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\begin{titlepage}
\hspace{9cm} SISSA/ISAS/100/93/EP
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\begin{centering}
{\huge Quantum mechanics and quantum Hall effect
on Riemann surfaces}\\
\vspace{1cm}
{\large Roberto Iengo$^{a,b}$
and Dingping Li$^a$\\
International School for Advanced studies, SISSA, I-34014
Trieste, Italy$^a$\\
%The address\\
Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Trieste,
Trieste, Italy$^b$} \\
%Continue of the address}\\
\end{centering}
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\begin{abstract}
The quantum mechanics of a system of charged
particles interacting with a magnetic field
on Riemann surfaces is studied.
We explicitly construct the wave functions of ground states
in the case of a metric proportional to the Chern form
of the $\theta$-bundle, and the wave functions of
the Landau levels in the case of the the Poincar{\' e} metric.
The degeneracy of the the Landau levels is obtained by using
the Riemann-Roch theorem. Then we construct the Laughlin wave function
on Riemann surfaces and discuss the mathematical structure
hidden in the Laughlin wave function.
Moreover the degeneracy of the Laughlin states is also
discussed.
\end{abstract}
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%{\footnotesize ($^\dagger$)email\\
%($^*$)email}
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\section{Introduction}
We study here the quantum mechanics of a system of charged particles living
on a two dimensional surface and interacting with a magnetic field
orthogonal to the surface. When the surface is an infinite plane and the
magnetic field is constant, we have the well known problem of the
Landau levels. This problem has received a renewed interest in the context
of the quantum Hall effect (QHE)(for a review, see Ref.\ \cite{pg}).
In fact, the QHE appears to be related to a rich physical
and also mathematical structure,
which is worthwhile to investigate
in various possible configurations.
A particular intriguing and
interesting case occurs when the two dimensional
surface is a Riemann surface of high genus.
Although not directly accessible to experiments, the problem
of the physics on the Riemann surface happens to have deep relations
with modern investigations on some interesting problems, like
the occurrence of chaos in the surface with a negative curvature
\cite{gutzw}, and recent developments
in the theory of Riemann surfaces,
for example, the moduli of
the surface and the vector bundles defined on
the moduli \cite{fay}.
In this paper, we will explore the problem directly from the
point of view of quantum mechanics,
by defining the Hamiltonian and then constructing
eigenfunctions of the Hamiltonian.
In Ref.\ \cite{comtet}, the problem in the case of
the surface being an open hyperbolic plane
with a constant negative curvature
was considered and Ref.\ \cite{antoine}
discussed the scattering on
a hyperbolic torus.
Instead, here we will mainly consider
compact Riemann surfaces.
Ref.\ \cite{avron} obtained
some interesting results about eigenvalues
and their multiplicity of a particle interacting with the
magnetic field, in the case of Riemann surfaces of high genus with
a constant curvature by using some results from the mathematical
literature, for example by using Selberg trace formula
(for a review on Selberg trace formula, see Ref.\ \cite{hejhal}).
See also Ref.\ \cite{asory} for a related
discussions.
We pursue this investigation by explicit
construction, which allows us to derive, and in some sense,
generalize all known results in a straightforward way and
provides us also the wave functions of Landau levels.
The wave functions turn out to be the holomorphic line
bundle defined on the surface, or for the high Landau levels,
they are obtained by applying some covariant derivatives on
the holomorphic line bundle.
Actually, the holomorphic line
bundle can be defined without reference to a particular metric.
Thus although we consider mainly
two cases for the metric,
we can make some possible interesting
generalizations to metrics of other kinds,
mainly for the case of the ground states.
It is known that the fractional quantum Hall effect (FQHE)
is related to the properties of the ground states, through
the Laughlin wave function. We will show the construction of the Laughlin
wave function on Riemann surfaces and indicate some interesting
relations with recent results appeared in the mathematical
literature\cite{fay}.
We organize the paper as follows.
In section 2, we use the Riemann-Roch theorem to derive
the degeneracy of the ground
states of a particle interacting
with a constant magnetic field
%(the definition of
%the constant magnetic field is also given there).
%Then we work out explicitly
%the ground states in the case of the Bergman metric.
As we will see, the natural definition of a constant magnetic
field is to take it proportional to the area form.
We consider first the case of a metric which is proportional
to the Chern form of the $\theta$-function line bundle
\cite{gaume}, defined explicitly
in section $2$. Since it is expressed in terms
of the canonical holomorphic one forms, we will call
it "canonical $\theta$-metric", abbreviated
as C$\theta$M\footnote{we understand there is not a standard
name for this metric in the literature. It is
proportional to the so called Bergman kernel,
which is also proportional to the curvature of
the Arakelov metric.}.
In section 3, we construct the Landau levels on
the surface with the Poincar{\' e} metric.
The eigenvalues of the Landau levels and their multiplicity
are obtained.
In section 4, we continue the discussion of section 3
to construct the wave functions of the
the Landau levels.
In section 5, we present the Laughlin wave function
on high genus surfaces with particular metrics.
The mathematical structure hidden in the Laughlin wave function
is pointed out.
In particular, we also discuss the degeneracy of the
Laughlin states.
\section{The lowest Landau level on
Riemann surfaces with the C$\theta$M metric}
%We are going to discuss quantum mechanics on Riemann surfaces.
We consider a particle on a Riemann surface
interacting with a "monopole" field,
that is the integral of the field strength out of the surface is different
from zero. We use the metric $ds^2=g_{z\bar z}dzd{\bar z}$
in complex coordinates and the volume form is
$dv=[ig_{z\bar z} / 2]dz\wedge d{\bar z}
= g_{z\bar z}dx\wedge dy$.
We apply a constant magnetic field on the surface.
The natural definition
of the constant magnetic field to the high genus Riemann
surface\cite{antoine,bolte} is
$$F=Bdv=(\partial_zA_{\bar z}-
\partial_{\bar z}A_z)dz\wedge d{\bar z},$$
with constant $B$.
Thus $ \partial_zA_{\bar z}-
\partial_{\bar z}A_z =ig_{z\bar z}B / 2$.
The flux $\Phi$ is given by $2\pi \Phi = \int F =BV$,
where $V$ is the area of the surface and
we assume here $B>0$ ($\Phi >0$).
The Hamiltonian of an electron on the surface
under the magnetic field is given
by the Laplace-Beltrami operator,
\begin{eqnarray}
H & = & [1/ 2m \sqrt{g}]
(P_{\mu}-A_{\mu})g^{\mu \nu}\sqrt{g}(P_{\nu}-A_{\nu})
\nonumber \\
& = & [ g^{z\bar z} / m]
[(P_z-A_z)(P_{\bar z}-A_{\bar z}) \nonumber \\
& & + (P_{\bar z}-A_{\bar z})(P_z-A_z)] \\
\label{hamil}
& = & [2g^{z\bar z}/ m]
(P_z-A_z)(P_{\bar z}-A_{\bar z})+[B/ 2m] \nonumber
\end{eqnarray}
where $g^{z\bar z}=[1 / g_{z\bar z}]$ and
$P_{z}=-i\partial_z ,\, P_{\bar z}=-i\partial_{\bar z}$
($\partial_z =(\partial_x -i\partial_y)/2$).
The inner product of two wave functions is defined as
$<\psi_1 | \psi_2 >=\int dv {\bar \psi_1 }\times \psi_2$.
$H^{\prime}=[2g^{z\bar z} / m]
(P_z-A_z)(P_{\bar z}-A_{\bar z})$
is a positive definite hermitian operator
because $<\psi | H^{\prime} |\psi > \, \geq 0$ for any $\psi$.
Thus if $H^{\prime} \psi =0$,
then $\psi$ satisfies
$(P_{\bar z}-A_{\bar z})\psi =0$.
The solutions of this equation
are the ground states of the Hamiltonian $H$ or $H^{\prime}$,
i.e. the lowest Landau level (LLL).
The existence of the solutions of this equation
is guaranteed by the Riemann-Roch
theorem\cite{griff,griff1}.
The solutions belong to the sections of the
holomorphic line bundle under the gauge field.
The Riemann-Roch theorem tells us that
$h^0(L)-h^1(L)=deg(L)-g+1$,
where $h^0(L)$ is the dimension of the sections of
the holomorphic line bundle
or the degeneracy of the ground states of the Hamiltonian $H$,
$h^1(L)$ is the dimension of the holomorphic differential
$(L^{-1}\times K)$ where $K$ is the canonical bundle and
$deg(L)$ is the degree of the line bundle which is equal to
the first Chern number of the gauge field, or the magnetic flux
out of the surface, $\Phi$.
When $deg(L) >2g-2$,
%(the magnetic field is very strong in the FQHE),
$h^1(L)$ is equal to zero \cite{griff}
and $h^0(L)=\Phi-g+1$.
As a consistent check, $h^0(L)$ indeed
gives the right degeneracy
of the ground states in the case of
a particle on the sphere and torus interacting
with a magnetic-monopole field.
If the Riemann surface $\Sigma$ has $g$ ($g>0$)
handles, there exist
abelian differentials, $g$ holomorphic and $g$
anti-holomorphic closed $1$-forms, $\omega_i$ and
$\bar \omega_i$.
They are normalized by
\begin{equation}
\int_{A_i} \omega_j= \delta_{ij} \, \, ,
\int_{B_i} \omega_j=\Omega_{ij}
\end{equation}
where $A_i, B_i$ are a canonical homology basis or
closed loops around handles on $\Sigma$, and the imaginary part
of $\Omega$ is a positive matrix.
We consider the C$\theta$M metric given by
\begin{equation}
g_{z\bar z}=
\bar \omega (Im \Omega )^{-1}\omega,
\end{equation}
which is always greater than zero.
This metric is proportional to
the Bergman reproducing Kernel
and it is also proportional to
the curvature of the Arakelov metric\cite{fay}.
Its most interesting feature (for our study here)
is that it is proportional to
the Chern form of the $\theta$-function line bundle\cite{gaume},
implying the covariant derivatives match the transformation
properties of the $\theta$-function.
We note that \cite{gaume}
\begin{equation}
V=\int_{\Sigma} dxdy
\bar \omega (Im \Omega )^{-1} \omega =g.
\end{equation}
Let us take $\Phi$ equal to $\gamma g$. Because of Dirac
quantization,
$\gamma g$ must be an integer and here we assume
that $\gamma g$ is a positive integer.
We will explicitly construct the ground states
in the case of $\gamma $ being a positive integer here
and the case of fractional $\gamma $
will be discussed in section $5.2$.
Now we have $B=2\pi \gamma$ and
$F_{z \bar z}=\partial_zA_{\bar z}-
\partial_{\bar z}A_z
=i\pi \gamma \bar \omega (Im \Omega )^{-1}
\omega$. We can take $A_{\bar z}=i
\pi \gamma \bar \omega (Im \Omega )^{-1}
u /2 $, where
$u^i
= \int^z_{z_0}\omega^i $
and $A_{\bar z}=\bar A_z$
are the gauge potentials in a certain gauge.
The ground states satisfy the equation,
\begin{equation}
\bar D \Psi =
(\bar \partial +{\pi\over 2}
\gamma \bar \omega (Im \Omega )^{-1}
u )\Psi=0.
\end{equation}
Because
\begin{equation}
D(u+\Omega n+m)= K(n,m)^{-1}D(u)K(n,m)
\end{equation}
with
$$K(n,m)=\exp \gamma [S(n,m)-\bar S(n,m)]$$
and
$$S(n,m)=(\pi /2) \bar u (Im \Omega)^{-1} (\Omega n+m),$$
thus we can choose
the boundary condition as \cite{iengole},
\begin{equation}
K(n,m)\Psi (u+\Omega n+m)=
\exp (i\alpha (n,m)) \Psi (u).
\label{boun}
\end{equation}
Generally we can take $\alpha (n,m)$
as
\begin{equation}
\alpha (n,m)=i \gamma \pi nm +2\pi i \gamma a_0m
-2\pi i b_0n.
\end{equation}
We define the function
\begin{equation}
\Psi =\exp [-(\pi /2)
\gamma \bar u(Im \Omega )^{-1}u
+(\pi /2)
\gamma u (Im \Omega )^{-1}u ]F.
\end{equation}
Then the function $F$ satisfies the equation,
\begin{equation}
F(u+\Omega n+m)=
\exp (-i\pi \gamma n\Omega n
-2\pi \gamma nu-i \gamma \pi nm+i\alpha (n,m))
F(u).
\label{theta}
\end{equation}
The solutions of Eq.\ (\ref{theta})
are
\begin{equation}
F_i(u)=\theta \left[ \begin{array}{c}
a \\ b
\end{array} \right]
\left(\gamma u|\gamma \Omega \right)
\label{nindp}
\end{equation}
with $b=b_0$ and $a_i=(a_0)_i+l_i / \gamma$,
$l_i=0, \cdots , \gamma -1$ and $i=1, \cdots , g$.
It seems that there are $\gamma^g$ solutions.
However, from the Riemann-Roch theorem,
we know that the degeneracy is
$\gamma g -g+1$ when $\gamma >1$ (remind that
$\gamma$ is a positive integer here, then we have
$\gamma g >2g-2$ and $h^1=0$).
We observe that
when $g=1$, $\gamma^g = \gamma g -g+1$ but
$\gamma^g > \gamma g -g+1$
for $\gamma >1$ and $g>1$. In fact, generally
the solutions given by Eq.\ (\ref{nindp})
are not linear-independent. Take, for example,
$F_1$ given by
\begin{equation}
F_1=\theta \left[ \begin{array}{c}
a_1 \\ b
\end{array} \right]
\left(\gamma u|\gamma \Omega \right),
\label{nindp1}
\end{equation}
then consider
$F_i / F_1$ which are the meromorphic functions on
the Riemann surface $\Sigma$.
Because $F_1$ has $\gamma g$ zeros, for example,
at $z_i, i=1, \cdots , \gamma g$,
the meromorphic functions will have possible poles at
points $z_i$. The dimension of such
meromorphic functions is
given by the Riemann-Roch theorem, the number of the
possible poles, $\gamma g$,
minus $g-1$ in the case of $\gamma >1$,
which is equal to $\gamma g -g+1$
and is the degeneracy of the ground states.
If $\gamma =1$, according to Eq.\ (\ref{nindp})
there is only one solution (it is possible that this solution
is identical to zero by the Riemann vanishing theorem and
thus there will be no solutions).
We remark that if $(a_0)_i=1/ 2 \, , (b_0)_i= \gamma / 2$,
the wave functions are transformed covariantly by
the modular transformations (for the case of $g=1$,
see Ref.\ \cite{iengole} ).
\section{The Landau levels on
Riemann surfaces with the Poincar{\' e} metric}
When $g>1$, the simply connected covering
space of the surface $\Sigma$ is the upper half plane $H$
(for example, Refs.\ \cite{griff1,gaume}), and
$\Sigma$ is equal to $H / \Gamma $, where
$\Gamma$ is the discreet subgroup of the isometry group
of $H$, isomorphic to the first homotopy of $\Sigma$.
$\Gamma$ is generated by the Fuchsian transformations around
a canonical homology basis, $\Gamma_{A_i}, \Gamma_{B_i}$
with
\begin{equation}
\prod^g_{i=1}\Gamma_{A_i}\Gamma_{B_i}
\Gamma^{-1}_{A_i} \Gamma^{-1}_{B_i}=1.
\end{equation}
The metric is given by the Poincar{\' e} metric,
$ds^2=y^{-2} (dx^2+dy^2)$, and we note that
$\int dv =\int y^{-2} dxdy =2\pi (2g-2)$ (without punctures).
In the case of the Poincar{\' e} metric $g_{z\bar z}=y^{-2}$,
the curvature is constant:
$$
g^{z\bar z}R_{z\bar z}= -2g^{z\bar z}\partial \bar \partial ln g_{z\bar z}=
-1.
$$
We take $F=Bdv$, and thus $A_z=-i B\partial (\ln g_{z\bar z}) /2$
and the flux $\Phi$ is equal to $2B(g-1)$.
Then we define $D=\partial - (B/2) \partial \ln g_{z\bar z}$
and $\bar D= \bar \partial + (B/2) \bar \partial
\ln g_{z\bar z}$.
The Hamiltonian is (we take $m=2$ in
Eq.\ (\ref{hamil}) for the simplicity),
\begin{equation}
H = -g^{z\bar z} D \bar D +(B/ 4).
\label{hamil1}
\end{equation}
The eigenfunctions satisfy
\begin{equation}
H \Psi = E \Psi .
\end{equation}
If $\tilde z$ is another local coordinate on $\Sigma$
and the domain of $\tilde z$ intersects non-trivially the
domain of $z$, $ g_{z\bar z}dz d\bar z$ is
invariant under coordinate changes, or
\begin{equation}
g_{z\bar z}dz d\bar z =
g_{\tilde z \tilde {\bar z} }d{\tilde z}
d\tilde {\bar z}
\end{equation}
on the intersection of the domains of $z$ and $\tilde z$.
$D$ and $\bar D$ are transformed as
\begin{equation}
{\tilde D}=(d z/d
{\tilde z})U^{-1} D U \, \, ,
\tilde {\bar D}=(d\bar z/d \tilde
{\bar z}) U^{-1} \bar D U
\end{equation}
where $U(z, \tilde z)= (dz / d \tilde z)^{-B/2}
(d \bar z / d \tilde {\bar z} )^{B/2}$.
The Hamiltonian is transformed as
\begin{equation}
\tilde H =U^{-1} H U,
\end{equation}
thus the wave function is transformed as
\begin{equation}
\tilde \Psi =U^{-1} \Psi
\end{equation}
or $\Psi (dz)^{B/2}(d\bar z)^{-B/2}$
is invariant under the transformation.
So we conclude that $\Psi$ is a differential form of type
$T^{\bar B/2}_{B/2}$.
Furthermore, the wave function is transformed
under the Fuchsian transformations as
\begin{equation}
\Psi (\gamma z) =u(\gamma,z) \Psi (z),
\, \, u(\gamma,z)=\nu (B, \gamma)
(cz+d)^{2B}/|cz+d|^{2B}
\label{boundaryp}
\end{equation}
where $\gamma$ is a Fuchsian group element
$\left( \begin{array}{cc}
a & b\\ c & d
\end{array} \right)$
and $\gamma z=(az+b)/(cz+d)$.
$\nu (B, -1)=e^{-i2\pi B}$ and
$u(\gamma_1 \gamma_2, z)=u(\gamma_1 , \gamma_2 z)
u(\gamma_2, z)$ are the consistency
conditions ensuring univaluedness
of the wave function on the universal
covering space \cite{hejhal}.
The boundary condition is twisted if $\nu (B, \gamma) \not= 1$.
The ground states are given by the solutions of the following
equation,
\begin{equation}
\bar D \Psi_0 =0 .
\end{equation}
The solutions of this equation are
$\Psi_0 =g^{(-B/2)}_{z\bar z} \tilde \Psi_0 $ with
$\bar \partial \tilde \Psi_0 =0$.
$\tilde \Psi_0$ belongs to a differential form of type
$T_{B}$. According to Eq.\ (\ref{boundaryp}),
$\tilde \Psi_0$ is transformed
under the Fuchsian transformations as
\begin{equation}
\tilde \Psi_0 (\gamma z) =u^{\prime}
(\gamma,z) \tilde \Psi_0 (z),
\, \, u^{\prime} (\gamma,z)=\nu (B, \gamma)
(cz+d)^{2B}
\label{boundaryp1}
\end{equation}
with $\nu (B, \gamma)$ defined in
Eq.\ (\ref{boundaryp}).
When $B=1$, $T_{B}$ is the canonical holomorphic
line bundle and $\tilde \Psi_0$ is given by
the sections of the canonical holomorphic
line bundle.
By the Riemann-Roch theorem,
we have $dim T_{B} -dim T_{1-B}=(2B-1)(g-1)$,
where $dim T_{B}$ is the dimension of the sections
of the holomorphic bundle $T_{B}$.
The dimension (or the degeneracy of the
ground states) of $T_1$ is $g$ for
the non-twisted boundary condition and
is $g-1$ for the twisted boundary condition
($\nu \not= 1$), because $dim T_0 =1$ for
the non-twisted boundary condition and
$dim T_0 =0$ for the twisted boundary condition.
When $B$ is an positive integer which is greater than one,
the dimension of $T_B$ is $(2B-1)(g-1)$ by
the Riemann-Roch theorem ($dim T_{1-B} =0$, as $1-B$ is negative).
When $B=1/2$, $T_{1/2}$ is $1/2$-differentials
(the spin bundle).
The dimension of $T_{1/2}$ is generically one
for the odd-spin structures and zero
for the even-spin structures.
The energy of the ground states is $B/4$. An expression for the wave
functions will be described in the next section. There we will also
indicate generalizations to the case of fractional $B$, provided that
$(2B-1)(g-1)$ is integer, and also possible generalizations to surfaces
with punctures.
Here and in the following, we call $g=g_{z\bar z}$ for short.
We introduce the covariant derivative,
$\nabla_z$, and its Hermitian conjugate
$(\nabla_z)^{\dagger}=-\nabla^z$,
\begin{eqnarray}
& \nabla_z & : T^{l}_k \to
T^{l}_{k+1} \, \, , \nabla_z =g^k\partial g^{-k}, \nonumber \\
& (\nabla_z)^{\dagger} &: T^{l}_k \to T^{l}_{k-1} \, \, ,
(\nabla_z)^{\dagger} =-
g^{-l-1} \bar \partial g^{l}.
\end{eqnarray}
Note that $D$ is the covariant operator
$\nabla_z$ acting on
$T^{ \bar B /2}_{B/2}$
($\bar D = g \nabla^z$ where $\nabla^z$
acts on $T^{ \bar B /2}_{B/2}$).
Let us next discuss the higher Landau levels.
By writing
$$
H-B/4 =-\nabla_z \nabla^z ,
$$
%where $\nabla^z$ is the covariant derivative and
%$-\nabla_z$ is the Hermitian conjugate of the covariant derivative
%$\nabla^z$.
we notice that if
$\Psi_1$ is an eigenfunction of $H$ with eigenvalue $E_1>B/4$,
then $\Psi_1 =-\nabla_z \nabla^z \Psi_1 /\epsilon_1 $
(where $\epsilon_1 =E_1-B/4\not= 0$). Therefore
$ \Psi_1 =\nabla_z \Phi$ for some $\Phi$. Of course, since $\Psi_1$
is of the form $T^{\bar B/2}_{B/2}$, then $\Phi$ will be of the form
$T^{\bar B/2}_{B/2-1}$. Thus we have, more explicitly,
$$
\Psi_1 = (\partial - (B/2-1)\partial ln g)\Phi .
$$
Due to the property of the Poincar{\' e} metric
$\partial \bar \partial ln g=g/2$, one can easily show that
$$
-\nabla_z \nabla^z \Psi_1 ={B-1\over 2}\Psi_1 +
\nabla_z (-\nabla_z \nabla^z \Phi ).
%\label{level1}
$$
%Notice that the notation is somewhat implicit since
%$$\nabla_z \nabla^z \Psi_1
%=g^{z\bar z}(\partial -B/2\partial ln g)
%(\bar \partial +B/2\bar \partial ln g)\Psi_1$$ whereas
%$$\nabla_z \nabla^z \Phi
%=(\partial -(B/2-1)\partial ln g)(\bar \partial +(B/2-1)\bar \partial ln g)
%\Phi $$
When $B\geq 1$, one can show that
$<\Psi_1|\nabla_z (-\nabla_z \nabla^z \Phi )>\geq 0$.
It is thus clear that the states of
the lowest excited level are obtained, if
there exist $\Phi$ such that $\nabla_z \nabla^z \Phi =0$,
i.e. $\bar D\Phi= 0$. This means that $\Phi = \Phi_0 =g^{-B/2}
\tilde \Phi_0$ with $\bar \partial \tilde \Phi_0 =0$. Since $\tilde \Phi_0$
is of the form $T_{B-1}$, there exist
solutions of the equation $\bar \partial \tilde \Phi_0 =0$
for $B\geq 1$.
The energy of the
lowest excited states is thus
$$
E_1={3\over 4}B-{1\over 2}.
$$
The degeneracy of this Landau level is the dimension of the
sections of the holomorphic bundle $T_{B-1}$
(which is equal to $(2B-3)(g-1)$ if $B>2$).
When $B<1$, there is only the $zero'th$ "Landau level"
(the lowest Landau level).
Beyond the "Landau levels", little is know about the spectrum.
We will make a comment about this point at the end of the
present section.
We can generalize the above discussion to high Landau levels.
The wave function of the $k'th$ Landau level is given by
\begin{eqnarray}
\Psi_k & = & (\nabla_z)^{k}\Phi_0 \nonumber \\
&=& (\partial-(B/2-1)\partial \ln g)
(\partial-(B/2-2)\partial \ln g) \\
& & \cdots
(\partial-(B/2-k)\partial \ln g)\Phi_0 \nonumber
\end{eqnarray}
with $\tilde \Phi_0=g^{B/2} \Phi_0$ and
$\bar \partial \tilde \Phi_0=0$. $\tilde \Phi_0$
is a differential form of the type $T_{B-k}$. Notice that this construction
generalizes the standard construction for the harmonic oscillator.
By using the relation, which holds for the Poincar{\' e} metric,
\begin{equation}
[\nabla^z \, \, \nabla_z]T^m_n
= -(m+n)/2,
\end{equation}
one can explicitly check that $\Psi_k$ is the eigenfunction
of the Hamiltonian, with the eigenvalue as
\begin{equation}
E_k=[B(2k+1)-k(k+1)]/4.
\end{equation}
The degeneracy of the $k'th$ Landau level
is given by the dimension of the sections of the holomorphic
bundle of the type $T_{B-k}$, which is equal to
$(2B-2k-1)(g-1)$ when $B-k>1$.
Because the dimension of
$T_n$ is zero when $n$ is negative,
$k$ must not be greater than $B$.
Hence there is only a finite number of "Landau levels".
When $B$ is an integer, $k$ can take value
from $0$ to $B$. When $k=B$,
the corresponding $\tilde \Psi_0$
is the (holomorphic)
differential form of the type $T_0$.
$T_0$ is a constant function on the surface. We can also include twisted
boundary conditions, which would physically correspond to the presence of
some magnetic flux through the handles.
If the boundary condition of the wave function is
the twisted one, %the dimension of $T_0$ is zero
there does not exist a non-zero constant function
which satisfies the twisted boundary condition.
Thus the dimension of $T_0$ is zero in this case
and there is not the $B'th$ Landau level.
When $k=B-1$, the degeneracy of this Landau level
is the dimension of the canonical bundle
$T_1$, which is equal to $g$
for the non-twisted boundary condition
and is equal to $g-1$
for the twisted boundary condition.
$B$ can be also an half-integer.
Then $k$ can take value from $0$ to $B-(1/2)$.
When $k=B-(1/2)$, the degeneracy of this Landau level
is the dimension of the spin bundle
$T_{1/2}$. The dimension of the
holomorphic sections of the spin bundle
generically is zero for the even-spin structures
and one for the odd ones (or for twisted ones).
In the next section,
we will show a construction of the wave functions
and we will see that it is possible to generalize the present scheme also
to
the case of $B$ fractional, provided a condition is satisfied, and to
include also "punctures" on the surface. We will
also discuss the resulting
spectrum of the Landau levels in the general case.
Of course, the "Landau levels" that we have found by the above method do not
exhaust the spectrum. In fact, when $k$ has reached the maximum value for
which $s=B-k$ is positive or zero, we can still express
$\Psi =\nabla_z^k g^{-B/2} \tilde \Phi$
%with $\Phi_n=g^{-B/2}\Phi_n$
where $\tilde \Phi$
is a $T_s$ differential and get an additional infinity of
levels and corresponding wave
functions by the solutions of the eigenvalue
equation for $-\nabla_z \nabla^z \tilde \Phi_n
=E_n \tilde \Phi_n$.
%where $\Phi_n$ belongs to $g^{-B/2}\Phi$.
The corresponding
eigenvalues for $\Psi$ will be
$E={1\over 4}(B(2k+1)-k(k+1))+E_n$.
In particular, for an integer $B$,
this will relate the general solution of
our problem to the eigenvalues and eigenfunctions of the Laplacian on the
scalar (i.e. the zero forms) on the Riemann surface with the Poincar{\' e}
metric, a problem which is not completely solved and for which there exists
a vast literature ( for a recent review see Ref.\ \cite{buser}).
\section{The wave functions of the Landau levels}
In order to complete the construction of the wave functions of the
"Landau levels" of the last section,
we would like to describe $\tilde \Phi_0$, that is
the holomorphic sections of the bundle corresponding to the differential
of the type $T_s$. We will present a formula for the determinant
$\det h_i(z_j)$,
where $h_i$ are the independent holomorphic sections and
$i,j$ run over the degeneracy of the Landau level. To get a particular
wave function, it is of course enough to consider this determinant as a
function of a particular $z$, fixing arbitrarily the remaining ones.
We have anticipated from the Riemann-Roch theorem that the degeneracy
is $N=(2s-1)(g-1)$, for $s$ integer or half integer greater than $1$,
the cases $s=1$, $s=1/2$, $s=0$ corresponding respectively to the
$g$ abelian differential, to the holomorphic spin structure(s) and
to the constant respectively, as recalled above.
The following formula does not make
reference to any particular metric, as the notion of holomorphic
differentials is introduced in a metric independent fashion.
The formula can be read from Ref.\ \cite{iengo}, and it has been
obtained in a contest of String theory, following the work of
Knizhnik\cite{kn}. It is:
\pagebreak
\begin{eqnarray}
\det h_i(z_j) & = & \theta \left[ \begin{array}{c}
a \\ b
\end{array} \right]
\left(\sum u_i -(2s-1)\sum_{i=1}^{g-1}r_i \right)
\prod^{g-1}_{i=1}
\left( \nu_0(z_i) \right)^{2s-1} \nonumber \\
& & \times \prod^N_{i1/2$. The dual line bundle will be
a form of the type $T_v$ with $v=1-s<1/2$. If $s>1$, since there
are no holomorphic $T_v$ forms with $v<0$, $N$ is the multiplicity for any
characteristic. In the case where $1>s>1/2$,
then $1/2>v>0$, and for the generic
moduli of the Riemann surface,
there will be one holomorphic $T_v$ form
for some characteristics.
Thus, for these characteristics, the multiplicity
of the level corresponding to $T_s$
will be $N+1$. The formula for the wave function would be in
this case a generalization of
Eq.\ (\ref{det})\cite{fay}. For those characteristics
there exists also a Landau level, generically with multiplicity $1$,
for those $s<1/2$ for which $(1-2s)(g-1)$ is a positive integer.
Since $s=B-k$ (recalled from the last section),
this means that we find
Landau levels provided that $(2B-1)(g-1)$
is a positive integer,
or, with some characteristics,
that $(1-2B)(g-1)$ is a nonnegative integer (remember that we
assume $B>0$). Therefore the general condition
for the existing of Landau levels
is that $2B(g-1)$ is
integer, that is the Dirac quantization condition.
We remind that the energy of the level is
$$
E_k={1\over 4}(B(2k+1)-k(k+1))
$$
corresponding to $s=B-k$. Thus we see that
the Landau level of maximal energy is obtained
for the value of $k$ which is nearest to $B-1/2$. If $B$ is integer then
the maximal energy is $B^2/4$, if $B$ is half integer the maximal energy is
${1\over 4}(B^2+1/4)$. If $B$ is another allowed
fraction, the maximal energy
is intermediate between the previous two.
This discussion about Eq.\ (\ref{det})
can be further generalized to the case of "punctures",
which formally corresponds
to the possibility of allowing poles for
$\Phi_0$ at some points of the surface, with the understanding that those
points are infinitely far (with the Poincar{\' e} metric) from any other
points. This means that the puncture
can be taken at infinity or on the real
axis in the upper half-plane, the surface making a narrow cusp there such
that the area is still finite. Thus this discussion makes
now use of a particular metric on the Riemann surface.
Quantum mechanically we require the wave function to be normalizable
and (taking the puncture at infinity) this implies for a differential
$T_s$ requiring $(y^2)^{s-2}|T_s|^2$ to be integrable in $y$
for $y \to \infty$. This means
that the poles of $T_s$ can be of order $s$ at most,
since a pole of order $r$ at $z=\infty$ would imply
$\lim_{y\to \infty} T_s \sim y^{r-2s}$ as it is seen by the appropriate
change of chart.
If we allow for punctures at say $w_1,...,w_n$, we have the freedom of
generalizing
Eq.\ (\ref{det}) by multiplying the r.h.s by
$\prod_i \prod_l (E(z_i,w_l))^{-s}$ and subtracting $(s\sum w_l)$ from
the argument of the theta function (this insures that (\ref{bound1})
continues to hold). This construction gives again a $T_s$ differential
provided now $N=(2s-1)(g-1)+ns$, which is the new multiplicity of the level
(we should be aware that $N=(2s-1)(g-1)+ns$ may not be
true in the case of $s\leq 1$. See the previous discussion).
Finally, a further generalization
could consist in allowing for some twists
on the punctures, corresponding to considering
branch points rather than poles at $w_l$.
\section{The Laughlin wave function
on Riemann surfaces}
\subsection{The Laughlin wave functions
in the constant field on the surface with the
Poincar{\' e} metric}
%in the magnetic field
%which is proportional to the curvature
%on the Riemann surface}
In the present section, we will discuss the Laughlin wave functions
in the constant field on the surface with the
Poincar{\' e} metric.
%proportional to the curvature. Usually, the magnetic field
%is not a constant magnetic field in this case.
%The magnetic field is a constant one
%if the curvature is constant, for example, in the case of
%the Poincar{\' e} metric (see the last section).
In the next subsection, the Laughlin states
in the constant magnetic field will be worked out in the case
of the magnetic field which is proportional to the C$\theta$M
metric. The mathematical structure behind
the Laughlin wave functions will be discussed in the end of this
subsection. We shall remark that the following discussions
can be generalized to the case of the magnetic field being
proportional to the curvature, if we take the Hamiltonian
in a special ordering,
\begin{equation}
H = -g^{z\bar z} D \bar D
%\label{hamil1}
\end{equation}
with $A_z=-i B\partial (\ln g) /2$ and $g$ is an arbitrary metric.
The magnetic field is a constant one
if the curvature is constant, for example, in the case of
the Poincar{\' e} metric (see the last section).
%We take $F_{z\bar z}=iB\bar \partial \partial \ln g$,
%$A_z=-i B\partial (\ln g) /2$ and $g>1$.
%The magnetic flux is $ \int F =2\pi B(2g-2)$.
%We consider the Hamiltonian as
%\begin{equation}
%H = -g^{z\bar z} D \bar D .
%\label{hamil1}
%\end{equation}
The generalization is straight forward and we will not discuss
it here.
Following section $3$,
we take $F_{z\bar z}=iB\bar \partial \partial \ln g$,
$A_z=-i B\partial (\ln g) /2$ and $g>1$.
The ground states satisfy
the equation, $\bar D \Psi_0=0$ and the solution of the equation
is $\Psi_0 =g^{-B/2} \tilde \Psi_0 $ with
$\partial \tilde \Psi_0 =0$. $\Psi_0$ is
$T_B$ differentials.
$h_i(z)$ are the solutions of the equation
$\partial \tilde \Psi_0 =0$.
In the FQHE,
the magnetic field applied is very strong.
Thus $B$ is a very large number and
the number of the sections of the
holomorphic $T_B$ differentials
is equal to $(2B-1)(g-1)$.
If the ground states are completely filled,
which corresponds to the case of the quantum Hall state
with filling $\nu =1$, the wave function of the
quantum Hall state is given by
$\Psi_{JL} =\prod_{i=1}
g^{-B/2}(z_i\bar z_i)
\det h_i(z_j)$ where $i=1, \cdots , (2B-1)(g-1)$
($\Psi_{JL}$ stands for the Jastrow-Laughlin type wave function).
A formula for $\det h_i(z_j)$ has
been shown and discussed in the previous
section, see Eq.\ (\ref{det}).
For any quantum Hall state, we write
$\Psi_{JL} =\prod_{i=1} g^{(-B/2)}(z_i\bar z_i)\Psi^{\prime}_J$
and $\Psi^{\prime}_J$ is a holomorphic function of
the coordinates of any electrons, as $\Psi_{JL}$ satisfies
the equation $\bar D_{z_i}\Psi_{JL}=0$
(we have this equation because every electron stays in the LLL).
The boundary condition,
Eq.\ (\ref{boundaryp}) or Eq.\ (\ref{boundaryp1})
for the single particle implies a boundary condition
on the many-body wave function (here is the wave function of the Hall
state),
\begin{equation}
\Psi^{\prime}_{JL} (\gamma z_j)
=u^{\prime} (\gamma,z) \Psi^{\prime}_{JL} (z_j),
\label{boundpp}
\end{equation}
with $u^{\prime} (\gamma,z)$ given by
by Eq.\ (\ref{boundaryp1}).
Furthermore, by using Eq.\ (\ref{boundpp}), the characteristics
$a$ and $b$ in Eq.\ (\ref{det})
can be fixed uniquely. Hence the degeneracy
of the Hall state at filling $\nu =1$ is one
(from the physical points of view, there is only one way
to completely fill the lowest Landau levels).
If the filling is equal to $1/m$,
we make an Ansatz for the wave function,
$\Psi_{JL} =\prod_{i=1}
g^{(-B/2)}(z_i\bar z_i)\Psi^{\prime}_{JL}$,
\begin{eqnarray}
\Psi^{\prime}_{JL} & = & \theta \left[ \begin{array}{c}
a \\ b
\end{array} \right]
\left(m\sum u_i -Q\sum_{i=1}^{g-1}r_i |
m\Omega \right)
\prod^{g-1}_{i=1}
\left( \nu_0(z_i) \right)^{Q_1} \nonumber \\
& & \times \prod^N_{iFrom the discussion of the previous section,
$B$ can be fractional in the above formulae
(the above Laughlin wave functions make sense
even $B$ is fractional), but $N$ and $\Phi$ must be integers.
We comment that the construction of the Laughlin wave function
by the determinant of the sections of the holomorphic vector bundle
can also apply to the case of $g=0,1$.
It seems to us that the degeneracy of Laughlin wave functions
is related to the different choice of
the basis of the sections of the
holomorphic vector bundle.
Finally we shall point out that the hierarchical wave function
on the Riemann surface can also be constructed
by following the method developed in Ref.\ \cite{lidpt}
\subsection{The Laughlin wave functions
in the constant field on the surface
with the C$\theta$M metric}
Following the first section, we take
$F_{z \bar z}=\partial_zA_{\bar z}-
\partial_{\bar z}A_z
=i\pi \gamma \bar \omega (Im \Omega )^{-1}
\omega$. $A_{\bar z}=i
\pi \gamma \bar \omega (Im \Omega )^{-1}U /2 $.
The ground states are given by
\begin{eqnarray}
\Psi_i & = & X(u)F_i(u) , \nonumber \\
X(u)& = & \exp [-(\pi /2)
\gamma \bar u(Im \Omega )^{-1}u
+(\pi /2) \gamma u (Im \Omega )^{-1}u ],
\end{eqnarray}
where $i=1, \cdots , \gamma g-g+1$ and we assume here that
$\gamma >1$ and $\gamma$ is an integer.
We remark that the following formula for the wave function
is true also in the case of fractional $\gamma$
and $\gamma g>2g-2$ (remind that $\gamma g$
is always an integer).
$F_i$ are the linear independent solutions
of Eq.\ (\ref{theta}).
The wave function of the electrons when the first Landau
levels (or ground states) are completely filled
(the filling is equal to $1$ in this case) is then
\begin{equation}
\Psi_{\nu =1} =\det \left( X(u_j)F_i(u_j) \right)
=\prod_i X(u_i)\det \left( F_i(u_j) \right).
\end{equation}
$\det \left( F_i(u_j) \right)$ can be
calculated even we do not know
how to select $F_i(u)$, the linear independent solutions
of Eq.\ (\ref{theta}).
According to Ref.\ \cite{fay},
$\det \left( F_i(u_j) \right)$ is equal to
\begin{eqnarray}
\theta \left[ \begin{array}{c}
a \\ b
\end{array} \right]
\left(\sum u_i \right)
\prod^N_{i2g-2$ and $\gamma$ being fractional,
the degeneracy of the ground states is still given by
$N=\gamma g -g+1$ and
the wave function of the electrons
at filling $\nu =1$ is still given by
Eq.\ (\ref{det}) multiplied by $\prod_i X(u_i)$.
The wave function of a single particle
can be obtained by fixing the coordinates of other particles
in the wave function of the electrons at filling $\nu =1$
in this case.
\subsection{The degeneracy of the Laughlin state,
a general discussion}
We have show that the degeneracy of some Laughlin states
in the last two subsections is $mg-g+1$ and we will try
to show here that, generally, the degeneracy of
Laughlin states
is $mg-g+1$ under some
reasonable assumptions.
The Laughlin type wave function is the
many particle wave function which looks like
$\Psi_{JL}=F(z_1, \cdots, z_N)\prod_{iFrom the previous discussion,
we make an assumption that $G_i$ is a function of the center
coordinate $\sum_i z_i$.
Because the meromorphic function can be always given by one
$\theta$ function divided by another
$\theta$ function, so the meromorphic function
with poles at the points which lie on some zeros
%(there are $mg$ zeros)
of the function $F_1^{\prime}$ is given by
\begin{equation}
G_i={\frac {\theta {a_i\brack b_i}(mu|m\Omega)}
{\theta {a_1\brack b_1}(mu|m\Omega)}}
\end{equation}
where now $u=\sum_i \int^{z_i} \omega$ and
${\theta {a_1\brack b_1}(mu|m\Omega)}$
has same zeros as $F_1^{\prime}$.
By Riemann-Roch theorem, the number of such
linear independent meromorphic functions
is $mg-g+1$ for $m > 1$ ($m=1$ is the case of the integer
QHE and the degeneracy of the Hall state is one).
Thus the number of the linear
independent functions $F_i^{\prime}$
or the degeneracy of the Laughlin states is
$mg-g+1$. However in Refs.\ \cite{gread,niu},
it was pointed out that the degeneracy of the Laughlin
states is s $m^g$ on the surface with $g$ handles.
In Ref.\ \cite{niu}, Wen and Niu analyzed the Chern-Simons effective theory
of the FQHE to get the degeneracy of the Laughlin states.
In the Chern-Simons theory, there are so called large components
of gauge fields (for example, Ref.\ \cite{bos})
and one part of the wave function
is $F^{\prime}_l({\cal A})=
{\theta {a_0+{l\over m} \brack b_0}(mu+m{\cal A}|m\Omega)}$,
where ${\cal A}$ is the large component of the gauge field,
$a_0+{l\over m}, b_0$ take values in ${\bf R}^g / {\bf Z}^g$
and $a_0, b_0$ is dependent on the boundary condition.
The phase space of ${\cal A}$ is Jacobian
variety ${\bf C}^g / {\bf Z}^g+\Omega {\bf Z}^g$.
Because ${\cal A}$ is now a dynamical variable,
all $m^g$ functions $F^{\prime}_l({\cal A})$ are independent
with each other. Thus the degeneracy of the Laughlin states is
$m^g$. However if we suppose that ${\cal A}$ is a constant
vector, the number of the linear independent functions
among $F^{\prime}_l({\cal A})$ is $mg-g+1$.
%\begin{equation}
%\end{equation}
%\begin{equation}
%\end{equation}
%\begin{eqnarray}
%\end{eqnarray}
\section{Acknowledgements}
We would like to thank Professors B. Dubrovin and
K.S. Narain for many useful conversations.
The work is partially supported by EEC, Science
Project SC1$^*$-CT92-0789.
\pagebreak
\font\pet=cmr10 at 10truept
\font\bf=cmbx10 at 10truept
\pet
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\end{document}