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\markboth{\quad \small \sl Ian I. Kogan \hfill}{\hfill\small
\sl Vortices on the World Sheet of a String: Critical Dynamics  \quad }
\setcounter{page}{2218}
\begin{document}

\title{Vortices on the World Sheet of a String{\Large :}
Critical Dynamics\,\footnote{\,{\uppercase{T}}he preprint
version of this paper was entitled ``\uppercase{V}ortices
\uppercase{O}n \uppercase{T}he \uppercase {W}orld \uppercase{S}heet
\uppercase{A}nd \uppercase{S}tring's \uppercase{C}ritical
\uppercase{D}ynamics'' -- editors' note.}}

\author{I.~I. KOGAN}


\address{Institute of Theoretical and Experimental Physics}

\maketitle

%\centerline{\small (Submitted 13 May 1987)}

\centerline{\small Pis'ma Zh. Eksp. Teor. Fiz. {\bf 45}, 556 (1987)
[\,JETP Lett.\ {\bf 45}, 709 (1987)\,]}
\centerline{\small (Submitted 13 May 1987)}
\bigskip
\abstracts{
Vortices on the world sheet of a string which is moving in a multiply 
connected space lead to a continuous dependence of the critical indices, in
particular, the critical dimensionality, on the parameters of cycles which
cannot be contracted. There exist limiting values of the latter quantities
because of a phase transition associated with a loss of conformal symmetry.
Numerous applications are duscussed.}

Let us examine a string in a formalism of summations over surfaces
\cite{Polyakov}, in which string dynamics is determined by a two-dimensional
conformal theory on a world sheet\cite{Belavin} which is propogating in a
multiply connected space $M$. For simplicity,
we will speak exclusively in terms of boson strings here. The generalization to
the case of superstrings requires some slight modifications. It will be
published separately, along with detailed explanations of the assertions made
below. A nontrivial $\pi_{1}(M)$ arises in a broad range of problems, e.g.,
(a) compactification onto tori generated by lattices, in particular, on the
maximum tori of Lie groups generated by root lattices in heterotic-string
model \cite{Greund}, where
\begin{equation}
\pi_{1}(M) = \underbrace{Z\times\cdots\times Z}_{r}\,,\nonumber
\end{equation}
where $r$ is the rank of the group; (b) compactification on multiply
connected Calabi-Yao manifolds or orbifolds \cite{Candelas}, where $\pi_1 (M)$
is a finite group, possibly non-Abelian; and (c) strings at finite
temperatures \cite{Hagedorn}, with a space which is periodic in an imaginary 
time with the period $T^{-1}\,,$ with the topology
$M\!=\!R^{D-1}\!\times S^1$,
and $\pi_1(M)=Z$\,. The radius $R$ is related to the temperature by
$R={(2\pi T)}^{-1}$. This example may also be thought of as  an extremely
simple compactification onto $S^1$.

In addition, a nontrivial $\pi_1$ arises in a description of metrics of the
black-hole type, with identified right and left worlds, against the background
of cosmological strings, and so forth. We will begin our analysis of vortices with case (c).

1. The action of the string is
\begin{equation}
S=\frac{1}{4\pi\alpha'}\int\!d^2\!z\,
\sqrt{g}\,g^{ab}(z)\,\partial_a x^{\mu}
\partial_b x^{\nu}g_{\mu\nu}(x)\,,\nonumber
\end{equation}
where $g_{ab}(z)$ and $g_{\mu\nu}(x)$ are the metrics on the world sheet and
in a nested space. In the critical dimensionality we can avoid $g_{ab}(z)$ and
obtain the action of a two-dimensional conformal theory of the fields $x^\mu $
alone. Corresponding to the compact dimension in which we are interested is
the action
\begin{eqnarray}
\varphi&\in&[\,0, 2\pi]\quad\qquad \nonumber\\[-4mm]
S=\frac{1}{4\pi\alpha'}\int\!d^2\!z\,\partial_a x\,\partial_a x=
\frac{R^2}{4\pi\alpha'}
\int\!d^2\!z\,\partial_a \varphi\,\partial_a \varphi\,;\hspace{8mm}&&\\[-4mm]
a&=&1,2\;\;\;,\qquad\qquad\nonumber
%\end{eqnarray}
\eea
which describes a continuous limit of the $XY$ model or of a planar magnet.
Berezinskii\,\cite{Berez} undertook a detailed study of this model, which was
continued by Kosterlitz and Thouless\,\cite{Koster} (the BKT model). Two phases
exist: a low-temperature phase with conformal invariance and a high-temperature
phase with exponentially decreasing correlation functions. The string familiar
to us is described by the first phase alone. The reason for the BKT phase
transition is the creation of vortices: nonperturbative fluctuations with a
logarithmically large action. The appearance of such fluctuations is not an
artifact of the regularization (on a lattice in
\mbox{Refs.\,\cite{Berez}\,and\cite{Koster}}\,).
In particular, the naive Pauli-Villars regularization does not work, since the
mass term breaks the symmetry
$\varphi\!\Rightarrow\!\varphi+2\pi n \,,$ while a potenital of the type
$M^2\!\cos\varphi $
leads to an infinite self-effect in the limit $M\!\to\infty \,$. That result is
a manifestation of the same vortex effects.

In the nonconformal phase the vortices form a two-dimensional Coulomb plasma
in which Debye screening leads to a loss of the conformal properties, while
in the conformal phase the vortices and antivortices are bound into dipoles.
The field and the action of a vortex with a charge $q$ [mapping class
$\pi_1 (S^1)\,$] are
\begin{equation}
\varphi(z)=-\frac{i}{2}\,q
\ln\!\!\left(\frac{z-z_0}{\bar{z}-\bar{z_0}}\right),
\quad S_q=\pi\beta\,q^2\ln\!\left(\frac{A}{a^2}\right),
\end{equation}
where $z=x+iy\,,\beta=R^2/4\pi\alpha^i$, $A$ is the area of the world sheet,
and $a$ is the spacing in lattice -- the ultraviolet cutoff. A phase transition
occurs when the creation of a vortex is thermodynamically favorable. Its free
energy is zero, and since the entropy of the vortex is obviously $\ln(A/a^2)$,
we can write
\begin{equation}
F_q=(\pi\beta\,q^2\!- 1)\ln\bigl(A/a^2\bigr)\,,
\;\quad |q|=1\,,
\;\quad \beta_c=\pi^{-1},
\;\quad R_c=2\sqrt{\alpha'}\,.\nonumber
\end{equation}

It is for specifically this $R_c$ that the soliton sector contains additional
massless states, while at $R<R_c$ the theory would have some additional
tachyons, so that we can see the relationship between the loss of conformal
properties on the world sheet and the appearance of additional tachyons. The
mass spectrum in string theory is ($L$ is the  soliton number, and $m$ is
the index of the momentum $p_\varphi$):
\begin{equation}
\frac{\alpha'M^2}{2}=2\,(N-1)+\frac{\alpha'm^2}{2R^2}+
\frac{L^2R^2}{2\alpha'}+mL\,;
\;\quad m,L=0\,,\pm1\,,\,\ldots\;.\label{masspec}
\end{equation}
If we now switch from $R_c$ to a critical temperature
$T_c={(2\pi R_c)}^{-1}\!$,
we find $T_c={(4\pi)}^{-1}{(\alpha')}^{-1/2}\!$. Comparing it with the limiting
temperature in string theory, i.e., the Hagedorn temperature $T_H$, which
would be given in the case of a boson string
[\refcite{Hagedorn}] by
$T_H={(2\pi)}^{-1}{[(D-2)\,\alpha'/6\,]}^{-1/2}$,
where $D$ is the dimensionality of the space-time, we find
\begin{equation}
D=26\,,\;\;T_c=T_H\,;\qquad D<26\,,\;\;T_c<T_H\,;\qquad D>26\,,\;\;T_c>T_H\,!
\nonumber
\end{equation}
The actual reason for the divergence in the Gibbs ensemble of strings is a BKT
phase transition on a world sheet and the loss of conformal symmetry!

In the conformal dipole phase, corrections to the correlation functions which
are nonperturbative in $\alpha'$ arise, and as a result there are corrections
to all the critical parameters of the two-dimensional conformal theory.
In the one-dipole approximation we have
\begin{equation}
\langle\,\varphi(z)\varphi(0)\,\rangle=
%-{(8\,\pi\beta)}^{-1}
-\,\frac{1}{8\,\pi\beta}\,
\Bigl(1+8\pi^2\frac{\beta}{\beta-\beta_c}\exp\,(-\mu\beta)\Bigr)\ln z^2 ,
\label{corfun}
\end{equation}
where $\mu$ is the minimal energy of the dipole -- the chemical potential. From
Eq.\,(\ref{corfun}) we find corrections to the coefficients in the operator
expansions which determine, for example, the mass spectrum of the
string and the cri- \linebreak[3] tical dimensionality. In the former case,
we deal with the product
%\marginpar{\small ``vortex'' or ``vertex''?}
of the ver- \linebreak[3]tex operator $V\!(z)$ and the two-dimensional
energy-momentum tensor $T(z)$
\beq
V(z)\,T(z')=\Delta_V{(z-z')}^{-2}V(z)+{(z-z')}^{-1}\partial_z V+\,\ldots\;.
\nonumber
\end{equation}
From conformal symmetry we have $\Delta_V=2\,$; finding $\Delta_V$ from
(\ref{corfun}), we then find the spectrum:
\beq
\quad
\frac{M^2\alpha'}{2}=2\,(N-1)+\frac{m^2\alpha'}{2R^2}
\Bigl(1+8\pi^2\frac{R^2}{R^2-4\alpha'}\,
\exp\,\bigl(-\mu R^2/4\pi\alpha'\bigr)\Bigr)\,.
\,\quad(\mbox{\ref{masspec}}')\nonumber
\end{equation}
From (\ref{masspec}$'$) we find a renormalization of $R_c$ due to the
dipole-dipole interaction; this renormalization has been found previously
by renormalization-group methods [\refcite{Koster},\,\refcite{Patash}\,]. The
corrections to the central charge of the Virasoro algebra are found from
\beq
T(z)T(z')=\frac{c}{2}{(z-z')}^{-4}+\,\ldots\;.\nonumber
\end{equation}
We find
\beq
c_\varphi=
{\Bigl(1+8\pi^2\frac{\beta}{\beta-\beta_c}\exp\,(-\mu\beta)\Bigr)}^2\,;
\qquad\sum_{i=1}^{D}c_i=26\,.\label{csubphi}
\end{equation}
This result tells us that the critical dimensionality deviates from its
standard value in the presence of a cycle (or cycles) which cannot be
contracted! It thus becomes possible to construct a theory in other critical
dimensionalities. Corrections to the equations of motion of massless fields
(the graviton, the dilaton, etc.) also arise; in particular,
a cosmological term which depends on the parameters, e.g., $T\,$, is generated:
\beq
\Lambda= \frac{8\pi^2}{3\alpha'}\,
\exp\bigl(-\mu/16{\pi}^3\alpha'\,T^2\bigr)\,.\nonumber
\end{equation}

2. We turn now to lattices. The interesting action in this case is
\beq
S=\beta\!\int\!\!d^2\!z\,a_{ij}\partial_a\varphi^i\partial_a\varphi^j\,;
\qquad i\,,\,j=1\,,\,\ldots\,,\,r\,,\nonumber
\end{equation}
where $a_{ij}$ is the matrix of the lattice. This model generalizes the $XY$
model to the case of several ``colors.'' It again has vortices, but in this
case the colors interact. The correlation function in this model is of the form
$\langle\,\varphi^i(z)\varphi^j(0)\,\rangle=
-(8\pi\beta)^{-1}a_{ij}^{-1}\ln z^2$. In the one-dipole approximation the
correction is proportional to $\delta_{ij}\,$, while in the two-dipole
approximation it is proportional to $a_{ij}$ (to the correlation energy of
dipoles with the colors $i$ and $j$). The higher-order corrections reduce to
either $\delta_{ij}$ or powers of $a_{ij}\,$:
\beq
\langle\,\varphi^i(z)\varphi^j(0)\,\rangle=
-(8\pi\beta)^{-1}(a_{ij}^{-1}+c_1\delta_{ij}+c_2\,a_{ij}+\,\ldots\,)\ln z^2.
\label{higherorder}
\end{equation}
It follows from (\ref{higherorder}) that only self-dual lattices with
$a_{ij}\!=a_{ij}^{-1}$ retain their structure in the face of dipole
renormalizations, and it is specifically these lattices which are used in the
model of a heterotic string! If we are studying a lattice with Lorentzian
signature, then the components which have the incorrect sign of the action
correspond to an antiferromagnet. Again in this case, nonperturbative effects
are important.

3. We conclude with a few words about compactification. In addition to the
possibility of other dimensionalities, which we have already mentioned
[see Eq.\,(\ref{csubphi})\,],
there is a mechanism for the generation of the condensates
of external fields with decreasing $R<R_c\,$. This situation gives rise to the
formation of a vortex phase described by the action \cite{Coleman}
\beq
S=\int\!d^2\!z(\beta\,\partial_a\chi\partial_a\chi+\lambda\cos\beta\chi)\,.
\nonumber
\end{equation}
If we now consider a sine-Gordon model in the external field of a tachyon, it
can contract the potential and restore conformal symmetry. In the supercase,
a Yang-Mills field will precipitate in a condensate, since the sine-Gordon
superpotential is $\bar\psi\psi\cos\beta\chi$. This is precisely the form
of the required part of the vortex operator of a Yang-Mills field. This
mechanism may explain the subsequent  breaking of gauge symmetry in the
superstring model [\refcite{Hosot},\,\refcite{Candelas}\,].

I wish to thank A.\,A.~Abrikosov (Jr.) and K.\,A.~Ter-Martirosyan for some
interesting discussions. After this paper had been prepared for publication,
A.\,A.~Tseytlin called my attention to the preprint by Sathiapalan
\cite{Sath}, who also discusses the role played by vortices for explaining
the Hagedorn temperature but does not mention a change in critical parameters.

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\end{thebibliography}

\end{document}
