TITLE: How to write a certain expression as a total derivative?
QUESTION [1 upvotes]: I would like to write the expression $g'(g'')^2$ as a total derivative, i.e. I would like to find some (analytical) expression, call it $h$, so that $h' = g'(g'')^2$. The expression $h$ can involve $g$ and any of its derivatives, as well as special functions, logarithms, etc.
Disclaimer: I don't know if this is even possible. If it's impossible, a reasonable argument why that's the case would be appreciated.
Thanks so much!
EDIT:
Editing to include the entire expression:
$\frac{1}{8}[2(g')^2g'']'-g'(g'')^2$
I would like to interpret this as the gradient of some potential — if this is impossible I will likely need to find an alternative way to look at the expression.

REPLY [0 votes]: You are after some $h$ such that
\begin{align}
h'&=g'(g'')^2.
\end{align}
Well, $h$ would have to be some function of $x$, $g$, $g'$, and $g''$: $h(x,g,g',g'')$. We can then write the total derivative out using the chain rule
\begin{align}
h'&=\frac{\partial h}{\partial x}+\frac{\partial h}{\partial g}\frac{\partial g}{\partial x}+\frac{\partial h}{\partial g'}\frac{\partial g'}{\partial x}+\frac{\partial h}{\partial g''}\frac{\partial g''}{\partial x}\\
&=\frac{\partial h}{\partial x}+\frac{\partial h}{\partial g}g'+\frac{\partial h}{\partial g'}g''+\frac{\partial h}{\partial g''}g'''.
\end{align}
Now we can match up the two equations to see if such a function exists.
\begin{align}
h'=g'(g'')^2=\frac{\partial h}{\partial x}+\frac{\partial h}{\partial g}g'+\frac{\partial h}{\partial g'}g''+\frac{\partial h}{\partial g''}g'''.
\end{align}
Unless $g=ae^x$ or $a$, $g'\neq g''\neq g'''$, so then
\begin{align}
0=\frac{\partial h}{\partial g''}g'''.
\end{align}
One must equal zero, so
\begin{align}
g'''=0 \ \ \ \ \ \text{or} \ \ \ \ \ \frac{\partial h}{\partial g''}=0.
\end{align}
Again if $g$ doesn't satisfy this condition,
Remaining is that
\begin{align}
g'(g'')^2=\frac{\partial h}{\partial x}+\frac{\partial h}{\partial g}g'+\frac{\partial h}{\partial g'}g''.
\end{align}
Perhaps the last factor will allow us to find $h$?
\begin{align}
g'(g'')^2&=\frac{\partial h}{\partial g'}g''\\
g'g''&=\frac{\partial h}{\partial g'}\\
\frac{1}{2}(g')^2g''+f(x,g)&=h(x,g,g').
\end{align}
$h$ is cannot be a function of $g''$, yet that is the only way for $h'=g'(g'')^2$, the other two terms don't help. So there is no function $h$ such that $h'=g'(g'')^2$ in general. (I didn't check the three cases I eliminated $g=ae^x$, $g=a$, $g'''=0$).


TITLE: Limit $\lim_{x \to 0} \left(\cot x-\frac{1}{\sin x}\right)$
QUESTION [1 upvotes]: Im having trouble calculating this limit:
$$\lim_{x \to 0} \left(\cot x-\frac{1}{\sin x}\right)$$
I've tried factoring out $\frac{1}{\sin x}$, $\cos x$, $\cot x$ and it doesn't lead me anywhere. I also tried looking at alternative form $\frac{\cos x-1}{\sin x}$, with no luck. I can't use l'Hospitals rule. Can anyone help? I think this is super easy but somehow I'm stuck.

REPLY [1 votes]: $$\cot x-\frac{1}{\sin x}=\frac{\cos x}{\sin x}-\frac{1}{\sin x}=\frac{\cos x-1}{\sin x}$$
The Taylor series expansion for $\sin(x)$ and $\cos(x)$ at $x=0$ is
$$\sin(x)=x-\frac{x^3}{3!}+\mathcal{O}(x^5)$$
$$\cos(x)=1-\frac{x^2}{2!}+\mathcal{O}(x^4)$$
therefore
$$\cos(x)-1=-\frac{x^2}{2!}+\mathcal{O}(x^4)$$
hence
$$\frac{\cos x-1}{\sin x}=\frac{-\frac{x^2}{2!}+\mathcal{O}(x^4)}{x-\frac{x^3}{3!}+\mathcal{O}(x^5)}=\frac{-\frac{x}{2!}+\mathcal{O}(x^3)}{1-\frac{x^2}{3!}+\mathcal{O}(x^4)}$$
then $\frac{\cos x-1}{\sin x} \to \frac{0}{1}=0$ as $x\to 0$.


text:
\begin{document}
\title{Hedlund Metrics and the Stable Norm}
\author{Madeleine Jotz}
\ead{madeleine.jotz@epfl.ch}
\address{Section de Math{\'e}matiques, Ecole Polytechnique F{\'e}d{\'e}rale de Lausanne,
 1015 Lausanne, Switzerland.}

\begin{abstract}
The real homology of a compact Riemannian manifold $M$ is
  naturally endowed with the stable norm. The stable norm on $H_1(M,\mathbb{R})$ arises
  from the Riemannian length functional by homogenization. It is difficult and
  interesting to decide which norms on the finite-dimensional vector space
  $H_1(M,\mathbb{R})$ are stable norms of a Riemannian metric on $M$. If the dimension of
  $M$ is at least three, I. Babenko and F. Balacheff proved in \cite{baba}
  that every polyhedral norm ball in $H_1(M,\mathbb{R})$, whose vertices are rational
  with respect to the lattice of integer classes in $H_1(M,\mathbb{R})$, is the stable
  norm ball of a Riemannian metric on $M$. This metric can even be chosen to
  be conformally equivalent to any given metric. In \cite{baba}, the stable
  norm induced by  the constructed metric is computed by  comparing the metric with a polyhedral one.  Here we
  present an alternative construction for the  metric, which remains in the
  geometric framework of smooth Riemannian metrics.
\end{abstract}

\begin{keyword}
 Riemannian metrics, stable norm, polytopes.
\MSC[2008] 53C22, 53C38, 58A10, 58F17, 53B21
\end{keyword}

\maketitle
\section{Introduction}
On every compact Riemannian manifold $M$ the real homology vector spaces
$H_m(M;\mathbb{R})$ are endowed with a natural norm $\|\cdot\|_s$, called \emph{stable
  norm}. This concept appeared  for the first time in \citet{federer} and was
named \emph{stable norm} in \citet{gromov}.
The stable norm on $H_1(M;\mathbb{R})$ arises directly from the Riemannian metric on the
manifold $M$.
The following equality for an integral class $v\in H_1(M;\mathbb{R})$ (see \cite{gromov})
\begin{equation*}
\|v\|_s:=\inf\{n^{-1}L(\gamma)| \gamma \text{ is a closed curve representing }nv,\,n \in \mathbb{N}\}
\end{equation*} allows a description of this object that is geometrically very
intuitive: the stable norm describes the geometry of the Abelian covering
$\bar{M}$ of $M$ from a point of view from which fundamental domains look
arbitrarily small.
Knowing the unit ball of this norm, one can decide on existence and properties
of some of the minimal geodesics relative to the Riemannian Abelian covering of the
manifold; these are curves in $M$ whose lifts to the Riemannian
Abelian  covering minimize arc length between each two of their points.
Bangert has presented  in \citep{ba} a Riemannian metric on the $3$-torus
$\mathbb{T}^3$, such that the unit ball of the induced stable norm on $H_1(\mathbb{T}^3;\mathbb{R})\simeq \mathbb{R}^3$ is a symmetric
octahedron. Furthermore, Babenko and Balacheff have shown in \citep{baba} that,
given a compact Riemannian manifold $(M,\rho)$ of dimension greater than $2$,
for every centrally symmetric and convex polytope in $H_1(M;\mathbb{R})$ with nonempty interior, such that the directions of its vertices
are rational, there is a Riemannian metric on
$M$ that is conformal to $\rho$ and induces the given polytope as unit ball
of the stable norm.
Here we propose an alternative Riemannian metric, satisfying the same
conditions. Our construction is a generalization of the Hedlund metric in
\citet{ba}. The idea, that can be already found in the original paper of
\citet{hedlund} and is also used in \cite{baba}, is to construct a metric that is ``small'' in tubular
neighborhoods of disjoint closed curves representing the vertices of the
polytope, and much ``bigger'' everywhere else.
The convexity properties of the
polytope play a decisive role in our computation of the stable norm induced
by the Hedlund metric.

Bangert and Hedlund use such metrics
in order to illustrate their results on minimal geodesics. Here we
focuse only on the proof of the theorem of \citet{baba}. In fact, if we wanted
to show results on minimal geodesics, we would need to specify the definition
of the Hedlund metric we give here. A discussion of the minimal geodesics for
such metrics (with additional assumptions) was made in \citet{jotz}.

\paragraph{Outline of the paper:} in the next section the construction of
tubular neighborhoods of curves will be recalled. There a lemma on existence
of representatives for cohomology classes with ``good''
properties on the tubular neighborhood will be stated. In the following section
the construction of the Riemannian metric will be given and the formula for
the corresponding stable norm will be computed.

\paragraph{Notations:} In the following $M$ will denote a compact smooth manifold with $\dim M\geq 3$ and $\rho$  a Riemannian metric on $M$. Let $\bar M$
denote the Abelian covering of $M$. More precisely $\bar M$ is the subcovering
of the universal covering whose group of deck transformations is the set
$H_1(M;\mathbb{Z})_{\mathbb{R}}$ of integer classes in
$H_1(M;\mathbb{R})$. We denote by $p:\bar M\to M$
the covering map and by  $\bar \rho:= p^*\rho$ the pull-back metric.
If $h:\pi_1(M)\to H_1(M;\mathbb{Z})$ denotes the Hurewicz homomorphism
(\citep[see][]{leetop}) and $T$  the torsion subgroup of
$H_1(M;\mathbb{Z})$, then the Abelian covering
can be described as the quotient manifold of the action of the normal
subgroup $h^{-1}(T)\subseteq \pi_1(M)$ of the fundamental group on the universal
cover $\tilde M$  of $M$. Hence the operation
\begin{align*}
\begin{array}{lccc}
\Phi:&H_1(M;\mathbb{Z})_{\mathbb{R}}\times\bar{M}&\to&\bar{M}\\
&(v,m)&\mapsto&\Phi(v,m)=:m+v
\end{array}
\end{align*}
of $H_1(M;\mathbb{Z})_{\mathbb{R}}$ on $\bar{M}$ is abelian and
torsionfree (that is why we choose to use this $+$-notation).\\
The de Rham cohomology
vector space $H^1_{\bf dR}(M)$ is isomorphic to the dual of $H_1(M;\mathbb{R})$
\citep[de Rham theorem]{lee}. In the following, we will use this isomorphism
without mentioning it.

Given a Riemannian metric $g$ on $M$, we will write $g^*$ for its dual metric.
The space of $1$-forms on $M$ (respectively on $\bar{M}$) will be denoted by
$\Omega^1(M)$ (respectively $\Omega^1(\bar{M})$).  We will denote by
$\|\cdot\|_x$ (or also simply $\|\cdot\|$) the norm on $T_xM$ induced by the
considered metric on $M$ (we will also use this notation for the norm on
$T_{\bar{x}}\bar{M}$, $\bar{x}\in\bar{M}$ induced from the corresponding
metric on $\bar{M}$). For a curve $\gamma:I\to M$, $L(\gamma)$ will be the
length  induced from the given metric on $M$ and for a curve $\bar\gamma:I\to
\bar{M}$, $\bar{L}(\bar\gamma)$ the length induced from the corresponding
periodic metric on $\bar{M}$.

Given a polytope $P$, we will call the set
$\{\sum_{i=1}^k\alpha_iv_i\mid \alpha_i\geq 0\}$
the \emph{cone over the face $S$}  of the polytope, where $v_1,\dots,v_k$ are the
vertices of $P$ lying in this face (i.e. $S=\{\sum_{i=1}^k\alpha_iv_i\mid
\alpha_i\geq 0 \text{ and }  \sum_{i=1}^k\alpha_i=1\}$).

An integer class $v$ in $H_1(M;\mathbb{Z})_{\mathbb{R}}$ will be called \emph{indivisible} if the equation
$v=n\cdot v'$, $ n\in \mathbb{Z}$ and $v'\in H_1(M;\mathbb{Z})_{\mathbb{R}}$ yields $n=\pm 1$.

\paragraph{Acknowledgment:} I would like to thank Prof. Victor
Bangert who supervised my diploma thesis and gave me much advice for
this paper. I am also very grateful that he gave me the possibility to stay at the University of
Freiburg during a few months after my diploma.

I also thank the referees for many useful comments.
\section{Tubular neighborhoods of curves, adapted one-forms}

\paragraph{Tubular neighborhoods and semi-geodesic coordinates.}
Let  $\gamma:[0,1]\to M$ be a regular simple closed curve. In the following, such a curve will be
called \emph{admissible}. We can write  $\gamma:\mathbb{S}^1\to
M$ and assume the curve $\gamma$ is parametrized proportionally to
arc length.

For $\varrho>0$ let $V_\varrho(\Gamma)$ denote the bundle of balls of radius
$\varrho$ in the normal bundle $\pi:N\Gamma\to\Gamma$ of the embedded
submanifold $\Gamma:=\gamma(\mathbb{S}^1)$ in
$M$. Analogously, if $I\subseteq\mathbb{S}^1$ is an interval, then
$V_\varrho(\gamma(I))=V_\varrho(\Gamma)\cap\pi^{-1}(\gamma(I))$. We choose $\varrho>0$
small enough such that the normal exponential map $E$ restricted to $V_\varrho(\Gamma)$
is a diffeomorphism onto an open neighborhood $U_\varrho(\Gamma)\subseteq M$
of $\Gamma$ (and similarly $U_\varrho(\gamma(I))=E(V_\varrho(\gamma(I)))$\,).
Such an open set  $U_\varrho(\Gamma)$ is called the \emph{tubular neighborhood} (of radius $\varrho$) of $\Gamma$.

Choose an orthogonal frame $(E_1,\ldots,E_m)$ on $U \subseteq M$ open, such
that for all $x=\gamma(t)$ in  $\Gamma\cap U$,
\[E_1\arrowvert_{x}=\dot{\gamma}(t)\] and, consequently,
$(E_2\arrowvert_{x},\ldots,E_m\arrowvert_{x})$ forms a basis for $N_x\Gamma$. Assume the open set
$U$ is such that $U_\varrho(\Gamma)\cap U=U_\varrho(\gamma(I))$ for an
open interval $I\subseteq \mathbb{S}^1$. The diffeomorphism \[\begin{array}{rlcc}\varphi:&U_\varrho(\gamma(I))&\to
&I\times B_\varrho^{m-1}\subseteq \mathbb{R}^m\\
&x&\mapsto&(s(x),\varphi_2(x),\dots,\varphi_m(x)),\end{array} \]
where $\varphi_j(x)$ and  $s(x)$ are such that
\begin{equation*}
E^{-1}(x)=\sum_{j=2}^m\varphi_j(x)\cdot
E_j\arrowvert_{\gamma(s(x))}\in \mathcal{V}_\varrho,
\end{equation*}
will be called a
  \emph{semigeodesic chart for $U_\varrho(\Gamma)$}. A particularity of this chart is that $\partial^\varphi_1\arrowvert_{x}=\dot{\gamma}(t)$ and, for $j=2,\ldots,m$,
$\partial^\varphi_j\arrowvert_{x}=E_j\arrowvert_{x}$ holds for all $x=\gamma(t)\in
  \Gamma\cap U$ (note that $\Gamma\cap U=\gamma(I)$\,).

The map $s$ is defined globally on $U_\varrho(\Gamma)$ and we have the
identity
\begin{equation}
ds\arrowvert_{\gamma(t)}(\dot{\gamma}(t))=\frac{d}{dt}s\circ\gamma(t)=\frac{d}{dt}t=1\label{ds}
\end{equation}
for all $t$ in $\mathbb{S}^1$.

Let $\gamma_1,\dots,\gamma_N$ be  disjoint admissible loops and choose
$\varrho>0$ so that the construction above is possible  for all the curves
$\gamma_1,\dots,\gamma_N$ simultaneously. Choose furthermore $\varepsilon$
with $\varrho>\varepsilon>0$  such that  the tubular neighborhoods with radius
$\varepsilon$ of the curves
 are disjoint. Set $\Gamma_j=\gamma_j(\mathbb{S}^1)$, $\Gamma=\cup_{j=1}^N\Gamma_j$, and
 $U_\varepsilon(\Gamma):=\cup_{j=1}^NU_\varepsilon(\Gamma_j)$. Then there exists a bump-function $\zeta$
on $M$ for the tubular neighborhoods, i.e., $\zeta$ is a smooth function such
that the following holds:

\begin{align}\label{stern}
          \zeta(y)&=\left\{\begin{array}{ll}
             1&,\ y\in U_\varepsilon(\Gamma)\\
              0      & ,\ y \in M\setminus U_\varrho(\Gamma).\end{array}\right.
\end{align}

\paragraph{``Good'' one-forms.}
Choose a connected fundamental domain $F_0$ for the action of $H_1(M;\mathbb{Z})_{\mathbb{R}}$ on $\bar{M}$. Denote by $\bar{\gamma}_j$ the lift
of $\gamma_j$ to $\bar{M}$ such that $\bar{\gamma}_j(0)\in F_0$ (note that
$\gamma_j$ is here considered as a smooth $1$-periodic curve
$\gamma_j:\mathbb{R}\to M$). Write
$\bar{\Gamma_i}=\bar{\gamma_i}(\mathbb{R})$ and $U_\varrho(\bar{\Gamma_i})$ the
corresponding lift to $\bar{M}$ of
$U_\varrho(\Gamma_i)$. Hence $U_\varrho(\bar{\Gamma_i})$ is the tubular
neighborhood of radius $\rho$ of $\bar{\Gamma_i}$. Thus the notion of a semigeodesic chart for $U_\varrho(\bar{\Gamma_i})$
makes also sense here, and $\bar{s}_i:U_\varrho(\bar{\Gamma_i})\to\mathbb{R}$ exists with
$\bar{s}_i(\bar{\gamma_i}(t))=t$ for all $t\in\mathbb{R}$.\label{barsi}
Since the covering map $p:\bar M\to M$ is a local isometry,
\[\bar x\in \exp_{\bar{M}}(N_{\bar{\gamma_i}(t)}\bar{\Gamma_i})
\Leftrightarrow p(\bar x)
\in \exp_{M}(N_{p\circ\bar{\gamma}_i(t)}\Gamma_i)\] holds for all $\bar x\in
U_\varrho(\bar{\Gamma_i})$
  and
\begin{equation}\label{blab}
(p^*ds_i)\arrowvert_{U_\varrho(\bar{\Gamma}_i)}=d\bar{s}_i.
\end{equation}

Define  $L_i=\bar{\Gamma_i}+H_1(M;\mathbb{Z})_{\mathbb{R}}$ and
$U_\varrho(L_i)=U_\varrho(\bar{\Gamma_i})+H_1(M;\mathbb{Z})_{\mathbb{R}}$, as well as
$L=\cup_{j=1}^N L_j$ and $U_\varrho(L)=\cup_{j=1}^N
U_\varrho(L_j)$. Choose $\varepsilon$ with $0<\varepsilon<\varrho$ and define
$U_\varepsilon(\bar{\Gamma_i})$,
$U_\varepsilon(L_i)$ and $U_\varepsilon(L)$ as above. The connected
components of $L$ will be called \emph{lines} in the following.

In the following, a regular simple closed
curve will be called an \emph{admissible} curve.

\begin{proposition}\label{funk}
Let $v_1, \dots, v_N$ be indivisible integer classes in
$H_1(M;\mathbb{Z})_{\mathbb{R}}$, that span $H_1(M;\mathbb{R})$ as a real
vector space. Let $\gamma_1, \dots,
\gamma_N$ be disjoint admissible representatives of those
classes,  and
$U_\varepsilon(\Gamma_1),\dots,U_\varepsilon(\Gamma_N)$ disjoint
tubular neighborhoods of these curves.
Furthermore let $\lambda \in H^1_{dR}(M)$ be an arbitrary cohomology class.
Then there exists a one-form $\omega$ representing $\lambda$ such that:
\begin{equation*}
\omega\arrowvert_{x}=\lambda(v_i)ds_i\arrowvert_{x} \text{ for } x \in
U_\varepsilon(\Gamma_i), \quad i=1,\dots, N.
\end{equation*}
\end{proposition}

\begin{proof}
 For $j=1,\dots,N$, the function $\bar{s}_j$ is defined on
  $U_\varrho(\bar{\Gamma}_j)$. Set $\bar{s}_j=0$ on
  $U_\varrho(\bar{\Gamma}_i)$ for $i\neq j$ and define:
\begin{align*}
s_\lambda:U_\varrho(L) & \to  \mathbb{R}\\
x=x_0+v_0 &\mapsto  \sum_{i=1}^N\lambda(v_i)\bar{s}_i(x_0)+\lambda(v_0).
\end{align*}
Doing so, each element $U_\varrho(L_j)$ is written  $x=x_0+v_0$ with $x_0 \in
U_\varrho(\bar{\Gamma}_j)\cap F_0$  and
$v_0 \in H_1(M;\mathbb{Z})_{\mathbb{R}}$.
For $x \in
U_\varrho(\bar{\Gamma}_j)\cap F_0$ holds:
$s_\lambda(x)=\lambda(v_j)\bar{s}_j(x)$. Thus, with the definition of $s_\lambda$, for
$v=z\cdot v_j$ with $z \in \mathbb{Z}$:
\begin{align*}
s_\lambda(x+v)=\lambda(v_j)\bar{s}_j(x)+\lambda(v)
   =\lambda(v_j)\cdot(\bar{s}_j(x)+z)
\overset{\eqref{blab}}=\lambda(v_j)\cdot\bar{s}_j(x+v).
\end{align*}
This leads to
$s_\lambda\arrowvert_{U_\varrho(\bar{\Gamma}_j)}=\lambda(v_j)\bar{s}_j$,
and analogously:
$s_\lambda\arrowvert_{U_\varrho(\bar{\Gamma}_j)+v}=\lambda(v_j)\bar{s}_j\circ\Phi(-v,\cdot)+\lambda(v)$.
Thus, $s_\lambda$ is a smooth function.

Choose an arbitrary representative $\omega'$ for
  $\lambda$. Since $\omega'$ is closed, the $1$-form
  $\tilde{p}^*\omega'\in \Omega^1(\tilde{M})$ is also closed, where
  $\tilde{p}:\tilde M\to M$ is the universal covering of $M$.
  Since each closed $1$-form on $\tilde M$ is exact, there exists
  $\tilde{f}\in C^\infty(\tilde{M})$ such that   $\tilde{p}^*\omega'=d\tilde{f}$. One can
  show easily that $\tilde{f}$ is invariant under the action of $h^{-1}(T)$ on
  $\tilde{M}$ and descends to $\bar f\in C^\infty(\bar{M})$,
  i.e., $\tilde{f}=\bar f\circ
  q$ where $q:\tilde{M}\to\tilde{M}/h^{-1}(T)=\bar{M}$ is the projection. We
  have $p\circ q=\tilde{p}$ and $q^*d\bar f=d\tilde{f}=\tilde{p}^*\omega'=
  q^*(p^*\omega')$ and hence $d\bar f=p^*\omega'$. Let
  $\bar{g}:=s_\lambda-\bar f\arrowvert_{U_\varrho(L)}:U_\varrho(L)\to\mathbb{R}$. A computation shows that for all $x \in U_\varrho(L)$ and $v
  \in H_1(M;\mathbb{Z})_{\mathbb{R}}$, we have $\bar{g}(x+v)=\bar{g}(x)$
and the existence of $g: U_\varrho(\Gamma)\to \mathbb{R}$ with $\bar{g}=g\circ p$ follows.

The map $g$ is smooth and we have on $U_\varrho(L)$:
\begin{align*}
p^*dg=d\bar{g}
=\sum_{i=1}^N\lambda(v_i)d\bar{s}_i-d\bar{f}
=p^*\left(\sum_{i=1}^N\lambda(v_i)ds_i-\omega'\right).
\end{align*}
Since $p$ is a surjective local diffeomorphism, the equality
$dg=\sum_{i=1}^N\lambda(v_i)ds_i-\omega'$ follows.

 Define now the smooth $1$-form
 \[\omega:=gd\zeta+(1-\zeta)\omega'+\zeta\sum_{i=1}^N\lambda(v_i)ds_i\]
 with $\zeta$ as in \eqref{stern}.
Using the fact that $\omega'$ is closed on $U_\varrho(\Gamma)$ and the
properties of $\zeta$, one can easily
verify that $\omega$ is smooth and closed.
Furthermore, for $x \in U_\varepsilon(\Gamma_j)$:
\begin{align*}
\omega\arrowvert_{x}=g(x)d\zeta\arrowvert_{x}+\left(1-\zeta(x)\right)\omega'\arrowvert_{x}+\zeta(x)\cdot\sum_{i=1}^N\lambda(v_i)ds_i\arrowvert_{x}
               =\lambda(v_j)ds_j\arrowvert_{x},
\end{align*}as claimed. We get
\begin{align*}
[\omega](v_j)=\int_{\gamma_j}\omega
=\lambda(v_j)\int_0^1ds_j\arrowvert_{\gamma_j(t)}\left(\dot{\gamma_j}(t)\right)dt\overset{(\ref{ds})}=\lambda(v_j)
\end{align*}
for $j=1,\dots,N$. With $\operatorname{span}\{v_1,\dots,v_N\}=H_1(M;\mathbb{R})$, this yields that
$\omega$ is a representative for $\lambda$.
\end{proof}
In the following, such a representative $\omega$ will be called a \emph{good
  representative} of $\lambda$ with respect to the family $\{v_1,\ldots,v_N\}$.

\section{Hedlund metrics}
Let $P$ be a centrally symmetric and convex polytope in $H_1(M;\mathbb{R})$ with nonempty interior, such that the directions of its vertices
are rational. Such a polytope will be called \emph{admissible}.  We call $\tilde{V}_P=\{\tilde{v}_1,\dots, \tilde{v}_N,-\tilde{v}_1,\dots,
-\tilde{v}_N\}$ the set of vertices  of $P$.

Let
$v_1,\dots,v_N$ be indivisible integer classes such that $v_i=\varepsilon_i\tilde{v}_i$ with $\varepsilon_i>0$,
$i=1,\dots,N$.
Define $V_P:=\{v_1,\dots,v_N,-v_1,\dots,-v_N\}$ and let
$J_i$ be the subset of $V_P$ consisting of the indivisible integer classes corresponding to the
vertices belonging to the $i$-th
face $S_i$ of $P$. In order to simplify the notation, we assume without loss
of generality that
$J_1=\{v_1,\ldots,v_k\}$ for an integer $k\leq N$.
The norm $|\cdot|$ on $H_1(M;\mathbb{R})$, whose unit ball is $P$, is given as follows
(for vectors lying in the cone over the face $S_1$):
\begin{equation}
v=\sum_{j=1}^{k}\alpha_j\tilde{v}_j \text{ with }\sum_{j=1}^{k}\alpha_j=1
\text{ and all }\alpha_j\geq0 \Rightarrow
|v|=1
\end{equation}
or generally
\begin{equation*}
v=\sum_{j=1}^{k}\alpha_j\tilde{v}_j \text{ with all }\alpha_j\geq0 \Rightarrow
|v|=\sum_{j=1}^{k}\alpha_j
\end{equation*}
and likewise for every other face of $P$.

Since $P$ is convex, for each face $S_i$ of $P$  exists  an element $\lambda_i$ of $H^1_{dR}(M)\simeq H^1(M,\mathbb{R})$
such that
\begin{align*}
\lambda_i(\tilde{v}_j)\left\{
\begin{array}{lll}
=&1,&\quad v_j=\varepsilon_j\tilde{v}_j\in J_i\\
<&1,&\quad v_j=\varepsilon_j\tilde{v}_j\not\in J_i\\
\end{array}\right.
\end{align*}
(i.e. $\lambda_i\equiv 1$ on the plane defined by the face $S_i$ and
$\lambda_i$ is
smaller on the rest of the polytope).
Now, since $P$ is symmetric, $-\lambda_i$ is the $1$-form corresponding to
$-S_i$ and we get in fact:
\begin{align}\label{lambda}
-1<\lambda_i(\tilde{v}_j)<1\text{ for }\pm v_j\not\in J_i.
\end{align}
We get an alternative definition for the norm:
\begin{equation}\label{defnorm}
v\in \bigoplus_{j=1}^k\mathbb{R}_{\geq 0}\cdot v_j\Rightarrow |v|=\lambda_1(v),
\end{equation}
and likewise for every other face of $P$.

The metrics defined below will be called \emph{Hedlund metrics} since such a
metric first appears in Hedlund's paper \cite{hedlund} in the case $M=\mathbb{T}^3$:

\begin{definition}\label{def}
Let  $P$ admissible polytope with vertices $\{\tilde
v_1,\ldots,\tilde v_N,-\tilde v_1,\ldots,-\tilde v_N\}$.
Let $v_1,\ldots,v_N\in H_1(M,\mathbb{Z})_{\mathbb{R}}$ be
the indivisible integer
classes such that
$\varepsilon_i\tilde{v}_i=v_i$ for some $\varepsilon_i>0$, $i=1,\dots,N$.
Choose disjoint admissible
curves $\gamma_1,\dots,\gamma_N$ representing the
classes  $v_1,\dots,v_N$.
For each face $S_i$ of $P$, let $\eta_i$ be a good representative of
$\lambda_i$ with respect to the family $\{v_1,\ldots,v_N\}$.
A \emph{Hedlund metric} associated to $P$ on $(M,\rho)$ is a Riemannian metric $g$ that is conformal to $\rho$ and such that its dual metric $g^*$ satisfies:
\begin{enumerate}
\renewcommand{\theenumi}{\arabic{enumi}}
\renewcommand{\labelenumi}{(H\,\theenumi)}
\item
 $g^*_{\gamma_i(t)}(ds_i\arrowvert_{\gamma_i(t)},ds_i\arrowvert_{\gamma_i(t)})
=\underset{x\in U_\varepsilon(\Gamma_i)}\max\,
g^*_x(ds_i\arrowvert_x,ds_i\arrowvert_x)=\frac{1}{\varepsilon_i^2}$ for all
$t\in [0,1]$\newline
and
$g_x^*(ds_i\arrowvert_x,ds_i\arrowvert_x)<\frac{1}{\varepsilon_i^2}$
for $x \in U_\varepsilon(\Gamma_i)\setminus\Gamma_i$ and all $i \in \{1,\dots,N\}$.
\item $g^*_x(\eta_i\arrowvert_x,\eta_i\arrowvert_x)\leq 1$ for all $i=1,\ldots,N$ and $x\not\in U_\varepsilon(\Gamma)$.
\end{enumerate}
\end{definition}
Remark that for orientable compact surfaces of positive genus, it is not possible to choose disjoint loops
representing the vertices of the polytope. In fact, it is shown in \citet{ba}
that in the case of the $2$-torus, the stable norm induced by a Riemannian
metric on $\mathbb{T}^2$ has always a strictly convex unit ball.
Yet, Massart shows in \cite{massart97} that this is not true in general:
the stable norm induced by a smooth Finsler metric on a closed, orientable
surface has neither to be strictly convex, nor smooth.
For a
\emph{non-orientable} surface, the analogon to Theorem \ref{main} can be found in
\citet{BaMa2008}: they show that if $M$ is a closed non-orientable
surface equipped with a Riemannian metric, then there exists in
every conformal class  a metric on $M$ whose stable norm has a polyhedron
as its unit ball.

\paragraph{Existence and properties of such a metric}
\begin{proposition}\label{propex}
On every compact Riemannian manifold $(M,\rho)$ with $\dim M\geq 3$ and for
every admissible $P$ in $H_1(M,\mathbb{R})$
there exists a Hedlund metric associated to $P$ on $(M,\rho)$.
\end{proposition}

\begin{proof}
Given the admissible polytope $P$, choose disjoint admissible
curves $\gamma_1,\dots,\gamma_N$ representing the indivisible integer classes
$v_1,\dots,v_N$ corresponding to its vertices \linebreak
$\tilde{v}_1,\dots,\tilde{v}_N$.
Let $\varepsilon_1,\dots,\varepsilon_N$ be the coefficients as in Definition
\ref{def}.
For each face $S_i$ of $P$, $i=1,\dots,l$, let $\eta_i$ be a good representative for $\lambda_i$.
Set \[\Omega:=\underset{\substack{j=1,\dots,l\\
                                 x \in M\setminus U_\varepsilon(\Gamma)}}\max
                             \rho^*_x(\eta_j\arrowvert_{x},\eta_j\arrowvert_{x})\] and
 \[\Omega_i:=\max\{\underset{\substack{j=1,\dots,l\\x \in U_{\varrho}(\Gamma_i)}}\max
                             \frac{\rho^*_x(\eta_j\arrowvert_{x},\eta_j\arrowvert_{x})}{\rho^*_x(ds_i\arrowvert_{x},ds_i\arrowvert_{x})},\varepsilon_i^2\}\] for $i=1,\dots,N$. Define:
\begin{align*}
h_i: U_{\varrho}(\Gamma_i)&\to(0,\infty)\\
    x&\mapsto
    \frac{1}{\varepsilon_i^2\rho^*_x(ds_i\arrowvert_{x},ds_i\arrowvert_{x})}\cdot\exp(-C_i\cdot\ell(x)^2)
\end{align*}
 where
 \[
 C_i:=\ln\left(\frac{\Omega_i}{\varepsilon_i^2}\right)\cdot\frac{1}{\varepsilon^2}>0\]
and $\ell(x)$ is the distance from to $x$ to its ``projection''
$\gamma_i(s_i(x))\in \Gamma_i$.
Define the smooth function  $F:M\to(0,\infty)$ by
\begin{eqnarray*}
F(x)= \zeta(x)\cdot\sum_{i=1}^bh_i(x)+(1-\zeta(x))\cdot\frac{1}{\Omega},
\end{eqnarray*}
where $\zeta$ is a smooth bump function as in \eqref{stern}.  It is then easy to verify that the metric
$g$ defined by
\begin{equation*}
g^*_x=F(x)\rho^*_x  \quad \text{ for all } x \in M
\end{equation*}
is a Hedlund metric associated to $P$.
\end{proof}

\begin{proposition}
It results  immediately from Definition \ref{def} and from the properties of
an admissible polytope that
\begin{equation}\label{normeta}
\|\eta_i\|^*:= \underset{x\in M}\max\big\|\eta_i\arrowvert_{x}\big\|^*_x=1
\end{equation}
for each face $S_i$ of $P$.
\end{proposition}

\begin{proof} Here again, we assume that $i=1$. The arguments are the same for
  every other face of $P$.
Outside of  $U_\varepsilon(\Gamma)$, Definition \ref{def} yields
$\big\|\eta_1\arrowvert_{x}\big\|^*_{x}\leq 1$. With
\begin{eqnarray*}
\big\|\eta_1\arrowvert_{x}\big\|^*_{x}=\left\{
\begin{array}{ll}
\varepsilon_j\big\|ds_j\arrowvert_{x}\big\|^*_{x}=1 & ,x\in \Gamma_j \\
&\text{ and }j=1,\dots k\\
\varepsilon_j\big\|ds_j\arrowvert_{x}\big\|^*_{x}< 1& ,x\in
U_\varepsilon(\Gamma_j)\setminus\Gamma_j\\
&\text{ and }j=1,\dots k\\
\left|\lambda_1(v_j)\right|\cdot\big\|ds_j\arrowvert_{x}\big\|^*_x=  \varepsilon_j\left|\lambda_1(\tilde{v}_j)\right|\cdot\frac{1}{\varepsilon_j}\overset{(\ref{lambda})}<1\, &,x\in
U_\varepsilon(\Gamma_{j})\\
 &\text{ and }j>k,\\
\end{array}\right.
\end{eqnarray*}
this proves the statement.
\end{proof}

For the proof of the following lemma, we need to compute the lengths of the chosen
admissible curve $\gamma_1,\dots,\gamma_N$ relative to the new metric. Choose
$x=\gamma_i(t)\in\Gamma_i$ and a semi-geodesic chart $\varphi$ around
$x$. Recall the construction of such a chart; the matrix representing $\rho$
relative to the orthogonal basis
$(\dot{\gamma}_i(t),\partial_2^\varphi\arrowvert_{x},\dots,\partial_m^\varphi\arrowvert_{x})$
of $T_xM$ is diagonal. Hence, because $g$ is conformal to $\rho$, the
matrix representing $g$ relative to this basis is diagonal, too. Since the covectors
$(ds_i\arrowvert_{x},d\varphi_2\arrowvert_{x},\dots,d\varphi_m\arrowvert_{x})$
form a dual basis of $T_x^*M$, we obtain
\[g_x(\dot{\gamma}_i(t),\dot{\gamma}_i(t))=\frac{1}{g^*_x(ds_i\arrowvert_{x},ds_i\arrowvert_{x})},\]
using the fact that the matrice representing $g_x$ in the basis
$(\dot{\gamma}_i(t),\partial_2^\varphi\arrowvert_{x},\dots,\partial_m^\varphi\arrowvert_{x})$
is inverse to the matrix representing $g^*_x$ in the dual basis.
But because of (H $1$) in Definition \ref{def}, we have
$g^*_x(ds_i\arrowvert_{x},ds_i\arrowvert_{x})=\frac{1}{\varepsilon_i^2}$.
Hence, this leads to:
\begin{align}
L(\gamma_i)=\int_0^1\varepsilon_{i}dt=\varepsilon_{i}.\label{length}
\end{align}
It is possible to show that $\gamma_i$ is even the shortest curve representing
$v_i$: Assume, without loss of generality, that $v_i\in J_1$ and choose an
arbitrary curve $c:[0,1]\to M$ representing $v_i$. We have
$\lambda_1(v_i)=\varepsilon_i$ and hence
\begin{align*}
\varepsilon_i=&\int_c\eta_1=\int_0^1\eta_1\arrowvert_{c(t)}(\dot{c}(t))dt\leq\int_0^1\|\eta_1\arrowvert_{c(t)}\|^*\|\dot{c}(t)\|dt\\
&\overset{\eqref{normeta}}\leq\int_0^11\cdot\|\dot{c}(t)\|dt=L(c).
\end{align*}

\begin{lemma}\label{obenab}
There is a constant $C=C(M,P)$ such that for each face $S_i$ of $P$, every $w\in\bigoplus_{v\in J_i}
\mathbb{N}\cdot v$
and every $x\in \bar{M}$, the distance from $x$ to $x+w$ is bounded abowe by $\lambda_i(w)+C$.
\end{lemma}

\begin{proof}
Recall the definitions of $\gamma_i$, $\Gamma_i$, $\bar{\gamma}_i$, $\bar{\Gamma}_i$, $i=1,\dots,N$, $L$ and $F_0$.
Define
\[D:=\underset{1\leq i,j\leq
  N}\max\,\,\underset{\substack{x\in\Gamma_i\\y\in\Gamma_j}}\min\,
d(x,y),\]

\[\operatorname{diam}(M):=\underset{x,y\in M}\max d(x,y)\]
and choose a real positive number $e$ such that $e>\underset{i=1,\dots,N}\max\varepsilon_i$.
Let
\begin{equation}\label{constant}
C:=2\cdot\operatorname{diam}(M) +\kappa\cdot (D+e)
\end{equation}
where $d$ is the distance on $M$ induced from the Hedlund metric $g$ and
$\kappa=\kappa(P)$ is the maximal number of vertices lying on a common face of
$P$.

Without loss of generality, we assume that $w\in\bigoplus_{v\in J_1}
\mathbb{N}\cdot v$, i.e. we can write $w=\sum_{i=1}^kn_iv_i$ with
$n_1,\dots,n_k\in\mathbb{N}$.
We give a path from $x$ to $x+w$ that has length
bounded above by $\lambda_1(w)+C=\sum_{i=1}^k\varepsilon_in_i+C$. Assume that
$x\in F_0$ (otherwise, if $x\in F_0+u$ with $u\in H_1(M;\mathbb{Z})_{\mathbb{R}}$, we can replace the
path with startpoint $x-u$ as constructed below  with its image under $\Phi_u$). We join $x$ with $x+w$
by a path that runs \emph{as much as possible} in $L$ with ``changes of lines''
that are \emph{as short as possible}:

Choose $i_1\in\{j\mid 1\leq j\leq k,\, n_j\neq 0\}$ such that the point $x_1$ in $L\cap F_0$ with minimal
distance from $x$ lies in $\bar\Gamma_{i_1}$. Let $\tau_1$ be the corresponding geodesic segment from
$x$ to $x_1$ with minimal length. This length $\bar{L}(\gamma_1)$ is smaller
than $\operatorname{diam}(M)$. Let $c_1$ be the segment of
$\bar{\gamma}_{i_1}$ connecting $x_1$ and $x_1+n_{i_1}v_{i_1}$. This segment
  has length equal to
\begin{align}
\bar L(c_1)=n_{i_1}\cdot  L(\gamma_{i_1})\overset{\eqref{length}}=n_{i_1}\cdot\varepsilon_{i_1}.\nonumber
\end{align}
Now choose $i_2\in \{j\mid 1\leq j\leq k,\, n_j\neq 0\}\setminus{i_1}$ and $x_2\in
\bar{\Gamma}_{i_2}+n_{i_1}v_{i_1}$ such that $x_2$ is the point of
$(L\setminus \bar{\Gamma}_{i_1})\cap
(F_0+n_{i_1}v_{i_1})$ having minimal distance from $\bar{\Gamma}_{i_1}\cap
(F_0+n_{i_1}v_{i_1})$. Let $x_1'$ be the point in $\bar{\Gamma}_{i_1}\cap
(F_0+n_{i_1}v_{i_1})$ at this minimal distance from $x_2$.
Let $c_1'$ be the section of  $\bar{\gamma}_{i_1}$ connecting $x_1$ and
$x_1'$; the length of $c_1'$ lies in
$[n_{i_1}\cdot\varepsilon_{i_1}-e,n_{i_1}\cdot\varepsilon_{i_1}+e]$. Let
$\tau_2$ be the minimal geodesic segment joining $x_1'$ and $x_2$, it has
length smaller than $D$. Now
continue in this way; choose $i_3\in \{j\mid 1\leq j\leq k,\, n_j\neq 0\}\setminus\{i_1,i_2\}$ and $x_3\in
\bar{\Gamma}_{i_3}+n_{i_1}v_{i_1}+n_{i_2}v_{i_2}$ such that $x_3$ is the point
of $(L\setminus(\bar{\Gamma}_{i_1}\cup\bar{\Gamma}_{i_2})) \cap
(F_0+n_{i_1}v_{i_1}+n_{i_2}v_{i_2})$ having minimal distance from $\bar{\Gamma}_{i_2}\cap
(F_0+n_{i_1}v_{i_1}+n_{i_2}v_{i_2})$. Let $x_2'$ be the point in $\bar{\Gamma}_{i_2}\cap
(F_0+n_{i_1}v_{i_1}+n_{i_2}v_{i_2})$ at this minimal distance from $x_3$. The curve $c_2'$
joining $x_2$ and $x_2'$ on $\bar{\Gamma}_{i_2}+n_{i_1}v_{i_1}$ has length
smaller than $n_{i_2}\cdot\varepsilon_{i_2}+e$.

If $n_j\neq 0$ for $j=1,\dots,k$, our path will be the composition
\[\gamma:=\tau_1*c_1'*\tau_2*c_2'*\dots*c_k'*\tau_{i_{k+1}}\]
where $\tau_{k+1}$ is the path joining the last point in
$L\cap(F_0+\sum_{i=1}^kn_iv_i)$ with minimal distance from $x+w$ to $x+w$ and
has length smaller than $\operatorname{diam}(M)$. Summing all
the lengths of those segments we get
\begin{align*}
\bar L(\gamma)\leq&\operatorname{diam}(M)+n_{i_1}\cdot\varepsilon_{i_1}+e+D+n_{i_2}\cdot\varepsilon_{i_2}+e+D\\&+\dots+n_{i_k}\cdot\varepsilon_{i_k}+e+\operatorname{diam}(M)\\
&=\lambda_1(w)+k\cdot
e+k\cdot D+2\cdot \operatorname{diam}(M)\leq \lambda_1(w)+C.
\end{align*}
Finally, if $n_j=0$ for some $j\in\{1,\dots,k\}$, we need to make fewer
changes of lines, and the inequality can be shown the same way.
\end{proof}

\paragraph{The stable norm and the main theorem.}
In the introduction of this paper, we gave the definition of the stable
norm induced from a Riemannian metric $g$ on $M$. Here we give a way to compute
 the stable norm of a vector lying in $H_1(M;\mathbb{Z})_{\mathbb{R}}$:
Define \begin{eqnarray*}
f:H_1(M;\mathbb{Z})_{\mathbb{R}}&\to&\mathbb{R}_{\geq0}\\
v&\mapsto&\inf\{L(\gamma)|\gamma \text{ closed curve representing }
v\}
\end{eqnarray*}
and $f_n:n^{-1}H_1(M;\mathbb{Z})_{\mathbb{R}}\to
\mathbb{R}_{\geq0}$, $f_n(v)=n^{-1}f(nv)$.
In \citet{ba} it  is shown that $f_n$ converges uniformly on compact sets to the
stable norm $\|\cdot\|_s$. Especially, we have: if $(v_n)_{n \in \mathbb{N}}$ is a sequence in
$H_1(M;\mathbb{Z})_{\mathbb{R}}$ with
$\lim_{n\to\infty}\frac{v_n}{n}=v \in H_1(M;\mathbb{R})$ (relative to the
standard topology on the vector space $H_1(M;\mathbb{R})\simeq\mathbb{R}^b$), then we have for
the norm of $v$:
\begin{equation*}
\|v\|_s=\underset{n\to\infty}\lim \frac{f(v_n)}{n}.
\end{equation*}
If $\bar{d}$ is the distance on $\bar{M}$ induced from $p^*g$,  we have
for $v \in H_1(M;\mathbb{Z})_{\mathbb{R}}$:
\begin{equation*}f(v)=\underset{x\in\bar{M}}\inf
\bar{d}(x,x+v)=\underset{x\in F_0}\min\, \bar{d}(x,x+v)
\end{equation*} because $p^*g$ is a periodic metric and the closure of $F_0$
is a compact set.
With $\underset{n\to\infty}\lim \frac{nv}{n}=v$, this yields:
\begin{eqnarray*}
\|v\|_s&=&\underset{n\to\infty}\lim \frac{f(nv)}{n}\\
     &=&\underset{n\to\infty}\lim\frac{\underset{x\in F_0}\min\, \bar{d}(x,x+nv)}{n}.
\end{eqnarray*}

\begin{theorem}\label{main}
The polytope $P$ is the unit ball of the stable norm on $H_1(M;\mathbb{R})$
induced by  an arbitrary Hedlund metric associated to $P$ on $M$.
\end{theorem}

Note that by Definition \ref{def}, the Hedlund metric is chosen in
the conformal class of  the given Riemannian
metric $\rho$ on $M$.
\begin{proof}
Let $g$ be a Hedlund-metric associated to $P$. We show that for each $w\in
\bigoplus_{j=1}^k\mathbb{N}\cdot v_j$, the stable norm of $w$ is given by
$\|w\|_s=\lambda_1(w)$. The proof of this works analogously for every other
face of $P$.  Consequently, this holds for all vectors in $H_1(M;\mathbb{R})$
that can be written as linear combinations of the vectors $v_1,\dots,v_N$ with
rational coefficients, and then, by continuity, this holds for all vectors in $H_1(M;\mathbb{R})$.
Let $x$ be an arbitrary point in $F_0$ and let $n\in \mathbb{N}$. Let
$\gamma:[0,1]\to \bar{M}$ be an arbitrary path from $x$ to $x+nw$. We have
\begin{align*}
\lambda_1(nw)&=\int_\gamma\eta_1=\int_0^1\eta_1\arrowvert_{\gamma(t)}(\dot{\gamma}(t))dt\leq\int_0^1\|\eta_1\arrowvert_{\gamma(t)}\|^*\|\dot{\gamma}(t)\|dt\\
&\overset{\eqref{normeta}}\leq\int_0^11\cdot\|\dot{\gamma}(t)\|dt=\bar{L}(\gamma)
\end{align*}
With this and Lemma \ref{obenab} we get
\begin{equation*}
\lambda_1(n\cdot w)\leq \bar{d}(x,x+nw)\leq \lambda_1(n\cdot w)+C.
\end{equation*}
Thus
\begin{equation*}
\lambda_1(n\cdot w)\leq \underset{x\in F_0}\min\,\bar{d}(x,x+nw)\leq \lambda_1(n\cdot w)+C,
\end{equation*}
and
\begin{equation*}
\lambda_1(w)\leq \frac{\underset{x\in F_0}\min\,\bar{d}(x,x+nw)}{n}\leq \lambda_1(w)+\frac{C}{n}.
\end{equation*}
Letting $n$ go to infinity, this yields $\|w\|_s=\lambda_1(w)$, as claimed.
\end{proof}

config: arxiv


TITLE: Continuous functions proof
QUESTION [1 upvotes]: Let $f$ and $g$ be continuous functions on R with the absolute value metric, and let $S$ $\subset$ R be countable. Show that if $f(x)=g(x)$ for all $x\in S^c$ (complement of $S$), then $f(x)=g(x)$ for all $x\in$ R.
Now since $f$ and $g$ are continuous on R, then they are obviously continuous on $S$ and $S^c$, considering R is the union of both. For $f(x)=g(x)$ to be true for all $x$ in R, then it should be sufficient to show that it holds in $S$, because it is given that it is true for $S^c$. I know I have to use the fact that $S$ is countable, but I can't think of how and why. Any hints?
Thank you.

REPLY [2 votes]: Since $S$ is countable, $S^c$ is dense. This means $\overline {S^c} = \mathbb R$. Now proof: if two continuous functions coincide on a dense set, they are equal: For that, we can use, that every metric space is also Hausdorff and given that, we know, that the diagonal $\Delta = \{ (x,x) \in \mathbb R \times \mathbb R \, \mid \, x \in \mathbb R \}$ is closed in $\mathbb R\times \mathbb R$. Consider the function $h\colon \mathbb R \to \mathbb R \times \mathbb R, \: h(x) = \bigl(f(x), g(x)\bigr)$. $h$ is continuous, because $f$ and $g$ are. Thus $h^{-1}(\Delta) = \{ x \in \mathbb R \, \mid \, f(x) = g(x) \}$ is closed. Because of $S^c \subset h^{-1}(\Delta)$ we get $\overline {S^c} = \mathbb R \subset h^{-1}(\Delta)$.
I like this proof because it holds for all Hausdorff spaces. I think i've read it in Munkres Topology book.
Hope this helps,
Maikel

REPLY [1 votes]: What you want is the following:
DEFINITION A set $D\subseteq \Bbb R$ is dense over $\Bbb R$ if for every $x\in \Bbb R$ there exists a sequence $\{x_n\}\subset D$ such that $x_n\to x$.
DEFINITION Given a function $f:X\to Y$, the restriction of $f$ to $D\subset X$ is the function $f\mid _D:D\to Y$ such that $f\mid_D(x)=f(x)$ for each $x\in D$.
THEOREM Let $f$ and $g$ be continuous over $\Bbb R$. Let $D$ be dense over $\Bbb R$. If $f\mid_D=g\mid_D$, then $f=g$. That is, if $g$ and $f$ agree on every point of $D$, then they agree on every point of $\Bbb R$.
PROOF Let $x\in \Bbb R\setminus D$. Then there exists a sequence $\{x_n\}\subset D$ such that $x_n\to x$. And for this sequence $\lim\; f(x_n)=f(x)=g(x)=\lim\; g(x_n)$, so the claim follows. Note that since $\{x_n\}\subset D$, $x_n\neq x$ for each $n$.
The proof is rather simple if you have already characterized continuity in terms of sequences, that is
PROPOSITION Let $f:[a,b]\to \Bbb R$ be continuous. Then $f$ is continuous if, and only if, for each sequence $\{x_n\}\subset [a,b]$ such that $x_n\to x$ (with $x\neq x_n$ for each $n$) we have that $f(x_n)\to f(x)$.
PROOF Suppose $f$ is continuous, and $x_n\to x$. Then for each $ \epsilon>0$ we get a $\delta >0$ such that for each $y$, whenever $|x-y|<\delta$, $|f(x)-f(y)|<\epsilon$. Since $x_n\to x$, we can obtain an $N$ such that if $n>N$, then $|x-x_n|<\delta$, so that for $n>N$; $|f(x)-f(x_n)|<\epsilon$, and one direction follows. Can you prove the other direction?


TITLE: Notation question for finite simple groups
QUESTION [0 upvotes]: Looking at the article on finite simple groups in Wikipedia ("List of finite simple groups") it uses a notation for the order of the groups that I am not familiar with.  For example, the order of the Steinberg groups has a factor of $\frac{q^{(n(n-1))}}{(4, q^{n+1})}$. What is the meaning of the comma in the parenthesis in the denominator?

REPLY [3 votes]: $(a,b)$ is the greatest common divisor of $a$ and $b$.


TITLE: Evaluating stochastic integral with exponential function
QUESTION [0 upvotes]: After solving for the Ornstein-Uhlenbeck process, i'm still stuck with a stochastic integral that I can't solve.
The Ornstein-Uhlenbeck process:
$X_{t} = X_{0} e^{-\theta t}+ \mu(1-e^{-\theta t}) + \sigma e^{-\theta t} \int_{0}^{t} e^{\theta s}dW_{s} $
Does anyone have an idea on how to solve this stochastic integral? ($\int_{0}^{t} e^{\theta s}dW_{s} $)
*Note that i'm trying to evaluate this expression for a Monte-Carlo simulation. I have already tried discretizing the integral but I would like to improve my results by using the exact solution.
Thanks in advance!

REPLY [0 votes]: If you only want to simulate that integral, then nothing is mysterious here.
It's just a normal variable of law $ \mathcal{N}\left( 0, \int_{0}^t e^{2 \theta s}.ds \right)$.
If you want to so look at $(W_s)$ as a stochastic process then you can see easily that $ (U_t ):= \left(  \int_{0}^t e^{ \theta s}.dW_s \right) $ is just a change time Brownian process. To be more precise,
Letting $g(t)=  \int_{0}^t e^{2 \theta s}.ds$.
Roughly speaking, there is a Brownian process $\tilde{W}$ such that:
$(U_t)=\left( \tilde{W}_{g(t)} \right)$


TITLE: Yet another Integral involving $e^{ax} +1$ and $e^{bx} + 1$.
QUESTION [2 upvotes]: A variation on: Another Integral involving $e^{ax} +1$ and $e^{bx} + 1$
Evaluate the integral  $$I(a,b)=\int_{0}^{1}\frac{(e^{ax})(e^{bx})}{\left(e^{ax}+1\right)\left(e^{bx}+1\right)}dx$$  for $a>b>0$.
Attempt
I suppose, like before, I have to simplify the integrand to seperate the $a$ and $b$ into different integrals. So far I have managed to do this:
$$1 - \frac{1}{\left(e^{bx}+1\right)} - \frac{(e^{bx})}{\left(e^{ax}+1\right)\left(e^{bx}+1\right)}$$
But I am stuck. Not sure if the question is unsolvable this way or if I can't see the trick.
Additional Info
This is not an "official" question from a textbook, course or quiz. There might be no nice solutions.

REPLY [0 votes]: Hint: $$I(a,b)=\int_{0}^{1}1-\frac{1}{\left(e^{bx}+1\right)}-\frac{1}{\left(e^{ax}+1\right)}+\frac{1}{\left(e^{ax}+1\right)\left(e^{bx}+1\right)}dx.$$


TITLE: Estimating $\sum_{n\leq x: n \in A} d(n)^a$ from below for large sets $A\subset \{1,2,\ldots,x\}.$
QUESTION [5 upvotes]: I apologise for the long-windedness of this question.
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S_a(x)=\sum_{n\leq x} d(n)^a.
$$
For $a=1,$ the following is well known
$$
S_1(x)=\sum_{n\leq x} d(n)=x \log x + (2 \gamma -1) x +{\cal O}(\sqrt{x})
$$
while for more general $a$, one has
$S_a(x) \sim C(a) x (\log x)^{2^a -1}$ where
$$
C(a) = \Gamma(2^a)^{-1} \prod_p \left( 1 - \frac{1}{p} \right)^{2^a} \left( \sum_{k \geq 0} \frac{(k+1)^a}{p^k}\right).
$$
More accurate estimates are available via the Selberg-Delange method, though the details get quite technical.
My question is the following. Let the subset $A$ be defined by $A\subset \{1,2,\ldots,x\},$ (assume $x$ is integer or use the floor function) with $$\#A\geq \frac{x}{2}+ c \frac{\log x}{\log \log x}:=Z(x).$$
Now define
$$
S^{A}_{a}(x)=\sum_{n\leq x:n \in A} d(n)^a.
$$
I want to lower bound this sum, as $A$ varies subject to the size condition, i.e., to derive a lower bound to
$$
M:=\min \left\{ S_a^A(x): A \subset \{1,\ldots,x\}, \#A =Z(x) \right\}.
$$
For $a=1,$ I hope a lower bound of the form $M\gg x \log x$ may be possible if the Poisson approximation is good enough. Essentially this would say that half the points achieve a constant factor of the full sum, even when restricted to those points with the fewest number of divisors.
From a Poisson approximation point of view, it would seem that approximately half the points $n\in \{1,\ldots,x\}$ have $\omega(n)\leq \log\log n,$ with the relevant Poisson distribution having mean $\log\log n.$
I am unsure if the known techniques are strong enough to address such a delicate size specification, compared to say $\lceil x/2 \rceil,$ but some comments related to Selberg-Delange have a discussion of how a multiplicative $O(1+(\log x)^{-c'})$ factor can be shown for various sums, which may imply that it is indeed possible.

REPLY [4 votes]: Even if you take $Z(x)=(1-\varepsilon)x$ for some fixed $0<\varepsilon<1$, you are going to get
$$
M=x(\ln x)^{a\ln 2+o(1)}.
$$
To prove this, observe that both $\omega(n)$ and $\Omega(n)$ have normal order $\ln\ln n$ (here $\Omega$ and $\omega$ are numbers of prime factors with and without multiplicity respectively). Also note that
$$
2^{\omega(n)}\leq d(n)\leq 2^{\Omega(n)}
$$
for every $n$, which implies that for any $\delta>0$ for all but $o(x)$ numbers $n\leq x$ we have
$$
(\ln x)^{\ln 2-\delta}\leq d(n) \leq (\ln x)^{\ln 2+\delta}.
$$
Therefore, for any $A\subset [1,x]\cap \mathbb N$ with $|A|\geq cx$ for some $c>0$ we have
$$
S_a^A\geq (|A|-o(x))(\ln x)^{a(\ln 2-\delta)},
$$
so that $M\gg x(\ln x)^{a(\ln 2-\delta)}$. On the other hand, if you throw all the $n$ with $d(n)>(\ln x)^{\ln 2+\delta}$ out of your interval, you will still have $x-o(x)$ numbers left. This implies that for any $\varepsilon>0$ and $x$ large enough there is $A$ with $|A|\geq (1-\varepsilon)x$ such that
$$
S_a^A\leq |A|(\ln x)^{a(\ln 2+\delta)}.
$$


TITLE: What's the reason behind Earth's gravity?
QUESTION [0 upvotes]: How can Earth attract any object irrespective of what it is made?

REPLY [4 votes]: The reason is the mass of the Earth. Its mass creates a gravitational field around it so anything that is inside the gravitational field of the Earth, would feel the gravitational force the Earth is exerting on the object. In the same way because of Newton's third law, the Earth would feel a gravitational force towards the object of the same magnitude it exerts over it.
The core of the Earth contains convecting currents due to highly conductive fluids, this creates electric currents which are the cause of the Earth`s magnetic field.The force of gravity acts between two objects of any mass, and since the mass of the Earth is bigger compared to a simple object as a car, then the Earth attracts any object.
The reason is that any body creates a gravitational field around it, so the massive the object, the stronger the gravitational force it exerts. Also since the gravitational force is inversely proportional to the square of the distance, then the closer it is the strongest is the gravitation force. E=MC2


text:
\begin{document}

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   \end{flushleft}\vspace{-1.3cm}
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     {\tt hep-th/yymmnnn} \\ December 2006
   \end{flushright}}
\newcommand{\Title}[1]{{\baselineskip=26pt
   \begin{center} \Large \bf #1 \\ \ \\ \end{center}}}
\newcommand{\Author}{\begin{center}
   \large \bf
Wen-Li Yang ${}^{a,b}$,
  Yao-Zhong Zhang ${}^{b,c}$ and  Xin Liu ${}^b$\end{center}}
\newcommand{\Address}{\begin{center}

${}^a$ Institute of Modern Physics, Northwest University,
       Xian 710069, P.R. China\\
${}^b$ Department of Mathematics, University of Queensland, Brisbane,
       QLD 4072, Australia\\
${}^c$ Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama
351-0198, Japan\\
E-mail: \,wenli@maths.uq.edu.au,\,\,
yzz@maths.uq.edu.au,\,\,xinliu@maths.uq.edu.au

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\Title{Free field realization of current superalgebra $gl(m|n)_k$}
\Author

\Address
\vspace{1cm}

\begin{abstract}
We construct the free field representation of the affine currents,
energy-momentum tensor and screening currents of
the first kind of the current superalgebra $gl(m|n)_k$  uniformly for
$m=n$ and $m\neq n$.  The energy-momentum tensor is
given by a linear combination of two Sugawara tensors associated
with the two independent quadratic Casimir elements of $gl(m|n)$.

\vspace{1truecm} \noindent {\it PACS:} 11.25.Hf; 02.20.Tw

\noindent {\it Keywords}:  Current algebra; free field
realization.
\end{abstract}
\newpage

\section{Introduction}
\label{intro} \setcounter{equation}{0}

Current superalgebras or affine superalgebras have emerged in a wide
range of physical areas ranging from
high energy physics to condensed matter physics. In high energy theoretical
physics, sigma models with supermanifold target spaces naturally appear
in the quantization of superstring theory in the AdS-type backgrounds.
It was argued in \cite{Bers99} that even without a WZNW term
the sigma model on $PSL(n|n)$ supergroup is already conformally
invariant. A WZNW term with any integer coefficient may then be added
whenever necessary, without violating the conformal invariance.
In \cite{Ber99},  the $PSU(1,1|2)$ sigma model was used to quantize superstring
theory on the AdS$_3\times S^3$ background with Ramond-Ramond (RR) flux.
In condensed matter physics, the supersymmetric treatment of quenched disorders
leads to current superalgebras of zero superdimension. It is believed
that critical behaviours of certain disordered systems
such as the integer quantum Hall transition are described by
sigma models or their WZNW generalizations
based on supergroups of zero superdimension
\cite{Ber95,Mud96,Maa97,Bas00,Gur00}.

As can be seen from the work \cite{Ber00}, in most above-mentioned applications
one expects to work with sigma models on some kind of coset supermanifolds
\cite{Met98,Roi00} with a WZNW term, and thus models of interest
are more complicated than WZNW models on non-coset supergroups. However,
even for such (non-coset) supergroup WZNW models, little has been known
in general \cite{Sch06}, largely due to the technical difficulties in
dealing with atypical and indecomposable representations which are common
features for most superalgebras.

The Wakimoto free field realization \cite{Wak86,Fra97} has been
proved to be a powerful method in the study of CFTs such as WZNW
models. Motivated by the above-mentioned applications, in this
paper  we construct free field representations of the current
superalgebra $gl(m|n)_k$ associated with the $GL(m|n)$ WZNW model
for $m=n$ and $m\neq n$ in a unified way.

Free field realization of the $gl(m|n)$ currents, {\it in
principle\/}, can be obtained by a general method outlined in
\cite{Fei90,Bou90,Ito90,Fre94,Ras98}, where differential
realizations of the corresponding finite dimensional Lie (super)
algebras play a key role. However, {\it in practice}, such
constructions for an {\it explicit} expression of the currents
become very complicated for higher-rank algebras \cite{Ito90,
Ras98, Din03, Zha05}. In this paper, we find a way to overcome the
complication. In our approach, the construction of the
differential operator realization becomes much simpler (c.f.
\cite{Ras98, Din03}). We demonstrated this by working out the
differential realization of $gl(4|4)$ in \cite{Yan06}. Here we
provide the complete results of $gl(m|n)$ for any $m$ and $n$.

This paper is organized as follows. In section 2, we briefly
review the definitions of finite-dimensional superalgebra
$gl(m|n)$ and the associated current algebra, which also services
as introducing our notation and some basic ingredients. In section
3, we construct explicitly the differential operator realization
of $gl(m|n)$ in the standard basis. In section 4 and section 5, we
construct the free field representation of the affine currents
associated with $gl(m|n)$ at a generic level $k$, and the
corresponding energy-momentum tensor. We moreover construct, in
section 6, the screening currents of the first kind. Section 7 is
for conclusions.

\section{Notation and prelimilaries}
\label{CUR} \setcounter{equation}{0}

Let us first fix our notation for the underlying non-affine
superalgebra $gl(m|n)$  which is non-semisimple for both $m=n$ and
$m\neq n$ \cite{Kac77,Fra96}.

$gl(m|n)$ is $\Zb_2$ graded and is
generated by the elements $\{E_{i,j}|\,i,j=1,\ldots,m+n\}$ which satisfy
the following (anti)commutation relations:
\bea [E_{i,j}, E_{k,l}]=
\d_{j\,k}E_{i,l}-(-1)^{([i]+[j])([k]+[l])}\d_{i\,l}E_{k,j}.
\label{Def-1}\eea
Here and throughout, we adopt the convention:
$[a,b]=ab-(-1)^{[a][b]}ba$.  The $\Zb_2$ grading of the generators
is $[E_{i,j}]=[i]+[j]$ with $[1]=\ldots=[m]=0,\,[m+1]=\ldots=[m+n]=1$.
$E_{i,j},\; 1\leq i\neq j\leq m+n$, are raising/lowering generators.
For a unified treatment of the $m=n$ and $m\neq n$ cases, we have chosen
$E_{i,i},\; i=1,\cdots, m+n$, to be the elements of the
Cartan subalgebra (CSA) of $gl(m|n)$.

Let us remark that other bases of the CSA widely used by most physicists
do not seem suitable for the unified treatment because the CSA elements
for $m=n$ and $m\neq n$ in those bases are different. This is seen as follows.
Let
$$ I=\sum_{i=1}^{m+n}E_{ii},~~~~~~ J=\sum_{i=1}^{m+n}(-1)^{[i]}E_{ii}.$$
In the fundamental representation of $gl(m|n)$, $I$ is the $(m+n)\times
(m+n)$ identity matrix and $J$ is the diagonal matrix
$J=diag(1,\cdots,1,-1,\cdots,-1)$. For $m\neq n$ the usual choice
of the $gl(m|n)$ CSA elements is $\{I, H_i=(-1)^{[i]}E_{i,i}-(-1)^{[i+1]}
E_{i+1,i+1}, \; i=1,\cdots,m+n-1\}$. However, this choice is inappropriate
for $m=n$ because in this case $I$ and $H_i,\;i=1,\cdots,2n-1$
become dependent:
$$ H_n=\frac{1}{n}\lt\{I-\sum_{i=1}^{n-1}[iH_i+(n-i)H_{n+i}]\rt\}.$$
That is, for $m=n$ one can not simutaneously choose, say $H_n$ and $I$ as part
of the $gl(n|n)$ CSA elements, in contrast to the $m\neq n$ case.
One popular choice of $gl(n|n)$ CSA elements is
$\{I, J, H_i,\;1\leq i\neq n\leq 2n-1\}$. On the other hand, for $m\neq n$,
$J$ is a linear combination of $I$ and $H_i,\;1\leq i\leq m+n-1$.
Note also that for $m=n$, $I\in sl(n|n)$ and both $sl(n|n)$ and $gl(n|n)$
are non-semisimple; for $m\neq n$, $I$ is in $gl(m|n)$ but not in $sl(m|n)$.

With our above choice of generators, it is easy to check that the usual
quadratic Casimir element of $gl(m|n)$ is
\bea
C_1=\sum_{i,j=1}^{m+n}\,(-1)^{[j]}E_{i,j}E_{j,i}.\label{Casim-1}
\eea
Since $gl(m|n)$ is non-semisimple, there exists another independent quadratic
Casimir element
\bea
C_2=\sum_{i,j=1}^{m+n}\,E_{i,i}E_{j,j}=\lt(\sum_{i=1}^{m+n}E_{i,i}\rt)^2.
\label{Casim-2}
\eea These two Casimir elements are useful in the
following for the construction of the correct energy-momentum tensor.

For any $m$ and $n$, $gl(m|n)$ has a non-degenerate and invariant metric
or bilinear form given by \cite{Kac77,Fra96} \bea
\lt(E_{i,j},\,E_{k,l}\rt)=str\lt(e_{i,j}e_{k,l}\rt).
\label{Inner-product}\eea
Here $e_{i,j}$, which is the $(m+n)\times(m+n)$ matrix with entry
$1$ at the $i$-th row and $j$-th column and zero elsewhere, is the
fundamental or defining representation of $E_{i,j}$;
$str$ denotes the supertrace, i.e., $str(a)=\sum_{i}(-1)^{[i]}\,a_{i\,i}$.

The $gl(m|n)$ current algebra is generated by the currents
$E_{i,j}(z)$  associated with the generators $E_{i,j}$ of
$gl(m|n)$. The current algebra at a general level $k$ obeys the
following OPEs \cite{Fra97}, \bea E_{i,j}(z)\,E_{l,m}(w)=
k\frac{\lt(E_{i,j},E_{l,m}\rt)}{(z-w)^2}+\frac{1}{(z-w)}
\lt(\d_{j\,l}\,E_{i,m}(w)-(-1)^{([i]+[j])([l]+[m])}\d_{i\,m}\,
E_{l,j}(w)\rt) .\label{OPE}\eea

\section{Differential operator realization of $gl(m|n)$}
 \label{DIR} \setcounter{equation}{0}

As mentioned in the introduction, practically it would be very
involved (if not impossible) to obtain the {\it explicit} free
field realization of  higher-rank algebras such as $gl(m|n)$ for a
larger value of $m+n$ by the general method outlined in
\cite{Fei90,Bou90,Ito90,Fre94,Ras98}. We have found a way to
overcome the complication. In our approach, the construction of
the differential operator realization becomes much simpler.

Let us introduce $\frac{1}{2}((m(m-1)+n(n-1))$ bosonic coordinates
$\{x_{i,j},\,x_{m+k,m+l}|\,1\leq i<j\leq m,\,1\leq k<l\leq n\}$
with the $\Zb_2$-grading: $[x_{i,j}]=0$, and $m\times n$ fermionic
coordinates $\{\t_{i,m+j}|\,1\leq i\leq m,\, 1\leq j\leq n\}$ with
the $\Zb_2$-grading: $[\t_{i,m+j}]=1$. These coordinates satisfy
the following (anti)commutation relations: \bea
&&[x_{i,j},x_{k,l}]=0,\,\,
[\partial_{x_{i,j}},x_{k,l}]=\d_{ik}\d_{jl},\no\\
&&[\t_{i,m+j},\t_{k,m+l}]=0,\,\,
[\partial_{\t_{i,m+j}},\t_{k,m+l}]=\d_{ik}\d_{jl},\no\eea and the
other (anti)commutation relations are vanishing. Let $\langle\L|$
be the lowest weight vector in the associated representation of
$gl(m|n)$, satisfying the following conditions: \bea
&&\langle\L|E_{j+1,j}=0,\quad 1\leq j\leq
m+n-1,\label{Lowestweight-1}\\
&&\langle\L|E_{i,i}=\l_i\,\langle\L|,\quad 1\leq i\leq
m+n.\label{Lowestweight-2}\eea An arbitrary vector in this
representation space is parametrized by $\langle\L|$ and the
coordinates ($x$ and $\t$)  as \bea
\langle\L,x,\t|=\langle\L|G_{+}(x,\t),\label{States}\eea where
$G_{+}(x,\t)$ is given by (c.f. \cite{Ras98}) \bea G_{+}(x,\t)=
(G_{m+n-1,m+n})\ldots (G_{j,m+n}\ldots G_{j,j+1})\ldots \,
(G_{1,m+n}\ldots G_{1,2}).\eea Here, for $i<j$, $G_{i,j}$ are
given by \bea
G_{i,j}=\lt\{\begin{array}{ll}e^{x_{i,j}E_{i,j}},&{\rm
if}\,\, [E_{i,j}]=0,\\
e^{\t_{i,j}E_{i,j}},&{\rm if}\,\, [E_{i,j}]=1.\end{array}\rt.\eea
One can define a differential operator realization $\rho^{(d)}$ of
the generators of $gl(m|n)$ by \bea
\rho^{(d)}(g)\,\langle\L,x,\t|\equiv \langle\L,x,\t|\,g,\qquad
\forall g\in gl(m|n).\label{definition} \eea Here $\rho^{(d)}(g)$
is a differential operator of the bosonic and ferminic coordinates
associated with the generator $g$, which can be obtained from the
defining relation (\ref{definition}). Moreover, the defining
relation also assure the differential operator realization is
actually a representation of $gl(m|n)$. Therefore it is sufficient
to obtain the associated differential operators which are related
to the simple roots, and the others can be constructed through the
simple ones by (anti)commutation relations
(\ref{Non-simple-1})-(\ref{Non-simple-2}) (see below). By using
the relation (\ref{definition}) and the Baker-Campbell-Hausdorff
formula, after some algebraic manipulations, we obtain the following
differential operator representation of the simple
generators in the standard (distinguished) basis \cite{Fra96}:
\bea \rho^{(d)}(E_{j,j+1})&=&\sum_{k\leq
j-1}x_{k,j}\,\partial_{x_{k,j+1}}+
\partial_{x_{j,j+1}},\,\,1\leq j\leq m-1, \label{Diff-1}\\
\rho^{(d)}(E_{m,m+1})&=&\sum_{k\leq
m-1}x_{k,m}\,\partial_{\t_{k,m+1}}
+\partial_{\t_{m,m+1}}, \\
\rho^{(d)}(E_{m+j,m+1+j})&=&\sum_{k\leq
m}\t_{k,m+j}\,\partial_{\t_{k,m+1+j}}+ \sum_{k\leq
j-1}x_{m+k,m+j}\,\partial_{x_{m+k,m+1+j}}+
\partial_{x_{m+j,m+1+j}},\no\\
&&\qquad 1\leq j\leq n-1,\\
\rho^{(d)}(E_{j,j})&=&\sum_{k\leq j-1}x_{k,j}\,\partial_{x_{k,j}}-
\sum_{j+1\leq k\leq m}x_{j,k}\,\partial_{x_{j,k}} -\sum_{k\leq
n}\t_{j,m+k}\,\partial_{\t_{j,m+k}}+\l_j,\no\\
&&\qquad 1\leq j\leq m-1,\\
\rho^{(d)}(E_{m,m})&=&\sum_{k\leq
m-1}x_{k,m}\,\partial_{x_{k,m}}-\sum_{k\leq
n}\t_{m,m+k}\,\partial_{\t_{m,m+k}}+\l_m,\\
\rho^{(d)}(E_{m+j,m+j})&=&\sum_{k\leq
m}\t_{k,m+j}\,\partial_{\t_{k,m+j}}+
\sum_{k\leq j-1}x_{m+k,m+j}\,\partial_{x_{m+k,m+j}}\no\\
&&\qquad\qquad- \sum_{j+1\leq k\leq
n}x_{m+j,m+k}\,\partial_{x_{m+j,m+k}}+\l_{m+j},\,\,1\leq j\leq n,\\
\rho^{(d)}(E_{j+1,j})&=&\sum_{k\leq j-1}
x_{k,j+1}\,\partial_{x_{k,j}}-\sum_{j+2\leq k\leq m} x_{j,k}\,
\partial_{x_{j+1,k}}-\sum_{k\leq n}\t_{j,m+k}\,\partial_{\t_{j+1,m+k}} \no\\
&&\quad -x_{j,j+1}\lt(\sum_{j+1\leq k\leq
m}x_{j,k}\,\partial_{x_{j,k}}+\sum_{k\leq n}
\t_{j,m+k}\,\partial_{\t_{j,m+k}}\rt)\no\\
&&\quad +x_{j,j+1}\lt(\sum_{j+2\leq k\leq
m}x_{j+1,k}\,\partial_{x_{j+1,k}}+\sum_{k\leq
n}\t_{j+1,m+k}\,\partial_{\t_{j+1,m+k}}\rt)\no\\
&&\quad+x_{j,j+1} \lt(\l_j-\l_{j+1}\rt),\,\,1\leq j\leq m-1,\\
\rho^{(d)}(E_{m+1,m})&=&\sum_{k\leq
m-1}\t_{k,m+1}\,\partial_{x_{k,m}}+\sum_{2\leq
k\leq n}\t_{m,m+k}\partial_{x_{m+1,m+k}}\no\\
&&\quad-\t_{m,m+1}\lt(\sum_{2\leq k\leq
n}\lt(\t_{m,m+k}\,\partial_{\t_{m,m+k}}
+x_{m+1,m+k}\,\partial_{x_{m+1,m+k}}\rt)\rt)\no\\
&&\quad +\t_{m,m+1}\lt(\l_{m}+\l_{m+1}\rt),\\
\rho^{(d)}(E_{m+1+j,m+j})&=&\sum_{k\leq
m}\t_{k,m+1+j}\,\partial_{\t_{k,m+j}} +\sum_{k\leq
j-1}x_{m+k,m+1+j}\,\partial_{x_{m+k,m+j}}\no\\
&&\quad -\sum_{j+2\leq k\leq n}
x_{m+j,m+k}\,\partial_{x_{m+1+j,m+k}}\no\\
&&\quad -x_{m+j,m+1+j}\sum_{j+1\leq
k\leq n}x_{m+j,m+k}\,\partial_{x_{m+j,m+k}}\no\\
&&\quad +x_{m+j,m+1+j}\sum_{j+2\leq k\leq
n}x_{m+1+j,m+k}\,\partial_{x_{m+1+j,m+k}}\no\\
&&\quad +x_{m+j,m+1+j}\lt(\l_{m+j}-\l_{m+1+j}\rt),\,\,1\leq j\leq
n-1.\label{Diff-2}\eea
The generators associated with the
non-simple roots can be constructed through the simple ones
by the (anti)commutation relations, \bea
\rho^{(d)}(E_{i,j})&=&[\rho^{(d)}(E_{i,k}),\rho^{(d)}(\,E_{k,j})],\,\,
1\leq i<k<j\leq m+n\,\,{\rm and}\,2\leq j-i,\label{Non-simple-1}\\
\rho^{(d)}(E_{j,i})&=&[\rho^{(d)}(E_{j,k}),\,\rho^{(d)}(E_{k,i})],\,\,
1\leq i<k<j\leq m+n \,\,{\rm and}\,2\leq j-i.\label{Non-simple-2}
\eea
A direct computation shows that the differential
realization (\ref{Diff-1})-(\ref{Non-simple-2}) of $gl(m|n)$
satisfies the commutation relations (\ref{Def-1}). Alternatively, one
may check that (\ref{Diff-1})-(\ref{Diff-2})
satisfy the (anti)commutation relations corresponding to
the simple roots together with the Serre relations \cite{Fra96}.

\section{Free field realization of $gl(m|n)_k$}
 \label{FFR} \setcounter{equation}{0}

With the help of the differential realization obtained in the last
section we can construct the free field representation
of the $gl(m|n)$ current algebra in
terms of $\frac{1}{2}(m(m-1)+n(n-1))$ bosonic $\b$-$\g$ pairs
$\{(\b_{i,j},\,\g_{i,j}),\, 1\leq i<j\leq m;
(\bar{\b}_{i,j}\bar{\g}_{i,j}),\, 1\leq i<j\leq n\}$,
$m\times n$ fermionic $b$-$c$ pairs $\{(\psi^{\dagger}_{i,j},\,\psi_{i,j}),\,
1\leq i\leq m,\, 1\leq j\leq n\}$ and $m+n$ free scalar fields
$\phi_i$, $i=1,\ldots,m+n$. These free fields obey the
following OPEs:\bea
&&\hspace{-0.8truecm}\b_{i,j}(z)\,\g_{k,l}(w)=-\g_{k,l}(z)\,\b_{i,j}(w)=
\frac{\d_{ik}\d_{jl}}{(z-w)},\,\,1\leq i<j\leq m,\,\,1\leq
k<l\leq m,\label{OPE-F-1}\\
&&\hspace{-0.8truecm}\bar{\b}_{i,j}(z)\,\bar{\g}_{k,l}(w)=-\bar{\g}_{k,l}(z)\,
\bar{\b}_{i,j}(w)= \frac{\d_{ik}\d_{jl}}{(z-w)},\,\,1\leq i<j\leq
n,\,\,1\leq k<l\leq n,\\
&&\hspace{-0.8truecm}\psi_{i,j}(z)\psi_{k,l}^{\dagger}(w)
=\psi_{k,l}^{\dagger}(z)
\psi_{i,j}(w)=\frac{\d_{ik}\d_{jl}}{(z-w)},\,\,\,\,\,\,1\leq
i,k\leq m,\,\,1\leq j,l\leq n,\\
&&\hspace{-0.8truecm}\phi_i(z)\phi_j(w)=(E_{i,i}, E_{j,j})\ln(z-w)=
   (-1)^{[i]}\,\d_{ij}\,
\ln(z-w),\,\,\,\,\,1\leq i,j\leq m+n,\label{OPE-F-2}\eea
and the other OPEs are trivial.

The free field realization of the $gl(m|n)$ current algebra
(\ref{OPE}) is obtained by the substitution in the
differential realization (\ref{Diff-1})-(\ref{Diff-2}) of $gl(m|n)$,
\bea
&&x_{i,j}\longrightarrow \g_{i,j}(z),\quad \partial_{x_{i,j}}
\longrightarrow \b_{i,j}(z),\quad 1\leq i<j\leq m,\no\\
&&x_{m+i,m+j}\longrightarrow \bar{\g}_{i,j}(z),\quad
\partial_{x_{m+i,m+j}} \longrightarrow \bar{\b}_{i,j}(z),
\quad 1\leq i<j\leq n,\no\\
&&\t_{i,m+j}\longrightarrow \psi^{\dagger}_{i,j}(z),\quad
\partial_{\t_{i,m+j}} \longrightarrow \psi_{i,j}(z),\quad 1\leq i\leq
m\,{\rm and}\,1\leq j\leq n, \no\\
&&\l_j\longrightarrow \sqrt{k+m-n}\partial\phi_j(z)-
\frac{(-1)^{[j]}(1+\a)}{2\sqrt{k+m-n}}\sum_{l=1}^{m+n}\phi_l(z),\,\,
1\leq j\leq m+n,\no\eea  with
$\a=1+\frac{2k}{m-n}-\frac{2\sqrt{k(k+m-n)}}{m-n}$,
followed by the addition of anomalous terms linear in
$\partial \psi^{\dagger}(z)$, $\partial\g(z)$ and
$\partial\bar{\g}(z)$ in the expressions of the currents. It is
remarked that for $m=n$, $\a$ is $\a=\lim_{m\rightarrow
n}(1+\frac{2k}{m-n}-\frac{2\sqrt{k(k+m-n)}}{m-n})=0$. Here we
present the realization of the currents associated with the simple roots,
\bea E_{j,j+1}(z)&=&\sum_{l\leq
j-1}\g_{l,j}(z)\b_{l,j+1}(z)+\b_{j,j+1}(z),\,\,1\leq j\leq
m-1,\label{Fre-currents-1}\\
E_{m,m+1}(z)&=&\sum_{l\leq m-1}\g_{l,m}(z)\psi_{l,1}(z)+\psi_{m,1}(z),\\
E_{m+j,m+j+1}(z)&=&\sum_{l\leq
m}\psi^{\dagger}_{l,j}(z)\psi_{l,j+1}(z)+\sum_{l\leq
j-1}\bar{\g}_{l,j}(z)\bar{\b}_{l,j+1}(z)+\bar{\b}_{j,j+1}(z),\no\\
&&\qquad 1\leq j\leq n-1,\\
E_{j,j}(z)&=&\sum_{l\leq j-1}\g_{l,j}(z)\b_{l,j}(z)- \sum_{j+1\leq
l\leq m}\g_{j,l}(z)\b_{j,l}(z) -\sum_{l\leq
n}\psi^{\dagger}_{j,l}(z)\psi_{j,l}(z)\no\\
&&\quad+\sqrt{k+m-n}\partial\phi_j(z)-\frac{1+\a}{2\sqrt{k+m-n}}
\sum_{l=1}^{m+n}\partial\phi_l(z),\no\\
&&\qquad 1\leq j\leq m,\label{CSA-currents1}\\

E_{m+j,m+j}(z)&=&\sum_{l\leq
m}\psi^{\dagger}_{l,j}(z)\psi_{l,j}(z)+ \sum_{l\leq
j-1}\bar{\g}_{l,j}(z)\bar{\b}_{l,j}(z)- \sum_{j+1\leq l\leq
n}\bar{\g}_{j,l}(z)\bar{\b}_{j,l}(z)\no\\
&&\quad+\sqrt{k+m-n}\partial\phi_{m+j}(z)+\frac{1+\a}{2\sqrt{k+m-n}}
\sum_{l=1}^{m+n}\partial\phi_l(z),\no\\
&&\qquad 1\leq j\leq n, \label{CSA-currents2}\\
E_{j+1,j}(z)&=&\sum_{l\leq j-1}
\g_{l,j+1}(z)\b_{l,j}(z)-\sum_{j+2\leq l\leq m} \g_{j,l}(z)
\b_{j+1,l}(z)-\sum_{l\leq n}\psi^{\dagger}_{j,l}(z)\psi_{j+1,l}(z) \no\\
&&\quad -\g_{j,j+1}(z)\lt(\sum_{j+1\leq l\leq
m}\g_{j,l}(z)\b_{j,l}(z)+\sum_{l\leq n}
\psi^{\dagger}_{j,l}(z)\psi_{j,l}(z)\rt)\no\\
&&\quad +\g_{j,j+1}(z)\lt(\sum_{j+2\leq l\leq
m}\g_{j+1,l}(z)\b_{j+1,l}(z)+\sum_{l\leq
n}\psi^{\dagger}_{j+1,l}(z)\psi_{j+1,l}(z)\rt)\no\\
&&\quad+\sqrt{k+m-n}\g_{j,j+1}(z)
\lt(\partial\phi_j(z)-\partial\phi_{j+1}(z)\rt)
+(k+j-1)\partial\g_{j,j+1}(z),\no\\
&&\quad\qquad\,\,1\leq j\leq m-1,\\
E_{m+1,m}(z)&=&\sum_{l\leq
m-1}\psi^{\dagger}_{l,1}(z)\b_{l,m}(z)+\sum_{2\leq
l\leq n}\psi^{\dagger}_{m,l}(z)\bar{\b}_{1,l}(z)\no\\
&&\quad-\psi^{\dagger}_{m,1}(z)\lt(\sum_{2\leq l\leq
n}\lt(\psi^{\dagger}_{m,l}(z)\,\psi_{m,l}(z)
+\bar{\g}_{1,l}(z)\bar{\b}_{1,l}(z)\rt)\rt)\no\\
&&\quad +\sqrt{k+m-n}\psi^{\dagger}_{m,1}(z)
\lt(\partial\phi_{m}(z)+\partial\phi_{m+1}(z)\rt)\no\\
&&\quad+(k+m-1)\partial\psi^{\dagger}_{m,1}(z),\\
E_{m+j+1,m+j}(z)&=&\sum_{l\leq
m}\psi^{\dagger}_{l,j+1}(z)\psi_{l,j}(z) +\sum_{l\leq
j-1}\bar{\g}_{l,j+1}(z)\bar{\b}_{l,j}(z)-\sum_{j+2\leq
l\leq n} \bar{\g}_{j,l}(z)\bar{\b}_{j+1,l}(z)\no\\
&&\quad -\bar{\g}_{j,j+1}(z)\lt(\sum_{j+1\leq l\leq
n}\bar{\g}_{j,l}(z)\bar{\b}_{j,l}(z)-\sum_{j+2\leq l\leq
n}\bar{\g}_{j+1,l}(z)\bar{\b}_{j+1,l}(z)\rt)\no\\
&&\quad +\sqrt{k+m-n}\bar{\g}_{j,j+1}
\lt(\partial\phi_{m+j}(z)-\partial\phi_{m+j+1}(z)\rt)\no\\
&&\quad-(k+m+1-j)\partial\bar{\g}_{j,j+1}(z), \,\,1\leq j\leq
n-1.\label{Fre-currents-2}\eea
Here and throughout normal ordering of free
fields is implied whenever necessary. The free field realization of
currents associated with the non-simple roots can be obtained from
the OPEs of the simple ones, similar to
(\ref{Non-simple-1})-(\ref{Non-simple-2}). It is straightforward
to check that the above free field realization of the currents
satisfy the OPEs of the $gl(m|n)$ current algebra.
Moreover, for the case $n=0$  our results reduce to those in
\cite{Bou90}, giving the free field realization of the $gl(m)$ current algebra.

Some remarks are in order. We have obtained the free field realization of
$gl(m|n)$ current algebra uniformly for any $m$ and $n$ for the CSA basis
we have chosen. It is easy to make simple basis transforms of the CSA
to get expressions for the more familar CSA bases. This is seen as follows.
Introduce new free scalar fields through linear combinations of the
original free scalar fields $\phi_i(z)$,
\bea
\phi_I(z)&=&\sum_{i=1}^{m+n}\phi_i(z),~~~~~\phi_J(z)=\sum_{i=1}^{m+n}
   (-1)^{[i]}\phi_i(z),\no\\
\phi_{H_i}(z)&=&(-1)^{[i]}\phi_i(z)-(-1)^{[i+1]}\phi_{i+1}(z).
\eea
In terms of the new scalar fields, the currents associated with $I,J$ and
$H_i$ take the form, as can easily be seen from (\ref{CSA-currents1}) and
(\ref{CSA-currents2}),
\bea
I(z)&=&\sum_{i=1}^{m+n}E_{i,i}(z)=\sqrt{k}\partial\phi_I(z),\no\\
J(z)&=&\sum_{i=1}^{m+n}(-1)^{[i]}E_{i,i}(z)=\sqrt{k+m-n}\partial\phi_J(z)
  -\frac{(m+n)(1+\alpha)} {2\sqrt{k+m-n}}\partial\phi_I(z),\no\\
& &-\sum_{j=1}^n\sum_{l\leq n}\psi^\dagger_{jl}(z)\psi_{jl}(z)
   -\sum_{j=1}^n\sum_{l\leq m}\psi^\dagger_{lj}(z)\psi_{lj}(z),\no\\
H_i(z)&=&(-1)^{[i]}E_{i,i}(z)-(-1)^{[i+1]}E_{i+1,i+1}(z)\no\\
&=&\tilde{H}_i(z)+\sqrt{k+m-n}\partial\phi_{H_i}(z),~~~1\leq i\leq m+n-1,
\eea
where $\tilde{H}_i(z)$ are functions of the $\beta$-$\gamma$ and
$b$-$c$ pairs only. Now for $m=n$, replacing the $2n$ original free scalar
fields $\phi_i(z)$ by $\{\phi_{H_i}(z),\;1\leq i\neq n\leq 2n-1, \phi_I(z),
\phi_J(z)\}$ and moreover using the relation,
$$\phi_{H_n}(z)=\frac{1}{n}\lt[\phi_I(z)-\sum_{i=1}^{n-1}\lt(i\phi_{H_i}(z)
   +(n-i)\phi_{H_{n+i}}(z)\rt)\rt]$$
to eliminate $\phi_{H_n}(z)$, then we obtain the $gl(n|n)$ currents
$\{E_{i,j}(z),\;1\leq i\neq j\leq 2n-1; I(z), J(z), H_l(z),\;1\leq l
 \neq n\leq 2n-1\}$ in the new basis in terms of the new free scalar fields
defined above together with the original $\beta$-$\gamma$ and $b$-$c$ pairs.
Similarly for $m\neq n$, we replace the $m+n$ original free scalar fields
$\phi_i(z)$ by  $\{\phi_{H_i}(z),\;1\leq i\leq m+n-1, \phi_I(z)\}$ to
obtain the $gl(m|n)$ currents $\{E_{i,j}(z),\;1\leq i\neq j\leq 2n-1; I(z),
H_l(z),\;1\leq l\leq m+n-1\}$ in the new basis.

Note that for $m=n$, $\phi_J(z)$ only appears in $J(z)$. Thus the free
field realization of $sl(n|n)$ current algebra may be obtained from
that of the $gl(n|n)$ current algebra  by simply dropping $J(z)$.
The free field realization of $psl(n|n)=sl(n|n)/I$ current algebra
is obtained by setting $\phi_I(z)=0$ and thus $I(z)=0$ in the realization
of the $sl(n|n)$ current algebra. For $m\neq n$, since
$\phi_I(z)$ only appears in $I(z)$ in the new basis,
one may obtain the free field realization of the $sl(m|n)$ current algebra
by simply dropping $I(z)$ in the realization of the $gl(m|n)$ current algebra.

\section{Energy-momentum tensor}
\label{EMT} \setcounter{equation}{0}

In this setion we construct the free field realization of the Sugawara
energy-momentum tensor associated with the $gl(m|n)$ current algebra.
After a tedious calculation, we find that the Sugawara tensor corresponding
to the quadratic Casimir $C_1$ is given by \bea
T_1(z)&=&\frac{1}{2(k+m-n)}\sum_{i,j=1}^{m+n}(-1)^{[j]}:
E_{i,j}(z)E_{j,i}(z):\no\\
&=&\frac{1}{2}\sum_{l=1}^{m+n}(-1)^{[l]}\partial\phi_l(z)\partial\phi_l(z)\no\\
&&\quad-\frac{1}{2\sqrt{k+m-n}}\partial^2
\lt(\sum_{i=1}^m(m-n-2i+1)\phi_i(z)-\sum_{j=1}^n(m+n-2j+1)\phi_{m+j}(z)\rt)\no\\
&&\quad+\sum_{i<j}^m\partial\g_{i,j}(z)\b_{i,j}(z)
+\sum_{i<j}^n\partial\bar{\g}_{i,j}(z)\bar{\b}_{i,j}(z)\no\\
&&\quad+\sum_{i=1}^m\sum_{j=1}^n\partial\psi^{\dagger}_{i,j}(z)\psi_{i,j}(z)
-\frac{1}{2(k+m-n)}\partial\phi_I(z)\,\partial\phi_I(z).\eea

On the other hand,  the Sugawara tensor corresponding to the
quadratic Casimir $C_2$ is  \bea
T_2(z)&=&\frac{1}{2(k+m-n)}\sum_{i,j=1}^{m+n}:E_{i,i}(z)E_{j,j}(z):\no\\
&=&\frac{k}{2(k+m-n)}\partial\phi_I(z)\,\partial\phi_I(z).\eea In
order that all currents $E_{i,j}(z)$ are primary fields with
conformal dimensional one, we define the energy-momentum tensor
$T(z)$ as follow: \bea T(z)&=&T_1(z)+\frac{1}{k}T_2(z)\no\\
&=&\frac{1}{2}\sum_{l=1}^{m+n}(-1)^{[l]}\partial\phi_l(z)\partial\phi_l(z)\no\\
&&\quad-\frac{1}{2\sqrt{k+m-n}}\partial^2
\lt(\sum_{i=1}^m(m-n-2i+1)\phi_i(z)-\sum_{j=1}^n(m+n-2j+1)\phi_{m+j}(z)\rt)\no\\
&&\quad+\sum_{i<j}^m\partial\g_{i,j}(z)\b_{i,j}(z)
+\sum_{i<j}^n\partial\bar{\g}_{i,j}(z)\bar{\b}_{i,j}(z)
+\sum_{i=1}^m\sum_{j=1}^n\partial\psi^{\dagger}_{i,j}(z)\psi_{i,j}(z).
\label{Energy-Momentum}\eea
It is straightforward to check that
$T(z)$ satisfy the following OPE, \bea
T(z)T(w)=\frac{c/2}{(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial
T(w)}{(z-w)},\eea
where the  central charge $c=0$ for $m=n$, and
\bea
c=
\frac{k\lt((m-n)^2-1\rt)}{k+m-n}+1
\label{Center-charge}\eea
for $m\neq n$. Moreover, we find that with regard to
the energy-momentum tensor $T(z)$ defined by
(\ref{Energy-Momentum}) all currents $E_{i,j}(z)$ are indeed
primary fields with conformal dimensional one, namely, \bea
T(z)E_{i,j}(w)=\frac{E_{i,j}(w)}{(z-w)^2}+\frac{\partial
E_{i,j}(w)}{(z-w)},\,\,1\leq i,j\leq m+n.\eea Therefore, $T(z)$ is
the very energy-momentum tensor of the $gl(m|n)$ current algebra.

\section{Screening currents}
\label{SC}  \setcounter{equation}{0}

Important objects in applying the free field realization to the
computation of correlation functions  of the associated CFT are
screening currents. A screening current is a primary field with
conformal dimension one and has the property that the singular
part of its OPE with the affine currents is a total derivative.
These properties ensure that integrated screening currents
(screening charges) may be inserted into correlators while the
conformal or affine Ward identities remain intact. This in turn
makes them very useful in the computation of correlation functions
\cite{Dos84,Ber90}.

Free field realization of the $gl(m|n)$ screening
currents of the first kind may be constructed from certain differential
operators $\{s_{i,j}|1\leq i<j\leq n+m\}$ \cite{Bou90,Ras98} which we
define as given from the relation
\bea s_{i,j}\,
\langle\L,x,\t|\equiv\langle\L|\,E_{i,j}\,G_{+}(x,\t),\qquad {\rm
for}\, 1\leq i,j\leq m+n.\label{Def-2}\eea
The above-defined operators $s_{i,j}$ give a differential
operator realization of a subalgebra of $gl(m|n)$.
Again it is sufficient to construct $s_{j,j+1}$ related to the
simple generators $E_{j,j+1}$, $1\leq j\leq m+n-1$ of $gl(m|n)$.
Let us denote those differential operators by $s_{j}$. Using
(\ref{Def-2}) and the Baker-Campbell-Hausdorff formula,
after some algebraic manipulations, we obtain the following explicit
expressions of $s_j$:
\bea s_j&=& \sum_{j+2\leq l\leq
m}x_{j+1,l}\partial_{x_{j,l}}+\sum_{l\leq
n}\theta_{j+1,m+l}\partial_{\theta_{j,m+l}}+\partial_{x_{j,j+1}},\quad
1\leq j\leq m-1,\\
s_m&=&\sum_{2\leq l\leq n}
x_{m+1,m+l}\partial_{\theta_{m,m+l}}+\partial_{\theta_{m,m+1}},\\
s_{m+j}&=&\sum_{j+2\leq l\leq
n}x_{m+j+1,m+l}\partial_{x_{m+j,m+l}}+\partial_{x_{m+j,m+j+1}},\quad
1\leq j\leq n-1.\eea
{}From (anti)communication relations similar to
(\ref{Non-simple-1}), one may obtain the differential operators $s_{i,j}$
associated with the non-simple generators of $gl(m|n)$.
Following the procedure similar to \cite{Bou90,Ras98},
we find the free field realization of the screening currents $S_j$
corresponding to the differential operators $s_j$,
\bea S_j(z)&=&\lt(\sum_{j+2\leq l\leq
m}\g_{j+1,l}(z)\b_{j,l}(z)
+\sum_{l=1}^n\psi^{\dagger}_{j+1,l}(z)\psi_{j,l}(z)+\b_{j,j+1}(z)\rt)
\tilde{s}_j(z),\no\\
&&\qquad\qquad\,\, 1\leq j\leq m-1,\label{Screening-1}\\
S_m(z)&=&\lt(\sum_{2\leq
l\leq n}\bar{\g}_{1,l}(z)\psi_{m,l}(z)+\psi_{m,1}(z)\rt)\tilde{s}_m(z),\\
S_{m+j}(z)&=&\lt(\sum_{j+2\leq l\leq
n}\bar{\g}_{j+1,l}(z)\bar{\b}_{j,l}(z)+\bar{\b}_{j,j+1}(z)\rt)
\tilde{s}_{m+j}(z) ,\,\,1\leq j\leq n-1,\eea where \bea
&&\tilde{s}_j(z)=e^{-\frac{1}{\sqrt{k+m-n}}(\phi_j(z)-\phi_{j+1}(z))},\quad
1\leq j\leq m-1,\\
&&\tilde{s}_m(z)=e^{-\frac{1}{\sqrt{k+m-n}}(\phi_{m}(z)+\phi_{m+1}(z))},\\
&&\tilde{s}_{m+j}(z)=e^{\frac{1}{\sqrt{k+m-n}}(\phi_{m+j}(z)-\phi_{m+j+1}(z))},
\quad 1\leq j\leq n-1. \label{Screening-2}\eea
The OPEs of the screening currents with the energy-momentum tensor
and the $gl(m|n)$ currents (\ref{Fre-currents-1})-(\ref{Fre-currents-2}) are
\bea &&
T(z)S_j(w)=\frac{S_j(w)}{(z-w)^2}+\frac{\partial S_j(w)}{(z-w)}
=\partial_w\lt\{\frac{S_j(w)}{(z-w)}\rt\},\,\,1\leq j\leq m+n-1,\\
&&E_{i+1,i}(z)S_j(w)=(-1)^{[i]+[i+1]}\d_{ij}\,
\partial_{w}\lt\{\frac{k \,\tilde{s}_j(w)}{(z-w)}\rt\},\,\,
1\leq i,j\leq m+n-1,\\
&&E_{i,i+1}(z)S_j(w)=0,\,\, 1\leq i,j\leq m+n-1,\\
&&E_{i,i}(z)S_j(w)=0,\,\, 1\leq i\leq m+n,\,\, 1\leq j\leq
m+n-1.\eea
The screening currents obtained this way are screening currents
of the first kind \cite{Ber86}. Moreover, $S_m(z)$ is
fermionic and the others are bosonic.

\section{Discussions}
\label{Con} \setcounter{equation}{0}

We have studied the $gl(m|n)$ current algebra at general level
$k$. We have constructed its Wakimoto free field realization
(\ref{Fre-currents-1})-(\ref{Fre-currents-2}) for $m=n$ and
$m\neq n$ in a unified way, and the
corresponding  energy-momentum tensor (\ref{Energy-Momentum})
which is a linear combination of two Sugawara tensors associated
with two quadratic Casimir elements of $gl(m|n)$.
We have also found $m+n-1$ screening currents,
(\ref{Screening-1})-(\ref{Screening-2}), of the first kind.
Our results reduce to those in \cite{Bou90} for $n=0$ (i.e. in the
bosonic case), and recover those in \cite{Yan06} for $m=n=4$, thus
providing a complete proof of the results in that paper.

To fully take the advantage of the free field approach in applications
mentioned in the introduction, explicit construction of primary fields
in terms of free fields is needed. It is well-known that there exist
two types of representations for the underlying finite dimensional
superalgebra $gl(m|n)$: typical and atypical representations.
Atypical representations, which are often indecomposable,
have no counterparts in the bosonic
algebra setting and the understanding of such representations is
still very much incomplete. Although the construction of the
primary fields associated with typical representations are similar
to the bosonic algebra cases, it is a highly non-trivial task to
construct the primary fields associated with atypical
representations \cite{Zha05}.

\section*{Acknowledgements}
We are happy to thank Akira Furusaki for helpful discussions. YZZ would
like to thank the Condensed Matter Theory Laboratory,
especially Akira Furusaki, of the RIKEN Discovery Research Institute
for warm hospitality and financial support.
This work has been supported by the Australian Research Council.
XL gratefully acknowledges the support of the IPRS and UQGSS
scholarships of the University of Queensland.

\vspace{1.00truecm}

\noindent{{\large \it Note added}}: \, We became aware that free
field realization of $sl(m|n)$  current algebra was investigated
previously in \cite{Bar91} (for $gl(n|n)$ case see also
\cite{Isi94}). There, the $sl(m|n)$ currents were expressed in
terms of  $sl(m)$ and $sl(n)$ currents with different levels and
some $b$-$c$ pairs. As part of the results of our paper, we give
the {\em explicit} expressions of $gl(m|n)$ currents in terms of
free fields, by using a different method.