace of the heat kernel tends to the number
of the zero modes

$\lim_{t\to\infty}\tilde{h}(t)=\lim_{t\to\infty}\sum_{n}e^{-t\lambda_{n}}=\dim\ker\Delta_{V}\,.$
(4.33)

On the general $D$-dimensional space-time, the trace of the heat kernel is
proportional to $1/t^{D/2}$ for the small $t$. So, if we investigate the small
$t$ behavior of the heat kernel for the graph Laplacian, we can confirm how
the dimensionality of the graph discretization is close to the Riemann
surface.

We construct the heat kernels of the graph Laplacian for several geodesic
polyhedrons with genus 0 (subdivisions of tetrahedron and octahedron) and plot
in Fig. 3. We find that the behavior of the heat kernel approached that of the
two-dimensional sphere ($1/t$ behavior) as the number of the vertices
increases and the discretization becomes finer. We also find that the trace of
the heat kernel represents the number of the zero modes in the large $t$
limit.

Figure 3: Plots of the heat kernel. If the number of vertices (or faces) of
the geodesic polyhedra is large, the behaviour of the heat kernel becomes very
close to that of the smooth two-dimensional sphere. (The “div” in the legend
stands for the number of times the triangular faces of the tetrahedron or
octahedron is divided into smaller triangles.)

###  4.4 Uplifting the fermion zero modes

Due to the existence of these zero modes, the discretized theory with the
action (4.2) is not well-defined since the partition function trivially
vanishes. we expand the fermion fields by the eigenvectors of the Laplacian
matrices as

$\displaystyle\boldsymbol{\eta}$
$\displaystyle=\eta_{0}\boldsymbol{v}_{0}+\sum_{i=1}^{n_{V}-1}\eta_{i}\boldsymbol{v}_{i},\quad\boldsymbol{\chi}=\chi_{0}\boldsymbol{u}_{0}+\sum_{a=1}^{n_{F}-1}\chi_{a}\boldsymbol{u}_{a},$
$\displaystyle\boldsymbol{\lambda}$
$\displaystyle=\sum_{I=1}^{2h}\lambda_{0}^{I}\boldsymbol{e}_{0}^{I}+\sum_{i=1}^{n_{V}-1}\lambda_{i}\boldsymbol{e}_{i}+\sum_{a=1}^{n_{F}-1}\lambda_{a}\boldsymbol{e}_{a}\,,$
(4.34)

and write the integration measure for the modes as $d{\cal B}d{\cal F}d{\cal
F}_{0}$, where

$\displaystyle d{\cal B}$
$\displaystyle=\left(\prod_{v=1}^{n_{V}}d\phi_{v}d\bar{\phi}_{v}\right)\left(\prod_{e=1}^{n_{E}}dU_{e}\right),$
(4.35) $\displaystyle d{\cal F}$
$\displaystyle=\left(\prod_{i=1}^{n_{V}-1}d\eta_{i}d\lambda_{i}\right)\left(\prod_{a=1}^{n_{F}-1}d\chi_{a}d\lambda_{a}\right),$
(4.36) $\displaystyle d{\cal F}_{0}$
$\displaystyle=\left(\prod_{I=1}^{2h}d\lambda_{0}^{2h-I+1}\right)d\chi_{0}d\eta_{0}\,.$
(4.37)

Evaluating the vev of a (not necessarily local) operator ${\cal O}[X]$, which
is a functional of the collective expression of all the fields $X$ and does
not include any fermion zero mode, the integration $\int d{\cal B}d{\cal
F}d{\cal F}_{0}\,{\cal O}[X]e^{-S[X]}$ trivially vanishes.

To avoid it, we have to insert all the fermion zero modes in the background
like

${\cal I}_{\cal O}\equiv\int d{\cal B}d{\cal F}d{\cal
F}_{0}\left(\eta_{0}\chi_{0}\prod_{I=1}^{2h}\lambda_{0}^{I}\right)\,{\cal{O}}[X]e^{-S[X]}\,.$
(4.38)

The inserted zero modes are integrated by the measure of $d{\cal F}_{0}$, then

${\cal I}_{\cal O}=\int d{\cal B}d{\cal F}\,{\cal O}[X]e^{-S[X]}\,,$ (4.39)

becomes well-defined since the measure does not include the fermion zero modes
anymore.

The straightforward way to achieve it automatically is to add mass terms of
the fermion zero modes to the action as

$\displaystyle S_{\mu}$
$\displaystyle=\frac{\mu_{0}}{g^{2}}\left(\eta_{0}\chi_{0}\right)+\frac{\mu_{1}}{g^{2}}\sum_{k=1}^{h}\lambda_{0}^{2k-1}\lambda_{0}^{2k}\equiv\frac{1}{2g^{2}}{\boldsymbol{\Psi}}^{T}{\cal
M}{\boldsymbol{\Psi}}\,.$ (4.40)

These terms not only make the Dirac matrix invertible but also supply the
necessary fermion zero modes in evaluating correlation functions as

$\int d{\cal B}d{\cal F}d{\cal
F}_{0}\,{\cal{O}}[X]e^{-S[X]-S_{\mu}}=\frac{\mu_{0}\mu_{1}^{h}}{g^{2h+2}}\,{\cal
I}_{\cal O}\,.$ (4.41)

The necessary fermion zero modes are supplied by expanding $e^{-S_{\mu}}$ by
the parameters $\mu_{0}$ and $\mu_{1}$ as the term with the coefficient
$\mu_{0}\mu_{1}^{h}$, and the other terms vanish as lack or excess of the
fermion zero modes. Note that the situation is the same when the operator
${\cal O}$ includes all or a part of the fermion zero modes, where only the
term including all the fermion zero modes survives.

Here we note that the supplied zero modes from $e^{-S_{\mu}}$ break the
$U(1)_{A}$ symmetry unless $h=1$ reflecting the quantum anomaly discussed soon
later. We also note that the mass terms (4.40) also break the $Q$-symmetry
softly since $\chi_{0}$ is not $Q$-invariant as shown in (4.18). As we will
discuss in the next section, it is possible to construct such mass terms that
cancel the fermion zero modes while preserving the $Q$-symmetry. In this
sense, although the mass term constructed here is simple, it is not the only
option. In particular, when discussing situations where supersymmetry plays an
important role, we should use the $Q$-invariant mass terms. However the choice
of mass term is not so important to discuss the anomalous $U(1)_{A}$ current,
as long as the partition function is well-defined. Therefore, in this section,
we will use the regularized action,

$\displaystyle S$
$\displaystyle=\frac{1}{2g^{2}}\left\\{\bar{\boldsymbol{\phi}}^{T}\Delta_{V}\boldsymbol{\phi}+|\boldsymbol{\mu}|^{2}+{\boldsymbol{\Psi}}^{T}\left(i\not{D}+{\cal
M}\right){\boldsymbol{\Psi}}\right\\}\,,$ (4.42)

for a while.

###  4.5 Chiral anomaly on the graph

We next consider the WT identity corresponding to the $U(1)_{A}$ symmetry. To
this end, we consider the following local $U(1)_{A}$ transformation,

$\eta^{v}\to e^{-i\theta_{\\!A}^{v}}\eta^{v},\quad\lambda^{e}\to
e^{i\theta_{\\!A}^{e}}\lambda^{e},\quad\chi^{f}\to
e^{-i\theta_{\\!A}^{f}}\chi^{f}.$ (4.43)

This local transformation is not the symmetry of the theory in general, but it
makes the action invariant if the transformation is global (independent of the
positions), namely $\theta_{\\!A}^{v}=\theta_{\\!A}^{e}=\theta_{\\!A}^{f}$. We
are now dealing with the graph $\Gamma$ and the dual graph $\check{\Gamma}$ on
an equal footing, where one edge is shared by two vertices and two faces. To
respect this structure, we take the transformation parameters
$\theta_{\\!A}^{e}$ of the edge $e$ to be the average of the transformation
parameters of the vertices $\theta_{\\!A}^{v}$ and faces $\theta_{\\!A}^{f}$
as

$\theta_{\\!A}^{e}=\frac{1}{4}(\theta_{\\!A}^{s(e)}+\theta_{\\!A}^{t(e)}+\theta_{\\!A}^{f_{+}(e)}+\theta_{\\!A}^{f_{-}(e)})\,,$
(4.44)

where $s(e)$, $t(e)$ and $f_{\pm}(e)$ are defined in Sec. 3.1. Then the
infinitesimal transformation of the action becomes

$\displaystyle\delta S$ $\displaystyle=\sum_{e\in E}\Bigl{\\{}(L^{e}_{\
v}\theta_{\\!A}^{v})J^{(V)}_{e}+(\check{L}^{e}_{\
f}\theta_{\\!A}^{f})J^{(F)}_{e}+\left((K^{e}_{\
v}\theta_{\\!A}^{v})-(\check{K}^{e}_{\
f}\theta_{\\!A}^{f})\right)G_{e}\Bigr{\\}}\,,$ (4.45)

where $K$ and $\check{K}$ are the matrices given by (3.5) and (3.8),
respectively, and

$\displaystyle J^{(V)}_{e}$
$\displaystyle\equiv\frac{i}{g^{2}}\Bigl{(}-\bar{\phi}^{t(e)}\phi^{s(e)}+\bar{\phi}^{s(e)}\phi^{t(e)}+\frac{1}{2}\lambda_{e}\sum_{v\in
V}K^{e}_{\ v}\eta^{v}\Bigr{)}\,,$ (4.46) $\displaystyle J^{(F)}_{e}$
$\displaystyle\equiv\frac{i}{2g^{2}}\lambda_{e}\sum_{f\in F}\check{K}^{e}_{\
f}\hat{\chi}^{f}\,,$ (4.47) $\displaystyle G_{e}$
$\displaystyle\equiv\frac{i}{4g^{2}}\lambda_{e}\left(\sum_{v\in V}(L^{e}_{\
v}\eta^{v})-\sum_{f\in F}(\check{L}^{e}_{\ f}\hat{\chi}^{f})\right)\,,$ (4.48)

which are defined on each edge $e\in E$ and thus the index $e$ is not
contracted.

Recalling that the mass terms (4.40) effectively supply all the fermion zero
modes to the integration, we have also to evaluate the transformation of the
integral measure $d{\cal B}d{\cal F}d{\cal F}_{0}$ and the inserted fermion
zero modes. The infinitesimal transformation of the measure becomes

$d{\cal B}d{\cal F}d{\cal
F}_{0}\to\left(1+\sum_{v}(i\theta_{\\!A}^{v})(1-\frac{1}{4}{\rm
deg}(v))+\sum_{f}(i\theta_{\\!A}^{f})(1-\frac{1}{4}{\rm deg}(f))\right)d{\cal
B}d{\cal F}d{\cal F}_{0}\,,$ (4.49)

and that of each fermion zero mode is

$\displaystyle\eta_{0}$ $\displaystyle\to\eta_{0}-i\sum_{v\in
V}\theta_{\\!A}^{v}(\boldsymbol{v}_{0})_{v}\eta^{v},$ $\displaystyle\chi_{0}$
$\displaystyle\to\chi_{0}-i\sum_{f\in
F}\theta_{\\!A}^{f}(\boldsymbol{u}_{0})_{f}\hat{\chi}^{f},$ (4.50)
$\displaystyle\lambda_{0}^{I}$ $\displaystyle\to\lambda_{0}^{I}+i\sum_{e\in
E}\theta_{\\!A}^{d}(\boldsymbol{e}_{0}^{I})_{e}\lambda^{e}\,.$

So the zero mode integral is evaluated as

$\displaystyle\int d{\cal
F}_{0}\,\delta\left(\eta_{0}\chi_{0}\prod_{I=1}^{2h}\lambda_{0}^{I}\right)=$
$\displaystyle\sum_{v\in
V}(i\theta_{\\!A}^{v})\left(-(\boldsymbol{v}_{0})_{v}^{2}+\frac{1}{4}\sum_{I=1}^{2h}\sum_{e\in
E}K^{e}_{\ v}(\boldsymbol{e}_{0}^{I})_{e}^{2}\right)$ $\displaystyle+$
$\displaystyle\sum_{f\in
F}(i\theta_{\\!A}^{f})\left(-(\boldsymbol{u}_{0})_{f}^{2}+\frac{1}{4}\sum_{I=1}^{2h}\sum_{e\in
E}\check{K}^{e}_{\ f}(\boldsymbol{e}_{0}^{I})_{e}^{2}\right)\,.$ (4.51)

Combining (4.45) and (4.51), we obtain the identities,

$\displaystyle\biggl{\langle}\sum_{e\in E}\left(L^{e}_{\
v}J_{e}^{(V)}+K^{e}_{\ v}G_{e}\right)\biggr{\rangle}$
$\displaystyle=-\left(1-\frac{1}{4}{\rm
deg}(v)\right)+\left((\boldsymbol{v}_{0})_{v}^{2}-\frac{1}{4}\sum_{I=1}^{2h}\sum_{e\in
E}K^{e}_{\ v}(\boldsymbol{e}_{0}^{I})_{e}^{2}\right)\,,$ (4.52)
$\displaystyle\biggl{\langle}\sum_{e\in E}\left(\check{L}^{e}_{\
f}J^{(F)}_{e}-\check{K}^{e}_{\ f}G_{e}\right)\biggr{\rangle}$
$\displaystyle=-\left(1-\frac{1}{4}{\rm
deg}(f)\right)+\left((\boldsymbol{u}_{0})_{f}^{2}-\frac{1}{4}\sum_{I=1}^{2h}\sum_{e\in
E}\check{K}^{e}_{\ f}(\boldsymbol{e}_{0}^{I})_{e}^{2}\right)\,,$ (4.53)

on each $v\in V$ and $f\in F$, respectively. These identities are the local WT
identities corresponding to (2.41) in the continuous theory. Note that we have
two local WT identities for the single $U(1)_{A}$ symmetry because we are
considering the graph and the dual graph at the same time: The transformation
parameters $\theta_{\\!A}^{v}$ and $\theta_{\\!A}^{f}$ can be assigned
independently on the vertices of the graph and the dual graph, respectively.

Interestingly, the quantities $1-{\rm deg}(v)/4$ and $1-{\rm deg}(f)/4$
appearing in the first parentheses of the right-hand sides correspond to the
scalar curvature of the continuum geometry. We can justify it from the fact
that they can be regarded as the deficit angle, which is proportional to the
scalar curvature in two-dimensional geometry. To see it, suppose that all the
faces are regular $n$-polygon of the same size. In this case ${\rm deg}(f)=n$
and the deficit angle around the vertex $v$ is given by
$\theta_{v}=2\pi\left(1-\frac{n-2}{2n}{\rm deg}(v)\right)$. Then the average
of the quantities around a vertex $v$ becomes the deficit angle as announced;

$1-\frac{1}{4}{\rm deg}(v)+\sum_{f\in F_{v}}\frac{1}{{\rm
deg}(f)}\left(1-\frac{1}{4}{\rm deg}(f)\right)=\frac{\theta_{v}}{2\pi}\,,$
(4.54)

where $F_{v}$ denotes the faces touching at the vertex $v$.

The second parentheses of the right-hand sides of (4.52) and (4.53) are the
contribution of the inserted fermion zero modes, which are necessary so that
these identities are consistent. We can see it as follows: If we take
summation over all the vertices of (4.52) and all the faces of (4.53) followed
by summing these two identities, the terms with $G_{e}$ trivially cancel and
the remaining terms $\sum_{v,e}L^{e}_{\ v}J^{(V)}_{e}$ and
$\sum_{f,e}{\check{L}}^{e}_{\ f}J^{(F)}_{e}$ vanish because of the structure
of the matrices $L$ and $\check{L}$. This corresponds to the fact that the
integration over a total derivative vanishes in the continuous theory. On the
other hand, after the same operation, the first term of the right-hand side
gives

$\displaystyle-n_{V}-n_{F}+\frac{1}{4}\sum_{v}{\rm
deg}(v)-\frac{1}{4}\sum_{f}{\rm deg}(f)=-n_{V}-n_{F}+n_{E}=-\chi_{h},$ (4.55)

which is a reproduction of the anomalous WT identity (2.42) in the continuous
theory with $G=U(1)$. However, if the second terms of the right-hand sides are
absent, we obtain an inconsistent expression unless $\chi_{h}=0$ because the
left-hand side vanishes. The second terms cure the situation such that

$\displaystyle\sum_{v\in
V}\left((\boldsymbol{v}_{0})_{v}^{2}-\frac{1}{4}\sum_{I=1}^{2h}\sum_{e\in
E}K^{e}_{\ v}(\boldsymbol{e}_{0}^{I})_{e}^{2}\right)+\sum_{f\in
F}\left((\boldsymbol{u}_{0})_{f}^{2}-\frac{1}{4}\sum_{I=1}^{2h}\sum_{e\in
E}\check{K}^{e}_{\ f}(\boldsymbol{e}_{0}^{I})_{e}^{2}\right)=2-2h=\chi_{h}\,,$
(4.56)

which cancels the contribution from the first terms (4.55). This is consistent
with the fact that the left-hand side vanishes.

##  5 Localization in the Abelian Theory

###  5.1 Saddle point equation

We now apply the localization method to perform the path integral of the
Abelian theory exactly.

Let $f[X]$ be a $Q$-closed operator. Recalling the action is written in the
$Q$-exact form such that $S=Q\Xi$, we see that the integration

$\left\langle f[X]\right\rangle_{t}\equiv\int dX\,f[X]e^{-tS}$ (5.1)

is independent of $t$. If we differentiate it by $t$, we obtain the vev of a
$Q$-exact operator, which vanishes at a supersymmetric vacuum;

$\frac{\partial}{\partial t}\langle f[X]\rangle_{t}=-\left\langle
Q(f[X]\,\Xi)\right\rangle_{t}=0\,.$ (5.2)

The path integral is localized at the saddle points in the limit of
$t\to\infty$. Thus we can evaluate the vev of a $Q$-closed operator, which
includes the partition function itself, exactly by the saddle point
approximation of the Abelian theory.

From the bosonic action (4.7), we can see the saddle points are given by
equations

$L\boldsymbol{\phi}=0,\quad\boldsymbol{\mu}=0.$ (5.3)

On the connected graph, the former equation always has a solution

$\boldsymbol{\phi}=\phi_{0}\boldsymbol{v}_{0}$ (5.4)

where $\phi_{0}\in\mathbb{C}$ and $\boldsymbol{v}_{0}$ is the zero mode of $L$
given by (3.3). We then expand the scalar fields around the saddle point as

$\begin{split}\boldsymbol{\phi}&=\phi_{0}\boldsymbol{v}_{0}+\frac{1}{\sqrt{t}}\tilde{\boldsymbol{\phi}},\\\
\bar{\boldsymbol{\phi}}&=\bar{\phi}_{0}\boldsymbol{v}_{0}+\frac{1}{\sqrt{t}}\tilde{\bar{\boldsymbol{\phi}}},\\\
\end{split}$ (5.5)

where we have rescaled the fluctuation by $1/\sqrt{t}$ for the later purpose.

For the moment map $\mu(P^{f})$, the latter saddle point equation just means
$P^{f}=1$. Since the plaquette variable is given by

$P^{f}=\exp\left\\{i{{L^{T}}^{f}}_{e}A^{e}\right\\},$ (5.6)

the moment map constraint is solved by

${{\check{L}^{T}}{}^{f}}_{e}\hat{A}^{e}=2\pi k^{f},$ (5.7)

where $k^{f}$ are integer numbers. As same as the scalar fields, we expand the
gauge field around the fixed point as

$\boldsymbol{A}=\hat{\boldsymbol{A}}+\frac{1}{\sqrt{t}}\tilde{\boldsymbol{A}}\,.$
(5.8)

In particular, around the fix points, the plaquette variable approximately
behaves as

$\boldsymbol{\mu}\sim\frac{1}{\sqrt{t}}\check{L}^{T}\tilde{\boldsymbol{A}},$
(5.9)

up to a linear order of a fluctuation $\tilde{\boldsymbol{A}}$. Then the face
part of the bosonic action around the saddle point becomes

$\boldsymbol{Y}^{T}\cdot\left(\boldsymbol{Y}-\frac{2}{\sqrt{t}}\check{L}^{T}\tilde{\boldsymbol{A}}\right)\,.$
(5.10)

Looking at this expression, we see that the trace mode in the auxiliary field
$\boldsymbol{Y}$ is decoupled from the gauge field at the saddle point. We
then separate it from the others as

$\boldsymbol{Y}=\frac{1}{\sqrt{t}}\left(Y_{0}\boldsymbol{u}_{0}+\tilde{\boldsymbol{Y}}\right)\,,$
(5.11)

and rewrite (5.10) as

$\frac{1}{t}\left\\{Y_{0}^{2}+\tilde{\boldsymbol{Y}}^{T}\cdot\left(\tilde{\boldsymbol{Y}}-2\check{L}\tilde{\boldsymbol{A}}\right)\right\\}\,.$
(5.12)

Note that we have put the factor $\frac{1}{\sqrt{t}}$ also to $Y_{0}$ because
it is not the zero mode in the action but is just a free mode as we have seen
above.

After integrating out $\tilde{\boldsymbol{Y}}$, the 1-loop effective bosonic
action becomes

$S_{B}^{\text{1-loop}}\equiv\lim_{t\to\infty}tS_{B}=\frac{1}{2g^{2}}\left\\{-Y_{0}^{2}+\tilde{\bar{\boldsymbol{\phi}}}^{T}\Delta_{V}\tilde{\boldsymbol{\phi}}+\tilde{\boldsymbol{A}}^{T}\check{L}\check{L}^{T}\tilde{\boldsymbol{A}}\right\\}\,.$
(5.13)

We can ignore the first term in the present localization argument, since the
integral of $Y_{0}$ is just Gaussian (by a rotation $Y_{0}\to iY_{0}$) and
irrelevant if any operator coupled with $\boldsymbol{Y}$ is not inserted.
However, we will see later that the existence of $Y_{0}$ becomes important
when inserting suitable operators including the auxiliary field.

For the fermions, we expand the fields around the fermion zero modes (if they
exist) as

$\displaystyle\boldsymbol{\eta}={\eta_{0}}\boldsymbol{v}_{0}+\frac{1}{\sqrt{t}}\tilde{\boldsymbol{\eta}},\quad\boldsymbol{\chi}={\chi_{0}}\boldsymbol{u}_{0}+\frac{1}{\sqrt{t}}\tilde{\boldsymbol{\chi}},\quad\boldsymbol{\lambda}=\sum_{I=1}^{2h}{\lambda_{0}^{I}}\boldsymbol{e}_{0}^{I}+\frac{1}{\sqrt{t}}\tilde{\boldsymbol{\lambda}}\,.$
(5.14)

Then the 1-loop effective action for the fermions reduces to

$S_{F}^{\text{1-loop}}\equiv\lim_{t\to\infty}tS_{F}=-\frac{1}{g^{2}}\left\\{\tilde{\boldsymbol{\eta}}^{T}L^{T}\tilde{\boldsymbol{\lambda}}+\tilde{\boldsymbol{\chi}}^{T}\check{L}^{T}\tilde{\boldsymbol{\lambda}}\right\\}.$
(5.15)

Note here that the fermion zero modes do not appear at all in the effective
action since they are defined as the kernels of $L$ and $\check{L}$.

For the inserted operator $f[X]$, by inserting the expansion (5.5) and (5.14),
we obtain

$\lim_{t\to\infty}f[X]=f[X_{0}]\,,$ (5.16)

where $X_{0}$ is the collective expression of the variables of the saddle
points. Therefore, in evaluating the integration, we have only to consider the
inserted operator only in integrating over $X_{0}$.

###  5.2 Gauge fixing and 1-loop determinant

The gauge transformation of the edge variable $A^{e}$ is given by

$\boldsymbol{A}^{\prime}=\boldsymbol{A}-L\boldsymbol{\xi},$ (5.17)

where $\boldsymbol{\xi}\in{\cal V}_{V}$ is a gauge transformation parameter.
We can see immediately (the exponent of) the plaquette variable

$\boldsymbol{P}=\exp\left[i\check{L}^{T}\boldsymbol{A}\right]$ (5.18)

is invariant under this transformation because of the orthogonality of the
incidence matrices (3.13). This gauge invariance still keeps in the 1-loop
effective action (5.13) and (5.15).

To proceed with the quantization (path integral) of the 1-loop effective
theory, we introduce a gauge fixing condition for the fluctuations

$L^{T}\tilde{\boldsymbol{A}}=0,$ (5.19)

which is an analogy with the Lorentz gauge in the continuous theory.

Introducing the Faddeev-Popov (FP) ghosts
$(\boldsymbol{c},\bar{\boldsymbol{c}})$ and the Nakanishi-Lautrup (NL) field
$\boldsymbol{B}$ defined on $V$ obey the BRST transformations

$\delta_{B}\boldsymbol{c}=0,\quad\delta_{B}\bar{\boldsymbol{c}}=2\boldsymbol{B},\quad\delta_{B}\boldsymbol{B}=0,$
(5.20)

the gauge fixing and FP term is written by a BRST exact form

$\begin{split}S_{\rm{GF+FP}}&=-\frac{1}{4g^{2}}\delta_{B}\left\\{\bar{\boldsymbol{c}}^{T}\cdot\left(\boldsymbol{B}-2L^{T}\tilde{\boldsymbol{A}}\right)\right\\}\\\
&=\frac{1}{2g^{2}}\left\\{\bar{\boldsymbol{c}}^{T}\Delta_{V}\boldsymbol{c}-\boldsymbol{B}^{T}\cdot\left(\boldsymbol{B}-2L^{T}\tilde{\boldsymbol{A}}\right)\right\\},\end{split}$
(5.21)

where $\Delta_{V}=L^{T}L$ and we have used the BRST transformation of the
gauge field

$\delta_{B}\tilde{\boldsymbol{A}}=-L\boldsymbol{c}.$ (5.22)

Combining the original action and gauge fixing terms, we obtain the total
action

$\begin{split}S^{\prime}&=S^{\text{1-loop}}_{B}+S^{\text{1-loop}}_{F}+S_{\rm{GF+FP}}\\\
&=\frac{1}{2g^{2}}\Big{\\{}\tilde{\bar{\boldsymbol{\phi}}}^{T}\Delta_{V}\tilde{\boldsymbol{\phi}}+\tilde{\boldsymbol{A}}^{T}\check{L}\check{L}^{T}\tilde{\boldsymbol{A}}-2\tilde{\boldsymbol{\eta}}^{T}L^{T}\tilde{\boldsymbol{\lambda}}-2\tilde{\boldsymbol{\chi}}^{T}\check{L}^{T}\tilde{\boldsymbol{\lambda}}\\\
&\hskip
105.2751pt+\bar{\boldsymbol{c}}^{T}\Delta_{V}\boldsymbol{c}-\boldsymbol{B}^{T}\cdot\left(\boldsymbol{B}-2L^{T}\tilde{\boldsymbol{A}}\right)\Big{\\}}.\end{split}$
(5.23)

After eliminating the NL field $\boldsymbol{B}$, we get

$S^{\prime}=\frac{1}{2g^{2}}\Big{\\{}\tilde{\bar{\boldsymbol{\phi}}}^{T}\Delta_{V}\tilde{\boldsymbol{\phi}}+\bar{\boldsymbol{c}}^{T}\Delta_{V}\boldsymbol{c}+\tilde{\boldsymbol{A}}^{T}\Delta_{E}\tilde{\boldsymbol{A}}-2\tilde{\boldsymbol{\eta}}^{T}L^{T}\tilde{\boldsymbol{\lambda}}-2\tilde{\boldsymbol{\chi}}^{T}\check{L}^{T}\tilde{\boldsymbol{\lambda}}\Big{\\}},$
(5.24)

where $\Delta_{E}=LL^{T}+\check{L}\check{L}^{T}$.

From this quadratic 1-loop effective action, we can perform the path integral
for the non-zero modes explicitly and the 1-loop determinant becomes

$\begin{split}(\text{1-loop
det})&=\frac{\det^{\prime}\Delta_{V}}{\det^{\prime}\Delta_{V}}\frac{\left(\det^{\prime}\Delta_{V}\det^{\prime}\Delta_{F}\det^{\prime}\Delta_{E}\right)^{1/4}}{\sqrt{\det^{\prime}\Delta_{E}}}\\\
&=\left(\frac{\det^{\prime}\Delta_{V}\det^{\prime}\Delta_{F}}{\det^{\prime}\Delta_{E}}\right)^{1/4}=1\,,\end{split}$
(5.25)

up to an irrelevant sign factor due to the Pfaffian from the fermions, where
the prime denotes that the zero modes are omitted in the evaluation of the
determinant and we have used the fact that the non-zero eigenvalues of
$\Delta_{E}$ are identical with those of $\Delta_{V}$ and $\Delta_{F}$:

$\operatorname{Spec}^{\prime}\Delta_{V}\oplus\operatorname{Spec}^{\prime}\Delta_{F}=\operatorname{Spec}^{\prime}\Delta_{E}\,.$
(5.26)

Since the zero modes are dropped from the 1-loop effective action
$S_{B}^{\text{1-loop}}$ and $S_{F}^{\text{1-loop}}$, there exist residual
integrals over the zero modes after integrating out the non-zero modes,
namely, the vev reduces to the integral over the zero modes

$\left\langle f[X]\right\rangle={\cal N}\int
d\phi_{0}d\bar{\phi}_{0}dY_{0}\left(\prod_{I=1}^{2h}d\lambda_{0}^{2h-I+1}\right)d\chi_{0}d\eta_{0}\,f[X_{0}]e^{\frac{1}{2g^{2}}Y_{0}^{2}}\,,$
(5.27)

up to an irrelevant normalization constant ${\cal N}$ of the path integral
measure. Note that this reduction works only when $f[X]$ is $Q$-closed. In
particular, by setting $f[X]=1$, we again see that the partition function
vanishes due to the Grassmann integral as pointed out in the previous
subsection.

###  5.3 Compensating for the zero modes

As we have discussed above, the vev of the operator reduces to the residual
integral over the zero modes after integrating out the non-zero modes.
Therefore, so that the operator has non-trivial vev, we need to insert at
least suitable fermion zero modes. In the previous subsection, we simply added
the mass term of the fermion zero modes (4.40), but such a not $Q$-invariant
operator is not appropriate in the present situation. Instead, we here
consider $Q$-closed operators in order not to spoil the above localization
argument. Such $Q$-closed operators behave as mass terms for the fermions,
namely an exponential of the fermion bi-linear term, including all the fermion
zero modes. We call this kind of physical operator the compensator in the
following [54].

First, we define a $Q$-exact operator which includes $\boldsymbol{\eta}$ and
$\boldsymbol{\chi}$ as

${\cal O}_{\eta\chi}\equiv Q\left(\chi_{f}\bar{\cal
W}^{\prime}(\bar{\phi}^{f})\right)=Y_{f}\bar{\cal
W}^{\prime}(\bar{\phi}^{f})+2\bar{\cal
W}^{\prime\prime}(\bar{\phi}_{f})\eta_{f}\chi^{f},$ (5.28)

where $\bar{\cal W}^{\prime}(\bar{\phi}^{f})$ is an analytic function of
$\bar{\phi}^{f}$ only555 Since we can regard $\bar{\cal W}$ as the
superpotential, the first and second derivatives of $\bar{\cal W}$ appear
here. . If we insert the exponential of this operator into the path integral,
all the fields are effectively replaced by their zero modes as mentioned in
the previous subsection and the inserted operator reduces to

$e^{-\frac{1}{g^{2}}{\cal O}_{\eta\lambda}}\to-\frac{2}{g^{2}}\bar{\cal
W}^{\prime\prime}(\bar{\phi}_{0})\eta_{0}\chi_{0}e^{-\frac{1}{g^{2}}Y_{0}\bar{\cal
W}^{\prime}(\bar{\phi}_{0})}$ (5.29)

in the integrand. It compensates for the zero modes of $\boldsymbol{\eta}$ and
$\boldsymbol{\chi}$ as expected.

We next define an operator which includes bi-linear terms of
$\boldsymbol{\lambda}$ to compensate for the zero-mode integral of
$\lambda_{0}^{I}$. As one of the candidates of the fermion bi-linear term, we
now consider an operator

$\frac{1}{2}\boldsymbol{\lambda}^{T}\Omega\boldsymbol{\lambda},$ (5.30)

where $\Omega$ is an anti-symmetric $n_{E}\times n_{E}$ matrix, namely
$\Omega^{T}=-\Omega$. The supercharge $Q$ is acting on this by

$\begin{split}\frac{1}{2}Q\left(\boldsymbol{\lambda}^{T}\Omega\boldsymbol{\lambda}\right)&=-\boldsymbol{\phi}^{T}L^{T}\Omega\boldsymbol{\lambda}.\end{split}$
(5.31)

On the other hand, we consider the operator piece

$\check{\boldsymbol{\phi}}^{T}\cdot\check{L}^{T}\boldsymbol{A},$ (5.32)

where $\check{\boldsymbol{\phi}}$ is a dual scalar field on the face $F$,
which is linearly constructed from the original scalar field
$\boldsymbol{\phi}$ on $V$ via

$\check{\boldsymbol{\phi}}=M\boldsymbol{\phi},$ (5.33)

where $M$ is an $n_{F}\times n_{V}$ matrix. Note here that there are multiple
candidates for $M$; typically $\check{\phi}^{f}$ is determined by $\phi^{v}$
on the representative point at the boundary of the face.

Substituting (5.33) into (5.32) and applying the $Q$-transformation, we find

$Q\left(\boldsymbol{\phi}^{T}M^{T}\check{L}^{T}\boldsymbol{A}\right)=\boldsymbol{\phi}^{T}M^{T}\check{L}^{T}\boldsymbol{\lambda}.$
(5.34)

Then, using (5.31) and (5.34), we see that the combined operator

${\cal
O}_{\lambda\lambda}=\check{\boldsymbol{\phi}}^{T}\check{L}^{T}\boldsymbol{A}+\frac{1}{2}\boldsymbol{\lambda}^{T}\Omega\boldsymbol{\lambda},$
(5.35)

is $Q$-closed if there is a relation between the matrices

$\Omega L+\check{L}M=0.$ (5.36)

So the operator ${\cal O}_{\lambda\lambda}$ could become a non-trivial
observable. We here note that the operator (5.35) is $Q$-closed but is not
$Q$-exact.

As well as ${\cal O}_{\eta\chi}$, if we insert $e^{-i{\cal
O}_{\lambda\lambda}}$ in the path integral, it reduces to an integrand of the
zero modes as a summation over the saddle points

$e^{-i{\cal O}_{\lambda\lambda}}\to\sum_{k\in{\mathbb{Z}}}e^{-i\left(2\pi
k\phi_{0}-\frac{1}{2}\lambda_{0}^{I}(\boldsymbol{e}_{0}^{I})^{T}\Omega\boldsymbol{e}_{0}^{J}\lambda_{0}^{J}\right)}={\cal
C}\left(\prod_{I=1}^{2h}\lambda_{0}^{I}\right)\sum_{n\in{\mathbb{Z}}}\delta(\phi_{0}-n),$
(5.37)

where ${\cal C}$ is an irrelevant constant and we have used that
$\boldsymbol{A}$ satisfies

${{\check{L}^{T}{}}^{f}}_{e}\hat{A}^{e}=2\pi k^{f},$ (5.38)

at the saddle point with the total magnetic flux $k=\sum_{f\in F}k^{f}$, and
the Poisson summation formula $\sum_{k\in{\mathbb{Z}}}e^{2\pi
ikx}=\sum_{n\in{\mathbb{Z}}}\delta(x-n)$. It compensates for the zero modes of
$\boldsymbol{\lambda}$ as expected.

Then, if we insert the compensator $e^{\frac{1}{g^{2}}{\cal
O}_{\eta\chi}}e^{-i{\cal O}_{\lambda\lambda}}$ in the path integral, we can
compensate for all the fermion zero modes and make the integration well-
defined. However, this is not the only effect of the compensator: it can
handle not only fermion zero modes but also the boson zero modes at the same
time.

To see this, let us consider the vev

$\left\langle f[\boldsymbol{\phi}]\,e^{-\frac{1}{2g^{2}}{\cal
O}_{\eta\chi}}\,e^{-i{\cal O}_{\lambda\lambda}}\right\rangle$ (5.39)

using the localization method, where $f[\boldsymbol{\phi}]$ is an analytic
function constructed by $\boldsymbol{\phi}$ only and trivially $Q$-closed.
Using (5.27) with (5.29) and (5.37), we obtain

$\displaystyle\left\langle f[\boldsymbol{\phi}]\,e^{-\frac{1}{2g^{2}}{\cal
O}_{\eta\chi}}\,e^{-i{\cal O}_{\lambda\lambda}}\right\rangle$
$\displaystyle=-\frac{2}{g^{2}}{\cal N}{\cal
C}\int\left(\prod_{I=1}^{2h}d\lambda_{0}^{2h-I+1}\right)d\chi_{0}d\eta_{0}\,\left(\eta_{0}\chi_{0}\prod_{I=1}^{2h}\lambda_{0}^{I}\right)$
$\displaystyle\qquad\times\int d\phi_{0}d\bar{\phi}_{0}dY_{0}\,\bar{\cal
W}^{\prime\prime}(\bar{\phi}_{0})f(\phi_{0})e^{\frac{1}{2g^{2}}(Y_{0}^{2}-2\bar{\cal
W}^{\prime}(\bar{\phi}_{0})Y_{0})}\sum_{n\in{\mathbb{Z}}}\delta(\phi_{0}-n)$
$\displaystyle=-\frac{\sqrt{8\pi}}{g}{\cal N}{\cal
C}\sum_{n\in{\mathbb{Z}}}f(n)\,,$ (5.40)

where, after eliminating $Y_{0}$, the integration over $\bar{\phi}_{0}$
becomes just a Gaussian integral by changing the variable from
$\bar{\phi}_{0}$ to $\bar{\cal W}^{\prime}(\bar{\phi}_{0})$. This result of
course holds for $f[X]=1$, and thus the partition function is also regularized
by inserting the compensators666 We also have to regularize
$\sum_{n\in{\mathbb{Z}}}1$ in some way though. . In this sense, the
compensators give a regularization of the zero modes in the Abelian theory.

##  6 Non-Abelian Gauge Theory

In this section, we generalize our discussion to the non-Abelian gauge group
$G=U(N)$. The bosonic variables (fields) on $V$, $E$, and $F$ are denoted by
$\Phi^{v}$, $U^{e}$, and $Y^{f}$, respectively. Also, the fermionic fields
$\eta^{v}$, $\lambda^{e}$, and $\chi^{f}$ exist on $V$, $E$, and $F$,
respectively. For the non-Abelian gauge group, all the fields except for
$U^{e}$ are extended to the adjoint representation of $U(N)$, namely $N\times
N$ matrices, while the edge variables $U^{e}$ are $N\times N$ unitary
matrices.

The supersymmetry transformation is given by

$\begin{array}[]{lcl}Q\Phi^{v}=0,&&\\\
Q\bar{\Phi}^{v}=2\eta^{v},&&Q\eta^{v}=\frac{i}{2}[\Phi^{v},\bar{\Phi}^{v}],\\\
QU^{e}=i\lambda^{e}U^{e},&&Q\lambda^{e}=-{{L_{U}}^{e}}_{v}\Phi^{v}+i\lambda^{e}\lambda^{e},\\\
QY^{f}=i[\check{\Phi}^{f},\chi^{f}],&&Q\chi^{f}=Y^{f},\end{array}$ (6.1)

where $L_{U}$ is defined as a gauge covariant incidence matrix;

${{L_{U}}^{e}}_{v}\Phi^{v}\equiv
U^{e}\Phi^{t(e)}{U^{e}}^{{\dagger}}-\Phi^{s(e)},$ (6.2)

and $f$ of $\check{\Phi}^{f}$ denotes the representative vertex of the face
$f$.

It is sometimes useful to define

$\Lambda^{e}\equiv\lambda^{e}U^{e}\,.$ (6.3)

The supersymmetry transformations for $U^{e}$ and $\Lambda^{e}$ become

$\begin{split}QU^{e}=i\Lambda^{e},\quad
Q\Lambda^{e}=-(U^{e}\Phi^{t(e)}-\Phi^{s(e)}U^{e}).\end{split}$ (6.4)

We can construct the supersymmetric action in the $Q$-exact form;

$S=-\frac{1}{2g^{2}}Q\operatorname{Tr}\left\\{\frac{i}{2}\eta_{v}[\Phi^{v},\bar{\Phi}^{v}]+(\bar{\Phi}_{t(e)}U_{e}^{\dagger}-U_{e}^{\dagger}\bar{\Phi}_{s(e)})\Lambda^{e}+\chi_{f}\left(Y^{f}-2\mu(P^{f})\right)\right\\}.$
(6.5)

The moment map $\mu(P^{f})$ in the action is a function of the plaquette
variable on each face labeled by $f$, which is defined by an ordered product
around a face

$P^{f}\equiv\left(U^{e_{1}}\right)^{{{\check{L}^{T}{}}^{f}}_{e_{1}}}\left(U^{e_{2}}\right)^{{{\check{L}^{T}{}}^{f}}_{e_{2}}}\cdots\left(U^{e_{n}}\right)^{{{\check{L}^{T}{}}^{f}}_{e_{n}}},$
(6.6)

where $\\{e_{1},e_{2},\cdots,e_{n}\\}$ are edges that surround the face $f$ in
this order. We choose the function $\mu(P^{f})$ so that it has a unique vacuum
at $P^{f}=1$ and asymptotically behaves as the field strength of the gauge
field around the vacuum in order to induce the gauge kinetic term after
eliminating the auxiliary field $Y^{f}$ [5, 50].

Using the supersymmetry transformations, the bosonic part becomes

$S_{B}=\frac{1}{2g^{2}}\operatorname{Tr}\left\\{\frac{1}{4}[\Phi^{v},\bar{\Phi}^{v}]^{2}+|U^{e}\Phi^{t(e)}-\Phi^{s(e)}U^{e}|^{2}-Y_{f}\left(Y^{f}-2\mu(P^{f})\right)\right\\},$
(6.7)

and the fermionic part becomes

$S_{F}=-\frac{1}{2g^{2}}\operatorname{Tr}\Big{\\{}-i\eta_{v}[\Phi^{v},\eta^{v}]+2(\eta_{t(e)}U_{e}^{\dagger}-U_{e}^{\dagger}\eta_{s(e)})\Lambda^{e}\\\
-i(\bar{\Phi}_{t(e)}U_{e}^{\dagger}\Lambda_{e}U_{e}^{\dagger}-U_{e}^{\dagger}\Lambda_{e}U_{e}^{\dagger}\bar{\Phi}_{s(e)})\Lambda^{e}\\\
-i\chi_{f}[\check{\Phi}^{f},\chi^{f}]+2\chi_{f}Q\mu(P^{f})\Big{\\}}.$ (6.8)

Using the $Q$-exact action, the partition function of the non-Abelian gauge
theory is given by

$Z=\int\prod_{v\in
V}\mathcal{D}\Phi^{v}\mathcal{D}\bar{\Phi}^{v}\mathcal{D}\eta^{v}\prod_{e\in
E}\mathcal{D}U^{e}\mathcal{D}\lambda^{e}\prod_{f\in
F}\mathcal{D}Y^{f}\mathcal{D}\chi^{f}\,e^{-S}.$ (6.9)

Under the $U(1)_{A}$ rotation, each field transforms as

$\begin{split}&\Phi^{v}\to e^{2i\theta_{A}}\Phi^{v},\quad\bar{\Phi}^{v}\to
e^{-2i\theta_{A}}\bar{\Phi}^{v},\quad U^{e}\to U^{e},\quad Y^{f}\to Y^{f},\\\
&\eta^{v}\to e^{-i\theta_{A}}\eta^{v},\quad\lambda^{e}\to
e^{i\theta_{A}}\lambda^{e},\quad\chi^{f}\to
e^{-i\theta_{A}}\chi^{f}.\end{split}$ (6.10)

Then, the path integral measure of the fermions has a $U(1)_{A}$ anomaly

$\prod_{v\in V}\mathcal{D}\eta^{v}\prod_{e\in
E}\mathcal{D}\lambda^{e}\prod_{f\in F}\mathcal{D}\chi^{f}\\\ \to e^{i\dim
U(N)\times(n_{V}-n_{E}+n_{F})\theta_{A}}\prod_{v\in
V}\mathcal{D}\eta^{v}\prod_{e\in E}\mathcal{D}\lambda^{e}\prod_{f\in
F}\mathcal{D}\chi^{f}\\\ =e^{iN^{2}\chi_{h}\theta_{A}}\prod_{v\in
V}\mathcal{D}\eta^{v}\prod_{e\in E}\mathcal{D}\lambda^{e}\prod_{f\in
F}\mathcal{D}\chi^{f}.$ (6.11)

We will later see that this $U(1)_{A}$ anomaly essentially comes from the
fermion zero modes.

As usual localization arguments, we can show that the $Q$-exact action (6.5)
is independent of an overall coupling constant $t$ of the rescaled action
$S\to tS$. So the saddle point approximation in the limit of $t\to\infty$
becomes exact and the path integral is localized at the saddle (fixed) points.
Form the bosonic part of the action $S_{B}$, we find that the saddle points
(localization fixed points) are given by the equations

$\displaystyle[\Phi^{v},\bar{\Phi}^{v}]$ $\displaystyle=$ $\displaystyle 0,$
(6.12) $\displaystyle{{L_{U}}^{e}}_{v}\Phi^{v}$ $\displaystyle=$
$\displaystyle 0,$ (6.13) $\displaystyle\mu(P^{f})$ $\displaystyle=$
$\displaystyle 0.$ (6.14)

The first equation (6.12) shows that $\Phi^{v}$ are diagonal. We denote this
diagonal solution by

$\hat{\Phi}^{v}=\operatorname{diag}(\hat{\phi}^{v,1},\hat{\phi}^{v,2},\ldots,\hat{\phi}^{v,N}).$
(6.15)

Using this diagonal expression, we can solve the second equation (6.13) by

$\displaystyle\hat{\phi}^{t(e),\pi_{e}(i)}=\hat{\phi}^{s(e),i},$ (6.16)
$\displaystyle U^{e}=\hat{U}^{e}\Pi^{e},$ (6.17)

where $\hat{U}^{e}$ is a diagonal matrix,

$\hat{U}^{e}=\operatorname{diag}(\hat{U}^{e,1},\hat{U}^{e,2},\cdots,\hat{U}^{e,N})\,,\quad\left(|\hat{U}^{e,i}|=1\right)$
(6.18)

and $\Pi^{e}$ is a permutation matrix which represents the order $N$
permutation $\pi_{e}\in\mathfrak{S}_{N}$ on the edge $e$.

Since the permutation belongs to the Weyl group of $U(N)$, we can choose
$\Pi^{e}=1$ without loss of generality by using a gauge transformation. So the
diagonal element of $\Phi^{v}$ on each vertex is written by a common diagonal
element independent of $v$, that is,

$\hat{\phi}^{v,i}=\phi_{0}^{i},$ (6.19)

which stands for a “constant” zero mode.

Finally, Eq. (6.14) implies a constraint on $U^{e,i}$;

$\prod_{e\in f}\left(\hat{U}^{e,i}\right)^{{{\check{L}^{T}}{}^{f}}_{e}}=1,$
(6.20)

for each $f$ and $i$.

Now let us consider an effective action near the saddle point. We expand
$\Phi^{v}$ and $U^{e}$ around the solution to the saddle point equation as

$\begin{split}\Phi^{v}&=\hat{\Phi}+\frac{1}{\sqrt{t}}\tilde{\Phi}^{v},\\\
U^{e}&=e^{\frac{i}{\sqrt{t}}\tilde{A}^{e}}\hat{U}^{e}\simeq\left(1+\frac{i}{\sqrt{t}}\tilde{A}^{e}\right)\hat{U}^{e},\end{split}$
(6.21)

where
$\hat{\Phi}=\operatorname{diag}(\phi_{0}^{1},\phi_{0}^{2},\ldots,\phi_{0}^{N})$.
All other fields including the fermions are treated as the fluctuations and
rescaled by $1/\sqrt{t}$ and we omit tilde for these fluctuations. Using the
Cartan-Weyl basis (see Appendix B), the Cartan parts are written as

$\hat{\Phi}=\phi_{0}^{i}\mathtt{H}_{i},\quad\hat{\bar{\Phi}}=\bar{\phi}_{0}^{i}\mathtt{H}_{i},\quad\hat{U}^{e}=\hat{U}^{e,i}\mathtt{H}_{i},$
(6.22)

and the fluctuations and fermions are expanded as follows;

$\begin{split}&\tilde{\Phi}^{v}=\tilde{\phi}^{v,i}\mathtt{H}_{i}+\tilde{\phi}^{v,\alpha}\mathtt{E}_{\alpha},\quad\tilde{\bar{\Phi}}^{v}=\tilde{\bar{\phi}}^{v,i}\mathtt{H}_{i}+\tilde{\bar{\phi}}^{v,\alpha}\mathtt{E}_{\alpha},\quad\tilde{A}^{e}=\tilde{A}^{e,i}\mathtt{H}_{i}+\tilde{A}^{e,\alpha}\mathtt{E}_{\alpha},\\\
&\eta^{v}=\eta^{v,i}\mathtt{H}_{i}+\eta^{v,\alpha}\mathtt{E}_{\alpha},\quad\chi^{f}=\chi^{f,i}\mathtt{H}_{i}+\chi^{f,\alpha}\mathtt{E}_{\alpha},\quad\lambda^{e}=\lambda^{e,i}\mathtt{H}_{i}+\lambda^{e,\alpha}\mathtt{E}_{\alpha},\end{split}$
(6.23)

where the upper and lower indices of $i$ and $\alpha$ are contracted.

Using these expansions, we find

$\begin{split}[\Phi^{v},\bar{\Phi}^{v}]&=\frac{1}{\sqrt{t}}\left([\hat{\Phi},\tilde{\bar{\Phi}}^{v}]+[\tilde{\Phi}^{v},\hat{\bar{\Phi}}]\right)+{\cal
O}(1/t),\\\
&=\frac{1}{\sqrt{t}}\left(\alpha(\phi_{0})\tilde{\bar{\phi}}^{v,\alpha}-\alpha(\bar{\phi}_{0})\tilde{\phi}^{v,\alpha}\right)\mathtt{E}_{\alpha}+{\cal
O}(1/t),\\\
{{L_{U}}^{e}}_{v}\Phi^{v}&=\frac{1}{\sqrt{t}}\left({L^{e}}_{v}\tilde{\phi}^{v,i}\mathtt{H}_{i}+{{L_{\hat{U}}}^{e}}_{v}\tilde{\phi}^{v,\alpha}\mathtt{E}_{\alpha}-i\alpha(\phi_{0})\tilde{A}^{e,\alpha}\mathtt{E}_{\alpha}\right)+{\cal
O}(1/t),\\\
\mu(P^{f})&=\frac{1}{\sqrt{t}}\left({{\check{L}^{T}{}}^{f}}_{e}\tilde{A}^{e,i}\mathtt{H}_{i}+{{\check{L}^{\dagger}_{\hat{U}}{}}^{f}}_{e}\tilde{A}^{e,\alpha}\mathtt{E}_{\alpha}\right)+{\cal
O}(1/t),\end{split}$ (6.24)

up to the leading order, where

$\alpha(\phi_{0})\equiv\alpha_{i}\phi_{0}^{i},\quad\alpha(\bar{\phi}_{0})\equiv\alpha_{i}\bar{\phi}_{0}^{i}.$
(6.25)

When we expand the moment map $\mu(P^{f})$, we require the covariant dual
incidence matrix ${{\check{L}^{\dagger}_{\hat{U}}{}}^{f}}_{e}$, which is
defined by

$\left.\delta\mu(P^{f})\right|_{U^{e}=\hat{U}^{e}}=i\sum_{e\in
f}{{\check{L}^{\dagger}_{\hat{U}}{}}^{f}}_{e}\delta A^{e}\equiv i\sum_{e\in
f}{{\check{L}^{\dagger}{}}^{f}}_{e}{\hat{X}^{f}}_{e}\delta
A^{e}{\hat{X}^{{\dagger}f}}_{e},$ (6.26)

with

${\hat{X}^{f}}_{e_{i}}=\begin{cases}\hat{U}^{\check{L}^{f}_{e_{1}}}_{e_{1}}\hat{U}^{\check{L}^{f}_{e_{2}}}_{e_{2}}\cdots\hat{U}^{\check{L}^{f}_{e_{i-1}}}_{e_{i-1}}&\text{if
}{\check{L}^{f}{}}_{e_{i}}=+1\\\
\hat{U}^{\check{L}^{f}_{e_{1}}}_{e_{1}}\hat{U}^{\check{L}^{f}_{e_{2}}}_{e_{2}}\cdots\hat{U}^{\check{L}^{f}_{e_{i}}}_{e_{i}}&\text{if
}{\check{L}^{f}{}}_{e_{i}}=-1\end{cases}.$ (6.27)

Using the above expansions up to quadratic order of fluctuations, we obtain
the rescaled 1-loop effective bosonic action;

$\displaystyle S_{B}^{\text{1-loop}}\equiv$
$\displaystyle\lim_{t\to\infty}tS_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2g^{2}}\Bigg{[}\sum_{i=1}^{N}\Big{\\{}|{L^{e}{}}_{v}\tilde{\phi}^{v,i}|^{2}-Y^{i}_{f}(Y^{f,i}-2{{\check{L}^{T}{}}^{f}}_{e}\tilde{A}^{e,i})\Big{\\}}$
$\displaystyle\hskip
14.22636pt+\sum_{\alpha}\Big{\\{}\frac{1}{4}\left|\alpha(\phi_{0})\tilde{\bar{\phi}}^{v,\alpha}-\alpha(\bar{\phi}_{0})\tilde{\phi}^{v,\alpha}\right|^{2}+|{{L_{\hat{U}}}^{e}{}}_{v}\tilde{\phi}^{v,\alpha}|^{2}+|\alpha(\phi_{0})\tilde{A}^{e,\alpha}|^{2}$
$\displaystyle\hskip
56.9055pt-Y^{-\alpha}_{f}(Y^{f,\alpha}-2{{\check{L}^{\dagger}_{\hat{U}}{}}^{f}}_{e}\tilde{A}^{e,\alpha})\Big{\\}}\Bigg{]}.$
(6.28)

Similarly, the fermionic part of the 1-loop effective action becomes

$\displaystyle S_{F}^{\text{1-loop}}\equiv$
$\displaystyle\lim_{t\to\infty}tS_{F}\ $ $\displaystyle=$
$\displaystyle-\frac{1}{2g^{2}}\Bigg{[}\sum_{i=1}^{N}\Big{\\{}2\eta^{i}_{v}{{L^{T}{}}^{v}}_{e}\lambda^{e,i}+2\chi^{i}_{f}{{\check{L}^{T}{}}^{f}}_{e}\lambda^{e,i}\Big{\\}}$
$\displaystyle\hskip
22.76219pt+\sum_{\alpha}\Big{\\{}2\eta^{-\alpha}_{v}{{L_{\hat{U}}^{\dagger}{}}^{v}}_{e}\lambda^{e,\alpha}+2\chi^{-\alpha}_{f}{{\check{L}_{\hat{U}}^{\dagger}{}}^{f}}_{e}\lambda^{e,\alpha}$
$\displaystyle\hskip
56.9055pt-i\alpha(\phi_{0})\eta^{-\alpha}_{v}\eta^{v,\alpha}-i\alpha(\phi_{0})\chi^{-\alpha}_{f}\chi^{f,\alpha}+i\alpha(\bar{\phi}_{0})\lambda^{-\alpha}_{e}\lambda^{e,\alpha}\Big{\\}}\Bigg{]}.$
(6.29)

This effective action gives the same path integral as the original action
owing to the $Q$-exactness of the action.

We first integrate over the components of the root vectors. To this end, we
fix the gauge symmetry $U(1)^{N}$ in the 1-loop actions (6.28) and (6.29) by
introducing the FP ghost $(c^{v,\alpha},\bar{c}^{v,\alpha})$ and NL field
$B^{v,\alpha}$. The corresponding BRST transformations are given by

$\begin{split}&\delta_{B}c^{v,\alpha}=0,\quad\delta_{B}\bar{c}^{v,\alpha}=2B^{v,\alpha},\quad\delta_{B}B^{v,\alpha}=0,\\\
&\delta_{B}\tilde{\phi}^{v,\alpha}=-i\alpha(\phi_{0})c^{v,\alpha},\quad\delta_{B}\tilde{\bar{\phi}}^{v,\alpha}=-i\alpha(\bar{\phi}_{0})c^{v,\alpha},\\\
&\delta_{B}\tilde{A}^{e,\alpha}=-{{L_{\hat{U}}}^{e}}_{v}c^{v,\alpha},\end{split}$
(6.30)

where we assume that the FP ghost and NL field are the same order in $t$ as
the fluctuations. We define the gauge fixing function for the root vectors by

$f^{v,\alpha}\equiv
B^{v,\alpha}-2{{L_{\hat{U}}^{\dagger}{}}^{v}}_{e}\tilde{A}^{e,\alpha}+i\alpha(\phi_{0})\tilde{\bar{\phi}}^{v,\alpha}+i\alpha(\bar{\phi}_{0})\tilde{\phi}^{v,\alpha}.$
(6.31)

Then the gauge fixing and FP ghost term is given in the BRST exact form by

$\begin{split}S^{\text{root}}_{\text{GF+FP}}&=-\frac{1}{4g^{2}}\delta_{B}\sum_{\alpha}\bar{c}_{v}^{-\alpha}f^{v,\alpha}\\\
&=\frac{1}{2g^{2}}\sum_{\alpha}\Bigg{[}-B_{v}^{-\alpha}f^{v,\alpha}+\bar{c}_{v}^{-\alpha}\left({{\Delta^{V}_{\hat{U}}{}}^{v}}_{v^{\prime}}+|\alpha(\phi_{0})|^{2}{\delta^{v}}_{v^{\prime}}\right)c^{v^{\prime},\alpha}\Bigg{]},\end{split}$
(6.32)

where

${{\Delta^{V}_{\hat{U}}{}}^{v}}_{v^{\prime}}\equiv{{L_{\hat{U}}^{\dagger}{}}^{v}}_{e}{{L_{\hat{U}}{}}^{e}}_{v^{\prime}}.$
(6.33)

After eliminating the NL filed $B^{v,\alpha}$, we get the action for
$(\tilde{\phi}^{v,\alpha},\tilde{\bar{\phi}}^{v,\alpha})$

$S^{\text{root}}_{(\tilde{\phi},\tilde{\bar{\phi}})}=\frac{1}{2g^{2}}\sum_{\alpha}\tilde{\bar{\phi}}_{v}^{-\alpha}\left({{\Delta^{V}_{\hat{U}}{}}^{v}}_{v^{\prime}}+|\alpha(\phi_{0})|^{2}{\delta^{v}}_{v^{\prime}}\right)\tilde{\phi}^{v^{\prime},\alpha},$
(6.34)

whose 1-loop determinant is completely canceled with the contribution from the
ghost part in (6.32).

In addition, integrating out the auxiliary field $Y^{f,\alpha}$, the action
for the gauge boson reduces to

$S^{\text{root}}_{\tilde{A}}=\frac{1}{2g^{2}}\sum_{\alpha}\tilde{A}_{e}^{-\alpha}\left({{\Delta^{E}_{\hat{U}}{}}^{e}}_{e^{\prime}}+|\alpha(\phi_{0})|^{2}{\delta^{e}}_{e^{\prime}}\right)\tilde{A}^{e^{\prime},\alpha},$
(6.35)

where

${{\Delta^{E}_{\hat{U}}{}}^{e}}_{e^{\prime}}\equiv{{L_{\hat{U}}{}}^{e}}_{v}{{L_{\hat{U}}^{\dagger}{}}^{v}}_{e^{\prime}}+{{\check{L}_{\hat{U}}{}}^{e}}_{v}{{\check{L}_{\hat{U}}^{\dagger}{}}^{v}}_{e^{\prime}}.$
(6.36)

Then we obtain the 1-loop determinant for the gauge boson
$\tilde{A}^{e,\alpha}$

$\prod_{\alpha>0}\frac{1}{|\alpha(\phi_{0})|^{2n_{E}^{0}}\det^{\prime}\left(\Delta^{E}_{\hat{U}}+|\alpha(\phi_{0})|^{2}\right)},$
(6.37)

where $n_{E}^{0}$ is the number of the zero eigenstates for the edge Laplacian
and the prime on the determinant stands for omitting the zero eigenvalues.

Next, let us consider the integral of the fermions. We need to care about the
Laplacian zero modes for the fermions, but we get the 1-loop determinant

$\prod_{\alpha>0}\alpha(\phi_{0})^{n^{0}_{V}+n^{0}_{F}}\alpha(\bar{\phi}_{0})^{n_{E}^{0}}\\\
\times\sqrt{{\det}^{\prime}\left(\Delta^{V}_{\hat{U}}+|\alpha(\phi_{0})|^{2}\right){\det}^{\prime}\left(\Delta^{F}_{\hat{U}}+|\alpha(\phi_{0})|^{2}\right){\det}^{\prime}\left(\Delta^{E}_{\hat{U}}+|\alpha(\phi_{0})|^{2}\right)}\\\
=\prod_{\alpha>0}\alpha(\phi_{0})^{n^{0}_{V}+n^{0}_{F}}\alpha(\bar{\phi}_{0})^{n_{E}^{0}}{\det}^{\prime}\left(\Delta^{E}_{\hat{U}}+|\alpha(\phi_{0})|^{2}\right),$
(6.38)

where we have used that the non-zero eigenvalues of $\Delta^{E}_{\hat{U}}$ is
a combination of the non-zero eigenvalues of $\Delta^{V}_{\hat{U}}$ and
$\Delta^{F}_{\hat{U}}$, namely777 We can see that the condition (6.20)
guarantees the orthogonality like (3.13) for $L_{\hat{U}}$ and
$\check{L}_{\hat{U}}$ in the concrete examples. See Appendix A.

$\operatorname{Spec}^{\prime}\Delta^{V}_{\hat{U}}\oplus\operatorname{Spec}^{\prime}\Delta^{F}_{\hat{U}}=\operatorname{Spec}^{\prime}\Delta^{E}_{\hat{U}}.$
(6.39)

Combining the 1-loop determinant of the bosons (6.37) and fermions (6.38), we
finally obtain the total 1-loop determinant for the root vector components

$\prod_{\alpha>0}\alpha(\phi_{0})^{n^{0}_{V}+n^{0}_{F}-n^{0}_{E}}=\prod_{\alpha>0}\alpha(\phi_{0})^{\chi_{h}},$
(6.40)

where we have used the index theorem on the graph Laplacians as well as the
Abelian theory. Note here that the above 1-loop determinant has an anomaly
phase under $U(1)_{A}$ symmetry as

$\prod_{\alpha>0}\alpha(\phi_{0})^{\chi_{h}}\to
e^{iN(N-1)\chi_{h}\theta_{A}}\prod_{\alpha>0}\alpha(\phi_{0})^{\chi_{h}}.$
(6.41)

Let us next think about the Cartan part of the fluctuations and fermions. The
Cartan part is nothing but $N$ copies of the Abelian theory discussed in Sec.
5. Introducing the FP ghost $(c^{v,i},\bar{c}^{v,i})$, NL field $B^{v,i}$ and
gauge fixing function for the Cartan modes,

$f^{v,i}=B^{v,i}-2{{L^{T}}^{v}}_{e}\tilde{A}^{e,i},$ (6.42)

the gauge fixing term for the Cartan modes is given by

$\begin{split}S^{\text{Cartan}}_{\text{GF+FP}}&=-\frac{1}{4g^{2}}\delta_{B}\sum_{i=1}^{N}\bar{c}_{v}^{i}f^{v,i}\\\
&=\frac{1}{2g^{2}}\sum_{i=1}^{N}\Bigg{[}-B_{v}^{i}f^{v,i}+\bar{c}_{v}^{i}{{\Delta_{V}{}}^{v}}_{v^{\prime}}c^{v^{\prime},i}\Bigg{]}.\end{split}$
(6.43)

We find that the 1-loop determinant of the bosons
$(\tilde{\phi}^{v,i},\tilde{\bar{\phi}}^{v,i})$ and $(c^{v,i},\bar{c}^{v,i})$
are canceled with each other. After eliminating the auxiliary field $Y^{f,i}$
and NL field $B^{v,i}$, the integral of the gauge boson $\tilde{A}^{e,i}$
gives the 1-loop determinant

$\frac{1}{\left({\det}^{\prime}\Delta_{E}\right)^{N}}.$ (6.44)

This 1-loop determinant for the bosons is canceled with a 1-loop determinant
for the fermions

$\left({\det}^{\prime}\Delta_{V}{\det}^{\prime}\Delta_{F}{\det}^{\prime}\Delta_{E}\right)^{N/2},$
(6.45)

by using the same fact as Eq. (5.26) for the Cartan part.

Thus we finally obtain the path integral measure over the zero modes in the
Cartan subalgebra

$Z={\cal
N}\int\prod_{i=1}^{N}d\phi_{0}^{i}d\bar{\phi}^{i}_{0}dY_{0}^{i}\left(\prod_{I=1}^{2h}d\lambda_{0}^{i,2h-I+1}\right)d\chi_{0}^{i}d\eta_{0}^{i}\,e^{\frac{1}{2g^{2}}(Y_{0}^{i})^{2}}\prod_{\alpha>0}\alpha(\phi_{0})^{\chi_{h}},$
(6.46)

up to a normalization constant ${\cal N}$. The zero-mode integral is a
multiple of the Abelian gauge theory except for the Vandermonde type
determinant $\prod_{\alpha>0}\alpha(\phi_{0})^{\chi_{h}}$. This phenomenon is
the same as what occurs in the continuum field theory localization and is
called “diagonalization” or “Abelianization”. (See for review [61].) This
integral measure has the $U(1)_{A}$ anomaly

$\prod_{i=1}^{N}d\phi_{0}^{i}d\bar{\phi}^{i}_{0}\left(\prod_{I=1}^{2h}d\lambda_{0}^{i,2h-I+1}\right)d\chi_{0}^{i}d\eta_{0}^{i}\,\prod_{\alpha>0}\alpha(\phi_{0})^{\chi_{h}}\\\
\to
e^{iN^{2}\chi_{h}\theta_{A}}\prod_{i=1}^{N}d\phi_{0}^{i}d\bar{\phi}^{i}_{0}\left(\prod_{I=1}^{2h}d\lambda_{0}^{i,2h-I+1}\right)d\chi_{0}^{i}d\eta_{0}^{i}\,\prod_{\alpha>0}\alpha(\phi_{0})^{\chi_{h}},$
(6.47)

as expected.

Due to the existence of the fermion zero modes, the partition function itself
is ill-defined. So we need to insert an operator which compensates for the
fermionic zero modes. As mentioned above, the path integral reduces to the
multiple integrals of the Abelian ones. So we can compensate the fermionic
zero modes by inserting the compensator discussed in Sec. 5.3 for each Cartan
part. It however seems to be difficult to construct a compensator, which is
invariant under the non-Abelian gauge group and supersymmetric ($Q$-closed),
prior to the Abelianization. We leave the construction of the complete
compensator in the non-Abelian gauge theory as a future problem.

##  7 Conclusion and Discussion

In this paper, the properties of the discretized two-dimensional
supersymmetric gauge theory (the generalized Sugino model) given in [52] were
studied analytically by using the techniques of graph theory.

From a graph theory point of view, the model is defined on a two-dimensional
graph and its dual graph, and the action can be efficiently described using
the so-called incidence matrix $L$ and the dual incidence matrix $\check{L}$.

The incidence and the dual incidence matrix map from a vector on the vertices
to a vector on the edges and form a vector on the faces to a vector on the
edges, respectively, and obey the property such that $L\check{L}^{T}=0$.
Therefore, if we consider the vectors on the vertices, edges, and faces as
analogs of 0-form, 1-form, and 2-form, respectively, we can regard $L$ and
${\check{L}}^{T}$ as the exterior derivatives, and $L^{T}$ and ${\check{L}}$
as its dual. The cohomology can be defined using $L$ and $\check{L}$, and a
parallel argument of Hodge’s theorem on the Riemann surfaces can be developed
on the graphs. In particular, we found that the structure of the kernel of $L$
and $\check{L}$ is completely determined by the topology of the graph.

We used the properties of these matrices to examine the generalized Sugino
model with gauge group $U(1)$ and found that the number of fermion zero modes
depends on the topology of the graph. Since these zero modes make the
partition function ill-defined, it is necessary to insert an appropriate
operator including zero modes in the background. We proposed a mass term for
the fermions so that this operation is done automatically. We confirmed that
the introduction of this mass term regularizes the Dirac matrix and makes the
theory itself well-defined. We also derived anomalous WT identities on the
graph corresponding to the classical global $U(1)$ symmetry, which is broken
by quantum mechanically unless the topology of the graph is the torus. In the
continuous theory, this anomaly appears as the scalar curvature in the WT
identity. On the other hand, in the theory on a graph, a quantity related to
the degrees of the vertex and face arises instead of the scalar curvature.
This corresponds to the fact that the scalar curvature of a two-dimensional
surface is given by the deficit angle.

We examined the generalized Sugino model from the viewpoint of topological
field theory by restricting the physical quantity to $Q$-cohomology. We used
the so-called localization technique to compute the expected value of a
general $Q$-closed operator. As a result of localization, the vev can be
expressed in terms of the usual integration by the zero modes. In that case,
unless the operator contains all fermion zero modes, the vev trivially
vanishes. We constructed $Q$-invariant operators (compensators) that cancel
out the fermion zero modes, and gave a prescription for computing the vev for
nontrivial values. The compensators introduced here regularize the theory
properly.

We also extended the graph-theoretic description to the non-Abelian theory.
Reflecting the non-commutativity of gauge groups, the incidence and dual
incidence matrices are transformed to be covariant differences instead of
ordinary differences. This transformation eliminates the orthogonality of the
incidence matrix and the dual incidence matrix unless all the plaquette
variables are unity ($P^{f}=1$) and the fermion zero modes that appeared in
the Abelian theory are lifted in most configurations. Therefore, in most
configurations, the Dirac matrix is regular and the inverse exists. However,
the situation is different around the saddle points of the $Q$-transformation.
Using the localization technique with an appropriate gauge fixing, the non-
Abelian generalized Sugino model can be effectively reduced to an Abelian
theory. As a result, the evaluation of the partition function is completely
parallel to the calculation of that of the Abelian theory, and the fermion
zero modes arising at the saddle points make an important contribution. In
particular, it is confirmed that the partition function becomes ill-defined
unless these fermion zero modes on the saddle points are properly treated.

The fact that the non-Abelian generalized Sugino model is also affected by the
fermion zero modes is quite important when carrying out numerical simulations.
As mentioned above, the fermion zero modes appear only on the saddle points of
the $Q$-transformation and thus the Dirac matrix is regular in almost all
configurations. Therefore the numerical simulation proceeds even without any
special treatment for the zero modes. However, since the saddle points of the
$Q$-transformation are a part of classical configurations, the zero modes
would affect the computation especially in the region close to the continuum
limit, and there is a possibility that reliable results cannot be obtained.
This conclusion holds even in the case of a torus background where the anomaly
is canceled because the fermion zero modes still exist at the saddle points.

In the case of the torus, the fermion zero modes are completely lifted up by
imposing the anti-periodic boundary condition in the temporal direction to the
fermions. Therefore the numerical simulations for the system with finite
temperature are expected to work well. All the simulations imposing the anti-
periodic boundary conditions for the fermions in the temporal direction have
been successfully carried out [30, 31, 32, 33, 34, 35].

On the other hand, it was reported in [32] that numerical calculations did not
yield the expected results for two-point functions in the continuum limit888
See also [36] where the WT identity is analytically examined by using a ”semi-
perturbative” treatment. , while it was reported in [35] that the vevs of
Yukawa terms consistently degenerate in the continuum limit.

At first glance, it seems that the simulation will not work of the presence of
the fermion zero modes, but the situation is slightly more complicated. The
point is that the theory is expected to have (at least) two phases; the phase
where the eigenvalues of the scalar field form a bound state and the phase
where they run freely [35]. In the numerical simulation, the flat directions
of the scalar field are controlled by introducing a mass term, and thus the
configurations are all in the phase with the bound state. On the other hand,
the partition function (6.46) is obtained by integrating out all the
configurations including both phases. Therefore, although it is one of the
possibilities, if the fermion zero modes are effectively lifted up in the
phase with the bound state, the simulation would work well even if one takes
the periodic boundary condition for the fermions. It will be interesting to
check whether the situation changes or not if we deal with the fermion zero
modes in an appropriate way.

It is only when the background is torus that the boundary condition can
eliminate all the fermion zero modes. This is because changing the boundary
condition is equivalent to transforming $D=(L,\check{L})$. In the case of the
torus, $D$ is a square matrix, so all zero modes will be eliminated if we
transform it in such a way that the zero modes of $L$ and $\check{L}$ are
eliminated Imposing the anti-periodic boundary condition is an example of this
kind of modification. On the other hand, in non-torus cases, there are always
zero modes no matter how much $D$ is deformed since the rank of the
rectangular matrix $D$ is at most $\min(n_{E},n_{V}+n_{F})$. Therefore it is a
peculiarity of the torus that the zero modes can be dealt with just by
considering the finite temperature.

In the non-toric cases, we have to insert some corrections to the diagonal
blocks of $\not{D}$ to eliminate all the fermion zero modes. This is
equivalent to introducing a mass term to the fermions and the (4.40) is an
example of this kind of modification. In the case of non-Abelian theories, it
is necessary to introduce such a mass term that appropriately lifts the zero
modes arising on the $Q$-fixed points without breaking the gauge symmetry and
with respecting the $Q$-symmetry if possible. For the zero modes of $\eta$ and
$\chi$, we can simply extend the compensator (5.28) as

$Q{\rm tr}\left(\chi_{f}\bar{\cal W}^{\prime}(\bar{\phi}^{f})\right)\,.$ (7.1)

For the zero modes of $\lambda$, however, it is still an open problem to
construct such an operator that is $Q$-closed (not $Q$-exact) and includes bi-
linear terms of $\lambda$ like (5.35). Instead, $Q$-exact operators like

$Q{\rm tr}\left(\lambda_{l}P_{f}\right)$ (7.2)

may work. It is also interesting to analyze the property of non-Abelian
compensators.

One problem that has not yet been achieved in previous studies is the
introduction of matter fields. In supersymmetric gauge theories, in order to
introduce the matter field as a chiral multiplet, it is essential to consider
chiral fermions.

The chiral fermions have been constructed on the regular square lattice by
various methods, but how to construct chiral fermions on a discrete space
arbitrarily partitioned by a graph is completely unknown. However, in this
paper, it was clarified that the incidence matrix in graph theory has a deep
connection with the Dirac operator, so it seems possible to define chiral
fermions using graph theory. We could use graph theory to introduce chiral
fermions on discrete spaces, to construct supersymmetric gauge theories
including matter fields, and to analyze and understand chiral anomalies
induced by chiral fermions. These are also important future issues.

Once the introduction of the matter field is achieved, the interaction with
the gauge field also allows the construction of solitons such as vortices on
the discretized Riemann surface. At present, the construction of solitons on
the graph is a novel problem. It is also very interesting to understand the
non-perturbative effects by such solitons in supersymmetric gauge theories on
the graph.

As mentioned in the introduction, the continuum limit of the generalized
Sugino model is a topologically twisted ${\cal N}=(2,2)$ supersymmetric gauge
theory, which is a theory on the Riemann surface with a $U(1)_{A}$ background
field balanced with the spin connection of the background space-time and the
fermions behave as fields with integer spins. Interestingly, this property is
similar to the Kähler-Dirac fermion. In fact, it is argued in [62, 63] that
systems with Kähler-Dirac fermion have an anomaly proportional to the
background Euler number as well. These approaches would be compatible with the
lattice gravity, which realizes gravity via a random triangulation. For
example, in the formulation given in [64], it is essential to place the gauge
field on the edge of the triangulation and the spin connection on the dual
edge. It is remarkable that, in this formulation, the action of the gravity is
written in terms of the deficit angle of the dual plaquette, whereas the
anomaly in the local WT identities appears as the deficit angle as well. It
will be interesting to consider lattice gravity from the viewpoint of graph
theory. In particular, it is suggestive that the construction of the
discretized theory with the supersymmetry is possible only when the spin
connection and the background gauge field are properly balanced. Through
research in this direction, we expect to obtain new insights from graph theory
for lattice gravity in higher dimensions [64, 65, 66].

## Acknowledgments

We would like to thank S. Kamata and T. Misumi for useful discussions in the
early stage of this work. We would like to thank N. Kawamoto for useful
discussions at Hokkaido University and for teaching us stimulating references.
S.M would like to thank M. Hanada and D. Kadoh for useful discussion. This
work is supported in part by Grant-in-Aid for Scientific Research (KAKENHI)
(C) Grant Number 17K05422 (K. O.) and Grant-in-Aid for Scientific Research
(KAKENHI) (C), Grant Number 20K03934 (S. M.). A part of this work is motivated
by the results of numerical simulations carried out on Oakforest-PACS
computational system provided by Tokyo University through the HPCI System
Research Project (Project ID: hp200052).

##  Appendix A Examples of the Graph Data

In this Appendix, we give concrete examples of the graph data and objects and
check some properties. We also give plaquette variables on each face and
covariantized version of the (dual) incidence matrix in non-Abelian gauge
theory.

###  A.1 Tetrahedron

Figure 4: A directed graph for a tetrahedron. There are 4 vertices and 6
directed edges.

A directed graph associated with a tetrahedron is shown in Fig. 4. There 4
vertices which is labeled by $V(\Gamma)=\\{v_{1},v_{2},v_{3},v_{4}\\}$. The
directed connectivity for 6 edges is given by
$E(\Gamma)=\\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\\}=\\{v_{1}\to
v_{2},v_{1}\to v_{3},v_{1}\to v_{4},v_{2}\to v_{3},v_{3}\to v_{4},v_{4}\to
v_{2}\\}$.

For this directed graph, the incidence matrix is given by

$L=\left(\begin{array}[]{cccc}-1&1&0&0\\\ -1&0&1&0\\\ -1&0&0&1\\\ 0&-1&1&0\\\
0&0&-1&1\\\ 0&1&0&-1\\\ \end{array}\right).$ (A.1)

We can construct the Laplacian matrix for the vertex from the incidence matrix

$\Delta_{V}=L^{T}L=\left(\begin{array}[]{cccc}3&-1&-1&-1\\\ -1&3&-1&-1\\\
-1&-1&3&-1\\\ -1&-1&-1&3\\\ \end{array}\right).$ (A.2)

Then the adjacency matrix becomes

$K=\left(\begin{array}[]{cccc}0&1&1&1\\\ 1&0&1&1\\\ 1&1&0&1\\\ 1&1&1&0\\\
\end{array}\right).$ (A.3)

The vertex Laplacian has the eigenvalues $\\{4,4,4,0\\}$, which contain one
zero.

Four faces are defined by $f_{1}=\\{e_{1},e_{4},\bar{e}_{2}\\}$,
$f_{2}=\\{e_{2},e_{5},\bar{e}_{3}\\}$, $f_{3}=\\{e_{3},e_{6},\bar{e}_{1}\\}$
and $f_{3}=\\{\bar{e}_{4},\bar{e}_{6},\bar{e}_{5}\\}$. Then the dual incidence
matrix is given by

$\check{L}=\left(\begin{array}[]{cccc}1&0&-1&0\\\ -1&1&0&0\\\ 0&-1&1&0\\\
1&0&0&-1\\\ 0&1&0&-1\\\ 0&0&1&-1\\\ \end{array}\right).$ (A.4)

We can see that $L^{T}\check{L}=\check{L}^{T}L=0$.

Using the dual incidence matrix, we can construct the Laplacian for the face
and edges as

$\begin{split}&\Delta_{F}=\check{L}^{T}\check{L}=\left(\begin{array}[]{cccc}3&-1&-1&-1\\\
-1&3&-1&-1\\\ -1&-1&3&-1\\\ -1&-1&-1&3\\\ \end{array}\right),\\\
&\Delta_{E}=LL^{T}+\check{L}\check{L}^{T}=\left(\begin{array}[]{cccccc}4&0&0&0&0&0\\\
0&4&0&0&0&0\\\ 0&0&4&0&0&0\\\ 0&0&0&4&0&0\\\ 0&0&0&0&4&0\\\ 0&0&0&0&0&4\\\
\end{array}\right),\end{split}$ (A.5)

which have the eigenvalues $\\{4,4,4,0\\}$ and $\\{4,4,4,4,4,4\\}$,
respectively.

An non-Abelian generalization for the incidence matrix acting on the adjoint
representation is given by

$L_{U}=\left(\begin{array}[]{cccc}-1&U^{1}\cdot U^{1{\dagger}}&0&0\\\
-1&0&U^{2}\cdot U^{2{\dagger}}&0\\\ -1&0&0&U^{3}\cdot U^{3{\dagger}}\\\
0&-1&U^{4}\cdot U^{4{\dagger}}&0\\\ 0&0&-1&U^{5}\cdot U^{5{\dagger}}\\\
0&U^{6}\cdot U^{6{\dagger}}&0&-1\\\ \end{array}\right),$ (A.6)

where “ $\cdot$ ” stands for an insertion position of matrices in the adjoint
representation when this covariant incidence matrix acts from the right;

$(X\cdot Y)A=XAY,$ (A.7)

for example.

A conjugate of the dual incidence matrix is derived from 4 plaquette
variables;

$P^{1}=U^{1}U^{4}U^{2{\dagger}},\quad P^{2}=U^{2}U^{5}U^{3{\dagger}},\quad
P^{3}=U^{3}U^{6}U^{1{\dagger}},\quad
P^{4}=U^{4{\dagger}}U^{6{\dagger}}U^{5{\dagger}}.$ (A.8)

It becomes

$\check{L}^{\dagger}_{U}=\begin{pmatrix}\cdot P^{1}&-P^{1}\cdot&0&U^{1}\cdot
U^{1{\dagger}}P^{1}&0&0\\\ 0&\cdot P^{2}&-P^{2}\cdot&0&U^{2}\cdot
U^{2{\dagger}}P^{2}&0\\\ -P^{3}\cdot&0&\cdot P^{3}&0&0&U^{3}\cdot
U^{3{\dagger}}P^{3}\\\ 0&0&0&-U^{4{\dagger}}\cdot
U^{4}P^{4}&-P^{4}\cdot&-P^{4}U^{5}\cdot U^{5{\dagger}}\end{pmatrix}.$ (A.9)

Then we find $L_{U}^{\dagger}\check{L}_{U}=\check{L}^{\dagger}_{U}L_{U}=0$ iff
$P^{f}=1$.

We can also define the covariant Laplacians by

$\Delta^{V}_{U}=L_{U}^{\dagger}L_{U},\quad\Delta^{F}_{U}=\check{L}_{U}^{\dagger}\check{L}_{U},\quad\Delta^{E}_{U}=L_{U}L_{U}^{\dagger}+\check{L}_{U}\check{L}_{U}^{\dagger},$
(A.10)

which have the same eigenvalues as $\Delta_{V}$, $\Delta_{F}$ and $\Delta_{E}$
iff $P^{f}=1$.

###  A.2 Torus

Figure 5: A directed graph for a $3\times 3$ torus. There are 9 vertices, 9
faces and 18 directed edges.

A directed graph for a torus is depicted in Fig. 5. The torus is divided into
$3\times 3$ square faces (9 faces in total) and the associated graph has a
periodicity for two directions.

We first provide the covariant incidence matrix for this graph. We can
immediately reproduce a usual incidence matrix by setting $U^{e}=1$ for all.

${\scriptsize L_{U}=\begin{pmatrix}-1&U^{1}\cdot
U^{1{\dagger}}&0&0&0&0&0&0&0\\\ 0&-1&U^{2}\cdot U^{2{\dagger}}&0&0&0&0&0&0\\\
U^{3}\cdot U^{3{\dagger}}&0&-1&0&0&0&0&0&0\\\ -1&0&0&U^{4}\cdot
U^{4{\dagger}}&0&0&0&0&0\\\ 0&-1&0&0&U^{5}\cdot U^{5{\dagger}}&0&0&0&0\\\
0&0&-1&0&0&U^{6}\cdot U^{6{\dagger}}&0&0&0\\\ 0&0&0&-1&U^{7}\cdot
U^{7{\dagger}}&0&0&0&0\\\ 0&0&0&0&-1&U^{8}\cdot U^{8{\dagger}}&0&0&0\\\
0&0&0&U^{9}\cdot U^{9{\dagger}}&0&-1&0&0&0\\\ 0&0&0&-1&0&0&U^{10}\cdot
U^{10{\dagger}}&0&0\\\ 0&0&0&0&-1&0&0&U^{11}\cdot U^{11{\dagger}}&0\\\
0&0&0&0&0&-1&0&0&U^{12}\cdot U^{12{\dagger}}\\\ 0&0&0&0&0&0&-1&U^{13}\cdot
U^{13{\dagger}}&0\\\ 0&0&0&0&0&0&0&-1&U^{14}\cdot U^{14{\dagger}}\\\
0&0&0&0&0&0&U^{15}\cdot U^{15{\dagger}}&0&-1\\\ U^{16}\cdot
U^{16{\dagger}}&0&0&0&0&0&-1&0&0\\\ 0&U^{17}\cdot
U^{17{\dagger}}&0&0&0&0&0&-1&0\\\ 0&0&U^{18}\cdot
U^{18{\dagger}}&0&0&0&0&0&-1\end{pmatrix}}.$ (A.11)

The Laplacian matrix on the vertex is

$\Delta_{V}={\footnotesize\left(\begin{array}[]{ccccccccc}4&-1&-1&-1&0&0&-1&0&0\\\
-1&4&-1&0&-1&0&0&-1&0\\\ -1&-1&4&0&0&-1&0&0&-1\\\ -1&0&0&4&-1&-1&-1&0&0\\\
0&-1&0&-1&4&-1&0&-1&0\\\ 0&0&-1&-1&-1&4&0&0&-1\\\ -1&0&0&-1&0&0&4&-1&-1\\\
0&-1&0&0&-1&0&-1&4&-1\\\ 0&0&-1&0&0&-1&-1&-1&4\\\ \end{array}\right)},$ (A.12)

which has the eigenvalues of $\\{6,6,6,6,3,3,3,3,0\\}$.

The covariant dual incidence matrix is made from the plaquette variables, but
it is a huge size matrix we can not typeset here. To display in the
manuscript, we give the incidence matrix by setting $U^{e}=1$ ;

$\check{L}^{T}={\tiny\left(\begin{array}[]{cccccccccccccccccc}-1&0&0&1&-1&0&1&0&0&0&0&0&0&0&0&0&0&0\\\
0&-1&0&0&1&-1&0&1&0&0&0&0&0&0&0&0&0&0\\\
0&0&-1&-1&0&1&0&0&1&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&-1&0&0&1&-1&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&-1&0&0&1&-1&0&1&0&0&0&0\\\
0&0&0&0&0&0&0&0&-1&-1&0&1&0&0&1&0&0&0\\\
1&0&0&0&0&0&0&0&0&0&0&0&-1&0&0&1&-1&0\\\
0&1&0&0&0&0&0&0&0&0&0&0&0&-1&0&0&1&-1\\\
0&0&1&0&0&0&0&0&0&0&0&0&0&0&-1&-1&0&1\end{array}\right)}.$ (A.13)

The Laplacian matrix on the face $\Delta_{F}$ is the same as $\Delta_{V}$ in
(A.12). The Laplacian matrix on the edge is given by

$\Delta_{E}={\tiny\left(\begin{array}[]{cccccccccccccccccc}4&-1&-1&0&0&0&-1&0&0&0&0&0&-1&0&0&0&0&0\\\
-1&4&-1&0&0&0&0&-1&0&0&0&0&0&-1&0&0&0&0\\\
-1&-1&4&0&0&0&0&0&-1&0&0&0&0&0&-1&0&0&0\\\
0&0&0&4&-1&-1&0&0&0&-1&0&0&0&0&0&-1&0&0\\\
0&0&0&-1&4&-1&0&0&0&0&-1&0&0&0&0&0&-1&0\\\
0&0&0&-1&-1&4&0&0&0&0&0&-1&0&0&0&0&0&-1\\\
-1&0&0&0&0&0&4&-1&-1&0&0&0&-1&0&0&0&0&0\\\
0&-1&0&0&0&0&-1&4&-1&0&0&0&0&-1&0&0&0&0\\\
0&0&-1&0&0&0&-1&-1&4&0&0&0&0&0&-1&0&0&0\\\
0&0&0&-1&0&0&0&0&0&4&-1&-1&0&0&0&-1&0&0\\\
0&0&0&0&-1&0&0&0&0&-1&4&-1&0&0&0&0&-1&0\\\
0&0&0&0&0&-1&0&0&0&-1&-1&4&0&0&0&0&0&-1\\\
-1&0&0&0&0&0&-1&0&0&0&0&0&4&-1&-1&0&0&0\\\
0&-1&0&0&0&0&0&-1&0&0&0&0&-1&4&-1&0&0&0\\\
0&0&-1&0&0&0&0&0&-1&0&0&0&-1&-1&4&0&0&0\\\
0&0&0&-1&0&0&0&0&0&-1&0&0&0&0&0&4&-1&-1\\\
0&0&0&0&-1&0&0&0&0&0&-1&0&0&0&0&-1&4&-1\\\
0&0&0&0&0&-1&0&0&0&0&0&-1&0&0&0&-1&-1&4\\\ \end{array}\right)},$ (A.14)

which has the eigenvalues $\\{6,6,6,6,6,6,6,6,3,3,3,3,3,3,3,3,0,0\\}$.

We can also show that the orthogonality
$L_{U}^{\dagger}\check{L}_{U}=\check{L}^{\dagger}_{U}L_{U}=0$ iff $P^{f}=1$,
explicitly.

##  Appendix B Cartan-Weyl basis and properties

We denote the generators in the Cartan subalgebla ${\mathfrak{u}}(N)$ by
$\mathtt{H}_{i}$ ($i=1,\ldots,N$) and the root vectors (Weyl generators) by
$\mathtt{E}_{\alpha}$. These generators obey the following commutation
relations;

$\begin{split}&[\mathtt{H}_{i},\mathtt{H}_{j}]=0,\\\
&[\mathtt{H}_{i},\mathtt{E}_{\pm\alpha}]=\pm\alpha_{i}\mathtt{E}_{\pm\alpha},\\\
&[\mathtt{E}_{\alpha},\mathtt{E}_{-\alpha}]=\sum_{i=1}^{N}\alpha_{i}\mathtt{H}_{i},\qquad[\mathtt{E}_{\alpha},\mathtt{E}_{\beta}]=N_{\alpha,\beta}\mathtt{E}_{\alpha+\beta},\end{split}$
(B.1)

and has the properties;

$\mathtt{E}_{\alpha}^{\dagger}=\mathtt{E}_{-\alpha},\quad\operatorname{Tr}\mathtt{E}_{\alpha}\mathtt{E}_{\beta}=\delta_{\alpha+\beta,0},\quad\operatorname{Tr}\mathtt{H}_{i}\mathtt{H}_{j}=\sum_{\alpha}\alpha_{i}\alpha_{j}=\delta_{ij}.$
(B.2)

Any adjoint representation of $U(N)$ group (generators of the Lie algebra) can
be expanded by

$X=\sum_{i=1}^{N}x^{i}\mathtt{H}_{i}+\sum_{\alpha}x^{\alpha}\mathtt{E}_{\alpha}.$
(B.3)

We use these conventions in the manuscript.

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