%Date: Wed, 26 Mar 1997 18:32:05 -0500 (EST)
%From: Pavel Etingof
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\topmatter
\title Lecture II-6: Gauge theory in 2 dimensions with
self-interacting bosons, the Wilson line operator, and confinement
\endtitle
\author {\rm {\bf Edward Witten} }\endauthor
\endtopmatter
\centerline{Notes by Pavel Etingof and David Kazhdan}
\vskip .1in
In the last lecture we considered 2-d gauge theories with fermions.
Today we will consider 2-dimensional gauge theories with bosons.
As before, we will work with Euclidean Lagrangians.
{\bf 6.1. Infrared behavior of $U(1)$ gauge theories with
bosons in 2-dimensions.}
We consider a $U(1)$ gauge theory with bosons in 2 dimensions.
Our spacetime is a Riemann surface $\Sigma$.
Our fields are $A$ -- a $U(1)$ connection in some line bundle $L$,
and $\phi_1,...,\phi_N$ -- complex scalar fields, which are sections of $L$.
Our Lagrangian is
$$
\int d^2x\left(\frac{(*F)^2}{4e^2}+\frac{1}{2}|d\phi|^2+\frac{\l}{4}
(|\phi|^2-a^2)^2\right)-\frac{i\theta}{2\pi}\int F,\tag 6.1
$$
where $|\phi|^2=\sum |\phi_i|^2, |d\phi|^2=\sum |d\phi_i|^2$.
We want to understand the infrared behavior of this theory.
Clasisically, the ``mass'' of the bosons is given by $\phi_i$ is $m^2=-
\l a^2$.
If $\l\to 0$ and $m^2>0$ remains fixed, this Lagrangian becomes the Lagrangian
of a gauge theory with ``free'' massive bosons (i.e. bosons interacting
with the gauge field only and not interacting with each other), which was
considered in Lecture II-4.
The parameters of the theory are $m,\theta,\l,e$ (with $\l>0$). We will
now keep $\l,e$ fixed, and vary $m,\theta$, to see how the behavior
of the theory depends on them.
First consider the situation $m^2\to +\infty$ (i.e. $m^2>>e^2$). In this
situation the quartic term in the Lagrangian becomes unimportant,
and the situation is essentially the same as in Lecture II-4. Namely,
the theory has a mass gap, and the interaction between bosons is approximated
by the 1-dimensional Coloumb
potential, which causes their confinement:
all states in the theory are of total charge zero.
Also, the space reversal symmetry is preserved for $\theta=0$,
but broken for $\theta=\pi$, thus producing two vacua and
a cut at $\theta=\pi$. For example, the vacuum energy density,
which is a $2\pi$-periodic function of $\theta$
given for $-\pi\le \theta\le \pi$ by formula (6.2) below,
has a discontinuous derivative at $\theta=\pi$.
Now consider the situation when $m^2\to -\infty$ (i.e. $m^2<<-e^2$).
In this case, the situation is totally different: because of the presence
of the quartic term, the classical potential for bosons has a minimum at
$|\phi|^2=a^2$, so the space of minima is $S^{2N-1}$. The gauge group
$U(1)$ acts freely on this space, so the space of classical vacua
is $\C P^{N-1}=S^{2N-1}/U(1)$. Thus, the low energy effective theory
is the theory of maps from $\Sigma$ to $\C P^{N-1}$ with the round
metric (the sigma-model).
{\bf Remark.} In Lecture 3 we saw that the theory
with the Lagrangian density $\frac{1}{2}(d\phi)^2+\l(\phi^2-a^2)^2$
flows in the infrared to the sigma-model of maps to the sphere. We
remarked that this means that this (superrenormalizable) theory
is a good cutoff for the sigma-model to the sphere. Similarly,
the superrenormalizable (and hence rigorously existing) theory
defined by Lagrangian (6.1) is a good UV cutoff for the theory of maps to
$\C P^{N-1}$.
For Large $N$, we found that the sigma-model of maps
to $\C P^{N-1}$ behaves in the infrared as the gauge theory
of massive bosons. Today we will consider in more detail the case $N=1$.
In this case $\C P^{N-1}$ is one point, so the theory defined by
(6.1) has a mass gap and a unique vacuum.
{\bf 6.2. The vacuum energy density.}
Recall that for $m^2>>e^2$ in Lecture II-4 we found the following value
of the vacuum energy density:
$$
E_{vac}(\theta)=\frac{e^2}{2}\left(\frac{\theta}{2\pi}\right)^2.\tag 6.2
$$
So the $\theta$-derivative of the energy density
is proportional to the first power of the coupling $e^2$.
In the case $m^2<<-e^2$, the situation is totally different. Namely, now
the contributions to the path integral from nontrivial line bundles
(relative to infinity)
is exponentially small in $e^2$, since any section of such a bundle
has to vanish somewhere, and this will have a big action due to the presense
of the quartic potential. Thus, to any finite order of perturbation
theory (in $e^2$), the path integral,
and hence the vacuum energy density are independent of $\theta$.
More precisely, the theta-dependence is exponentially small, and comes from
the sum over {\it instantons} -- lowest action configurations of nontrivial
first Chern class.
{\bf 6.3. Instantons.}
Let us compute the coefficient of $e^{i\theta}$ in the Fourier expansion
of the path integral with Lagrangian (6.1). This is equivalent to taking
the path integral over connections in line bundles with $c_1=1$.
As usual, the main contribution to the path integral comes from field
configurations which are close to the configuration minimizing
the action. Such a configuration $(\phi,A)$ is called an instanton.
Let $(\phi,A)$ be an instanton.
In order for its action to be finite, we must have
$$
|\phi(x)|\to a, F_A(x)\to 0, x\to \infty,\tag 6.3
$$
where $F_A$ is the curvature of $A$. Moreover,
since $\phi$ is classically massive, this convergence
is exponentially fast. Thus, an instanton in our problem
has to be a highly localized field configuration.
In our further discussion of instantons, we will assume that $\Sigma=\R^2$,
and $c_1$ is measured with respect to trivialization at infinity.
In this case, we expect that instantons are
rotationally symmetric, with respect to some center of rotation $x_0$.
Without loss of generality we can assume that $x_0=0$, so that the instanton
has the form $\phi=ae^{i\alpha}f(r)$, $A=g(r)d\alpha$, where
$f(r)$, $g(r)$ are some functions of the radius $r$ (where $r,\alpha$
are polar coordinates on $\R^2$). It is easy to show
that in this case one must have $f(0)=0$, $f(\infty)=1$,
and $g(0)=0$, $g(\infty)=1$. Thus, we get a boundary value problem for
ordinary differential equations. It can be proved by considering
the corresponding ODE that this problem
has a unique solution. Thus, the instanton is unique, up to translations.
In particular, it has a definite size, and its action is
a well-defined positive constant $I$.
{\bf Remark.} In this respect, our instanton is different from
the instantons of the $\C P^{N-1}$ model, which could be
transformed by any conformal automorphism, and therefore had no
definite size.
By dimensional analysis
it is clear that $I$ is of the form $a^2h(\frac{\l}{e^2})$,
where $h$ is a dimensionless function. It is possible to show that
$h(z)\sim Cz$ as $z\to \infty$, where $C$ is a constant,
so for small $e$ (compared to $\l$), $I\sim \frac{C\l a^2}{e^2}$.
This calculation illustrates
the fact that the contribution of the instanton to the path integral
is exponentially small with respect to $e^2$.
{\bf 6.4. Instanton gas.}
Now we want to understand how to compute the contribution of the instantons
to the path integral, and when such a computation gives a good approximation.
First of all consider line bundles with $c_1=2$. It can be shown that
there is no instanton in this topological class if $e^2/\lambda$ is
sufficiently large. We will assume that there is no instanton
for $c_1>1$, but what we will say can be generalized to the case
when there is one (for small $e^2/\lambda$). In the case when there
is no instanton with $c_1>1$,
the problem of minimization of action in the case $c_1=2$ has no global
minimum (the infinum is not attained). However,
we can consider approximate instantons, whose action approaches
the infinum arbitrarily closely. More precisely, if we take
$A(x)=A_*(x-x_1)+A_*(x-x_2),\phi(x)=\phi_*(x-x_1)\phi_*(x-x_2)/a$, where
$(A_*,\phi_*)$ is the instanton centered at $0$, then the action
of $(A,\phi)$ equals $2I$, plus a correction which is of order
$e^{-c|x_1-x_2|}$, where $c$ is a positive constant.
Thus, the infinum of actions for $c_1=2$ is $2I$.
Similarly, the infinum of actions for $c_1=n$ is $nI$, for any
$n\in \Z_+$. This follows from the fact that the action of
the field configuration
$$\sum_{i=1}^n A_*(x-x_i), a\prod_{i=1}^n\frac{\phi_*(x-x_i)}{a}$$
has action which is exponentially close to $nI$ when $|x_i-x_j|$ are big.
Now consider the situation when $c_1<0$. It is easy to see that
$(-A_*,\bar \phi_*)$ is the instanton for $c_1=-1$.
It is called the antiinstanton. Thus,
the situation for $n<0$ is symmetric to the situation for $n\in \Z_+$:
the field configuration $$-\sum_{i=1}^{|n|}A_*(x-y_i),
a\prod_{i=1}^n\frac{\bar\phi_*(x-y_i)}{a}$$ has action exponentially
close to $|n|I$ when $|y_i-y_j|$ is big.
More generally, one can consider field configurations
$$\sum_{i=1}^n A_*(x-x_i)-\sum_{i=1}^m A_*(x-y_i),
a\prod_{i=1}^n\frac{\phi_*(x-x_i)}{a}
\prod_{i=1}^m\frac{\bar\phi_*(x-y_i)}{a},$$
with $c_1=n-m$ which have action exponentially
close to $(n+m)I$ when $x_i,y_j$ are distant from each other.
{\bf Remark.}
Such field configurations are called ``instanton gas''.
The term ``instanton gas'' refers to a gas with long range
Coulomb forces (like the instantons we studied later in the Polyakov
model in 2+1 dimensions). This gas with exponentially small forces
at long range is more like an ordinary gas of atoms, for instance
the air in the atmosphere. It is an almost ideal gas, the ideal gas law
of thermodynamics is the case that the forces are exactly zero. Any
real gas (hydrogen, oxygen) behaves as an almost ideal gas if the density
is small enough, because then the particles are generally at big distances
where the interactions are small. That is the case for the instantons
in this model because the instanton action is big (and the instantons
have a definite size)
{\bf 6.5. Summing over instantons.}
As we remarked, the perturbation series with respect to powers of $e^2$
for the path integral with Lagrangian (6.1) does not involve contributions
from instantons, as they are exponentially small. Let us introduce
a refined perturbation series, which will take instantons into account.
For this purpose we will work on a Riemann surface of volume $V$, and
introduce a new perturbation parameter $W=Ve^{-I}$.
We will consider the perturbation expansion with respect to both $W$ and $e$.
The key fact is that in this
refined perturbation expansion, the only contributions of finite order
in $W$ and $e$ are from the instanton gas. Thus, the approximation
to the partition function obtained this way has the form
$$
Z(e^2,\theta,W)=e^{VP_0}\sum_{n,m=0}^{\infty}
\frac{W^{m+n}P_+^nP_-^me^{i\theta(n-m)}}{m!n!},\tag 6.4
$$
where $P_0,P_+,P_-$ are the perturbation series around the zero
solution, the instanton, and the antiinstanton, respectively.
Here the term with indices $m,n$ comes from a field configuration
with $n$ instantons and $m$ antiinstantons, distant from each other.
The factorials in the denominator arise from the fact that instantons
and antiinstantons are not labeled, and their permutation does not change
the configuration.
Summing (6.4) by ordinary calculus, we get
$$
Z(e^2,\theta,W)=e^{VP_0+W(P_+e^{i\theta}+P_-e^{-i\theta})}.\tag 6.5
$$
In our case, $P_+=P_-=P$, so we get
$$
Z(e^2,\theta,W)=e^{VP_0+2WP\cos\theta}.\tag 6.6
$$
Now we can compute the energy density of the vacuum in this approximation.
This can be done from the equality $Z=e^{-VE_{vac}}$, which is
the definition of the vacuum energy density $E_{vac}$. Thus,
$$
E_{vac}=-P_0-2Pe^{-I}\cos\theta.\tag 6.7
$$
As we expected, the theta-dependence is exponentially suppressed, but we were
able to compute the main term of this dependence.
The next order correction to (6.7) is of order $e^{-2I}$, but it
is hard to calculate. But if we had an instanton for $c_1=2$ with
$I'<2I$, then the correction would be of the form $e^{-2I'}\cos 2\theta$.
We see that unlike the case $m^2>>0$, where $E_{vac}=e^2\theta^2/8\pi^2$
is a non-smooth function of $\theta$ (it has a cut at $\theta=\pm \pi$),
in the case $m^2<<0$ the function $E_{vac}(\theta)=-P_0+2Pe^{-I}\cos\theta$
is smooth in the first approximation. Thus, for $m^2<<0$ there is no cut
at $\theta=\pm \pi$, and there is a mass gap and a unique vacuum for any
$\theta$.
Thus, the cut at $\theta=\pm \pi$ has to add at some point $m^2=m^2_c\in \R$.
At this point the effective potential of the theory (if it makes sense)
has a quartic critical point at the origin, so that the theory has no
mass gap. It is conjectured that this theory is conformal, and is the
continuous limit of the 2-dimensional Ising model.
{\bf 6.6. The Wilson line operator.}
In this and subsequent sections we will define the Wilson line operator,
and try to understand its physical and formal properties.
Suppose we have a gauge theory with gauge group $G$ on a spacetime $M$.
Let $A$ denote the corresponding gauge field with values in the Lie algebra
$\g$ of $G$. Let $C$ be a closed loop in $M$, and $Hol(A,C)\in G$ be the
holonomy of the connection $A$ along $C$ (it is only defined up to
conjugation). Let $R$ be a finite-dimensional representation of $G$.
Define the classical Wilson line (or Wilson loop) functional to be
$$
W_R(C)(A)=\Tr|_R(Hol(A,C)).\tag 6.8
$$
It is clear that this functional is gauge invariant.
If $C$ is a union of disconnected loops $C_i$ labeled with representations
$R=(R_i)$, then by definition $W_R(C)=\prod W_{R_i}(C_i)$.
If $G$ is abelian, and $C$ is cotractible, then
$W_R(C)=e^{i\int_D F}$, where $F$ is the curvature of $A$, and
$D$ is a disk such that $\d D=C$.
An important generalization of this is the following:
$\hat G$ is the simply-connected cover of $G$, and $R$ is
a representation of the universal covering
$\hat G$ of $G$. In this case it is easy to see that
$W_R(C)$ is still well-defined when $C$ bounds a disk $D$.
(in the abelian case, it follows from the above integral representation
of $W$).
Now we want to define an analogue of $W_R(C)$ in quantum theory.
For this purpose we need to renormalize the classical functional
$W_R(C)$. This can be done by expanding $W_R(C)$ in powers of $A$
(like the exponential is expanded in Taylor series):
$$
W_R(C)=\text{dim}(R)+\int dl\Tr|_R(A(l))+
\frac{1}{2}\int dl dl'\Tr|_R(A(l)A(l'))+...,\tag 6.9
$$
where $l$ is some parameter on $C$, and
$A(l)$ is the evaluation of the form $A$ on the
tangent vector $\frac{d}{dl}$ to $C$ at the point $l$, with respect to this
parametrization. Each term of this expansion
is polynomial and can be renormalized as usual.
In fact, one can show in most cases that
the operator $W_R(C)$ has only multiplicative renormalization.
This follows from the fact that classically,
$W_R(C)$ is the trace of the monodromy of the differential equation
$x'=Ax$ along the loop. This equation is renormalized to
$x'=(A+c)x$, where $c$ is an operator in the theory, invariant
under the same symmetries as $A$, in the adjoint representation of
the gauge group, of dimension the same and lower than $A$. If there is
no such operators except for constants
(constants come up for the $U(1)$-case)
then this equation will change to $x'=(A+c)x$ under
renormalization, where $c$ is a scalar operator. This shows that $W_R(C)$
will have multiplicative renormalization.
More precisely, one can show
that in critical dimension 4, the divergent renormalization
factor has the form $e^{L(C)\Lambda f(e^2)+o(\L)}$,
where $\L$ is the cutoff and $L(C)$ is the length of $C$, while
in the superrenormalizable case (in less than 4 dimensions),
the divergent factor has power growth with respect to $\L$.
A more physical way of thinking of the Wilson loop operator
is the following. We will use the Hamiltonian picture.
Thus, $M=M_s\times \R$, where $M_s$ is the space manifold.
The Hilbert space of the theory is then $\Cal H$ the space of functions on
$\Cal A$, where
$\Cal A$ is the space of gauge classes of connections on
$M_s$. The space $\Cal H$ has the form $\Cal H=(\Cal H_0)^{\tilde G}$,
where $\Cal H_0$ is the space of functions on all connections,
and $\tilde G$ is the group of gauge transformations.
Now let $C$ be a loop in $M$.
Suppose first that the loop $C$ is in the submanifold
$t=0$. In this case, the Wilson loop operator is obviously just
the operator of multiplication by the function $W_R(C)$.
Now consider the situation when $C$ is not in a horizontal section
of the spacetime, but a general curve. In this case, $W_R(C)$
is no longer multiplication by a function.
{\bf Remark. }
We think of $C$ as a worldline of a ``charge''
which transforms in a representation $R$. The best is to think
of two charges of type $R,R^*$ which are born at some time $t_0$ from nothing
at the same point $x_0$, then fly around for a while (until $t_1$), and finally
recombine at the time $t_1$, back into nothing, at a point $x_1$.
We think of these charges as classical, external objects. That is,
the expectations in the presence of $C$ are conditional
expectations, given that the worldlines of the charges form the loop $C$.
In this setting, we can regard $e^{iH(t_1-t_0)}W_R(C)$
as the evolution operator from time $t_0$ to time $t_1$, in the
system with presence of $C$ (here $H$ is the Hamiltonian
of the system).
The operator $W_R(C)$ allows us to define the time ordered correlation
functions in the presence of $C$ for any set of local operators at points
$(x,t)$ such that $t\notin [t_0,t_1]$, simply as correlation functions with
the insertion of $W_R(C)$ in the right place.
But what should we do for $t_0>L$, and then
annihilate each other. Then
The expectation value $\$ of the Wilson loop operator
(in the Euclidean setting) has the following meaning: it is approximately
equal to $Ce^{-TV(L)}$,
where $V(L)$ is the energy of interaction of the charges
at distance $L$, and $C$ is a constant.
This is clear from the interpretation of the Wilson
loop operator as an evolution operator, which is given above.
So the asymptotics of $\$ depends on the asymptotics
of $V(L)$ as $L\to \infty$.
Physicists believe that above 2 dimensions, in gauge theories
with a mass gap, there are
two possibilities:
1. Higgs regime: $V{L}\to \text{const}, L\to \infty$.
In this regime, charges can separate from each other.
2. Confinement regime: $V(L)\sim \text{const}L$ as $L\to \infty$,
where the constant is positive.
In this regime charges are confined, and cannot separate without
spending an arbitrarily large amount of energy.
Now let us consider a Wilson loop $C$ of any shape, with circumference $S$ and
minimal area of the spanning surface $A$.
There is a general conjecture that patterns 1 and 2, if they hold
for $T>>L>>0$, hold for an arbitrary $C$ (large in all directions)
in the following form:
the Higgs regime corresponds
to the asymptotics
$\\sim e^{-wS}$ (the circumference law),
and the confinement regime corresponds
to the asymptotics $\\sim e^{-wA}$ (the area law), where $w>0$.
{\bf Remark.} Actually, confinement can only occur if
$R$ is a representation of the universal cover $\hat G$ of $G$ and not
of $G$ itself. Indeed, if $R$ is a representation of $G$,
there are physical processes contributing to $\$
in which large portions of the Wilson line have zero charge
(i.e. carry the trivial representation of $G$)
(i.e. some particles have annihilated the charges on the Wilson line).
These processes have an amplitude which is bigger than that
predicted by the area law.
As a toy example, let us consider the 2-dimensional theory with Lagrangian
(6.1), which we studied in the first part of the lecture.
In this case, $G=U(1)$. Let $R$ be the representation of $\hat G$ defined by
$\l\in \R$: $x\to e^{i\l x}$. Then, classically,
$W_R(C)=e^{i\l\int_D F_A}$.
Thus, quantum mechanically
$$
\=\int DAD\phi D\bar\phi e^{-\Cal L}e^{i\l\int_D F}. \tag 6.16
$$
Let $V$ be the volume of spacetime, and $A_C$ be the area
enclosed by $C$. We will split the path integral (6.16)
into a product of two -- the integral over values
of fields inside the loop $C$ and over the values outside.
Observe that the last factor in (6.16)
(the holonomy factor) is
of the same type as the topological term in (6.1).
Therefore, according to the results
of the first part of this lecture, (6.16) yields
$$
\= \frac{1}{Z}e^{-(V-A_C)E(\theta)-A_CE(\theta+\l)},\tag 6.17
$$
up to boundary terms, which we will neglect here.
(Here $E(\theta)$ is the vacuum energy density).
Since $Z=e^{-VE(\theta)}$, we get
$$
\=e^{A_C(E(\theta)-E(\theta+\l))}.\tag 6.18
$$
According to our calculations, for $\theta=0$
$$
E(\theta+\l)-E(\theta)=E(\l)-E(0)=2(1-cos \l)e^{-I/e^2}P. \tag 6.19
$$
This is always positive when $\l$ is not a multiple of $2\pi$. Thus,
theory (6.1) for $\theta=0$ exhibits the confinement regime.
In more than 2 dimensions, one expects that this theory obeys Higgs
regime.
{\bf 6.9. The confinement conjecture.}
The following conjecture is central in quantum field theory.
\proclaim{Conjecture} Let $G$ be a simple compact Lie group,
and $R$ be a representation of $G$ which is not a
representation of the adjoint group $G_{ad}$. In 3 and 4 dimensions,
the pure gauge theory with gauge group $G$ and Lagrangian
$\int \Tr F\wedge *F$ exhibits confinement
for charges with values in $R$ and $R^*$.
\endproclaim
The physically interesting case of this theory is $G=SU(3)$, $R=\C^3$.
This special case of the conjecture would explain confinement of quarks.
In 2 dimensions, this conjecture is true, as we saw in the previous lectures.
\end