date23062021

# Deconstructing finite temperature pure gauge theory.

Herbert Neuberger<EMAIL_ADDRESS>Department of Physics and
Astronomy, Rutgers University,
Piscataway, NJ 08854, U.S.A

(date)

###### Abstract

A deconstructed finite temperature gauge theory has the Euclidean time
direction kept discrete and finite. The ultraviolet behavior is that of one
dimension less than at zero temperature. One can add to the action a gauge
invariant term dependent on Polyakov loops, without worrying about
nonlocality. The deformation of Ünsal and Yaffe is best analyzed in this
framework. It is shown that turning it on causes a dramatic change in the
Hilbert space on which the transfer matrix in the temperature direction acts.
The undeformed Hilbert space is everywhere a gauge singlet while the deformed
one includes all states transforming locally, anywhere, under any $SU(N)/Z(N)$
irreducible representation. Implications of this in the context of large $N$
reduction are discussed.

###### pacs:

11.15.Ha, 11.15.PG

##  I Introduction.

Pure $SU(N)$ lattice gauge theory with Wilson action on a periodic four torus
is well defined by an integral over group elements. There is a local gauge
invariance group and nothing special needs to be done about it. The
integration includes all gauge copies. In numerical simulations one does not
gauge fix because locality is useful for importance sampling. Theoretically,
one is free to partially or maximally fix a gauge. Only nonperturbative,
exact, gauge fixing will be given consideration in this article. Perturbative
BRST gauge fixing cannot be extended to exactly work on the lattice brs2  in
any simple way.

A transfer matrix $\mathfrak{T}=e^{-H}$ is needed in order to construct the
finite temperature Gibbs ensemble. This paper is not concerned with $H$ itself
because there is no continuous time. $\mathfrak{T}$ is constructed from the
action in a Euclidean path integral. For a purely bosonic system consisting of
only site variables, finite temperature amounts to the imposition of periodic
boundary conditions. For gauge theories, $\mathfrak{T}$ is typically extracted
from the path integral in a temporal gauge. At zero temperature one can set
all time-like links to unity and $\mathfrak{T}$ is easily read off. At finite
temperature the anticipated periodicity clashes with this gauge fixing because
one cannot gauge away winding parallel transport.

When read off directly from the path integral without fixing a gauge
$\mathfrak{T}$ 111Equation (3.144) in MoMubook , first paperback edition,
1997, Cambridge University Press, 1994. does not come out in explicit form.
One needs to integrate out the time component of the gauge field exactly. For
periodic boundary conditions one cannot avoid having to do this in order to
get an explicit formula for $\mathfrak{T}$.

A deformation of the finite temperature periodic path integral was introduced
in UY  adding a term nonlocal in the euclidean evolution direction to the
standard, local action. This complicates somewhat the extra integral needed to
get $\mathfrak{T}$ explicitly. It will be shown that the deformation has a
major effect on the Hilbert space $\mathfrak{T}$ acts in.

It was also claimed in UY  that the deformation can be generalized to allow
large $N$ Eguchi-Kawai ek  reduction down to a single lattice site, all the
way to continuum. Other proposals to achieve this end appear, superficially,
to involve more drastic alterations. I shall argue that the modification of
Hilbert space makes this newer proposal also drastic.

In order to make the proposals more comparable, the quenching prescription of
qek  will be adapted to provide a way alternative to that of UY  to replace
the canonical deconfined high temperature phase by a reconfined one.

##  II Notation and generalities.

The Einstein summation convention won’t be used and no meaning will be
attached to whether an index is upper or lower. Summations are often implied
by matrix products and trace operations.

Imagine a finite periodic four dimensional hypercubic lattice embedded in and
aligned with a continuum ${\cal T}^{4}$. Its sites are labeled
$x=(x_{1},x_{2},x_{3},x_{4})$ where $x_{\mu}=1,...,l_{\mu}$ are integer
coordinates and the $l_{\mu}$ are lengths of sides in the $\mu$ direction.
Directed links connect nearest neighboring sites from $x$ to $x+\mu$. In this
context $\mu$ also denotes a unit vector in the $\mu$ positive direction. To
each positively oriented link a variable $U_{x,\mu}$ is assigned. $U_{x,\mu}$
is an $N\times N$ unitary matrix of unit determinant. To a link from $x$ to
negative $\mu$ direction we attach $U^{\dagger}_{x-\mu,\mu}\equiv U_{x,-\mu}$.
When the site index $x$ is not present the reference will be to the collection
of all links in the $\mu$ direction. Each link matrix is a faithful
representation of one abstract element in $SU(N)$ and shall be identified with
it. This is the fundamental representation of the group. To each site $x$ we
attach an $N$-dimensional complex vector, $\phi_{x}$. The $\phi_{x}$ denote
elements in a $C^{N}$ and collectively provide a space $\mathcal{V}$ on which
matrices with $(Nl_{1}l_{2}l_{3}l_{4})^{2}$ entries act. These large matrices
will be introduced later. The $\phi_{x}$ serve only this purpose and otherwise
are “unphysical”. Here we deal only with gauge theory without site anchored
matter.

The matrix entries $U_{x,\mu}^{i,j},i,j=1,...N$ of $U_{x,\mu}$ depend on
$N^{2}-1$ real variables, but shall be viewed as a set of $N^{2}$ independent
complex variables. The constraint to $SU(N)$ is implemented by a restrictions
on the domain of integration over $U_{x,\mu}^{i,j},i,j=1,...N$ which is
embedded in $R^{N^{2}-1}$. The integration measure is left and right invariant
Haar, and the group volume is set to unity.

One-link parallel transporters $T_{\mu}(U_{\mu})$ act on $\mathcal{V}$ by

$[T_{\mu}(U_{\mu})\phi]_{x}=U_{x-\mu,\mu}\phi_{x-\mu}$ (II.1)

The result is another vector with entries defined by the equation employing
implicit matrix-vector multiplication. A $T_{\mu}$ has two indices, each of
the form $(x,i)$ where $x$ is a site and $1\leq i\leq N$ is a group index. The
action of $T_{\mu}$ can be visualized as picking up the $N$ vector
$\phi_{x}^{j}$ and then parallel transporting it along the positively oriented
link in the $\mu$-direction to site $x+\mu$. Parallel transport makes the $N$
vector rotate in group space under the action of $U_{x,\mu}$. $T_{\mu}$ does
this for all sites $x$. Equation (II.1) is short-hand for

$[T_{\mu}(U_{\mu})\phi]_{(x,i)}=\sum_{y}\delta_{y,x-\mu}\sum_{j=1}^{N}U_{y,\mu}^{ij}\phi_{(y,j)}$
(II.2)

$T_{\mu}$ can also be written as

$T_{\mu}=T_{\mu}^{0}\mathbf{U^{\mu}},$ (II.3)

which separates group from site indices:
$[\mathbf{U^{\mu}}\phi]_{x}^{i}=\sum_{j}U_{x,\mu}^{ij}\phi_{x}^{j}$.
$T_{\mu}^{0}$ generates cyclic rotation of sites in the $\mu$ direction,
$[T_{\mu}^{0}\phi]_{x}^{i}=\phi_{x-\mu}^{i}$. $T_{\mu}^{0}$’s for different
$\mu$’s commute and can be simultaneously diagonalized. An action polynomial
in norms of $[T_{\mu},T_{\nu}]$ commutators can be easily arranged to attain
its minimum at $T_{\mu}=T_{\mu}^{0}$. In object oriented language one can
think of $T_{\mu}$ as a group-action overloaded cyclic shift $T_{\mu}^{0}$.

###  II.1 Discrete Fourier Transform.

Consider a circle made out of $L$ sites, labeled by $q=0,...L-1$ in
consecutive order in the positive direction. Let $v_{q}$ be the components of
a site column vector $v$. The $T^{0}_{\mu}$ matrices with no group indices are
replaced in this subsection by an $L\times L$ matrix $T^{0}$. $T^{0}$ is
diagonalized by the Discrete Fourier Transform (DFT) matrix $\Omega$:

$\Omega^{pq}=\frac{1}{\sqrt{L}}\omega^{-pq},\;\;p,q=0,1...,L-1\;\;\;\omega=e^{-\frac{2\pi
i}{L}}$ (II.4)

$\Omega$ is explicitly symmetric. The columns $p$ of $\Omega$ are orthonormal
by:

$\frac{1}{L}\sum_{p}\omega^{qp}\omega^{-pq^{\prime}}=\delta_{qq^{\prime}}$
(II.5)

$\Omega$ is unitary, in addition to being symmetric. The columns of $\Omega$
are eigenvectors of $T^{0}$:

$\sum_{q}T^{0}_{q^{\prime}q}\omega^{-pq}=\omega^{p}\omega^{-pq^{\prime}}\;\;{\rm
because}\;\sum_{q}T^{0}_{q^{\prime}q}v_{q}=v_{q^{\prime}-1}$ (II.6)

Hence

$T^{0}\Omega=\Omega\;{\rm
diag}(1,\omega,..,\omega^{L-1}),\;\;\Omega^{\dagger}T^{0}\Omega={\rm
diag}(1,\omega,..,\omega^{L-1}),\;\;T^{0}=\Omega\;{\rm
diag}(1,\omega,..,\omega^{L-1})\Omega^{\dagger}$ (II.7)

$\Omega$ takes a site vector with components $v_{q}$ to ${\hat{v}}_{p}$, its
DFT, and $\Omega^{\dagger}$ inverts with $v=\Omega^{\dagger}{\hat{v}}$:

$(\Omega
v)_{p}={\hat{v}}_{p}=\frac{1}{\sqrt{L}}\sum_{q}\omega^{-pq}v_{q},\;\;(\Omega^{\dagger}{\hat{v}})_{q}=v_{q}=\frac{1}{\sqrt{L}}\sum_{p}\omega^{qp}{\hat{v}}_{p}$
(II.8)

Also,

$[\Omega\;{\rm
diag}(v_{0},v_{1},...,v_{L-1})\;\Omega^{\dagger}]_{pp^{\prime}}=\frac{1}{L}\sum_{q}\omega^{-pq}v_{q}\;\omega^{qp^{\prime}}={\hat{v}}_{p-p^{\prime}}$
(II.9)

Taking the trace, one gets $\sum_{q}v_{q}=L{\hat{v}}_{0}$.

When the $v_{q}$ are complex and independent, so are the ${\hat{v}}_{p}$; if
the $v_{q}$ are real and independent, the complex ${\hat{v}}_{p}$ are
constrained by ${\hat{v}}_{p}={\hat{v}}^{*}_{L-1-p}$.

Arithmetic in the $p,q$ indices is modulo $L$ with the representative set
$(0,1,...,L-1)$. All indices out of range are to be replaced by their
representative in this set. Defining the $L\times L$ matrix $\hat{V}$ by
$({\hat{V}})_{pp^{\prime}}\equiv{\hat{v}}_{p-p^{\prime}}$ shows that
${\hat{V}}$ is a circulant matrix, ${\hat{V}}={\rm
circ}({\hat{v}}_{0},{\hat{v}}_{1},....{\hat{v}}_{L-1})$. Hence, conjugation by
$\Omega...\Omega^{\dagger}$ takes a diagonal matrix in site space to a
circulant matrix in momentum space. Conjugation of $T^{0}$ by
$\Omega^{\dagger}...\Omega$ produces a diagonal matrix in momentum space. The
norm squared in matrix space of ${\hat{V}}$ is

${\rm
Tr}{\hat{V}}{\hat{V}}^{\dagger}=\sum_{q}|v_{q}|^{2}=\sum_{p,p^{\prime}}|{\hat{v}}_{p-p^{\prime}}|^{2}=L\sum_{p}|{\hat{v}}_{p}|^{2}$
(II.10)

In general, summation over $p$ and $p^{\prime}$ of a summand that depends only
on $r=p-p^{\prime}$ gives a factor of $L$ times a summation over $r\equiv
p-p^{\prime}$ in the range $0$ to $L-1$.

Circulant matrices are a natural way to ensure momentum conservation in
diagrams. The indices $p$ and $p^{\prime}$ provide a ’t Hooft double lines
notation, with Feynman propagator momenta given by $p-p^{\prime}$. This is
particularly useful in the case of planar diagrams, where planarity is a
consequence of another index family, like color.

On the full lattice introduced one employs a direct product of four DFT
matrices $\Omega_{\mu}$ each factor operating on an index
$q_{\mu}=0,1,....,l_{\mu}-1$. The arithmetic in each $q_{\mu}$ is modulo
$l_{\mu}$.

###  II.2 Gauge group.

To each site one also attaches a gauge transformation matrix $g_{x}\in SU(N)$.
A collection of $g_{x}$ defines a gauge transformation on $\mathcal{V}$ given
by:

$[^{g}\mathfrak{G}\phi]_{x}=g_{x}\phi_{x}$ (II.11)

Then,

$[(^{g}\mathfrak{G})^{\dagger}T(U_{\mu})(^{g}\mathfrak{G})\phi]_{x}=g_{x}^{\dagger}U_{x-\mu}g_{x-\mu}\phi_{x-\mu}$
(II.12)

We define the action of a gauge transformation on the link variables as the
replacement of $U_{x,\mu}$ by ${}^{g}U_{x,\mu}$:

${}^{g}U_{x,\mu}=g_{x+\mu}^{\dagger}U_{x,\mu}g_{x}$ (II.13)

Then, the action of gauge transformations on parallel transporters is by
conjugation:

$(^{g}\mathfrak{G})^{\dagger}(T_{\mu}(U_{\mu})^{g}\mathfrak{G}=T_{\mu}(^{g}U_{\mu})$
(II.14)

The matrices ${}^{g}\mathfrak{G}$ make up the gauge group:

${}^{gh}\mathfrak{G}={{}^{g}\mathfrak{G}}{{}^{h}\mathfrak{G}},(^{g}\mathfrak{G})^{\dagger}={{}^{g^{\dagger}}\mathfrak{G}}=({{}^{g}\mathfrak{G}})^{-1}$
(II.15)

The norm of some $X$ in the algebra generated by the $T_{\mu}$,
$||X||^{2}=TrXX^{\dagger}$, is gauge invariant.

###  II.3 Action in path integral.

The Wilson partition function is the simplest possible on a hypercubic
lattice:

$Z_{W}=\int[\prod_{x,\mu}dU_{x,\mu}]e^{-\frac{\beta}{2}\sum_{\mu<\nu}Tr[[T_{\mu}(U_{\mu}),T_{\nu}(U_{\nu})][T_{\mu}(U_{\mu}),T_{\nu}(U_{\nu})]^{\dagger}]}$
(II.16)

$\beta$ is the lattice coupling constant. The trace is a sum over site and
group indices. Before the trace is taken the expressions (dropping the
arguments) read:

$[T_{\mu},T_{\nu}][T_{\mu},T_{\nu}]^{\dagger}=2-T_{\mu}T_{\nu}T_{\mu}^{\dagger}T_{\nu}^{\dagger}-T_{\nu}T_{\mu}T_{\nu}^{\dagger}T_{\mu}^{\dagger}=(1-T_{\mu}T_{\nu}T_{\mu}^{\dagger}T_{\nu}^{\dagger})+[{\rm
h.c.}]$ (II.17)

The term proportional to the unit matrix can be dropped from the action. The
remaining two terms represent parallel transports round local small closed
square circuits in the $\mu-\nu$ plane, diagonal in lattice sites, circulating
in opposite directions in the $\mu-\nu$ plane. In general, a circuit which
does not close gives a parallel transport matrix with zeros on the diagonal
because the site components of the indices are unequal. Eight distinct choices
for the first commutator (which then determines the second) are possible after
choosing one $\mu$–$\nu$ plane (if $\mu=\nu$ the commutator vanishes and there
is no plane.) The first term has two possible choices for direction and
independently two choices for sense. The second term has a direction fixed by
the first and two choices for sense. The direction is $\mu$ or $\nu$ and the
sense is positive (no dagger) or negative (with dagger). Everyone of the eight
choices produces the same action. These eight possibilities split into two
groups of four, according to the sign of circulation in the $\mu$–$\nu$ plane.

###  II.4 Connection to continuum.

We assume a continuum $su(N)$ matrix valued one form
$A=A_{\mu}(\mathpzc{x})d\mathpzc{x}_{\mu}$ is given and define a continuum
$\phi(\mathpzc{x})$ on the ${\cal T}^{4}$ torus the lattice is embedded in.
The spacing between nearest neighboring lattice points is uniformly equal to
$a$. Let $D_{\mu}(A_{\mu})$ be the covariant derivative associated with $A$ in
the direction $\mu$. It acts on fields $\phi(\mathpzc{x})$. Consider the
action of $e^{aD_{\mu}(A_{\mu})}$ on $\phi(\mathpzc{x})$. Observe that
$[e^{aD_{\mu}(A)}](\phi)(\mathpzc{x})$ restricts to $\phi_{x}$ with
$x=\mathpzc{x}$: if we know the continuum function $\phi(\mathpzc{x})$ at the
sites $\mathpzc{x}=x$ the action of $e^{aD_{\mu}(A_{\mu})}$ on
$\phi(\mathpzc{x})$ will produce values again localized on sites. The in-
between points do not enter directly. The action of $e^{aD_{\mu}(A_{\mu})}$
restricts to sites where it becomes the action of $T_{\mu}(A_{\mu})$. With
this restriction we have

$T_{\mu}(A_{\mu})=e^{aD_{\mu}(A_{\mu})}$ (II.18)

Consequently, formally,
$[T_{\mu}(U_{\mu}),T_{\nu}(U_{\nu})]=a^{2}[D_{\mu}(A_{\mu}),D_{\nu}(A_{\nu})]+...\equiv
a^{2}F_{\mu\nu}(A)+...$, establishing the formal connection of the Wilson
action to the continuum one. Note that the commutator $[T_{\mu},T_{\nu}]$ is
not simply related to its hermitian conjugate, nor is it site diagonal. Thus,
in itself, it is not a good choice for representing the local non-abelian
field strength on the lattice. Combining terms coming from the alternative
forms mentioned after equation (II.17) provides a set of eight circuits which
have just one site index shared by all eight and keeping them untraced, one
generates eight objects transforming under the adjoint representation of the
gauge group at the common site. Subtracting the sum of four terms with one
circulation from the sum of terms of opposite circulation provides a good
lattice version of the continuum nonabelian field strength. It is known as the
SW sw  (or clover) term.

###  II.5 Large $N$ reduction.

Gauge transformations act by the same conjugation simultaneously on all four
$T_{\mu}$’s. Any function of the link variables that is expressible as the
trace of a product of $T_{\mu}$’s and their conjugates is gauge invariant.
These observables are summed over all lattice sites. Their average expectation
values are obtained after division by $\prod_{\mu}l_{\mu}$. When the number of
$T_{\mu}$’s minus that of $T_{\mu}^{\dagger}$’s does not vanish modulo
$l_{\mu}$ for all $\mu$ every diagonal entry of the product matrix vanishes
and the trace is zero.

Eguchi-Kawai ek  reduction is the statement that in the large $N$ limit of the
theory the expectation values of these variables are identical to those of the
theory with $l_{\mu}=1$ for all $\mu$. With a single available site, the
$T_{\mu}$’s become just complete unitary matrices. If the number of sites is
much larger than $N$ the $T_{\mu}$ matrices are sparse. In the regime that
Eguchi-Kawai reduction holds this spareness becomes irrelevant and only a
trivial dependence on the volume remains. This is vaguely reminiscent of the
eigenvalue distributions of single random banded Hermitian matrices bm .

In quenched Eguchi-Kawai reduction qek , where there is only one site,
$T_{\mu}=U_{\mu}$ transforms in the adjoint under the gauge group, so its
eigenvalues are gauge invariant. The minimum of the action is attained when
all $[U_{\mu},U_{\nu}]=0$ which means they are simultaneously diagonalizable.
This allows to separate out the eigenvalues and by quenching them they are
capable of playing the role of lattice momenta in perturbation theory.

In the clever model of twisted Eguchi-Kawai reduction twist , the action
minimum is arranged to be attained at $\theta_{\mu\nu}$-twisted configurations
– $U_{\mu}U_{\nu}=e^{i\theta_{\mu\nu}}U_{\nu}U_{\mu}$ – where a choice of
twists $e^{i\theta_{\mu\nu}}=e^{-i\theta_{\nu\mu}}$ is made from the start.
Solutions to these relations produce a system on noncommutative Euclidean
space in the continuum limit. In perturbation theory this is detectable only
at sub-leading order in $1/N$, because in planar-diagram the twists cancel out
and the untwisted series is recovered. Quenched and twisted models are drastic
alternatives to ek  alluded to earlier.

##  III Transfer matrix and Hilbert space.

In a quantization that starts from a path integral one foliates the space of
configurations into co-dimension one leaves of identical structure which are
connected in the action only in sequential pairs. The (Euclidean) evolution
takes one leaf to the next one. The kernel is expressed in terms of the same
variables as the path integral. Thus, the Hilbert space is defined as the
space of integrable functions of the variables resident on one leaf. For
lattice gauge theory the functions are on a compact space – there is a
discrete countable basis. The inner product of the Hilbert space comes from
the ${\cal{L}}_{2}$ norm on complex functions on the space-link space.

On our hypercubic lattice we pick the evolution direction as $\mu=4$. The
space-leaves are three dimensional tori of sizes $l_{1}\times l_{2}\times
l_{3}$ at fixed $t\equiv x_{4}$. The kernel is homogeneous along $t$ with its
two arguments determined by variables appearing in one $(t,t+1)$ pair. The $t$
variables is shared with the preceding kernel and the $t+1$ ones with the
following one. Integration over shared variables is factorized by the
completeness identity of the countable basis in which $\mathfrak{T}$ has
discrete indices. This basis provides discretely valued indices for the
transfer matrix. Sometimes a more formal labeling by continuous variables is
used. Integration in the path integral is ordered ascending in $t$. Periodic
boundary conditions close the circle cyclically. The path integral becomes a
trace of the transfer matrix $\mathfrak{T}$ raised to integral power.

For a theory with only site anchored variables and nearest neighbor
interactions one can read $\mathfrak{T}$ off the path integral and the Hilbert
space is determined. In gauge theory, one needs to first integrate out all
link variables connecting adjacent leaves. Only then can $\mathfrak{T}$ be
extracted explicitly; this will be shown in great detail.

For a general action this integral may be complicated. In gauge theory, there
is gauge invariance under transformations parametrized by site variables in
four space. We only consider gauge invariant observables, and we could just
integrate over link variables modulo gauge transformations. The number of
intra leaf link matrices is equal to that of gauge variables. As expected, the
integration over leave–connecting links giving $Z_{W}$ (eq. II.16) is doable
and restricts to a gauge invariant ecosystem.

With $\mathfrak{T}$ in hand, one can deal with excitations by inserting
operators in the trace. Some observables may contain time-link variables,
which were eliminated when $\mathfrak{T}$ was constructed. In order to
reproduce answers for all expectation values accessible to the path integral
formalism one must revisit the integration over link variables appearing in
observables.

###  III.1 Explicit construction

The separation between time-link and space-link variables requires some new
notation. The positive space directions emanating from site
${\vec{x}}=(x_{1},x_{2},x_{3})$ on one leaf will be denoted by
$\alpha,\beta...$. The distinct space leaves will be labeled by $t$. Space
link variables on leaf $t$ will be denoted $U^{t}_{{\vec{x}},\alpha}$; when
the entire set of a leaf is intended, the index $\vec{x}$ will be
dropped.222The notation $X^{t}$ for the transpose of a matrix $X$ shall not be
employed in this paper: superscript $t$ refers to the leaf at “time” $t$. The
action of $T^{t}_{\alpha}$ is restricted to leaf $t$; they are inter leaf
operators. The link-matrix arguments of the in-leaf parallel transporters
$T^{t}_{\alpha}(U^{t}_{\alpha})$ will be dropped. Three dimensional space
gauge transformations are similarly restricted to act only on leaf $t$. They
shall be denoted by ${}^{g}{\mathfrak{G}}_{t}$ and act by
$(^{g}{\mathfrak{G}}_{t})^{\dagger}T_{\alpha}(U^{t}_{\alpha})^{g}{\mathfrak{G}}_{t}=T^{t}_{\alpha}(^{g}U^{t}_{\alpha})$.

The action term $S$ in the exponent of (II.16) is split into an inter leaf and
an intra leaf part:

$\displaystyle\begin{split}S=&\sum_{\mu<\nu}Tr{\big{[}}[T_{\mu},T_{\nu}][T_{\mu},T_{\nu}]^{\dagger}{\big{]}}=\\\
\sum_{t}&\\{\sum_{\alpha<\beta}Tr{\big{[}}[T^{t}_{\alpha},T^{t}_{\beta}][T^{t}_{\alpha},T^{t}_{\beta}]^{\dagger}{\big{]}}+\sum_{\alpha}Tr{\big{[}}[T_{t+1/2},T^{t}_{\alpha}][T_{t+1/2},T^{t}_{\alpha}]^{\dagger}{\big{]}}\\}\end{split}$
(III.1)

$T_{t+1/2}$ is the parallel transporter from leaf $t$ to $t+1$ in the $\tau$
direction. Transport in the opposite direction is $(T_{t+1/2})^{\dagger}$. The
(suppressed) argument of $T_{t+1/2}$ is the collection over $\vec{x}$ in leaf
$t$ of $U^{t}_{\vec{x},t}$ and that of $(T_{t+1/2})^{\dagger}$ is the
collection over $\vec{x}$ in leaf $t+1$ of $U^{t+1}_{\vec{x},t}$. It is
convenient to replace $U^{t}_{\vec{x},t}$ by $g_{t+1/2}({\vec{x}})$ because
one can view these matrices as generating an inter leaf $t+1/2$ gauge
transformation by
$U^{t+1}_{{\vec{x}},\alpha}\rightarrow[g_{t+1/2}({\vec{x}})]U^{t+1}_{{\vec{x}},\alpha}[g_{t+1/2,\alpha}({{\vec{x}}+{\hat{\alpha}}})]^{\dagger}$
or, equivalently,
$U^{t}_{{\vec{x}},\alpha}\rightarrow[g_{t+1/2}({\vec{x}})]^{\dagger}U^{t}_{\vec{x},\alpha}g_{t+1/2}({\vec{x}}+\hat{\alpha})$.
We shall use $T^{0}_{t+1/2}$ to denote the gauge field independent lattice
shift forwards. Its dagger is the shift backwards.

$S=\sum_{t}S(t,t+1)$ where each term is associated with the pair ($t$,$t+1$).
$S(t,t+1)$ is made out of two inter leaf terms, $S(t)$, $S(t+1)$ and one intra
leaf term, $S(t+1/2)$.

$S(t)=\frac{1}{2}\sum_{\alpha<\beta}Tr{\big{[}}[T^{t}_{\alpha},T^{t}_{\beta}][T^{t}_{\alpha},T^{t}_{\beta}]^{\dagger}{\big{]}}$
(III.2)
$S(t+1/2)=\sum_{\alpha}Tr{\big{[}}[T^{0}_{t+1/2},T^{t}_{\alpha}(U^{t}_{\alpha})][T^{0}_{t+1/2},T^{t}_{\alpha}(^{g_{t+1/2}}U^{t+1}_{\alpha})]^{\dagger}]{\big{]}}$
(III.3)

$S(t)$ is a ${}^{g}{\mathfrak{G}}_{t}$ invariant function
$\mathfrak{S}_{1}(U^{t}_{{\vec{x}},\alpha})$. $S(t+1/2)$ is a function
${\mathfrak{S}}_{2}(U^{t}_{{\vec{x}},\alpha},U^{t+1}_{{\vec{x}},\alpha},g_{t+1/2}({\vec{x}}))$
which can be viewed as a function
${\mathfrak{F}}^{A}_{2}(^{g_{t+1/2}({\vec{x}})}{U^{t+1}_{{\vec{x}},\alpha}},U^{t}_{{\vec{x}},\alpha})$
or as function
${\mathfrak{F}}^{B}_{2}(^{[g_{t+1/2}({\vec{x}})]^{\dagger}}U^{t}_{{\vec{x}},\alpha},U^{t+1}_{{\vec{x}},\alpha})$.

The partition function can be rewritten as:

$\begin{split}Z_{W}=&\int[dg^{1}]....[dg^{l_{4}}]\int[dU^{1}]....[dU^{l_{4}}]\;e^{-\frac{\beta}{4}[\mathfrak{S}_{1}(U^{1}_{\vec{x},\alpha})+\mathfrak{S}_{1}(U^{2}_{\vec{x},\alpha})]}e^{-\frac{\beta}{4}[\mathfrak{S_{2}}(U^{1}_{\vec{x},\alpha},U^{2}_{\vec{x},\alpha},g^{2}_{\vec{x}})]}\\\
&....e^{-\frac{\beta}{4}[\mathfrak{S}_{1}(U^{t}_{\vec{x},\alpha})+\mathfrak{S}_{1}(U^{t+1}_{\vec{x},\alpha})]}e^{-\frac{\beta}{4}[\mathfrak{S_{2}}(U^{t}_{\vec{x},\alpha},U^{t+1}_{\vec{x},\alpha},g^{t+1}_{\vec{x}})]}e^{-\frac{\beta}{4}[\mathfrak{S}_{1}(U^{t+1}_{\vec{x},\alpha})+\mathfrak{S}_{1}(U^{t+2}_{\vec{x},\alpha})]}e^{-\frac{\beta}{4}[\mathfrak{S_{2}}(U^{t+1}_{\vec{x},\alpha},U^{t+2}_{\vec{x},\alpha},g^{t+2}_{\vec{x}})]}\\\
&....e^{-\frac{\beta}{4}[\mathfrak{S}_{1}(U^{l_{4}-1}_{\vec{x},\alpha})+\mathfrak{S}_{1}(U^{l_{4}}_{\vec{x},\alpha})]}e^{-\frac{\beta}{4}[\mathfrak{S_{2}}(U^{l_{4}-1}_{\vec{x},\alpha},U^{l_{4}}_{\vec{x},\alpha},g^{l_{4}}_{\vec{x}})]}e^{-\frac{\beta}{4}[\mathfrak{S}_{1}(U^{l_{4}}_{\vec{x},\alpha})+\mathfrak{S}_{1}(U^{1}_{\vec{x},\alpha})]}e^{-\frac{\beta}{4}[\mathfrak{S_{2}}(U^{l_{4}}_{\vec{x},\alpha},U^{1}_{\vec{x},\alpha},g^{1}_{\vec{x}})]}\end{split}$
(III.4)

The $U$ integrals are manipulated first, at fixed $g$. We make a change of
integration variables
$U^{t}_{\alpha}\rightarrow^{g_{t+1/2}}\\!U^{t}_{\alpha}$. This does not affect
the $\mathfrak{S}_{1}$ terms. In the $\mathfrak{S}_{2}$ terms it replaces
$g^{t}_{\vec{x}}$ by ${\mathbf{1}}$. One has to keep the
${}^{g_{t+1/2}}U^{t}_{\alpha}$’s in the integration measure because the
$\mathfrak{S}_{2}$ are not invariant under the change of integration
variables. This is a round about way of fixing a gauge in the four dimensional
sense, however, it holds also on the torus, despite the fact that parallel
transport along a winding curve cannot be gauged away. The integration over
the $g$ variables has been moved from the kernel to the measure.

###  III.2 Gauge averaging per leaf.

One now factorizes the integrals over link variables with the completeness
relation shared between consecutive pairs of leaves and ends up with a product
over infinitely dimensional matrices replacing equation (III.4). Periodicity
implies that a trace is taken.

For pure $SU(N)$ gauge theory the space is a tensor product over one Hilbert
space for each matrix variable, $V$, consisting of complex valued functions of
$V$:

$(\Psi,\Phi)=\int[dV]\overline{\Psi(V)}\Phi(V)$ (III.5)

The single link Hilbert space has a basis consisting of multivariate
polynomials of increasing degrees in the entries of $V$ and their complex
conjugates. Elements of the basis are labeled by a composite index
$\mathfrak{R}=(R,i,j)$ where $R$ denotes a unique irreducible representation
of dimension $d_{R}$ and $1\leq i,j\leq d_{R}$ label the entries:
$\Psi_{\mathfrak{R}}(V)=\sqrt{d_{R}}[{\cal D}_{R}(V)]_{ij}$. The polynomials
${\cal D}_{R}(V)_{ij}$ are determined by the associated Young diagram. The
completeness relation is

$\sum_{\mathfrak{R}}\Psi_{\mathfrak{R}}(V)\overline{\Psi_{\mathfrak{R}}(U)}=\delta_{SU(N)}(V,U)=\delta(VU^{\dagger},\mathbf{1}),$
(III.6)

where the delta-function is on group elements. Hence the full Hilbert space is
spanned by a basis of multinomials, consisting of products of representation
polynomials, one for each $R$ per link. $U$ and $V$ now denote collections
over all link variables in a leaf. On the left hand side the sum over
$\mathfrak{R}$ is a sum over all possible combinations of
${\mathfrak{R}}_{\vec{x},\alpha}$, one for each $(\vec{x},\alpha)$ link on a
leaf. Similarly, $\Psi_{\mathfrak{R}}(V)$ and $\Psi_{\mathfrak{R}}(U)$ each
are a product over such factors, one for each link. Finally, he right hand
side of the equation is a product of one $\delta_{SU(N)}$ for each link on the
leaf. Equation (III.6) replaces each integrations over shared space-like link
variables by a factorized form and the path becomes a matrix product of
identical factors $\mathfrak{T}$.

I now turn to the more economical Dirac notation with states $\ket{U}$ where
$U$ denotes the entire inter link configuration of a space leaf. Then the
generalized $\delta_{SU(N)}(V,U)$ is $\braket{V}{U}$ with $\bra{V}$ in the
dual space to $\ket{U}$ and the decomposition of unity is
$\int[dU]\ket{U}\\!\bra{U}$.

Our task is to evaluate the integration over the $g$ variables at fixed $U$’s.
The dependence on $g$’s resides only in the measure. Consider $\mathfrak{P}$,
an operator given below:

$\mathfrak{P}=\int[dg]\sum_{\mathfrak{R}}\Psi_{\mathfrak{R}}(V)\overline{\Psi_{\mathfrak{R}}(^{g}U)}$
(III.7)

The integration variables consist of one $g_{\vec{x}}$ for each site on one
leaf. Acting with $\mathfrak{P}$ on a state $\Psi(^{g}U)$ we obtain a state
$\Phi(U)$:

$\Phi(V)=(\mathfrak{P}\Psi)(V)=\int[dg]\int[d^{g}\\!U]\braket{V}{{}^{g}\\!U}\Psi(U)=\int[dg]\Psi(^{g^{\dagger}}\\!V)=\int[dg]\Psi(^{g}V)$
(III.8)

Obviously, $(\mathfrak{P}\Phi)(V)=\Phi(V)$, so
$\mathfrak{P}^{2}=\mathfrak{P}$. The operator $\mathfrak{P}$ projects on the
gauge singlet space of the leaf gauge group. The same projector can be
represented by using the tangent vector fields associated with infinitesimal
space-leaf gauge transformations. This would give the Gauss law constraints on
states appearing in the Kogut-Susskind (KS) continuous time Hamiltonian
defined on a spatial lattice. The KS construction was built from scratch,
without relying directly on the Wilson path integral. The intra leaf term is
now given by a group manifold Laplacian for each leaf–link variable. It is
constructed from the same vector fields entering Gauss law constraints. Gauss’
law has nothing to do with continuous, as opposed to discrete, time.

We could first introduce an unrestricted transfer matrix $\mathfrak{T}_{0}$
starting from the exponentiated Kogut-Susskind Hamiltonian and forgetting
about Gauss’ law. The restricted $\mathfrak{T}$ we get from the requirement to
reproduce exactly the path integral is then
$\mathfrak{P}\;\mathfrak{T}_{0}\;\mathfrak{P}$. $\mathfrak{T}_{0}$ is given
by:

$\begin{split}\bra{V}\mathfrak{T}_{0}\ket{U}=&e^{-\frac{\beta}{4}\sum_{\alpha<\beta}Tr[[T_{\alpha}(V_{\alpha}),T_{\beta}(V_{\beta})][T_{\alpha}(V_{\alpha}),T_{\beta}(V_{\beta})]^{\dagger}]}\\\
&e^{-frac{N}{8\lambda}\sum_{\alpha<\beta}Tr[[T_{\alpha}(U_{\alpha}),T_{\beta}(U_{\beta})][T_{\alpha}(U_{\alpha}),T_{\beta}(U_{\beta})]^{\dagger}]}\\\
&e^{-\frac{\beta}{2}\sum_{\alpha}Tr[[T_{\alpha}(V_{\alpha}),T_{t}^{0}][T_{\alpha}(U_{\alpha}),T_{t}^{0}]^{\dagger}]]}\\\
&e^{-\frac{\beta}{4}\sum_{\alpha<\beta}Tr[[T_{\alpha}(U_{\alpha}),T_{\beta}(U_{\beta})][T_{\alpha}(U_{\alpha}),T_{\beta}(U_{\beta})]^{\dagger}]}=\mathfrak{T}_{0}(V,U)=\mathfrak{T}_{0}(U^{\dagger},V^{\dagger})\end{split}$
(III.9)

In terms of $\mathfrak{T}$, $Z_{W}$ is given by $Tr\mathfrak{T}^{l_{4}}$.

Alternatively, start with an $l_{4}=\infty$ system, fix the gauge
$U_{\vec{x},t}=\mathbf{1}$ and adopt $\mathfrak{T}_{0}$ from there. Next,
prove that $[\mathfrak{T}_{0},\;^{g}G_{t}]=\mathbf{0}$ for any $t$. For finite
$l_{4}$, with periodic boundary conditions, one needs to keep a nontrivial
$U_{\vec{x},t}$ at one $t$, say at $t=l_{4}$. Integrating over the nontrivial
$U_{\vec{x},l_{4}}$ forces the insertion $\mathfrak{P}$ only at $t=l_{4}$.
$\mathfrak{P}$ can then be propagated to all other locations along the $t$
span by using ${\mathfrak{P}}^{n}=\mathfrak{P}$ for any positive integer $n$
and $[\mathfrak{P},\;{\mathfrak{T}}_{0}]=0$. So, fixing the gauge correctly
produces the same result as not picking any gauge at all, as it should be.

###  III.3 Observables containing time-like link matrices.

The obvious observable set we have access to consists of multinomial functions
of link variables that are invariant under $\mathfrak{P}$. These are traces of
parallel transport round closed loops, both contractible and not. Let
$\operatorname{\mathcal{O}}$ be such an observable on one leaf. Then,

${\braket{\operatorname{\mathcal{O}}}}_{path-
integral}=\operatorname{\mathbf{Tr}}(\mathfrak{T}\operatorname{\mathcal{O}}).$
(III.10)

So far, the set of observables does not include Wilson loops with time-like
links. Time like links in a gauge invariant observable are fine in the path
integral. Hence, observables containing time like links must be allowed also
in transfer matrix language. The presence of time like links in an observable
affects the integration over time like link matrices. The latter were absorbed
into three dimensional gauge transformations acting on individual links
appearing in the action. Equation III.10 does not hold anymore. The time links
present in the observable would be washed out by specific factors in
$\mathfrak{P}$.

This easy to correct. Let us take the simple example of a localized
rectangular Wilson loop in the $x_{1}-x_{4}$ plane with an extent $l$ in the
$x_{1}$ direction and $t$ in the $x_{4}$ one. In this case one no longer
averages over all space anchored gauge transformations. The factors in
$\mathfrak{P}$ acting at the spatial locations of the corners separated only
in the $x_{4}$ direction are replaced by projectors on polynomials of space
link variables which transform as ${\mathfrak{R}}=F$ and its conjugate
respectively. Here, $F$ denotes the fundamental representation. To work this
out explicitly one uses the basis of equation III.6. One can then generalize
and replace the two lines in each leaf by any contour connecting the same
endpoints. If the separation $t$ is large enough (but still smaller than
$l_{4}/2$ ) the dominating state will contain a superposition of many paths
connecting the corners and arbitrary extra closed contours. For large enough
$t$ the state of lowest energy is in the sector of Hilbert space selected by
the amended projectors will dominate. This state, with fixed ends and in the
limit of large separations is an object effective open string theory could
describe approximately.

There is a qualitative difference between observables and action: while both
are functions of space links, the observables consists of a relatively small
number of local “disturbances” over the vacuum, which is the ground state of
$\mathfrak{T}$. More precisely, when the spatial volume is taken into
infinity, the observable depends on only a finite number of spatial links.
This is analogous to particle excitations above the vacuum in non-gauge
theories.

###  III.4 Finite temperature deconfinement transition.

Consider again the open string state described in the previous section. Its
energy relative to the sourceless vacuum state grows linearly with the
separation $l$ at large enough $l$. This is an asymptotic statement: one would
need to take the space directions to infinity for a precise formulation. This
will hold at low enough temperatures, but ceases to be true above a transition
temperature, where the theory deconfines. The presence of such a transition
for stringy states has been known before lattice gauge theory was invented
hagedorn . The point is that the number of different contributions (orthogonal
states entering additively in the trace) increases exponentially with $l$
because a meandering curve anchored at 2 widely separated states can change
its shape anywhere locally along it while still obeying the constraint to
fixed end points. The entropy can overcome the suppression coming from the
energy at high enough temperature. When this happens the string “dissolves”.

The clearest symptom of this has to do with an extra global symmetry. For zero
temperature this symmetry is part of the gauge group; for non-zero temperature
it is separate. Every spatial location defines a minimal length parallel
transporter round the periodic direction which can be multiplied by a space
independent $N^{\rm th}$-root of unity, $z$, without changing the action. In
equations, $T_{4}\rightarrow zT_{4}$, with $z\in Z(N)\subset SU(N)$. This is a
global (in space) $Z(N)$ symmetry. The string state has sources in conjugate
fundamentals at its distinct ends, so also remains invariant. After
deconfinement no correlation should be left between the two sources at the
string ends when they are widely separated. The expectation value of the
observable should cluster into a product of two expectation values, each
associated with one of the strings ends. Each one of these factors no longer
is invariant under the $Z(N)$. So, unless the correlation is exactly zero, the
global $Z(N)$ is spontaneously broken at infinite spatial volume.

The symmetry breaking feature of the finite temperature transition has led to
an industry of effective $Z(N)$ models whose variables could be thought of as
stand-ins for traces of winding (Polyakov-) loops. Approximate renormalization
group methods ogil  indicated that such effective models have some validity
and this physical picture has accumulated support over the years and is widely
accepted today.

One can also consider an observable consisting of two parallel minimal length
thermal loops separated in space and winding once in opposite directions. Now,
the partition function in the presence of sources depends on their separation
and the energy–entropy argument applies again. $Z(N)$ breaking by clustering
is directly evident.

###  III.5 Eguchi Kawai reduction and its connection to finite temperature.

Eguchi Kawai reduction has already been introduced. The four dimensional
Euclidean system consists of just four $SU(N)$ unitary matrices associated
with four loops, all beginning and ending at the same single point. Each loop
has an associated $Z(N)$ symmetry. The original Eguchi Kawai claim was that
this model exactly reproduces infinite $N$ and arbitrary volume $SU(N)$
lattice gauge theory with Wilson action. For their argument to hold,
preservation of the $Z(N)$ symmetry at is crucial. It turned out that the
$Z(N)$’s will break also as $N\to\infty$, once the physical gauge coupling is
weak enough.

It was understood that the role of momentum space might be taken over by the
eigenvalue sets of the four link matrices, but that would work only if these
eigenvalue sets were individually and independently uniformly distributed,
which, would ensure the preservation of the four $Z(N)$’s also at infinite
$N$. The failure of this to happen could be understood from a one loop
calculation of the effective potential governing the dynamics of these
eigenvalues. In the quenched version this was handled by redefining the model
requiring the eigenvalues to be quenched and thus neutralizing the induced
effective potential. For small $\beta$ the $Z(N)$’s did not break at infinite
$N$ and quenching had no effect. Hence, the validity of the original Eguchi
Kawai analysis was preserved in the strong coupling regime. The extension to
weak coupling by quenching reproduced normal infinite volume lattice
perturbation theory and was conjectured to “fix” EK-reduction. The amended
model is called the “quenched Eguchi-Kawai” (QEK) model.

While the finite temperature transition occurs only at infinite spatial volume
with any finite $N$, the one here takes place, because $N$ was taken to
infinity, even at finite spatial volume. Nevertheless, the explicit
calculation of the effective potential made it obvious that the mechanisms
were similar. This was shown in fintempnpb . I present this derivation below
in a somewhat more general case than in the original paper, because it is
related to “deconstruction” deconstruction . Also, it is more detailed,
emphasizing the mechanism by which taking $N\to\infty$ first eliminates volume
dependence.

###  III.6 Summary of section.

The restriction to space-leaf gauge invariant states in the transfer matrix
formalism is a direct consequence of periodicity in the path integral. If the
time extent is infinite one can use a partially fixed gauge
$U_{{\vec{x}},t}=\mathbf{1}$ and read off the transfer matrix with ease. For a
finite periodic extent the same transfer matrix works but now the projector
$\mathfrak{P}$ must be introduced in the trace. This can be obtained by now
fixing all $U_{{\vec{x}},t}=\mathbf{1}$ but one, and integrating over the
unfixed link variable. This framework extends to the inclusion of all
observables accessible to the path integral by exploring local alterations of
the projectors $\mathfrak{P}$.

The transfer matrix formalism leads naturally to a physical picture predicting
deconfinement above some finite temperature. The transition reflects itself in
spontaneous breaking of a global $Z(N)$ symmetry in the path integral
formalism.

Eguchi-Kawai reduction has all four directions compact and therefore four
independent $Z(N)$’s. They need to all be preserved for it to work. It turns
out that these symmetries do break spontaneously at large $N$, even though the
four volume is minimal. It was later found numerically that if all four
directions are large enough in physical, continuum units (but far from
absurdly large) the desired symmetry remains preserved in the continuum limit
contek . So, Eguchi-Kawai volume independence is likely valid in continuum,
but only for volumes whose sides are large enough.

##  IV Finite temperature at weak gauge coupling.

In this section we slightly change notation and focus: We single out one
direction whose periodicity is $N_{t}$, the lattice inverse temperature. In
the spatial directions all periodicities are $L$. The spatial lattice has $d$
dimensions; of special interest are the ultraviolet complete cases $d\leq 4$.

###  IV.1 Complete gauge fixing at finite temperature.

This time, the gauge will be completely fixed. All directions in the space of
remaining variables will have bounded fluctuations in the Gaussian
approximation, eliminating the need for ghosts, without abandoning validity
outside perturbation theory, where BRST gauge fixing does not apply brs2 .

Consider a ring of $N_{t}$ sites connected by links on which $SU(N)$ unitary
matrices $g_{0},g_{1},...,g_{N_{t}-1}$ reside. This ring is perpendicular to
each leaf at the site located at ${\vec{x}}$ on that leaf. Start at site $2$
and make a gauge transformation there by $g_{1}$. $g_{1}$ acts backwards on
the link $1-2$ and replaces $g_{1}$ there by $\mathbf{1}$. It also acts on the
leaf, but the action does not change and neither does the Haar integration
measure. It acts forward on link $2-3$, replacing $g_{2}$ that resided there
by $g_{1}g_{2}$. Now we make a gauge transformation by $g_{1}g_{2}$ at site
$3$ which puts unity on link $2-3$ amd $g_{1}g_{2}g_{3}$ on link $3-4$. When
we reach site $N_{t}-1$ this process is done one last time, with the product
$g_{1}g_{2}....g_{N_{t}-1}$ being placed on link $N_{t}-1$. We cannot go
further, because this would affect link $0$. This is as it should be because
the product $g_{1}g_{2}....g_{N_{t}-}$ transforms by conjugation under the
gauge group, so its set of eigenvalues is gauge invariant: a multiplication
from only one side would change these. It is still possible though to do a
gauge transformation by an $SU(N)$ matrix $W$ at all sites on the ring. This
leaves all the links we set to unity unchanged because they get a $W$ from one
end and a $W^{\dagger}$ from the other. On link $(N_{t}-1)-0$ the variable
$g_{0}g_{1}....g_{N_{t}-1}$ gets conjugated. We can choose $W$ to be the
diagonalizing matrix. We fix $W$ up to a discrete element of $Z(N)$ to produce
a diagonal matrix on link $(N_{t}-1)-0$ whose angles increase from $-\pi$ to
$\pi$ as their index increases from $0$ to $N_{t}-1$. The remaining $Z(N)$ can
be fixed by forcing one entry of $W$ into a chosen interval of length
$\frac{2\pi}{N}$ on the unit circle. The change of variables from $g\equiv
g_{0}g_{2}....g_{N_{t}-1}$ to $W$ and $\theta_{j}$ that takes place at each
$x$ is accompanied by the usual Jacobian. There is one such factor for each
spatial site and it has preference for an equally spaced distribution of the
$\theta_{j}$’s round the unit circle. One can move all $W$’s along the unit
links until they completely disappear. Only the Jacobian factors, one for each
$x$, remain.

In perturbation theory we want translational invariance in all directions
because then we can simultaneously diagonalize all $T_{\mu}$’s at the
expansion point using the DFT matrices, as discussed. To achieve translational
invariance in the $\tau$ direction we allow the introduction of diagonal
unitary link matrices not in $SU(N)$: their determinant is not unity. There is
no impediment to doing that because one can multiply a unitary matrix by any
complex one. In our case matters are even simpler since the “foreign” matrices
are diagonal with $U(1)$ elements on their diagonals, so they make group
theoretical sense in terms of $U(N)$. To be sure, all integration variables
are $SU(N)$ matrices.

Let the ordered diagonal matrices consisting of eigenvalues of
$g=g_{0}g_{2}....g_{N_{t}-1}$ at each spatial site ${\vec{x}}$ be:

$D_{g}({\vec{x}})={\rm
diag}(e^{i\theta_{1}({\vec{x}})},e^{i\theta_{2}({\vec{x}})},....,e^{i\theta_{N}({\vec{x}})}),\;\;\;\sum_{j=1}^{N}\theta_{j}({\vec{x}})=0{\rm
mod}(2\pi).$ (IV.1)

We choose an $N_{t}^{\rm th}$ root of $D_{g}(x)$, $D_{P}(x)$,

$D_{P}(x)={\rm
diag}{\Big{(}}e^{i\frac{\theta_{1}(x)}{N_{t}}},............,e^{i\frac{\theta_{N}(x)}{N_{t}}}{\Big{)}}$
(IV.2)

By a further variable transformation (acting like an ordinary gauge
transformations, but with phase elements not in $SU(N)$), each link in the
time direction can be made to carry one copy of $D_{P}(x)$. Note that there
are only $L^{d}$ $D_{P}({\vec{x}})$ variables; they do not depend on $t$. In
the path integral one integrates over $L^{d}$ sets of $\theta({\vec{x}})$’s.
They are nonlocal in $t$, constrained to be identical on all $N_{t}$ links in
the $t$ direction.

###  IV.2 Finite temperature effective potential to one loop.

The calculation in this subsection is a generalization of an old result
fintempnpb . The presentation here will be more detailed. The objective is to
derive in leading order perturbation theory the effective potential governing
the distribution of the eigenvalues of an untraced Polyakov loop in a manner
similar to the derivation of the effective potential for all angles in the
Eguchi Kawai model qek . By “effective potential” I mean the familiar term in
the generating function of one-particle-irreducible diagram. It is needed here
for fields taking values in a curved compact space, the $SU(N)$ group
manifold. The ultraviolet regularization is the discrete lattice and the
infrared one comes by keeping only a finite number of sites. There is no
“large field problem”.

I work in space-dimension $d$ on a torus of $L^{d}\times N_{t}$ sites. The
limit $L\to\infty$ will also be considered. The spatial directions are labeled
by $\mu=1,2,...,d$. The (d+1)-direction is $\tau$. A site location on the
$d+1$ dimensional lattice is labeled by $(x,t)$. $x$ has $d$ components and
$t$ is the $d+1$ component of the site index. $\tau$ is the unit vector in the
$d+1$ direction. The partition function, $Z$, up to normalizations which are
allowed to change as the calculation progresses, is:

$\begin{split}Z=&\int\prod_{x}\prod_{t=0}^{N_{t}-1}{\Big{\\{}}\prod_{\mu=1}^{d}[dU_{\mu}(x,t)]dU_{0}(x,t){\Big{\\}}}e^{S(U_{\mu},U_{\tau})}\\\
S(U_{\mu},U_{\tau})=&\frac{\beta}{2}\sum_{x}\sum_{t=0}^{N_{t}}{\Big{\\{}}\sum_{1\leq\mu,\nu\leq
d}tr[U_{\mu}(x,t)U_{\nu}(x+\mu,t)U_{\mu}^{\dagger}(x+\nu,t)U_{\nu}^{\dagger}(x,t)]+\\\
&c\sum_{\mu=1}^{d}tr[U_{\mu}(x,t)U_{\tau}(x+\mu,t)U_{\mu}^{\dagger}(x,t+\tau)U_{\tau}^{\dagger}(x,t)+h.c.]{\Big{\\}}}\\\
=&-\frac{\beta}{2}{\big{\\{}}\sum_{1\leq\mu<\nu\leq
d}Tr[T_{\mu},T_{\nu}][T_{\mu},T_{\nu}]^{\dagger}+c\sum_{\mu=1}^{d}[Tr[T_{\mu},T_{\tau}][T_{\mu},T_{\tau}]^{\dagger}+Tr[T_{\mu},T_{\tau}^{\dagger}][T_{\mu},T_{\tau}^{\dagger}]^{\dagger}]{\big{\\}}}\end{split}$
(IV.3)

The gauge is completely fixed as before. There are $L^{d}$ angle variable
sets, $\theta_{j}(x)\in[-\pi,\pi)$, $j=1,...,N$ ordered in increasing order in
$j$. They do not depend on $t$. The effective potential for the angles
$\theta$ is computed to leading order in $\frac{1}{\beta}$, at fixed ratio
$\frac{\beta_{t}}{\beta}=c$.

$\begin{split}Z=&{\rm Cst}\int\prod_{x}\prod_{k}[d\theta_{k}(x)]\prod_{1\leq
k<j\leq
N}[\sin^{2}\frac{\theta_{j}(x)-\theta_{k}(x)}{2}]\prod_{x}[{\delta}_{{\rm
mod}(2\pi)}(\sum_{k}\theta_{k}(x))]\\\
&\int\prod_{\mu,x,t}[dU_{\mu}(x,t)]e^{S(U_{\mu}(x,t),U_{\tau}(x,t)=D_{P}(x))}\end{split}$
(IV.4)

The $\sin^{2}$ term is a Jacobian resulting from changing integration
variables. The original ones were $SU(N)$ matrices $W(x)$, one for spatial
site $x$, the new ones are their eigenvalues. The Jacobian at each $x$ shows
that they repel. The unitary diagonalizing matrices are absorbed as spatial
gauge transformations.

The finite temperature global $Z(N)$ symmetry acts by
$\theta_{j}(x)\rightarrow\theta_{j}(x)+\frac{2k\pi}{N},k=0,1,2,..,N-1$. If the
$e^{i\theta_{j}}(x)$ phases consist of all distinct $N^{\rm th}$ roots of
unity for each $x$, elements of $Z(N)$ cyclically rotate one angle into the
next, indicating a preferred $Z(N)$ invariant state compatible with repulsion.
At $\beta=0$, the $Z(N)$ is preserved by the vacuum. When $\beta$ becomes
large, the action must be taken into account. The following leading order
calculation in $1/\beta$ shows that the Jacobian domination will be reversed
when $\beta\to\infty$. Somewhere on the positive $\beta$ axis a $Z(N)$
breaking phase transition will occur.

The calculation proceeds on very similar lines as in qek . One expands around
a classical configuration that maximizes $S$ and dominates at infinite
$\beta$:

$U_{\mu}^{ij}(x,t)=\delta_{ij},\;\;\;\theta_{i}(x)=\theta_{i}^{0}\;\;\;i,j=1,...,N$
(IV.5)

The corresponding $T$-parallel transporters are denoted by $T_{\mu}^{0}$ and
$T_{\tau}^{0}(\theta)$. They mutually commute. Dependence on $\theta_{i}^{0}$
enters only through $T_{\tau}^{0}(\theta)$.

One needs to integrate over fluctuations $\delta\theta^{j}(x)$ and $\delta
U_{\mu}(x,t)$ at quadratic order in the exponents. They induce fluctuations
$\delta T_{\mu},\delta T_{\tau}$. Since the background is commutative and the
action consists of a commutator times its adjoint and traced over, second
order contributions arise solely from first order variation of the
commutators. The remaining $T$ factors are $T^{0}$’s.

There will be no quadratic cross terms containing $\delta T_{\mu}$ and $\delta
T_{\tau}$:

$Tr(\delta T_{\mu}\delta
T_{\tau}{T_{\mu}^{0}}^{\dagger}{T_{\tau}^{0}(\theta)}^{\dagger})+Tr(\delta
T_{\mu}\delta
T_{\tau}^{\dagger}{T_{\mu}^{0}}^{\dagger}T_{\tau}^{0}(\theta))=Tr[\delta
T_{\mu}(\delta T_{\tau}{T_{\tau}^{0}(\theta)}^{\dagger}+\delta
T_{\tau}^{\dagger}T_{\tau}^{0}(\theta)){T_{\mu}^{0}}^{\dagger}]=0,$ (IV.6)

because, due to the $t$-independence of $U_{\tau}(x,t)$,
$[T_{\tau},T_{\tau}^{0}(\theta)]=0$ and $[\delta
T_{\tau},{T_{\tau}^{0}(\theta)}^{\dagger}]=0$. Also,
$\delta(T_{\tau}T_{\tau}^{\dagger})=0$.

Contributions from the measure term are at lower order in $\frac{1}{\beta}$
because there was no $\beta$ dependence in the Jacobian to start with.

The decoupling of $\delta T_{t}$ and $\delta T_{\mu}$ fluctuations can also be
seen directly at the level of plaquettes, by cancellations of cross terms
coming from $t$-consecutive time-like plaquettes.

Integration over fluctuation of the $\delta T_{\tau}-\delta T_{\tau}$ type
will make a contribution that is independent of the $\theta_{i}^{0}$ because
the $T_{\tau}^{0}(\theta)$ factors will cancel out. In summary, for the
purpose of computing the effective potential, we only need to integrate over
fluctuations in $U_{\mu}$. Two types of terms will be generated: one
associated with terms coming from $[T_{\mu},T_{\nu}]$ – leaf plaquettes – and
the other from $[T_{\mu},T_{\tau}]$ – plaquettes in the $\tau$ direction.
Fluctuations in $T_{\mu}$ that are diagonal in color indices can also be
ignored because they do not couple to the $\theta^{0}_{i}$; only off-diagonal
fluctuation need to be taken into account. To be sure, the integration over
diagonal fluctuations is under control: they are explicitly compact.

####  IV.2.1 Contribution from the $[T_{\mu},T_{\tau}]$ term.

The quadratic contribution comes only from the first order variations $\delta
F_{\mu,\tau}$ because the action is proportional to
$Tr[F_{\mu,\tau}F_{\mu,\tau}^{\dagger}]$ and at zeroth order all matrices
commute. The variation in $T_{\mu}$ is

$\delta
T_{\mu}=\delta(T_{\mu}^{0}\mathbf{U}^{\mu})=T_{\mu}^{0}\delta\mathbf{U}^{\mu}=T_{\mu}^{0}(\mathbf{1}+i\delta
A_{\mu})$ (IV.7)

where $\delta A_{\mu}$ is hermitian. The second term in the commutator,
$T_{\tau}(\theta)$, is a square matrix with $NN_{t}L^{d}$ columns whose
eigenvalues are $L^{d}$-degenerate and given by $e^{\frac{2\pi
i}{N_{t}}l+i\theta_{j}},l=0,1,...,N_{t}-1,j=1,...,N$. Conjugating by DFT
$\Omega$ simultaneously for all $d+1$ directions,
$\Omega^{*}T_{\tau}^{0}\delta A_{\mu}\Omega\equiv\delta{\hat{A}}_{\mu}$,
yields matrices circulant in all $d+1$ momentum indices. The $\delta A_{\mu}$
contain all the integration variables in matrix form. The integration measure
is associated with the norm $Tr\delta A_{\mu}\delta{A_{\mu}}^{\dagger}$. This
measure is invariant under conjugation of $\delta A_{\mu}$ by a unitary
matrix. We can integrate over $\delta{\hat{A}}_{\mu}$ instead of
$\delta{A}_{\mu}$. The leaf index $t$ goes into an index $l$ and the
$d$-component site index $x$ goes into a $d$ component momentum index $p$.

The basic structure is identical to the one in qek : Let $X$ be a complex
matrix and $D$ a diagonal unitary one. Then
$[X,D]_{a,b}=X_{a,b}(D_{b}-D_{a})$. With $D_{a}=e^{i\gamma_{a}}$, we find

$Tr[X,D][X,D]^{\dagger}=2\sum_{a<b}|X_{a,b}|^{2}\sin^{2}\frac{\gamma_{a}-\gamma_{b}}{2}$
(IV.8)

This simple formula is used in the intermediate steps from equation (7) to (9)
in qek . In our case $a,b$ are compound indices, $a=(t,x,j)$ and
$b=(t^{\prime},x^{\prime},k)$.

$\displaystyle\begin{split}&Tr[\delta T_{\mu},T_{\tau}^{0}(\theta)][\delta
T_{\mu},T_{\tau}^{0}(\theta)]^{\dagger}=2\sum_{l,p,l^{\prime},p^{\prime},k<j}|(\delta{\hat{A}_{\mu}})_{(l,p,j),(l^{\prime},p^{\prime},k)}|^{2}\sin^{2}\frac{2\pi(l-l^{\prime})+\theta^{0}_{j}-\theta^{0}_{k}}{2N_{t}}\\\
&=2N_{t}L^{d}\sum_{l,p,k<j}|(\delta{\hat{A}_{\mu}})_{(0,0,j),(l,p,k)}|^{2}\sin^{2}\frac{2\pi
l+\theta^{0}_{j}-\theta^{0}_{k}}{2N_{t}}\end{split}$ (IV.9)

The angles $\theta^{0}$ only enter through the differences
$\theta^{0}_{i}-\theta^{0}_{j}$. Hence, we can ignore contributions coming
from diagonal color components
$(\delta{\hat{A}}_{\mu})_{(l,p,j),(l^{\prime},p^{\prime},j)}$ as they produce
terms independent of $\theta^{0}$.

####  IV.2.2 Contribution from the $[T_{\mu},T_{\nu}]$ term.

The angles $\theta^{0}_{j}$ do not enter directly, but the commutator
contributes to similar terms in the action which do. In isolation, the
$[T_{\mu},T_{\nu}]$ terms are $N_{t}$ decoupled $d$-dimensional Wilson action
lattice theories. Hence, second order contribution must contain “zero modes”
that correspond to a gauge transformation acting independently on leaves. The
contribution evaluated before provides masses to all these modes. To be sure,
the full system has no leaf-restricted gauge symmetry. The leafs are coupled
through the angles: had we integrated over them fully, we would have returned
to the projector on gauge invariant leaf states we discussed earlier.

Let us pick one pair $\mu\neq\nu$.

$\delta[T_{\mu},T_{\nu}]=[\delta T_{\mu},T_{\nu}^{0}]-[\delta
T_{\nu},T_{\mu}^{0}]$ (IV.10)

We need to evaluate the second order term
$Tr[\delta[T_{\mu},T_{\nu}](\delta[T_{\mu},T_{\nu}])^{\dagger}$. The
$\delta{\hat{T}}_{\mu}$ – $\delta{\hat{T}}^{\dagger}_{\mu}$ and
$\delta{\hat{T}}_{\nu}$ – $\delta{\hat{T}}^{\dagger}_{\nu}$ terms have
familiar structures. The $\delta{\hat{T}}_{\mu}$ –
$\delta{\hat{T}}^{\dagger}_{\nu}$ and $\delta{\hat{T}}_{\nu}$ –
$\delta{\hat{T}}^{\dagger}_{\mu}$ come with a projector in the $d$-dimensional
space indexed by $\mu,\nu$ and proportional to
$\sin\frac{2\pi(p_{\mu}-{p^{\prime}}_{\mu})}{2L}\sin\frac{2\pi(p_{\nu}-{p^{\prime}}_{\nu})}{2L}$.
This form is obtained after absorbing some momentum dependent phases in the
$\delta{\hat{A}}_{\mu}$ variables, which drop out. The quadratic form
connecting all $\delta{\hat{A}}_{\mu}$, $\delta{\hat{A}}_{\nu}^{*}$ variables
has zero modes. All of them get masses from the $\mu-\tau$ plaquettes. So,
there are no divergences in the path integral, even when truncating to this
order in $\beta^{-1}$. If none of the momentum components are zero, one can
choose a new basis of $d$ component vectors, different at each momentum, which
separates out the zero modes explicitly. There will be unresolved degeneracies
when some or all components of the momenta vanish. This is not a problem, one
just needs to linearly complete the choices in some way. Traditionally, in the
continuum one ignores this, and goes to longitudinal and transverse
components. For notational simplicity I’ll do this below, with the
understanding that nothing is fudged even at these degeneracy points.

${\hat{A}}_{\mu}$, viewed as a $d$-vector, is decomposed into $d-1$
$d$-component vectors transverse to the vector
$\sin\frac{\pi(p_{\mu}-p^{\prime}_{\mu})}{L}$, $e^{(\alpha)}$, and a
longitudinal $e^{\rm long}$. These vectors depend on $p-p^{\prime}$.
${\hat{A}}_{\mu}=\sum_{\alpha=1}^{d-1}{\hat{A}}^{(\alpha)}e^{(\alpha)}_{\mu}+{\hat{A}}^{\rm
long}e^{\rm long}_{\mu}$. The $e^{(\alpha)}$ and $e^{\rm long}$ make up an
orthonormal set under the inner product
$\braket{e}{e^{{}^{\prime}}}=\sum_{\mu=1}^{d}e^{*}_{\mu}e^{{}^{\prime}}_{\mu}$.
The ${\hat{A}}^{(\alpha)}$ carry the same indices as ${\hat{A}}_{\mu}$ and so
does ${\hat{A}}^{\rm long}$, which does show up explicitly in the term coming
from spatial plaquettes.

In summary, the total contribution to the action from spatial plaquettes,
ignoring additive constants, is

$Tr[T_{\mu},T_{\nu}][T_{\mu},T_{\nu}]^{\dagger}=2N_{t}L^{d}\sum_{1\leq k,j\leq
N}\sum_{l,p}\sum_{\alpha=1}^{d-1}|(\delta{\hat{A}}^{(\alpha)}_{(0,0,j)(l,p,k)}|^{2}[\sum_{\nu=1}^{d}\sin^{2}\frac{2\pi
p_{\nu}}{2L}]$ (IV.11)

####  IV.2.3 Total contribution.

The integration measure is
$\prod_{\mu=1}^{d}\prod_{l=0}^{N_{t}-1}\prod_{p}\prod_{1\leq k<j\leq
N}d{(\delta{\hat{A}}_{\mu})}_{(0,0,j),(l,p,k)}d[{(\delta{\hat{A}}_{\mu})^{*}}_{(0,0,j),(l,p,k)}]$.
One can replace $\mu$ and the associated $\mu$ indexed variables by the set
$\alpha$ and “${\rm long}$”, because the transformation relating them is
orthonormal. (The diagonal $j=k$ terms would be real, but they can be ignored
as explained.) This expression contains the list of all independent (complex)
variables which are integrated over. The integrand is factorized into
elementary Gaussian terms. Each independent integration variable can be
rescaled without altering the $\theta^{0}$ dependent contribution to the
effective potential. This changes an overall normalization which is irrelevant
for the probability distribution of the angles $\theta^{0}$. With
$c^{-1}=\rho$ the integrand’s exponent is:

$\displaystyle\begin{split}&-L^{d}N_{t}\sum_{1\leq k<j\leq
N}{\sum}_{l,p}[\sum_{\alpha=1}^{d-1}|(\delta{\hat{A}}^{(\alpha)})_{(0,0,j),(l,p,k)}|^{2}[\sin^{2}\frac{2\pi
l+\theta^{0}_{j}-\theta^{0}_{k}}{2N_{t}}+\rho\sum_{\nu=1}^{d}\sin^{2}\frac{2\pi
p_{\nu}}{2L}]+\\\ &|(\delta{\hat{A}}_{\mu}^{\rm
long})_{(0,0,j),(l,p,k)}|^{2}\sin^{2}\frac{2\pi
l+\theta^{0}_{j}-\theta^{0}_{k}}{2N_{t}}]\end{split}$ (IV.12)

Above, ${\sum}_{p}$ is over all $p$, in ranges $0,..,L-1$ for each component.
The constant $\rho>0$ is of order $1$. The $\theta^{0}_{j}$ integration
measure factor needs to be added to $v(\theta^{0})$, the result obtained by
integration over fluctuations.

This expression can be understood as follows: We are working in an axial gauge
corrected for compactness in the direction that was gauge fixed, to leading
order in perturbation theory around a configurations that breaks $d+1$ gauge
invariance. This correction endows the transverse gauge field components with
distinct masses, explicitly depending on group indices. In this deconstructed
$d$-dimensional framework we have $\frac{N(N-1)}{2}N_{t}$ massive gauge bosons
represented by as many complex fields. There also are $(N-1)N_{t}$ real bosons
whose masses are not dependent on the color index. $N-1$ out of the latter
have zero mass.

The term $\sin^{2}\frac{2\pi l+\theta^{0}_{j}-\theta^{0}_{k}}{2N_{t}}$ above,
appearing in sums over the discrete momenta in the $\tau$ direction, indicates
how fluctuating values of the ordered $\theta$ angles fill in the gaps of
length $\frac{2\pi}{N_{t}}$ created by the integral values of $l$, reflecting
finite temperature and periodicity in the $\tau$ direction. If the spacings
are uniformly filled, the dependence on temperature disappears. At infinite
$N$ the distribution of eigenvalues fills the gaps continuously for large
enough (but finite) $N_{t}$. If the global $Z(N)$ is preserved the
distribution is flat and the fill is “perfect” for any $N_{t}$.

Before carrying out the integral over fluctuations we wish to eliminate the
products over $l$. This leaves one with a purely $d$-dimensional point of
view, as desired in a deconstruction.

The sum over $l$ can be performed starting from the identity (known to
Mathematica):

$\displaystyle\begin{split}2^{n-1}&\prod_{k=0}^{n-1}[\cos
x-\cos(y+\frac{2k\pi}{n})]=\cos nx-\cos ny\\\
2^{2n}&\prod_{k=0}^{n-1}[\sin^{2}(\frac{y}{2}+\frac{k\pi}{n})-\sin^{2}\frac{x}{2}]=\sin^{2}\frac{ny}{2}-\sin^{2}\frac{nx}{2}\end{split}$
(IV.13)

This identity is between polynomials in $e^{ix}$ and $e^{iy}$. It can be
proved by matching zeros and normalizations. Let us first focus on the
$\delta{\hat{A}}_{\mu}^{\rm long}$ term. Using the second form of the identity
the integration over the longitudinal component produces a factor

$\prod_{p}\prod_{l}\prod_{k<j}\prod_{l=0}^{N_{t}}\sin^{2}(\frac{\theta_{j}-\theta^{0}_{k}}{2}+\frac{\pi
l}{N_{t}})={\rm
Cst}\prod_{k<j}(\sin^{2}\frac{\theta_{k}-\theta_{j}}{2})^{L^{d}}$ (IV.14)

in the denominator. This exactly cancels the Jacobian factor evaluated at the
uniform configuration $\theta_{j}(x)=\theta^{0}_{j}$. The upshot is that only
the transverse components on the spatial leaves contribute to the effective
potential. Here, this is a consequence of combining restricted gauge
invariance with the projection on gauge singlets.

We now turn to this remaining contribution of the transverse components of the
vector potential on the spatial leaves. Replacing $x$ by $ix$ in the above
identity we get

$\displaystyle\begin{split}2^{n-1}\prod_{k=0}^{n-1}[\cosh
x-\cos(y+\frac{2k\pi}{n})]=&\cosh nx-\cos ny\\\
2^{2n}\prod_{k=0}^{n-1}[\sin^{2}(\frac{y}{2}+\frac{k\pi}{n})+\sinh^{2}\frac{x}{2}]=&\sin^{2}\frac{ny}{2}+\sinh^{2}\frac{nx}{2}\end{split}$
(IV.15)

To use this variant of the identity we define the quantity $R(p)$ by:

$R(p)=\cosh^{-1}[1+\rho\sum_{\nu}\sin^{2}\frac{\pi p_{\nu}}{L}],\;\;R(p)\geq
1.$ (IV.16)

Then,

$v(\theta^{0})=-(d-1)\sum_{1\leq i<j\leq
N}{\sum}_{p}\log(\sin^{2}\frac{\theta^{0}_{i}-\theta^{0}_{j}}{2}+\frac{\cosh(N_{t}R(p))-1}{2}).$
(IV.17)

Using the definition of Chebyshev polynomials for arguments outside the
interval $(-1,1)$,

$T_{n}(\cosh x)=\cosh(nx),$ (IV.18)

we obtain:

$v(\theta^{0})=-(d-1)\sum_{1\leq i<j\leq
N}{\sum}_{p}\log[\sin^{2}\frac{\theta^{0}_{i}-\theta^{0}_{j}}{2}+\frac{T_{N_{t}}(1+\rho\sum_{\nu=1}^{d}\sin^{2}\frac{\pi
p_{\nu}}{L})-1}{2}].$ (IV.19)

Discarding another constant, the formula becomes

$v(\theta^{0})=-(d-1)\sum_{1\leq i<j\leq
N}{\sum}_{p}\log[T_{N_{t}}(1+\rho\sum_{\nu=1}^{d}\sin^{2}\frac{\pi
p_{\nu}}{L})-\cos(\theta^{0}_{j}-\theta^{0}_{i})].$ (IV.20)

The effective $V_{\rm eff}(\theta^{0})$ potential is defined per unit spatial
volume with a minus sign so that its minimum gives the vacuum energy density.
We discard an $N_{t}$ dependent constant which carries no $\theta^{0}$
dependence. The discarded term is a temperature dependent vacuum energy
correction.

$V_{\rm eff}(\theta^{0})=\sum_{1\leq i<j\leq
N}{\mathfrak{e}}(\theta^{0}_{j}-\theta^{0}_{i}),\;\;\;{\mathfrak{e}}(\gamma)=\frac{d-1}{L^{d}}\sum_{p}\log[1-\frac{\cos\gamma}{T_{N_{t}}(1+\rho\sum_{\nu=1}^{d}\sin^{2}\frac{\pi
p_{\nu}}{L})}]$ (IV.21)

It is now obvious that $\gamma=0$ is a minimum of $\mathfrak{e}(\gamma)$
showing that the angles $\theta^{0}_{i}$ now attract and could therefore cause
spontaneous $Z(N)$ symmetry breakdown.

The formula holds at finite $N$, $N_{t}$, $L$ and lattice spacing (set to 1,
but easily formally replaced by “$a$”). The
$L\to\infty,N_{t}\to\infty,N\to\infty,a\to 0$ limits can be taken in different
orders or various correlated ways. Exploring this systematically would take us
too far afield. As an example, I shall take the infinite $L$ limit followed by
the formal “$a\to 0$” limit, taken only in the spatial directions. This is the
“deconstruction” limit. It amounts to $\rho\to\infty$, because the spatial
lattice spacing becomes infinitely smaller than the separation among leaves.
The infinite $L$ limit (assuming it converges) is:

${\mathfrak{e}}^{L=\infty}(\gamma)=(d-1)\int_{|k_{\mu}|<\pi}\frac{d^{d}k}{(2\pi)^{d}}\log[1-\frac{\cos\gamma}{T_{N_{t}}(1+\rho\sum_{\nu=1}^{d}\sin^{2}\frac{k_{\nu}^{2}}{2})}]$
(IV.22)

A finite continuum limit requires $N_{t}>\frac{d}{2}$. Thinking about momenta
in units of mass, we need to give $\rho$ inverse mass squared units. Then,
using a mass dimensional parameter to measure all momenta we take the
$\rho\to\infty$ limit, expressing everything with the help of one mass scale,
$f$:

$[{\mathfrak{e}}^{L=\infty}]_{\rho=\infty}(\gamma)=\frac{(d-1)f^{d}}{\Gamma(\frac{d}{2})(2\pi)^{\frac{d}{2}}}\int_{x=0}^{\infty}x^{d-1}dx\log[1-\frac{\cos\gamma}{T_{N_{t}}(1+x^{2})}]$
(IV.23)

The Chebyshev polynomial in the denominator increases rapidly with $x^{2}$
since its argument is outside $[-1,1]$, so expanding the logarithm to leading
order captures the relevant qualitative properties for reasonable values of
$N_{t}$ and $\cos\gamma$.

While now $\cos\gamma$ goes to 1 at the minimum of the effective potential,
and a $Z(N)$ symmetric probability distribution for the $\theta^{0}_{j}$
variables is disfavored, one needs to keep in mind that the formula only holds
as a leading order computation. In a simulation, one should not expect to see
degeneracies of the $\theta_{0}^{j}$: the repulsion term in the measure is
exact, and therefore there always will be peaks of a width $\sim\frac{1}{N}$
centered at $N^{\rm th}$ roots of unity. The height of peaks would be
decreasing away from the central peak at $\gamma=0$ in both directions for
$\gamma\in[-\pi,\pi]$. For large enough $\beta$ the $Z(N)$ would be
spontaneously broken. One could extend the calculation to include a spatial
angle-derivative term to allow consideration of angles varying slowly on the
leaves.

To get at the standard continuum finite temperature effective potential the
simplest would be to keep $\rho=1$ and take $N_{t}$ and $L$ to infinity in a
correlated way, producing a finite temperature and an infinite spatial volume
in the end. The effective potential here is deconstructed and different.

The $N_{t}=1$ case violates the bound $N_{t}>\frac{d}{2}$ at the physical
point $d=3$, which blocks taking continuum limit in the spatial directions.
One still can take $L\to\infty$. The continuum limit for the effective
potential can be taken after appropriate subtractions. In this
superrenormalizable case one has good control over the ultraviolet. One has a
gauge fixed three dimensional $SU(N)$ gauge theory interacting with a compact,
$SU(N)$-valued Higgs field in the adjoint.

##  V The ÜY deformation.

In the last equation of qek  (eq.(18)) it was pointed out that the quantity
$P(\theta)=\frac{1}{N^{2}}\sum_{i,j=1}^{N}\sin^{2}\frac{\theta_{i}-\theta_{j}}{2}$
is bounded from above by $\frac{1}{2}$ for any set of angles and that for a
uniform distribution the bound was saturated. In the numerical simulations of
that paper the departure from saturation was used to estimate the location of
$Z(N)$ symmetry breaking transitions. The bound is easy to prove:

$\frac{1}{N^{2}}|\sum_{i=1}^{N}e^{i\theta_{i}}|^{2}=1-2P(\theta)$ (V.1)

It is not true that saturation of the bound implies uniformity: If $N=mn$ with
$m,n$ integers larger than 1, the index $i$ can be replaced by a double index
$(a,b)$ with $1\leq a\leq m$ and $1\leq b\leq n$. In a manipulation familiar
from FFT (fast Fourier transform), the sum over $i$ can be reorganized

$\sum_{i=1}^{N}e^{i\theta_{i}}=\sum_{b=1}^{n}\sum_{a=1}^{m}e^{i\theta_{a,b}}$
(V.2)

To saturate the bound it is enough that the sum over $a$ vanish for all $b$.
Such an angle set could preserve only a $Z(m)$ subgroup of $Z(mn)$: the sum
over $n$ at fixed $m$ does not need to vanish. It is for this reason that all
recent large $N$ simulations that relied on a qualitative large $N$
suppression of finite volume effects I have been involved in have been
restricted to prime $N$.

In its simplest form, the ÜY deformation is a new term in the action given by
a sum over spatial sites $\sum_{x}P(\theta(x))$ (up to an irrelevant
constant). It changes the dependence on time link variables of the action
itself, not just of some observables. When only the latter are impacted, the
change is limited. The observables can be viewed as probes of the theory’s
vacuum. They act as a limited number of excitations over a homogeneous vacuum,
similar to particle states (in this case allowing also configurations extended
in space, like strings).

The new term is taken with a sign promoting angle repulsion. It counteracts
the cancellation of the Jacobian by fluctuations in the spatial link variables
and, beyond a certain strength, will restore $Z(N)$ symmetry. In this phase
the system reconfines delia .

###  V.1 The ÜY term biases gauge averaging.

I now turn back to the derivation of a transfer matrix and the associated
projector on gauge invariant states starting from the original action. It is
much simpler to first fix the gauge such that all time like link variables are
fixed to unity except one, whose link variable is the open Polyakov loop.

Obviously, the deformation affects gauge averaging. With the term present the
derivation of the transfer matrix will not produce a projector on singlet
states because the measure in the averaging procedure is changed from Haar by
a weight dependent on the character of the adjoint representation of $SU(N)$.
This would still prohibit $Z(N)$ charge carrying representations but would
allow all $SU(N)/Z(N)$ representations (in other words, only the $Z(N)$
subgroup of $SU(N)$ still is averaged over). The projectors we obtained in the
undeformed theory get replaced by an operator $\mathfrak{Y}$, which is not a
projector. For example, a spatial open (untraced) Wilson loop will have a
deformation dependent expectation value, while it would vanish in the absence
of deformation. The deformed gauge averaging operator $\mathfrak{Y}$ is
inserted once for each site index $x$. Denoting the extra term in the action
by a $Z(N)$ invariant function of a group element, $\mathfrak{U}(g)$, we find:

$\mathfrak{Y}=\int[dg]e^{\mathfrak{U}(g)}\sum_{\mathfrak{R}}\Psi_{\mathfrak{R}}(V_{1})\overline{\Psi_{\mathfrak{R}}(^{g}V_{2})}$
(V.3)

The action is:

$\mathfrak{Y}\Psi(V)=\Psi_{Y}(V)=\int[dg]e^{\mathfrak{U}(g)}\Psi(^{g}V)$ (V.4)

Convolving once we get

$[\mathfrak{Y*Y}]\Psi(U)=\Psi_{Y*Y}(\mathfrak{U})=\int[dg^{\prime}]e^{\mathfrak{U}(g^{\prime})}\int[dg]e^{\mathfrak{U}(g)}\Psi(^{g^{\prime}}{{}^{{g}}U})$
(V.5)

$\mathfrak{Y*Y}$ clearly is not the same as $\mathfrak{Y}$: it is not a
projector. Each application applies the same weighted average and there is a
cumulative effect to multiple applications. When $\mathfrak{Y}$ acts on an
overall gauge singlet state, it gets reproduced times some constant. In other
words, up to a (calculable) multiplicative constant. $\mathfrak{Y}$ is a
projector if acting on an eigenstate of $\mathfrak{P}$.

The ÜY deformation alters the entire Hilbert space of the undeformed theory.
So long as the system is finite in all directions the distinction is only
qualitative and one can imagine how the vacuum gets deformed adiabatically as
the deformation is slowly turned on. This assumes that the infinite sums over
representations converge. At infinite volume though, the deformed Hilbert
space becomes orthogonal to the undeformed one. Something similar will happen
also when one takes $N\to\infty$, while the system size stays finite.

Intuitively, the fraction of states propagating with the $\mathfrak{Y}$
insertion and that are gauge singlets is small because there are infinitely
more states carrying all possible $SU(N)/Z(N)$ irreducible representations of
the gauge group. So far, I have not been able to turn this into a conclusion
that in a standard Monte Carlo simulation the reconfined results obtained with
the ÜY deformation one obtains are overwhelmingly unrelated to those obtained
in the absence of the deformation.

If we follow $\mathfrak{Y}$ immediately by the projector $\mathfrak{P}$, only
the singlets remain. In this case the deformation decouples and we can
integrate it out. This result may be of some use in the future, but does not
close the issue.

The one time link partition function for the simplest form of the deformation
is given by:

${\cal
J}(h)=\prod_{i=1}^{N}[\int_{-\pi\leq\theta_{i}<\pi}\frac{d\theta_{i}}{2\pi}]{\prod_{i<j}}[e^{2h\cos(\theta_{i}-\theta_{j})}\sin^{2}\frac{\theta_{i}-\theta_{j}}{2}]$
(V.6)

Expand each factor in a Fourier series:

$\begin{split}&e^{2h\cos(\phi)}\sin^{2}(\frac{\phi}{2})=\frac{1}{2}p_{0}+\sum_{n=1}^{\infty}p_{n}\cos(n\phi)\;\;\;with\;\;\;p_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}d\phi
e^{2h\cos(\phi)}[\frac{1-\cos(\phi)}{2}]\cos(n\phi)=\\\
&\frac{1}{\pi}\int_{0}^{\pi}d\phi
e^{2h\cos(\phi)}[\cos(n\phi)-\frac{\cos((n+1)\phi)+\cos((n-1)\phi)}{2}]=\\\
&I_{n}(2h)-\frac{I_{n+1}(2h)+I_{n-1}(2h)}{2}\;\;\;\;\;\;with\;\;\;\;\;\;I_{k}(z)\equiv\frac{1}{\pi}\int_{0}^{\pi}e^{z\cos\theta}\cos(k\theta)=I_{-k}(z)\end{split}$
(V.7)

Inserting the Fourier series in each factor of the integral defining ${\cal
J}(2h)$, we get

$\begin{split}{\cal
J}(h)=&\prod_{i=1}^{N}[\int_{-\pi\leq\theta_{i}<\pi}\frac{d\theta_{i}}{2\pi}]\\\
{\prod_{i<j}}&{\big{\\{}}\frac{1}{2}[I_{0}(2h)-I_{1}(2h)]+[\sum_{k_{ij}=1}^{\infty}{\big{[}}I_{k_{ij}}(2h)-\frac{I_{k_{ij}+1}(2h)+I_{k_{ij}-1}(2h)}{2}{\big{]}}e^{{ik_{ij}(\theta_{i}-\theta_{j})}}\big{\\}}\end{split}$
(V.8)

Now consider all terms containing at least one $\theta_{1}$ in the exponent.
To get a nonzero answer when carrying out the $\theta_{1}$ integration one
would need them to cancel each other, but the $k_{ij}\geq 1$ for all $i<j$.
Continuing to the $\theta_{2}$ integral eliminates all $k_{2j}$ terms, and so
on. The expression collapses to:

$\begin{split}{\cal
J}(2h)=&{\big{[}}\frac{I_{0}(2h)-I_{1}(2h)}{2}{\big{]}}^{\frac{N(N-1)}{2}}{\rm{~{}with}}\\\
{\cal
J}(0)=&\frac{1}{2}\;\;\;{\rm{because}}\;\;\;I_{0}(0)=1\;\;{\rm{and}}\;\;I_{1}(0)=0\\\
\frac{{\cal J}(2h)}{{\cal
J}(0)}=&[I_{0}(2h)-I_{1}(2h)]^{\frac{N(N-1)}{2}}\end{split}$ (V.9)

For all $x>0$, $1>I_{0}(x)-I_{1}(x)>0$, while $I_{0}(x)-I_{1}(x)>1$ for all
$x<0$. So, ${\cal{J}}(2h)>0$ for all $h$. At $N=\infty$, $\frac{{\cal
J}(2h)}{{\cal J}(0)}$ is $+\infty$ when $x<0$ and $0$ at $x>0$. ${\cal J}(2h)$
is more properly viewed as an integral over the group $SU(N)/Z(N)$ as nothing
changes under the shift $\theta_{i}\rightarrow\theta_{i}+r$ for all $i$ where
$e^{2i\pi{r}}$ is an $N$-th root of unity. Since we normalized group
integration to unity for $SU(N)$ and only look at ratios, this makes not
difference.

In summary, the fraction of gauge invariant states relatively to the total
number of states allowed for $h>0$ (the relevant range for reconfinement) is
suppressed by a power of spatial volume and $N(N-1)/2$. When either or both
are infinite, the theory seems very different from ordinary, all scale
confining pure $SU(N)$ gauge theory. It is the latter that is of interest in
the quest of total large $N$ reduction.

In UY  it is argued that in the absence of $Z(N)$ breaking the Schwinger Dyson
equations at leading order in $N$ remain independent of $h$. I see nothing
wrong with the formal argument, nor with the claim that a $h$ of right size
will restore global $Z(N)$. At strong couplings, $\beta,\beta_{t},h$ small,
this argument ought to be convincing. The convergence radius of the strong
coupling series for $h$ (the usual counting rules say $h$ is not scaled by any
power of $N$) may be just smaller than the value needed for reconfinement. For
larger $h$ the dramatic change in the Hilbert space structure takes the system
to a phase different from standard QCD in a major way. Strictly speaking, the
lattice Schwinger-Dyson loop equations loose their validity outside the strong
coupling regime. Here it seems that it is a phase transition that blocks their
validity. I suspect that at infinite $N$ the reconfined regime of finite
temperature pure gauge theory describes something very remote from ordinary
finite temperature QCD and therefore the generalization meant to solve the
problem of full one site large $N$ reduction will not work.

It also should be kept in mind that large $N$ reduction does work in continuum
without any deformations, for box sizes larger than about $\frac{1}{T_{c}}$
where $T_{c}$ is the finite temperature transition in pure gauge theory in
terms of the string tension at infinite volume contek . There have been
numerical tests of the ÜY deformation for $N=3,4$ and temperatures above
$T_{c}$ by some modest amount and it was found that some physical quantities
seem consistent with an adiabatic deformation of the same in the undeformed
theory, taken at temperatures in the regular confined phase. For one site
reduction to work automatically in the deformed theory one would need to see
results at larger values of $N$ and an analysis of their approach to the
$N=\infty$ limit.

When testing whether ÜY is an adiabatic deformation at large $N$ it should
also be kept in mind that there is a lattice bulk transition at large $N$ in
Wilson lattice pure $SU(N)$ gauge theory that invalidates strong coupling
results before one gets into the perturbative regime. That bulk transition has
to be reproduced at the same coupling under any prescription that seeks to
achieve single site reduction on the lattice at infinite $N$. Any prescription
for single site reduction if it holds, has to do so on the lattice.

##  VI Reconfinement by quenching.

Getting back to finite temperature, I ask whether there is a quenching
alternative to deformation reconfinement.

Using the fully gauge fixed form above it is natural to ask what happens if we
quench the $\theta_{i}(x)$ variables. The same arguments that applied in the
Eguchi-Kawai case would apply also here at large $N$. The angles are gauge
invariant variables and there are only $N-1$ of them per site. This would
support temperature independence in the E-K sense.

It might be of interest to consider this quenched deconstructed model and do
simulations directly in the axial gauge. The split into two stages one of
doing the gauge field integral at fixed angles, and the next of averaging the
averages of the first step over the angles distributed with the Jacobian
measure might make it easier to separate out physically relevant effects, and
at the same time, hopefully have a more clear understanding at the Feynman
planar diagram level of perturbation theory grosskitazawa . In ordinary, not
deconstructed finite temperature QCD, perturbation theory gets stuck at some
order because of infrared divergences. In the deconstructed case, the theory
is strictly three dimensional and the infrared divergences definitely don’t go
away but ultraviolet divergences are simpler.

At the technical level, I prefer the deconstructed view: the angle repulsion
term comes in explicitly, because one never works with vector potentials in
the Lie Algebra for the time direction. Taking the continuum limit in all four
directions makes it awkward to keep the repulsion term, the remnant of gauge
averaging which is crucial in deriving a Hilbert space view of what the
standard periodic gauge invariant lattice path integral tells us. Gauge
averaging is a procedure well defined for compact groups. In the continuum it
may get mixed up with ordinary gauge fixing and the manipulations seem to me
less reliable.

Local deformations to an action are typically expressed in terms of the same
elementary fields. As a start, their renormalization can be borrowed from the
renormalization of the observable corresponding to the undeformed theory which
is well understood. In four dimensions the Polyakov loop is a composite
nonlocal operator, which requires non-perturbative renormalization. The single
gauge invariant method I know of is by continuum smearing smear  which comes
with a dimensional parameter determined in the undeformed case when the
operator is just an observable. When the latter is promoted to a term in the
action it isn’t even clear to me what the dimensions of the coupling constant
$h$ should be.

So long as one relies on some undiscriminating version of effective field
theory I think the deconstructed framework is better suited, because then the
operator is not really nonlocal.

##  VII Summary

My presentation has been very detailed. Only basic information about lattice
gauge theory would be required to understand the paper. I have not produced
hard answers to the questions that motivated me. I can only hope that there
are some new insights here that would be useful in the future.

The main points were:

(1) The periodic Euclidean time lattice (path) integral of Wilson lattice
gauge theory produces a transfer matrix acting in a Hilbert space consisting
of square integrable functions of spatial link variable configurations which
are singlets under a local spatial gauge group.

(2) There is an exactly equivalent fixed gauge integral construction which
has, in addition to the spatial link variables, $N$ dynamical angles per
spatial site and a repulsion term of a simple form dictated by the structure
of $SU(N)$.

(3) The ÜY deformation adds an additional, angle dependent, term which alters
the structure of the Hilbert space by allowing all $Z(N)$ invariant
representations of the gauge group.

(4) In the limit of $N\to\infty$ the entire ÜY Hilbert space is orthogonal to
the Hilbert space in the absence of the deformation.

(5) There is a quenched alternative to the deformation, which also would
restore $Z(N)$ symmetry.

(6) The ÜY deformation and the quenched version admit generalizations to all
directions and both were proposed as ways to define a single site matrix model
equivalent to the full Wilson pure gauge model. I believe that the quenched
version has a better chance to work than the deformation version, but neither
has been proven. In the continuum, on a torus of sides larger than the inverse
deconfinement temperature large $N$ reduction holds without either alteration.
I don’t think this is known for the ÜY case, and am skeptical that it would
hold for the same continuum $SU(N)$ theory that the ordinary Wilson action
produces.

(7) It was speculated that starting from a deconstructed finite temperature
$SU(N)$ gauge theory might be beneficial in that four dimensional ultraviolet
behavior is not fully turned on.

##  VIII Suspicions.

I am skeptical about the ÜY prescription for single lattice site reduction.
The deformation is not adiabatic even for smallest values of the its coupling.
I cannot rule out being proven wrong by a thorough numerical simulation.

The root of the problem is that the deformation is nonlocal in one direction
in the physical finite temperature system. The simultaneous addition of ÜY
deformations for all directions descends, from a system of $L^{3}N_{t}$ sites
arranged in a four dimensional hypercube, so I doubt it would work for extreme
large $N$ reduction.

All arguments about large $N$ stem from simple degrees of freedom counting. It
is this way of thinking that, seems to me, relies fundamentally on having a
local action. Even when perturbation theories of different types organized
themselves so that they admit large $N$ limits, conclusions drawn about
equivalences implicitly presuppose a unique vacuum. The way these conclusions
can fail comes by a nonanalytic change in the “vacuum”. Even the restricted ÜY
deformation to finite temperature seems subject to the same worry. Again, I
cannot rule out being proven wrong by a thorough numerical simulation.

Thorough numerical simulations are time and effort consuming and do not work
in one blow, rather, require projects extending years. I am not sure that this
problem will be deemed of sufficient interest to get embarked on by some
individual or group.

There exists also another motivation for the ÜY deformation, by introducing
adjoint massless matter. This theory is supersymmetric. Then, there is no
$Z(N)$ breaking and Eguchi-Kawai reduction formally holds susy  – in the
continuum limit. At one loop, an easy computation of massless adjoint fermion
feedback gives a contribution exactly canceling the bosonic one, preserving
the partition function = 1 rule in supersymmetry. This computation explicitly
shows how the $Z(N)$ breaking is avoided in Feynman perturbation theory. One
can break supersymmetry and give the fermions a large mass, so that the
effective low energy theory does not see the supersymmetry but hope that large
$N$ reduction is maintain at all volumes.

On the lattice one would need the overlap overlap  to permit an emergent
supersymmetry in the continuum because masslessness is protected. In this
arrangement decoupling the adjoints by a large mass will take them out of the
game completely and the hope is dashed.

###### Acknowledgements.

I thank Massimo D’Elia, Tony Gonzalez-Arroyo and Rajamani Narayanan for
comments.

##  IX References Cited

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