%Paper: hep-th/9209011
%From: wu@mail.physics.utah.edu (yong-shi wu)
%Date: Fri, 4 Sep 92 01:05:14 GMT
%Date (revised): Mon, 7 Sep 1992 23:31:22 -0600



\input phyzzx.tex

%%%%%%%%%%%%% Here is some simple definitions. %%%%%%%%%%%%%%%%%

\def\np{Nucl. Phys.}
\def\pl{Phys. Lett.}
\def\pr{Phys. Rev.}
\def\prl{Phys. Rev. Lett.}
\def\cmp{Comm. Math. Phys.}
\def\intmp{Intern. J. Mod. Phys.}
\def\mpl{Mod. Phys. Lett.}
\def\dd{\hbox{d}}
\def\DD{\hbox{D}}

%%%%%%%%%%%% End of definitions.  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Pubnum={UU-HEP-92/6}
\date{June, 1992 }

\titlepage
\title{\bf The Complete Structure of the Cohomology Ring and Associated
Symmetries in $D=2$ String Theory}

\author{Yong-Shi Wu}
\address{Department of Physics, University of Utah
\break Salt Lake City, Utah 84112, USA}
\author{Chuan-Jie Zhu}
\address{Department of Physics, University of Utah
\break Salt Lake City, Utah 84112, USA \break
{\rm and} \break International School for Advanced Studies (SISSA/ISAS)
\break {\rm and} \break INFN, Sezione di Trieste \break
via Beirut 2-4, I-34013 Trieste, Italy\footnote*{\rm Present address.} }

\vglue .5cm

\abstract\nobreak
{We determine explicitly all structure constants of the whole
chiral BRST cohomology ring in $D=2$ string theory including both
the discrete states and tachyon states. This is made possible by
establishing several identities for Schur polynomials with operator
argument and exploring associativity. Furthermore we find that
the (chiral) symmetry algebra of the charges obtained by using the descent
equations can actually be read off from the cohomology
ring structure by simple operation involving the ghost field $b$.
We also determine the enlarged symmetry algebra which contains the
charges having ghost number $-1$ and $1$. Finally the complete symmetry
transformation rules are derived for closed string discrete states by
carefully combining the left and right sectors. It turns out that
the new states introduced recently by Witten and Zwiebach are naturally
created when symmetries act on the old states.}

\vfill\eject
\endpage

\centerline{\bf 1. Introduction}

\REF\GMA{D. J. Gross and A. A. Migdal, \prl\ {\bf 64} (1990) 717. }

\REF\GMB{M. Douglas and S. Shenker, \np\ {\bf B335} (1990) 635. }

\REF\GMC{E. Brezin and V. Kazakov, \pl\ {\bf B236} (1990) 144. }

\REF\KMA{M. Douglas, \pl\ {\bf B238} (1990) 176. }

\REF\KMB{E. Witten, \np\ {\bf B340} (1990) 281. }

\REF\PA{ A. M. Polyakov, \pl\  {\bf B180} (1981) 207.}

\REF\GTW{ A. Gupta, S. Trivedi and M. Wise, \np\ {\bf B340} (1990) 475.}

\REF\GL{ M. Goulian and M. Li, \prl\ {\bf 66} (1991) 2051.}

\REF\KITA{ Y. Kitazawa, \pl\ {\bf B265} (1991) 262; \intmp\ {\bf A7}
(1992) 3403. }

\REF\DOT{ Vl. S. Dotsenko, \mpl\ {\bf A6} (1991) 3601. }

In the last three years or so we have witnessed a spectacular success
in exactly solving certain $D\leq 2$ string theories by exploring
first matrix models [\GMA, \GMB, \GMC], and then integrable models
[\KMA] and topological
field theory [\KMB]. However, many of the results obtained in these
approaches, though explicit and exact, can hardly be given a transparent
physical interpretation. A consensus of opinion in this field has been
that by understanding these results in the more conventional Liouville
theory approach [\PA], we may be able to gain better understanding of the
underlying physics, and to develop new methods and derive new results.
Recent achievements in this regard include: 1) exact computation
of some correlation functions in Liouville theory and explicit verification
of the agreement with matrix models [\GTW, \GL, \KITA, \DOT]; 2) better
understanding of the physical states and their symmetries in the Liouville
theory for the $D\leq 2$ string theories (see below for references). In this
paper we will report the progress we have made in the second direction.

\REF\KAC{J. Goldstone (unpublished); V. G. Kac, in { \it Group Theoretical
Methods in Physics}, Lecture Notes in Physics, vol 94.}

\REF\GKN{D. J. Gross, I. R. Klebanov and M. Newman, \np\ {\bf B350}
(1991) 621. }

\REF\PAA{A. M. Polyakov, \mpl\ {\bf A6} (1991) 635.}

\REF\APA{A. M. Polyakov, ``Singular States in 2D Quantum Gravity,'' preprint
PUPT-1289 (Sept., 1991).}

\REF\LZ{B. Lian and G. Zuckerman, \pl\ {\bf B254} (1991) 417; {\bf B266}
(1991) 21. }

\REF\BMP{
P. Bouwknegt, J. McCarthy and K. Pilch, \cmp\ {\bf 145} (1992) 541. }

\REF\BMPA{
P. Bouwknegt, J. McCarthy and K. Pilch, ``BRST Analysis of Physical
States for $2D$ (Super) Gravity Coupled to (Super) Conformal Matter,''
preprint CERN-TH-6279/91 (October, 1991). }

\REF\MM{S. Mukherji, S. Mukhi and A. Sen, \pl\ {\bf B266} (1991) 337. }

\REF\IO{K. Itok and N. Ohta, ``BRST Cohomology and Physical States in 2D
Supergravity Coupled to $\hat{c}\le 1$ Matter,'' preprint
FERMILAB-PUB-91/228-T, OS-GE 20-91, Brown-HET-834 (Sept., 1991). }

\REF\WITA{E. Witten, \np\ {\bf B373} (1992) 187. }

\REF\KMS{D. Kutasov, E. Martinec and N. Seiberg, \pl\ {\bf B276} (1992) 437. }

\REF\WITZ{E. Witten and B. Zwiebach, \np\ {\bf B377} (1992) 55.  }

Among the known soluble string theories the $D=2$ model is interesting
in many aspects: It has a simple two-dimensional space-time
interpretation, incorporates the interactions between gravity and a
massless scalar ``tachyon" field, and embraces intriguing space-time
physics such as the formation and evaporation of black holes. A
by-now-well-known feature of the $D=2$ string theory is the existence
of the unusual ``discrete states", in addition to the familiar tachyons.
Some discrete states were known [\KAC] for some time. Recently they
became popular because of their appearance first in the matrix-model
calculations of Gross, Klebanov, and Newman [\GKN] and then in the
Liouville-theory analysis by Polyakov [\PAA, \APA]. The subsequent rigorous
BRST analysis [\LZ, \BMP, \BMPA] (see also [\MM, \IO]) has indicated a
more complex pattern of the discrete states than one had expected.
Besides the states of conventional ghost number one, there are also
physical states of ghost number zero and two, appearing legitimately
as non-trivial BRST cohomology classes. The study of the discrete states
and the symmetries associated to them in $D=2$ string theory could be
a good starting point for discovering the fully fledged stringy symmetries
and formulating a background independent string theory. Moreover such a study,
combined with relevant Ward identities, could lead to systematic evaluation
of correlation functions in string theory in the Liouville theory approach.

It was Witten [\WITA] who first pointed out that the ghost number zero
states generate a ``ground ring'', which is characteristic of $D=2$ string
theory. By simple considerations involving the ground ring, he was able to
explain many aspects of the free fermion description that comes from the
matrix model (see also [\KMS]). Further developments along this line of
thought have been made in [\WITZ], where a careful treatment of closed
string BRST cohomology is given and a systematic construction of symmetry
charges from the discrete states via the ``descent equations" is proposed.
One of the main points in these works is that many of the structures they
found (at the $SU(2)$ radius) can be naturally described in terms of the
differential geometry of a certain three-dimensional cone, on which the ring
of functions is isomorphic to the ground ring of the ghost-number-zero
states.

In this paper we want to study the same problems, the discrete states and
associated symmetries in $D=2$ string theory, from somewhat different point
of view, purely algebraic and more explicit. We are motivated by
the observation that all the discrete states form a (graded) ring [\WITZ]
in the following sense in terms of BRST cohomology [\WITA]. Let $V_1(z)$
and $V_2(z)$ be two (chiral) BRST invariant operators. The operator product
$V_1(z)V_2(w)$ is also BRST invariant, so all the terms in its short distance
expansion for $z\to w$ are BRST invariant. Negative powers of $z-w$ may arise
in this short distance expansion, but the operators multiplying the negative
powers of $z-w$ are of negative dimension and thus must be BRST commutators
(as there is no nontrivial BRST cohomology class at negative dimensions).
Therefore, modulo the BRST commutators, the short distance limit of
$V_1(z)V_2(w)$ is some BRST invariant operator $V_3(w)$ (which may be zero):
$$V_1(z)V_2(w) \sim V_3(w) + \{ Q, \cdots\} . \eqn\qaq$$
This gives rise to the desired multiplication law, $V_1 \cdot V_2 = V_3$.
This procedure obviously defines an associative ring structure for the
set of all BRST cohomology classes, which is graded by ghost number.
Clearly the ground ring is only a subring of this ring, which we will call
the (chiral) BRST cohomology ring.

Instead of expressing the structure of the whole cohomology ring in terms
of the ground (sub)ring in differential geometric language [\WITA, \WITZ],
we want to directly attack the problem of explicitly calculating all the
structure constants of the whole chiral BRST cohomology ring in $D=2$
string theory, including both the discrete and tachyon states. This is
possible, since an explicit representation for all discrete states is
available in the literature [\BMPA], which involves Schur polynomials
with operator argument. Starting from this representation and using
several identities we have derived to rearrange Schur polynomials, we
found that the BRST cohomology ring is computable and have obtained explicit
results for the structure constants. We have been able to include the
tachyon states in our computation and find that the product of two negative
parity tachyon states can give rise to a discrete state of ghost number two,
or $P_{j, m}$ in the notation of [\WITZ]. This again shows the
incompleteness of the physical states of standard ghost number one.

With the structure constants of the (chiral) BRST cohomology ring
explicitly known, it is not too hard to derive the structure constants
for the associated (chiral) symmetry charge algebra or the transformation
rules for the discrete states. In doing so, the general procedure, proposed
in ref. [\WITZ], of constructing conserved charges from the discrete states
via descent equations is extremely powerful. Our results show that the
(chiral) symmetry charge algebra can be read off from the cohomology ring
structure by simple operation involving the ghost field $b$. In combining
the right and left (chiral) cohomology classes to obtain closed string
BRST cohomology, we have to pay attention to the subtleties pointed out
by Witten and Zwiebach [\WITZ], which allow the existence of more closed
string discrete states and associated symmetries than what had been
recognized before from naive combination of chiral BRST cohomologies.
With these subtleties taken into account, we have determined the
transformation rules in closed string theory. The rules obtained show
that the new states introduced by Witten and Zwiebach [\WITZ] are
naturally created when symmetry charges act on the old states.
Since the cohomology ring structure
implies the charge algebra but the converse is not true, we can say that
the cohomology ring is more fundamental than the symmetry charge algebra.

This paper is organized as follows. In the next section we review
the previous results about the classification of physical
states in the $D=2$ string theory at the $SU(2)$ point (with a vanishing
cosmological constant). In particular we present an explicit
representation of all the (chiral) discrete states in terms of BRST
invariant local operators [\WITA, \WITZ]. A very useful rearrangement
formula for $P_{j, m}$ is announced here and its proof will be deferred
to sec. 3.

\REF\PAK{I. Klebanov and A. M. Polyakov, \mpl\ {\bf A6} (1991) 3273. }

\REF\LI{ M. Li, ``Correlators of Special States in $c= 1$ Liouville Theory'',
preprint UCSBTH-91-47 (October, 1991).}

\REF\TAN{Y. Matsumura, N. Sakai and Y. Tanii, ``Coupling of Tachyons and
Discrete States in $c=1$ 2-D Gravity,'' preprint TIT/HE-186 or STUPP-92-124
(Jan., 1992); ``Interaction of Tachyons and Discrete States in $c=1$ 2-D
Quantum Gravity,'' preprint TIT/HE-187 or STUPP-124-125 (Jan., 1992).  }

In sec. 3 we explicitly determine all structure constants of the chiral BRST
cohomology ring. Part of the ring structure was already known [\WITA,
\PAK, \LI]. The new results are derived by using several useful identities
involving Schur polynomials with operator arguments, which are established
by us and proved in appendix A, and by exploring associativity. In this
section we will present some sample calculations and give the proof
of the rearrangement formula for $P_{j, m}$. Our new results include
the structure constants involving higher ghost number states, like
$O_{j_1m_1}P_{j_2m_2}$ and $Y^+_{j_1m_1}P_{j_2m_2}$ in the notation of
[\WITZ]. In particular, by explicit calculation we find
$Y^-_{j_1 m_1} Y^-_{j_2 m_2}$ is zero.

The complete chiral cohomology ring involves also the tachyon states. The
structure constants involving tachyon states are derived in sec. 4
and are compared with previous results [\TAN] about the OPE and couplings
of tachyons and discrete states in $D=2$ string theory. Amazingly we find
$T^{(-)}_pT^{(-)}_q$ is generally non-zero. Of particular interest is that
this product gives rise to a term proportional to $P_{jm}$ in addition
to the usual term $aY^-_{j+1, m}$.

In sec. 5 we first explain how one can construct conserved charges from
BRST invariant local operators. The procedure is just part of the general
procedure of constructing conserved charges via descent equations
(taking only one of the chiral sectors). Then we derive some general
formulas of getting transformation rules from the ring multiplication
laws. The complete enlarged (chiral) charge
algebra is explicitly computed.

In sec. 6 we derive the symmetry transformation rules for closed
string discrete states by combining the left and right sectors and using
the results of sec. 5. Here the relevant point is to show that the new
closed string states introduced by Witten and Zwiebach [\WITZ] through
the semi-relative cohomology are naturally created when symmetries act
on the old states. The sample calculations presented in this section
suggest some very simple relations involving the multiplication of the
operator $(a+\bar{a})$. The proof of these relations and also some
details about the calculations are presented in appendix C.

For the convenience of reader, this paper contains three appendices:
Appendix A collects relevant formulas for Schur polynomials and presents
the proof for our newly established identities which are useful in the
text. Appendix B presents the complete chiral transformation rules and the
chiral symmetry algebra by using the  chiral BRST cohomology ring structure
equations (sec. 3) and the general formulas derived in sec. 5.
Appendic C gives  some  details for the calculations of the
transformation rules in closed string theory and also the proof of two
general results in this regard in sec. 6.

\vfill\eject

\centerline{\bf 2. Representation of the Chiral Discrete States}

Like in refs. [\WITA, \WITZ], we consider the $D=2$ string theory at the
$SU(2)$ radius with vanishing world-sheet cosmological constant. We
will follow the notation of [\WITA, \WITZ] as much as possible. The
basic fields appearing in the construction of the physical states are
the world-sheet ``matter" field $X$, the Liouville field $\phi$
and the ghost fields $b$ and $c$. At the $SU(2)$ radius and vanishing
world-sheet cosmological
constant, both the fields $X$ and $\phi$ are free fields, and the
right and left movers are decoupled. Thus we may start with one (say,
the right one) of the chiral sectors. The OPE's for the basic fields are
$$\eqalign{
& X(z)X(w) \sim -\ln (z-w), \cr
& \phi(z) \phi(w) \sim -\ln (z-w), \cr
& b(z)c(w) \sim { 1\over z-w} .
} \eqn\xpb$$
The stress energy tensors are
$$ \eqalign{
& T_X = -{ 1\over 2} (\partial_z X)^2, \cr
& T_{\phi} = -{ 1\over 2} (\partial_z \phi)^2 +\sqrt{2} \partial^2_z\phi, \cr
& T_{bc} = 2 (\partial_z c) b + c\partial_z b.
} \eqn\str$$
In the following we also use the light-cone fields $X^{\pm} ={1\over \sqrt{2}}
(X\pm i\phi)$. The combined matter-Liouville stress energy tensor is
$$ T = T_X + T_{\phi} = -\partial_z X^+ \partial_z X^- - i(\partial^2_z X^+
 - \partial^2_z X^-) . \eqn\tt$$
The central charge of $T$ is 26 and we have the following nilpotent  BRST
charge
$$Q = \oint_0 [\dd z] :c(z) (T(z) + \partial_z c(z) b(z)) :  , \eqn\qq$$
where $[\dd z] \equiv { \dd z \over 2\pi i }$. Here and in what follows the
normal ordering for the ghost fields is such that the following is satisfied:
$$ b(z) c(w) = { 1\over z-w} + :b(z) c(w):  . \eqn\norbc$$

According to the general principle of BRST quantization, the physical states
are defined as $Q$-closed but not $Q$-exact states, i.e. $Q$-cohomology
classes.
By a one to one correspondence between states and fields, one can also define
the states in terms of local field operators. We will use the local field
operator representation exclusively in this paper so that a physical state is
represented as a BRST invariant but not exact local operator.

Before presenting the explicit representation for all the discrete states, a
technical remark is in order. It is easy to see that from the matter field
$X(z)$ one can construct an $SU(2)$ Kac-Moody algebra:
$$\eqalign{
J_{\pm}(z) = e^{\pm i\sqrt{2} X(z) } ,~~~
J_3(z) = { i\over \sqrt{ 2} } \partial_z X(z),
} \eqn\kac$$
with the following OPE's
$$\eqalign{
& J_+(z) J_-(w) \sim { 1\over (z-w)^2} + { 2\over z-w} J_3(w) ,\cr
& J_3(z) J_{\pm}(w) \sim \pm { 1\over z-w} J_{\pm} (w) , \cr
& J_3(z) J_3(w) \sim { 1/2 \over (z-w)^2} ,
} \eqn\kacc$$
Of particular importance is the fact that the zero modes of $J_i$,
$$\hat{J} _{\pm, 3} =\oint_0  [\dd z] J_{\pm, 3}(z), \eqn\zero$$
commutes with $Q$: $[Q, \hat{J}_{\pm, 3} ] = 0$. Thus all
the physical states should form $SU(2)$ multiplets. In fact all the discrete
physical states do fall into  finite dimensional multiplets. If there exists
a physical states $V_{j, j}(w)$ such that
$$\eqalign{
& [\hat{J}_+, V_{j, j} (w)] = 0, \cr
& [\hat{J}_3, V_{j, j} (w)] = j V_{j, j} (w),
} \eqn\deffj$$
one can get the remaining  states in the multiplet by using the
$\hat{J}_-$ operator
$$ \eqalign{
V_{j, m} (w) & =
\left( { (j+m)!\over (2j)!(j-m)!} \right)^{1/2}
\underbrace{[\hat{J}_-, [\hat{J}_-, \cdots, [\hat{J}_-}_{j-m},
V_{j, j}(w)\underbrace{]\cdots ]}_{j-m}\cr
&\equiv
\left( { (j+m)!\over (2j)!(j-m)!} \right)^{1/2}
(\hat{\hat{J}}_-)^{j-m} V_{j, j}(w), } \eqn\ddii$$
where
$$ \hat{\hat{J}}_- V_{j, m-1} =\oint_w[\dd z] J_-(z) V_{j, m} (w).
\eqn\raise$$

After all these preliminaries we can now summarize the results of
[\LZ, \BMP, \BMPA] in compact form. All the physical states are
divided into two categories: relative physical states that are
annihilated by $b_0$ and absolute physical states that are not
annihilated by $b_0$. Apart from the tachyon states (see sec. 4),
the other relative physical states have discrete momenta $(p_X,
p_\phi)$ and are created by following local vertex operators:
$$\eqalign{
& O_{j, m} =
\left( { (j+m)!\over (2j)!(j-m)!} \right)^{1/2}
(\hat{\hat{J}}_-)^{j-m} O_{j, j} , \cr
& O_{j, j} = :\left( -S_{2j}(-iX^-) + \sum_{q=1}^{2j} S_{2j-q}(-iX^-)
{ c\partial^{q-1} b \over (q-1)! } \right) e^{ 2ijX^+} :, \cr
& Y^{\pm}_{j, m}  = c:
\left( { (j+m)!\over (2j)!(j-m)!} \right)^{1/2}
(\hat{\hat{J}}_-)^{j-m}
e^{ (ij X +(1\mp j)\phi)\sqrt{2} } :, \cr
& P_{j, m} =
\left( { (j+m)!\over (2j)!(j-m)!} \right)^{1/2}
(\hat{\hat{J}}_-)^{j-m} P_{j, j} , \cr
& P_{j, j} =
\sum_{q = 1}^{2j + 1}: S_{2j+1-q}(-iX^+ - iX^-) {\partial^{q+1}
c\over (q+1)! } ce^{ -2iX^+ +2i(j+1)X^-}:  ,
} \eqn\res$$
where $m= -j, -j+1, \cdots, j$ and $j = 0, 1/2, 1, 3/2, \cdots$.
In the above equation $S_k(-iX^-)$ is a Schur polynomial of degree $k$
with the operator $-iX^-$ as its argument. We refer the reader to
appendix A for our notation and relevant properties of the Schur
polynomials. In Fig. 1 we show the $(p_X, p_\phi)$ momenta of these
discrete states. According to the type of Liouville dressing,
$O_{j,m}$ and $Y^+_{j,m}$ are called the ``plus'' states,  and
$Y^-_{j,m}$ and $P_{j,m}$ are called the ``minus'' states. Note that
the states $O_{j,m}$, $Y^{\pm}_{j,m}$ and $P_{j,m}$ have ghost number
$0$, $1$ and  $2$ respectively.

A remark is in order about the representation of $P_{j, j}$ (and all the
$P_{j, m}$'s by $SU(2)$ action). According to [\BMP], the non-triviality of
$P_{j, j}$ could be proved by changing the prefactor to a representation
in terms of only the light-cone variable $X^+$ by adding some BRST exact
terms. We find that it is quite difficult to prove the
non-triviality of $P_{j, j}$ with the representation given in $\res$.
In the process of seeking a general procedure of handling this rearrangement
problem we have found the following general representation for $P_{j, j}$:
$$ \eqalign{
P_{j, j} & \equiv
\sum_{q = 1}^{2j + 1} S_{2j+1-q}(-iX^+ - iX^-) { \partial^{q+1}
c\over (q+1)! }c e^{ -2iX^+ +2i(j+1)X^-}   \cr
& = { \Gamma(2j+2)\Gamma(\delta+1) \over \Gamma(2j+1+\delta) }
\sum_{q = 1}^{2j + 1} S_{2j+1-q}(-iX^+ - i\delta X^-) { \partial^{q+1}
c\over (q+1)! } ce^{ -2iX^+ +2i(j+1)X^-} \cr
& \equiv { \Gamma(2j+2)\Gamma(\delta+1) \over \Gamma(2j+1+\delta) }
 P_{j, j}(\delta) .  } \eqn\newf$$
with $\delta$ an arbitrary real number. Later we will see the usefulness of
this formula. It will be proved in the next section.

As for the absolute physical states, they can be obtained from the relative
physical states by ``multiplying'' the following local operator [\WITZ, \LI]
$$ a = {1\over \sqrt{2} } [Q, \phi] = \partial c +{1\over \sqrt{2} }c
\partial \phi, \eqn\aaa$$
In the usual sense $\phi$ is not a conformal field, but $a$ is. Obviously
$a$ is BRST invariant and is not BRST exact in the usual space of conformal
fields. This being so,
we can get absolute physical states from the relative ones by
forming the local product  with $a$, i.e.
$$(aV_{jm})(w) = \oint_w[\dd z] { 1\over z-w} a(z) V_{jm}(w). \eqn\defab$$
The local multiplication with $a$ commutes with the $SU(2)$ action.
Thus, $(aV_{j,m})$, or simply $aV_{j,m}$ when there is no confusion,
has the same $SU(2)$ quantum numbers, as well as the same $(p_X, p_\phi)$
momenta, as $V_{j, m}$. This way of forming an absolute physical state
can be viewed as the multiplication law for the BRST invariant operator $a$
and a relative physical state in the ring. This makes it easy to derive
the ring multiplication involving absolute physical states from that of
relative physical operators, as exemplified in the following:
$$\eqalign{
& \qquad O_{j_1m_1} O_{j_2m_2} = O_{j_1+ j_2, m_1+ m_2} \to \cr
& aO_{j_1m_1} O_{j_2m_2} = O_{j_1m_1} aO_{j_2m_2}
= aO_{j_1+ j_2, m_1+ m_2}.}
\eqn\oooa$$
Note that the multiplication of $a$ with itself gives zero. So only the
ring structure of the relative physical states will be computed.

\vfill
\eject

\centerline{ \bf 3. Cohomology Ring of the Discrete Chiral States}

First let us summarize the known results on the explicit ring structure
of the discrete chiral states:

1) Witten [\WITA] derived the chiral ground ring as the ring of
polynomial functions in $x\equiv O_{1/2, 1/2}$ and $y\equiv O_{1/2, -1/2}$.
This implies that the operators $O_{jm}$ are closed under multiplication:
$$ O_{j_1m_1} O_{j_2m_2} =
\langle j_1m_1j_2m_2\mid j_1+j_2, m_1+m_2\rangle
O_{j_1+ j_2, m_1+ m_2}, \eqn\ground$$
up to normalization.  We will prove this result by explicit calculation
and find the rescaling of $O_{jm}$ such that all the structure
constants here become unity.( Another, perhaps more natural, normalization
will be presented at the end of this section.)

2) Polyakov and Klebanov [\PAA] derived the following OPE
($Y^{\pm}_{jm} = c\Psi^{(\pm)}_{jm}$):
$$\eqalign{
& \Psi^{(+)}_{j_1m_1}(z) \Psi^{(+)}_{j_2m_2}(w) \sim \cdots + { 1\over z-w}
(j_2m_1-j_1m_2)\Psi^{(+)}_{j_1+j_2-1, m_1+m_2}(w), \cr
&  \Psi^{(-)}_{j_1m_1}(z) \Psi^{(-)}_{j_2m_2}(w) \sim \cdots + { 1\over z-w}
\times 0, \cr
&   \Psi^{(+)}_{j_1m_1}(z) \Psi^{(-)}_{j_2m_2}(w) \sim \cdots + { 1\over z-w}
\times 0, \qquad j_1\ge j_2+1, \cr
& \Psi^{(+)}_{j_1m_1}(z) \Psi^{(-)}_{j_1+j_2-1, -m_1-m_2}(w) \sim \cdots -
{ 1\over z-w} (j_2m_1-j_1m_2)\Psi^{(-)}_{j_2, -m_2}(w).
} \eqn\poly$$
This gives a term $aY^{\pm}_{j_2\pm(j_1-1), m_1+m_2}$ in the product
$Y^{\pm}_{j_1m_1}Y^{\pm}_{j_2m_2}$. However, this is only part of the
multiplication law.  As we will see later, in addition, there is another
term proportional to $P_{j_2-j_1, m_1+m_2}$ in the product of $Y^+_{j_1m_1}
Y^-_{j_2m_2}$.  This is
due to the higher pole terms in $\poly$ which were not computed explicitly.

3) Li [\LI] obtained the following multiplication law:
$$O_{j_1m_1} Y^+_{j_2m_2} = {j_2\over j_1+j_2} Y^+_{j_1+j_2, m_1+m_2}
- {1\over j_1+j_2} (j_2m_1-j_1m_2)
aO_{j_1+j_2-1, m_1+m_2}. \eqn\lieq$$
It was obtained partly by explicit calculation and partly by using the fact
that $Y^+_{jm} $ acts on the ground ring as a differential operator [\WITA].
According to our experience the second term in $\lieq$ can be derived
from the associativity of the ring. We will explore associativity
in several places either to check our results or to derive the ring structure
constants which are not explicitly calculable.

To begin with, let us first compute $O_{j_1m_1}O_{j_2m_2}$. Counting the
momenta and ghost number, we know that this product must
be proportional to $O_{j_1+j_2, m_1+m_2}$:
$$ O_{j_1m_1}O_{j_2m_2}= g(j_1, j_2)
\langle j_1m_1j_2m_2\mid j_1+j_2, m_1+m_2\rangle
O_{j_1+j_2, m_1+m_2}, \eqn\ooo$$
where $\langle j_1m_1j_2m_2\mid j_1+j_2, m_1+m_2\rangle$ are the
Clebsch-Gordan coefficients.
To determine the unknown function $g(j_1, j_2)$, one may consider
some special values of $m_1$ and $m_2$ such that a
direct calculation is possible. Let us take $m_1=j_1$, $m_2=j_2$ and
$j_1= {1\over 2}$. The general case can be reached by induction.
We have\footnote{\S}{Some useful formulas about the OPEs
involving Schur polynomials are given in the appendix A. }
$$ O_{1/2,1/2}O_{j, j}(w) = \oint_w[\dd z] { 1\over z-w}
O_{1/2,1/2}(z) O_{j, j} (w),
\eqn\aaaa$$
with
$$\eqalign{
  O_{1/2,1/2}&(z) O_{j, j} (w)= :(c(z)b(z) +i\partial_zX^-(z)) e^{iX^+(z)} :\cr
&  \times :(-S_{2j} (-iX^-(w)) +
\sum_{q=1}^{2j}S_{2j-q}(-iX^-(w)) { c(w)\partial_w^{q-1}
b(w)\over (q-1)! } )e^{2ijX^+(w)}:   \cr
 = & :\left\{ -\left[ c(z)b(z)+i\partial_z X^-(z) + 2j{ 1\over z-w} \right]
\sum_{k=0}^{2j} S_{2j-k} (-iX^-(w)){ 1\over (z-w)^k } \right. \cr
&  + \sum_{q=1}^{2j}\left[(c(z)b(z) +i\partial_z X^-(z)+ 2j{ 1\over z-w} )
{ c(w)\partial_w^{q-1} b(w) \over (q-1)! }
\right.\cr
& \left.
+ { c(z)\partial_w^{q-1}b(w) \over (q-1)! } { 1\over z-w}
+ b(z) c(w) { 1\over (z-w)^q}
+ { 1\over (z-w)^{q+1} } \right]\cr
&  \left. \times \sum_{k=0}^{2j-q} S_{2j-q-k}
(-iX^-(w)) { 1\over (z-w)^k } \right\} e^{ iX^+(z) + 2ijX^+(w) } : .
} \eqn\expan$$
Let us compute the various terms appearing in $\expan$. First one can show
that the term containing four ghost fields is zero:\footnote*{The
exponential factor $  e^{ iX^+(z)+2ijX^+(w)}$ is suppressed.}
$$ \eqalign{
\sum_{q=1}^{2j} & :c(z)b(z) c(w) { \partial_w^{q-1} b(w) \over (q-1)!}
\sum_{k=0}^{2j-q}S_{2j-q-k} (-iX^-(w)) { 1\over (z-w)^k }
%e^{ iX^+(z)+2ijX^+(w)}
: \cr
& = \sum_{k=1}^{2j} :S_{2j-k}(-iX^-(w)) { 1\over (z-w) ^{k-1} } c(z)b(z)c(w)
\sum_{q=1}^k (z-w)^{q-1} { \partial_w^{q-1} b(w) \over (q-1)! }
% e^{ iX^+(z)+2ijX^+(w)}
: \cr
& = \sum_{k=1}^{2j} :S_{2j-k}(-iX^-(w)) { 1\over (z-w) ^{k-1} } c(z)b(z)c(w)
(b(z) + o( (z-w)^k ) ) : \cr
% e^{ iX^+(z)+2ijX^+(w)}
& = o(z-w) .
} \eqn\oko$$
The integration over $z$ then gives vanishing contribution. For the terms
containing no ghost field, we have
$$ \eqalign{
& \left[ -(i\partial_zX^-(z) + 2j{ 1\over z-w} )
\sum_{k=0}^{2j} S_{2j-k} (-iX^-(w)){ 1\over (z-w)^k }\right. \cr
&\qquad\qquad\qquad\quad
\left.+ \sum_{q=1}^{2j} \sum_{k=0}^{2j-q} S_{2j-q-k} (-iX^-(w))
{ 1\over (z-w)^{q+k-1}  } \right]
% e^{ iX^+(z) + 2ijX^+(w) }
\cr
& =  \left[  \sum_{k=0}^{2j} (k-2j) S_{2j-k} (-iX^-(w)){ 1\over (z-w)^{k+1}
 } + \sum_{k=0}^{2j} { 1\over (z-w)^{k}  }P(k)  + o(z-w) \right]
% e^{ iX^+(z) + 2ijX^+(w) }
 ,
} \eqn\okoa$$
where
$$P(k) = \sum_{l=k}^{2j} S_{2j-l} (-iX^-(w))
\Big(-i { \partial_w^{l-k+1}X^-(w) \over (l-k)! } \Big) . \eqn\okob$$
By using the last equation in appendix A we find that all the singular terms
in $\okoa$ are zero and the regular term $P(0)$ is $(2j+1)S_{2j+1} (-i
X^-(w) )$. A similar but a bit more tedious calculation of the remaining
terms (i.e. those containing two ghost fields) gives
$$ -(2j+1)\sum_{q=1}^{2j+1}: S_{2j+1-q} (-iX^-(w)) { c(w)\partial_w^{q-1}b(w)
\over (q-1)! }  e^{ iX^+(z) + 2ijX^+(w) } : +o(z-w). \eqn\okoc$$
It is remarkable that all singular terms in the OPE of
$O_{1/2, 1/2}(z)$ and $O_{j, j}(w)$ are zero exactly
without any BRST exact terms.

Combining the above results, we have
$$ O_{1/2, 1/2} O_{j, j} = -(2j+1) O_{j+1/2, j+1/2} . \eqn\okod$$
Using it recursively one obtains
$$ O_{j_1j_1} O_{j_2, j_2} = -{ (2(j_1+j_2))!\over (2j_1)!(2j_2)! }
O_{j_1+j_2, j_1+j_2} . \eqn\okoe$$
Recalling the explicit expression for the relevant Clebsch-Gordan coefficients
here:
$$\eqalign{
\langle j_1m_1j_2m_2&\mid jm\rangle \mid_{j=j_1+j_2, m=m_1+m_2}\cr
& = \left[
{ (2j_1)!\over (j_1+m_1)!(j_1-m_1)! }
{ (2j_2)!\over (j_2+m_2)!(j_2-m_2)! }
{ (j+m)!(j-m)! \over (2j)! } \right]^{1/2} , } \eqn\okof$$
and rescaling $O_{jm}$ to\footnote*{It is not quite natural to scale the
vertex operator $O_{jm}$'s (also $Y^{\pm}_{jm}$ and $P_{jm}$'s) by an $m$
dependent factor which spoil the $SU(2)$ structure among them. This
normalization (for $Y^{\pm}_{jm}$'s) was used in [\PAK] and is quite useful
as to make all the formulas explicit and simple. We will give the structure
equations in their $SU(2)$ covariant form at the end of this section. }
$$ -(2j)! \left[ { (j+m)!(j-m)!\over (2j)! }\right]^{1/2} O_{j, m}~~ ,
\eqn\okog$$
we get the following ring multiplication law:
$$O_{j_1m_1} O_{j_2m_2} = O_{j_1+j_2, m_1+m_2} . \eqn\okoh$$

As the second example, we derive the multiplication law for
$Y^+_{j_1m_1}Y^+_{j_2m_2}$ from the first equation of $\poly$ as follows.
First by momentum and ghost number counting we conclude that
$$Y^+_{j_1m_1}Y^+_{j_2m_2} = g(j_1, j_2)
\langle j_1m_1j_2m_2\mid j_1+j_2-1, m_1+m_2\rangle
aY^+_{j_1+j_2-1, m_1+m_2} . \eqn\poa$$
In order to determine the function $g(j_1, j_2)$ we consider the
special value $m_1=j_1$ and $m_2= j_2-1$. In this special case the higher
order pole terms in the first equation of $\poly$ vanish: Because
$Y^+_{jm}(z)=c(z)\Psi^{(+)}_{jm}(z)$, and $:c(z)c(w):\mid_{z=w}=0$,
the single pole term is the only term contributing to
$Y^+_{j_1m_1}Y^+_{j_2m_2}$. Noticing that
$aY^+_{jm} = j :\partial_zc(z)Y^+_{jm}(z):$
and rescaling $Y^+_{jm}$ as $\Psi^{(+)}_{jm}$ in [\PAA] to
$$ \sqrt{j/2} (2j-1)!\left[ { (j+m)!(j-m)!\over (2j-1)! } \right]^{1/2}
Y^+_{jm} , \eqn\poc$$
we get the following ring multiplication law by making use of $\poly$:
$$Y^+_{j_1m_1}Y^+_{j_2m_2} = {1\over j_1+j_2-1} (m_1j_2-m_2j_1)
 aY^+_{j_1+j_2-1, m_1+m_2}.\eqn\poaa$$

The other ring multiplication law involving the plus states only is
$O_{j_1m_1}Y^+_{j_2m_2}$.
By momentum and ghost number counting we have
$$\eqalign{ O_{j_1m_1}Y^+_{j_2m_2}= & g_1(j_1, j_2)
\langle j_1m_1j_2m_2\mid j_1+j_2, m_1+m_2\rangle Y^+_{j_1+j_2, m_1+m_2} \cr
& + g_2(j_1, j_2)\langle j_1m_1j_2m_2\mid j_1+j_2-1, m_1+m_2\rangle
aO_{j_1+j_2-1, m_1+m_2} .
} \eqn\poaf$$
The unknown function $g_1(j_1, j_2)$ can easily  be computed by setting
$m_1= j_1$ and $m_2=j_2$ because the second term then vanishes. After
appropriate rescaling as in $\okog$ and $\poc$ we fnd
$$O_{j_1m_1}Y^+_{j_2m_2}= {j_2\over j_1+j_2} Y^+_{j_1+j_2, m_1+m_2}
+ \tilde{g}_2(j_1,j_2)(m_1j_2-m_2j_1)aO_{j_1+j_2-1, m_1+m_2} .\eqn\xxx$$
To compute $\tilde{g}_2(j_1, j_2)$, we explore the associativity of the
ring multiplication as follows.
Multipling both sides of $\xxx$ by $Y^+_{j_2m_3}$ from the right,
$$\eqalign{
(O_{j_1m_1}Y^+_{j_2m_2})Y^+_{j_3m_3} = &{j_2\over j_1+j_2}
Y^+_{j_1+j_2, m_1+m_2}Y^+_{j_3m_3}  \cr
& + \tilde{g}_2(j_1,j_2)(m_1j_2-m_2j_1)
aO_{j_1+j_2-1, m_1+m_2}Y^+_{j_3m_3}  .
} \eqn\axxx$$
On one hand, we have
$$\eqalign{
aO_{j_1+j_2-1, m_1+m_2}Y^+_{j_3m_3}  = &
a( O_{j_1+j_2-1, m_1+m_2}Y^+_{j_3m_3}) \cr
=& { j_3 \over j_1+j_2+ j_3 -1} aY^+_{j_1+j_2+j_3-1, m_1+m_2+m_3} ,
} \eqn\axxxa$$
because of $a^2= 0$. On the other hand, for the LHS of $\axxx$ we have
$$\eqalign{
(O_{j_1m_1}Y^+_{j_2m_2})Y^+_{j_3m_3} =& O_{j_1m_1} (Y^+_{j_2m_2}
Y^+_{j_3m_3})  \cr
=& O_{j_1m_1} \Big( (m_2j_3 - m_3j_2) { 1\over j_2+ j_3 -1}
aY^+_{j_2+j_3-1, m_2+m_3} \Big) \cr
=&  (m_2j_3 - m_3j_2) { 1\over j_2+ j_3 -1} a(O_{j_1m_1} Y^+_{j_2+j_3-1,
m_2+m_3}) \cr
=&  (m_2j_3 - m_3j_2) { 1\over j_1 + j_2+ j_3 -1}
aY^+_{j_1+j_2+j_3-1, m_1+m_2+m_3} .
} \eqn\pad$$
Substituting $\axxxa$ and $\pad$ into $\axxx$ we get
$\tilde{g}_2(j_1, j_2) = - 1/ (j_1+j_2)$.

To compute the ring structure constants involving the minus states like
$Y^-_{jm}$, one can not simply use $\poly$ because of the higher order
pole terms. Their existence can be seen from the product
$Y^-_{j_1m_1}Y^-_{j_2m_2}$. By momentum and ghost number counting we have
$$ Y^-_{j_1m_1}Y^-_{j_2m_2} =g(j_1, j_2)
\langle j_1m_1j_2m_2\mid j_1+j_2, m_1+m_2\rangle P_{j_1+j_2, m_1+m_2} .
\eqn\new$$
Setting $m_1=j_1$ and $m_2=j_2$ we get
$$ \eqalign{
 Y^-_{j_1j_1} Y^-_{j_2j_2}(w)= &\oint_w[\dd z]
:c(z)e^{-iX^+(z)+i(2j_1+1)X^-(z)} :
:c(w)e^{-iX^-(w)+i(2j_2+1)X^-(w)} : \cr
 =&\sum_{q=0}^{2(j_1+j_2)+1} :{ \partial^{q+1}c \over (q+1)!} c
S_{2(j_1+j_2)+1-q}(-iX^++i(2j_1+1)X^-)  \cr
& \times e^{-2iX^++2i(j_1+j_2+1)X^- }  : .
} \eqn\newa$$
 From $\new$ we see that there should be  no $aY^-_{j_1+j_1+1, j_1+j_2}$
term in $\newa$. So the $q=0$ term in $\newa$ must be BRST equivalent to
$P_{j_1+j_2, j_1+j_2}$ (up to a proportional constant). In fact one can
prove that
$cS_{2j+1}(-iX^+-i\delta X^-) e^{-2iX^+ + 2i(j+1)X^- } $
is BRST invariant for any $\delta$. For $\delta=-1$, $-2$, $\cdots$, $-(2j+1)$,
it is actually BRST exact. We have (modulo BRST exact terms)
$$cS_{2j+1}(-iX^+-i\delta X^-) e^{-2iX^+ + 2i(j+1)X^- }  = { \Gamma(2j+2+
\delta ) \over (2j+2)! \Gamma(\delta + 1) } \sqrt{ 2(j+1)}
Y^+_{j+1, j}. \eqn\newb$$
This can easily be proved by using the raising operator $\hat{\hat{T}}_+$. The
action of $a$ on
$cS_{2j+1}(-iX^+-i\delta X^-) e^{-2iX^+ + 2i(j+1)X^- } $
gives the following:
$$\eqalign{
a\Big(  cS_{2j+1}&(-iX^+-i\delta X^-) e^{-2iX^+ + 2i(j+1)X^- }\Big) \cr
 = & \Big\{
{1\over 2}(\delta -1)
\sum_{q=1}^{2j+1} {\partial^{q+1}c\over (q+1)! } c
S_{2j+1-q}(-iX^+-i\delta X^-) \cr
&  -(j+1) \partial c c S_{2j+1}(-iX^+-i\delta X^-)
\Big\} e^{-2iX^++ 2i(j+1)X^- } .
} \eqn\newc$$
By using the above formula in $\newa$ ($\delta= -(2j_1+1)$ and $j=j_1+j_2$)
we get then
$$ \eqalign{Y^-_{j_1j_1} Y^-_{j_2j_2}  = &- {j_2\over j_1+j_2+1}
\sum_{q=1}^{2(j_1+j_2)+1} {\partial^{q+1}c\over (q+1)! } c \cr
& \times
S_{2(j_1+j_2)+1-q}(-iX^+-i(2j_1+1) X^-) e^{-2iX^++ 2i(j+1)X^- } .} \eqn\newd$$
This vanishes by making use of $\newf$. Recalling $\new$, we have
$$ Y^-_{j_1m_1}Y^-_{j_2m_2}= 0, \eqn\newe$$

There is little difficulty to extend the above calculation to determine the
ring multiplication law for $Y^+_{j_1m_1} Y^-_{j_2m_2}$. The result is
$$\eqalign{
Y^+_{j_1m_1}Y^-_{j_2m_2} = & {j_1\over j_2-j_1+1} P_{j_2-j_1, m_1+m_2} \cr
& +  { 1\over j_2-j_1+1} (m_1j_2+m_2j_1+m_1) aY^-_{j_2-j_1+1, m_1+m_2},
} \eqn\xaa$$
if we rescale $P_{jm}$ to
$$(-1)^{j+m} { 1\over (2j+1)!} \left[ { (2j)!\over (j+m)!(j-m)!}\right]^{1/2}
P_{jm} , \eqn\xab$$
and $Y^-_{jm}$ to
$$(-1)^{j+m} {1\over \sqrt{j/2} } { 1\over (2j-1)! }
\left[ { (2j-1)!\over (j+m)!(j-m)! } \right]^{1/2} Y^-_{jm} .\eqn\xac$$

Finally let us calculate the product $O_{j_1m_1} P_{j_2m_2}$. In the
course of this calculation we will find a proof of $\newf$.
Generally we should have
$$\eqalign{
O_{j_1m_1}P_{j_2m_2} =& g_1(j_1, j_2)
\langle j_1m_1j_2m_2\mid j_2-j_1,m_1+m_2\rangle
P_{j_2-j_1, m_1+m_2} \cr
& + g_2(j_1, j_2) \langle j_1m_1j_2m_2\mid j_2-j_1+1, m_1+m_2\rangle
aY^-_{j_2-j_1+1, m_1+m_2}  .
} \eqn\xad$$
We will compute $g_1(j_1,j_2)$ only, which can be deduced from
$g_1(1/2, j_2)$. The other unknown function can be derived by using
associativity. For $j_1=-m_1=1/2$ and $m_2=j_2=j$ we have
$$ \eqalign{
O_{1/2, -1/2} P_{j,j}(\delta) =&
\left\{ (2j+\delta) \sum_{q=1}^{2j} { \partial^{q+1} c \over (q+1)! }
 c S_{2j-q} (-iX^+-i\delta X^-) \right.\cr
 &\left.  + \sum_{q=0}^{2j} { \partial^{q+1} c \over (q+1)! } c S_{2j-q}
(-iX^+-i\delta X^-)  \right\}
e^{-2iX^++ i(2j-1)X^- } \cr
=& { (2j+1)(\delta + 2j) \over (2j+1) } P_{j-1/2, j-1/2} (\delta) + \cdots ,
} \eqn\xae$$
where $\cdots$ denotes the term proportional to $aY^-_{j+1/2, j-1/2}$.
Setting $P_{j,j}(\delta) =F_j(\delta)P_{j,j}$, one knows that $F_j(\delta)$
is a polynomial function with the highest degree $2j$ and $F_j(1)$ is 1. From
$\xae$ we get
$$\eqalign{
F_j(\delta) & = { \delta + 2j \over 2j+ 1}F_{j-1/2}(\delta) \cr
& = { \delta + 2j\over 2j+1 } { \delta + 2j-1 \over 2j } \cdots
{\delta+1 \over 2} F_0(\delta) = { (\delta +2j)!\over (2j+1)! \delta !} .
} \eqn\xaf$$
This proves $\newf$. Substituting this result back into $\xae$ one has
$$ O_{1/2, -1/2} P_{j,j} = (2j+2)P_{j-1/2, j-1/2} + \cdots . \eqn\xag$$
For $O_{j_1,-j_1} = -(-1)^{2j_1} (O_{1/2, -1/2})^{2j_1}/(2j_1)^2$,
we have then
$$ O_{j_1, -j_1} P_{j_2, j_2} = -(-1)^{2j_1} { (2(j_2+1))! \over (2j_1)!
(2(j_2-j_1+1))! } P_{j_2-j_1, j_2-j_1} + \cdots . \eqn\xah$$
 From this result it follows
$$\eqalign{
O_{j_1m_1}P_{j_2m_2} = & { j_2+1\over j_2-j_1 + 1} P_{j_2-j_1, m_1+m_2} \cr
& + { \tilde{ g} _2({j_1,j_2}) 1\over j_2-j_1 + 1}
(m_1j_2+m_2j_1+m_1) aY^-_{j_2-j_1+1, m_1+m_2} ,
} \eqn\xai$$
after rescaling $O_{j_1,m_1}$ and $P_{j_2,m_2}$ as in $\okog$ and $\xab$.
The unknown function $\tilde{g}_2(j_1, j_2)$ will be determined in a moment.

Let us summarize the ring structures we have determined. First we have
$$\eqalign{
O_{j_1m_1} O_{j_2m_2} = & O_{j_1+j_2, m_1+m_2} ,  \cr
O_{j_1m_1}Y^+_{j_2m_2}= & {j_2\over j_1+j_2} Y^+_{j_1+j_2, m_1+m_2}
- { 1\over j_1+j_2} (m_1j_2-m_2j_1)aO_{j_1+j_2-1, m_1+m_2} , \cr
Y^+_{j_1m_1}Y^+_{j_2m_2} =& {1\over j_1+j_2-1} (m_1j_2-m_2j_1)
 aY^+_{j_1+j_2-1, m_1+m_2},  \cr
Y^+_{j_1m_1}Y^-_{j_2m_2} = & {j_1\over j_2-j_1+1} P_{j_2-j_1, m_1+m_2} \cr
& + { 1\over j_2-j_1+1} (m_1j_2+m_2j_1+m_1) aY^-_{j_2-j_1+1, m_1+m_2}.
}\eqn\xax$$
The other non-vanishing  products are
$$ \eqalign{
O_{j_1m_1} Y^-_{j_2m_2} = & g_1(j_1, j_2) Y^-_{j_2-j_1, m_1+m_2} , \cr
Y^+_{j_1m_1} P_{j_2m_2} = & { g_2(j_1, j_2) \over j_2-j_1+2}
(m_1j_2+m_2j_1+m_1) aP_{j_2-j_1 +1, m_1+m_2} , \cr
O_{j_1m_1}P_{j_2m_2} = & { j_2+1\over j_2-j_1 + 1} P_{j_2-j_1, m_1+m_2} \cr
& + { \tilde{ g}_2({j_1,j_2}) \over j_2-j_1 + 1} (m_1j_2+m_2j_1+m_1)
aY^-_{j_2-j_1+1, m_1+m_2} ,
} \eqn\xaj$$
with
$$g_1(j_1,j_2)=g_2(j_1,j_2)=\tilde{g}_2(j_1,j_2)=1. \eqn\wuadd$$
while the vanishing products are
$$ \eqalign{
Y^-_{j_1m_1} Y^-_{j_2m_2} =&  0 , \cr
Y^-_{j_1m_1}P_{j_2m_2}  = & 0, \cr
P_{j_1m_1} P_{j_2m_2} = & 0.
} \eqn\xak$$

The three unknown functions in $\xaj$ are determined from associativity.
Multiplying both sides of $\poaa$ from the right by $Y^-_{j_3m_3}$
and using the known result $\xaa$ and the second equation of $\xaj$, we get
$g_2(j_1,j_3-j_2)=1$ or $g_2(j_1, j_2)=1$. To determine
$\tilde{g}_2(j_1, j_2)$ we multiply  the third equation of $\xaj$ by
$Y^+_{j_3m_3}$ and using $\xxx$ and the second equation of $\xaj$. The
result is $\tilde{g}_2(j_1, j_2)= 1$. Finally $g_1(j_1, j_2)$ is
determined by multiplying the first equation of $\xaj$  by
$aY^+_{j_3m_3}$, resulting in $g_1(j_1, j_2)=1$. This result is also
confirmed by explicit calculation.

\def\cgj{\langle j_1m_1 j_2m_2 |j_1+j_2,m_1+m_2 \rangle }
\def\cgjp{\langle j_1m_1 j_2m_2 |j_1+j_2-1,m_1+m_2 \rangle }
\def\cgpj{\langle j_1m_1 j_2m_2 |j_2-j_1,m_1+m_2 \rangle }
\def\cgpjp{\langle j_1m_1 j_2m_2 |j_2-j_1+1,m_1+m_2 \rangle }

To conclude this section let us note that we have chosen an $m$ dependent
normalization for all the vertex operator $O_{jm}$, $Y^{\pm}_{jm}$ and
$P_{jm}$'s. Here we also give the structure equations in their $SU(2)$
covariant form, which are obtained from $\xax$ and $\xaj$ by the following
replacement:
$$\eqalign{
O_{jm}\quad & \to \quad C(jm)O_{jm}, \cr
Y^+_{jm}\quad & \to \quad N(jm) Y^+_{jm},  \cr
Y^-_{jm}\quad & \to \quad
(-1)^{j+m} {\sqrt{j(2j+1)} \over N(jm) } Y^-_{jm}, \cr
P_{jm}\quad & \to \quad
(-1)^{j+m} {\sqrt{j(2j+1)} \over C(jm) } P_{jm}, } \eqn\nokl$$
where $C(jm) = \left[ { (j+m)!(j-m)!\over (2j)! } \right]^{1/2}$ and
$N(jm) = \left[ { (j+m)!(j-m)!\over (2j-1)! } \right]^{1/2}$.
We have then
$$\eqalign{
O_{j_1m_1} O_{j_2m_2} = & \cgj O_{j_1+j_2, m_1+m_2} ,  \cr
O_{j_1m_1}Y^+_{j_2m_2}= & \sqrt{ j_2\over j_1+j_2} \cgj
 Y^+_{j_1+j_2, m_1+m_2} \cr
& -\sqrt{j_1\over j_1+j_2} \cgjp aO_{j_1+j_2-1, m_1+m_2} , \cr
Y^+_{j_1m_1}Y^+_{j_2m_2} =& \sqrt{j_1+j_2\over j_1+j_2-1} \cgjp
 aY^+_{j_1+j_2-1, m_1+m_2},  \cr
Y^+_{j_1m_1}Y^-_{j_2m_2} = & \sqrt{ j_1\over j_2-j_1+1}\cgpj
P_{j_2-j_1, m_1+m_2} \cr
& -\sqrt{ j_2+1\over j_2-j_1+1} \cgpjp aY^-_{j_2-j_1+1, m_1+m_2},  \cr
O_{j_1m_1} Y^-_{j_2m_2} = & \cgpj Y^-_{j_2-j_1, m_1+m_2} , \cr
Y^+_{j_1m_1} P_{j_2m_2} = & -\sqrt{ j_2-j_1+1\over j_2-j_1+2 } \cgpjp
aP_{j_2-j_1 +1, m_1+m_2} , \cr
O_{j_1m_1}P_{j_2m_2} = & \sqrt{j_2+1\over j_2-j_1+1} \cgpj
 P_{j_2-j_1, m_1+m_2} \cr
& - \sqrt{ j_1\over j_2-j_1+1} \cgpjp aY^-_{j_2-j_1+1, m_1+m_2}.
} \eqn\xppaj$$
The other three vanishing products remain unchanged.

\vfill\eject

\centerline{ \bf 4. Chiral Ring Structure Including the Tachyon States}

In this section we incorporate the tachyon states into the chiral
cohomology ring. The tachyon states are labelled by a continuous momentum
$p$ and there are two chiralities:
$$ T^{ (\pm) }_p = ce ^{ ipX+ (\sqrt{2} \mp p) \phi } . \eqn\qaa$$
As in the case of discrete states, there also exist absolute tachyon states
$$ aT^{ (\pm) }_p = \pm { p \over \sqrt{2} } \partial c
ce ^{ ipX+ (\sqrt{2} \mp p) \phi } . \eqn\qab$$
To simplify our presentation we will consider only those tachyon states
which are not included in the discrete states, i.e. $p\neq j\sqrt{2}$, where
$j$ is an integer or half integer. Let us first consider the product of
discrete states with tachyon states. By momentum addition this product must
give rise to another tachyon state. Since any tachyon states must be of
the form given by either $\qaa$ or $\qab$, there are only four
non-vanishing products of this type:
$$ \eqalign{
O_{j, \pm j} T^{(\pm)}_p  & \sim  T^{(\pm )} _{p\pm j\sqrt{2}} , \cr
Y^+_{j, \pm (j-1) } T^{(\pm)}_p  & \sim  a  T^{(\pm )}_{p\pm (j-1)\sqrt{2}} .
} \eqn\qac$$
The explicit calculation of these products is easy and we get, in
agreement with [\WITZ],
$$ \eqalign{
O_{j,  j} T^{(+)}_p  =& { p \over p + \sqrt{2}j }
T^{(+ )}_{p+ j\sqrt{2}} , \cr
O_{j, -j} T^{(-)}_p = & T^{(-)}_{p-\sqrt{2} j} , \cr
Y^+_{j , (j-1) } T^{(+)}_p  =&  - { p\over p +\sqrt{2}(j-1) }
 a  T^{(+ )}_{p+\sqrt{2}(j-1)} , \cr
Y^+_{j , -(j-1) } T^{(-)}_p  =&  -
 a  T^{(- )}_{p- \sqrt{2}(j-1)} , \cr
 } \eqn\qad$$
if we rescale $T^{(\pm)}$ to
$$  (\Gamma(1+\sqrt{2}p) ) ^{\pm 1}  T^{ (\pm)} _p . \eqn\pae$$

Now let us consider the product of two tachyons. For the product
$T^{(-)}_p T^{(-)}_q$ the exponential factor is $e^{ i(p+q)X + (2\sqrt{2} +
(p+q))\phi} $ which can not be the exponential factor of any tachyon state. To
be able to obtain a discrete state we must have
$$ p+ q = j \sqrt{2}, \eqn\qaf$$
where $j $ is an integer or half integer. For $j\ge -1/2$ we have
$$  T^{(-)}_p T^{(-)}_q = d_1aY^-_{j+1, j} + d_2P_{j, j} ,  \eqn\qag$$
where $d_1$ and $d_2$ are two constants.
For $j \le -1$ the product is zero, for there is no $aY^+_{-j -1, j}$. By
explicit calculation we have
$$ \eqalign{
T^{(-)}_pT^{(-)}_q = & \sum_{k=0} ^{2j+1} { \partial^{k+1} c \over (k+1)! } c
S_{2j+1-k } (-iX^+ +i(\sqrt{2} p +1) X^- ) e^{ (ijX+(2+j)\phi)\sqrt{2} } \cr
=& { \Gamma(2j+1-\sqrt{2} p) \over 2(j+1) \Gamma(-\sqrt{2} p) }
\Big(P_{j, j} + aY^-_{j +1, j} \Big),
} \eqn\qah$$
or
$$
T^{(-)}_p T^{(-)}_q = { \sin \sqrt{2}\pi p \over 2(j+1) \pi }
\Big(P_{j, j} + aY^-_{j +1, j} \Big),
\eqn\qai$$
after making rescaling $\pae$. Similarly we have, with $p+q= -j\sqrt{2}$,
$$ T^{(+)}_pT^{(+)}_q ={  \pi pq \over (j+1) \sin\sqrt{2}\pi q}
\Big(P_{j, -j} - aY^-_{j +1, -j} \Big).
\eqn\qaj$$

The product $T^{(+)}_p T^{(-)}_q$ gives no new result. It either reduces
to $\qad$ or to $Y^+_{j_1, \pm j_1} Y^-_{j_2, \pm j_2}$,
which is a special case of $\xaa$.

Finally let us point out that part of the above results was also
obtained in [\TAN]. There they computed the single pole term in the OPE of
$\Psi^{(\pm)}_p \Psi^{(\pm)}_q$, where $T_p ^{(\pm)} = c\Psi^{(\pm)}_p$.
This only gives the term $aY^-_{j+1, \pm j} $ in $\qai$ or $\qaj$.
The other term $P_{j, \pm j} $ comes from the higher order
pole terms in the OPE of $\Psi^{(\pm)}_p \Psi^{(\pm)}_q$.

\vfill\eject

\centerline{ \bf 5. Chiral Transformation Rules and the Chiral
Charge  Algebra}

In this section we will use the above results to study the (chiral)
symmetry charge algebra associated to the chiral states. We will follow
the general strategy of constructing BRST invariant charges
from BRST invariant operators [\WITZ]. We consider only the holomorphic
part in this section. In the next section we will use the results to
derive the transformation rules for the discrete closed-string states by
combining the left and right sectors.

Starting from a generic BRST invariant operator $V_{jm}$, one defines
$V^{(1)}_{jm}$ by the OPE
$$ b(z)V_{jm}(w) \sim \cdots + { 1\over z-w} V_{jm}^{(1)} , \eqn\raa$$
or
$$ V^{(1)}_{jm}(w) = \oint_0 [\dd z] b(z)V_{jm}(w),   \eqn\rab$$
up to total derivatives and BRST exact terms. Then one can prove
$$ \partial_w V_{jm}  (w) = \{ Q, V^{(1)}_{jm} (w) \} . \eqn\rac$$
Indeed,
$$\eqalign{
\{ Q, V^{(1)}_{jm}& (w) \} =  \oint_w [\dd z]
:c(z) \big( T(z) +\partial_z c(z)
b(z) \big):  \oint_w [\dd \tilde{z} ] b(\tilde{z})V_{jm}(w) \cr
= & \Big( \oint_w[\dd \tilde{z}]\oint_w [\dd z] + \oint_w[\dd \tilde{z}]
\oint_{\tilde{z}} [\dd z] \Big)
:c(z) \big( T(z) +\partial_z c(z)
b(z) \big):  b(\tilde{z})V_{jm}(w) \cr
= & \oint_w[\dd z] \Big( T(z) + :2\partial_z c(z)b(z)+ c(z)\partial_zb(z)
:\Big) V_{jm}(w) \cr
= & \partial_w V_{jm}(w) .
}\eqn\rad$$
In terms of $V^{(1)}_{jm}$ one can define a BRST invariant charge
$${ \hat{V}} _{jm} = \oint_0[\dd z] V^{(1)}_{jm} (z). \eqn\rae$$
The (anti-)commutator of $\hat{V}_{jm}$ with any BRST invariant local operator
gives another BRST invariant operator, i.e.
$$ [\hat{V}_{j_1m_1} , V_{j_2m_2}(z) ] = \sum_{jm} g_(j_1, j_2; j)
\langle j_1m_1j_2m_2\mid jm\rangle V_{jm}(z) . \eqn\raf$$

Let us see how one can compute the RHS of $\raf$ explicitly from
the knowledge of the cohomology ring structure. Let us consider first
the relative physical states. For them the double pole term in $\raa$
actually vanishes, because there is no $\partial c$ term in $V_{jm}$.
By definition, the ring multiplication law is given by
$$
(V_{j_1m_1}V_{j_2m_2}) (w)
= \oint_w[\dd z] { 1\over z-w} V_{j_1m_1} (z) V_{j_2m_2} (w).
\eqn\rag$$
Applying $\oint_w[\dd \tilde{z}] (\tilde{z} -w) b(\tilde{z}) $
on both sides from the left, we get
$$ \eqalign{
{\rm RHS} = &  \Big( \oint_w[\dd z] \oint_w[\dd \tilde{z} ] +
\oint_w[\dd z] \oint_z[\dd \tilde{z} ]\Big) { \tilde{z} -w \over z-w}
b(\tilde{z} ) V_{j_1m_1}(z) V_{j_2m_2}(w) \cr
= & \oint_w[\dd z] V^{(1)}_{j_1m_1} (z) V_{j_2m_2}(w) = [\hat{V}_{j_1m_1},
V_{j_2m_2}(w) ] ;
} \eqn\rah$$
Thus we have derived the important formula
$$ [\hat{V}_{j_1m_1} , V_{j_2m_2} ] = \oint_w [\dd z] (z-w)b(z)(V_{j_1m_1}
V_{j_2m_2})(w). \eqn\rahh$$
This formula allows us to compute the action of a charge on a physical
states from the knowledge of the cohomology ring structure, exhibited by
$$
(V_{j_1m_1}V_{j_2m_2}) (w)
= \sum_{jm } \tilde{g}(j_1, j_2; j)\langle j_1m_1j_2m_2\mid jm\rangle
V_{jm}(w). \eqn\wuadd $$
In fact, by using $\wuadd$ the RHS of $\rahh$ is simply given by
$$ \hbox{ RHS of}~~ \rahh = \sum_{jm} \tilde{g}(j_1, j_2; j)
\langle j_1m_1 j_2m_2\mid
jm\rangle \oint_w[\dd z] (z-w)b(z) V_{jm}(w). \eqn\rai$$
We note that the last integration selects only terms with a factor
$\partial c$ in $V_{jm}$. In this way we see that the transformation
rules for the discrete charges acting on the discrete states can be
simply read off from the structure equation (i.e. the multiplication law)
of the cohomology ring.

Now let us apply the above results to the discrete states
$O_{jm}$, $Y^{\pm}_{jm}$ and $P_{jm}$. From these discrete states
one can construct BRST invariant charges of ghost number $-1$, $0$
and $1$. We denote these charges as
$\hat{X}_{jm}$, $Q^{\pm}_{jm}$ and $\hat{Z}_{jm}$:
$$ \eqalign{
\hat{X}_{jm} \equiv &  \oint_0[\dd w] \oint_w[\dd z] b(z) O_{jm}(w)
\equiv \oint_0 [\dd w] X_{jm}(w), \cr
Q^{\pm}_{jm} \equiv &  \oint_0[\dd w] \oint_w[\dd z] b(z) Y^{\pm}_{jm}(w)
\equiv \oint_0 [\dd w] W^{\pm}_{jm}(w), \cr
\hat{Z}_{jm} \equiv &  \oint_0[\dd w] \oint_w[\dd z] b(z) P_{jm}(w)
\equiv \oint_0 [\dd w] Z_{jm}(w).} \eqn\epxd$$
Their action on discrete states can be directly read off from the
ring structure by using $\rahh$. We have
$$ \eqalign{
& [\hat{X}_{j_1m_1} , O_{j_2m_2} ] = 0, \qquad
\{ \hat{X}_{j_1m_1} , Y^-_{j_2m_2}\} = 0, \cr
& \{ \hat{X}_{j_1m_1} , Y^+_{j_2m_2} \}
= -(m_1j_2- m_2j_1)O_{j_1+j_2-1, m_1+m_2}, \cr
& [\hat{X}_{j_1m_1}, P_{j_2m_2}]
= -(m_1j_2+m_2j_1+m_1)Y^-_{j_2-j_1+1, m_1+m_2}, \cr
& [Q^+_{j_1m_1}, Y^+_{j_2m_2} ]
= (m_1j_2-m_2j_1)Y^+_{j_1+j_2-1, m_1+m_2}, \cr
& [Q^+_{j_1m_1}, Y^-_{j_2m_2} ]
= -(m_1j_2+m_2j_1+m_1)Y^-_{j_2-j_1+1, m_1+m_2} , \cr
& [Q^+_{j_1m_1}, P_{j_2m_2}]
= -(m_1j_2+m_2j_1+m_1)P_{j_2-j_1+1, m_1+m_2}, \cr
& \{ Q^-_{j_1m_1}, Y^-_{j_2m_2} \} = 0, \quad
[Q^-_{j_1m_1}, P_{j_2m_2} ] = 0, \quad
[ \hat{Z}_{j_1m_1} , P_{j_2m_2} ] =0 .
}\eqn\raj$$
Applying once more
$\oint_0[\dd w] \oint_w[\dd z] b(z)$ from the
left on these equations\footnote*{This operation changes all the discrete
states into the corresponding charges.}, we get the following chiral charge
algebra
$$ \eqalign{
& [ \hat{X}_{j_1m_1} , Q^+_{j_2m_2}]
= (m_1j_2- m_2j_1)\hat{X}_{j_1+j_2-1, m_1+m_2}, \cr
& \{\hat{X}_{j_1m_1}, \hat{Z}_{j_2m_2}\}
= (m_1j_2+m_2j_1+m_1)Q^-_{j_2-j_1+1, m_1+m_2}, \cr
& [Q^+_{j_1m_1}, Q^+_{j_2m_2} ]
= (m_1j_2-m_2j_1)Q^+_{j_1+j_2-1, m_1+m_2}, \cr
& [Q^+_{j_1m_1}, Q^-_{j_2m_2} ]
= -(m_1j_2+m_2j_1+m_1)Q^-_{j_2-j_1+1, m_1+m_2} , \cr
& [Q^+_{j_1m_1}, \hat{Z}_{j_2m_2}]
= -(m_1j_2+m_2j_1+m_1)\hat{Z}_{j_2-j_1+1, m_1+m_2}. \cr
}\eqn\rak$$
The remaining (anti-)commutators are zero.

Similarly one can also construct BRST invariant charges from absolute physical
states. We simply denote these absolute charges as $a\hat{X}_{jm}$,
$aQ^{\pm}_{jm}$ and
$a\hat{Z}_{jm}$\footnote{**}{$a\hat{X}_{jm}\equiv \oint_0[\dd w]\oint_w[\dd z]
b(z)(aO_{jm})(w)\neq a\cdot \hat{X}_{jm},$ etc. }. Then one easily proves the
following general relations
$$\eqalign{
[\hat{V}_{j_1m_1}, aV_{j_2m_2}(w)] = & -(-1)^{[V_{j_1}]} g_V(j_2)
(V_{j_1m_1}V_{j_2m_2})(w) \cr
& + \oint_w[\dd z](z-w)b(z)(V_{j_1m_1}aV_{j_2m_2})(w) , \cr
[a\hat{V}_{j_1m_1}, V_{j_2m_2}(w)] = & - g_V(j_1)
(V_{j_1m_1}V_{j_2m_2})(w) \cr
& + \oint_w[\dd z](z-w)b(z)(aV_{j_1m_1}V_{j_2m_2})(w) , \cr
[a\hat{V}_{j_1m_1}, aV_{j_2m_2}(w)] = & -(-1)^{[V_{j_1}]}
(g_V(j_1)- g_V(j_2) (aV_{j_1m_1}V_{j_2m_2})(w) , \cr } \eqn\zza$$
just by deforming the integration contour as in $\rah$.  Here in the first
and third equations $[V_{j_1}]$ denotes the ghost number of $V_{j_1m_1}$.
The function $g_V(j_a)$ ($a=1,2$) is defined as
$$ g_O(j_a) = -  g_P(j_a) = j_a+1, \qquad g_{Y^{\pm}} (j_a) = \pm j_a ,
 \eqn\zzb$$
depending on the type of the operator $V_{j_a m_a}$.

Now we present some examples to see how the above formulas can be
used to derive the chiral transformation rules and chiral charge
algebra involving absolute physical states or absolute charges. We have,
e.g.,
$$ \eqalign{
[\hat{X}_{j_1m_1} , aY^+_{j_2m_2} ]  = &
{j_1j_2\over j_1 + j_2} Y^+_{j_1+j_2, m_1+m_2}
+ {j_2 \over j_1+j_2}(m_1j_2-m_2j_1) aO_{j_1+j_2-1, m_1+m_2} , \cr
[aQ^+_{j_1m_1}, O_{j_2m_2} ] = &
{j_1j_2\over j_1 + j_2} Y^+_{j_1+j_2, m_1+m_2}
- {j_1 \over j_1+j_2}(m_1j_2-m_2j_1) aO_{j_1+j_2-1, m_1+m_2} , \cr
[Q^+_{j_1m_1}, aO_{j_2m_2}] = &
-{j_1(j_1-1) \over j_1+j_2} Y^+_{j_1+j_2, m_1+m_2}
\cr
& + {j_2+1\over j_1+j_2} (m_1j_2 -m_2j_1)aO_{j_1+j_2-1, m_1+m_2} , \cr
[a\hat{X}_{j_1m_1}, Y^+_{j_2m_2} ] = &
{j_2(j_2-1) \over j_1+j_2} Y^+_{j_1+j_2, m_1+m_2}
\cr
& + {j_1+1\over j_1+j_2} (m_1j_2 -m_2j_1)aO_{j_1+j_2-1, m_1+m_2}
. } \eqn\zzc$$
Applying $\oint_0[\dd w] \oint_w [\dd z] b(z)$ from the
left to the above equations, we get the following (chiral)
(anti-)commutation relations
$$ \eqalign{
\{\hat{X}_{j_1m_1} , aQ^+_{j_2m_2} \} = &
-{j_1j_2 \over j_1 + j_2} Q^+_{j_1+j_2, m_1+m_2}
\cr
& + {j_1\over j_1 +j-2}(m_1j_2 - m_2j_1)a\hat{X}_{j_1+j_2 -1, m_1+m_2} , \cr
[Q^+_{j_1m_1}, a\hat{X}_{j_2m_2}] = &
-{j_1(j_1-1)\over j_1+j_2} Q^+_{j_1+j_2, m_1+m_2}
\cr
& + {j_2\over j_1 +j-2}(m_1j_2 - m_2j_1)a\hat{X}_{j_1+j_2 -1, m_1+m_2} . }
\eqn\zzd$$

We have derived the complete set of the chiral transformation rules
and the chiral charge algebra and collect them in appendix B.

\vfill\eject

\centerline{ \bf 6. Symmetry Transformation of the Discrete String States}

All of our discussions up to now deal with the right-moving sector only.
In this section we will combine the left and right sectors to derive
the symmetry transformation rules for the discrete states in closed string
theory. Before doing this let us recall briefly the physical states
in $D=2$ closed string theory.

As discussed in [\WITZ], the physical closed string states in $D=2$
should satisfy an extra condition
$(b_0-\bar{b}_0 ) |\hbox{phys.} \rangle = 0$.
We will not go into any details for solving this condition
but simply quote their results. By matching the Liouville momenta,
the physical closed string discrete states that obey
the $b_0-\bar{b}_0$ condition are as follows ($G$ is the total
ghost number):

\noindent 1) the plus states
$$\eqalign{
& G=0: \quad O_{jm}\bar{O}_{jm'} ;\cr
& G=1: \quad O_{jm}\bar{Y}^+_{j+1, m'},\quad Y^+_{j+1, m}\bar{O}_{jm'} ,
\quad (a+ \bar{a}) O_{jm}\bar{O}_{jm'} ;\cr
& G=2: \quad Y^+_{jm}\bar{Y}^+_{jm'},
\quad (a+\bar{a}) O_{jm}\bar{Y}^+_{j+1, m'},
\quad (a+\bar{a})Y^+_{j+1, m}\bar{O}_{jm'} ; \cr
& G=3: \quad (a+ \bar{a})Y^+_{jm}\bar{Y}^+_{jm'}; } \eqn\zze$$

\noindent 2) the minus states
$$\eqalign{
& G=2: \quad Y^- _{jm}\bar{Y}^-_{jm'} ;\cr
& G=3: \quad P_{jm}\bar{Y}^-_{j+1, m'},\quad Y^-_{j+1, m}\bar{P}_{jm'} ,
\quad (a+ \bar{a}) Y^-_{jm}\bar{Y}^-_{jm'} ;\cr
& G=4: \quad P_{jm}\bar{P}_{jm'}, \quad (a+\bar{a}) P_{jm}\bar{Y}^-_{j+1, m'},
\quad (a+\bar{a})Y^-_{j+1, m}\bar{P}_{jm'} ; \cr
& G=5: \quad (a+ \bar{a})P_{jm}\bar{P}_{jm'}. } \eqn\zzf$$
 From the above list we see that half of the states (the states without $(a+
\bar{a})$) are just the products of left and right relative physical states
which satisfy the stronger condition:
$b_0|\hbox{phys.}\rangle=\bar{b}_0|\hbox{phys.}\rangle=0$. For later
convenience we call these states relative physical (string) states.
The other half of the states contain $(a +\bar{a})$ and do not
satisfy either $b_0|\hbox{phys.}\rangle=0$ or
$\bar{b}_0|\hbox{phys.}\rangle=0$. We will call them absolute
physical (string) states. For closed string states, all the absolute
physical states can be obtained from the relative physical states
simply by multiplying $(a+\bar{a})$.

The general framework for constructing conserved charges in $D=2$
closed string theory again has been given in sec. 4 of [\WITZ].
Summarizing their results, one starts with any BRST invariant zero
form $\Omega^{(0)}$ and constructs the one and two forms $\Omega^{(1)}$
and $\Omega^{(2)}$ satisfying the descent equations
$$ \eqalign{
0 = & \{ Q_T,  \Omega^{(0) } \} , \cr
\dd\Omega^{(0)} = & \{ Q_T, \Omega^{(1)} \} , \cr
\dd\Omega^{(1)} = & \{ Q_T, \Omega^{(2)} \} . } \eqn\zzg$$
These equations imply that the charge
$ A = { 1\over 2\pi i} \oint_0 \Omega^{(1)} $
is a BRST invariant charge (the second equation)
conserved up to BRST trivial operators (the third equation). $\Omega^{(1)}$
and  $\Omega^{(2)}$ are given by\footnote*{ $\oint_w[\dd z] { 1\over
z-w} = -\oint_w[\dd \bar{z}] { 1\over \bar{z}-\bar{w} } = 1.$ }
$$\eqalign{
 \Omega^{(1)}(w, \bar{w}) = & \Big(\dd w \oint_w [\dd z] b(z) - \dd \bar{w}
\oint_w[\dd \bar{z}] \bar{b}(\bar{z})\Big)  \Omega^{(0)}(w, \bar{w}), \cr
 \Omega^{(2)}(w, \bar{w}) = & - \dd w\wedge \dd \bar{w} \oint_w[\dd z']
b(z') \oint_w[\dd \bar{z} ] \bar{b} (\bar{z}) \Omega^{(0)} (w,\bar{w}) . }
\eqn\zzh$$

In what follows we take $\Omega^{(0)}$ to be $Y^+_{j+1, m}\bar{O}_{jm'}$
as an example and derive the symmetry transformations for discrete string
states. We have then
$$ \eqalign{
  \Omega^{(1)}(w, \bar{w}) = & \dd w W^+_{j+1, m } \bar{O}_{jm'}(w, \bar{w})
- \dd \bar{w} Y^+_{j+1, m} \bar{X}_{jm'}(w, \bar{w}), \cr
  \Omega^{(2)}(w, \bar{w}) = & - \dd w\wedge \dd \bar{w} W^+_{j+1, m}
\bar{X}_{jm'} (w, \bar{w}) . } \eqn\zzi$$
The conserved charge $A_{j; mm'}\equiv{ 1\over 2 \pi i }
\oint_0  \Omega^{(1)}(w, \bar{w}) $ acting on the (plus) relative discrete
states gives rise to
$$\eqalign{
[A_{j_1;m_1m_1'},& O_{j_2m_2}\bar{O}_{j_2m_2'}] \cr
=  & (m_1j_2-m_2(j_1+1))O_{j_1+j_2,m_1+m_2} \bar{O}_{j_1+j_2, m_1'+m_2'}, \cr
[A_{j_1;m_1m_1'},& O_{j_2m_2}\bar{Y}^+_{j_2+1, m_2'}]\cr
= & -{1\over j_1+j_2+1}(m_1j_2-m_2(j_1+1))(m_1'(j_2+1)-m_2'j_1) \cr &\times
(a +\bar{a} )O_{j_1+j_2, m_1+m_2}\bar{O}_{j_1+j_2, m_1'+m_2'}  \cr
& + {j_2 + 1\over j_1+j_2 + 1 }(m_1j_2 -m_2(j_1+1))O_{j_1+j_2, m_1+m_2}
\bar{Y}^+_{j_1+j_2+1, m_1'+m_2'}\cr
&-{ j_1+1\over j_1+j_2+1} (m_1'(j_2+1)-m_2'j_1)Y^+_{j_1+j_2+1,m_1+m_2}
\bar{O}_{j_1+j_2, m_1'+m_2'}, \cr
[A_{j_1;m_1m_1'},& Y^+_{j_2+1,m_2}\bar{O}_{j_2m_2'}] \cr = &
(m_1(j_2+1)-m_2(j_1+1)) Y^+_{j_1+j_2+1, m_1+m_2}
\bar{O}_{j_1+j_2, m_1'+m_2'},\cr
[A_{j_1;m_1m_1'},& Y^+_{j_2m_2}\bar{Y}^+_{j_2m_2'} ] \cr = &
{1\over j_1+j_2} (m_1j_2-m_2(j_1+1))  (m_1'j_2-m_2'j_1)\cr &\times
(a +\bar{a}) Y^+_{j_1+j_2, m_1+m_2}\bar{O}_{j_1+j_2-1, m_1'+m_2'}\cr
& + { j_2\over j_1+j_2} (m_1j_2-m_2(j_1+1)) Y^+_{j_1+j_2,m_1+m_2}\bar{Y}^+_
{j_1+j_2, m_1'+m_2'} .}
\eqn\zzp$$
Note that the action of the charge $A_{j_1; m_1m_1'}$ on the relative
discrete states also give rise to absolute physical states.
This feature was noted by Witten and Zwiebach [\WITZ] too.
The example given by them is a special case of the last
equation in $\zzp$. Here we are able to systematically
derive the complete set of transformation rules with the help of the
knowledge of the cohomology ring structure. In appendix C we will show that
the action of any charge on the physical closed string discrete states gives
only (a linear combination of) physical closed string discrete states,
i.e., those listed in $\zze$ and $\zzf$.

The action of the charge $A_{j_1; m_1m_1'}$ on the (plus) absolute discrete
states can also be derived. We have
$$\eqalign{
[A_{j_1;m_1m_1'},& (a+\bar{a})O_{j_2m_2}\bar{O}_{j_2m_2'}]\cr = &
(m_1j_2-m_2(j_1+1))(a+\bar{a})
O_{j_1+j_2,m_1+m_2} \bar{O}_{j_1+j_2, m_1'+m_2'}, \cr
[A_{j_1;m_1m_1'},& (a+ \bar{a}) O_{j_2m_2}\bar{Y}^+_{j_2+1, m_2'}] \cr =
&-{ j_1+1\over j_1+j_2+1} (m_1'(j_2+1)-m_2'j_1)
(a +\bar{a}) Y^+_{j_1+j_2+1,m_1+m_2}
\bar{O}_{j_1+j_2, m_1'+m_2'} \cr
& + {j_2 + 1\over j_1+j_2 + 1 }(m_1j_2 -m_2(j_1+1))
(a+\bar{a}) O_{j_1+j_2, m_1+m_2}
\bar{Y}^+_{j_1+j_2+1, m_1'+m_2'}, \cr
[A_{j_1;m_1m_1'},& (a+\bar{a})Y^+_{j_2+1,m_2}\bar{O}_{j_2m_2'}] \cr = &
(m_1(j_2+1)-m_2(j_1+1)) (a+\bar{a}) Y^+_{j_1+j_2+1, m_1+m_2}
\bar{O}_{j_1+j_2, m_1'+m_2'},\cr
[A_{j_1;m_1m_1'},& (a+\bar{a}) Y^+_{j_2m_2}\bar{Y}^+_{j_2m_2'} ] \cr = &
{ j_2\over j_1+j_2} (m_1j_2-m_2(j_1+1)) (a +\bar{a})
Y^+_{j_1+j_2,m_1+m_2}\bar{Y}^+_ {j_1+j_2, m_1'+m_2'}. \cr } \eqn\zzj$$
Comparing $\zzp$ and $\zzj$, we observe that the left sides of all
the equations in $\zzj$ can be obtained from $\zzp$ simply by
the multiplication of $(a+\bar{a})$. (Note that $(a+\bar{a})^2=0$.)
This suggests the following general formula:
$$ [ \hat{V}_{j_1; m_1m_1'}, (a+\bar{a})V_{j_2;m_2m_2'}] = \pm
(a+\bar{a}) [ \hat{V}_{j_1; m_1m_1'}, V_{j_2;m_2m_2'}] , \eqn\pqq$$
where $\hat{V}_{j_1; m_1m_1'}$ is the charge derived from the relative physical
state $V_{j_1;m_1m_1'}$ and $V_{j_2; m_2m_2'}$ is a relative physical state.
Here one takes the ``$-$'' sign if $ \hat{V}_{j_1; m_1m_1'}$ has odd (total)
ghost number. The general validity of $\pqq$ will be proved in Appendix C.

\def\pxxx{ (a+\bar{a}) }

Quite similarly  one can construct conserved charges from absolute discrete
states and study their action on discrete states.  The corresponding charges
will simply be denoted as $  (a+\bar{a})\hat{V}_{j;mm'}$. Now take
$\Omega^{(0)}$, for example, to be $(a+\bar{a})Y^+_{j+1,m}O_{j,m}$.
By explicit calculation we have
$$\eqalign{
[\pxxx &A_{j_1;m_1m_1'}, O_{j_2m_2}\bar{O}_{j_2m_2'}] \cr
=  &- (m_1j_2-m_2(j_1+1))\pxxx
O_{j_1+j_2,m_1+m_2} \bar{O}_{j_1+j_2, m_1'+m_2'}, \cr
[\pxxx &A_{j_1;m_1m_1'}, O_{j_2m_2}\bar{Y}^+_{j_2+1, m_2'}]\cr
= & -{j_2 + 1\over j_1+j_2 + 1 }(m_1j_2 -m_2(j_1+1))\pxxx O_{j_1+j_2, m_1+m_2}
\bar{Y}^+_{j_1+j_2+1, m_1'+m_2'}\cr
&+{ j_1+1\over j_1+j_2+1} (m_1'(j_2+1)-m_2'j_1)\pxxx Y^+_{j_1+j_2+1,m_1+m_2}
\bar{O}_{j_1+j_2, m_1'+m_2'}, \cr
[\pxxx &A_{j_1;m_1m_1'}, Y^+_{j_2+1,m_2}\bar{O}_{j_2m_2'}] \cr = &
-(m_1(j_2+1)-m_2(j_1+1)) \pxxx Y^+_{j_1+j_2+1, m_1+m_2}
\bar{O}_{j_1+j_2, m_1'+m_2'},\cr
[\pxxx &A_{j_1;m_1m_1'}, Y^+_{j_2m_2}\bar{Y}^+_{j_2m_2'} ] \cr
= & - { j_2\over j_1+j_2} (m_1j_2-m_2(j_1+1))
\pxxx Y^+_{j_1+j_2,m_1+m_2}\bar{Y}^+_
{j_1+j_2, m_1'+m_2'} , }
\eqn\zzp$$
and
$$\eqalign{ & \{ \pxxx A_{j_1;m_1m_1'}, \pxxx O_{j_2m_2}\bar{O}_{j_2m_2'}\} =0,
\cr
& [\pxxx A_{j_1;m_1m_1'}, \pxxx O_{j_2m_2}\bar{Y}^+_{j_2+1, m_2'}]= 0 \cr
& [\pxxx A_{j_1;m_1m_1'}, \pxxx Y^+_{j_2+1,m_2}\bar{O}_{j_2m_2'}]=  0, \cr
& \{ \pxxx A_{j_1;m_1m_1'},
\pxxx Y^+_{j_2m_2}\bar{Y}^+_{j_2m_2'} \} = 0. } \eqn\pxqq$$
 From the above results we would like to guess the following general formulas
$$ \eqalign{& [ \pxxx \hat{V}_{j_1;m_1m_1'} , V_{j_2;m_2m_2'}] = -\pxxx
[ \hat{V}_{j_1;m_1m_1'} , V_{j_2;m_2m_2'}], \cr &
[ \pxxx \hat{V}_{j_1;m_1m_1'} , \pxxx V _{j_2;m_2m_2'}] = 0. } \eqn\pyqq$$
Again the general validity of these equations will be proved in Appendix C.

A remark is in order about the transformation rules in closed string theory.
For the chiral transformation rules we see quite clearly from the general
formulas $\rahh$ and $\zza$ and also the
explicit results in appendix B that the
multiplication by $a$ is not (anti-)commutative with the operation of taking
(anti-) commutator between charge and discrete state,  and also the
operation of getting the charges from the states, i.e. in general
$$\eqalign{ & [\hat{V}_{j_1m_1}, aV_{j_2m_2}] \neq -(-1)^{[V_{j_1}]} a
[\hat{V}_{j_1m_1}, V_{j_2m_2}] , \cr
& [a\hat{V}_{j_1m_1}, V_{j_2m_2}] \neq - a
[\hat{V}_{j_1m_1}, V_{j_2m_2}] , \cr
& [a \hat{V}_{j_1m_1}, aV_{j_2m_2}] \neq 0 .
} \eqn\zhuadda$$
This is in sharp contrast with the transformation rules for closed string
theory. In closed string theory, the $(a+\bar{a})$ plays the role of $a$
played in chiral theory. The eqs. $\pqq$ and $\pyqq$ shows clearly that
the multiplication by $(a+\bar{a})$ can be moved freely and    can be
taken out of the commutator. At the moment we do not have a deep
understanding of this result.

\vglue .5cm

\centerline{ \bf 7. Discussion and Conclusions}

In this paper we have explicitly determined the complete set of
structure constants of the chiral BRST cohomology ring
for $D=2$ string theory. As we demonstrated in sec. 5, this ring
structure encodes  the full chiral charge algebra
and the corresponding symmetry transformation rules. The latter in turn
can be used to derive the full symmetry algebra and transformation rules
for closed string states, as shown in sec. 6. We alos found that the
operator $(a+\bar{a})$ in closed string theory obeys a quite simple rule
as given in eqs. $\pqq$ and $\pyqq$.

\REF\GK{D. Gross and I. Klebanov, \np\ {\bf B352} (1991) 671.}

\REF\MS{G. Moore and N. Seiberg, Rutgers University preprint No.
RU-91-29, 1991.}

\REF\AJ{J. Avan and A. Jevicki, \pl\ {\bf B266} (1991) 35; {\bf B272}
(1991) 17.}

\REF\DDMW{S.R. Das, A. Dhar, G. Mandal and S.R. Wadia, IAS preprint
IASSNS-HEP-91/52.}

\REF\PSR{C. Pope, L. Romans and X. Shen, \pl\ {\bf B236} (1990) 236.}

\REF\WXY{E. Witten, Phys. Rev. {\bf D44} (1991) 314.}

\REF\YWW{F. Yu and Y.-S. Wu, \prl\ {\bf 68} (1992) 2996.}

\REF\BKK{I. Bakas and E. Kiritsis, Maryland/Berkeley/LBL preprint
No. UCB-PTH-91/44, LBL-31213, and UMD-PP-92/37, 1991.}

\REF\YYWW{F. Yu and Y.-S. Wu, \np\ {\bf B373} (1992) 713.}


The search for the infinite symmetries in $D=2$ string theory
has been done recently in various approaches, with apparently
different outcomes. In the $c=1$ matrix model [\GK, \MS, \AJ, \DDMW],
it is the linear $W_{\infty}$ symmetry [\PSR], which is realized through
its well-known free-fermion realization. In the gauged WZW model approach
[\WXY], the first-quantized $D=2$ string theory is described by the
world-sheet $SL(2,R)/U(1)$ model at level $k=9/4$. Recently it has been
shown [\YWW, \BKK, \YYWW] that for generic $k$ there exists a hidden
non-linear $\hat{W}_{\infty}$ [\YWW] current algebra, whose generators
are higher-spin currents formed from the basic coset currents through
an elegant generating function [\YWW]. In this paper we are discussing
the symmetries in the Liouville approach, initiated in [\WITA, \WITZ],
for physical string states. Through we have been able to work out
explicit symmetry transformation rules, the relatioship with the
$W_{\infty}$ symmetries appearing in other approaches as mentioned above
remains to be clarified.

\REF\KAPP{I. R. Klebanov, \mpl\ {\bf A7} (1992) 723.}

\REF\KACH{S. Kachru, \mpl\ {\bf A7} (1992) 1419. }

\REF\PAS{I. R. Klebanov and A. Pasquinucci, ``Correlation Function from
Two Dimensional String Ward Identities,'' preprint PUPT-1313 (April, 1992).}

\REF\BER{M. Bershadsky and D. Kutasov, ``Scattering of Open and Closed
Strings in $1+1$ Dimensions,'' preprint PUPT-1315 or HUTP-92/A016 (April,
1992). }

\REF\FRA{P. Di Francesco and D. Kutasov, \np\ {\bf B375} (1992) 119. }

\REF\CHA{N. Chair, V. K. Dobrev and H. Kanno, \pl\ {\bf B283} (1992) 194.}

\REF\DOTB{Vl. S. Dotsenko, ``Remarks on the Physical States and the Chiral
Algebra of 2D Gravity Coupled to $c\le 1$ Matter,'' preprint PRA-LPTHE 92-4
(Jan., 1992). }

\REF\KACHA{S. Kachru, ``Extra States and Symmetries in $D<2$ Closed String
Theory,'' preprint PUPT-1314 (April, 1992). }

\REF\VER{E. Verlinde, ``The Master Equation of 2D String Theory,''
preprint IASSNS-HEP-92/5 (Feb., 1992). }

As shown in several recent papers [\KACH, \PAS, \BER], the symmetry
algebra and transformation rules can be used to derive Ward identities
relating different correlation functions. In fact the sub-algebra of
charges with zero ghost number can be exploited to completely
determine the multi-tachyon amplitudes. Similarly one expects
that the amplitudes containing more than one discrete states may be
determined from amplitudes containing less number of discrete states,
by exploiting Ward identities derived from charges with non-zero
ghost number. Let us close this paper with some remarks about this
problem.

First, if one tries to compute the four-point amplitudes containing
the exotic discrete states ($O_{jm}$ or $P_{jm}$) directly,
one always gets zero or $\infty$. This is because the
cosmological constant has been taken to be $\mu=0$ and the momenta been taken
at discrete values. One can circumvent
the problems by taking a generic $\mu$ and compute the amplitudes in
the presence of a cosmological constant [\LI] and regularizing the
$D=2$ string theory by $D<2$
string theory or equivalently the $c<1$ matter coupled to gravity
[\FRA]. At least in certain cases, taking the limit $c\to 1 $ could
lead to some meaningful results.  In principle there seems no problem
to extend our approach to $D<2$ string theory, but in practice this
extension may be confronted with some serious technical problems. (See
refs. [\CHA, \DOTB, \KACHA].)

Second, while it is not too difficult to derive the Ward identities
[\VER, \PAS], it is highly non-trivial to use them in computing
correlation functions. The difficulty lies in choosing the proper charges
to derive a proper set of Ward identities that is soluble to give
more-point amplitude in terms of less-point amplitudes.  There seems
no guarantee that all the more-point amplitudes can be obtained from
less-point amplitudes by Ward identities [\VER].
Combined with the regularization
problem mentioned above, the practical use of Ward identities in computing
more correlation functions remains a topic to be attacked.



\vglue .6cm

\centerline{Acknowledgement}

The authors would like to thank Dr. Miao Li for interesting discussions.
The work was supported in part by U.S. NSF through grant No. PHY-9008452
and the work of Zhu at SISSA/ISAS was supported by an INFN post-doctoral
fellowship.

\vfill\eject

\centerline{\bf  Appendix A.  Schur Polynomials}

The elementary Schur polynomials $S_k(x)$ are defined through the generating
function
$$ \sum_{k\ge 0} S_k(x)z^k = \exp\{ \sum_{k\ge 1}  x_k  z^k\}. \eqn\apa$$
where the $x_k$'s, with $k=1$, 2, $\cdots$, are the arguments of the
polynomials. What we are using in this paper is the Schur polynomials having
$x_k$'s to be
$$ x_k = - i \delta^-{ \partial^k X^+\over k!}
         - i \delta^+{ \partial^k X^-\over k!} , \eqn\apb$$
where $\delta^+$ and $\delta^-$ are two real numbers. To
simplify our notations we denote
$$ S_k(x) = S_k(-i \delta^-{ \partial^j X^+\over j!}
- i \delta^+{ \partial^j X^-\over j!}) \equiv
S_k(-i\delta^-X^+-i\delta^+X^-) .
\eqn\apc$$
These special Schur polynomials are related with the exponential function

\noindent $\exp\{iX^{\pm}(z+w)\}$ as follows
$$ \eqalign{ \sum_{k\ge 0} S_k(-i&\delta^-X^+(w)-i\delta^+X^-(w)) z^k
\cr & = \exp\{
-i \delta^-(X^+(z+w)-X^+(w)) -i \delta^+(X^-(z+w)- X^-(w) ) \} ,} \eqn\apd$$
or
$$ S_k(-i\delta^-X^+-i\delta^+X^-)  = { 1\over k!}
\partial^k\Big( \exp\{ -i \delta^-X^+ -i \delta^+X^- \}\Big).  \eqn\ape$$
We have been able to prove the following identities for OPE's containing
Schur polynomials:
$$ \partial X^+(z) :S_k(-i\delta X^-(w) ) : \sim
i\delta \sum_{j=1 }^k { 1\over (z-w)^{j+1} } :S_{k-j} (-i\delta X^-(w)):,
\eqn\apf$$
$$ : S_k(-i\delta X^-(z) ) :\partial X^+(w) \sim
i\delta\sum_{j=1}^k { (-1)^{j+1} \over (z-w)^{j+1} }
:S_{k-j}(-i\delta X^-(z)):, \eqn\apg$$
$$ \eqalign{ : S_k(-i\delta X^-(z) ) : & : e^{isX^+(w)}:\cr
&  \sim \sum_{n=1}^k
{ \Gamma({\delta s +n}) \over n! \Gamma({\delta s}) }{(-1)^n \over (z-w)^n }
:S_{k-n} (-i\delta X^-(z) ) e^{isX^+(w)  } : ,}  \eqn\api$$
$$ \eqalign{  : e^{isX^+(z)} : & : S_k(-i\delta X^-(w) ) : \cr
& \sim \sum_{n=1}^k  { \Gamma({\delta s +n}) \over n! \Gamma({\delta s}) }
{1 \over (z-w)^n } :S_{k-n} (-i\delta X^-(w) ) e^{isX^+(z)  } : .} \eqn\apj$$
These identities are crucial in our explicit determination of the structure
constants of the chiral BSRT cohomology ring.  Let us prove eqs. $\apf$ and
$\api$. The other two equations can be proved similarly.

Writing the Schur polynomial $S_k(-i\delta X^-(w))$ in a contour
integration form
$$S_k(-i\delta X^-(w)) = \oint_0[\dd Z] {1 \over Z^{k+1}}
e^{-i\delta\sum_{j=1}^\infty
{Z^j\over j!} \partial^jX^-(w) } , \eqn\appa$$
we have
$$\eqalign{
\partial X^+(z)& :S_k(-i\delta X^-(w) ) : =\oint_0[\dd Z]
{1 \over Z^{k+1}} \partial X^+(z)
 :e^{-i\delta\sum_{j=1}^{\infty} { Z^j \over j! } \partial^j X^-(w) } : \cr
& \sim - i\delta \oint_0[\dd Z]
{1 \over Z^{k+1}} \sum_{j=1}^{\infty} {Z^j\over j!} \langle
\partial X^+(z) \partial^j X^-(w) \rangle :e^{-i\delta\sum_{j=1}^{\infty}
{ Z^j \over j! } \partial^j X^-(w) } :\cr
& = i\delta \oint_0[\dd Z]
{1 \over Z^{k+1}} \sum_{j=1}^{\infty} {Z^j\over (z-w)^{j+1} }
:e^{-i\delta\sum_{j=1}^{\infty}
{ Z^j \over j! } \partial^j X^-(w) } :\cr
& = i\delta \sum_{j=1}^{k}{1\over (z-w)^{j+1}} :S_{k-j}(-i\delta X^-(w) ). }
\eqn\appb$$
This proves $\apf $. To prove $\api$ we use the following equation for
the normal ordering of two exponential functions:
$$ :e^{-i\delta \partial^j X^-(z)}: :e^{isX^+(w)}:  \sim
e^{\delta s \langle \partial^j X^-(z) X^+(w) \rangle }
: e^{-i\delta \partial^j X^-(z) + isX^+(w) } : . \eqn\appc$$
We have
$$\eqalign{
 :S_k&(-i\delta X^-(w) ) ::e^{isX^+(w)}: =
\oint_0[\dd Z] { 1\over Z^{k+1} }
:e^{-i\delta\sum_{j=1}^{\infty} { Z^j \over j! } \partial^j X^-(w) } :
:e^{isX^+(w)}: \cr
& \sim  \oint_0[\dd Z] { 1\over Z^{k+1} }
e^{\delta s \sum_{j=1}^{\infty} { Z^j\over j!}
\langle \partial^j X^-(z) X^+(w) \rangle }
:e^{ -i\delta \partial^j X^-(z) + isX^+(w) } : \cr
& = \oint_0[\dd Z] { 1\over Z^{k+1} }
{ 1\over  ( 1+ { Z\over z-w} )^{\delta s} }
:e^{ -i\delta \partial^j X^-(z) + isX^+(w) } :  \cr
& =  \sum_{n=1}^k
{ \Gamma({\delta s +n}) \over n! \Gamma({\delta s}) }{(-1)^n \over (z-w)^n }
:S_{k-n} (-i\delta X^-(z) ) e^{isX^+(w)  } : . } \eqn\cccccc$$

The other formula we use in the text is [\BMP]
$$ \sum_{m=j+1 } ^k (m-j)x_{m-j}S_{k-m}(x) = (k-j) S_{k-j}(x) ,  \eqn\apk$$
which can easily be proved through the generating function:
$$ \eqalign{
\sum_{k\ge j} (k-j)S_{k-j}z^{k-j} =&  z{ \dd\over \dd x} \Big( \sum_{k\ge j}
S_{k-j}(x)z ^{k-j} \Big) \cr
 = & z{ \dd\over \dd z} \big( \exp (\sum x_kz^k) \big) = \sum_{k\ge 1} kx_k
z^k\exp(\sum_{k}x_kz^k ) \cr
= \sum_{k\ge 1} \sum_{l\ge 0} kx_k & S_l(z) z^{k+l} = \sum_{k\ge j } \Big(
\sum_{m= j+1}^k (m-j)x_{m-j} S_{k-m}(x) \Big) z^{k-j} . } $$

\vglue .5cm

\centerline{\bf  Appendix B. The Chiral Transformation Rules
and Chiral Charge Algebra}

\def\jja{j_1m_1}
\def\jjb{j_2m_2}
\def\cjja{j_1+j_2,m_1+m_2}
\def\cjjb{j_1+j_2-1,m_1+m_2}
\def\cjjc{j_2-j_1, m_1+m_2}
\def\cjjd{j_2-j_1+1, m_1+m_2}
\def\cjje{j_1-j_2, m_1+m_2}
\def\cjjf{j_1-j_2+1, m_1+m_2}
\def\jjaa{(m_1j_2-m_2j_1)}
\def\jjab{(m_1j_2+m_2j_1+m_1)}
\def\jjac{(m_1j_2+m_2j_1+m_2)}

In this appendix we present the full set of chiral transformation rules.
All of them are derived from  $\rahh$ and $\zza$ by using our results
on the cohomology ring structure. We have
$$ \eqalign{
\{ \hat{X}_{j_1m_1} , Y^+_{j_2m_2} \} =&
-(m_1j_2- m_2j_1)O_{j_1+j_2-1, m_1+m_2}, \cr
[\hat{X}_{j_1m_1}, P_{j_2m_2}]
=& -(m_1j_2+m_2j_1+m_1)Y^-_{j_2-j_1+1, m_1+m_2}, \cr
} $$
$$\eqalign{
\{\hat{X}_{j_1m_1} , aO_{j_2m_2} \} =& j_1O_{j_1+j_2,m_1+m_2}, \cr
[ \hat{X}_{j_1m_1} , aY^+_{j_2m_2}]  =&{j_1j_2\over j_1+j_2}Y^+_{j_1+j_2,
m_1+m_2}+{j_2\over j_1+j_2}(m_1j_2-m_2j_1)aO_{j_1+j_2-1, m_1+m_2}, \cr
[ \hat{X}_{j_1m_1} , aY^-_{j_2m_2}] =& j_1Y^-_{j_2-j_1, m_1+m_2}, \cr
\{\hat{X}_{j_1m_1}, aP_{j_2m_2}\} =& {j_1(j_2+1)\over j_2-j_1+1}
P_{j_2-j_1, m_1+m_2}\cr
& +{j_2+1\over j_2-j_1+1}(m_1j_2+m_2j_1+m_1)aY^-_{j_2-j_1+1, m_1+m_2}, \cr
} $$
$$\eqalign{
 [Q^+_{j_1m_1}, O_{j_2m_2}] =& (m_1j_2-m_2j_1)O_{j_1+j_2-1, m_1+m_2}, \cr
[Q^+_{j_1m_1}, Y^+_{j_2m_2} ] = &(m_1j_2-m_2j_1)Y^+_{j_1+j_2-1, m_1+m_2}, \cr
 [Q^+_{j_1m_1}, Y^-_{j_2m_2} ]
=& -(m_1j_2+m_2j_1+m_1)Y^-_{j_2-j_1+1, m_1+m_2} , \cr
 [Q^+_{j_1m_1}, P_{j_2m_2}]
=& -(m_1j_2+m_2j_1+m_1)P_{j_2-j_1+1, m_1+m_2}, \cr
} $$
$$\eqalign{
[Q^+_{\jja}, aO_{\jjb}] = & -{j_1(j_1-1)\over j_1+j_2} Y^+_{\cjja}
 \cr & + {j_2+1\over j_1+j_2} \jjaa aO_{\cjjb}, \cr
   [Q^+_{\jja}, aY^+_{\jjb}] =& {j_2\over j_1+j_2-1} \jjaa aY^+_{\cjjb}, \cr
   [Q^+_{\jja}, aY^-_{\jjb}] =& -{j_1(j_1-1)\over j_2-j_1+1} P_{\cjjc}
 \cr & -{j_2\over j_2-j_1+1}\jjab aY^-_{\cjjd}, \cr
   [Q^+_{\jja}, aP_{\jjb}] =& -{j_2+1\over j_2-j_1+2}\jjab aP_{\cjjd}, \cr
} $$
$$\eqalign{
   [Q^-_{\jja}, Y^+_{\jjb}] =& \jjac Y^-_{\cjjf}, \cr
   [Q^-_{\jja}, aO_{\jjb}] =& (j_1+1)Y^-_{\cjje}, \cr
   [Q^-_{\jja}, aY^+_{\jjb}] =& -{j_2(j_1+1)\over j_1-j_2+1} P_{\cjje}
 \cr & -{j_2\over j_1-j_2+1}\jjac aY^-_{\cjjf}, \cr
} $$
$$\eqalign{
   [\hat{Z}_{\jja}, O_{\jjb}] =& -\jjac Y^-_{\cjjf}, \cr
   \{\hat{Z}_{\jja}, Y^+_{\jjb}\} =& -\jjac P_{\cjjf}, \cr
} $$
$$\eqalign{
   \{\hat{Z}_{\jja}, aO_{\jjb}\} =& -{(j_1+1)(j_1+2)\over j_1-j_2+1}P_{\cjje}
 \cr & -{j_2+1\over j_1-j_2+1}\jjac aY^-_{\cjjf}, \cr
   [\hat{Z}_{\jja}, aY^+_{\jjb}] =& -{j_2\over j_1-j_2+2}\jjac aP_{\cjjf}, \cr
} $$
$$\eqalign{
   [a\hat{X}_{\jja}, O_{\jjb}] =& j_2O_{\cjja}, \cr
   [a\hat{X}_{\jja}, Y^+_{\jjb}] =& {j_2(j_2-1)\over j_1+j_2} Y^+_{\cjja}
 \cr & +{j_1+1\over j_1+j_2}\jjaa aO_{\cjjb}, \cr
   [a\hat{X}_{\jja}, Y^-_{\jjb}] =& -(j_2+1)Y^-_{\cjjc}, \cr
   [a\hat{X}_{\jja}, P_{\jjb}] =& -{(j_2+1)(j_2+2)\over j_2-j_1+1} P_{\cjjc}
 \cr & -{j_1+1\over j_2-j_1+1}\jjab aY^-_{\cjjd}, \cr
} $$
$$\eqalign{
   [a\hat{X}_{\jja}, aO_{\jjb}] =& -(j_1-j_2)aO_{\cjja}, \cr
   [a\hat{X}_{\jja}, aY^+_{\jjb}] =& -{j_2(j_1-j_2+1)\over j_1+j_2}
aY^+_{\cjja}, \cr
   [a\hat{X}_{\jja}, aY^-_{\jjb}] =& -(j_1+j_2+1)aY^-_{\cjja}, \cr
   [a\hat{X}_{\jja}, aP_{\jjb}] =& -{(j_2+1)(j_1+j_2+2)\over j_2-j_1+1}
aP_{\cjjc}, \cr
} $$
$$\eqalign{
   [aQ^+_{\jja}, O_{\jjb}] =& {j_1j_2\over j_1+j_2}Y^+_{\cjja}
 -{j_1\over j_1+j_2} \jjaa aO_{\cjjb}, \cr
   \{aQ^+_{\jja}, Y^+_{\jjb}\} =& -{j_1\over j_1+j_2-1}\jjaa aY^+_{\cjjb}, \cr
   \{aQ^+_{\jja}, Y^-_{\jjb}\} =& { j_1(j_2+1) \over j_2-j_1+1}P_{\cjjc}
 \cr & -{j_1\over j_2-j_1+1}\jjab aY^-_{\cjjd}, \cr
   [aQ^+_{\jja}, P_{\jjb}] =& -{j_1\over j_2-j_1+2} \jjab aP_{\cjjd}, \cr
} $$
$$\eqalign{
   \{aQ^+_{\jja}, aO_{\jjb}\} =& {j_1(j_1-j_2-1)\over j_1+j_2}
aY^+_{\cjja}, \cr
   [aQ^+_{\jja}, aY^-_{\jjb}] =& {j_1(j_1+j_2)\over j_2-j_1+1} aP_{\cjjc}, \cr
} $$
$$\eqalign{
   [aQ^-_{\jja}, O_{\jjb}] =& j_2Y^-_{\cjje}, \cr
   \{aQ^-_{\jja}, Y^+_{\jjb}\} =& -{j_2(j_2-1)\over j_1-j_2+1} P_{\cjje}
 \cr & -{j_1\over j_1-j_2+1} \jjac aY^-_{\cjjf}, \cr
   \{aQ^-_{\jja}, aO_{\jjb}\} =& -(j_1+j_2+1)aY^-_{\cjje}, \cr
   [aQ^-_{\jja}, aY^+_{\jjb}] =& {j_2(j_1+j_2)\over j_1-j_2+1} aP_{\cjje}, \cr
} $$
$$\eqalign{
   [a\hat{Z}_{\jja}, O_{\jjb}] =& {j_2(j_1+1)\over j_1-j_2+1} P_{\cjje}
 \cr & +{j_1+1\over j_1-j_2+1} \jjac aY^-_{\cjjf},  \cr
   [a\hat{Z}_{\jja}, Y^+_{\jjb}] =& {j_1+1\over j_1-j_2+1} \jjac
aP_{\cjjf}, \cr
   [a\hat{Z}_{\jja}, aO_{\jjb}] =& {(j_1+1)(j_1+j_2+2)\over j_1-j_2+1}
aP_{\cjje}, \cr
}$$
$$\eqalign{
[\hat{X}_{j_1m_1} , O_{j_2m_2} ] =& 0, \cr
   [Q^-_{\jja}, O_{\jjb}] =& 0, \cr
   [Q^-_{\jja}, P_{\jjb}] =& 0, \cr
   [Q^-_{\jja}, aP_{\jjb}] =&0, \cr
   [aQ^-_{\jja}, P_{\jjb}] =& 0, \cr
   [\hat{Z}_{\jja}, P_{\jjb}] =& 0, \cr
   \{\hat{Z}_{\jja}, aP_{\jjb}\} =& 0, \cr
   \{aQ^+_{\jja}, aP_{\jjb}\} =& 0, \cr
   \{aQ^-_{\jja}, aP_{\jjb}\} =& 0, \cr
   [a\hat{Z}_{\jja}, P_{\jjb}] =& 0, \cr
   [a\hat{Z}_{\jja}, aY^-_{\jjb}] =& 0, \cr }\qquad
\eqalign{
\{ \hat{X}_{j_1m_1} , Y^-_{j_2m_2}\} =& 0, \cr
   [Q^-_{\jja}, Y^-_{\jjb}] =& 0, \cr
   [Q^-_{\jja}, aY^-_{\jjb}] =& 0, \cr
   \{aQ^-_{\jja}, Y^-_{\jjb}\} =& 0, \cr
   \{\hat{Z}_{\jja}, Y^-_{\jjb}\} =& 0, \cr
   [\hat{Z}_{\jja}, aY^-_{\jjb}] =& 0, \cr
   [aQ^+_{\jja}, aY^+_{\jjb}] =& 0, \cr
   [aQ^-_{\jja}, aY^-_{\jjb}] =& 0, \cr
   [a\hat{Z}_{\jja}, Y^-_{\jjb}] =& 0, \cr
   [a\hat{Z}_{\jja}, aY^+_{\jjb}] =& 0, \cr
   [a\hat{Z}_{\jja}, aP_{\jjb}] =& 0. \cr } $$
The chiral charge algebra can be obtained from the above equations by acting
with $\oint_0[\dd w]$  $\oint_w[\dd z] b(z)$ from the left. This action
transforms all the local fields into the corresponding charges, as shown
in sec. 5.

\vglue .5cm

\centerline{ \bf Appendix C. Proof of Eqs. $\pqq$ and $\pyqq$ }

In this appendix we will give some details about the computation of the
transformation rules in closed string theory and also the proof of eqs.
$\pqq$ and $\pyqq$. To simplify the presentation we omit the ``magnetic''
index of the (chiral) discrete states and write the chiral cohomology ring
structure equation as follows:
$$V_{j_1}V_{j_2} = A_j + aB_{j'} , \eqn\cappa$$
where $V_{j_1}$, $ V_{j_2}$, $A_j$ and $B_{j'}$ are all chiral relative
discrete states. By using $\rahh$ and $\zza$ we can write the chiral
transformation rules as follows:
$$ \eqalign{
& [\hat{V}_{j_1}, V_{j_2}] = g_B(j')B_{j'}, \cr
& [\hat{V}_{j_1}, aV_{j_2}]
= (-1)^{[V_{j_1}]} \big( (g_V(j_1) -1)A_j -g_V(j_2)aB_{j'} \big), \cr
& [a\hat{V}_{j_1}, V_{j_2}] = (g_V(j_2) - 1)A_j - g_V(j_1)aB_{j'}, \cr
& [a\hat{V}_{j_1}, aV_{j_2}]
= -(-1)^{[V_{j_1}]} \big( g_V(j_1) - g_V(j_2) \big) aA_j.
} \eqn\cappb$$
Note that the relation $g_A(j)= g_B(j')= g_V(j_1)+g_V(j_2)-1$ has been used
in our derivation of the above rules.

A relative closed string discrete state is just the product a left relative
discrete state and a right relative discrete states: $V_{j_1\bar{j}_1} =
V_{j_1}\bar{V}_{\bar{j}_1 } $.  A necessary and sufficient condition for the
pairing (at the $SU(2)$ point, of course) is $g_V(j_1)=g_{\bar{V}}(\bar{j}_1)$.
The charge $\hat{V}_{j_1\bar{j}_1}$ derived from the discrete state
$V_{j_1\bar{j}_1}$ can be written symbolically  (for the purpose of doing
calculation) as follows
$$\hat{V}_{j_1\bar{j}_1} = \hat{V}_{j_1} \bar{V}_{\bar{j}_1} -
(-1)^{[V_{j_1}]} {V}_{j_1} \hat{\bar{V}}_{\bar{j}_1} . \eqn\cappc$$
The action of $\hat{V}_{j_1\bar{j}_1}$ on $V_{j_2\bar{j}_2} =
V_{j_2}\bar{V}_{\bar{j}_2 } $ is computed as follows:
$$\eqalign{
[ \hat{V}_{j_1\bar{j}_1} &,V_{j_2\bar{j}_2} ] =
  [ \hat{V}_{j_1} \bar{V}_{\bar{j}_1} -
(-1)^{[V_{j_1}]} {V}_{j_1} \hat{\bar{V}}_{\bar{j}_1},
V_{j_2}\bar{V}_{\bar{j}_2 }] \cr
= & (-1)^{[\bar{V}_{\bar{j}_1} ][V_{j_2}]} \Big( [ \hat{V}_{j_1}, V_{j_2}]
\bar{V}_{\bar{j}_1} \bar{V}_{\bar{j}_2} - (-1)^{[V_{j_1}] + [V_{j_2}] }
V_{j_1}V_{j_2}[\hat{\bar{V}}_{\bar{j}_1}, \bar{V}_{\bar{j}_2} ]\Big)  \cr
= & (-1)^{[\bar{V}_{\bar{j}_1} ][V_{j_2}] } g_B(j')
\Big(B_{j'}\bar{A}_{\bar{j}}
- (-1)^{[V_{j_1}] + [V_{j_2}] } A_j \bar{B}_{\bar{j}'} \cr
& \qquad\qquad\qquad \quad
- (-1)^{[V_{j_1}] + [V_{j_2}] } (a+\bar{a}) B_{j`}\bar{B}_{\bar{j}'} \Big) .
} \eqn\cappd$$
 From the above result we see that the action of any charge on the closed
string discrete state gives only physical closed string states. No states
other than  those listed in $\zze$ and $\zzf$ are created.

The action of $\hat{V}_{j_1\bar{j}_1}$ on the absolute state $(a+\bar{a})
V_{j_2\bar{j}_2}$ can also be computed. We have
$$ \eqalign{
[ \hat{V}_{j_1\bar{j}_1} &,(a+\bar{a})V_{j_2\bar{j}_2} ] =
[ \hat{V}_{j_1} \bar{V}_{\bar{j}_1} -
(-1)^{[V_{j_1}]} {V}_{j_1} \hat{\bar{V}}_{\bar{j}_1},
aV_{j_2}\bar{V}_{\bar{j}_2 }+(-1)^{[V_{j_2}]}
V_{j_2}\bar{a}\bar{V}_{\bar{j}_2 }] \cr
= & (-1)^{[\bar{V}_{\bar{j}_1} ][V_{j_2}] + [\bar{V}_{\bar{j}_1} ]
+ [V_{j_1}] -1} g_B(j')(a+\bar{a}) \Big(B_{j'}\bar{A}_{\bar{j}}
-   (-1)^{[V_{j_1}] + [V_{j_2}] } A_j \bar{B}_{\bar{j}'}  \Big) \cr
= & (-1)^{ [\bar{V}_{\bar{j}_1} ]
+ [V_{j_1}] -1} (a+\bar{a})
[ \hat{V}_{j_1\bar{j}_1} ,V_{j_2\bar{j}_2} ],  \cr
} \eqn\cappe$$
by using $\cappd$. This proves $\pqq$.

Eq. $\pyqq$ can be proved similarly. We have
$$ \eqalign{
[ (a+\bar{a})&\hat{V}_{j_1\bar{j}_1} ,V_{j_2\bar{j}_2} ] \cr
= & [ a\hat{V}_{j_1} \bar{V}_{\bar{j}_1}
 - {V}_{j_1}\bar{a} \hat{\bar{V}}_{\bar{j}_1}
 + (-1)^{[V_{j_1}]} a{V}_{j_1} \hat{\bar{V}}_{\bar{j}_1}
 + (-1)^{[V_{j_1}]} \hat{V}_{j_1}\bar{a} \bar{V}_{\bar{j}_1},
V_{j_2}\bar{V}_{\bar{j}_2 }] \cr
= & (-1)^{[\bar{V}_{\bar{j}_1} ][V_{j_2}] }
g_A(j)(a+\bar{a}) \Big(-B_{j'}\bar{A}_{\bar{j}}
 +   (-1)^{[V_{j_1}] + [V_{j_2}] } A_j \bar{B}_{\bar{j}'}  \Big) \cr
= & - (a+\bar{a})
[ \hat{V}_{j_1\bar{j}_1} ,V_{j_2\bar{j}_2} ],  \cr
} \eqn\cappf$$
and
$$ \eqalign{
[ (a+\bar{a})\hat{V}_{j_1\bar{j}_1},(a+\bar{a})V_{j_2\bar{j}_2} ] = &
[ a\hat{V}_{j_1} \bar{V}_{\bar{j}_1}
 - {V}_{j_1}\bar{a} \hat{\bar{V}}_{\bar{j}_1}
 + (-1)^{[V_{j_1}]} a{V}_{j_1} \hat{\bar{V}}_{\bar{j}_1}\cr
& +  (-1)^{[V_{j_1}]} \hat{V}_{j_1}\bar{a} \bar{V}_{\bar{j}_1},
aV_{j_2}\bar{V}_{\bar{j}_2 }+(-1)^{[V_{j_2}]}
V_{j_2}\bar{a}\bar{V}_{\bar{j}_2 }] \cr
= & \quad 0.  \cr
}\eqn\cappg$$
This concludes the proof of $\pyqq$.

\vfill\eject

\refout

\bye



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2161 1961 n
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2156 1944 n
2156 1956 m
2168 1944 n
2165 1950 m
1771 1961 m
1766 1959 n
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1761 1951 n
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1766 1941 n
1771 1939 n
1774 1939 n
1778 1941 n
1781 1944 n
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1783 1951 n
1781 1956 n
1778 1959 n
1774 1961 n
1771 1961 n
1778 1956 m
1766 1944 n
1766 1956 m
1778 1944 n
1775 1950 m
1381 1961 m
1376 1959 n
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1371 1951 n
1371 1949 n
1372 1944 n
1376 1941 n
1381 1939 n
1384 1939 n
1388 1941 n
1391 1944 n
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1393 1951 n
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1384 1961 n
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1376 1944 n
1376 1956 m
1388 1944 n
1385 1950 m
1576 2221 m
1571 2219 n
1567 2216 n
1566 2211 n
1566 2209 n
1567 2204 n
1571 2201 n
1576 2199 n
1579 2199 n
1583 2201 n
1586 2204 n
1588 2209 n
1588 2211 n
1586 2216 n
1583 2219 n
1579 2221 n
1576 2221 n
1583 2216 m
1571 2204 n
1571 2216 m
1583 2204 n
1580 2210 m
1966 2221 m
1961 2219 n
1957 2216 n
1956 2211 n
1956 2209 n
1957 2204 n
1961 2201 n
1966 2199 n
1969 2199 n
1973 2201 n
1976 2204 n
1978 2209 n
1978 2211 n
1976 2216 n
1973 2219 n
1969 2221 n
1966 2221 n
1973 2216 m
1961 2204 n
1961 2216 m
1973 2204 n
1970 2210 m
1771 2481 m
1766 2479 n
1762 2476 n
1761 2471 n
1761 2469 n
1762 2464 n
1766 2461 n
1771 2459 n
1774 2459 n
1778 2461 n
1781 2464 n
1783 2469 n
1783 2471 n
1781 2476 n
1778 2479 n
1774 2481 n
1771 2481 n
1778 2476 m
1766 2464 n
1766 2476 m
1778 2464 n
1775 2470 m
1966 2741 m
1961 2739 n
1957 2736 n
1956 2731 n
1956 2729 n
1957 2724 n
1961 2721 n
1966 2719 n
1969 2719 n
1973 2721 n
1976 2724 n
1978 2729 n
1978 2731 n
1976 2736 n
1973 2739 n
1969 2741 n
1966 2741 n
1973 2736 m
1961 2724 n
1961 2736 m
1973 2724 n
1970 2730 m
1576 2741 m
1571 2739 n
1567 2736 n
1566 2731 n
1566 2729 n
1567 2724 n
1571 2721 n
1576 2719 n
1579 2719 n
1583 2721 n
1586 2724 n
1588 2729 n
1588 2731 n
1586 2736 n
1583 2739 n
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1576 2741 n
1583 2736 m
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1580 2730 m
1381 3001 m
1376 2999 n
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2360 3250 m
1966 3261 m
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1966 3261 n
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1961 3256 m
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1576 3261 m
1571 3259 n
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1583 3256 m
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1580 3250 m
1186 3261 m
1181 3259 n
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991 3521 m
986 3519 n
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2555 3510 m
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2161 1441 m
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2161 1419 n
2164 1419 n
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2164 1441 n
2161 1441 n
2162 1438 m
2162 1422 n
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1771 1441 m
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1772 1438 m
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1381 1441 m
1376 1439 n
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1371 1431 n
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1381 1419 n
1384 1419 n
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1381 1441 n
1382 1438 m
1382 1422 n
1374 1430 m
1389 1430 n
1385 1430 m
991 1441 m
986 1439 n
982 1436 n
981 1431 n
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991 1419 n
994 1419 n
998 1421 n
1001 1424 n
1003 1429 n
1003 1431 n
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994 1441 n
991 1441 n
992 1438 m
992 1422 n
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999 1430 n
995 1430 m
1186 1701 m
1181 1699 n
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1181 1681 n
1186 1679 n
1189 1679 n
1193 1681 n
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1196 1696 n
1193 1699 n
1189 1701 n
1186 1701 n
1187 1698 m
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1179 1690 m
1194 1690 n
1190 1690 m
1576 1701 m
1571 1699 n
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1566 1691 n
1566 1689 n
1567 1684 n
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1579 1679 n
1583 1681 n
1586 1684 n
1588 1689 n
1588 1691 n
1586 1696 n
1583 1699 n
1579 1701 n
1576 1701 n
1577 1698 m
1577 1682 n
1569 1690 m
1584 1690 n
1580 1690 m
1966 1701 m
1961 1699 n
1957 1696 n
1956 1691 n
1956 1689 n
1957 1684 n
1961 1681 n
1966 1679 n
1969 1679 n
1973 1681 n
1976 1684 n
1978 1689 n
1978 1691 n
1976 1696 n
1973 1699 n
1969 1701 n
1966 1701 n
1967 1698 m
1967 1682 n
1959 1690 m
1974 1690 n
1970 1690 m
2356 1701 m
2351 1699 n
2347 1696 n
2346 1691 n
2346 1689 n
2347 1684 n
2351 1681 n
2356 1679 n
2359 1679 n
2363 1681 n
2366 1684 n
2368 1689 n
2368 1691 n
2366 1696 n
2363 1699 n
2359 1701 n
2356 1701 n
2357 1698 m
2357 1682 n
2349 1690 m
2364 1690 n
2360 1690 m
2161 1961 m
2156 1959 n
2152 1956 n
2151 1951 n
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2152 1944 n
2156 1941 n
2161 1939 n
2164 1939 n
2168 1941 n
2171 1944 n
2173 1949 n
2173 1951 n
2171 1956 n
2168 1959 n
2164 1961 n
2161 1961 n
2162 1958 m
2162 1942 n
2154 1950 m
2169 1950 n
2165 1950 m
1771 1961 m
1766 1959 n
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1762 1944 n
1766 1941 n
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1764 1950 m
1779 1950 n
1775 1950 m
1381 1961 m
1376 1959 n
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1371 1951 n
1371 1949 n
1372 1944 n
1376 1941 n
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1389 1950 n
1385 1950 m
1576 2221 m
1571 2219 n
1567 2216 n
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1566 2209 n
1567 2204 n
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1583 2201 n
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1588 2209 n
1588 2211 n
1586 2216 n
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1576 2221 n
1577 2218 m
1577 2202 n
1569 2210 m
1584 2210 n
1580 2210 m
1966 2221 m
1961 2219 n
1957 2216 n
1956 2211 n
1956 2209 n
1957 2204 n
1961 2201 n
1966 2199 n
1969 2199 n
1973 2201 n
1976 2204 n
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1969 2221 n
1966 2221 n
1967 2218 m
1967 2202 n
1959 2210 m
1974 2210 n
1970 2210 m
1771 2481 m
1766 2479 n
1762 2476 n
1761 2471 n
1761 2469 n
1762 2464 n
1766 2461 n
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1778 2461 n
1781 2464 n
1783 2469 n
1783 2471 n
1781 2476 n
1778 2479 n
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1771 2481 n
1772 2478 m
1772 2462 n
1764 2470 m
1779 2470 n
1775 2470 m
1966 2741 m
1961 2739 n
1957 2736 n
1956 2731 n
1956 2729 n
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1969 2719 n
1973 2721 n
1976 2724 n
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1978 2731 n
1976 2736 n
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1969 2741 n
1966 2741 n
1967 2738 m
1967 2722 n
1959 2730 m
1974 2730 n
1970 2730 m
1576 2741 m
1571 2739 n
1567 2736 n
1566 2731 n
1566 2729 n
1567 2724 n
1571 2721 n
1576 2719 n
1579 2719 n
1583 2721 n
1586 2724 n
1588 2729 n
1588 2731 n
1586 2736 n
1583 2739 n
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1576 2741 n
1577 2738 m
1577 2722 n
1569 2730 m
1584 2730 n
1580 2730 m
1381 3001 m
1376 2999 n
1372 2996 n
1371 2991 n
1371 2989 n
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1376 2981 n
1381 2979 n
1384 2979 n
1388 2981 n
1391 2984 n
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1393 2991 n
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1384 3001 n
1381 3001 n
1382 2998 m
1382 2982 n
1374 2990 m
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1771 3001 m
1766 2999 n
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1778 2981 n
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1772 2998 m
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2161 3001 m
2156 2999 n
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2161 2979 n
2164 2979 n
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2164 3001 n
2161 3001 n
2162 2998 m
2162 2982 n
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2169 2990 n
2165 2990 m
2356 3261 m
2351 3259 n
2347 3256 n
2346 3251 n
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2347 3244 n
2351 3241 n
2356 3239 n
2359 3239 n
2363 3241 n
2366 3244 n
2368 3249 n
2368 3251 n
2366 3256 n
2363 3259 n
2359 3261 n
2356 3261 n
2357 3258 m
2357 3242 n
2349 3250 m
2364 3250 n
2360 3250 m
1966 3261 m
1961 3259 n
1957 3256 n
1956 3251 n
1956 3249 n
1957 3244 n
1961 3241 n
1966 3239 n
1969 3239 n
1973 3241 n
1976 3244 n
1978 3249 n
1978 3251 n
1976 3256 n
1973 3259 n
1969 3261 n
1966 3261 n
1967 3258 m
1967 3242 n
1959 3250 m
1974 3250 n
1970 3250 m
1576 3261 m
1571 3259 n
1567 3256 n
1566 3251 n
1566 3249 n
1567 3244 n
1571 3241 n
1576 3239 n
1579 3239 n
1583 3241 n
1586 3244 n
1588 3249 n
1588 3251 n
1586 3256 n
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1576 3261 n
1577 3258 m
1577 3242 n
1569 3250 m
1584 3250 n
1580 3250 m
1186 3261 m
1181 3259 n
1177 3256 n
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1177 3244 n
1181 3241 n
1186 3239 n
1189 3239 n
1193 3241 n
1196 3244 n
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1196 3256 n
1193 3259 n
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1186 3261 n
1187 3258 m
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1179 3250 m
1194 3250 n
1190 3250 m
991 3521 m
986 3519 n
982 3516 n
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991 3499 n
994 3499 n
998 3501 n
1001 3504 n
1003 3509 n
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994 3521 n
991 3521 n
992 3518 m
992 3502 n
984 3510 m
999 3510 n
995 3510 m
1381 3521 m
1376 3519 n
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1381 3499 n
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1382 3518 m
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1374 3510 m
1389 3510 n
1385 3510 m
1771 3521 m
1766 3519 n
1762 3516 n
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1772 3518 m
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1764 3510 m
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1775 3510 m
2161 3521 m
2156 3519 n
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2161 3499 n
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2168 3501 n
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2164 3521 n
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2162 3518 m
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2154 3510 m
2169 3510 n
2165 3510 m
2551 3521 m
2546 3519 n
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2558 3501 n
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2552 3518 m
2552 3502 n
2544 3510 m
2559 3510 n
2555 3510 m
2554 1392 m
2550 1388 n
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2541 1366 n
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2550 1388 m
2546 1381 n
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2580 1365 m
2564 1384 m
2576 1365 n
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2565 1384 m
2578 1365 n
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2591 1384 m
2578 1365 n
2560 1384 m
2571 1384 n
2583 1384 m
2594 1384 n
2571 1347 m
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2618 1365 m
2614 1396 m
2614 1374 n
2603 1385 m
2625 1385 n
2644 1386 m
2603 1352 m
2604 1351 n
2603 1350 n
2602 1351 n
2602 1352 n
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2604 1356 n
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2606 1341 n
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2602 1335 n
2602 1332 n
2613 1357 m
2615 1356 n
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2608 1343 n
2602 1334 m
2603 1335 n
2606 1335 n
2612 1333 n
2615 1333 n
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2606 1335 m
2612 1332 n
2616 1332 n
2618 1333 n
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2637 1344 m
2628 1332 m
2627 1333 n
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2630 1333 n
2630 1331 n
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2649 1344 m
2639 1352 m
2640 1351 n
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2638 1351 n
2638 1352 n
2639 1355 n
2640 1356 n
2644 1357 n
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2652 1356 n
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2655 1352 n
2655 1350 n
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2650 1345 n
2644 1343 n
2642 1341 n
2639 1339 n
2638 1335 n
2638 1332 n
2649 1357 m
2651 1356 n
2652 1355 n
2654 1352 n
2654 1350 n
2652 1347 n
2649 1345 n
2644 1343 n
2638 1334 m
2639 1335 n
2642 1335 n
2648 1333 n
2651 1333 n
2654 1334 n
2655 1335 n
2642 1335 m
2648 1332 n
2652 1332 n
2654 1333 n
2655 1335 n
2655 1338 n
2673 1344 m
2664 1392 m
2667 1388 n
2671 1383 n
2675 1375 n
2676 1366 n
2676 1360 n
2675 1351 n
2671 1344 n
2667 1338 n
2664 1335 n
2667 1388 m
2671 1381 n
2673 1375 n
2675 1366 n
2675 1360 n
2673 1351 n
2671 1346 n
2667 1338 n
2555 1365 m
2164 1392 m
2160 1388 n
2156 1383 n
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1980 1357 n
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1985 1332 n
2006 1344 m
1997 1332 m
1996 1333 n
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1998 1333 n
1998 1331 n
1997 1328 n
1996 1327 n
2018 1344 m
2014 1357 m
2010 1356 n
2008 1352 n
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2010 1333 n
2014 1332 n
2016 1332 n
2020 1333 n
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2020 1356 n
2016 1357 n
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2010 1334 n
2012 1333 n
2014 1332 n
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2020 1355 n
2019 1356 n
2016 1357 n
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1667 1604 m
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2033 1625 m
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1413 1904 m
1424 1904 n
1401 1867 m
1413 1867 n
1448 1885 m
1444 1916 m
1444 1894 n
1433 1905 m
1455 1905 n
1474 1906 m
1436 1872 m
1438 1873 n
1442 1877 n
1442 1852 n
1440 1876 m
1440 1852 n
1436 1852 m
1446 1852 n
1467 1864 m
1458 1852 m
1457 1853 n
1458 1854 n
1460 1853 n
1460 1851 n
1458 1848 n
1457 1847 n
1479 1864 m
1469 1863 m
1491 1863 n
1510 1864 m
1503 1872 m
1505 1873 n
1509 1877 n
1509 1852 n
1508 1876 m
1508 1852 n
1503 1852 m
1514 1852 n
1534 1864 m
1525 1912 m
1529 1908 n
1532 1903 n
1536 1895 n
1538 1886 n
1538 1880 n
1536 1871 n
1532 1864 n
1529 1858 n
1525 1855 n
1529 1908 m
1532 1901 n
1534 1895 n
1536 1886 n
1536 1880 n
1534 1871 n
1532 1866 n
1529 1858 n
1385 1885 m
1579 2172 m
1575 2168 n
1571 2163 n
1568 2155 n
1566 2146 n
1566 2140 n
1568 2131 n
1571 2124 n
1575 2118 n
1579 2115 n
1575 2168 m
1571 2161 n
1570 2155 n
1568 2146 n
1568 2140 n
1570 2131 n
1571 2126 n
1575 2118 n
1605 2145 m
1589 2164 m
1601 2145 n
1601 2127 n
1590 2164 m
1603 2145 n
1603 2127 n
1616 2164 m
1603 2145 n
1585 2164 m
1596 2164 n
1608 2164 m
1619 2164 n
1596 2127 m
1608 2127 n
1643 2145 m
1639 2176 m
1639 2154 n
1628 2165 m
1650 2165 n
1669 2166 m
1630 2144 m
1631 2145 n
1634 2146 n
1634 2130 n
1633 2145 m
1633 2130 n
1630 2130 m
1637 2130 n
1650 2138 m
1625 2123 m
1643 2123 n
1652 2109 m
1628 2115 m
1629 2114 n
1628 2113 n
1627 2114 n
1627 2115 n
1628 2117 n
1629 2117 n
1631 2118 n
1635 2118 n
1637 2117 n
1638 2117 n
1639 2115 n
1639 2113 n
1638 2112 n
1635 2110 n
1631 2109 n
1630 2108 n
1628 2106 n
1627 2104 n
1627 2101 n
1635 2118 m
1636 2117 n
1637 2117 n
1638 2115 n
1638 2113 n
1637 2112 n
1635 2110 n
1631 2109 n
1627 2103 m
1628 2104 n
1630 2104 n
1634 2102 n
1636 2102 n
1638 2103 n
1639 2104 n
1630 2104 m
1634 2102 n
1637 2102 n
1638 2102 n
1639 2104 n
1639 2105 n
1651 2110 m
1647 2112 m
1646 2113 n
1647 2114 n
1648 2113 n
1648 2111 n
1647 2108 n
1646 2107 n
1667 2124 m
1658 2123 m
1679 2123 n
1699 2124 m
1691 2144 m
1692 2145 n
1695 2146 n
1695 2130 n
1694 2145 m
1694 2130 n
1691 2130 m
1698 2130 n
1711 2138 m
1686 2123 m
1703 2123 n
1713 2110 m
1689 2115 m
1690 2114 n
1689 2114 n
1688 2114 n
1688 2115 n
1689 2117 n
1690 2118 n
1692 2118 n
1695 2118 n
1698 2118 n
1699 2117 n
1699 2115 n
1699 2113 n
1699 2112 n
1696 2110 n
1692 2109 n
1690 2108 n
1689 2106 n
1688 2104 n
1688 2101 n
1695 2118 m
1697 2117 n
1698 2117 n
1698 2115 n
1698 2113 n
1698 2112 n
1695 2110 n
1692 2109 n
1688 2103 m
1689 2104 n
1690 2104 n
1694 2102 n
1697 2102 n
1699 2103 n
1699 2104 n
1690 2104 m
1694 2102 n
1698 2102 n
1698 2102 n
1699 2104 n
1699 2105 n
1711 2110 m
1707 2172 m
1711 2168 n
1714 2163 n
1718 2155 n
1720 2146 n
1720 2140 n
1718 2131 n
1714 2124 n
1711 2119 n
1707 2115 n
1711 2168 m
1714 2161 n
1716 2155 n
1718 2146 n
1718 2140 n
1716 2131 n
1714 2126 n
1711 2119 n
1580 2145 m
1969 2172 m
1965 2168 n
1961 2163 n
1958 2155 n
1956 2146 n
1956 2140 n
1958 2131 n
1961 2124 n
1965 2118 n
1969 2115 n
1965 2168 m
1961 2161 n
1960 2155 n
1958 2146 n
1958 2140 n
1960 2131 n
1961 2126 n
1965 2118 n
1995 2145 m
1979 2164 m
1991 2145 n
1991 2127 n
1980 2164 m
1993 2145 n
1993 2127 n
2006 2164 m
1993 2145 n
1975 2164 m
1986 2164 n
1998 2164 m
2009 2164 n
1986 2127 m
1998 2127 n
2033 2145 m
2029 2176 m
2029 2154 n
2018 2165 m
2040 2165 n
2059 2166 m
2020 2144 m
2021 2145 n
2024 2146 n
2024 2130 n
2023 2145 m
2023 2130 n
2020 2130 m
2027 2130 n
2040 2138 m
2015 2123 m
2033 2123 n
2042 2109 m
2018 2115 m
2019 2114 n
2018 2113 n
2017 2114 n
2017 2115 n
2018 2117 n
2019 2117 n
2021 2118 n
2025 2118 n
2027 2117 n
2028 2117 n
2029 2115 n
2029 2113 n
2028 2112 n
2025 2110 n
2021 2109 n
2020 2108 n
2018 2106 n
2017 2104 n
2017 2101 n
2025 2118 m
2026 2117 n
2027 2117 n
2028 2115 n
2028 2113 n
2027 2112 n
2025 2110 n
2021 2109 n
2017 2103 m
2018 2104 n
2020 2104 n
2024 2102 n
2026 2102 n
2028 2103 n
2029 2104 n
2020 2104 m
2024 2102 n
2027 2102 n
2028 2102 n
2029 2104 n
2029 2105 n
2041 2110 m
2037 2112 m
2036 2113 n
2037 2114 n
2038 2113 n
2038 2111 n
2037 2108 n
2036 2107 n
2057 2124 m
2049 2144 m
2051 2145 n
2053 2146 n
2053 2130 n
2053 2145 m
2053 2130 n
2049 2130 m
2057 2130 n
2070 2138 m
2045 2123 m
2062 2123 n
2072 2110 m
2048 2115 m
2049 2114 n
2048 2114 n
2047 2114 n
2047 2115 n
2048 2117 n
2049 2118 n
2051 2118 n
2054 2118 n
2057 2118 n
2057 2117 n
2058 2115 n
2058 2113 n
2057 2112 n
2055 2110 n
2051 2109 n
2049 2108 n
2048 2106 n
2047 2104 n
2047 2101 n
2054 2118 m
2056 2117 n
2056 2117 n
2057 2115 n
2057 2113 n
2056 2112 n
2054 2110 n
2051 2109 n
2047 2103 m
2048 2104 n
2049 2104 n
2053 2102 n
2056 2102 n
2057 2103 n
2058 2104 n
2049 2104 m
2053 2102 n
2056 2102 n
2057 2102 n
2058 2104 n
2058 2105 n
2070 2110 m
2066 2172 m
2069 2168 n
2073 2163 n
2077 2155 n
2078 2146 n
2078 2140 n
2077 2131 n
2073 2124 n
2069 2119 n
2066 2115 n
2069 2168 m
2073 2161 n
2075 2155 n
2077 2146 n
2077 2140 n
2075 2131 n
2073 2126 n
2069 2119 n
1970 2145 m
1774 2432 m
1770 2428 n
1766 2423 n
1763 2415 n
1761 2406 n
1761 2400 n
1763 2391 n
1766 2384 n
1770 2378 n
1774 2375 n
1770 2428 m
1766 2421 n
1765 2415 n
1763 2406 n
1763 2400 n
1765 2391 n
1766 2386 n
1770 2378 n
1800 2405 m
1784 2424 m
1796 2405 n
1796 2387 n
1785 2424 m
1798 2405 n
1798 2387 n
1811 2424 m
1798 2405 n
1780 2424 m
1791 2424 n
1803 2424 m
1814 2424 n
1791 2387 m
1803 2387 n
1838 2405 m
1834 2436 m
1834 2414 n
1823 2425 m
1845 2425 n
1864 2426 m
1829 2397 m
1826 2396 n
1823 2392 n
1822 2386 n
1822 2383 n
1823 2377 n
1826 2373 n
1829 2372 n
1832 2372 n
1835 2373 n
1838 2377 n
1839 2383 n
1839 2386 n
1838 2392 n
1835 2396 n
1832 2397 n
1829 2397 n
1829 2397 m
1827 2396 n
1826 2395 n
1824 2392 n
1823 2386 n
1823 2383 n
1824 2377 n
1826 2374 n
1827 2373 n
1829 2372 n
1832 2372 m
1834 2373 n
1835 2374 n
1836 2377 n
1838 2383 n
1838 2386 n
1836 2392 n
1835 2395 n
1834 2396 n
1832 2397 n
1857 2384 m
1848 2372 m
1847 2373 n
1848 2374 n
1850 2373 n
1850 2371 n
1848 2368 n
1847 2367 n
1869 2384 m
1865 2397 m
1862 2396 n
1859 2392 n
1858 2386 n
1858 2383 n
1859 2377 n
1862 2373 n
1865 2372 n
1868 2372 n
1871 2373 n
1874 2377 n
1875 2383 n
1875 2386 n
1874 2392 n
1871 2396 n
1868 2397 n
1865 2397 n
1865 2397 m
1863 2396 n
1862 2395 n
1860 2392 n
1859 2386 n
1859 2383 n
1860 2377 n
1862 2374 n
1863 2373 n
1865 2372 n
1868 2372 m
1870 2373 n
1871 2374 n
1872 2377 n
1874 2383 n
1874 2386 n
1872 2392 n
1871 2395 n
1870 2396 n
1868 2397 n
1893 2384 m
1929 2405 m
1923 2412 m
1918 2410 n
1914 2406 n
1913 2402 n
1913 2398 n
1914 2393 n
1918 2389 n
1923 2387 n
1927 2387 n
1932 2389 n
1936 2393 n
1938 2398 n
1938 2402 n
1936 2406 n
1932 2410 n
1927 2412 n
1923 2412 n
1923 2412 m
1920 2410 n
1916 2406 n
1914 2402 n
1914 2398 n
1916 2393 n
1920 2389 n
1923 2387 n
1927 2387 m
1931 2389 n
1934 2393 n
1936 2398 n
1936 2402 n
1934 2406 n
1931 2410 n
1927 2412 n
1965 2405 m
1952 2412 m
1952 2387 n
1954 2412 m
1954 2387 n
1954 2402 m
1956 2406 n
1959 2410 n
1963 2412 n
1968 2412 n
1970 2410 n
1970 2408 n
1968 2406 n
1967 2408 n
1968 2410 n
1947 2412 m
1954 2412 n
1947 2387 m
1959 2387 n
1995 2405 m
2024 2405 m
2008 2424 m
2021 2405 n
2021 2387 n
2010 2424 m
2022 2405 n
2022 2387 n
2035 2424 m
2022 2405 n
2004 2424 m
2015 2424 n
2028 2424 m
2039 2424 n
2015 2387 m
2028 2387 n
2062 2405 m
2048 2425 m
2069 2425 n
2088 2426 m
2054 2397 m
2050 2396 n
2048 2392 n
2046 2386 n
2046 2383 n
2048 2377 n
2050 2373 n
2054 2372 n
2056 2372 n
2060 2373 n
2062 2377 n
2063 2383 n
2063 2386 n
2062 2392 n
2060 2396 n
2056 2397 n
2054 2397 n
2054 2397 m
2051 2396 n
2050 2395 n
2049 2392 n
2048 2386 n
2048 2383 n
2049 2377 n
2050 2374 n
2051 2373 n
2054 2372 n
2056 2372 m
2058 2373 n
2060 2374 n
2061 2377 n
2062 2383 n
2062 2386 n
2061 2392 n
2060 2395 n
2058 2396 n
2056 2397 n
2081 2384 m
2073 2372 m
2072 2373 n
2073 2374 n
2074 2373 n
2074 2371 n
2073 2368 n
2072 2367 n
2093 2384 m
2090 2397 m
2086 2396 n
2084 2392 n
2082 2386 n
2082 2383 n
2084 2377 n
2086 2373 n
2090 2372 n
2092 2372 n
2096 2373 n
2098 2377 n
2099 2383 n
2099 2386 n
2098 2392 n
2096 2396 n
2092 2397 n
2090 2397 n
2090 2397 m
2087 2396 n
2086 2395 n
2085 2392 n
2084 2386 n
2084 2383 n
2085 2377 n
2086 2374 n
2087 2373 n
2090 2372 n
2092 2372 m
2094 2373 n
2096 2374 n
2097 2377 n
2098 2383 n
2098 2386 n
2097 2392 n
2096 2395 n
2094 2396 n
2092 2397 n
2117 2384 m
2108 2432 m
2112 2428 n
2115 2423 n
2119 2415 n
2121 2406 n
2121 2400 n
2119 2391 n
2115 2384 n
2112 2378 n
2108 2375 n
2112 2428 m
2115 2421 n
2117 2415 n
2119 2406 n
2119 2400 n
2117 2391 n
2115 2386 n
2112 2378 n
1775 2405 m
2554 3472 m
2550 3468 n
2546 3463 n
2543 3455 n
2541 3446 n
2541 3440 n
2543 3431 n
2546 3424 n
2550 3418 n
2554 3415 n
2550 3468 m
2546 3461 n
2545 3455 n
2543 3446 n
2543 3440 n
2545 3431 n
2546 3426 n
2550 3418 n
2580 3445 m
2564 3464 m
2576 3445 n
2576 3427 n
2565 3464 m
2578 3445 n
2578 3427 n
2591 3464 m
2578 3445 n
2560 3464 m
2571 3464 n
2583 3464 m
2594 3464 n
2571 3427 m
2583 3427 n
2618 3445 m
2603 3465 m
2625 3465 n
2644 3466 m
2603 3432 m
2604 3431 n
2603 3430 n
2602 3431 n
2602 3432 n
2603 3435 n
2604 3436 n
2608 3437 n
2613 3437 n
2616 3436 n
2618 3435 n
2619 3432 n
2619 3430 n
2618 3427 n
2614 3425 n
2608 3423 n
2606 3421 n
2603 3419 n
2602 3415 n
2602 3412 n
2613 3437 m
2615 3436 n
2616 3435 n
2618 3432 n
2618 3430 n
2616 3427 n
2613 3425 n
2608 3423 n
2602 3414 m
2603 3415 n
2606 3415 n
2612 3413 n
2615 3413 n
2618 3414 n
2619 3415 n
2606 3415 m
2612 3412 n
2616 3412 n
2618 3413 n
2619 3415 n
2619 3418 n
2637 3424 m
2628 3412 m
2627 3413 n
2628 3414 n
2630 3413 n
2630 3411 n
2628 3408 n
2627 3407 n
2649 3424 m
2639 3432 m
2640 3431 n
2639 3430 n
2638 3431 n
2638 3432 n
2639 3435 n
2640 3436 n
2644 3437 n
2649 3437 n
2652 3436 n
2654 3435 n
2655 3432 n
2655 3430 n
2654 3427 n
2650 3425 n
2644 3423 n
2642 3421 n
2639 3419 n
2638 3415 n
2638 3412 n
2649 3437 m
2651 3436 n
2652 3435 n
2654 3432 n
2654 3430 n
2652 3427 n
2649 3425 n
2644 3423 n
2638 3414 m
2639 3415 n
2642 3415 n
2648 3413 n
2651 3413 n
2654 3414 n
2655 3415 n
2642 3415 m
2648 3412 n
2652 3412 n
2654 3413 n
2655 3415 n
2655 3418 n
2673 3424 m
2664 3472 m
2667 3468 n
2671 3463 n
2675 3455 n
2676 3446 n
2676 3440 n
2675 3431 n
2671 3424 n
2667 3418 n
2664 3415 n
2667 3468 m
2671 3461 n
2673 3455 n
2675 3446 n
2675 3440 n
2673 3431 n
2671 3426 n
2667 3418 n
2555 3445 m
2164 3472 m
2160 3468 n
2156 3463 n
2153 3455 n
2151 3446 n
2151 3440 n
2153 3431 n
2156 3424 n
2160 3418 n
2164 3415 n
2160 3468 m
2156 3461 n
2155 3455 n
2153 3446 n
2153 3440 n
2155 3431 n
2156 3426 n
2160 3418 n
2190 3445 m
2174 3464 m
2186 3445 n
2186 3427 n
2175 3464 m
2188 3445 n
2188 3427 n
2201 3464 m
2188 3445 n
2170 3464 m
2181 3464 n
2193 3464 m
2204 3464 n
2181 3427 m
2193 3427 n
2228 3445 m
2213 3465 m
2235 3465 n
2254 3466 m
2213 3432 m
2214 3431 n
2213 3430 n
2212 3431 n
2212 3432 n
2213 3435 n
2214 3436 n
2218 3437 n
2223 3437 n
2226 3436 n
2228 3435 n
2229 3432 n
2229 3430 n
2228 3427 n
2224 3425 n
2218 3423 n
2216 3421 n
2213 3419 n
2212 3415 n
2212 3412 n
2223 3437 m
2225 3436 n
2226 3435 n
2228 3432 n
2228 3430 n
2226 3427 n
2223 3425 n
2218 3423 n
2212 3414 m
2213 3415 n
2216 3415 n
2222 3413 n
2225 3413 n
2228 3414 n
2229 3415 n
2216 3415 m
2222 3412 n
2226 3412 n
2228 3413 n
2229 3415 n
2229 3418 n
2247 3424 m
2238 3412 m
2237 3413 n
2238 3414 n
2240 3413 n
2240 3411 n
2238 3408 n
2237 3407 n
2259 3424 m
2252 3432 m
2254 3433 n
2258 3437 n
2258 3412 n
2256 3436 m
2256 3412 n
2252 3412 m
2262 3412 n
2283 3424 m
2277 3427 m
2276 3429 n
2277 3431 n
2279 3429 n
2279 3426 n
2277 3422 n
2276 3420 n
2308 3445 m
2337 3445 m
2324 3464 m
2324 3427 n
2326 3464 m
2326 3427 n
2319 3464 m
2340 3464 n
2346 3463 n
2348 3461 n
2349 3457 n
2349 3452 n
2348 3448 n
2346 3446 n
2340 3445 n
2326 3445 n
2340 3464 m
2344 3463 n
2346 3461 n
2348 3457 n
2348 3452 n
2346 3448 n
2344 3446 n
2340 3445 n
2319 3427 m
2331 3427 n
2376 3445 m
2364 3432 m
2367 3433 n
2370 3437 n
2370 3412 n
2369 3436 m
2369 3412 n
2364 3412 m
2375 3412 n
2396 3424 m
2387 3412 m
2386 3413 n
2387 3414 n
2388 3413 n
2388 3411 n
2387 3408 n
2386 3407 n
2408 3424 m
2400 3432 m
2403 3433 n
2406 3437 n
2406 3412 n
2405 3436 m
2405 3412 n
2400 3412 m
2411 3412 n
2432 3424 m
2423 3472 m
2426 3468 n
2430 3463 n
2433 3455 n
2435 3446 n
2435 3440 n
2433 3431 n
2430 3424 n
2426 3418 n
2423 3415 n
2426 3468 m
2430 3461 n
2432 3455 n
2433 3446 n
2433 3440 n
2432 3431 n
2430 3426 n
2426 3418 n
2165 3445 m
1774 3472 m
1770 3468 n
1766 3463 n
1763 3455 n
1761 3446 n
1761 3440 n
1763 3431 n
1766 3424 n
1770 3418 n
1774 3415 n
1770 3468 m
1766 3461 n
1765 3455 n
1763 3446 n
1763 3440 n
1765 3431 n
1766 3426 n
1770 3418 n
1800 3445 m
1784 3464 m
1796 3445 n
1796 3427 n
1785 3464 m
1798 3445 n
1798 3427 n
1811 3464 m
1798 3445 n
1780 3464 m
1791 3464 n
1803 3464 m
1814 3464 n
1791 3427 m
1803 3427 n
1838 3445 m
1823 3465 m
1845 3465 n
1864 3466 m
1823 3432 m
1824 3431 n
1823 3430 n
1822 3431 n
1822 3432 n
1823 3435 n
1824 3436 n
1828 3437 n
1833 3437 n
1836 3436 n
1838 3435 n
1839 3432 n
1839 3430 n
1838 3427 n
1834 3425 n
1828 3423 n
1826 3421 n
1823 3419 n
1822 3415 n
1822 3412 n
1833 3437 m
1835 3436 n
1836 3435 n
1838 3432 n
1838 3430 n
1836 3427 n
1833 3425 n
1828 3423 n
1822 3414 m
1823 3415 n
1826 3415 n
1832 3413 n
1835 3413 n
1838 3414 n
1839 3415 n
1826 3415 m
1832 3412 n
1836 3412 n
1838 3413 n
1839 3415 n
1839 3418 n
1857 3424 m
1848 3412 m
1847 3413 n
1848 3414 n
1850 3413 n
1850 3411 n
1848 3408 n
1847 3407 n
1869 3424 m
1865 3437 m
1862 3436 n
1859 3432 n
1858 3426 n
1858 3423 n
1859 3417 n
1862 3413 n
1865 3412 n
1868 3412 n
1871 3413 n
1874 3417 n
1875 3423 n
1875 3426 n
1874 3432 n
1871 3436 n
1868 3437 n
1865 3437 n
1865 3437 m
1863 3436 n
1862 3435 n
1860 3432 n
1859 3426 n
1859 3423 n
1860 3417 n
1862 3414 n
1863 3413 n
1865 3412 n
1868 3412 m
1870 3413 n
1871 3414 n
1872 3417 n
1874 3423 n
1874 3426 n
1872 3432 n
1871 3435 n
1870 3436 n
1868 3437 n
1893 3424 m
1887 3427 m
1886 3429 n
1887 3431 n
1889 3429 n
1889 3426 n
1887 3422 n
1886 3420 n
1918 3445 m
1947 3445 m
1934 3464 m
1934 3427 n
1936 3464 m
1936 3427 n
1929 3464 m
1950 3464 n
1956 3463 n
1958 3461 n
1959 3457 n
1959 3452 n
1958 3448 n
1956 3446 n
1950 3445 n
1936 3445 n
1950 3464 m
1954 3463 n
1956 3461 n
1958 3457 n
1958 3452 n
1956 3448 n
1954 3446 n
1950 3445 n
1929 3427 m
1941 3427 n
1986 3445 m
1974 3432 m
1977 3433 n
1980 3437 n
1980 3412 n
1979 3436 m
1979 3412 n
1974 3412 m
1985 3412 n
2006 3424 m
1997 3412 m
1996 3413 n
1997 3414 n
1998 3413 n
1998 3411 n
1997 3408 n
1996 3407 n
2018 3424 m
2014 3437 m
2010 3436 n
2008 3432 n
2007 3426 n
2007 3423 n
2008 3417 n
2010 3413 n
2014 3412 n
2016 3412 n
2020 3413 n
2022 3417 n
2024 3423 n
2024 3426 n
2022 3432 n
2020 3436 n
2016 3437 n
2014 3437 n
2014 3437 m
2012 3436 n
2010 3435 n
2009 3432 n
2008 3426 n
2008 3423 n
2009 3417 n
2010 3414 n
2012 3413 n
2014 3412 n
2016 3412 m
2019 3413 n
2020 3414 n
2021 3417 n
2022 3423 n
2022 3426 n
2021 3432 n
2020 3435 n
2019 3436 n
2016 3437 n
2042 3424 m
2033 3472 m
2036 3468 n
2040 3463 n
2043 3455 n
2045 3446 n
2045 3440 n
2043 3431 n
2040 3424 n
2036 3418 n
2033 3415 n
2036 3468 m
2040 3461 n
2042 3455 n
2043 3446 n
2043 3440 n
2042 3431 n
2040 3426 n
2036 3418 n
1775 3445 m
1384 3472 m
1380 3468 n
1376 3463 n
1373 3455 n
1371 3446 n
1371 3440 n
1373 3431 n
1376 3424 n
1380 3418 n
1384 3415 n
1380 3468 m
1376 3461 n
1375 3455 n
1373 3446 n
1373 3440 n
1375 3431 n
1376 3426 n
1380 3418 n
1410 3445 m
1394 3464 m
1406 3445 n
1406 3427 n
1395 3464 m
1408 3445 n
1408 3427 n
1421 3464 m
1408 3445 n
1390 3464 m
1401 3464 n
1413 3464 m
1424 3464 n
1401 3427 m
1413 3427 n
1448 3445 m
1433 3465 m
1455 3465 n
1474 3466 m
1433 3432 m
1434 3431 n
1433 3430 n
1432 3431 n
1432 3432 n
1433 3435 n
1434 3436 n
1438 3437 n
1443 3437 n
1446 3436 n
1448 3435 n
1449 3432 n
1449 3430 n
1448 3427 n
1444 3425 n
1438 3423 n
1436 3421 n
1433 3419 n
1432 3415 n
1432 3412 n
1443 3437 m
1445 3436 n
1446 3435 n
1448 3432 n
1448 3430 n
1446 3427 n
1443 3425 n
1438 3423 n
1432 3414 m
1433 3415 n
1436 3415 n
1442 3413 n
1445 3413 n
1448 3414 n
1449 3415 n
1436 3415 m
1442 3412 n
1446 3412 n
1448 3413 n
1449 3415 n
1449 3418 n
1467 3424 m
1458 3412 m
1457 3413 n
1458 3414 n
1460 3413 n
1460 3411 n
1458 3408 n
1457 3407 n
1479 3424 m
1469 3423 m
1491 3423 n
1510 3424 m
1503 3432 m
1505 3433 n
1509 3437 n
1509 3412 n
1508 3436 m
1508 3412 n
1503 3412 m
1514 3412 n
1534 3424 m
1529 3427 m
1527 3429 n
1529 3431 n
1530 3429 n
1530 3426 n
1529 3422 n
1527 3420 n
1559 3445 m
1588 3445 m
1575 3464 m
1575 3427 n
1577 3464 m
1577 3427 n
1570 3464 m
1592 3464 n
1597 3463 n
1599 3461 n
1601 3457 n
1601 3452 n
1599 3448 n
1597 3446 n
1592 3445 n
1577 3445 n
1592 3464 m
1595 3463 n
1597 3461 n
1599 3457 n
1599 3452 n
1597 3448 n
1595 3446 n
1592 3445 n
1570 3427 m
1583 3427 n
1628 3445 m
1616 3432 m
1618 3433 n
1622 3437 n
1622 3412 n
1620 3436 m
1620 3412 n
1616 3412 m
1626 3412 n
1647 3424 m
1638 3412 m
1637 3413 n
1638 3414 n
1640 3413 n
1640 3411 n
1638 3408 n
1637 3407 n
1659 3424 m
1649 3423 m
1671 3423 n
1690 3424 m
1683 3432 m
1685 3433 n
1689 3437 n
1689 3412 n
1688 3436 m
1688 3412 n
1683 3412 m
1694 3412 n
1714 3424 m
1705 3472 m
1709 3468 n
1712 3463 n
1716 3455 n
1718 3446 n
1718 3440 n
1716 3431 n
1712 3424 n
1709 3418 n
1705 3415 n
1709 3468 m
1712 3461 n
1714 3455 n
1716 3446 n
1716 3440 n
1714 3431 n
1712 3426 n
1709 3418 n
1385 3445 m
994 3472 m
990 3468 n
986 3463 n
983 3455 n
981 3446 n
981 3440 n
983 3431 n
986 3424 n
990 3418 n
994 3415 n
990 3468 m
986 3461 n
985 3455 n
983 3446 n
983 3440 n
985 3431 n
986 3426 n
990 3418 n
1020 3445 m
1004 3464 m
1016 3445 n
1016 3427 n
1005 3464 m
1018 3445 n
1018 3427 n
1031 3464 m
1018 3445 n
1000 3464 m
1011 3464 n
1023 3464 m
1034 3464 n
1011 3427 m
1023 3427 n
1058 3445 m
1043 3465 m
1065 3465 n
1084 3466 m
1043 3432 m
1044 3431 n
1043 3430 n
1042 3431 n
1042 3432 n
1043 3435 n
1044 3436 n
1048 3437 n
1053 3437 n
1056 3436 n
1058 3435 n
1059 3432 n
1059 3430 n
1058 3427 n
1054 3425 n
1048 3423 n
1046 3421 n
1043 3419 n
1042 3415 n
1042 3412 n
1053 3437 m
1055 3436 n
1056 3435 n
1058 3432 n
1058 3430 n
1056 3427 n
1053 3425 n
1048 3423 n
1042 3414 m
1043 3415 n
1046 3415 n
1052 3413 n
1055 3413 n
1058 3414 n
1059 3415 n
1046 3415 m
1052 3412 n
1056 3412 n
1058 3413 n
1059 3415 n
1059 3418 n
1077 3424 m
1068 3412 m
1067 3413 n
1068 3414 n
1070 3413 n
1070 3411 n
1068 3408 n
1067 3407 n
1089 3424 m
1079 3423 m
1101 3423 n
1120 3424 m
1110 3432 m
1112 3431 n
1110 3430 n
1109 3431 n
1109 3432 n
1110 3435 n
1112 3436 n
1115 3437 n
1120 3437 n
1124 3436 n
1125 3435 n
1126 3432 n
1126 3430 n
1125 3427 n
1121 3425 n
1115 3423 n
1113 3421 n
1110 3419 n
1109 3415 n
1109 3412 n
1120 3437 m
1122 3436 n
1124 3435 n
1125 3432 n
1125 3430 n
1124 3427 n
1120 3425 n
1115 3423 n
1109 3414 m
1110 3415 n
1113 3415 n
1119 3413 n
1122 3413 n
1125 3414 n
1126 3415 n
1113 3415 m
1119 3412 n
1124 3412 n
1125 3413 n
1126 3415 n
1126 3418 n
1144 3424 m
1135 3472 m
1139 3468 n
1142 3463 n
1146 3455 n
1148 3446 n
1148 3440 n
1146 3431 n
1142 3424 n
1139 3418 n
1135 3415 n
1139 3468 m
1142 3461 n
1144 3455 n
1146 3446 n
1146 3440 n
1144 3431 n
1142 3426 n
1139 3418 n
995 3445 m
1189 3212 m
1185 3208 n
1181 3203 n
1178 3195 n
1176 3186 n
1176 3180 n
1178 3171 n
1181 3164 n
1185 3158 n
1189 3155 n
1185 3208 m
1181 3201 n
1180 3195 n
1178 3186 n
1178 3180 n
1180 3171 n
1181 3166 n
1185 3158 n
1215 3185 m
1199 3204 m
1211 3185 n
1211 3167 n
1200 3204 m
1213 3185 n
1213 3167 n
1226 3204 m
1213 3185 n
1195 3204 m
1206 3204 n
1218 3204 m
1229 3204 n
1206 3167 m
1218 3167 n
1253 3185 m
1238 3205 m
1260 3205 n
1279 3206 m
1238 3184 m
1239 3183 n
1238 3182 n
1237 3183 n
1237 3184 n
1238 3185 n
1239 3185 n
1241 3186 n
1245 3186 n
1247 3185 n
1248 3185 n
1248 3182 n
1247 3181 n
1245 3180 n
1242 3180 n
1245 3186 m
1246 3185 n
1247 3185 n
1247 3182 n
1246 3181 n
1245 3180 n
1245 3180 m
1246 3179 n
1248 3177 n
1249 3176 n
1249 3173 n
1248 3172 n
1247 3171 n
1245 3170 n
1241 3170 n
1239 3171 n
1238 3172 n
1237 3173 n
1237 3174 n
1238 3175 n
1239 3174 n
1238 3173 n
1247 3178 m
1248 3176 n
1248 3173 n
1247 3172 n
1246 3171 n
1244 3170 n
1260 3178 m
1235 3163 m
1253 3163 n
1262 3149 m
1238 3155 m
1239 3154 n
1238 3153 n
1237 3154 n
1237 3155 n
1238 3157 n
1239 3157 n
1241 3158 n
1245 3158 n
1247 3157 n
1248 3157 n
1249 3155 n
1249 3153 n
1248 3152 n
1245 3150 n
1241 3149 n
1240 3148 n
1238 3146 n
1237 3144 n
1237 3141 n
1245 3158 m
1246 3157 n
1247 3157 n
1248 3155 n
1248 3153 n
1247 3152 n
1245 3150 n
1241 3149 n
1237 3143 m
1238 3144 n
1240 3144 n
1244 3142 n
1246 3142 n
1248 3143 n
1249 3144 n
1240 3144 m
1244 3142 n
1247 3142 n
1248 3142 n
1249 3144 n
1249 3145 n
1261 3150 m
1257 3152 m
1256 3153 n
1257 3154 n
1258 3153 n
1258 3151 n
1257 3148 n
1256 3147 n
1277 3164 m
1268 3163 m
1289 3163 n
1309 3164 m
1299 3184 m
1300 3183 n
1299 3182 n
1298 3183 n
1298 3184 n
1299 3185 n
1300 3185 n
1302 3186 n
1305 3186 n
1308 3185 n
1309 3185 n
1309 3182 n
1308 3181 n
1305 3180 n
1303 3180 n
1305 3186 m
1307 3185 n
1308 3185 n
1308 3182 n
1307 3181 n
1305 3180 n
1305 3180 m
1307 3179 n
1309 3178 n
1309 3176 n
1309 3174 n
1309 3172 n
1308 3171 n
1305 3170 n
1302 3170 n
1300 3171 n
1299 3172 n
1298 3174 n
1298 3174 n
1299 3175 n
1300 3174 n
1299 3174 n
1308 3178 m
1309 3176 n
1309 3174 n
1308 3172 n
1307 3171 n
1305 3170 n
1321 3178 m
1296 3163 m
1313 3163 n
1323 3150 m
1299 3155 m
1300 3154 n
1299 3154 n
1298 3154 n
1298 3155 n
1299 3157 n
1300 3158 n
1302 3158 n
1305 3158 n
1308 3158 n
1309 3157 n
1309 3155 n
1309 3153 n
1309 3152 n
1306 3150 n
1302 3149 n
1300 3148 n
1299 3146 n
1298 3144 n
1298 3141 n
1305 3158 m
1307 3157 n
1308 3157 n
1308 3155 n
1308 3153 n
1308 3152 n
1305 3150 n
1302 3149 n
1298 3143 m
1299 3144 n
1300 3144 n
1304 3142 n
1307 3142 n
1309 3143 n
1309 3144 n
1300 3144 m
1304 3142 n
1308 3142 n
1308 3142 n
1309 3144 n
1309 3145 n
1321 3150 m
1317 3212 m
1321 3208 n
1324 3203 n
1328 3195 n
1330 3186 n
1330 3180 n
1328 3171 n
1324 3164 n
1321 3159 n
1317 3155 n
1321 3208 m
1324 3201 n
1326 3195 n
1328 3186 n
1328 3180 n
1326 3171 n
1324 3166 n
1321 3159 n
1190 3185 m
1579 3212 m
1575 3208 n
1571 3203 n
1568 3195 n
1566 3186 n
1566 3180 n
1568 3171 n
1571 3164 n
1575 3158 n
1579 3155 n
1575 3208 m
1571 3201 n
1570 3195 n
1568 3186 n
1568 3180 n
1570 3171 n
1571 3166 n
1575 3158 n
1605 3185 m
1589 3204 m
1601 3185 n
1601 3167 n
1590 3204 m
1603 3185 n
1603 3167 n
1616 3204 m
1603 3185 n
1585 3204 m
1596 3204 n
1608 3204 m
1619 3204 n
1596 3167 m
1608 3167 n
1643 3185 m
1628 3205 m
1650 3205 n
1669 3206 m
1628 3184 m
1629 3183 n
1628 3182 n
1627 3183 n
1627 3184 n
1628 3185 n
1629 3185 n
1631 3186 n
1635 3186 n
1637 3185 n
1638 3185 n
1638 3182 n
1637 3181 n
1635 3180 n
1632 3180 n
1635 3186 m
1636 3185 n
1637 3185 n
1637 3182 n
1636 3181 n
1635 3180 n
1635 3180 m
1636 3179 n
1638 3177 n
1639 3176 n
1639 3173 n
1638 3172 n
1637 3171 n
1635 3170 n
1631 3170 n
1629 3171 n
1628 3172 n
1627 3173 n
1627 3174 n
1628 3175 n
1629 3174 n
1628 3173 n
1637 3178 m
1638 3176 n
1638 3173 n
1637 3172 n
1636 3171 n
1634 3170 n
1650 3178 m
1625 3163 m
1643 3163 n
1652 3149 m
1628 3155 m
1629 3154 n
1628 3153 n
1627 3154 n
1627 3155 n
1628 3157 n
1629 3157 n
1631 3158 n
1635 3158 n
1637 3157 n
1638 3157 n
1639 3155 n
1639 3153 n
1638 3152 n
1635 3150 n
1631 3149 n
1630 3148 n
1628 3146 n
1627 3144 n
1627 3141 n
1635 3158 m
1636 3157 n
1637 3157 n
1638 3155 n
1638 3153 n
1637 3152 n
1635 3150 n
1631 3149 n
1627 3143 m
1628 3144 n
1630 3144 n
1634 3142 n
1636 3142 n
1638 3143 n
1639 3144 n
1630 3144 m
1634 3142 n
1637 3142 n
1638 3142 n
1639 3144 n
1639 3145 n
1651 3150 m
1647 3152 m
1646 3153 n
1647 3154 n
1648 3153 n
1648 3151 n
1647 3148 n
1646 3147 n
1667 3164 m
1658 3163 m
1679 3163 n
1699 3164 m
1691 3184 m
1692 3185 n
1695 3186 n
1695 3170 n
1694 3185 m
1694 3170 n
1691 3170 m
1698 3170 n
1711 3178 m
1686 3163 m
1703 3163 n
1713 3150 m
1689 3155 m
1690 3154 n
1689 3154 n
1688 3154 n
1688 3155 n
1689 3157 n
1690 3158 n
1692 3158 n
1695 3158 n
1698 3158 n
1699 3157 n
1699 3155 n
1699 3153 n
1699 3152 n
1696 3150 n
1692 3149 n
1690 3148 n
1689 3146 n
1688 3144 n
1688 3141 n
1695 3158 m
1697 3157 n
1698 3157 n
1698 3155 n
1698 3153 n
1698 3152 n
1695 3150 n
1692 3149 n
1688 3143 m
1689 3144 n
1690 3144 n
1694 3142 n
1697 3142 n
1699 3143 n
1699 3144 n
1690 3144 m
1694 3142 n
1698 3142 n
1698 3142 n
1699 3144 n
1699 3145 n
1711 3150 m
1711 3168 m
1709 3169 n
1711 3171 n
1712 3169 n
1712 3166 n
1711 3162 n
1709 3160 n
1741 3185 m
1770 3185 m
1799 3185 m
1786 3204 m
1786 3168 n
1788 3204 m
1788 3168 n
1781 3204 m
1802 3204 n
1808 3203 n
1810 3201 n
1811 3197 n
1811 3192 n
1810 3188 n
1808 3186 n
1802 3185 n
1788 3185 n
1802 3204 m
1806 3203 n
1808 3201 n
1810 3197 n
1810 3192 n
1808 3188 n
1806 3186 n
1802 3185 n
1781 3168 m
1793 3168 n
1838 3185 m
1826 3184 m
1827 3185 n
1830 3186 n
1830 3170 n
1829 3185 m
1829 3170 n
1826 3170 m
1833 3170 n
1846 3178 m
1821 3163 m
1838 3163 n
1848 3150 m
1824 3155 m
1825 3154 n
1824 3154 n
1823 3154 n
1823 3155 n
1824 3157 n
1825 3158 n
1827 3158 n
1830 3158 n
1833 3158 n
1834 3157 n
1835 3155 n
1835 3153 n
1834 3152 n
1831 3150 n
1827 3149 n
1826 3148 n
1824 3146 n
1823 3144 n
1823 3141 n
1831 3158 m
1832 3157 n
1833 3157 n
1834 3155 n
1834 3153 n
1833 3152 n
1831 3150 n
1827 3149 n
1823 3143 m
1824 3144 n
1826 3144 n
1830 3142 n
1832 3142 n
1834 3143 n
1835 3144 n
1826 3144 m
1830 3142 n
1833 3142 n
1834 3142 n
1835 3144 n
1835 3145 n
1847 3150 m
1843 3152 m
1842 3153 n
1843 3154 n
1844 3153 n
1844 3151 n
1843 3148 n
1842 3147 n
1863 3164 m
1854 3163 m
1875 3163 n
1895 3164 m
1887 3184 m
1888 3185 n
1891 3186 n
1891 3170 n
1890 3185 m
1890 3170 n
1887 3170 m
1894 3170 n
1907 3178 m
1882 3163 m
1899 3163 n
1909 3150 m
1885 3155 m
1886 3154 n
1885 3154 n
1884 3154 n
1884 3155 n
1885 3157 n
1886 3158 n
1888 3158 n
1891 3158 n
1894 3158 n
1895 3157 n
1895 3155 n
1895 3153 n
1895 3152 n
1892 3150 n
1888 3149 n
1887 3148 n
1885 3146 n
1884 3144 n
1884 3141 n
1891 3158 m
1893 3157 n
1894 3157 n
1895 3155 n
1895 3153 n
1894 3152 n
1891 3150 n
1888 3149 n
1884 3143 m
1885 3144 n
1887 3144 n
1891 3142 n
1893 3142 n
1895 3143 n
1895 3144 n
1887 3144 m
1891 3142 n
1894 3142 n
1895 3142 n
1895 3144 n
1895 3145 n
1907 3150 m
1903 3212 m
1907 3208 n
1910 3203 n
1914 3195 n
1916 3186 n
1916 3180 n
1914 3171 n
1910 3164 n
1907 3159 n
1903 3155 n
1907 3208 m
1910 3201 n
1912 3195 n
1914 3186 n
1914 3180 n
1912 3171 n
1910 3166 n
1907 3159 n
1580 3185 m
1969 3212 m
1965 3208 n
1961 3203 n
1958 3195 n
1956 3186 n
1956 3180 n
1958 3171 n
1961 3164 n
1965 3158 n
1969 3155 n
1965 3208 m
1961 3201 n
1960 3195 n
1958 3186 n
1958 3180 n
1960 3171 n
1961 3166 n
1965 3158 n
1995 3185 m
1979 3204 m
1991 3185 n
1991 3167 n
1980 3204 m
1993 3185 n
1993 3167 n
2006 3204 m
1993 3185 n
1975 3204 m
1986 3204 n
1998 3204 m
2009 3204 n
1986 3167 m
1998 3167 n
2033 3185 m
2018 3205 m
2040 3205 n
2059 3206 m
2018 3184 m
2019 3183 n
2018 3182 n
2017 3183 n
2017 3184 n
2018 3185 n
2019 3185 n
2021 3186 n
2025 3186 n
2027 3185 n
2028 3185 n
2028 3182 n
2027 3181 n
2025 3180 n
2022 3180 n
2025 3186 m
2026 3185 n
2027 3185 n
2027 3182 n
2026 3181 n
2025 3180 n
2025 3180 m
2026 3179 n
2028 3177 n
2029 3176 n
2029 3173 n
2028 3172 n
2027 3171 n
2025 3170 n
2021 3170 n
2019 3171 n
2018 3172 n
2017 3173 n
2017 3174 n
2018 3175 n
2019 3174 n
2018 3173 n
2027 3178 m
2028 3176 n
2028 3173 n
2027 3172 n
2026 3171 n
2024 3170 n
2040 3178 m
2015 3163 m
2033 3163 n
2042 3149 m
2018 3155 m
2019 3154 n
2018 3153 n
2017 3154 n
2017 3155 n
2018 3157 n
2019 3157 n
2021 3158 n
2025 3158 n
2027 3157 n
2028 3157 n
2029 3155 n
2029 3153 n
2028 3152 n
2025 3150 n
2021 3149 n
2020 3148 n
2018 3146 n
2017 3144 n
2017 3141 n
2025 3158 m
2026 3157 n
2027 3157 n
2028 3155 n
2028 3153 n
2027 3152 n
2025 3150 n
2021 3149 n
2017 3143 m
2018 3144 n
2020 3144 n
2024 3142 n
2026 3142 n
2028 3143 n
2029 3144 n
2020 3144 m
2024 3142 n
2027 3142 n
2028 3142 n
2029 3144 n
2029 3145 n
2041 3150 m
2037 3152 m
2036 3153 n
2037 3154 n
2038 3153 n
2038 3151 n
2037 3148 n
2036 3147 n
2057 3164 m
2049 3184 m
2051 3185 n
2053 3186 n
2053 3170 n
2053 3185 m
2053 3170 n
2049 3170 m
2057 3170 n
2070 3178 m
2045 3163 m
2062 3163 n
2072 3150 m
2048 3155 m
2049 3154 n
2048 3154 n
2047 3154 n
2047 3155 n
2048 3157 n
2049 3158 n
2051 3158 n
2054 3158 n
2057 3158 n
2057 3157 n
2058 3155 n
2058 3153 n
2057 3152 n
2055 3150 n
2051 3149 n
2049 3148 n
2048 3146 n
2047 3144 n
2047 3141 n
2054 3158 m
2056 3157 n
2056 3157 n
2057 3155 n
2057 3153 n
2056 3152 n
2054 3150 n
2051 3149 n
2047 3143 m
2048 3144 n
2049 3144 n
2053 3142 n
2056 3142 n
2057 3143 n
2058 3144 n
2049 3144 m
2053 3142 n
2056 3142 n
2057 3142 n
2058 3144 n
2058 3145 n
2070 3150 m
2069 3168 m
2068 3169 n
2069 3171 n
2071 3169 n
2071 3166 n
2069 3162 n
2068 3160 n
2100 3185 m
2129 3185 m
2158 3185 m
2145 3204 m
2145 3168 n
2147 3204 m
2147 3168 n
2140 3204 m
2161 3204 n
2167 3203 n
2168 3201 n
2170 3197 n
2170 3192 n
2168 3188 n
2167 3186 n
2161 3185 n
2147 3185 n
2161 3204 m
2165 3203 n
2167 3201 n
2168 3197 n
2168 3192 n
2167 3188 n
2165 3186 n
2161 3185 n
2140 3168 m
2152 3168 n
2197 3185 m
2184 3184 m
2186 3185 n
2188 3186 n
2188 3170 n
2188 3185 m
2188 3170 n
2184 3170 m
2192 3170 n
2205 3178 m
2180 3163 m
2197 3163 n
2207 3150 m
2183 3155 m
2184 3154 n
2183 3154 n
2182 3154 n
2182 3155 n
2183 3157 n
2184 3158 n
2186 3158 n
2189 3158 n
2192 3158 n
2193 3157 n
2193 3155 n
2193 3153 n
2193 3152 n
2190 3150 n
2186 3149 n
2185 3148 n
2183 3146 n
2182 3144 n
2182 3141 n
2189 3158 m
2191 3157 n
2192 3157 n
2193 3155 n
2193 3153 n
2192 3152 n
2189 3150 n
2186 3149 n
2182 3143 m
2183 3144 n
2185 3144 n
2189 3142 n
2191 3142 n
2193 3143 n
2193 3144 n
2185 3144 m
2189 3142 n
2192 3142 n
2193 3142 n
2193 3144 n
2193 3145 n
2205 3150 m
2202 3152 m
2201 3153 n
2202 3154 n
2203 3153 n
2203 3151 n
2202 3148 n
2201 3147 n
2222 3164 m
2214 3184 m
2216 3185 n
2218 3186 n
2218 3170 n
2217 3185 m
2217 3170 n
2214 3170 m
2221 3170 n
2235 3178 m
2209 3163 m
2227 3163 n
2237 3150 m
2213 3155 m
2213 3154 n
2213 3154 n
2212 3154 n
2212 3155 n
2213 3157 n
2213 3158 n
2216 3158 n
2219 3158 n
2221 3158 n
2222 3157 n
2223 3155 n
2223 3153 n
2222 3152 n
2220 3150 n
2216 3149 n
2214 3148 n
2213 3146 n
2212 3144 n
2212 3141 n
2219 3158 m
2221 3157 n
2221 3157 n
2222 3155 n
2222 3153 n
2221 3152 n
2219 3150 n
2216 3149 n
2212 3143 m
2213 3144 n
2214 3144 n
2218 3142 n
2221 3142 n
2222 3143 n
2223 3144 n
2214 3144 m
2218 3142 n
2221 3142 n
2222 3142 n
2223 3144 n
2223 3145 n
2235 3150 m
2231 3212 m
2234 3208 n
2238 3203 n
2242 3195 n
2243 3186 n
2243 3180 n
2242 3171 n
2238 3164 n
2234 3159 n
2231 3155 n
2234 3208 m
2238 3201 n
2240 3195 n
2242 3186 n
2242 3180 n
2240 3171 n
2238 3166 n
2234 3159 n
1970 3185 m
2359 3212 m
2355 3208 n
2351 3203 n
2348 3195 n
2346 3186 n
2346 3180 n
2348 3171 n
2351 3164 n
2355 3158 n
2359 3155 n
2355 3208 m
2351 3201 n
2350 3195 n
2348 3186 n
2348 3180 n
2350 3171 n
2351 3166 n
2355 3158 n
2385 3185 m
2369 3204 m
2381 3185 n
2381 3167 n
2370 3204 m
2383 3185 n
2383 3167 n
2396 3204 m
2383 3185 n
2365 3204 m
2376 3204 n
2388 3204 m
2399 3204 n
2376 3167 m
2388 3167 n
2423 3185 m
2408 3205 m
2430 3205 n
2449 3206 m
2408 3184 m
2409 3183 n
2408 3182 n
2407 3183 n
2407 3184 n
2408 3185 n
2409 3185 n
2411 3186 n
2415 3186 n
2417 3185 n
2418 3185 n
2418 3182 n
2417 3181 n
2415 3180 n
2412 3180 n
2415 3186 m
2416 3185 n
2417 3185 n
2417 3182 n
2416 3181 n
2415 3180 n
2415 3180 m
2416 3179 n
2418 3177 n
2419 3176 n
2419 3173 n
2418 3172 n
2417 3171 n
2415 3170 n
2411 3170 n
2409 3171 n
2408 3172 n
2407 3173 n
2407 3174 n
2408 3175 n
2409 3174 n
2408 3173 n
2417 3178 m
2418 3176 n
2418 3173 n
2417 3172 n
2416 3171 n
2414 3170 n
2430 3178 m
2405 3163 m
2423 3163 n
2432 3149 m
2408 3155 m
2409 3154 n
2408 3153 n
2407 3154 n
2407 3155 n
2408 3157 n
2409 3157 n
2411 3158 n
2415 3158 n
2417 3157 n
2418 3157 n
2419 3155 n
2419 3153 n
2418 3152 n
2415 3150 n
2411 3149 n
2410 3148 n
2408 3146 n
2407 3144 n
2407 3141 n
2415 3158 m
2416 3157 n
2417 3157 n
2418 3155 n
2418 3153 n
2417 3152 n
2415 3150 n
2411 3149 n
2407 3143 m
2408 3144 n
2410 3144 n
2414 3142 n
2416 3142 n
2418 3143 n
2419 3144 n
2410 3144 m
2414 3142 n
2417 3142 n
2418 3142 n
2419 3144 n
2419 3145 n
2431 3150 m
2427 3152 m
2426 3153 n
2427 3154 n
2428 3153 n
2428 3151 n
2427 3148 n
2426 3147 n
2447 3164 m
2438 3184 m
2439 3183 n
2438 3182 n
2437 3183 n
2437 3184 n
2438 3185 n
2439 3185 n
2441 3186 n
2444 3186 n
2447 3185 n
2447 3185 n
2447 3182 n
2447 3181 n
2444 3180 n
2442 3180 n
2444 3186 m
2446 3185 n
2447 3185 n
2447 3182 n
2446 3181 n
2444 3180 n
2444 3180 m
2446 3179 n
2447 3178 n
2448 3176 n
2448 3174 n
2447 3172 n
2447 3171 n
2444 3170 n
2441 3170 n
2439 3171 n
2438 3172 n
2437 3174 n
2437 3174 n
2438 3175 n
2439 3174 n
2438 3174 n
2447 3178 m
2447 3176 n
2447 3174 n
2447 3172 n
2446 3171 n
2444 3170 n
2460 3178 m
2435 3163 m
2452 3163 n
2462 3150 m
2438 3155 m
2439 3154 n
2438 3154 n
2437 3154 n
2437 3155 n
2438 3157 n
2439 3158 n
2441 3158 n
2444 3158 n
2447 3158 n
2447 3157 n
2448 3155 n
2448 3153 n
2447 3152 n
2445 3150 n
2441 3149 n
2439 3148 n
2438 3146 n
2437 3144 n
2437 3141 n
2444 3158 m
2446 3157 n
2446 3157 n
2447 3155 n
2447 3153 n
2446 3152 n
2444 3150 n
2441 3149 n
2437 3143 m
2438 3144 n
2439 3144 n
2443 3142 n
2446 3142 n
2447 3143 n
2448 3144 n
2439 3144 m
2443 3142 n
2446 3142 n
2447 3142 n
2448 3144 n
2448 3145 n
2460 3150 m
2456 3212 m
2459 3208 n
2463 3203 n
2467 3195 n
2468 3186 n
2468 3180 n
2467 3171 n
2463 3164 n
2459 3159 n
2456 3155 n
2459 3208 m
2463 3201 n
2465 3195 n
2467 3186 n
2467 3180 n
2465 3171 n
2463 3166 n
2459 3159 n
2360 3185 m
2164 2952 m
2160 2948 n
2156 2943 n
2153 2935 n
2151 2926 n
2151 2920 n
2153 2911 n
2156 2904 n
2160 2898 n
2164 2895 n
2160 2948 m
2156 2941 n
2155 2935 n
2153 2926 n
2153 2920 n
2155 2911 n
2156 2906 n
2160 2898 n
2190 2925 m
2174 2944 m
2186 2925 n
2186 2907 n
2175 2944 m
2188 2925 n
2188 2907 n
2201 2944 m
2188 2925 n
2170 2944 m
2181 2944 n
2193 2944 m
2204 2944 n
2181 2907 m
2193 2907 n
2228 2925 m
2213 2945 m
2235 2945 n
2254 2946 m
2216 2912 m
2218 2913 n
2222 2917 n
2222 2892 n
2220 2916 m
2220 2892 n
2216 2892 m
2226 2892 n
2247 2904 m
2238 2892 m
2237 2893 n
2238 2894 n
2240 2893 n
2240 2891 n
2238 2888 n
2237 2887 n
2259 2904 m
2252 2912 m
2254 2913 n
2258 2917 n
2258 2892 n
2256 2916 m
2256 2892 n
2252 2892 m
2262 2892 n
2283 2904 m
2274 2952 m
2277 2948 n
2281 2943 n
2285 2935 n
2286 2926 n
2286 2920 n
2285 2911 n
2281 2904 n
2277 2898 n
2274 2895 n
2277 2948 m
2281 2941 n
2283 2935 n
2285 2926 n
2285 2920 n
2283 2911 n
2281 2906 n
2277 2898 n
2165 2925 m
1774 2952 m
1770 2948 n
1766 2943 n
1763 2935 n
1761 2926 n
1761 2920 n
1763 2911 n
1766 2904 n
1770 2898 n
1774 2895 n
1770 2948 m
1766 2941 n
1765 2935 n
1763 2926 n
1763 2920 n
1765 2911 n
1766 2906 n
1770 2898 n
1800 2925 m
1784 2944 m
1796 2925 n
1796 2907 n
1785 2944 m
1798 2925 n
1798 2907 n
1811 2944 m
1798 2925 n
1780 2944 m
1791 2944 n
1803 2944 m
1814 2944 n
1791 2907 m
1803 2907 n
1838 2925 m
1823 2945 m
1845 2945 n
1864 2946 m
1826 2912 m
1828 2913 n
1832 2917 n
1832 2892 n
1830 2916 m
1830 2892 n
1826 2892 m
1836 2892 n
1857 2904 m
1848 2892 m
1847 2893 n
1848 2894 n
1850 2893 n
1850 2891 n
1848 2888 n
1847 2887 n
1869 2904 m
1865 2917 m
1862 2916 n
1859 2912 n
1858 2906 n
1858 2903 n
1859 2897 n
1862 2893 n
1865 2892 n
1868 2892 n
1871 2893 n
1874 2897 n
1875 2903 n
1875 2906 n
1874 2912 n
1871 2916 n
1868 2917 n
1865 2917 n
1865 2917 m
1863 2916 n
1862 2915 n
1860 2912 n
1859 2906 n
1859 2903 n
1860 2897 n
1862 2894 n
1863 2893 n
1865 2892 n
1868 2892 m
1870 2893 n
1871 2894 n
1872 2897 n
1874 2903 n
1874 2906 n
1872 2912 n
1871 2915 n
1870 2916 n
1868 2917 n
1893 2904 m
1887 2907 m
1886 2909 n
1887 2911 n
1889 2909 n
1889 2906 n
1887 2902 n
1886 2900 n
1918 2925 m
1947 2925 m
1934 2944 m
1934 2907 n
1936 2944 m
1936 2907 n
1929 2944 m
1950 2944 n
1956 2943 n
1958 2941 n
1959 2937 n
1959 2932 n
1958 2928 n
1956 2926 n
1950 2925 n
1936 2925 n
1950 2944 m
1954 2943 n
1956 2941 n
1958 2937 n
1958 2932 n
1956 2928 n
1954 2926 n
1950 2925 n
1929 2907 m
1941 2907 n
1986 2925 m
1978 2917 m
1974 2916 n
1972 2912 n
1971 2906 n
1971 2903 n
1972 2897 n
1974 2893 n
1978 2892 n
1980 2892 n
1984 2893 n
1986 2897 n
1988 2903 n
1988 2906 n
1986 2912 n
1984 2916 n
1980 2917 n
1978 2917 n
1978 2917 m
1976 2916 n
1974 2915 n
1973 2912 n
1972 2906 n
1972 2903 n
1973 2897 n
1974 2894 n
1976 2893 n
1978 2892 n
1980 2892 m
1983 2893 n
1984 2894 n
1985 2897 n
1986 2903 n
1986 2906 n
1985 2912 n
1984 2915 n
1983 2916 n
1980 2917 n
2006 2904 m
1997 2892 m
1996 2893 n
1997 2894 n
1998 2893 n
1998 2891 n
1997 2888 n
1996 2887 n
2018 2904 m
2014 2917 m
2010 2916 n
2008 2912 n
2007 2906 n
2007 2903 n
2008 2897 n
2010 2893 n
2014 2892 n
2016 2892 n
2020 2893 n
2022 2897 n
2024 2903 n
2024 2906 n
2022 2912 n
2020 2916 n
2016 2917 n
2014 2917 n
2014 2917 m
2012 2916 n
2010 2915 n
2009 2912 n
2008 2906 n
2008 2903 n
2009 2897 n
2010 2894 n
2012 2893 n
2014 2892 n
2016 2892 m
2019 2893 n
2020 2894 n
2021 2897 n
2022 2903 n
2022 2906 n
2021 2912 n
2020 2915 n
2019 2916 n
2016 2917 n
2042 2904 m
2033 2952 m
2036 2948 n
2040 2943 n
2043 2935 n
2045 2926 n
2045 2920 n
2043 2911 n
2040 2904 n
2036 2898 n
2033 2895 n
2036 2948 m
2040 2941 n
2042 2935 n
2043 2926 n
2043 2920 n
2042 2911 n
2040 2906 n
2036 2898 n
1775 2925 m
1384 2952 m
1380 2948 n
1376 2943 n
1373 2935 n
1371 2926 n
1371 2920 n
1373 2911 n
1376 2904 n
1380 2898 n
1384 2895 n
1380 2948 m
1376 2941 n
1375 2935 n
1373 2926 n
1373 2920 n
1375 2911 n
1376 2906 n
1380 2898 n
1410 2925 m
1394 2944 m
1406 2925 n
1406 2907 n
1395 2944 m
1408 2925 n
1408 2907 n
1421 2944 m
1408 2925 n
1390 2944 m
1401 2944 n
1413 2944 m
1424 2944 n
1401 2907 m
1413 2907 n
1448 2925 m
1433 2945 m
1455 2945 n
1474 2946 m
1436 2912 m
1438 2913 n
1442 2917 n
1442 2892 n
1440 2916 m
1440 2892 n
1436 2892 m
1446 2892 n
1467 2904 m
1458 2892 m
1457 2893 n
1458 2894 n
1460 2893 n
1460 2891 n
1458 2888 n
1457 2887 n
1479 2904 m
1469 2903 m
1491 2903 n
1510 2904 m
1503 2912 m
1505 2913 n
1509 2917 n
1509 2892 n
1508 2916 m
1508 2892 n
1503 2892 m
1514 2892 n
1534 2904 m
1525 2952 m
1529 2948 n
1532 2943 n
1536 2935 n
1538 2926 n
1538 2920 n
1536 2911 n
1532 2904 n
1529 2898 n
1525 2895 n
1529 2948 m
1532 2941 n
1534 2935 n
1536 2926 n
1536 2920 n
1534 2911 n
1532 2906 n
1529 2898 n
1385 2925 m
1579 2692 m
1575 2688 n
1571 2683 n
1568 2675 n
1566 2666 n
1566 2660 n
1568 2651 n
1571 2644 n
1575 2638 n
1579 2635 n
1575 2688 m
1571 2681 n
1570 2675 n
1568 2666 n
1568 2660 n
1570 2651 n
1571 2646 n
1575 2638 n
1605 2665 m
1589 2684 m
1601 2665 n
1601 2647 n
1590 2684 m
1603 2665 n
1603 2647 n
1616 2684 m
1603 2665 n
1585 2684 m
1596 2684 n
1608 2684 m
1619 2684 n
1596 2647 m
1608 2647 n
1643 2665 m
1628 2685 m
1650 2685 n
1669 2686 m
1630 2664 m
1631 2665 n
1634 2666 n
1634 2650 n
1633 2665 m
1633 2650 n
1630 2650 m
1637 2650 n
1650 2658 m
1625 2643 m
1643 2643 n
1652 2629 m
1628 2635 m
1629 2634 n
1628 2633 n
1627 2634 n
1627 2635 n
1628 2637 n
1629 2637 n
1631 2638 n
1635 2638 n
1637 2637 n
1638 2637 n
1639 2635 n
1639 2633 n
1638 2632 n
1635 2630 n
1631 2629 n
1630 2628 n
1628 2626 n
1627 2624 n
1627 2621 n
1635 2638 m
1636 2637 n
1637 2637 n
1638 2635 n
1638 2633 n
1637 2632 n
1635 2630 n
1631 2629 n
1627 2623 m
1628 2624 n
1630 2624 n
1634 2622 n
1636 2622 n
1638 2623 n
1639 2624 n
1630 2624 m
1634 2622 n
1637 2622 n
1638 2622 n
1639 2624 n
1639 2625 n
1651 2630 m
1647 2632 m
1646 2633 n
1647 2634 n
1648 2633 n
1648 2631 n
1647 2628 n
1646 2627 n
1667 2644 m
1658 2643 m
1679 2643 n
1699 2644 m
1691 2664 m
1692 2665 n
1695 2666 n
1695 2650 n
1694 2665 m
1694 2650 n
1691 2650 m
1698 2650 n
1711 2658 m
1686 2643 m
1703 2643 n
1713 2630 m
1689 2635 m
1690 2634 n
1689 2634 n
1688 2634 n
1688 2635 n
1689 2637 n
1690 2638 n
1692 2638 n
1695 2638 n
1698 2638 n
1699 2637 n
1699 2635 n
1699 2633 n
1699 2632 n
1696 2630 n
1692 2629 n
1690 2628 n
1689 2626 n
1688 2624 n
1688 2621 n
1695 2638 m
1697 2637 n
1698 2637 n
1698 2635 n
1698 2633 n
1698 2632 n
1695 2630 n
1692 2629 n
1688 2623 m
1689 2624 n
1690 2624 n
1694 2622 n
1697 2622 n
1699 2623 n
1699 2624 n
1690 2624 m
1694 2622 n
1698 2622 n
1698 2622 n
1699 2624 n
1699 2625 n
1711 2630 m
1707 2692 m
1711 2688 n
1714 2683 n
1718 2675 n
1720 2666 n
1720 2660 n
1718 2651 n
1714 2644 n
1711 2639 n
1707 2635 n
1711 2688 m
1714 2681 n
1716 2675 n
1718 2666 n
1718 2660 n
1716 2651 n
1714 2646 n
1711 2639 n
1580 2665 m
1969 2692 m
1965 2688 n
1961 2683 n
1958 2675 n
1956 2666 n
1956 2660 n
1958 2651 n
1961 2644 n
1965 2638 n
1969 2635 n
1965 2688 m
1961 2681 n
1960 2675 n
1958 2666 n
1958 2660 n
1960 2651 n
1961 2646 n
1965 2638 n
1995 2665 m
1979 2684 m
1991 2665 n
1991 2647 n
1980 2684 m
1993 2665 n
1993 2647 n
2006 2684 m
1993 2665 n
1975 2684 m
1986 2684 n
1998 2684 m
2009 2684 n
1986 2647 m
1998 2647 n
2033 2665 m
2018 2685 m
2040 2685 n
2059 2686 m
2020 2664 m
2021 2665 n
2024 2666 n
2024 2650 n
2023 2665 m
2023 2650 n
2020 2650 m
2027 2650 n
2040 2658 m
2015 2643 m
2033 2643 n
2042 2629 m
2018 2635 m
2019 2634 n
2018 2633 n
2017 2634 n
2017 2635 n
2018 2637 n
2019 2637 n
2021 2638 n
2025 2638 n
2027 2637 n
2028 2637 n
2029 2635 n
2029 2633 n
2028 2632 n
2025 2630 n
2021 2629 n
2020 2628 n
2018 2626 n
2017 2624 n
2017 2621 n
2025 2638 m
2026 2637 n
2027 2637 n
2028 2635 n
2028 2633 n
2027 2632 n
2025 2630 n
2021 2629 n
2017 2623 m
2018 2624 n
2020 2624 n
2024 2622 n
2026 2622 n
2028 2623 n
2029 2624 n
2020 2624 m
2024 2622 n
2027 2622 n
2028 2622 n
2029 2624 n
2029 2625 n
2041 2630 m
2037 2632 m
2036 2633 n
2037 2634 n
2038 2633 n
2038 2631 n
2037 2628 n
2036 2627 n
2057 2644 m
2049 2664 m
2051 2665 n
2053 2666 n
2053 2650 n
2053 2665 m
2053 2650 n
2049 2650 m
2057 2650 n
2070 2658 m
2045 2643 m
2062 2643 n
2072 2630 m
2048 2635 m
2049 2634 n
2048 2634 n
2047 2634 n
2047 2635 n
2048 2637 n
2049 2638 n
2051 2638 n
2054 2638 n
2057 2638 n
2057 2637 n
2058 2635 n
2058 2633 n
2057 2632 n
2055 2630 n
2051 2629 n
2049 2628 n
2048 2626 n
2047 2624 n
2047 2621 n
2054 2638 m
2056 2637 n
2056 2637 n
2057 2635 n
2057 2633 n
2056 2632 n
2054 2630 n
2051 2629 n
2047 2623 m
2048 2624 n
2049 2624 n
2053 2622 n
2056 2622 n
2057 2623 n
2058 2624 n
2049 2624 m
2053 2622 n
2056 2622 n
2057 2622 n
2058 2624 n
2058 2625 n
2070 2630 m
2066 2692 m
2069 2688 n
2073 2683 n
2077 2675 n
2078 2666 n
2078 2660 n
2077 2651 n
2073 2644 n
2069 2639 n
2066 2635 n
2069 2688 m
2073 2681 n
2075 2675 n
2077 2666 n
2077 2660 n
2075 2651 n
2073 2646 n
2069 2639 n
1970 2665 m
/solid f
1 1 m
e
%%Trailer
EndPSPlot



