A comment on instantons and their fermion zero modes in adjoint $QCD_{2}$.

A.V. Smilga

SUBATECH, Université de Nantes, 4 rue Alfred Kastler, BP 20722, Nantes 44307,
France.

The adjoint 2-dimensional $QCD$ with the gauge group $SU(N)/Z_{N}$ admits
topologically nontrivial gauge field configurations associated with nontrivial
$\pi_{1}[SU(N)/Z_{N}]=Z_{N}$. The topological sectors are labelled by an
integer $k=0,\ldots,N-1$. However, in contrast to $QED_{2}$ and $QCD_{4}$,
this topology is not associated with an integral invariant like the magnetic
flux or Pontryagin index. These instantons may admit fermion zero modes, but
there is always an equal number of left-handed and right-handed modes, so that
the Atiyah-Singer theorem, which determines in other cases the number of the
modes, does not apply.

The mod. 2 argument [1] suggests that, for a generic gauge field
configuration, there is either a single doublet of such zero modes or no modes
whatsoever. However, the known solution of the Dirac problem for a wide class
of gauge field configurations [2, 3, 4] indicates the presence of $k(N-k)$
zero mode doublets in the topological sector $k$. In this note, we demonstrate
in an explicit way that these modes are not robust under a generic enough
deformation of the gauge background and confirm thereby the conjecture of Ref.
[1].

The implications for the physics of this theory (screening vs. confinement
issue) are briefly discussed.

##  1 Introduction

The Lagrangian of the massless 2-dimensional QCD with fermions lying in the
adjoint representation of the $SU(N)$ gauge group reads [5]

$\displaystyle{\cal L}\ =\ {\rm
Tr}\left\\{-\frac{1}{2}F_{\mu\nu}F_{\mu\nu}+i\bar{\psi}\gamma_{\mu}D_{\mu}\psi\right\\},$
(1.1)

where $A_{\mu}=A_{\mu}^{a}t^{a}$, $\psi=\psi^{a}t^{a}$ is a 2-component
Majorana spinor, $D_{\mu}\psi=\partial_{\mu}\psi-ig[A_{\mu},\psi]$, and the
coupling $g$ has the dimension of mass. We will not display the dependence on
$g$ in what follows. It can always be restored on dimensional grounds. In
2-dimensional Minkowski space, the gamma matrices can be chosen as
$\gamma^{0}_{M}=\sigma^{2},\gamma^{1}=i\sigma_{1}$, where $\sigma_{1,2,3}$ are
the standard Pauli matrices. Then $\gamma^{5}=\gamma^{0}_{M}\gamma^{1}_{M}\ =\
\sigma_{3}$. The Euclidean gamma matrices, which we will mostly need in the
following, are

$\displaystyle\gamma^{0}_{E}\ =\ \sigma_{2},\qquad\gamma^{1}\ =\
\sigma_{1}\,.$ (1.2)

In this case, the elements of the center of $SU(N)$ do not act faithfully on
the fields, and we are dealing with the group $SU(N)/Z_{N}$. Its fundamental
group is nontrivial, $\pi_{1}[SU(N)/Z_{N}]=Z_{N}$, which leads to the
existence of topologically nontrivial Euclidean field configurations—the
instantons [6, 2]. They are characterized by an integer $k=0,1,\ldots,N-1$.

In Refs. [2, 3], we imposed certain natural boundary conditions for the
fermion fields and solved the Dirac equation for a class of topologically
nontrivial backgrounds on a cylinder [2] and on the Euclidean plane [3]. We
found the existence of $k(N-k)$ left-handed and $k(N-k)$ right-handed zero
modes in the spectrum. It was, however, argued in Ref.[1] that, for a generic
field configuration, there are only $[k(N-k)]_{\rm mod.2}$ doublets of zero
modes.

The main motivation of this study was to reconcile these two seemingly
contradictive statements. As a result, we confirm the conjecture of Ref. [1]
and demonstrate how the naive zero modes disappear when the background is
deformed in a general enough way.

The plan of the paper is the following.

In the next section, we consider, following Ref. [1], the theory on a finite
torus and show that, even in a topologically nontrivial sector, the field
density can be brought to zero by a continuous deformation of the potential.

In Sect. 3, we study the spectrum of the Dirac operator and show that all its
eigenstates with nonzero eigenvalues $\lambda$ are split the quartets
including two degenerate states with the eigenvalue $|\lambda|$ and two
degenerate states with the eigenvalue $-|\lambda|$. The spectrum may include
also some number of the zero mode doublets. This number may be odd, and in
this case at least one such doublet must stay on zero for an arbitrary
deformation, or it can be even, and in this case the zero modes are not
protected. This generic mod. 2 argument is confirmed by the analysis of the
topogically nontrivial toroidal configuration with zero field density. The
number of the zero modes depends on the fermionic boundary conditions.
Choosing the conditions similar to the conditions in Refs. [2, 3, 4], one
finds in this case that the zero modes are absent if $N$ is odd and there are
gcd$(N,k)$ doublets of zero modes if $N$ is even. This does not coincide with
$k(N-k)$, but one can observe that the two estimates have the same parity.

In Sect. 4, we consider, following [2, 3, 4], the theory on a cylinder
$S^{1}\times R$, with $S^{1}$ being a finite spatial circle with antiperiodic
boundary conditions for the fermion fields and $R$ the infinite Euclidean time
axis, $t\in(-\infty,\infty)$. For a special Cartan instanton background, the
Dirac operator admits in this case $k(N-k)$ doublets of zero modes. The same
result is reproduced for the theory placed on a finite torus.

In Sect. 5, we first show that these zero modes are robust under a particular
class of deformations that do not modify the values of the potential at
$t=\pm\infty$. Then we consider more general deformations of the potential and
show that in this case most zero modes disappear, leaving only one doublet of
the modes when $k(N-k)$ is odd.

In Sect. 6, we discuss the impact of the zero analysis presented in the main
body of the paper on the physics of this theory — whether it exhibits
confinement with the area law for the fundamental Wilson loops or screening
characterized by the perimeter law.

##  2 Instantons on the torus

We consider the theory on an Euclidean torus,

$\displaystyle 0\leq x\leq L,\qquad 0\leq t\leq\beta\,.$ (2.1)

The generic toroidal boundary conditions for $A_{\mu}(x,t)$ read [7]

$\displaystyle A_{\mu}(x+L,t)$ $\displaystyle=$ $\displaystyle
A^{\Omega_{1}}_{\mu}(x,t)\equiv-i[\partial_{\mu}\Omega_{1}(x,t)]\Omega_{1}^{-1}(x,t)+\Omega_{1}(x,t)A_{\mu}(x,t)\Omega_{1}^{-1}(x,t)\,,$
$\displaystyle A_{\mu}(x,t+\beta)$ $\displaystyle=$ $\displaystyle
A^{\Omega_{2}}_{\mu}(x,t)\equiv-i[\partial_{\mu}\Omega_{2}(x,t)]\Omega_{2}^{-1}(x,t)+\Omega_{2}(x,t)A_{\mu}(x,t)\Omega_{2}^{-1}(x,t)\,.$
(2.2)

However, the gauge transformation matrices $\Omega_{1,2}\in SU(N)$ are not
quite arbitrary: we have to require that, going from the point $(x,t)$ to the
point $(x+L,t+\beta)$ in two different ways, we obtain the same potential
$A_{\mu}(x+L,t+\beta)$. Bearing in mind that the potential does not transform
under the action of the center, we obtain the self-consistency condition

$\displaystyle\Omega_{1}(x,t+\beta)\,\Omega_{2}(x,t)\ =\
\omega_{N}^{k}\,\Omega_{2}(x+L,t)\,\Omega_{1}(x,t)$ (2.3)

with

$\omega_{N}=e^{2\pi i/N}\,.$

An integer $k$ labels the topological sector.

We choose in the following $\Omega_{1}(x,t)=\mathbb{1}$ and the time-
independent $\Omega_{2}(x,t)\equiv\Omega(x)$ that interpolates between
$\Omega(0)=\mathbb{1}$ and $\Omega(L)=\omega_{N}^{k}\mathbb{1}$. This gives

$\displaystyle A_{\mu}(x+L,t)$ $\displaystyle=$ $\displaystyle A_{\mu}(x,t),$
$\displaystyle A_{\mu}(x,t+\beta)$ $\displaystyle=$
$\displaystyle-i[\partial_{\mu}\Omega(x)]\Omega^{-1}(x)+\Omega(x)A_{\mu}(x,t)\Omega^{-1}(x)\,,$
(2.4)

Note that $\Omega(x)$ represents a noncontractible loop in $SU(N)/Z_{N}$. It
is a “large” gauge transformation.

######  Theorem 1.

[8] The field density of any field configuration $A_{\mu}(x,t)$ satisfying (2)
can be brought to zero by a smooth deformation.

###### Proof.

Consider first the simplest case ${\bf N=2}$. By a topologically trivial gauge
transformation, we can bring $\Omega(x)$ to the form111$\tau^{a}=2t^{a}$ are
the Pauli matrices in the color space.

$\displaystyle\Omega(x)\ =\ \exp\left\\{\frac{i\pi x}{L}\tau^{3}\right\\}\,.$
(2.5)

Perform now a gauge transformation

$\displaystyle A_{\mu}\ \to\ B_{\mu}=-i(\partial_{\mu}U)U^{-1}+UA_{\mu}U^{-1}$
(2.6)

with $U(x,t)$ satisfying the following b.c. :

$\displaystyle U(x+L,t)$ $\displaystyle=$ $\displaystyle i\tau^{1}U(x,t),$
$\displaystyle U(x,t+\beta)$ $\displaystyle=$ $\displaystyle
i\tau^{3}U(x,t)\Omega^{-1}(x)\,.$ (2.7)

######  Lemma.

A continuous matrix function satisfying (2) exists.

###### Proof.

We choose $U(0,0)=\mathbb{1}$. Then the conditions (2) dictate the following
values of $U(x,t)$ at the edges of the square:

$\displaystyle U(x,0)\ $ $\displaystyle=$ $\displaystyle\
\exp\left\\{\frac{i\pi x}{2L}\tau^{1}\right\\}\,,$ $\displaystyle U(0,t)\ $
$\displaystyle=$ $\displaystyle\ \exp\left\\{\frac{i\pi
t}{2\beta}\tau^{3}\right\\}\,,$ $\displaystyle U(x,\beta)\ $ $\displaystyle=$
$\displaystyle\ i\tau^{3}\exp\left\\{\frac{i\pi
x}{2L}\tau^{1}\right\\}\exp\left\\{-\frac{i\pi x}{L}\tau^{3}\right\\}\,,$
$\displaystyle U(L,t)\ $ $\displaystyle=$ $\displaystyle\
i\tau^{1}\exp\left\\{\frac{i\pi t}{2\beta}\tau^{3}\right\\}\,.$ (2.8)

In the corners:

$U(0,0)=\mathbb{1},\quad U(0,\beta)=i\tau^{3},\quad U(L,0)=i\tau^{1},\quad
U(L,\beta)=i\tau^{2}\,.$

Capitalizing on the fact that $\pi_{1}[SU(2)]=0$, we can also continuously
define $U(x,t)$ in the interior of the square. ∎

If $A_{\mu}(x,t)$ satisfies (2) and $U(x,t)$ satisfies (2), then the gauge-
transformed field $B_{\mu}(x,t)$ satisfies

$\displaystyle B_{\mu}(x+L,t)\ $ $\displaystyle=$ $\displaystyle\
\tau^{1}B_{\mu}(x,t)\tau^{1}\,,$ $\displaystyle B_{\mu}(x,t+\beta)\ $
$\displaystyle=$ $\displaystyle\ \tau^{3}B_{\mu}(x,t)\tau^{3}\,.$ (2.9)

These b.c. allow for a continuous deformation $B_{\mu}\to 0$. Performing the
inverse gauge transformation with the matrix $U^{-1}(x,t)$, we obtain a pure
gauge configuration,

$\displaystyle A_{\mu}^{\rm deformed}(x,t)\ =\ iU^{-1}\partial_{\mu}U\,.$
(2.10)

The corresponding field density is zero.

Let now ${\bf N=3}$. There are two nontrivial topological sectors: with $k=1$
and with $k=2$. If $k=1$, the loop $\Omega(x)$, uncontractible in
$SU(3)/Z_{3}$, can be chosen in the form

$\displaystyle\Omega(x)\ =\ \exp\left\\{\frac{2i\pi x}{3L}{\rm
diag}(1,1,-2)\right\\}\,,$ (2.11)

so that $\Omega(L)=\omega_{3}\mathbb{1}$.

Perform the gauge transformation (2.6) with $U(x,t)$ satisfying the following
b.c. :

$\displaystyle U(x+L,t)\ $ $\displaystyle=$ $\displaystyle\ SU(x,t)\,,$
$\displaystyle U(x,t+\beta)\ $ $\displaystyle=$ $\displaystyle\
CU(x,t)\Omega^{-1}(x)\,,$ (2.12)

where $C,S\in SU(3)$ are the “clock and shift” matrices [9]:

$\displaystyle C\ =\ \left(\begin{array}[]{ccc}1&0&0\\\ 0&\omega_{3}&0\\\
0&0&\omega_{3}^{\,2}\end{array}\right),\qquad S\ =\
\left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\ 0&1&0\end{array}\right)\,.$ (2.19)

The identities

$S^{3}\ =\ C^{3}\ =\ 1,\quad SC=\omega_{3}^{2}CS$

hold.

One can represent $C=e^{iP},\ S=e^{iQ}$ with Hermitian $P,Q$. On the edges of
the square, we derive

$\displaystyle U(x,0)\ $ $\displaystyle=$ $\displaystyle\
\exp\left\\{\frac{ixQ}{L}\right\\}\,,$ $\displaystyle U(0,t)\ $
$\displaystyle=$ $\displaystyle\ \exp\left\\{\frac{iPt}{\beta}\right\\}\,,$
$\displaystyle U(x,\beta)\ $ $\displaystyle=$ $\displaystyle\
C\exp\left\\{\frac{iQx}{L}\right\\}\Omega^{-1}(x)\,,$ $\displaystyle U(L,t)\ $
$\displaystyle=$ $\displaystyle\ S\exp\left\\{\frac{iPt}{\beta}\right\\}\,.$
(2.20)

In the corners:

$U(0,0)=\mathbb{1},\quad U(0,\beta)=C,\quad U(L,0)=S,\quad U(L,\beta)\ =\
SC\,.$

Bearing in mind that $\pi_{1}[SU(3)]=0$, we can also continuously define
$U(x,t)$ in the interior of the square.

The gauge-transformed field (2.6) satisfies the conditions

$\displaystyle B_{\mu}(x+L,t)\ $ $\displaystyle=$ $\displaystyle\
SB_{\mu}(x,t)S^{-1}\,,$ $\displaystyle B_{\mu}(x,t+\beta)\ $ $\displaystyle=$
$\displaystyle\ CB_{\mu}(x,t)C^{-1}\,.$ (2.21)

It can be smoothly deformed to zero, which means that the original field
$A_{\mu}(x,t)$ satisfying the conditions (2) can be smoothly deformed to a
pure gauge form with zero field density.

In the sector $k=2$, we choose

$\displaystyle\Omega(x)\ =\ \exp\left\\{\frac{2i\pi x}{3L}{\rm
diag}(2,-1,-1)\right\\}\,,$ (2.22)

If we substitute $\omega_{3}\to\omega_{3}^{-1}=\omega_{3}^{2}$ in the
definition of the matrix $C$ and the subsequent formulas, the whole reasoning
can be repeated.

${\bf N\geq 4}$. The generalization is straightforward. In the sector with a
given $k$, one should choose $\Omega(x)$ in the form

$\displaystyle\Omega(x)\ =\ \exp\left\\{\frac{2i\pi x}{NL}{\rm
diag}(\underbrace{k,\ldots,k}_{N-k},\,\underbrace{k-N,\ldots,k-N}_{k})\right\\}\,,$
(2.23)

so that $\Omega(L)=\omega_{N}^{k}\mathbb{1}$ with $\omega_{N}=e^{2i\pi/N}$.

The matrices $C,S$ entering the boundary conditions (2) and (2) may now be
defined as

$\displaystyle C\ =\ e^{i\pi k(N+1)/N}{\rm
diag}\\{1,\omega_{N}^{k},\ldots,\omega_{N}^{k(N-1)}\\}\,,$ $\displaystyle S\
=\ e^{i\pi(N+1)/N}\left(\begin{array}[]{ccccc}0&0&\cdots&\cdots&1\\\
1&0&\cdots&\cdots&0\\\ \cdots&\cdots&\cdots&\cdots&\cdots\\\
0&\cdots&\cdots&1&0\end{array}\right)\,.$ (2.28)

The phase factors in (2) are chosen such that $C,S\in SU(N)$: $\det C=\det
S=1$.

As earlier, the field $B_{\mu}(x,t)$, untwined by the gauge transformation,
can be continuously deformed to zero.

∎

The theorem just proven clearly displays that a nontrivial topology is not
associated in our case with any topological charge like the magnetic flux in
the 2D Abelian gauge theory or the Pontryagin index in the 4D Yang-Mills
theory,222Here $F$ is the field density 2-form $F=F_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}/2$.

$\displaystyle q_{2}\ =\ \frac{1}{2\pi}\int F,\qquad q_{4}\ =\
\frac{1}{8\pi^{2}}\int{\rm Tr}\\{F\wedge F\\}\,.$ (2.29)

Indeed, the presence of the topological invariants (2.29) depends on the fact
that their integrands are exact forms: $F^{\rm Ab}=dA$ and ${\rm Tr}\\{F\wedge
F\\}\ =\ d\,{\rm Tr}\\{A\wedge dA+\frac{2i}{3}A\wedge A\wedge A\\}$. But in
$QCD_{2}$, the form $F$ is not exact and Tr$\\{F\\}$ is simply zero.

##  3 Dirac operator and its spectrum

Bearing in mind the chosen explicit form (1.2) of the Euclidean gamma
matrices, the Euclidean Dirac spectral problem reads

$\displaystyle i{\cal D}\psi=\left(\sigma_{2}\frac{\partial}{\partial
t}+\sigma_{1}\frac{\partial}{\partial
x}\right)\psi-i(\sigma_{2}[A_{0},\psi]+\sigma_{1}[A_{1},\psi])\ =\
i\lambda\psi\,,$ (3.1)

$\psi\equiv\psi^{a}t^{a}$. It has the following important symmetries:

  * •

For any eigenfunction $\psi$ with eigenvalue $\lambda$, the function

$\displaystyle\psi^{\prime}=\sigma_{3}\psi$ (3.2)

is an eigenfunction with eigenvalue $-\lambda$.

  * •

For any eigenfunction $\psi$ with eigenvalue $\lambda$, the function

$\displaystyle\psi^{\prime\prime}=\sigma_{2}\psi^{*}$ (3.3)

is an eigenfunction with the same eigenvalue.

Suppose $\lambda\neq 0$. Then all the states

$\psi_{1},\quad\psi_{2}=\sigma_{3}\psi_{1},\quad\psi_{3}=\sigma_{2}\psi_{1}^{*},\quad\psi_{4}=\sigma_{3}\sigma_{2}\psi_{1}^{*}$

are linearly independent. Indeed, the states $\psi_{1},\psi_{3}$ cannot
coincide with the states $\psi_{2},\psi_{4}$ because the latter have a
different eigenvalue of $\cal D$. On the other hand, the state $\psi_{1}$
cannot coincide with $\psi_{3}$. Indeed, the equality
$\sigma_{2}\psi_{1}^{*}=\kappa\psi_{1}$ can be spelled out in terms of the
spinor components of $\psi_{1}$ as

$\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\left(\begin{array}[]{c}a^{*}\\\ b^{*}\end{array}\right)\
=\ \kappa\left(\begin{array}[]{c}a\\\
b\end{array}\right)\quad\Longrightarrow\quad\left\\{\begin{array}[]{c}ia^{*}=\kappa
b\\\ -ib^{*}=\kappa a\end{array}\right.\,,$

which is possible only if $a=b=0$.

In other words, the spectrum of the excited states of the operator $H={\cal
D}^{2}$ in a generic gauge background is split into quartets of degenerate
states. $H$ is a second-order differential operator, which may be called a
Hamiltonian. The four-fold degeneracy of all the excited levels means that
this Hamiltonian enjoys extended ${\cal N}=2$ supersymmetry. Extended
supersymmetry implies the existence of two doublets of Hermitially conjugated
supercharges. One such doublet, associated with the symmetry (3.2) has a nice
local form,

$\displaystyle Q\ =\ {\cal D}(1+\sigma_{3}),\qquad Q^{\dagger}\ =\ {\cal
D}(1-\sigma_{3})\,.$ (3.4)

Another doublet associated with the symmetry (3.3) is nonlocal.

Two of the quartet states, $\psi_{1}+\psi_{2}$ and $\psi_{3}+\psi_{4}$ are
right-handed (they are eigenstates of the chirality operator
$\gamma^{5}=\sigma^{3}$ with eigenvalue $+1$) and two other states,
$\psi_{1}-\psi_{2}$ and $\psi_{3}-\psi_{4}$ are left-handed.

If $\lambda=0$, $\psi_{1}$ and $\psi_{2}$ are not necessarily linearly
independent. They are not if $\psi_{1}$ has a definite chirality. In this
case, we have a doublet of states, $\psi_{1}$ and $\psi_{3}$. Note that these
states have opposite chiralities, so that the Atiyah-Singer index
$n_{L}-n_{R}$ of the supersymmetric Hamiltonian $H$ is equal to zero.

Suppose that in a particular gauge background there is only one doublet of the
fermion zero modes. Then these modes cannot shift from zero under a smooth
deformation, because a doublet cannot become a quartet. But if there are two
such doublets, they can move from zero simultaneously, forming a quartet.
Similarly, if we have any even number $2n$ of doublets — they all can move
from zero forming $n$ quartets of excited states. And if there were an odd
number $2n+1$ of doublets, $2n$ of them can be shifted from zero, but one
doublet is bound to stay.

This is the mod. 2 index argument of Ref. [1]. It says that there are no
compelling reasons to expect the existence of more than one doublet of the
fermion zero modes of the Dirac operator (3.1) in a generic gauge background.

To find out whether this theoretical lower bound for the number of the zero
modes is saturated for particular backgrounds, one should perform an explicit
study of the solutions to the problem (3.1).

One possibility [1] is to perform a gauge transformation (2.6) of the gauge
fields and simultaneously of the fermion fields and then smoothly deform
$B_{\mu}(x,t)$ to zero, in which case we are simply dealing with the free
Dirac problem,

$\displaystyle\left(\sigma_{2}\frac{\partial}{\partial
t}+\sigma_{1}\frac{\partial}{\partial x}\right)\Psi(x,t)\ =\ 0$ (3.5)

Eq. (3.5) is equivalent to a doublet of equations

$\displaystyle\left(\frac{\partial}{\partial x}\pm i\frac{\partial}{\partial
t}\right)\Psi_{\pm}(x,t)\ =\ 0\,,$ (3.6)

for the upper and lower spinor components. The solution to (3.6) is very
simple: $\Psi_{+}$ must be holomorphic and $\Psi_{-}$ antiholomorphic in
$z=x+it$.

A nontriviality resides, however, in the boundary conditions to be imposed on
the fermion field. The result depends on their choice. We choose the
conditions

$\displaystyle\psi(x+L,t)$ $\displaystyle=$ $\displaystyle-\psi(x,t),$
$\displaystyle\psi(x,t+\beta)$ $\displaystyle=$
$\displaystyle\Omega(x)\psi(x,t)\Omega^{-1}(x)\,,$ (3.7)

These conditions are similar to (2) (periodicity in space and the periodicity
up to a large gauge transformation under the imaginary time shift), but note
the presence of the extra minus in the first line. We inserted it to make
contact with the settings of Ref. [4], where the fermion fields were also
chosen to be antiperiodic in spatial direction. This choice can be traced back
to earlier papers [10, 2] where the adjoint $QCD_{2}$ at finite temperature
was studied — as is well-known, a finite temperature amounts to a finite
Euclidean time extension with antiperiodic conditions for the fermion fields.
In [4], the whole picture was rotated by $\pi/2$ and the theory was considered
on a finite spatial circle, while keeping the antiperiodic fermion boundary
conditions.

Consider first the case ${\bf N=2}$. After the gauge transformation

$\displaystyle\psi(x,t)\to\Psi(x,t)\ =\ U(x,t)\psi(x,t)U^{-1}(x,t)\,,$ (3.8)

with $U(x,t)$ satisfying (2), the transformed field satisfies the conditions

$\displaystyle\Psi(x+L,t)$ $\displaystyle=$
$\displaystyle-\tau^{1}\Psi(x,t)\tau^{1},$ $\displaystyle\Psi(x,t+\beta)$
$\displaystyle=$ $\displaystyle\tau^{3}\Psi(x,t)\tau^{3}\,,$ (3.9)

The matrix functions $\Psi_{\pm}(x,t)$ are double periodic functions on the
large torus, $\\{0\leq x\leq 2L,\ 0\leq t\leq 2\beta\\}$. The only
(anti)holomorphic nonsingular double periodic function (the absence of the
poles follows from the normalizabily requirement) is a constant. We only have
to check now if the boundary conditions (3) admit constant solutions. The
answer is positive, the solution is $\Psi\propto\tau^{3}$. We have proven the
theorem:

######  Theorem 2.

In the $N=2$ theory with the fermion boundary conditions (3), the Dirac
operator on the background $A_{\mu}(x,t)$ representing a pure gauge (2.10)
[which corresponds to $B_{\mu}(x,t)=0$] has one left-handed and one right-
handed zero mode.

Note that, for the fermion boundary conditions with the positive sign in the
first line in (3) and (3), there would be no zero mode solutions whatsoever.

The next in complexity case is ${\bf N=3}$. Let $k=1$. We may repeat our
reasoning by imposing the fermion boundary conditions (3) with $\Omega(x)$
given by (2.11), performing the gauge transformation with the parameter
$U(x,t)$ satisfying (2) and choosing the background $B_{\mu}(x,t)=0$. The
problem boils down to the search of the (anti)holomorphic Hermitian matrix
functions $\Psi(x\pm it)$ that satisfy the conditions

$\displaystyle\Psi(x+L,t)$ $\displaystyle=$ $\displaystyle-S\Psi(x,t)S^{-1},$
$\displaystyle\Psi(x,t+\beta)$ $\displaystyle=$ $\displaystyle
C\Psi(x,t)C^{-1}\,.$ (3.10)

However, such functions do not exist. The conditions (3) imply the periodicity
in the imaginary time direction and antiperiodicity in the spatial direction
on the large torus, $\\{0\leq x\leq 3L,\ 0\leq t\leq 3\beta\\}$. The only
(anti)holomorphic nonsingular matrix that satisfies this condition is
$\Psi(x,t)=0$. Obviously, this reasoning applies to any odd $N$ with any $k$.

We have proven the theorem:

######  Theorem 3.

In the theory with odd $N$ and with the fermion boundary conditions (3) in any
topological sector $k$, the Dirac operator on the background $A_{\mu}(x,t)$
representing a pure gauge (2.10) does not admit zero modes.

Consider now the case of generic even $N\geq 4$.

We have to search constant Hermitian matrices $\Psi$ that commute with $C$ and
anticommute with $S$ in (2).

$\bullet$ Consider first the case $k=1$. Then the condition $\Psi C=C\Psi$
brings $\Psi$ in the Cartan subalgebra. There is only one (up to a factor)
diagonal real traceless matrix that anticommutes with $S$:

$\displaystyle\Psi\ =\ {\rm diag}(1,-1,\ldots,1,-1)\,,$ (3.11)

giving a single zero mode of the Dirac operator. Clearly, this also applies to
the theory with any $k$ that does not have common nontrivial divisors with
$N$.

$\bullet$ Let $k=2$. The matrix $C$ may now be chosen as

$C_{k=2}\ =\ {\rm
diag}(1,e^{4i\pi/N},\ldots,e^{-4i\pi/N},1,e^{4i\pi/N},\ldots,e^{-4i\pi/N})$

Each eigenvalue is repeated twice. The centralizer of such $C$ is the
subalgebra

$\displaystyle\mathbb{c}\ =\ \underbrace{su(2)\oplus\cdots\oplus
su(2)}_{N/2}\oplus\underbrace{u(1)\oplus\cdots\oplus u(1)}_{N/2-1}\,.$ (3.12)

This centralizer includes the Cartan subalgebra, which gives the zero mode
(3.11) as earlier, and also certain nondiagonal matrices depending on $N/2$
complex parameters. In the particular case $N=6$, these matrices have the form

$\displaystyle\Psi\ =\ \left(\begin{array}[]{cccccc}0&0&0&a_{1}&0&0\\\
0&0&0&0&a_{2}&0\\\ 0&0&0&0&0&a_{3}\\\ a_{1}^{*}&0&0&0&0&0\\\
0&a_{2}^{*}&0&0&0&0\\\ 0&0&a_{3}^{*}&0&0&0\end{array}\right)\,.$ (3.19)

The condition $\Psi S+S\Psi=0$ implies the chain of $N/2$ relations

$\displaystyle a_{1}+a_{2}\ =\dots\ =\ a_{N/2-1}+a_{N/2}\ =\
a_{N/2}+a_{1}^{*}\ =\ 0\,.$ (3.20)

The solution is

$\displaystyle\\{a_{1},\ldots,a_{N/2}\\}\ =\
\lambda\,\\{1,-1,\ldots,(-1)^{N/2-1}\\}$ (3.21)

with a real $\lambda$ if $N/2$ is even and an imaginary $\lambda$ if $N/2$ is
odd. This gives the second doublet of zero modes.

We obtain the same result (two zero modes) for any $k$ with gcd$(N,k)=2$. In
this case, the centralizer of $C$ is still the subalgebra (3.12), the
nondiagonal elements of the centralizer are still parametrized by $N/2$
complex numbers $a_{j}$ and we still have one Cartan zero mode doublet and one
doublet (3.21).

$\bullet$ Let now gcd$(N,k)=3$. In this case, the clock matrix $C$ includes
$N/3$ different eigenvalues that enter thrice. The centralizer is

$\displaystyle\mathbb{c}\ =\ \underbrace{su(3)\oplus\cdots\oplus
su(3)}_{N/3}\oplus\underbrace{u(1)\oplus\cdots\oplus u(1)}_{N/3-1}\,.$ (3.22)

Its nondiagonal elements include $N$ complex parameters organized in two
different “ladders”, as illustrated below for $N=6,k=3$.

$\displaystyle\Psi\ =\ \left(\begin{array}[]{cccccc}0&0&a_{1}&0&a_{5}&0\\\
0&0&0&a_{2}&0&a_{6}\\\ a_{1}^{*}&0&0&0&a_{3}&0\\\ 0&a_{2}^{*}&0&0&0&a_{4}\\\
a_{5}^{*}&0&a_{3}^{*}&0&0&0\\\
0&a_{6}^{*}&0&a_{4}^{*}&0&0\end{array}\right)\,.$ (3.29)

In contrast to (3.19), the ladders in (3.29) are “long” — they involve 6
complex parameters each. On the other hand, the elements in the left ladder
and in the right ladder are complex conjugated to each other.

The condition $\\{\Psi,S\\}=0$ that the matrix (3.29) should satisfy to
represent a zero mode gives the long chain of relations

$\displaystyle a_{1}+a_{2}\ =\ a_{2}+a_{3}\ =\ a_{3}+a_{4}\ =\
a_{4}+a_{5}^{*}\ =\ a_{5}^{*}+a_{6}^{*}\ =\ a_{6}^{*}+a_{1}\ =\ 0$ (3.30)

for the left ladder, and the right ladder gives nothing new. The solution is

$\displaystyle\\{a_{1},\ldots,a_{6}\\}\ =\ \\{a,-a,a,-a,a^{*},-a^{*}\\}$
(3.31)

with a complex a. For a generic $N$ with gcd$(N,k)=3$, the relations are

$\displaystyle a_{1}+a_{2}\ =\ \ldots\ =\ a_{2N/3-1}+a_{2N/3}\ =\
a_{2N/3}+a_{2N/3+1}^{*}\ =\ \ldots\ =\ a_{N-1}^{*}+a_{N}^{*}\ =\ 0\,,$ (3.32)

and their solution also involves a single complex parameter. This gives 2
nondiagonal zero mode doublets, to which the Cartan doublet should be added.

$\bullet$ When gcd$(N,k)=4$, the matrix $C$ includes 4 coinciding sets of
$N/4$ different eigenvalues, the centralizer is

$\displaystyle\mathbb{c}\ =\ \underbrace{su(4)\oplus\cdots\oplus
su(4)}_{N/4}\oplus\underbrace{u(1)\oplus\cdots\oplus u(1)}_{N/4-1}\,,$ (3.33)

and its nondiagonal elements depend on $3N/2$ complex parameters organized in
three ladders: two of them are complex conjugate to each other, depend on $N$
different complex parameters and the condition $\\{\Psi,S\\}=0$ gives a “long”
chain of relations like in (3.32), leaving only one complex parameter.
Besides, there is a ladder depending on $N/2$ complex parameters and their
complex conjugates. The requirement $\\{\Psi,S\\}=0$ gives a short chain of
relations like in (3.20), leaving only one real parameter. We obtain four
doublets of zero modes: a Cartan doublet and three nondiagonal doublets.

$\bullet$ This counting is easily generalized for an arbitrary $k$. When
gcd$(N,k)=r$ and $r$ is odd, the nondiagonal elements of the centralizer of
$C$ are parameterized by $N(r-1)/2$ complex numbers organized in $r-1$ “long”
ladders. Only a half of these ladders are relevant — the other half includes
complex conjugated parameters. After imposing the condition $\Psi S+S\Psi=0$,
only one complex parameter is left for each doublet of complex conjugate
ladders. This gives $r-1$ doublets of zero modes, to which the Cartan doublet
should be added.

If $r$ is even, the nondiagonal part of the centralizer still depends on
$N(r-1)/2$ complex parameters organized in $r-1$ ladders. But only $r-2$ of
these ladders [$(r-2)/2$ doublets of complex conjugated ladders) are long.
They originally include $N(r-2)/2$ complex parameters, of which only $(r-2)/2$
are left after imposing the condition $\\{\Psi,S\\}=0$. This gives $r-2$
doublets of zero modes. There is also a “short” ladder depending on $N/2$
complex parameters, of which only one real parameter is left after imposing
the anticommutation condition. This gives one zero mode doublet.

All together we obtain

$(r-2)_{\rm long}+1_{\rm short}+1_{\rm Cartan}\ =\ r$

doublets of zero modes — the same number as for odd $r$.

We have proven the theorem:

######  Theorem 4.

In the theory with even $N$ with the fermion boundary conditions (3), the
Dirac operator on the background $A_{\mu}(x,t)$ representing a pure gauge
(2.10) [which corresponds to $B_{\mu}(x,t)=0$] admits gcd$(N,k)$ right and
gcd$(N,k)$ left zero modes in the topological sector $k$.

As was mentioned above, the number of modes depends on the fermion boundary
conditions. For example, if the periodic boundary conditions in the both
directions are imposed, the number of the zero mode doublets is equal to [1]

$\displaystyle n_{0}^{\rm double\ periodic}\ =\ {\rm gcd}(N,k)-1\,.$ (3.34)

Indeed, we now have to count the elements of $\mathbb{c}$ that commute with
$S$. This excludes the elements of the Cartan subalgebra. Consider a generic
nondiagonal element of $\mathbb{c}$. It includes several ladders — long and
short. The condition $[\Psi,S]=0$ dictates that all the complex matrix
elements in a long ladder coincide. The matrix elements in a short ladder also
coincide with an additional constraint that they must be real. This gives the
same count of parameters as in the problem with the boundary conditions (3).
We arrive at (3.34).

If we choose the b.c. that are antiperiodic in imaginary time, but periodic in
space, we need to count the matrices that commute with $S$ and anticommute
with $C$. For $k=1$, $S$ is related to $C$ by a group conjugation,
$S=VCV^{-1}$ and the same is true for their centralizers. Thus, the
centralizer of $S$ also represents a Cartan subalgebra embedded in $su(N)$ in
a noncanonical way. Only one of its generators anticommutes with $C$, and we
obtain one single zero mode doublet.

If $k>1$, the matrices $S$ and $C$ belong to different conjugacy classes and
there is no reason for the same counting in the problem where $S$ and $C$ are
interchanged. And generically the counting is different, indeed. In the case
$N=4,k=2$, it happens to be the same, but for $N=6,k=2$ it is already
different: there are two matrices that satisfy $[\Psi,C]=\\{\Psi,S\\}=0$ and
no matrices satisfying $[\Psi,S]=\\{\Psi,C\\}=0$ whatsoever [1].

We presented the calculation of the number of fermion zero modes on a torus in
a particular gauge background. For $k=1$, there is one doublet of zero modes
or none depending on whether $N$ is even or odd. This conforms to the mod. 2
argument outlined above. But for higher $k$, the number of the zero mode
doublets is sometimes larger than 1.

In order to understand whether the counting $n_{0}=$gcd$(N,k)$ holds for a
generic field configurations, it is natural to choose some other handleable
gauge background and solve the problem in that case.

##  4 Cartan instantons and their zero modes

In this section, we will not unwind the gauge field by the gauge
transformation (2.6) and then deform it to zero, as we did before, but
consider the original boundary conditions (2), choose the simplest
topologically nontrivial gauge background $A_{\mu}(x,t)$ and study the Dirac
spectrum there.

###  4.1 On the cylinder

A similar problem was first solved in Ref. [2]. In that paper, we studied the
physics of the theory (1.1) at finite temperature, i.e. the theory was put on
the cylinder with the finite extension $\beta$ along the imaginary time $t$
and the infinite spatial extension. We imposed the boundary conditions

$\displaystyle A_{\mu}(x,t+\beta)\ $ $\displaystyle=$ $\displaystyle\
A_{\mu}(x,t)\,,$ $\displaystyle\psi(x,t+\beta)\ $ $\displaystyle=$
$\displaystyle\ -\psi(x,t)\,.$ (4.1)

and solved the Dirac equation in a topologically nontrivial background with
$k=1$. We obtained $N-1$ doublets of zero modes. This result was then
confirmed in [4], whose authors rotated the cylinder by 90o and considered the
theory on a finite spatial circle of length $L$ and the infinite extension in
$t$. In [3], we generalised the discussion for any $k$ and discussed also the
Dirac problem on an infinite Euclidean plane. We derived the presence of
$k(N-k)$ doublets of zero modes.

We reproduce here this derivation following the approach of [4], where the
physical instanton picture is somewhat more transparent. In the second half of
this section, we translate it onto a finite torus.

Thus, we impose the boundary conditions333The physical picture happens to be
more simple when the spatial fermionic boundary conditions are antiperiodic.
It would also be interesting to perform a systematic study of the theory with
periodic b.c. both for $A_{\mu}$ and $\psi$.

$\displaystyle A_{\mu}(x+L,t)\ $ $\displaystyle=$ $\displaystyle\
A_{\mu}(x,t)\,,$ $\displaystyle\psi(x+L,t)\ $ $\displaystyle=$ $\displaystyle\
-\psi(x,t)$ (4.2)

and impose the Hamilton gauge $A_{0}(x,t)=0$. The instanton $A_{1}(x,t)$ is
then interpreted as a topologically nontrivial tunneling transition trajectory
between the different vacua, similar to the interpretation of the familiar
BPST instanton [11].

We are in a position to study the vacuum structure of our theory. We consider
the case $N=2$ first.

In $QCD_{2}$, a classical vacuum with zero field density is the constant field
configuration $A_{1}=$ const. By a gauge rotation, one can bring $A_{1}$ to
the Cartan subalgebra. For $SU(2)$, we may pose $A_{1}=a\tau^{3}$.
Classically, the energy of all such constant configurations is zero. But
taking into account the quantum corrections due to fermion loops,444In two
dimensions, there are no physical degrees of freedom associated with the gauge
fields, and the latter do not contribute. the effective potential emerges
[10]. In Refs.[10], bearing in mind the finite temperature applications, the
effective potential for the zeroth component of the vector potential was
calculated assuming the antiperiodic boundary conditions for the fermions
under the Euclidean time shift. For $SU(2)$, this potential reads

$\displaystyle V^{\rm eff}(A_{0}=a\tau^{3})\ =\
\frac{\beta}{2\pi}\left[\left(2a+\frac{\pi}{\beta}\right)_{{\rm
mod.}\frac{2\pi}{\beta}}-\frac{\pi}{\beta}\right]^{2}\,.$ (4.3)

In our case, the dependence is the same:

$\displaystyle V^{\rm eff}(a)\ =\
\frac{L}{2\pi}\left[\left(2a+\frac{\pi}{L}\right)_{{\rm
mod.}\frac{2\pi}{L}}-\frac{\pi}{L}\right]^{2}\,.$ (4.4)

Figure 1: Effective potential (4.4)

It is periodic with the period $\pi/(L)$ (see Fig. 1). This periodicity is due
to the fact that any $a$ outside the interval

$\displaystyle 0\ \leq\ a\ \leq\ \frac{\pi}{L}$ (4.5)

can be brought into this interval by a topologically trivial gauge
transformation. There are two types of such transformations:

  1. 1.

The shift $a\to a+(2\pi)/L$ realized by the gauge transformation

$\displaystyle\tilde{\Omega}(x)\ =\ \exp\left\\{\frac{2\pi
ix}{L}\tau^{3}\right\\}\,.$ (4.6)

In contrast to the loop (4.7), this loop is contractible in $SU(2)/Z_{2}$.

  2. 2.

The Weyl reflection $a\tau^{3}\to-a\tau^{3}$ realized by the rotation by $\pi$
around the first or the second color axis.

The interval (4.5) may be called Weyl alcove.

We are left with only two vacuum states at $a=0$ and $a=\pi/(L)$. They are
related by a noncontractible gauge transformation

$\displaystyle\Omega(x)\ =\ \exp\left\\{\frac{i\pi x}{L}\tau^{3}\right\\}\,,$
(4.7)

The instanton that we are interested in (we will call it the Cartan instanton)
interpolates between these states along the path

$\displaystyle A_{1}(t)\ =\ a(t)\tau^{3}\quad{\rm with}\quad a(-\infty)=0,\
a(\infty)=\frac{\pi}{L}\,.$ (4.8)

We will assume that $a(t)$ tends to its asymptotic values exponentially fast.

Note that there are no antiinstantons: the configuration (4.8) with
$a(-\infty)=\pi/L$ and $a(\infty)=0$ belongs to the same topological class.

The Dirac equation (3.1) with $A_{0}=0$ and $A_{1}(t)$ given by (4.8) admits
the following doublet of zero modes satisfying the antiperiodicity condition:

$\displaystyle\Phi^{+}\ $ $\displaystyle=$ $\displaystyle\
\tau^{+}\left(\begin{array}[]{c}0\\\ 1\end{array}\right)e^{i\pi
x/L}e^{\phi(t)}\,,$ (4.11) $\displaystyle\Phi^{-}\ $ $\displaystyle=$
$\displaystyle\ \tau^{-}\left(\begin{array}[]{c}1\\\
0\end{array}\right)e^{-i\pi x/L}e^{\phi(t)}\,,$ (4.14)

where

$\displaystyle\frac{d\phi(t)}{dt}\ =\ \frac{\pi}{L}-2a(t)\,.$ (4.15)

When $t\to\pm\infty$, the solutions behave as

$\displaystyle\psi_{\pm}(t)\ \sim\ e^{-\pi|t|/L}\,.$ (4.16)

They are normalizable.

We see that there are two zero modes—with positive and negative chirality.
They have the opposite color structure, which corresponds to the fields of
positive and negative charge in the Abelian theory. In agreement with the
standard Atiyah-Singer theorem, the zero modes have negative chirality in the
former case and positive chirality in the latter case.

Consider now the case $N=3$. The Weyl alcove for $SU(3)$ represents a triangle
shown in Fig. 2. The effective potential for the classical vacuum,

$A_{1}\ =\ {\rm diag}(a_{1},a_{2},a_{3}),\qquad\sum_{j}a_{j}=0\,,$

is the sum of three terms:

$\displaystyle V^{\rm eff}(a_{j})\ =\
\frac{L}{2\pi}\sum_{j<k}\left[\left(a_{j}-a_{k}+\frac{\pi}{L}\right)_{{\rm
mod.}\frac{2\pi}{L}}-\frac{\pi}{L}\right]^{2}\,.$ (4.17)

Figure 2: The Weyl alcove for $N=3$. The vertices of the triangle mark the
topologically distinct vacua.

The potential (4.17) has the minima at the vertices of the triangle:

$\displaystyle A_{1}^{(0)}=0;\quad A_{1}^{(1)}=\frac{2\pi}{3L}{\rm
diag}(1,1,-2)=\frac{4\pi}{L\sqrt{3}}t^{8};\quad
A_{1}^{(3)}=\frac{2\pi}{3L}{\rm diag}(2,-1,-1)\,.$ (4.18)

The Cartan instantons interpolate between different vacua. There are in
principle 6 such configurations corresponding to three edges of the triangle
passed in two directions, but all “clockwise” instantons belong to the same
topological class $k=1$ and all “counterclockwise” instantons belong to the
class $k=2$. It is sufficient to consider only one of them, the configuration

$\displaystyle A_{1}(t)\ =\ \frac{2}{3}a(t)\,{\rm diag}(1,1,-2)\,,$ (4.19)

All other configurations have the same properties.

The Dirac equation (3.1) with $A_{0}=0$ and $A_{1}(t)$ given by (4.19) admits
two doublets of zero modes:

$\displaystyle\Phi^{+}_{1,2}$ $\displaystyle=$ $\displaystyle
E^{+}_{1,2}\left(\begin{array}[]{c}0\\\ 1\end{array}\right)e^{i\pi
x/L}e^{\phi(t)}\,,$ (4.22) $\displaystyle\Phi^{-}_{1,2}$ $\displaystyle=$
$\displaystyle E^{-}_{1,2}\left(\begin{array}[]{c}1\\\
0\end{array}\right)e^{-i\pi x/L}e^{\phi(t)}\,,$ (4.25)

where

$\displaystyle E^{+}_{1}\ =\ \left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
0&0&0\end{array}\right),\qquad E^{+}_{2}\ =\
\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\ 0&0&0\end{array}\right)$ (4.32)

and $E^{-}_{1,2}$ are Hermitially conjugated. $\phi(t)$ is related to $a(t)$
in the same way (4.15) as before.

Generically, for larger values of $N$, the instantons of different types are
labelled by an integer $k=1,\ldots,N-1$ and have the form

$\displaystyle A_{1}(t)\ =\ \frac{2}{N}a(t)\ {\rm
diag}(\underbrace{k,\ldots,k}_{N-k},\,\underbrace{k-N,\ldots,k-N}_{k})\,.$
(4.33)

Only the instantons with $k=1,\ldots,[N/2]$ are essentially different. For
example, for $SU(4)$, the Weyl alcove is an asymmetric tetrahedron with the
vertices representing the following elements of $su(4)$: $O=0$ and

$\displaystyle A=\frac{\pi}{2L}{\rm diag}(1,1,1,-3);\quad B=\frac{\pi}{2L}{\rm
diag}(2,2,-2,-2);\quad C=\frac{\pi}{2L}{\rm diag}(3,-1,-1,-1)\,.$ (4.34)

It has four short and two long edges and, correspondingly, there are
instantons of two types: with $k=1$ and with $k=2$.

The instanton (4.33) admits $k(N-k)$ zero mode doublets, given by the same
formulas as in (4.22) with $k(N-k)$ different matrices $E^{+}$’s and $k(N-k)$
different $E^{-}$’s. Only one component of these matrices somewhere in the
upper right (correspondingly, lower left) block $k\times(N-k)$ is equal to 1.
All other components are zeros.

###  4.2 On the torus

We now roll up into a ring also the Euclidean time dimension and impose the
boundary conditions (2) on the gauge fields and (3) on the fermion fields with
a topologically nontrivial $\Omega(x)$.

Consider first the case $N=2$. $\Omega(x)$ is given by (4.7). The instanton
configuration reads

$\displaystyle A_{1}(t)\ =\ \frac{\pi t}{L\beta}\tau^{3}\,.$ (4.35)

Again, the Dirac operator admits a doublet of zero modes, which have, however,
a somewhat more complicated form than on the cylinder,

$\displaystyle\Phi^{+}$ $\displaystyle=$
$\displaystyle\tau^{+}\left(\begin{array}[]{c}0\\\
1\end{array}\right)\sum_{n=-\infty}^{\infty}\exp\left\\{\frac{i\pi
x}{L}(2n+1)\right\\}\,\exp\left\\{-\frac{\pi\beta}{L}\left(\frac{t}{\beta}-n-\frac{1}{2}\right)^{2}\right\\}\,,$
(4.38) $\displaystyle\Phi^{-}$ $\displaystyle=$
$\displaystyle\tau^{-}\left(\begin{array}[]{c}1\\\
0\end{array}\right)\sum_{n=-\infty}^{\infty}\exp\left\\{-\frac{i\pi
x}{L}(2n+1)\right\\}\,\exp\left\\{-\frac{\pi\beta}{L}\left(\frac{t}{\beta}-n-\frac{1}{2}\right)^{2}\right\\}\,.$
(4.41)

These functions belong to the $\Theta$ family. The same functions describe the
toric fermion zero modes in the constant magnetic field of unit flux in the
Abelian theory.

It is also interesting to see what would happen if we imposed the periodic
boundary conditions in both directions:

$\displaystyle\psi(x+L,t)$ $\displaystyle=$ $\displaystyle\psi(x,t),$
$\displaystyle\psi(x,t+\beta)$ $\displaystyle=$
$\displaystyle\Omega(x)\psi(x,t)\Omega^{-1}(x)\,,$ (4.42)

In this case, one obtains a doublet of the charged zero modes given by the
similar expressions:

$\displaystyle\Phi^{+}_{\rm per.}$ $\displaystyle=$
$\displaystyle\tau^{+}\left(\begin{array}[]{c}0\\\
1\end{array}\right)\sum_{n=-\infty}^{\infty}\exp\left\\{\frac{2i\pi
nx}{L}\right\\}\,\exp\left\\{-\frac{\pi\beta}{L}\left(\frac{t}{\beta}-n\right)^{2}\right\\}\,,$
(4.45) $\displaystyle\Phi^{-}_{\rm per.}$ $\displaystyle=$
$\displaystyle\tau^{-}\left(\begin{array}[]{c}1\\\
0\end{array}\right)\sum_{n=-\infty}^{\infty}\exp\left\\{-\frac{2i\pi
nx}{L}\right\\}\,\exp\left\\{-\frac{\pi\beta}{L}\left(\frac{t}{\beta}-n\right)^{2}\right\\}\,.$
(4.48)

And on top of that, there is a doublet of constant neutral zero modes:

$\displaystyle\Phi_{1}\ =\ \tau^{3}\left(\begin{array}[]{c}1\\\
0\end{array}\right)\qquad{\rm and}\qquad\Phi_{2}\ =\
\tau^{3}\left(\begin{array}[]{c}0\\\ 1\end{array}\right)\,.$ (4.53)

Two doublets altogether.

For higher $N$, the cylindrical argumentation above is translated onto the
torus in a similar fashion. The toric zero modes have the same color structure
as the cylindrical ones, while their coordinate dependence is the same as in
(4.38).

We arrive at the conclusion:

Naive instantons of the $SU(N)$ theory belonging to the topological sector $k$
admit $k(N-k)$ doublets of the fermion zero modes.

##  5 Deformations

From the fact that different gauge background admit different number of zero
modes, it follows that this number is not a topological invariant and may
change under deformation. In this section, we confirm it in a quite explicit
way. We will study the deformations of cylindrical Cartan instantons — this is
essentially simpler than for the toric configurations.

###  5.1 Fixed asymptotics

To begin with, we will prove the following theorem:

######  Theorem 5.

In the gauge $A_{0}=0$, consider the gauge field background

$\displaystyle A_{1}(x,t)\ =\ A_{1}^{(0)}(t)+\alpha(x,t)\,,$ (5.1)

where $A_{1}^{(0)}(t)$ is the Cartan instanton configuration (4.33) and
$\alpha(x,t)=\alpha^{a}(x,t)t^{a}$ is periodic in $x$:

$\displaystyle\alpha(x,t)=\alpha^{a}(x,t)t^{a}\ =\
\sum_{m=-\infty}^{\infty}\alpha_{m}(t)e^{2\pi imx/L}$ (5.2)

with $\alpha_{n}(t)$ falling off to zero as $t\to\pm\infty$ in such a way that
the integral $\int\alpha_{n}(t)\,dt$ converges there.

Then the equation (3.1) still has $k(N-k)_{L}+k(N-k)_{R}$ normalized zero mode
solutions in any order in the perturbation $\alpha(x,t)$.

This statement and the idea of the proof can be found back in [2], but we did
not give there much details, which we are going to provide now.

###### Proof.

Consider first the $SU(2)$ theory.

Consider the fate of the positively charged mode [the first line in
Eq.(4.11)]. This mode carries the negative chirality, and we can replace in
this case $\sigma_{1}\to 1,\ \sigma_{2}\to-i$. We are going to solve the
equation

$\displaystyle\left(\frac{\partial}{\partial t}+i\frac{\partial}{\partial
x}\right)\psi+[a(t)\tau^{3}+\alpha,\psi]\ =\ 0$ (5.3)

by iterations. We pose

$\displaystyle\psi=\psi_{0}+\psi_{1}+\psi_{2}+\ldots\,,$ (5.4)

where $\psi_{0}$ is the Cartan zero mode in (4.11) and $\psi_{n}$ is of order
$\sim\alpha^{n}$. We obtain the chain of equations

$\displaystyle\left(\frac{\partial}{\partial t}+i\frac{\partial}{\partial
x}\right)\psi_{n}+a(t)[\tau^{3},\psi_{n}]\ =\ -[\alpha,\psi_{n-1}]\
\stackrel{{\scriptstyle\rm def}}{{=}}\ \ \gamma_{n-1}\,.$ (5.5)

Each equation in (5.5) is split into three equations for the different color
components:

$\displaystyle\left(\frac{\partial}{\partial t}+i\frac{\partial}{\partial
x}\right)\psi_{n}^{3}\ =\ \gamma_{n-1}^{3}\,,$ (5.6)

$\displaystyle\left(\frac{\partial}{\partial t}+i\frac{\partial}{\partial
x}+2a(t)\right)\psi_{n}^{+}\ =\ \gamma_{n-1}^{+}$ (5.7)

and

$\displaystyle\left(\frac{\partial}{\partial t}+i\frac{\partial}{\partial
x}-2a(t)\right)\psi_{n}^{-}\ =\ \gamma_{n-1}^{-}$ (5.8)

I. Let us assume first that the deformation $\alpha^{a}$ does not depend on
$x$. Then the $x$-dependence of all the terms in (5.4) and (5.6)-(5.8) is the
same as that of $\psi_{0}$

$\displaystyle\psi_{n}(x,t)\ =\ \psi_{n}(t)e^{i\pi
x/L}\,,\qquad\gamma^{a}_{n}(x,t)=\gamma_{n}(t)e^{i\pi x/L}\,.$ (5.9)

In this case, the chain (5.6) - (5.8) is reduced to the system of ordinary
differential equations,

$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}\right)\psi_{n}^{3}(t)\ =\ \gamma_{n-1}^{3}(t)\,,$ (5.10)
$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}+2a(t)\right)\psi_{n}^{+}(t)\ =\ \gamma_{n-1}^{+}(t)\,,$
(5.11)

and

$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}-2a(t)\right)\psi_{n}^{-}(t)\ =\ \gamma_{n-1}^{-}(t)\,.$
(5.12)

We will prove the existence of the normalized solutions to this system for all
$n$ by induction.

More exactly, we will prove that the correction $\psi_{n}(t)$ decays at
$t\to\pm\infty$ as $\sim e^{-\pi|t|/L}$ at any order $n$.

  * •

For $n=0$, this follows from the explicit solution in (4.11).

  * •

Suppose that $\psi_{n-1}(t)$ decays as $e^{-\pi|t|/L}$. Let us prove that
$\psi_{n}(t)$ also has this property. Note first that, if $\psi_{n-1}(t)\
\sim\ e^{-\pi|t|/L}$, $\gamma_{n-1}(t)$ decays faster than $e^{-\pi|t|/L}$.

Note also that, by our assumption about the behavior of $\alpha(t)$, the
integrals $\int^{\infty}e^{\pi t/L}\gamma_{n-1}(t)\,dt$ and
$\int_{-\infty}e^{-\pi t/L}\gamma_{n-1}(t)\,dt$ converge.

(i) Consider Eq. (5.10). Its formal solution is

$\displaystyle\psi_{n}^{3}(t)=-e^{\pi t/L}\int_{t}^{\infty}e^{-\pi
t^{\prime}/L}\,\gamma_{n-1}^{3}(t^{\prime})\,dt^{\prime}\,+C\,.$ (5.13)

Choose $C=0$. For $t\to\infty$, $\psi_{n}^{3}(t)$ decays faster than $e^{-\pi
t/L}$ together with $\gamma_{n-1}^{3}(t)$, and for $t\to-\infty$, the integral
is finite and $\psi_{n}^{3}(t)\sim e^{\pi t/L}=e^{-\pi|t|/L}$.

(ii)

Consider now equation (5.11). Choose its particular solution in the form

$\displaystyle\psi_{n}^{+}(t)\ =\
-e^{F(t)}\int_{t}^{\infty}\,e^{-F(t^{\prime})}\,\gamma_{n-1}^{+}(t^{\prime})\,dt^{\prime}\,,$
(5.14)

where

$\displaystyle F^{\prime}(t)=\frac{\pi}{L}-2a(t)\,.$ (5.15)

When $t\to\infty$, $F(t)\to-\pi t/L$ and

$\psi_{n}^{+}(t)\ \sim\ e^{-\pi t/L}\int_{t}^{\infty}e^{\pi
t^{\prime}/L}\,\gamma_{n-1}^{+}(t^{\prime})\,dt^{\prime}\ \sim\ e^{-\pi
t/L}\,,$

as the integral converges at the upper limit.

When $t\to-\infty$, $F(t)\to\pi t/L=-\pi|t|/L$ and

$\psi_{n}^{+}(t)\ \sim\ e^{-\pi|t|/L}\int_{-\infty}^{\infty}e^{\pi
t^{\prime}/L}\,\gamma_{n-1}^{+}(t^{\prime})\,dt^{\prime}\ \sim\
e^{-\pi|t|/L}\,,$

as the integral converges at both limits.

(iii)

For Eq. (5.12), the reasoning is analogous. We choose its solution as

$\displaystyle\psi_{n}^{-}(t)\ =\
-e^{G(t)}\int_{t}^{\infty}\,e^{-G(t^{\prime})}\,\gamma_{n-1}^{-}(t^{\prime})\,dt^{\prime}\,,$
(5.16)

where $G^{\prime}(t)=\frac{\pi}{L}[1+2a(t)]$.

When $t\to\infty$,

$\psi_{n}^{-}(t)\ \sim\ e^{3\pi t/L}\int_{t}^{\infty}e^{-3\pi
t^{\prime}/L}\,\gamma_{n-1}^{-}(t^{\prime})\,dt^{\prime}\ \sim\ e^{-\pi
t/L}\,.$

When $t\to-\infty$,

$\psi_{n}^{-}(t)\ \sim\
e^{-\pi|t|/L}\int_{-\infty}^{\infty}e^{-G(t^{\prime})}\,\gamma_{n-1}^{-}(t^{\prime})\,dt^{\prime}\
\sim\ e^{-\pi|t|/L}\,.$

II. Let us now take into account the higher Fourier modes in the expansion
(5.2).

Their presence entails the presence of higher Fourier modes in the expansion
for the correction $\psi_{n}(x,t)$:

$\displaystyle\psi_{n}(x,t)\ =\
\sum_{m}\psi_{n}^{(m)}(t)\exp\left\\{\frac{i\pi x}{L}(1+2m)\right\\}$ (5.17)

The equations (5.10) - (5.12) acquire the form

$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}(2m+1)\right)\psi_{n}^{(m)\,3}(t)\ =\
\gamma_{n-1}^{(m)\,3}(t)\,,$ (5.18)
$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}(2m+1)+2a(t)\right)\psi_{n}^{(m)\,+}(t)\ =\
\gamma_{n-1}^{(m)\,+}(t)\,,$ (5.19)

and

$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}(2m+1)-2a(t)\right)\psi_{n}^{(m)\,-}(t)\ =\
\gamma_{n-1}^{(m)\,-}(t)\,,$ (5.20)

where

$\displaystyle\gamma_{n-1}^{(m)\,a}(t)\ =\
-\sum_{p+q=m}[\alpha_{p}(t),\psi^{(q)}_{n-1}(t)]\,.$ (5.21)

We can prove now that the three components of $\psi_{n}^{(m)}(t)$
exponentially decay at large $|t|$ by induction in the same way as we did it
in the absence of the higher harmonics in (5.2). Consider e.g. Eq. (5.18). Let
$m>0$. Choose the particular solution of the equation in the form

$\displaystyle\psi_{n}^{(m)\,3}(t)=-e^{\pi t(2m+1)/L}\int_{t}^{\infty}e^{-\pi
t^{\prime}(2m+1)/L}\,\gamma_{n-1}^{(m)\,3}(t^{\prime})\,dt^{\prime}\,.$ (5.22)

By inductive assumption, $\psi_{n-1}^{(m)\,a}(t)$ and hence
$\gamma_{n-1}^{(m)\,a}(t)$ fall down $\sim e^{-\pi|t|/L}$. By the same
reasoning as before, it follows that $\psi_{n}^{(m)\,3}(t)$ falls down $\sim
e^{-\pi|t|/L}$; the presence of the factor $2m+1$ in the exponents in
Eq.(5.22) is irrelevant.

If $m<0$, we choose the solution in the form

$\displaystyle\psi_{n}^{(m)\,3}(t)=e^{\pi t(2m+1)/L}\int_{-\infty}^{t}e^{-\pi
t^{\prime}(2m+1)/L}\,\gamma_{n-1}^{(m)\,3}(t^{\prime})\,dt^{\prime}$ (5.23)

and, by exploring the limits $t\to\pm\infty$, deduce that it falls down $\sim
e^{-\pi|t|/L}$.

The equations (5.19) and (5.20) can be treated in a similar way.

####  5.1.1 $N>2$

This proof can be translated without much change to the theories with higher
$N$. Consider the $SU(3)$ theory. The Cartan instanton has the form (4.19). It
has two doublets of zero modes. Let us add the deformation (5.2) with the same
properties as before and explore the fate of one of the positive chirality
modes. For example, the fate of the mode

$\displaystyle\Phi^{+(0)}_{1}(t)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
0&0&0\end{array}\right)_{\rm color}\\!\left(\begin{array}[]{c}0\\\
1\end{array}\right)_{\rm spin}\Phi^{(4+i5)(0)}(t)\,,$ (5.29)

with

$\Phi^{(4+i5)(0)}(t)\ =\ e^{i\pi
x/L}e^{\phi(t)},\qquad\phi^{\prime}(t)=\pi[1-2a(t)]/L\,.$

We obtain the chain of the equations

$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}(2m+1)\right)\psi_{n}^{(m)\,1,2,3,8}(t)\ =\
\gamma_{n-1}^{(m)\,1,2,3,8}(t)$ (5.31)

and

$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}(2m+1)\mp\frac{2\pi}{L}a(t)\right)\psi_{n}^{(m)\,4\pm i5}(t)\
=\ \gamma_{n-1}^{(m)\,4\pm i5}(t)\,,$
$\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\pi}{L}(2m+1)\mp\frac{2\pi}{L}a(t)\right)\psi_{n}^{(m)\,6\pm i7}(t)\
=\ \gamma_{n-1}^{(m)\,6\pm i7}(t)$ (5.32)

with

$\displaystyle\gamma_{n-1}^{(m)\,a}(t)\ =\
if^{abc}\sum_{p+q=m}\alpha_{p}^{c}(t)\psi^{(q)\,b}_{n-1}(t)\,.$ (5.33)

The inductive proof that, at any order, the corrections $\psi_{n}^{(m)\,a}(t)$
fall down exponentially as $t\to\pm\infty$ is translated from the proof for
the $N=2$ theory without much change.

For an arbitrary $N$, we obtain a similar chain. Ii involves the equations for
the components $\psi_{n}^{(m)\,a}(t)$ where the index $a$ corresponds to the
centralizer $SU(k)\times SU(N-k)\times U(1)$ of the Cartan instanton
confuguration (4.33). These components do not “feel” the presence of the gauge
field. It involves also $k(N-k)$ doublets of the components corresponding to
the root vectors that do not commute with (4.33). The inductive proof
constructed above works also in this case.

∎

###  5.2 Generic deformations

To understand that the deformations considered above are not the most generic
ones, consider the $SU(2)$ theory with the gauge background

$\displaystyle A_{1}(t)\ =\ \tau^{3}b(t),\qquad{\rm with}\qquad
b(-\infty)=0,\quad b(\infty)=\frac{2\pi}{L}\,.$ (5.34)

In this case, the Dirac equation admits two doublets of normalized zero modes.
The positively charged modes are

$\displaystyle\Phi_{1}(x,t)\ =\ \tau^{+}\left(\begin{array}[]{c}0\\\
1\end{array}\right)e^{i\pi x/L}e^{\phi_{1}(t)}\,,$ (5.37)
$\displaystyle\Phi_{2}(x,t)\ =\ \tau^{+}\left(\begin{array}[]{c}0\\\
1\end{array}\right)e^{3i\pi x/L}e^{\phi_{2}(t)}\,,$ (5.40)

where

$\displaystyle\frac{d\phi_{1}}{dt}=\frac{\pi}{L}-2b(t),\qquad\frac{d\phi_{2}}{dt}=\frac{3\pi}{L}-2b(t)\,.$
(5.41)

In the Abelian theory, the existence of two zero modes follows from the
Atiyah-Singer theorem: the field (5.34) carries the double magnetic flux.
However, in the non-Abelian theory, the configuration (5.34) is topologically
trivial, it satisfies the boundary condition

$\displaystyle A_{1}(x,\infty)\ =\
-i\partial_{x}\tilde{\Omega}(x)\,\tilde{\Omega}^{-1}(x)+\tilde{\Omega}(x)A_{1}(x,-\infty)\tilde{\Omega}^{-1}(x)$
(5.42)

with a contractible loop (4.6). We expect that the zero modes (5.37) are not
robust under a generic non-Abelian deformation.

However, they are robust under the deformations of the same kind as in (5.1)
with the requirement that $\alpha(x,t)$ vanishes rapidly at $t=\pm\infty$. All
the steps in the proof of Theorem 5 can also be reproduced in this case. The
deformations of a more general nature such that the corresponding Dirac
operator does not sustain zero modes anymore must exist, and they do.

The loop (4.6) is contractible, i.e. there exists a continuous family
$\Omega_{\xi}(x)$ such that

$\displaystyle\Omega_{\xi}(0)=\Omega_{\xi}(L)=\mathbb{1},\quad\Omega_{0}(x)=\Omega(x),\qquad\Omega_{1}(x)=\mathbb{1}\,.$
(5.43)

Consider the corresponding family of field configurations,

$\displaystyle A_{1}^{(\xi)}(x,t)\ =\
-i\frac{b(t)L}{2\pi}\partial_{x}\Omega_{\xi}(x)\,\Omega_{\xi}^{-1}(x)\,.$
(5.44)

As $\xi$ changes from 0 to 1, the configuration (5.44) interpolates between
the field (5.34) sustaining two doublets of zero modes to the configuration
$A_{1}(x,t)=0$ where the zero modes are absent555A constant zero mode for the
free Dirac equation would be present if $\psi(x,t)$ satisfied periodic
boundary conditions on the spatial circle, but our b.c. are antiperiodic. A
very plausible guess is that the zero modes disappear as soon as $\xi\neq 0$
and the loop slides aside as in Fig. 3.

Figure 3: Sliding loops

Note that the configuration

$\displaystyle A_{1}^{(\xi)}(x,\infty)\ =\
-i\partial_{x}\Omega_{\xi}(x)\,\Omega_{\xi}^{-1}(x)$ (5.45)

is still a vacuum configuration with zero field density [we assumed that
$b(t\to\infty)$ approaches the value $2\pi/L$ exponentially fast]. It is
topologically trivial, being related to $A_{1}=0$ and to $A_{1}=2\pi\tau^{3}$
by contractible gauge transformations. But the configurations (5.44) are not
related to (5.34) by gauge transformations. They represent genuine
deformations of (5.34), and these deformations matter!

Note also that one can find a gauge transformation $A_{1}\to\tilde{A}_{1}$
that brings (5.44) to the form where $\tilde{A}_{1}(x,t)$ keeps its asymptotic
values: $\tilde{A}_{1}^{(\xi)}(x,-\infty)=0$ and
$\tilde{A}_{1}^{(\xi)}(x,\infty)=2\pi\tau^{3}$. This transformation is
realized by $\omega_{\xi}(x,t)$ such that

$\omega_{\xi}(x,-\infty)=\mathbb{1},\qquad\omega_{\xi}(x,\infty)=\tilde{\Omega}(x)\Omega_{\xi}^{-1}(x)\,.$

Thus, the fact that the configuration (5.44) carries no zero modes if $\xi\neq
0$ seems to contradict the statement above that the zero modes are robust
under the deformations that keep the asymptotics. There is no contradiction,
however, as the transformation $\omega_{\xi}(x,t)$ brings about a nonzero
$A_{0}(x,t)$. And if one imposes the Hamilton gauge, a generic deformed
instanton configuration does not keep the simple boundary condition assumed in
Sect. 5.1.

For $N=2$, a generic topologicallly nontrivial instanton keeps a single
doublet of zero modes. This is dictated by Theorem 2, which is also valid on a
cylinder. But the two doublets of zero modes (4.22) for $N=3$ theory are not
robust under a generic deformation. To understand that, consider along with
the field (4.19),

$\displaystyle A_{1}(t)\ =\ \frac{2}{3}a(t)\,{\rm diag}(1,1,-2)\,,$ (5.46)

the field

$\displaystyle B_{1}(t)\ =\ \frac{4}{3}a(t)\,{\rm diag}(2,-1,-1)$ (5.47)

These two fields satisfy the same boundary conditions and belong to the same
topological class. The vacua at $t=\infty$ are related as

$\displaystyle B_{1}(\infty)\ =\
-i\partial_{x}V(x)V^{-1}(x)+V(x)A_{1}(\infty)V^{-1}(x)\,,$ (5.48)

where

$\displaystyle V(x)=\exp\left\\{\frac{2i\pi x}{L}{\rm diag}(1,-1,0)\right\\}$
(5.49)

represents a contractible loop in $SU(3)/Z_{3}$.

Still, the configuration (5.46) has two doublets of zero modes and the
configuration (5.47) has four such doublets: the modes

$\displaystyle\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&0\\\
0&0&0\end{array}\right)_{\\!\\!\rm color}\left(\begin{array}[]{c}0\\\
1\end{array}\right)_{\\!\\!\rm spin}e^{i\pi
x/L}e^{\phi_{1}(t)},\qquad\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
0&0&0\end{array}\right)_{\\!\\!\rm color}\left(\begin{array}[]{c}0\\\
1\end{array}\right)_{\\!\\!\rm spin}e^{i\pi x/L}e^{\phi_{1}(t)}$ (5.60)

with $\dot{\phi}_{1}(t)=\pi/L-4a(t)$,

$\displaystyle\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&0\\\
0&0&0\end{array}\right)_{\\!\\!\rm color}\left(\begin{array}[]{c}0\\\
1\end{array}\right)_{\\!\\!\rm spin}e^{3i\pi
x/L}e^{\phi_{2}(t)},\qquad\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
0&0&0\end{array}\right)_{\\!\\!\rm color}\left(\begin{array}[]{c}0\\\
1\end{array}\right)_{\\!\\!\rm spin}e^{3i\pi x/L}e^{\phi_{2}(t)}$ (5.71)

with $\dot{\phi}_{2}(t)=3\pi/L-4a(t)$, and the Hermitially conjugated modes of
opposite chirality.

Consider now the family of configurations $A_{1}^{(\xi)}(x,t)$ that smoothly
interpolate between $A_{1}(t)$ and $B_{1}(t)$ as $\xi$ changes from 0 to 1.
For intermediate values of $\xi$, there probably are no zero modes at all.

##  6 Discussion: screening vs. confinement

The technical issue about the existence or non-existence of fermion zero modes
in adjoint $QCD_{2}$ is interesting by its relationship to the physics of this
model. The model is confining in the sense that its physical spectrum does not
involve states that carry color charge. But a nontrivial question is whether
we are dealing with confinement in the strong sense where the potential
between two static color sources grows linearly and the Wilson loop has the
area law or with the confinement in the weak sense or screening with the
perimeter law for the Wilson loop.

For example, in the pure Yang-Mills theory in 4 dimensions, we have666Or
rather we believe to have. strong confinement of fundamental heavy sources,
but adjoint sources are screened by the gluons. In $QCD_{4}$, the fundamental
sources are also screened due to the presence of dynamical fundamental quarks.
The ordinary $QCD_{2}$ with dynamical quarks has the same properties as
$QCD_{4}$: any heavy colored source is screened. The adjoint $QCD_{2}$ with
massive fermions has the same properties as 4-dimensional gluodynamics: strong
confinement for fundamental sources, whereas adjoint sources (and all other
sources with zero $n$-ality) are screened.

And the massless adjoint $QCD_{2}$ exhibits a nontrivial behavior. We showed
in Ref. [13] that, contrary to naive expectations, the massless adjoint
fermions may well screen the fundamental sources and all other sources with
nonzero $n$-ality.777A similar phenomenon is known to take place in the
massless Schwinger model where the massless fermions of charge one manage to
screen any heavy source of integer or fractional electric charge [12].  This
observation was based on the following conjecture: 888In Ref. [13], we also
presented several other arguments, but we only discuss here the most solid one
having an immediate relationship to the main subject of this paper.

######  Conjecture 1.

In the topological trivial sector of the massless adjoint $QCD_{2}$, the
fundamental Wilson loop average

$\displaystyle\langle W(C)\rangle_{\rm triv}\ =\ \left\langle\frac{1}{N}{\rm
Tr}\left\\{P\exp\left(ig\oint_{C}A_{\mu}(x)\,dx^{\mu}\right)\right\\}\right\rangle_{\rm
triv}$ (6.1)

satisfies the property

$\displaystyle\sigma\ =\ \lim_{{\cal A}\to\infty}\left[-\frac{\ln\langle
W(C)\rangle_{\rm triv}}{{\cal A}}\right]\ =\ 0\,,$ (6.2)

where ${\cal A}$ is the area embraced by the contour.

In other words, the averages of large Wilson loops fall down in this sector
according to the perimeter law rather than area law.

Together with most other experts, we believe this conjecture to be correct
though, to the best of our knowledge, its formal proof has not been given yet.
The difficulty lies in the non-Abelian nature of the theory. The Abelian
version of this conjecture is an exact theorem:

###### Proof.

Consider the Schwinger model at the infinite Euclidean plane in the trivial
topological sector,

$\displaystyle\Phi=\frac{e}{2\pi}\int F(x)\,d^{2}x\ =\ \,0\,,$ (6.3)

where $F=\varepsilon_{\mu\nu}\partial_{\mu}A_{\nu}/2$ is the Abelian field
density and the fermion charge $e$ is included in the definition of the flux.
Suppose that $F(x)$ vanishes at infinity fast enough. In view of (6.3), one
can then choose a gauge where $A_{\mu}(x)$ also vanishes there. And that means
that the Abelian Wilson loop,

$\displaystyle W^{\rm Ab}(C)\ =\
\exp\left(ie^{\prime}\oint_{C}A_{\mu}(x)\,dx^{\mu}\right)\,,$ (6.4)

is equal to 1 for the asymptotic contour $C$ embracing infinity, and this is
true for any charge $e^{\prime}$ of a heavy probe.

Note that in the sectors with nonzero flux $\Phi=n$, the vector potential
behaves at infinity as

$\displaystyle A_{\mu}(x)\ =\
-i[\partial_{\mu}e^{in\theta}]\,e^{-in\theta}\,,$ (6.5)

where $\theta$ is the polar angle, and the asymptotic Wilson loop takes the
value

$W^{\rm asympt}_{n}=\exp\left\\{\frac{2i\pi ne^{\prime}}{e}\right\\}.$

We go back to the sector $n=0$ and consider a large but not asymptotically
large loop. The Stokes theorem allows us to write

$\displaystyle\langle W^{\rm Ab}(C)\rangle_{n=0}\ =\
\left\langle\exp\left\\{ie^{\prime}\int_{D}F(x)\,d^{2}x\right\\}\right\rangle_{n=0}\,,$
(6.6)

where $D$ is the domain embraced by the loop. The Gaussian nature of the path
integral in the Schwinger model allows one to present it as

$\displaystyle\exp\left\\{-\frac{e^{\prime
2}}{2}\int_{D}\int_{D}d^{2}xd^{2}y\,\langle F(x)F(y)\rangle_{n=0}\right\\}\,.$
(6.7)

The correlator $\langle F(x)F(y)\rangle_{n=0}$ depends only on $x-y$. In the
massless Schwinger model, it is known exactly. It decays exponentially
$\propto e^{-\mu\sqrt{(x-y)^{2}}}$ at large separations ($\mu=e/\sqrt{\pi}$
being the mass of the Schwinger boson) and satisfies the property

$\displaystyle\int_{\rm whole\ plane}d^{2}x\,\langle F(x)F(0)\rangle_{n=0}\ =\
0\,.$ (6.8)

If the integral (6.8) did not vanish, the exponent in (6.7) would be
proportional to the area ${\cal A}$ of the domain $D$ giving a nonzero string
tension. But as it vanishes, the string tension vanishes too.

If the integral is done over a finite region $D$ rather than the whole plane,
the integrals in (6.8), (6.7) do not vanish. The double integral in (6.7) is
saturated by the values of $x$ that are close to the border of $D$ and $y$
that are within the distance $\sim\mu^{-1}$ from $x$. We obtain the perimeter
law [14],

$\displaystyle\langle W^{\rm Ab}(C)\rangle_{n=0}\ =\
\exp\left\\{-\frac{e^{\prime 2}P}{4\mu}\right\\}\,,$ (6.9)

implying that the potential between the external sources of charge $\pm
e^{\prime}$ levels off at large separations at the value

$\displaystyle V(\infty)\ =\ \frac{e^{\prime 2}}{2\mu}\,.$ (6.10)

∎

Consider now the non-Abelian theory of interest. Assuming as before that the
field density vanishes at infinity, the gauge potential in the topological
sector $k$ can be brought into the form

$\displaystyle A_{\mu}(x)\ =\
-i\left[\partial_{\mu}\exp\left\\{\frac{ik\theta}{N}\tau^{3}\right\\}\right]\,\exp\left\\{-\frac{ik\theta}{N}\tau^{3}\right\\}\,,$
(6.11)

This gives the values

$\displaystyle W_{k}\ =\ e^{2i\pi k/N}$ (6.12)

for the asymptotic Wilson loop [6]. For $k=0$, it is just the unity.

The property (6.2) could possibly be proven using the non-Abelian version of
the Stokes theorem. The latter reads [15]

$\displaystyle P\exp\left\\{i\oint_{C}A_{\mu}(x)\,dx^{\mu}\right\\}\ =\ {\cal
P}\exp\left\\{i\int_{D}{\cal F}\,d^{2}x\right\\}\,,$ (6.13)

where $P$ is the ordinary path ordering and ${\cal P}$ is the operator of
area-ordering, i.e. the infinitesimal loops should first be multiplied over
along the direction of $x$, and then along the direction of $y$. As for ${\cal
F}$, it is not simply the field density, but the object

$\displaystyle{\cal F}(x)\ =\ UF(x)U^{-1}\,,$ (6.14)

where

$\displaystyle U=P\exp\left\\{i\int_{O}^{x}A_{\mu}(x)\,dx^{\mu}\right\\}$
(6.15)

is the string operator connecting a reference point $O$ on the contour $C$ to
the point $x$ inside the contour along a certain prescribed path.

Anyway, if we assume that the conjecture above is correct, we can be sure that
the $N=2$ theory exhibits screening. Indeed, the topologically nontrivial
sector there has a doublet of zero modes, the fermion determinant vanishes and
hence this sector does not contribute to the path integral. Whatever is true
in the topologically trivial sector (like the perimeter law for the Wilson
loop), is true in the whole theory.

The situation is less clear for higher $N$ starting from $N=3$. In [13], the
existence of the zero modes for $k\neq 0$ was assumed also for higher $N$,
which entailed the conclusion that this theory exhibits screening. But we know
today that, e.g. for $N=3$, a generic Euclidean field configuration does not
sustain fermion zero modes and the contributions of the topologically
nontrivial sectors do not vanish. In the sum of the contributions of the
sectors $k=0,1,2$, cancellations might occur, so that $\langle W(C)\rangle$
exhibits the area rather than perimeter law and the theory is confining [1].

In recent [16], it was argued, however, that this does not happen and adjoint
massless $QCD_{2}$ with any unitary gauge group always exhibits screening. It
is associated with the existence of $\sim e^{2N}$ extra hidden symmetries,
which the theory defined on a large spatial circle enjoys. String operators
similar to (6.15) played an important role in this analysis.

An independent argument in favour of the screening scenario was given in [17].
These authors studied the theory involving the massless adjoint fermions and
massive fermions in the fundamental representation (call them quarks). Using
the technique of the discretized light-cone quantization, they could perform a
numerical study the spectrum of the quark-antiquark states and observed the
continuous spectrum of these states above some threshold. This suggests that
the quark-antiquark potential levels off at large separations [cf. Eq.
(6.10)], implying the existence of the states where a heavy quark is screened
by a cloud of an infinite number of massless adjoint fermions.999Any finite
number of adjoint fermions are not capable, of course, to screen a fundamental
source.

More studies in this direction are highly desirable.

## Acknowledgements

I am indebted to Igor Klebanov, Zohar Komargodski and Yuya Tanizaki for
illuminating discussions.

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