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\begin{document}

\centerline{\bf Lecture~4}
\centerline{\bf $O(3)$\-/model: mass generation by instantons}
\bigskip
Consider the $O(3)$\-/model:
\eq$$
S[\bn]={1\over2g}\int d^2x\,(\d_\mu\bn)^2,
\qquad
n_1^2+n_2^2+n_3^2=1.
\label{o3model}
$$
We will be interested in the functions $\bn(x)$ with finite action. These function correspond to processes of finite amplitude $e^{-S[\bn]}$. It follows from the finiteness of the action that the solution should tend to a constant at infinity:
\eq$$
\bn(x)\mathinner{\mathop\to\limits_{x\to\infty}}\bn_0.
\label{bcond}
$$
Thus, the infinity can be considered as a point on the two-dimensional sphere $S^2$. The continuous function $\bn(x)$, which satisfies the condition (\ref{bcond}), defines a continuous mapping
\eq$$
\bn:S^2\to S^2.
$$
Such mappings are classified by the {\it topological number\/} $q$, such that $\bn$ realizes a $|q|$\-/sheeted covering of a sphere by a sphere. If $q>0$, the mapping preserves orientation, while if $q<0$, reverses it. A representative of the class $q$ is, for example, a mapping:
\eq$$
\theta'=\theta,
\qquad
\varphi'=q\varphi,
$$
where $(\theta,\varphi)$ are the polar coordinates on a sphere.

Let us express the topological number in terms of the field $\bn$. We will consider it as a mapping of the sphere $S^2$ onto the sphere $S^{2\prime}$. On the sphere $S^2$ we set the coordinates $x=(x^1,x^2)$, while on the sphere $S^{2\prime}$ the coordinates $x'=(x^{\prime1},x^{\prime2})$ and the metric~$g'_{\mu\nu}$. The metric on the sphere $S^2$ is not essential. Let
$$
S=\int_{S^{2\prime}}d^2x'\sqrt{g'}
$$
is the area of the sphere~$S^{2\prime}$. Then the topological number can be expressed in terms of the area covered by the map $x'(x)$ when $x$ covers the whole sphere~$S^2$:
$$
q={1\over S}\int_{x'(S^2)}d^2x'\,\sqrt{g'}={1\over S}\int_{S^2}d^2x{\d(x')\over\d(x)}\sqrt{g'}.
$$
In spherical coordinates
$$
q={1\over4\pi}\int^{2\pi}_0d\varphi\int^\pi_0d\theta\,
{\d(\theta',\varphi')\over\d(\theta,\varphi)}\sin\theta'
={1\over4\pi}\int d^2x\,
{\d(\theta',\varphi')\over\d(x^1,x^2)}\sin\theta'.
$$
To find the Jacobian $\d(\theta',\varphi')/\d(x^1,x^2)=(\d\theta'/\d x^1)(\d\varphi'/\d x^2)-(\d\theta'/\d x^2)(\d\varphi'/\d x^1)$, rewrite the spherical variables in terms $\bn$ and $x$. Put
\eq$$
\bn=(\sin\theta'\cos\varphi',\sin\theta'\sin\varphi',\cos\theta').
\label{jacsubst}
$$
It can be checked by a direct calculation that
\eq$$
{1\over2}\bn\,(\d_\mu\bn\times\d_\nu\bn)\epsilon^{\mu\nu}
={\d(\theta',\varphi')\over\d(x^1,x^2)}\sin\theta'.
\label{jacobian}
$$
From this we obtain
\eq$$
q={1\over8\pi}\int d^2x\,\bn\,(\d_\mu\bn\times\d_\nu\bn)\epsilon^{\mu\nu}.
\label{qint}
$$

It is possible to derive (\ref{qint}) even easier. The vector $\d_1\bn\,dx^1$ is a small displacement vector on a sphere that corresponds to the displacement by $(dx^1,0)$ in the $x$\-/space. Similarly, $\d_2\bn\,dx^2$ is a displacement vector on a sphere that corresponds to the displacement by $(0,dx^2)$.Both vectors are perpendicular to the vector~$\bn$. Then $|\d_1\bn\times\d_2\bn\,dx^1\,dx^2|=\pm\bn\,(\d_1\bn\times\d_2\bn)\,d^2x$ is the area of a small parallelogram on the sphere. The ``$+$'' sign corresponds to the orientation preserving map, and the ``$-$'' sign corresponds to the orientation reversing map. Therefore,
$$
\int d^2x\,\bn\,(\d_1\bn\times\d_2\bn)
={1\over2}\int d^2x\,\bn\,(\d_\mu\bn\times\d_\nu\bn)\epsilon^{\mu\nu}
$$
is equal to the area on the sphere, covered during the integration over the entire $x$\-/space, which is equal to~$4\pi q$.

From the identity
\eq$$
\int d^2x\,(\d_\mu\bn+\epsilon_{\mu\nu}\bn\times\d^\nu\bn)^2
=2\int d^2x\,(\d_\mu\bn)^2
-2\int d^2x\,\bn\,(\d_\mu\bn\times\d_\nu\bn)\epsilon^{\mu\nu}
\label{intid}
$$
we obtain
\eq$$
S[\bn]
={4\pi q\over g}
+{1\over4g}\int d^2x\,(\d_\mu\bn+\epsilon_{\mu\nu}\bn\times\d^\nu\bn)^2.
\label{Sqplus}
$$
By the substitution $\bn\to-\bn$, $q\to-q$ we have
\eq$$
S[\bn]
=-{4\pi q\over g}
+{1\over4g}\int d^2x\,(\d_\mu\bn-\epsilon_{\mu\nu}\bn\times\d^\nu\bn)^2.
\label{Sqminus}
$$
It obviously follows from this that
\eq$$
S[\bn]\ge{4\pi|q|\over g}.
\label{Sqmin}
$$
Inequality (\ref{Sqmin}) turns into an equality on solutions of the first\-/order equations (the {\it self\-/duality equations\/})
\Align$$
\d_\mu\bn
&=-\epsilon_{\mu\nu}\bn\times\d^\nu\bn
\qquad(q\ge0),
\label{sdpos}
\\
\d_\mu\bn
&=\epsilon_{\mu\nu}\bn\times\d^\nu\bn
\hphantom{-}\qquad(q\le0).
\label{sdneg}
$$
The solutions of these equations minimize the action and, therefore, constitute a subclass of the solutions to the equations of motion. To solve the self\-/duality equations explicitly, we use the stereographic projection onto the plane. It is convenient to use the complex coordinate $w$ on this plane:
\eq$$
n_1+in_2={2w\over1+|w|^2},
\qquad
n_3={1-|w|^2\over1+|w|^2}.
\label{stereogr}
$$
Substituting (\ref{stereogr}) into (\ref{sdpos}), (\ref{sdneg}), we obtain
\Align$$
\bd w
&=0
\qquad(q\ge0),
\label{sdcpos}
\\
\d w
&=0
\qquad(q\le0).
\label{sdcneg}
$$
Consider the case $q>0$. The function $w(z)$ must be a meromorphic function on the sphere, that is, the only singularities of this function, including a singularity at infinity, are poles. The number of nodes and poles of such a function should be finite. The point $\bn=(0,0,1)$ corresponds to nodes, while the point $\bn=(0,0,-1)$ to poles. The most general such solution (\ref{sdcpos}) has the form
\eq$$
w(q,\va,\vb,c;z)=c\prod^q_{j=1}{z-a_j\over z-b_j},
\label{solution-pos}
$$
where $c\in\C\setminus\{0\}$, $a_j,b_j\in\C\cup\{\infty\}$, $a_i\ne b_j$ ($\forall i,j$). Besides, the limiting cases $a_i\to\infty$, $ca_i=\const$ and $b_i\to\infty$, $c/b_i=\const$ should be taken into account, but these cases have zero measure. It is easy to find a topological number corresponding to the solution (\ref{solution-pos}). Let $w_0\in\C\cup\{\infty\}$. Then the number of solutions to the equation $w(z)=w_0$ accounting for multiplicities is independent of~$w_0$. Let $w_0$ be equal, for example, to infinity. Then it is obvious that the number of solutions to this equation is equal to~$q$. This means that $w(z)$ provides a $q$\-/sheeted covering and the topological number is~$q$.

Similarly, in the case of a topological number $q<0$ we obtain
\eq$$
w(q,\va,\vb,c;\bz)=c\prod^{-q}_{j=1}{\bz-a_j\over\bz-b_j},
\label{solution-neg}
$$
The solutions (\ref{solution-pos}) and (\ref{solution-neg}) are called {\it (multi)instanton solutions}.

Let us try to use instanton solutions to approximately calculate the functional integral for the quantum $O(3)$\-/model. We will integrate over multi\-/instanton solutions and small fluctuations near them.

Rewrite (\ref{Sqplus}) and (\ref{Sqminus}) in terms of the field $w(z,\bz)$:
\Align$$
S[w,\bw]
&={4\pi q\over g}+{8\over g}\int d^2x\,{\bd w\d\bw\over(1+|w|^2)^2}
\label{Sqwplus}
\\
&=-{4\pi q\over g}+{8\over g}\int d^2x\,{\d w\bd\bw\over(1+|w|^2)^2}.
$$

Consider the case of $q\ge0$. Let
\eq$$
S_q[\varphi,\bvarphi]
=S[w(q,\va,\vb,c;z)(1+g^{1/2}\varphi(z,\bz)),
  w^*(q,\va,\vb,c;z)(1+g^{1/2}\bvarphi(z,\bz))].
\label{Sqactiondef}
$$
It is easy to see that
\eq$$
S_q[\varphi,\bvarphi]
={4\pi q\over g}
+8\int d^2x\,{|w|^2\over(1+|w|^2)^2}\bd\varphi\d\bvarphi
$$
in the quadratic approximation. Accordingly, the $q$\-/instanton contribution to the partition function has the form
\eq$$
Z_q
={e^{-4\pi q/g}\over(q!)^2}
\int d\mu(\va,\vb,c)\,
Z[w(q,\va,\vb,c;z)],
\qquad
Z[w]
=\int D\varphi\,D\bvarphi\,\exp\left(
-8\int d^2x\,{|w|^2\over(1+|w|^2)^2}\bd\varphi\d\bvarphi
\right).
\label{Zqdef}
$$
Here $d\mu(\va,\vb,c)$ is the integration measure over $\va$, $\vb$ and $c$, which is invariant under translations, extensions and inversions. This measure is easy to find:
\eq$$
d\mu(\va,\vb,c)
=k^q{d^2c\over|c|^2}\prod^q_{j=1}d^2a_j\,d^2b_j\,
\prod_{i<j}|a_i-a_j|^4|b_i-b_j|^4\prod_{i,j}|a_i-b_j|^{-4}
\label{instmeasure}
$$
with a certain constant~$k$.

Evidently the integral $Z[w]$ is independent of the coupling constant $g$. Due to the contribution of ultraviolet and infrared cutoffs, this integral will not be literally invariant with respect to the conformal transformations of the parameters $a_j$, $b_j$. It will be transformed by the rule:
$$
Z[w]\to Z[w']\prod^q_{j=1}\left|da'_j\over da_j\right|^{2\alpha}
\left|db'_j\over db_j\right|^{2\alpha}
$$
with a certain $\alpha$. It follows that
$$
Z[w]\sim f(c)\prod_{i<j}|a_i-a_j|^{-4\alpha}|b_i-b_j|^{-4\alpha}
\prod_{i,j}|a_i-b_j|^{4\alpha}.
$$
Certain rather complex calculations can show that $\alpha=1/2$ and $f(c)=|c|^2/(1+|c|^2)^2$, so the integral over $c$ gives just a finite factor. If you accept this, you obtain~\cite{BP}
\eq$$
Z_q\sim{\lambda^q\over(q!)^2}\int\prod^q_{j=1}d^2a_j\,d^2b_j\,
\prod_{i<j}|a_i-a_j|^2|b_i-b_j|^2\prod_{i,j}|a_i-b_j|^{-2},
\label{Zqfin}
$$
where $\lambda\sim e^{-4\pi/g}$.

It is easy to verify that this is a plasma described by the sine\-/Gordon model with $\beta^2=1$ or the Thirring model with $g=0$, that is, a free massive fermion. This means that the instantons in the model generate a mass. Detailed calculations were performed in~\cite{FFS}.

The fault of the above reasoning is that we only took into account the instanton solutions. We can, of course, take into account the anti-instanton solutions. However, a realistic description of the model can only be when using fluctuations near functions containing both instantons and anti-instantons. Unfortunately, such a description is not yet available.

There is one important consequence of the existence of a topological number (\ref{qint}). Indeed, the action of the $O(3)$\-/models can be modified:
\eq$$
S_\theta(\bn)=S(\bn)+i\theta q
={1\over2g}\int d^2x\,(\d_\mu\bn)^2
+i{\theta\over8\pi}
\int d^2x\,\bn(\d_\mu\bn\times\d_\nu\bn)\epsilon^{\mu\nu}.
\label{thetaaction}
$$
The last term is called the {\it $\theta$\-/term}. The physical properties of the model substantially depend on $\theta$, since the partition function (and the generating functional) are written as
$$
Z=\sum^\infty_{q=-\infty}e^{i\theta q}Z_q.
$$
In particular, it is known that for $\theta=\pi$ the $O(3)$\-/model is massless, but not scale\-/invariant.

\def\refname{\normalsize Bibliography}
\begin{thebibliography}{9}

\bibitem{BP}A.~A.~Belavin, A.~M.~Polyakov, {\it Metastable states of two-dimensional isotropic ferromagnets}, JETP Letters 22 (1975) 245--247

\bibitem{FFS}V.~A.~Fateev, I.~V.~Frolov, A.~S.~Shvarts, {\it Quantum Fluctuations of Instantons in the Nonlinear Sigma Model}, Nucl.\ Phys.\ B154 (1979) 1--20

\end{thebibliography}

\noindent[Both papers are available in Russian in the collection "Instantons, Strings and Conformal Field Theory", ed.\ A.~A.~Belavin, 2002].

\bigbreak
\bigskip
\centerline{\bf Problems}
\medskip
{\bf1.}~By using the substitution (\ref{jacsubst}) obtain~(\ref{jacobian}).

{\bf2.}~Derive the identity~(\ref{intid}).

{\bf3.}~Prove the equivalence of the equations (\ref{sdcpos}), (\ref{sdcneg}) to the equations (\ref{sdpos}), (\ref{sdneg}).

{\bf4.}~Express the topological charge $q$ in terms of the function $w(z,\bz)$ in the form of two\-/dimensional integral.

{\bf5$^*$.}~Let $\bn(x)=(n_1(x),\ldots,n_N(x))$ be \emph{complex} scalar fields on the plain restricted by the condition
$$
|\bn(x)|^2\equiv\sum^N_{i=1}|n_i(x)|^2=1,
$$
and $A_\mu(x)$ is a real (co)vector field. Consider the action
$$
S[\bn,A]={1\over2g}\int d^2x\,\overline{D_\mu\bn}\cdot D_\mu\bn,
\qquad
D_\mu=\d_\mu+A_\mu.
$$
Show that the quantity
$$
q={1\over2\pi}\int d^2x\,\epsilon_{\mu\nu}\,\d_\mu A_\nu
$$
is an integer topological charge, and in each topological sector we have
$$
S\ge{\pi\over g}|q|.
$$
Find the self\-/duality equations.



\end{document}
