\chapter{The Supergravity Solutions}

The type II solution for $N$ coincident D$n$-branes is, for $n<7$,
[\hor,\itmal,\huten]
$$
\eqalign{& ds^2=f_n^{-1/2}(-dt^2+dx_1^2+...+dx_n^2)+f_n^{1/2}
(dx_{n+1}^2+...+dx_9^2),
\cr &
e^
\phi=e^{\phi_{\infty}}
 f_n^{-(n-3)/4},
\cr &
 A_{0...n}=-{1 \over 2} (f_n^{-1}-1) ,
\cr}
\eqn\abc$$
where $f_n$ is a harmonic function of the transverse coordinates
$x_{n+1},...,x_9$, which, for  $N$ coincident branes at the origin $r=0$
(where the radial coordinate $r=
(x_{n+1}^2+...+x_9^2)^{1/2}$)
is
$$
 f_n= 1+ \left( {a \over r } \right) ^{7-n}
 \eqn\abc$$
 where
 $$ a^{7-n}= {g_S N d_n \over m_S ^{7-n}}
 ={R N d_n \over m_S ^{6-n}}=
 {\gamma  ^2  m_S ^{2(n-5)}}
 \eqn\abc$$
 Here $d_n$ is a constant given in [\itmal], and
 $$
 \gamma =\sqrt{d_n}g_N=g_{YM}\sqrt{d_n N}
 \eqn\abc$$
 In the asymptotic region $r/a >> 1$, the solution approaches
 flat space with $\phi \to \phi _ \infty $, giving string theory in flat space
with
coupling constant $g_S$, while in the near region $r/a <<1$, it is
appproximately
described by the   geometry
$$   ds^2=
\left( {a \over r } \right) ^{-(7-n)/2}
dx^2_{||}+
\left( {a \over r } \right) ^{(7-n)/2}(dr ^2 + r^2 d\Omega^2_{8-n})
\eqn\met$$
and is   singular as $ r \to 0$ (for $n \ne 3$).
The dilaton is given by
$$\phi = \phi _{\infty } + {(7-n) (n-3)\over 4} \log  {(r/a)}
\eqn\abc$$ so that
as $r\to 0$, $ e^\phi \to 0 $ for $3<n<7 $ and $ e^\phi \to \infty  $ for
$n<3$.

In the limits considered here, if $N$ is kept fixed then $a \to 0 $ for $n<6$,
and is finite ($a=d_6 RN$) for $n=6$, while if $N \to \infty  $ keeping $g_{YM}
^2 N$
fixed, $a \to \infty  $. If $r$ is rescaled at the same time, the limiting form
will
be the asymptotic flat geometry if $a/r$ remains small, but will be the near
geometry \met\ if $a/r$ becomes large.
The limiting forms of these geometries in the matrix limit have been considered
in [\mal,\itmal], in the limit in which the radial coordinate is also scaled so
that
$$U = m_S ^2 r
\eqn\abc$$
is held fixed; this corresponds to keeping fixed
the energy of a string joining D-branes separated by a distance $r$ in the
D-brane
description, and $U$ is the energy scale of the super Yang-Mills  description
[\mal,\itmal].
In terms of $U$,
$$
 f_n= 1+ \left( {b \over U } \right) ^{7-n}
 \eqn\abc$$
 where
 $$ b^{7-n}= {g_S N d_n   m_S ^{7-n}}
 ={R N d_n   m_S ^{8-n}}=
 {\gamma  ^2  m_S ^{4}}
 \eqn\abc$$

Then in any  limit in which $m_S \to \infty $ while $U$ and $g_N=g_{YM}\sqrt{  N}$
are kept fixed,
 the solution becomes [\itmal]
$$\eqalign{& ds^2=m_S ^{-2}\left(
 {{U^{(7-n)/2}}\over {\gamma }}dx^2_{||}+
 {{\gamma}\over {U^{(7-n)/2}}}
 dU^2+\gamma U^{(n-3)/2}
d\Omega^2_{8-n}
\right) ,
\cr & e^{\phi}=
g_{YM}^2\left(  {g_N^2  }\over {U^{7-n}}\right) ^{ {3-n}\over {4}} \sim
 { {g_{eff}^{(7-n)/2}}\over {N}}
  \cr }
\eqn\thro$$
where  $g_{eff}^2=g_N^2U^{n-3}$ is the effective coupling at energy scale $U$.
This is the geometry of the near region \met,
 rescaled by a factor $m_S ^2$ which diverges in the limit.
 It is the remarkable fact that the metric only depends on
 $g_{YM} $ and $N $ in the combination $g_{YM} ^2 N$ that leads to the structure of
the large $N$ theory [\itmal].

In particular, if $g_{YM} \to 0, N \to \infty $, as in the case here, we have the
metric \thro\ but
$e^\phi \to 0$.
Note that in this limit, although $g_S= e^{\phi _\infty } \to \infty $ for
$n>3$,
$e^\phi = e^{\phi _\infty } f_n^{-(n-3)/4}
\to 0$.
In terms of $g_{eff}$ the curvature {\cal R}
associated with the metric   is [\itmal]
$$
m_S ^{-2}{\cal  R} \approx {1\over g_{eff} } \sim \sqrt{{ U^{3-n} \over g_{YM}^2 N }}
\eqn\abc$$
This will be small if $g_{eff}$ is large, and this is a necessary condition for
the supergravity solution to be reliable.
As   the position-dependent string coupling
$e^\phi$ becomes
zero in this limit,  the weakly-coupled type II picture is reliable.
Then if $g_{eff}>>1$, i.e. at energy scales $U$ such that
$$U^{n-3} >> g_N ^ 2
\eqn\abc$$
the limiting theory is described by free type II superstrings in the
near-geometry background \thro, while for energies
$$U^{n-3} << g_N ^ 2
\eqn\abc$$
the effective coupling $g_{eff}$ is small, and the theory is described by the
large $N$ limit of super Yang-Mills  for $n=5,6$.

\chapter {The Role of the D0-branes  for $n=6$.}

We now return to the role of the $D0$-branes for $n=6$.
For $n=6$, the free limit or large $N$ limit of super Yang-Mills  was obtained
by taking
$R\to 0$
after the limits
described above. If $R$ is kept finite, there are D0-branes of finite mass
surviving in the bulk and
which might lead to extra
 degrees of freedom in the world-volume theory.
The interaction between D0 branes and D6 branes is not completely understood,
but  D0-branes do not
form a stable BPS bound state with the D6-branes, and there is a repulsive
force. If a D0-brane is
put on a D6-brane, it would tend to shrink to zero size and then escape from
the
brane into the
bulk. However, there are non-supersymmetric bound states that are quadratically
stable [\tay], so
that D0-brane states might be expected to play a role in the world-volume
theory
for finite $R$,
and this need not be superymmetric. In the free theory arising from the limit
$m_S \to \infty $ keeping
$N$  fixed, the D0-brane
could correspond to an extra free scalar moving in a compact dimension,
corresponding to the fibre of
the multi-Taub-NUT space.

In [\GravDu], it was argued that the supersymetric world-volume dynamics of the
M-theory Kaluza-Klein monopole $R^{6,1}\times (Taub-NUT)$ should be described
by the super Yang-Mills  multiplet
in 6+1 dimensions. In a covariant approach, the 6+1 world-volume theory should
include a vector field
and 11 scalars
$X^M$, of which $6+1$ scalars can be identified with the world-volume
coordinates by choosing a
static gauge, leaving 4 scalars taking values in the Taub-NUT space. The vector
multiplet in 7
dimensions has only 3 scalars however, so that we obtain a vector multiplet and
an extra scalar,
taking values in the  fibre of the Taub-NUT, which is a circle of radius $R$.
Translations in this fibre direction do not constitute true deformations, and
are pure gauge as
they constitute an isometry of the space-time; in particular, there is no
modulus associated with
them [\rub]. In [\kkmo], it was proposed that the effective description of the
4
scalars should be a
sigma-model whose target space is the Taub-NUT space, in which the fibre
isometry
is gauged, leaving a theory of 3 scalars, which fit into the vector multiplet.
In [\kkmo], a kinetic term was proposed for the bosonic degrees of freedom, and
shown to
have appropriate properties, and in particular the reduction to 5+1 dimensions
gives the D6-brane
action.
The general form of the Wess-Zumino term for    gauged sigma-models in any
dimension was given in
[\HS], and it is straightforward to add  such a term to the kinetic term of
[\kkmo] to obtain an
action that reduces to the full D6-brane action; Wess-Zumino terms for such
actions have also been
considered in [\berg].

The D0-brane degree of freedom clearly corresponds to the scalar taking values
in the Taub-NUT
fibre, and its zero-mode drops out because of the
isometry symmetry of the target space. The scalar can be decoupled from the
Taub-NUT sigma-model
by gauging, as in [\kkmo], or by taking
$R\to 0$. It cannot fit into a 7-dimensional supermultiplet with the vector and
other scalars.

To discuss whether the D0-brane survives in the  limit $N\to \infty $, we need
to
know how the energy
of the metastable D0-D6 bound state depends on $N$ and $m_S$.
In the next section, we will show that, at least in regimes in which
supergravity is reliable, the
bound state of $k$ D0-branes to the $N$ D6-branes does not decouple for matrix
theory with fixed $N$,
but does in the $N\to \infty $ limit of [\Mat]. For the limits proposed here
with $m_S \to \infty $ and
$R,R_i$ fixed, these decouple   both for the free limit with $N$ fixed and in
the  large $N$ limit.

\chapter {D-Brane Bound States}

The    D$n$-D$n-p$ brane bound state has been extensively studied for $p=2,4$.
For $p=2$, there is a true bound state preserving half the supersymmetries, and
the D$n$-D$n-2$ brane
bound state wrapped on an
$n$-torus gives a state of mass
$M$ satisfying
$$M^2=Q^2+P^2
\eqn\abc$$
For $p=4$, there is a marginally boud state preseserving $1/4$ of the
supersymmetry, and the wrapped
D$n$-D$n-4$ brane bound state has mass
$$M=|Q|+|P|
\eqn\abc$$
Here
$P$ is the mass of $N$ D$n$-branes,
$$P=g_S^{-1} m_S^{n+1} \Big(\prod_{i=1}^n R_i\Big) \eqn\pmass$$
and $Q$ is the mass of $k$ D$n-p$ branes
$$Q=kg_S^{-1} m_S^{n-p+1} \Big(\prod_{i=1}^{n-p} R_i\Big) \eqn\pmass$$
The   energy $E=M-|P|$ is then $E=|Q|$  for the marginally bound states with
$p=4$, while for
$p=2$,
$$E\sim {Q^2\over 2|P|}
\eqn\abc$$
if $Q/P <<1$, as will be the case for large $N$, so that $E\sim 1/N$.
The formulae are also the mass formulae for the black hole solution obtained by
dimensionally
reducing the brane solutions to $10-n$ dimensions (or less).
The black hole has
   two   electric charges $Q,P$ with respect to two different gauge fields and
preserves $1/2$
($p=2$) or $1/4$ ($p=4$) of the supersymmetry.

For $p=6$,  the D6-D0 brane configuration is only metastable and is not
supersymmetric.
Its dimensional reduction
should give a dyonic  black hole in 4 dimensions carrying an electric charge
$Q$ and a magnetic
charge
$P$ with respect to the same gauge field, which is the vector field obtained
from the dimensional
reduction of the Ramond-Ramond one-form.
Such dyonic black holes have been considered in this context in
  [\kur-\dar]. The relevant black hole solution is that of [\mai,\giba,\gib],
and, for the extreme black hole, the mass $M$ satisfies
$$M^2 =4 \left( Q^2+P^2 \right)-\Sigma ^2
\eqn\mas$$
where
$\Sigma$ is the dilaton charge defined by
$$
{1 \over 6}\Sigma = {Q^2\over \Sigma + \sqrt{ 3} M}+{P^2\over \Sigma - \sqrt{ 3} M}
\eqn\mart$$
 For the magnetically charged solution with $Q=0$,
$$P=M, \qquad  \Sigma =-
\sqrt 3M, \qquad Q=0
\eqn\abc$$
while
For the electrically charged solution with $P=0$,
$$Q=M, \qquad  \Sigma =
\sqrt 3M, \qquad P=0
\eqn\abc$$

The dilaton charge can be eliminated from these formulae (using e.g. the
parameterisation of [\mai]) to give
$$M=\left( Q^{2/3}+P^{2/3} \right) ^{3/2}
\eqn\abc$$
Then if $|Q|<<|P|$,
$$ E=M-P
\sim {3\over 2} P^{1/3}Q^{2/3}
\eqn\abc$$
plus corrections of order
$|Q|^{4/3}|P|^{-1/3}$

Defining the volume of the 6-torus in string units,
$$\hat V_6=m_S^{6}V_6 =m_S^{6} \Big(\prod_{i=1}^6 R_i\Big)
\eqn\abc$$
the energy can be written as
$$\eqalign{
E&\sim {3\over 2} N^{1/3}k^{2/3}{1\over R} \hat V_6^{1/3}
\cr &
= {3\over 2} N^{1/3}k^{2/3}{1\over g_{YM} ^2}
V_6^{1/3}
\cr &
= {3\over 2} N^{4/3}k^{2/3}{1\over g_N ^2}
V_6^{1/3}
\cr}
\eqn\eng$$
This formula will apply whenever
$Q << P$;
as $|Q/P| \sim k/(N\hat V_6) $, we have that $|Q/P| \to 0$  if $m_S \to \infty $
with
$R_i$ fixed, as is the case for the matrix and decoupling limits
discussed here. Then
the higher order corrections, which are suppressed by a factor of
$|Q/P|^{2/3}$,
vanish in the limit  and
the formula  \eng\ is exact.
 In the usual matrix limit in which $g_{YM},N,V_6$ are kept fixed, this gives
$E \propto N^{1/3} m_p^3$ and this remains finite, so that the finite $N$
matrix theory should
include these bound states, but if $N$ is taken to infinity (keeping $g_{YM}$
fixed, as in [\Mat,\kp])
these states decouple.
For the limits proposed here, $  V_6 $ is kept fixed
and $g_{YM} \to 0$, so that these states decouple either if $N$ is kept fixed or
if
$N\to \infty $, $g_N$ fixed, in which case
$E\sim N^{4/3}$.
If   $R$ is kept fixed, the states in which D0-branes bind to the D6-branes
decouple, even though
the bulk D0-branes remain at finite mass, while if $R\to 0$, the bulk D0-branes
decouple also.
 (Note that if $R$ and the string-scale volume $\hat V_6$ were kept fixed,
these states would remain for finite $N$ but decouple as $N\to \infty $, with
$E\sim N^{1/3}$.)

\chapter {Discussion}

We have seen that for $n=3,4,5,6$, there is a limit of the theory of $N$
D$n$-branes that gives
the $N \to \infty $ limit of $n+1$ dimensional super Yang-Mills, for which the Planck mass
and
string mass both
tend to infinity, so that the bulk and stringy degrees of freedom should all
decouple. For $n=4$,
this involves taking $R\to 0$, so that the extra dimension disappears,
while for
$n=3,5$ this involves
$\hat
g_S
\to 0,\hat
m_S
\to \infty $, so that  the stringy degrees of freedom decouple.
 The D$n$-brane action has higher-derivative terms such as those from the
expansion of the
Dirac-Born-Infeld kinetic term, but these are suppressed by powers of the
string
scale and
disappear in the limit. If there is a complete decoupling of all other degrees
of freedom, then
this limit should give a consistent quantum theory, and the simplest way in
which this might
occur would be if the large $N$ limits of these super Yang-Mills  theories were
finite.
As the theories with finite $N$   are not finite or renormalizable
for $n>3$, this would be quite
remarkable.

If one considers the situation with $N$ large but finite, the theory should
make
sense (assuming
M-theory does) and   there are extra degrees of freedom that become important
at
the short
distance scale    $l_{YM}= g_{YM} ^{2/(n-3)}$, with $g_{YM}$ small but finite.
For $n=4$, these are Kaluza-Klein modes asscociated with a circle of radius
$R=l_{YM}$, for $n=5$
these include string excitations of non-critical strings with string mass $\hat
m_S =1/l_{YM}$, and
for $n=6$ these include bulk modes coupling at the Planck scale $m_p=
1/l_{YM}$.
However, taking $N\to \infty $ takes the effective cut-off $l_{YM} \to 0$, and
it
appears that the extra
short-distance degrees of freedom decouple.
Some evidence for this is given in [\lar], where it is shown that for certain
processes and in
certain kinematical regimes, certain non-planar    super Yang-Mills  processes
correspond to
supergravity effects
that are suppressed by powers of the Planck scale.

For finite $N$, the divergence structure of super Yang-Mills  in $D=n+1$
dimensions
has been explicitly calculated to two loops by Marcus and Sagnotti [\mar], and
their results have
been confirmed in [\dix], while the implications of non-renormalisation
theorems
were investigated in
[\hs]. For $D=8,10$, there is a divergence occuring at one-loop, for $D=7,9$
there are divergences at
2-loops, calculated in [\mar]. For $D=5,6$, the theory is finite to two-loop
order [\mar], but there are  supersymmetric counterterms at 3-loops for $D=6$
and at 4-loops for
$D=5$ that are allowed by $N=2$ non-renormalisation theorems [\hs].
In [\dix], it was argued that for $D=5$,
the 4-loop and 5-loop $D=5$ divergences should be
absent also, and that  the first  ultra-violet divergence should occur at
6-loops (or higher).
Whenever counterterms are not forbidden, they are expected to actually occur.

The 2-loop counterterm in 7 dimensions is of the form
$C_{abcd}F^aF^bDF^cDF^d+....$ (with certain contractions of the  space-time
indices, suppressed here)
and there are two
possible group-theory factors    $C_{abcd} $, one of which is associated with
planar graphs and the
other with non-planar graphs [\mar,\dix].  The actual counterterm is a linear
combination of the two,
so that in the large $N$ limit the non-planar one  is suppressed, but the
planar
one survives.
Thus for $D=7$, the theory defined by taking the large $N$ limit while keeping
the ultra-violet
cut-off fixed remains ultra-violet divergent, i.e. there are still planar terms
which
diverge as the
cut-off is taken to infinity.
For $D=5,6$ the allowed counterterms can also have planar or non-planar
group-theory factors, and
there is no reason to expect that the coefficients of the planar ones should
vanish, although it
would be interesting to have explicit confirmation of this.

The likely persistance of ultra-violet divergences in the large $N$ limit at
first seems
disappointing. It could mean that there is not a complete decoupling in this
limit, but it is hard to see how this could be the case, since
the constants $1/m_S , 1/m_p$ etc governing these couplings have all gone to
zero. Alternatively,
it has been proposed that  the large $N$ limit can be used to give a finite
formulation
of perturbatively non-renormalisable
theories (there is an extensive literature, but see, for example, [\pari]), and
something similar may
play a role here.
However, finiteness would have meant that large $N$ super Yang-Mills  provides
a good
description at all
energy scales, which would be hard to reconcile with the proposed high energy
beahviour of
[\mal,\itmal] and section 4.
The picture that seems to fit best is that the large $N$ limit of the theory of
$N$
D-branes proposed here gives a limit in which the bulk and string modes
decouple, leaving a  theory that is consistent (since it is derived from
M-theory), is
described at low energies by the 't Hooft limit of  super Yang-Mills, and   the extra
degrees of freedom that
become relevant at short distances are
described by
the curved space string theories proposed
in [\mal,\itmal],
as we shall now argue.

Consider the case of $n=4$, which is the one that is best understood.
For finite $R=g_{YM} ^2$ and finite $N$, the theory is described by the (2,0)
U(N)
tensor theory in 6
dimensions, compactified on a circle of
radius
$R$. This is believed to be a consistent finite superconformally invariant
field
theory in 6
dimensions [\rozali-\seinotes]. The large $N$ limit has been argued to exist,
and to be described at
high energies by the type IIA string theory on  the limiting form of the
D4-brane background
[\itmal].
 Taking $R\to 0$ while keeping $N$ fixed, the Kaluza-Klein modes decouple,
leaving free
super Yang-Mills  theory in 4+1 dimensions, with $g_{YM} \to 0$.
Taking the 't Hooft limit of the $D=6$ theory, keeping $NR=Ng_{YM} ^2$ fixed,
should give a
consistent theory. As $U(N)$ super Yang-Mills  is expected to be a good
effective
description at length scales
that are large compared to $R$, the $N \to \infty , R\to 0$ limit might be
expected
to correspond to
the large $N$ limit of $4+1$ super Yang-Mills. However, the large $N$ limit of the
(2,0) theory should still be finite, while the large $N$ limit of the $4+1$
super Yang-Mills  may still be
divergent. If so, this would  suggest  that
the two theories are not the same, and that the large $N$ super Yang-Mills  is
only an
effective description.
The only scale surviving in the limit is that set by $g_N^2=RN$, which is kept
fixed in the limit,
so it seems reasonable to assume that this is the effective cut-off.
The resulting theory is  effectively described by super Yang-Mills  at length
scales large
compared with
$g_N^2$, but extra degrees of freedom must become important at distances small
compared to this.
However, at distances small compared to  $RN$, we can use the description
proposed in [\itmal]
and the extra degrees of freedom that enter are provided   by   type IIA
string
theory
on the limiting form of the D4-brane background. (For large enough $N$, the
other regime of [\itmal]
described by M-theory on an M5-brane background is absent.)

This suggests  the following picture. The $D=6$ (2,0) theory gives a finite
theory for all $N,R$.
If $R>0$ and $N$ is small, Kaluza-Klein modes are important at length scales
small compared to $R$
and provide the extra degrees of freedom needed at short distances in the
non-renormalisable $D=5$
super Yang-Mills. In the 't Hooft limit, $R \to 0$ so that the Kaluza-Klein modes decouple,
but it appears that
the resulting $D=5$ theory (the 't Hooft limit of $D=5$ super Yang-Mills) remains
non-renormalisable.
The effective cut-off must be $RN=g_N^2$, and the extra degrees of freedom
needed to regulate the
theory at shorter distance scales are provided by IIA string theory in the
near-region of a D4-brane
background.

A similar picture should then apply for the non-renormalisable super Yang-Mills
 theories in
$D=6,7$; for finite
$N,g_{YM}$, the extra degrees of freedom needed at distances small compared with
$l_{YM}$ are string
modes in
$D=6$ and certain bulk modes in $D=7$.   For the 't Hooft limit in which
$l_{YM}
\to 0$, the
resulting theory is again presumably not finite, even though the usual
effective cut-off $l_{YM} $
has been removed, and
the only surviving scale in the theory is that set by $g_N$, which should be
the
effective cut-off.
 The
planar limit of the
super Yang-Mills  is a good effective description at distance scales large
compared with
this, and the type II
string theory in the near region limit of the D$n$-brane background
is a good effective description at distance scales small compared with
the $g_N$ scale.
This generalises the duality between    $D=4$  super Yang-Mills  and type IIB
string theory in anti-de Sitter space
[\mal,\witt]: at weak coupling
$g_{eff}<<1$ (where
$g_{eff}^2=Ng_{YM}^2 U^{n-3} $ is the dimensionless effective coupling at
energy
$U$) there is a
super Yang-Mills  description, while for $g_{eff} >>1$ there is a description
in terms of
free type II string theory in the near
region limit of a D-brane background.

To summarise, the theory of $N$ D$n$-branes in the
large $N$ limit described here gives a theory
in which gravity and the bulk degrees of freedom decouple, along with the
string
excitations, to
leave a \lq matter theory'. If M-theory is consistent, these theories should be
also.
For $n\le 4$, such a decoupling can be obtained for finite $N$, but for $n>4$,
one either obtains a
free theory or one must
take $N\to \infty  $.
 For weak effective coupling $g_{eff}$ the theory is the large $N$ limit
of $U(N)$ super Yang-Mills, while for strong coupling $g_{eff}>>1$ it is described by the
free type II string
theory in the near region limit of the D$n$-brane space-time proposed in
[\mal,\itmal]. For $n=3$,
the
super Yang-Mills  theory is perturbatively finite, and the strong coupling
formulation  in
terms of
free type IIB strings in  anti-de Sitter space and the weak coupling
formulation
 in terms of large
$N$
super Yang-Mills  are dual formulations of the same theory [\mal,\witt]. For
$n>3$, there
are again two dual
formulations, but here the energy scale $U$ plays a role, so that the super
Yang-Mills
decription gives the
behaviour  at energies low compared to
$U_N=g_N^{-2/(n-3)}$ while the high energy behaviour is that of
the free string theories of [\mal,\itmal].
There are thus two scales that enter into the D-brane theory: $g_{YM}$ and $g_N$,
and
extra short distance degrees of freedom enter at whichever scale is the larger.
 In the limits in
which $N$ is kept fixed, extra degrees of freedom enter at the energy scale set
by $g_{YM}$ which are
Kaluza-Klein modes for $n=4$, string degrees of freedom for $n=5$ and bulk
degrees of freedom for
$n=6$. In the large $N$ limits considered here in which $g_{YM} \to 0$, these
degrees of freedom
associated with $g_{YM}$ decouple, but the new degrees of freedom that enter at
the energy scale set
by $g_N$ are associated with free string theories in D-brane space-times.

\ack
{I would like to thank Tom Banks, Mike Douglas, Juan Maldacena, Steve Shenker
and  Paul Townsend for helpful discussions. This work was partly supported by
the NSF, Grant No. PHY94-07194, and partly by the EPSRC. }

\refout

\bye

\\
Title: Decoupling Limits in M-Theory
Authors: C.M. Hull
Comments: 23 Pages, Tex, Phyzzx Macro.
Report-no: QMW-98-09, NSF-ITP-98-024
\\
Limits of a system of $N$ D$n$-branes in which the bulk and string
degrees of freedom decouple to leave a \lq matter' theory are investigated and,
for $n>4$, either
give a free theory or require taking $N\to \infty $. The decoupled matter
theory is described at low
energies by  the $N\to \infty $ limit of $n+1$ dimensional super Yang-Mills, and at high
energies by a free type
II string  theory in a curved space-time. Metastable  bound states of D6-branes
with mass $M$ and
D0-branes with mass $m$ are shown to have an energy proportional to
$M^{1/3}m^{2/3}$ and decouple,
whereas in matrix theory they only decouple in the large $N$ limit.
\\


\section{Introduction}

The $J/\Psi$ suppression has been proposed as one of the signals
for the quark-gluon plasma (QGP) which is expected to be formed in
nucleus-nucleus collisions at sufficiently high energies \cite{matsui}.
This suggestion has stimulated a number of experiments,
which indeed have  observed a significant reduction of the scaled
$J/\Psi$ yield when going from proton-nucleus to nucleus-nucleus
reactions \cite{NA38}. Especially the NA50 experiment has reported an
abrupt decrease in  $J/\Psi$ production in Pb~+~Pb collisions
at 158 GeV per nucleon \cite{gonin,NA50,NA50a} in going from peripheral to central
collisions.

A lot of theoretical effort has been spent
to understand the experimental results (see \cite{gerhuf} for
a recent review).
Beside the suggestion of a possible formation of a QGP
\cite{KhaQM96,blaizot,wong} various scenarios based
on $J/\Psi$ absorption by hadrons have also been proposed
\cite{gerschel,gavin,gavin1,Capella,cassing-jpsi,armesto}.
A part of the suppression can  be explained
by $J/\psi $--absorption on the surrounding nucleons
\cite{gerschel}, additional absorption
might be attributed to `comovers' (`mesons')
being produced as secondaries \cite{gavin,gavin1}.
It has been shown in \cite{cassing-jpsi}
within the microscopic covariant
transport approach HSD (Hadron String Dynamics) \cite{cassing} that the observed
suppression of the $J/\Psi$ yield in nuclear collisions is consistent with such
an hadronic absorption scenario.
However, the 'comover' models are still a matter of debate: The employed
absorption cross section of $c\bar{c}$ pairs on comovers
of 3 mb is treated as a free  parameter in order to explain the data.
On the other hand, this cross section might be overestimated
considerably \cite{antico}.

In our present work we therefore consider an alternative mechanism
for  $J/\Psi$ production in heavy ion collisions.
We will focus on the effect of $c\bar{c}$ dissociation in the prehadronic phase
and not on absorption by comovers.
This is motivated by the fact, that the very early collision phase is
not described by hadrons but by highly excited strings, which in the HSD model
are created by momentum transfer among target and projectile
nucleons . Their
production and decay is treated within the FRITIOF model \cite{LUND} and
describes the first few fm/c of the collision. The $c \bar{c}$ state, most likely
as a color octet, is also produced at the earliest state of the
reaction in a hard collision among the nucleons via gluon-gluon fusion.
Hence it is natural to ask
what happens if this colored state moves into such a temporary
environment of strings.
As the string carries a lot of internal energy (to produce the later secondaries)
in a small and localized space-time volume
the quarkonia state might get completely dissociated by the
intense color electric field inside a single string.
In this
respect Loh et. al.   \cite{loh} have investigated the possible $J/\Psi$
dissociation in a color electric flux tube in a semiclassical model
based on the Friedberg-Lee color dielectric Lagrangian. They find that
the $c \bar{c}$ state becomes dissociated rather immediately on a timescale
less than 1 fm/c.
In a heavy ion collision, especially for the more
heavy systems, the effective region (or volume) of all the strings
being produced is expected to be large so that the strings might
become rather closely packed.
If the $c\bar{c}$ states will get dissociated by the numerous
and individual strings this will lead to an additional suppression
of $J/\Psi$ in the early phase prior to hadronization.
To explore this intuitive idea we include the effect of
$c\bar{c}$ dissociation by strings into the HSD model and study the overall
$J/\Psi$ production and dissociation dynamically.

The HSD model is briefly presented in section 2, while  in section 3 we describe
the $c\bar{c}$ production process and show details of the string
evolution in  the HSD model. We then address the dissociation process
of the $c\bar{c}$ states on baryons and the chromoelectric flux tube of
the strings (section 4). We conclude with a summary of our investigations.

\section{The covariant transport approach}
In this work we perform our analysis along the line of the HSD approach
\cite{cassing} in the cascade modus
which is based on a coupled set of covariant transport
equations for the phase-space distributions $f_{h} (x,p)$ of hadron $h$
\cite{cassing}, i.e.
\begin{samepage}
\begin{eqnarray}  \label{g24}
\lefteqn{\left( {\partial \over \partial t} + {\vec{p}_1 \over m}
\vec{\nabla} \right)
f_1(x,p_1)  } \nonumber \\
&&= \sum_{2, 3, 4\ldots}\int d2 d3 d4 \ldots
 [G^{\dagger}G]_{12\to 34\ldots}
\delta^4(p_1^{\mu} +p_2^{\mu} -p_3^{\mu} -p_4^{\mu}\ldots )  \nonumber\\
&& \times \left\{ f_3(x,p_3)f_4(x,p_4)\bar{f}_1(x,p)
\bar{f}_2(x,p_2)\right.  \nonumber\\
&& -\left. f_1(x,p)f_2(x,p_2)\bar{f}_3(x,p_3)
\bar{f}_4(x,p_4) \right\} \ldots\ \ .
\end{eqnarray}
\end{samepage}
Here $ [G^{\dagger}G]_{12\to 34\ldots}
\delta^4(p_1^{\mu} +p_2^{\mu} -p_3^{\mu} -p_4^{\mu}\ldots )$
is the `transition rate' for the process
$1+2\to 3+4+\ldots$,
while the phase-space factors
\begin{equation}
\bar{f}_{h} (x,p)=1 \pm f_{{h}} (x,p)
\end{equation}
are responsible for fermion Pauli-blocking or Bose enhancement,
respectively, depending on the type of hadron in the final/initial
channel. The dots in eq.~(\ref{g24}) stand for further contributions in
the collision term with more than two hadrons in the final/initial
channels. The transport approach (\ref{g24}) is fully specified
by the transition rates $G^\dagger G\,\delta^4 (\ldots )$ in the collision
term, that describes the scattering and hadron production and absorption
rates. This transport approach
was found to describe reasonably well hadronic as well as dilepton
data from SIS to SPS energies \cite{cassing,cassing1}.

In the present approach we propagate explicitly -- apart from the
baryons (cf.~\cite{cassing}) -- pions, kaons, $\eta$'s, $\eta^\prime$'s,
the $1^-$ vector mesons $\rho, \omega, \phi$ and $K^*$'s as well as the
axial vector meson $a_1$.
The high energy hadron-hadron
collisions are described by the FRITIOF model \cite{LUND}
resulting in two excited
strings. The dynamical evolution of the strings is now
included explicitly. A string is characterized by the leading quark and
diquark (or antiquark in the case of a mesonic string) and by the energy
stored in between.
The fragmentation of the strings into hadrons starts after the
formation time, which is set to  $\tau_F=0.8$ fm/c
(see Fig. \ref{string_dyn}). $\tau_F$ controls the baryon and meson
rapidity  distribution $dN/dy$ in comparison to experimental data.
The length of the strings is given by the (center of mass)
collision time $t_0$  of the
hadrons, the formation time and the velocity of the
leading quarks/diquarks $\beta_i=|\vec{p}_i|/E_i$, $i=1,2$
\begin{eqnarray}  \label{stringlength}
l(t)=\left\{
\begin{array}{lcl}
|\beta_2-\beta_1| \cdot (t-t_0) &, & (t-t_0) \le \tau_F  \\
\left(|\beta_2-\beta_1| - 2 \cdot \sqrt{1-{\tau_F^2 \over (t-t_0)^2}} \right)
\cdot (t-t_0) &, & \tau_F < (t-t_0) \le t_{max} \\
0 &, & (t-t_0) > t_{max},
\end{array}
\right.
\end{eqnarray}
where $t_{max}=t_0+2 \tau_F / (\sqrt{ 4 - |\beta_2-\beta_1|^2})$ is the time
when the string is completely hadronized.
The radius of the flux tube is an unknown
parameter, which will be of particular significance for the $c\bar{c}$ dissociation
by strings, which we will address in section 4.
The cross section of the high energy secondary interactions of the leading
quarks/diquarks are treated within a simple additive quark model
$\sigma(q-B)=1/3 \,\sigma(B-B)$ and $\sigma(qq-B)=2/3 \, \sigma(B-B)$.

In our simulation the production and decay of strings dominates the very early
collision phase before the subsequent hadronic state of matter is formed.
In Fig. \ref{string-pic} a characteristic representation of the hadrons
and the strings during the high density phase in a central
Pb~+~Pb collision at 160 AGeV is shown.
As pointed out already in the introduction several
hundred strings are formed during a central Pb-Pb collision at SPS energies.
It turns out that most of them are in fact rather short due to
secondary interactions of the
leading quarks/diquarks. In Fig. \ref{string_number}
the number of strings and  the averaged string length
in a central Pb-Pb collision at 160GeV is shown as a function of
the center of mass time.

After the hadronization many low energy rescattering processes
are included. The  meson-baryon and baryon-baryon collisions are treated
similar to the BUU model \cite{Wolf90}.
As meson-meson channels we include the reactions $\pi \pi \rightarrow \rho,
\pi \pi \rightarrow K \bar{K}, \pi \rho \rightarrow \phi, \pi \rho
\rightarrow a_1$ as well as the time reversed reactions using Breit-Wigner
cross sections with parameters from the literature \cite{PDB} and
exploiting detailed balance. For the present analysis the low
energy rescattering processes do not play an essential role.
Within the philosophy outlined in the introduction the production
and absorption of the $c\bar{c}$ states happens in the first few fm/c
of the collision before secondary particles are produced.

\section{Charmonium production}

Since the probability of producing initially a $c \bar{c}$ pair is
very small, a perturbative approach is used for technical reasons
as described in \cite{cassing-jpsi}.
Whenever  two nucleons collide a $c\bar{c}$ pair is produced with a
probability factor $W$, which is given by the ratio of the $J/\Psi$
 to $NN$ cross section at a center-of-mass energy
$\sqrt{s}$ of the baryon-baryon collision,
\begin{equation}
W= \frac{\sigma_{BB \rightarrow J/\Psi + X} (\sqrt{s})}
{\sigma_{BB \rightarrow BB+X} (\sqrt{s})}.
\label{wratio}
\end{equation}
The parameterization used for the $J/\Psi$ cross section, the rapidity
and $p_t$ distribution of the $c\bar{c}$ states is taken as in
\cite{cassing-jpsi}, which fits reasonable well the experimental data.

It is known from experiment that the Drell-Yan cross section as a
hard process scales with $(A_P \times A_T)$ \cite{NA38,NA50,Carlos}.
This is in contrast to most of the soft hadronic observables, like
e.g. the pion multiplicity, which scale like $A_P + A_T$. This indicates
a strong difference in the production of hard and soft processes in
heavy ion collisions.
Since the production scheme for $c\bar{c}$ is similar to the production
of Drell-Yan pairs the total initial $c\bar{c}$ production
cross section also scales with $A_P \times A_T$. It
is important to stress that only with this stringent assumption
the ratio of the $J/\Psi$ to the Drell-Yan cross section is a direct measure
for the  $J/\Psi$ suppression. We want to note that the
$A_P \times A_T$ behavior of the initial
$c\bar{c}$ production cross section is still a matter of debate.
As pointed out by Frankel and Frati \cite{Frankel} and also by Tai An et al.
\cite{Tai} the strong $J/\Psi$  suppression in central Pb+Pb collisions might be
explained by a nonlinear  scaling of the
initial $c\bar{c}$ production cross section due to initial state
interactions. Such initial state interactions should also diminish the
Drell-Yan cross section, which, on the other side, is not seen
experimentally (for a comprehensive discussion on this issue we
refer to \cite{kharz}).
In any case, in our approach we stay with the same scaling behavior
and investigate
the subsequent absorption of $c\bar{c}$ pairs on baryons and strings.
For that reason we emphasize
that the $A_P \times A_T$ scaling of the initial $c\bar{c}$ production
is an input of our calculation and not a result. To implement this scaling we
separate the production of the
hard and soft processes: The space-time production vertices of the
$c\bar{c}$ pairs are calculated before every run neglecting the soft
processes. After that we follow the motion of the Charmonium pairs
within the full hadronic background by propagating it as a free particle.
Again, only with
this concept the $A_P \times A_T$ dependence of the $c\bar{c}$ production
can be reproduced in a hadronic transport simulation.

\section{$c\bar{c}$ dissociation}
The $c\bar{c}$ state as a rather heavy hadronic particle and a result
of a hard process is formed immediately
in comparison to the soft particle production
from string fragmentation. Thus first of all the $c\bar{c}$ states
move not in a hadronic environment but in an environment of color electric
strings of 'wounded' nucleons (see Fig. \ref{string-pic}).
As motivated in the introduction and in ref.\cite{loh} we now
assume that a $c\bar{c}$ state immediately
dissociates whenever it moves into the region of the color electric
field of a string. In this sense strings are completely
black for $c\bar{c}$ states. It was found in \cite{loh} that the additional force
acting on the charm quarks is given by $2 \times \sigma \approx 2 \,GeV/fm$,
where $\sigma$ denotes the phenomenological string constant of a
chromoelectric flux, which is sufficient to immediately break up a
$c\bar{c}$ state. One can also argue that the field energy density contained in a
string is given by $\sigma / (\pi R_S^2)$. For $R_S \approx 0.3 \,fm$ one
accordingly has a local high color electric energy density of
$\approx 4\, GeV/fm^3$, which substantially screens the binding potential
of the Charmonium state \cite{matsui}.

For practical reasons the dissociation by a string is modeled
when the center of mass of the $c\bar{c}$ state is located inside the string.
The length of the strings is given by eq. (\ref{stringlength}), while
the string radius $R_s$ is an unknown parameter.
Absorption by  strings spanned between the parent particles of the
$c\bar{c}$ pair are excluded, since this effect already
is included in the production cross section.

Additionally, the $c\bar{c}$ pair may be destroyed by collisions with incoming
baryons. For the collisions with baryons we use the minimum
distance concept described in Ref. \cite{Wolf90}.  For the actual cross
sections employed we assume that the $c\bar{c}$ pair initially is
produced in a color-octet
state and immediately picks up a soft gluon to form a color neutral
$c\bar{c}-g$ Fock state \cite{KhaQM96} (color dipole).  This extended
configuration in space is assumed to have a 6~mb absorption cross
section in collisions with baryons ($c \bar{c}+B\to\Lambda_c+\bar D$)
as in Refs.  \cite{gonin,gerschel,KhaQM96} during the lifetime $\tau$
of the $c\bar{c}-g$ state, for which we adopt $\tau =$ 0.3 fm/c as suggested
by Kharzeev \cite{KhaQM96}. One also has to specify the
absorption cross sections of the formed resonances $J/\Psi$
on baryons. For simplicity we use 3~mb following Refs.
\cite{KhSatz96,cassing-jpsi}.

\section{Results}
In proton-nucleus collision the $c\bar{c}$ dissociation by strings
is practically negligible. The absorption is dominated by
$c\bar{c}$-baryon collisions so that the proton nucleus data allow to
fix this absorption cross section. In Fig. \ref{pa_abs} we show
our results for the $J/\Psi$ survival probability  $S^{J/\Psi}$
using 6~mb for the
absorption cross section of the $c \bar{c}$-pairs on nucleons in
comparison to the data \cite{gonin}.
The experimental `survival
probabilities'  $S_{exp}^{J/\Psi}$ in this figure as well as in the
following comparisons are defined by the ratio of experimental
$J/\Psi$ to Drell-Yan cross sections as
\begin{eqnarray}
S_{exp}^{J/\Psi} = \left.{\left(B_{\mu\mu}\sigma^{J/\Psi}_{AB}\over
\sigma^{DY}_{AB}|_{2.9-4.5 \ {\rm GeV}}\right)} \right/
{\left(B_{\mu\mu}\sigma^{J/\Psi}_{pd}\over \sigma^{DY}_{pd}\right)},
\label{sexp}\end{eqnarray}
where $A$ and $B$ denote the target and projectile mass while
$\sigma^{J/\Psi}_{AB}$ and $\sigma^{DY}_{AB}$ stand for the $J/\Psi$ and
Drell-Yan cross sections from $AB$ collisions, respectively, and
$B_{\mu\mu}$ is the branching ratio of $J/\Psi$ to dimuons.
The theoretical ratio is defined as
\begin{eqnarray}
S_{theor}^{J/\Psi} = {M_{J/\Psi}\over N_{J/\Psi}},
\label{stheor}\end{eqnarray}
where $N_{J/\Psi}$ is the multiplicity of initially produced $J/\Psi$'s
while $M_{J/\Psi}$ is the multiplicity of $J/\Psi$'s that survive
the hadronic final state interactions.

In Fig. \ref{pa_abs} results are shown for two different string radii
$R_S=0.1fm$ and $R_S=0.4fm$. The  difference of the
curves is rather small, which indicates the small effect of
dissociation by strings.
For p~+~U and $R_S=0.4 fm$ only 2\% of the $J/\Psi$ are absorbed by strings.
This is completely different for heavy ion collisions,
where the absorption on strings becomes a much more important effect.

To compare our results for S~+~U and Pb~+~Pb to the NA38 and NA50 data,
the experimental trigger conditions must be included.
In these  experiments only events are recorded with a
$\mu^+\mu^-$ pair of invariant mass $M\ge 1.5$~GeV. This trigger
condition is obtained by
\begin{eqnarray}
{d\sigma_{theor}^{\mu\mu} \over E_T} = 2\pi N_0 \int\limits_0^\infty
b db \ {dN\over dE_T}(b) \ \sum\limits_{i} W_i^{\mu\mu} (b),
\label{ET-theor}\end{eqnarray}
where $W_i^{\mu\mu} (b)$ are the weights for produced $\mu\mu$ pairs
within the experimental cuts.
$N_0$ is a normalization factor to adjust to the experimental
number of events. We take the same weight factors $W_i^{\mu\mu} (b)$
as in \cite{cassing-jpsi} which reasonably reproduce the experimental
$E_t$ distribution as shown in \cite{cassing-jpsi}.

Qualitatively the results are not changed by this procedure. In Fig.
\ref{hi_sup} our results are shown for  S~+~U and Pb~+~Pb as a function of
the transverse energy and for four different string radii. A strong dependence
on the string radius $R_S$ is observed and $R_S=0.2 fm$ gives the best fit
to experimental data \cite{NA50,NA50a}. With this string radius 40\% of the absorbed
$J/\Psi$'s are dissociated by strings in central collisions
of Pb~+~Pb as shown in  Tab. \ref{tab1}.

\begin{table}
\centerline{\begin{tabular}{|c|c|c|c|}
\hline
$R_S [fm]$ & $A_{tot}$ & $A_B$ & $A_S$  \\
\hline\hline
0.1        & 0.65    &  0.53 & 0.12   \\ \hline
0.2        & 0.74    &  0.44 & 0.30   \\ \hline
0.3        & 0.82    &  0.36 & 0.46   \\ \hline
0.4        & 0.88    &  0.29 & 0.59   \\ \hline
\end{tabular}}
\caption{Total $J/\Psi$ absorption probability $A_{tot}=1-S_{theor}^{J/\Psi}$,
absorption by baryons
$A_B$ and by strings $A_S$ for different string radii in a central
Pb~+~Pb collision as obtained
in the HSD approach.}
\label{tab1}
\end{table}

\section{Summary}
In this work we have addressed the question of $c\bar{c}$ absorption
in relativistic heavy ion collisions in the microscopic
transport approach HSD.
Of particular interest was the effect of the prehadronic phase, which
in the HSD model is described by independent strings.
In principle the comover absorption
scenario could be included in addition.
However, since most of the $c\bar{c}$ pairs are already absorbed by
strings and baryons in the early stage of the reaction, the effect
of comovers is expected to be much less than found in
ref. \cite{cassing-jpsi}.

The dissociation mechanism by strings was treated in a
simple geometrical picture. Whenever a
$c\bar{c}$ pair moves into a string it dissociates and forms $D\bar{D}$
mesons.
Strings are completely 'black' for $c\bar{c}$ states.
This stringent condition is a first ansatz to investigate the effect
of the $c\bar{c}$ dissociation by strings. Better theoretical foundations
of the dissociation mechanism of charmonium ($J/\Psi$ and $\psi'$) and
preresonant states ($c\bar{c}g$) are definitely needed
to improve the understanding of this geometrical picture.

We found that the absorption by strings plays is an important effect in the
first few fm/c of the collision phase before secondary particles
are produced. Adopting a string radius of $0.2 fm$ we got a qualitative
agreement with p+A data and the NA38 and NA50 data. This radius seems to
be rather small, but for two reasons it should be seen as a lower bound
for $R_S$.
First of all, we have assumed that the strings are completely black for
$c\bar{c}$ states, and secondly it
dissociates whenever it moves into the region of the color electric
field of a string. With the requirement
that the total $c\bar{c}$ state should be inside the string when the
dissociation
process starts, one should add the $c\bar{c}$ radius to our value of $R_S$.
This would give a string radius of $\tilde{R}_S\approx 0.4-0.5 fm$.

With our results we can estimate the fraction of the reaction volume
which is filled by the flux tubes of the strings.
In the very central region with 1 fm thickness along the longitudinal
direction in the cms frame, we get $N_S\sim 420$ strings in a
central Pb~+~Pb collision generated within the HSD approach.
The averaged string length $<l> \sim 0.5 fm$ is rather small due to
secondary interaction of the leading quarks. Only a small amount
of strings reaches a length above 1.2 fm.
With these results we get a string volume of
\begin{eqnarray}
V_s=\pi R_S^2 \cdot <l> \cdot N_S \approx \left\{
\begin{array}{cc} 65 fm^3, & R_S=0.3 fm
\\ 115 fm^3, & R_S=0.4 fm
\end{array}
\right. .
\end{eqnarray}
The total volume in this central cylinder is given by
\begin{eqnarray}
V_t=\pi R_{Pb}^2 \cdot 1 fm \approx 130 fm^3,
\end{eqnarray}
where $R_{Pb}=6.6\,fm$ is the radius of the Pb nuclei.
Comparing $V_s$ and $V_t$ illustrates that 50-75\% of the central volume
in an ultrarelativistic heavy ion collision is filled by strings in
the first few $fm / c$ assuming a string radius of 0.3 - 0.4 fm.
The probability for a $c\bar{c}$ pair to pass the central
region thus is rather small.
This explains the strong dependence of the $J/\Psi$ results
on the string radius.

We conclude our study by noting that $c\bar{c}$ dissociation in the
prehadronic phase seems to be a dominant and likely charmonium  absorption
process in relativistic heavy ion collisions. The qualitative
behavior of the proton-nucleus and nucleus-nucleus data can be
described within this picture.
We note that the sudden jump in the $J/\Psi$ suppression for
Pb~+~Pb suggested by the data for $E_T\approx 50\,GeV$ in Fig.
\ref{hi_sup} cannot be described in our dynamical model as in Ref.
\cite{cassing-jpsi} based on string and hadron dynamics
(cf. also \cite{kharz} for a recent overview), because the predicted
suppression within our approach is always a rather smooth function
of atomic number and centrality of the collision.
If confirmed experimentally, this might indicate a new phase with other
degrees of freedom (partons) and reaction channels possibly
related to a deconfined QGP formation \cite{satz2}.


\section{Introduction}

Let $(M,g)$ be a complete Riemannian manifold and $G$ a compact Lie group acting on $M$ by isometries. This action is called polar if there exists a closed embedded submanifold $\Sigma \subseteq M$ meeting all orbits orthogonally (\cite{PalaisTerng}). Then $M$ is called a polar G-manifold and such a submanifold $\Sigma$ is called a section and comes with a natural action by a discrete group of isometries $\Pi = \Pi(\Sigma)$, called its generalized Weyl group. Recall that by definition, $\Pi(\Sigma):= N(\Sigma)/Z(\Sigma)$, where

\begin{eqnarray*}
N(\Sigma)&=&\{g\in G| g\Sigma=\Sigma \},\\
Z(\Sigma)&=&\{g\in G| gx=x,\ x\in\Sigma \}.
\end{eqnarray*}

Polar actions have nice properties and have been studied by many people, see for instance \cite{Dadok}, \cite{GZ}, {\cite{M2}}, {\cite{PalaisTerng}}, \cite{PT}.
Basic examples of polar actions are the adjoint action of a compact Lie group on its Lie algebra. More generally, isotropy representations of symmetric spaces are also polar. It's a classical theorem of Dadok \cite{Dadok} which shows that a polar representation is (up to orbit equivalence) the isotropy representation of a symmetric space. An important feature of polar actions is the following Chevalley Restriction Theorem \cite{PalaisTerng}.

\begin{theorem}
Let $(M,g)$ be a polar G-manifold with a section $\Sigma$ and generalized Weyl group $\Pi$. Then the following restriction to $\Sigma$ is an isomorphism:
$$|_\Sigma:\ \ C^\infty(M)^G \rightarrow C^\infty(\Sigma)^\Pi,$$
where $C^\infty(M)^G$ is the algebra of G-invariant smooth functions on $M$.

\end{theorem}

For a generalization of Chevalley Restriction Theorem to tensors, see \cite{M2}.

In this paper we study symplectic aspects of polar actions. More precisely, we are looking at the lifting action of $G$ on the cotangent bundle $T^*M$ with its canonical symplectic structure $\omega$. This action is a Hamiltonian action with a moment map given by $u: T^*M\rightarrow \mathfrak{g}^*$ with
\begin{eqnarray}
u_X(x,\xi) = \langle \xi, X^*(x)\rangle, {\label{1.1}}
\end{eqnarray}
where $\mathfrak{g}$ is the Lie algebra of $G$, $\mathfrak{g}^*$ is the dual of $\mathfrak{g}$ and $u_X(x,\xi)=\langle u$, $X\rangle(x,\xi),\ X\in \mathfrak{g}$. Moreover, $X^*$ is the vector field on $M$ generated by $X$.
The moment map satisfies the following equations:
\begin{eqnarray*}
 du_X&=&i_{X^{\#}}\omega,\\
u(g\cdot (x,\xi))&=&Ad^*_g\cdot u(x,\xi),\ \forall g\in G,
\end{eqnarray*}
where $X^\#$ is the vector field on $T^*M$ generated by $X$.

Our starting point is the following observation.

\begin{proposition}{\label{prop1}}
Let $(M,g)$ be a polar $G$-manifold with a section $\Sigma$. Then $T^*\Sigma$ meets all $G$-orbits of the action $G\times u^{-1}(0)\rightarrow u^{-1}(0)$.
\end{proposition}

Here $T^*\Sigma$ is seen as a submanifold of $T^*M$ under the natural isomorphism $T^*M \cong TM$ induced by the Riemannian metric $g$.
Note that in general $T^*\Sigma$ can not meet all orbits of the lifting action of $G$ on $T^*M$ as it is easy to see that $T^*\Sigma \subseteq u^{-1}(0)$ from ({\ref{1.1}}).

Let $C^\infty(T^*M)^G$ be the algebra of $G$-invariant smooth functions on $T^*M$. It carries a natural Poisson algebra structure with Poisson bracket $\{f,g\}:=\omega(X_f,X_g)$, where $X_f$ is the Hamiltonion vector field of $f$ satisfying $i_{X_f}\omega=df$.
The generalized Weyl group $\Pi$ of the $G$-action on $M$ also acts on $T^*\Sigma$.
Let $C^\infty(T^*\Sigma)^\Pi$ be the algebra of $\Pi$-invariant smooth functions on $T^*\Sigma$ with its natural Poisson algebra structure. Our main result is the following symplectic analogue of Chevalley Restriction Theorem.

\begin{theorem}{\label{mainthm}}
Let $(M,g)$ be a polar $G$-manifold with a section $\Sigma$ and generalized Weyl group $\Pi$. Then the following restriction to $T^*\Sigma$ is a surjective homomorphism of Poisson algebras:
$$|_{T^*\Sigma}:\ C^\infty(T^*M)^G\rightarrow C^\infty(T^*\Sigma)^\Pi.$$
\end{theorem}

The above restriction to $T^*\Sigma$ is not injective in general as $T^*\Sigma \subseteq u^{-1}(0) \neq T^*M$ unless $u\equiv 0$.

The symplectic reduction $T^*M\sslash G:=u^{-1}(0)/G$ is not a smooth manifold in general. However, it's a stratified symplectic space defined in \cite{SL}. The reader is referred to \cite{SL} for the precise definition of stratified symplectic spaces.
A basic example is given by
$$X_0=X\sslash G:= J^{-1}(0)/G,$$
where $X$ is a Hamiltonian $G$-space with a moment map $J: X\rightarrow \mathfrak{g}^*$. Following \cite{SL}, we define a function $f_0: X_0 \rightarrow \mathbb{R}$ to be smooth if there exists a function $F\in C^\infty(X)^G$ with $F|_{J^{-1}(0)}=\pi^*f_0$, where $\pi: J^{-1}(0)\rightarrow J^{-1}(0)/G$ is the projection map.
In other words, $C^\infty(X_0)$ is isomorphic to $C^\infty(X)^G/{I^G}$, where $I^G$ is the ideal of $G$-invariant smooth functions on $X$ vanishing on $J^{-1}(0)$. The algebra $C^\infty(X_0)$ inherits a Poisson algebra structure from $C^\infty(X)$.

Let $G$ and $H$ be Lie groups and $X$, resp. $Y$, be smooth manifolds on which $G$, resp. $H$ act properly. The stratified symplectic spaces $T^*X\sslash G$ and $T^*Y\sslash H$ are isomorphic if there exists a homeomorphism $\phi: T^*X\sslash G \rightarrow T^*Y\sslash H$
and the pullback map

\begin{eqnarray*}
\phi^*:\ C^\infty(T^*Y\sslash H) &\rightarrow& C^\infty(T^*X\sslash G),\\
f &\mapsto& f\circ \phi
\end{eqnarray*}
is an isomorphism of Poisson algebras.

In \cite{LMS} (Page 13, Conjecture 3.7), they made the following conjecture.

\begin{conjecture}{\label{conj}}
Let $G$ and $H$ be Lie groups and $X$, resp. $Y$ be smooth manifolds on which $G$, resp. $H$ act properly. Assume that the orbit spaces $X/G$ and $Y/H$ are diffeomorphic in the sense that there exists a homeomorphism $\phi: X/G\rightarrow Y/H$ such that the pullback map $\phi^*$ is an isomorphism from
$C^\infty(Y/H):= C^\infty(Y)^H$ to $C^\infty(X/G):= C^\infty(X)^G$. Then $T^*X\sslash G$ and $T^*Y\sslash H$ are isomorphic.
\end{conjecture}

Using Theorem $\ref{mainthm}$, we can give a partial answer to
conjecture $\ref{conj}$. More precisely, we have the following corollary.

\begin{corollary}{\label{coro1}}
Let $M$ be a polar $G$-manifold with a section $\Sigma$ and generalized Weyl group $\Pi$. Then $T^*M\sslash G$ and $T^*\Sigma\sslash \Pi$ are isomorphic.
\end{corollary}

Under a slightly different assumption, it was proved that $T^*M\sslash G$ is homomorphic to $T^*\Sigma\slash \Pi$ in \cite{LMS} (Proposition 3.8).

Proposition $\ref{prop1}$, Theorem $\ref{mainthm}$ and Corollary $\ref{coro1}$ will be proved in section 3.
A main ingredient of the proof is a characterization of symplectic slice representations of the lifting action $G$ on $T^*M$, which is done by using the natural Sasaki metric on $T^*M$. Then combining the Multi-variable Chevalley restriction theorem proved by Tevelev \cite{Tev} and other things, we are able to prove our results. For details, see section 3.

\section*{Acknowledgements} The first author is partially supported by Scientific Research Foundation 172119, Institute for Advanced Study, Tongji University. The second author is partially supported by the Project MYRG2015-00235-FST of the University of Macau. Part of this work was done when both authors were visiting the Institute of Mathematical Sciences in the Chinese University of Hong Kong. We also thanks Professors Huai-Dong Cao and Naichung Conan Leung for helpful discussion.

\section{Sasaki metrics on $TM$ and $T^*M$.}

In this section we describe the Sasaki metrics on $TM$ and $T^*M$ constructed in {\cite{SA}}. Given a Riemannian metric $g$ on $M$, its Levi-Civita connection determines a splitting $TTM= \mathcal{H}M \oplus \mathcal{V}M$, where $\mathcal{V}M= \ker d\pi$, $\pi: TM \rightarrow M$ is the projection and $\mathcal{H}M$ is spanned by $X^h$, $X$ is a smooth vector field on $M$.
To describe $X^h$, let $(x,v)\in TM$ and $\gamma(t): [0,1] \rightarrow M$ be a smooth curve such that $\gamma(0)=x$, $\gamma'(0)=X(x)$. Let $Y(t) \in T_{\gamma(t)}M$ such that
\begin{equation*}
\begin{cases}
\nabla_{\gamma'}Y=0\\
Y(0)=v.
\end{cases}
\end{equation*}
Then $X^h(x,v)=:\bar{\gamma}'(0)$, where $\bar{\gamma}\left(t\right)=\left(\gamma\left(t\right), Y\left(t\right)\right)$.
From the definition of $X^h$, we see that $d\pi (X^h(x,v))=X(x)$.
Let $I_g$ be the natural isomorphism $T^*M\cong TM$ induced by the Riemannian metric $g$.
Then using the splitting $TTM= \mathcal{H}M \oplus \mathcal{V}M{\cong}TM\oplus TM$, we define the Sasaki metric $\tilde{g}$ by
$$\tilde{g}\langle\left(X_1,X_2\right), \left(Y_1,Y_2\right)\rangle:=g\langle X_1,Y_1\rangle+g\langle X_2,Y_2\rangle.$$

Define an almost complex structure $J$ by setting $J(X, Y)=(-Y,X).$ Then $\tilde{g}\langle J\cdot, J\cdot\rangle=\tilde{g}\langle\cdot,\cdot\rangle$ and the symplectic form $\Omega := \tilde{g}(J\cdot,\cdot)$ is nothing but the pullback of $\omega$ by the isomorphism $I^{-1}_g: TM\cong T^*M$, where $\omega$ is the standard symplectic form on $T^*M$.

The Sasaki metric on $T^*M$ is the pullback of $\tilde{g}$ under the isomorphism $I_g: T^*M\rightarrow TM$.
The following Lemma will be important for us.

\begin{lemma}{\label{lemma2.1}}
If $\Sigma$ is a totally geodesic submanifold of $(M,g)$, then $T\Sigma$ is a totally geodesic submanifold of $(TM, \tilde{g})$, where $\tilde{g}$ is the Sasaki metric on $TM$.
\end{lemma}

$\bf{Proof:}$ Let $X$ be a smooth vector field on $M$ such that $X(x)\in T_x\Sigma$, $\forall x\in \Sigma$. As $\Sigma$ is totally geodesic, we see that $X^h|_{T\Sigma}$ is a smooth vector field on $T\Sigma$ from the construction of $X^h$.

The vector field $X$ also induces a vertical vector field $X^{\perp}$ on $TM$. We choose a local coordinate to describe $X^{\perp}$. Let $(x^1,...,x^n)$ be a local coordinate system at $x\in M$, where $n=\dim M$. Then any tangent vector $v\in T_xM$ can be decomposed as $v=v^i{\partial\over\partial x_i}$.
The set of parameters $\{x^1,...,x^n,v^1,...,v^n\}$ forms a natural coordinate system of $TM$. The natural frame in $T_{(x,v)}TM$ is given by $\tilde{\partial}_i={\partial\over\partial{x_i}}$ and $\tilde{\partial}_{n+i}={\partial\over\partial{v_i}}$.
Now if $X= X^i{\partial\over{\partial x_i}}$ is a vector field on $M$, then the vertical vector field $X^{\perp}$ on $TM$ is given by $X^{\perp}=X^i\tilde{\partial}_{n+i}$. As $X(x)\in T_x\Sigma,\ \forall x\in \Sigma$, by definition we see that $X^{\perp}|_{T\Sigma}$ is a vector field on $T\Sigma$.

To see that $T\Sigma$ is totally geodesic in $TM$, choose two vector fields $X$, $Y$ on $M$ such that $X(x),\ Y(x)\in T_x\Sigma,\ \forall x\in \Sigma$, then we have the following formula {\cite{Ko}}:

\begin{eqnarray}
\tilde{\nabla}_{X^{\perp}} Y^{\perp}&=&0,{\label{sasaki1}}\\
(\tilde{\nabla}_{X^h}Y^{\perp})(x,v)&=& (\nabla_X Y)^{\perp}(x,v)+{1\over2}R_x(v, Y_x, X_x)^h(x,v),{\label{sasaki2}}\\
(\tilde{\nabla}_{X^{\perp}}Y^h)(x,v)&=& {1\over2} (R_x(v,X_x,Y_x))^h(x,v),{\label{sasaki3}}\\
(\tilde{\nabla}_{X^h}Y^h)(x,v)&=& (\nabla_X Y)^h(x,v)-{1\over2}R_x(X_x,Y_x,v)^{\perp}(x,v),{\label{sasaki4}}
\end{eqnarray}
where $(x,v)\in T\Sigma$ and $\nabla$, resp. $\tilde{\nabla}$ are Levi-Civita connections of $g$, resp. $\tilde{g}$ and $R$ is the Riemann curvature tensor of $g$.

Since $\Sigma$ is totally geodesic, then
$\nabla_X Y(x),\ R_x(v,X_x,Y_x) \in T_x\Sigma.$
From ({\ref{sasaki1}}) $-$ ({\ref{sasaki4}}), it follows that $T\Sigma$ is totally geodesic.

{\hfill $\square$}\medskip

\section{A symplectic analogue of Chevalley Restriction Theorem}

In this section we prove Propositon {\ref{prop1}}, Theorem {\ref{mainthm}} and Corollary {\ref{coro1}}. A crucial property of polar actions we will use in the proof is the following result ({\cite{PalaisTerng}} Theorem 4.6):

\begin{proposition}{\label{prop3}}
 Let $M$ be a polar $G$-manifold with a section $\Sigma$. Then the slice representation at $x$ is polar with a section $T_x \Sigma$, $\forall x \in \Sigma$.
\end{proposition}

Given Propositon {\ref{prop3}}, we can now give a proof of Propositon {\ref{prop1}}.

Recall that $u: T^*M\rightarrow \mathfrak{g}^*$ is given by
$$u_X(x, \xi)= \langle\xi, X^*(x)\rangle.$$
Then for any $(x,\xi)\in u^{-1}(0)$, $\langle \xi, X^*(x)\rangle=0$, $\forall X \in \mathfrak{g}$.
Under the isomorphism $I_g: T^*M \cong TM$ induced by the Riemannian metric $g$, the the vector $\xi^\#:= I_g(\xi) \perp T_x(G\cdot x)$, i.e. $\xi^\# \in T_x(G\cdot x)^{\perp}$.

As the isometric action $G\times M \rightarrow M$ is polar with a section $\Sigma$, there exists $ h_1 \in G$ such that $h_1 x\in \Sigma$.
Then $h_1\xi^\# \in T_{h_1 x }(G\cdot x)^{\perp}$.

By Proposition $\ref{prop3}$, the slice representation
$$G_{h_1 x} \times T_{h_1 x }(G\cdot x)^{\perp} \rightarrow T_{h_1 x }(G\cdot x)^{\perp}$$
is polar. Hence there exists $h_2\in G_{h_1 x}$ such that
$h_2(h_1\xi^\#)\in T_{h_1 x}\Sigma$.

Let $h=h_2 h_1$, then
$$h(x,\xi^\#)=(h x, h{\xi^\#})=(h_1 x, h_2 h_1 {\xi^\#})\in T\Sigma.$$
So $T^*\Sigma$ meets all orbits of the action $G\times u^{-1}(0)\rightarrow u^{-1}(0).$

We proceed to give a proof of Theorem {\ref{mainthm}}. Recall that we have a splitting $TT^*M\cong \mathcal{H}M\oplus\mathcal{V}M$, which induces an isomorphism $TT^*M \overset{(d\pi, I_g)}{\cong} TM \oplus TM$, where $d\pi$ is the differential of the projection $T^*M\rightarrow M$ and $I_g$ is the natural isomorphism $T^*M\cong TM$ induced by the Riemannian metric $g$.

Let $\{x^1,...,x^n, \xi_1,...\xi_n\}$ be a local coordinate of $T^*M$ at $(x,\xi)$ and $\Gamma^k_{ij}$ be the Christoffel symbols of the Levi-Civita connection $\nabla$ induced by $g$.
Then the horizontal lift of $\partial\over{\partial x_i}$ at $(x,\xi)$ is given by
$${\tilde{\partial}\over{\partial x_i}}={\partial\over{\partial x_i}}+\Gamma^k_{il}\xi_k{\partial\over{\partial \xi^l}}.$$
Here a horizontal lift of a vector $X$ at $(x,\xi)$ is defined to be the unique vector $\tilde{X}\in \mathcal{H}M$ such that
$d\pi(x,\xi)(\tilde{X})=X$.

In terms of local coordinate system $\{x^1,...,x^n, \xi_1,...\xi_n\}$, the almost complex structure $J$ defined in section 2 can be rephrased as

\begin{eqnarray*}
J({\tilde{\partial}\over{\partial x_i}})&=&g_{ij}{\partial\over{\partial\xi^j}},\\
J({\partial\over{\partial \xi^i}})&=&-g^{ij}{\tilde{\partial}\over{\partial x_j}},
\end{eqnarray*}
where $g_{ij}=g({\partial\over{\partial x_i}},{\partial\over{\partial x_j}})$ and $(g^{ij})$ is the inverse matrix of $(g_{ij})$.

Let $X^*=X^i{\partial\over{\partial x_i}}$ be a vector field on $M$ generated by $X\in \mathfrak{g}$. Then the corresponding vector field on $T^*M$ generated by $X$ is
\begin{eqnarray*}
X^\#(x,\xi)=X^i{\partial\over{\partial x_i}}-\sum_{i,j}{\partial X^j\over{\partial x_i}}\xi_j{\partial\over{\partial \xi^i}},
\end{eqnarray*}
see {\cite{BV}} (Page 16 Lemma 11).

The Sasaki metric $\tilde{g}$ on $T^*M$ satisfies
\begin{eqnarray*}
\tilde{g}\langle{\tilde{\partial}\over\partial x_i},{\tilde{\partial}\over\partial x_j}\rangle&=&g_{ij},\\
\tilde{g}\langle{\tilde{\partial}\over\partial x_i},{{\partial}\over\partial \xi^j}\rangle&=&0,\\
\tilde{g}\langle{{\partial}\over\partial \xi^i},{{\partial}\over\partial \xi^j}\rangle&=& g^{ij}.
\end{eqnarray*}

\begin{lemma}
$\forall (x,\xi)\in T^*\Sigma$, the Sasaki metric $\tilde{g}$ on $T^*M$ induces an orthogonal splitting
$$T_{(x,\xi)}T^*M=T_{(x,\xi)}(G\cdot(x,\xi))\oplus JT_{(x,\xi)}(G\cdot(x,\xi))\oplus V$$
with $T_{(x,\xi)}T^*\Sigma\subseteq V$ and $V$ is the orthogonal complement of $T_{(x,\xi)}(G\cdot(x,\xi))\oplus JT_{(x,\xi)}(G\cdot(x,\xi))$.
\end{lemma}

{\bf{Proof:}} Let $X^\#_i$ be two vector fields on $T^*M$ generated by $X_i\in \mathfrak{g}$, $i=1,2$ respectively and $Y \in T_{(x,\xi)}T^*\Sigma$. Then $\tilde{g}\langle J X^\#_1, X^\#_2\rangle=\omega(X^\#_1,X_2^\#)=(i_{X_1^\#}\omega)(X^\#_2).$
Let $u$ be the moment map defined in ($\ref{1.1}$),
as $(x,\xi)\in T^*\Sigma\subseteq u^{-1}(0)$, by the $G$-equivalence of $u$, we get $G(x,\xi)\subseteq u^{-1}(0)$.
Hence $$\tilde{g}\langle JX_1^\#,X_2^\#\rangle=(i_{X_1^\#}\omega)(X^\#_2)=du_{X_1}(X_2^\#)=0.$$

By the definition of $J$, we get $JT_{(x,\xi)}T^*\Sigma\subseteq T_{(x,\xi)}T^*\Sigma.$
As $T^*\Sigma\subseteq u^{-1}(0),$ we get
$$\tilde{g}\langle X^\#_1, Y\rangle=\tilde{g}\langle JX_1^\#,JY\rangle=\omega(X_1^\#, JY)=(i_{X_1^\#}\omega)(JY)=du_{X_1}(JY)=0.$$
Similarly, $\tilde{g}\langle JX_1^\#, Y\rangle=0$. Hence $T_{(x,\xi)}T^*\Sigma \subseteq V.$
{\hfill $\square$}\medskip

The representation
$$G_{(x,\xi)} \times V\rightarrow V$$
is called the symplectic slice representation at $(x,\xi)$.
Note that $G_{(x,\xi)}=(G_x)_\xi=:\{h\in G_x|\ h\xi=\xi\}$.

The following Lemma will be crucial for us.

\begin{lemma}{\label{lemma3.1}}
Let $M$ be a polar $G$-manifold with a section $\Sigma$. Then the symplectic slice representation at $(x,\xi)\in T^*\Sigma$ is the diagonal action (up to identification)
$$(G_x)_{\xi^\#}\times (W\oplus W)\rightarrow W\oplus W,$$
where $W := (G_x \xi^\#)^{\perp}$ is the orthogonal complement of $G_x \xi^\#$ in the slice $(G\cdot x)^{\perp},$ i.e. we have
\begin{eqnarray*}
T_xM&=&T_x(G\cdot x)\oplus (T_x(G\cdot x)^{\perp}),\\
T_x(G\cdot x)^{\perp}&=& G_x \xi^\# \oplus (G_x \xi^\#)^{\perp}.
\end{eqnarray*}
\end{lemma}

${\bf{Proof:}}$ Let $G_{(x,\xi)}\times V\rightarrow V$ be the symplectic slice representation at $(x,\xi)$. Under the isomorphism $\Phi: \mathcal{H}M\oplus \mathcal{V}M \overset{(d\pi,I_g)}{\longrightarrow}TM\oplus TM$, we first claim that
$$\Phi(V)=W\oplus W.$$

Choose a local coodinate system $\{x^1,...,x^n,\xi_1...,\xi_n\}$
of $T^*M$ at $(x,\xi)$. Then we have
\begin{eqnarray*}
d\pi({\tilde{\partial}\over\partial x_i})&=&{\partial\over \partial x_i},\\
I_g({\partial\over\partial\xi^i})&=& g^{ij}{\partial\over \partial x_j}.
\end{eqnarray*}
Let $Z= a^i{\tilde{\partial}\over\partial x_i}+b_i{\partial\over\partial\xi^i}\ \in TT^*M$. Then $\Phi(Z)=(d\pi,I_g)(Z)=(a^i{\partial\over\partial x_i},g^{ij}b_i{\partial\over\partial x_j})=:(Y_1,Y_2).$

Let $X^*=X^i{\partial\over\partial x_i}$ be the vector field on $M$ generated by $X\in\mathfrak{g}$, then the corresponding vector field on $T^*M$ is
\begin{eqnarray*}
X^\#(x,\xi)&=& X^i{\partial\over\partial x_i}-\sum_{i,j}{\partial X^j\over\partial x_i}\xi_j{\partial\over\partial \xi^i}\\
&=&X^i{\tilde{\partial}\over\partial x_i}-X^i\Gamma^k_{il}\xi_k{\partial\over\partial\xi^l}-\sum_{ij}{\partial X^j\over\partial x_i}\xi_j{\partial\over\partial \xi^i}\\
&=&X^i{\tilde{\partial}\over\partial x_i}-g\langle{\nabla_{\partial\over\partial x_i}}X^*,\xi^\#\rangle{\partial \over \partial \xi^i}.
\end{eqnarray*}

Then we have

\begin{eqnarray}
\tilde{g}(X^\#,Z)&=&\tilde{g}\langle X^i{\tilde{\partial}\over\partial x_i}-g\langle{\nabla_{\partial\over\partial x_i}}X^*,\xi^\#\rangle{\partial \over \partial \xi^i}, a^j{\tilde{\partial}\over\partial x_j}+b_j{\partial\over\partial\xi^j}\rangle{\nonumber}\\
&=& g_{ij}X^ia^j-g^{ij}b_j g\langle{\nabla_{\partial\over\partial x_i}}X^*,\xi^\#\rangle{\nonumber}\\
&=&g\langle X^*, Y_1\rangle- g\langle \nabla_{Y_2}X^*, \xi^\#\rangle. {\label{3.1}}
\end{eqnarray}

We also have

\begin{eqnarray}
\tilde{g}\langle X^\#, JZ\rangle&=& \tilde{g}\langle X^i{\tilde{\partial}\over\partial x_i}-g\langle{\nabla_{\partial\over\partial x_i}}X^*,\xi^\#\rangle{\partial \over \partial \xi^i}, J(a^j{\tilde{\partial}\over\partial x_j}+b_j{\partial\over\partial\xi^j})\rangle{\nonumber}\\
&=&\tilde{g}\langle X^i{\tilde{\partial}\over\partial x_i}-g\langle{\nabla_{\partial\over\partial x_i}}X^*,\xi^\#\rangle{\partial \over \partial \xi^i}, a^jg_{jk}{{\partial}\over\partial \xi^k}-b_j g^{jk}{\tilde{\partial}\over\partial x_k}\rangle{\nonumber}\\
&=&-X^ib_i-a^i g\langle\nabla_{\partial\over \partial x_i}X^*,\xi^\#\rangle{\nonumber}\\
&=&-g\langle X^*, Y_2\rangle- g\langle\nabla_{Y_1}X^*,\xi^\#\rangle.
\end{eqnarray}

Now we proceed to prove $\Phi(V)=W\oplus W$.
Let $Z\in TT^*M$ such that $\Phi(Z)=(Y_1,Y_2)\in W\oplus W$.
We claim that $Z\in V$ and it follows that $W\oplus W\subseteq \Phi(V).$
In fact, as $(Y_1,Y_2)\in W\oplus W$, we get
\begin{eqnarray}
g\langle X^*, Y_1\rangle=0, {\label{3.3}}\\
g\langle X^*, Y_2\rangle=0. {\label{3.4}}
\end{eqnarray}

As $X^*$ is a Killing vector field, we get
\begin{eqnarray}
g\langle \nabla_{Y_2}X^*, \xi^\#\rangle=-g\langle\nabla_{\xi^\#}X^*, Y_2\rangle=g\langle\nabla_{\xi^\#}Y_2, X^*\rangle, {\label{3.5}}
\end{eqnarray}
and
\begin{eqnarray}
g\langle \nabla_{Y_1}X^*, \xi^\#\rangle=g\langle \nabla_{\xi^\#}Y_1,X^*\rangle. {\label{3.6}}
\end{eqnarray}

As $M$ is a polar $G$-manifold with a section $\Sigma$, By Proposition {\ref{prop3}}, the slice representation $G_x \times T_x(G\cdot x)^{\perp}\rightarrow T_x(G\cdot x)^{\perp}$ is polar with a section $T_x\Sigma$. Then by Proposition {\ref{prop3}} again, the slice representation
$(G_x)_{\xi^\#} \times W \rightarrow W$ is polar with a section $T_{\xi^\#}T_x\Sigma$.
As $Y_1\in W$, there exists $h\in (G_x)_{\xi^\#}$ such that $hY_1\in T_{\xi^\#}T_x\Sigma$. Hence $Y_1\in h^{-1}(T_{\xi^\#}(T_x\Sigma))=T_{\xi^\#}T_x(h^{-1}\Sigma)\cong T_x(h^{-1}\Sigma).$ We also have $\xi^\#=h^{-1}\xi^\#\in T_x(h^{-1}\Sigma),$ as $\Sigma$ is totally geodesic ({\cite{PalaisTerng}}, Theorem 3.2), so is $h^{-1}\Sigma$.

Then
\begin{eqnarray}
g\langle \nabla_{\xi^\#}Y_1, X^*\rangle=g\langle B(\xi^\#,Y_1),X^*\rangle=0 {\label{3.7}},
\end{eqnarray}

\begin{eqnarray}
g\langle \nabla_{\xi^\#}Y_2,X^*\rangle = g\langle B(\xi^\#, Y_2), X^*)=0 {\label{3.8}}
\end{eqnarray}
where $B(\ ,\ )$ is the second fundamental form of $h^{-1}\Sigma$.

By $(\ref{3.1}),\ (\ref{3.3}),\ (\ref{3.5})$ and $(\ref{3.8})$, we get
$$\tilde{g}\langle X^\#, Z\rangle=g\langle X^*, Y_1\rangle-g\langle\nabla_{Y_2}X^*,\xi^\#\rangle=0.$$

Similary we get
$\tilde{g}\langle X^\#, JZ\rangle=0$. Hence $\tilde{g}\langle JX^\#, Z\rangle=-g\langle X^\#, JZ\rangle=0$.
It follows that $Z\in V$, which implies that $W\oplus W\subseteq \Phi(V)$.

On the other hand, we claim that $\dim{(W\oplus W)}= \dim \Phi(V)$.
In fact,
\begin{eqnarray*}
\dim (W\oplus W) &=& 2\dim W\\
&=& 2(\dim (T_x G\cdot x)^{\perp}-\dim(G_x\cdot \xi^\#))\\
&=& 2(\dim M- \dim G\cdot x-(\dim G_x-\dim(G_x)_{\xi^\#}))\\
&=& 2 \dim M- 2(\dim G- \dim G_{(x,\xi)}),
\end{eqnarray*}
and
\begin{eqnarray*}
\dim \Phi(V)&=& \dim V\\
&=& \dim T^*M- 2 \dim G\cdot(x,\xi)\\
&=& 2(\dim M- \dim G\cdot(x,\xi))\\
&=& 2 (\dim M- (\dim G-\dim G_{(x,\xi)})).
\end{eqnarray*}
Hence $\dim \Phi(V)= \dim{(W\oplus W)}$ and we have $\Phi(V)= W\oplus W$.

Now Lemma {\ref{lemma3.1}} follows from the following commutative diagram

\[\begin{tikzcd}
V \arrow{r}{\Phi} \arrow[swap]{d}{G_{(x,\xi)}} & {W\oplus W} \arrow{d}{(G_x)_{\xi^\#}} \\
V \arrow{r}{\Phi} & W\oplus W
\end{tikzcd}
\]

{\hfill $\square$}\medskip

Given Lemma {\ref{lemma3.1}}, we can now give a proof of Theorem {\ref{mainthm}}. We first show that the following restriction map is surjective:
\begin{eqnarray*}
|_{T^*\Sigma}:\ C^\infty(T^*M)^G\rightarrow C^\infty(T^*\Sigma)^\Pi.
\end{eqnarray*}

$\forall (x,\xi)\in T^*\Sigma$, the Sasaki metric $\tilde{g}$ on $T^*M$ induces an orthogonal splitting
$$T_{(x,\xi)}T^*M=T_{(x,\xi)}G(x,\xi)\oplus J T_{(x,\xi)}G(x,\xi)\oplus V,$$
where $\Phi(V)\cong W\oplus W$ by Lemma {\ref{lemma3.1}}.

The Slice Theorem says that for an open $G$-invariant tubular neighborhood $U_{(x,\xi)}$ of the orbit $G(x,\xi)$, there is a $G$-equivalent diffeomorphism
\begin{eqnarray*}
\exp^{\perp}:\  G\times_{G_{(x,\xi)}}S^{\perp}_{(x,\xi)}(\epsilon)\rightarrow U_{(x,\xi)},
\end{eqnarray*}
where $S^{\perp}_{(x,\xi)}:= JT_{(x,\xi)}G(x,\xi)\oplus V$,
$S^{\perp }_{(x,\xi)}(\epsilon)$ is the $\epsilon$-ball in $S^{\perp}_{(x,\xi)}$
 and $\exp^{\perp}$ is the normal exponential map of $G(x,\xi)$.

Let $U= \underset{(x,\xi)\in T^*\Sigma}{\bigcup} U_{(x,\xi)}$. As $T^*\Sigma$ intersects all orbits in $u^{-1}(0)$ by Proposition {\ref{prop1}}, we see that $U$ is a $G$-invariant open neighborhood of $u^{-1}(0)$.
$\forall f\in C^\infty(T^*\Sigma)^\Pi$, we first show that there exists
$F_{\epsilon}\in C^\infty(U_{(x,\xi)})^G$ such that
\begin{eqnarray}
F_{\epsilon}|_{T^*\Sigma \cap U_{(x,\xi)}}=f|_{T^*\Sigma \cap U_{(x,\xi)}} {\label{3.7}}
\end{eqnarray}

By the existence of $G$-invariant partition of unity subject to the cover
$U= \underset{(x,\xi)\in T^*\Sigma}{\bigcup} U_{(x,\xi)}$,
then there exists $F\in C^\infty(U)^G$ such that $F|_{T^*\Sigma}=f$.
Extending $F$ to $\tilde{F}\in C^\infty(T^*M)^G$, we then prove our desired result.

To prove $(\ref{3.7})$, we first recall some facts on polar representations which we will use. Let $(G, K)$ be a symmetric pair and consider the isotropy representation of $K$ on $\mathfrak{p}= T_K{(G\slash K)}$. It is a polar action and any maximal abelian sub-algebra $\Sigma$ is a section. Its generalized Weyl group $\Pi$ is also called the "baby" Weyl group. Consider the diagonal action of $K$ on $\mathfrak{p}^m$ (respectively $\Pi$ on $\Sigma^m$) and the corresponding algebra of invarant ($m$-variable) polynomials $\mathbb{R}[\mathfrak{p}^m]^K$ (respectivelly $\mathbb{R}[\Sigma^m]^\Pi$). Then we have the following result due to Tevelev {\cite{Tev}}.

\begin{theorem}{\label{thm3.1}}
The restriction map $|_{\Sigma}:\ \mathbb{R}[\mathfrak{p}^m]^K\rightarrow \mathbb{R}[\Sigma^m]^\Pi$ is surjective.
\end{theorem}

As a polar representation is (up to orbit equivalence) the isotropy representation of a symmetric space {\cite{Dadok}}. Theorem {\ref{thm3.1}} generalizes to the class of polar representations {\cite{M2}} (Corollary 2).

\begin{corollary}{\label{coro2}}
Let $K\subseteq O(\mathfrak{p})$ be a linear representation which is also polar with a section $\Sigma$ and generalized Weyl group $\Pi$. Then the restriction is surjective:
$$|_{\Sigma}:\ \mathbb{R}[\mathfrak{p}^m]^K\rightarrow \mathbb{R}[\Sigma^m]^\Pi.$$
\end{corollary}

\begin{corollary}{\label{coro3}}
Let $\mathfrak{p}$ be a polar representation of a compact Lie group $K$ with a section $\Sigma$ and generalized Weyl group $\Pi$. Then the restriction to $\Sigma$ is surjective:
$$|_{\Sigma}:\ C^\infty(\mathfrak{p}^m)^K\rightarrow C^\infty(\Sigma^m)^\Pi.$$
\end{corollary}

{\bf{Proof:}} It's a classical result of Hillbert ({\cite{Sp}}, Proposition 2.4.14) that $\mathbb{R}[\mathfrak{p}^m]^K$ is finitely generated. Let $\rho_1,...,\rho_n$ be generators. By Corollary {\ref{coro2}}, $\rho_1|_{\Sigma},...,\rho_n|_{\Sigma}$ generate $\mathbb{R}[\Sigma^m]^\Pi$.

For any $f\in C^\infty(\Sigma^m)^\Pi$, apply Schwarz's Theorem {\cite{S}} to the action of $\Pi$ on $\Sigma^m$, we get $F\in C^\infty(\mathbb{R}^n)$ such that $f=F\circ \rho|_{\Sigma}$, where
$\rho|_{\Sigma}:\ \Sigma^m \rightarrow \mathbb{R}^n$ be the map whose coordinates are $\rho_1|_{\Sigma},...,\rho_n|_{\Sigma}$.
Then $\tilde{f}=F\circ \rho \in C^\infty(\mathfrak{p}^m)^K$ such that $\tilde{f}|_{\Sigma}=f$.

{\hfill $\square$}\medskip

We can now give a proof of ({\ref{3.7}}). By Lemma {\ref{lemma2.1}}, as $\Sigma$ is totally geodesic in $M$, then $T^*\Sigma$ is totally geodesic in $T^*M$. Hence the normal exponential map $\exp^{\perp}$ of the orbit $G\cdot(x,\xi)$ maps the $\epsilon$-ball $B_{\epsilon}$ in $T_{(x,\xi)}T^*\Sigma{\cong}T_x\Sigma\oplus T_x\Sigma$ diffeomorphically onto $T^*\Sigma \cap U_{(x,\xi)}$.
$\forall f \in C^{\infty}(T^*\Sigma)^\Pi$, $f\circ \exp^{\perp}:\ B_{\epsilon}\rightarrow \mathbb{R}$ is a $\Pi_{(x,\xi)}$-invariant smooth function, where $\Pi_{(x,\xi)}=\{h\in\Pi|\ h(x,\xi)=(x,\xi)\}$. Let $W$ be a polar representation of $K:= G_{(x,\xi)}$ with a section $T_{\xi^\#}T_x\Sigma\cong T_x\Sigma$ defined in Lemma {\ref{lemma3.1}}. By Corollary {\ref{coro3}}, we see that there exists $f_{\epsilon}\in C^\infty(W\oplus W)^K$ such that
$$f_{\epsilon}|_{B_{\epsilon}}=f\circ\exp^{\perp}$$
Hence $f_{\epsilon}\circ(\exp^{\perp})^{-1}=f$ on $T^*\Sigma\cap U_{(x,\xi)}$. Combined with Lemma {\ref{lemma3.1}} and the Slice theorem, then $f_{\epsilon}$ is pulled back to be a smooth function on $G\times S^{{\perp}}_{(x,\xi)}(\epsilon)$ which descends to $F_{\epsilon}\in C^\infty(U_{(x,\xi)})^G$ such that $F_{\epsilon}=f$ on $T^*\Sigma\cap U_{(x,\xi)}$. We finish the proof of the surjectivity part of Theorem {\ref{mainthm}}.

Let $\omega$ be the standard symplectic form on $T^*M$. We show that the restriction to $T^*\Sigma$ preserves Poisson brackets
$(C^\infty(T^*M)^G,\{\ ,\ \}_1)$ and $(C^\infty(T^*\Sigma)^\Pi,\{\ ,\ \}_2)$, where $\{\ ,\ \}_{i=1,2}$ are Poisson brackets induced by $\omega$ and $\omega|_{T^*\Sigma}$ respectively.

Let $\mathring{M}\subseteq M$ be the union of principal orbits and $\mathring{\Sigma}=\Sigma\cap \mathring{M}$.
Then $\mathring{\Sigma}$ is open and dense in $\Sigma$ ({\cite{GZ}} Propsition 1.3). It follows that $T^*\mathring{\Sigma}\subseteq T^*\Sigma$ is also open and dense. $\forall (x,\xi)\in T^*\mathring{\Sigma}$, we have the following orthogonal splitting with respect to the Sasaki metric $\tilde{g}$ on $T^*M$:

\begin{eqnarray}{\label{split}}
T_{(x,\xi)}T^*M\cong T_{(x,\xi)}G(x,\xi)\oplus JT_{(x,\xi)}G(x,\xi)\oplus T_{(x,\xi)}T^*\mathring{\Sigma}.
\end{eqnarray}
To see this, as $\mathring{\Sigma}$ consists of principal orbits, the slice representation at $x\in \mathring{\Sigma}$ is trivial. Hence $G_{(x,\xi)}=G_x$, $\forall(x,\xi)\in T^*\mathring{\Sigma}$. By {\cite{PalaisTerng}}, we also have $\dim G\cdot x+\dim\mathring{\Sigma}=\dim M$.
Then the dimension of the vector space on the right hand side of ({\ref{split}}) is equal to
$$2\dim G(x,\xi)+2\dim\mathring{\Sigma}=2(\dim G\cdot x+\dim\mathring{\Sigma})=2\dim M$$
which finishes the proof of (\ref{split}).

$\forall f \in C^\infty(M)^G$, at $(x,\xi)\in T^*\mathring{\Sigma}$, we can write $$X_f=X+JY+Z,$$ where $X,Y\in T_{(x.\xi)}G(x,\xi)$, $Z\in T_{(x,\xi)}T^*\mathring{\Sigma}$.

Recall that $i_{X_f}\omega=df$, $\omega$ is the standard symplectic form on $T^*M$.
Since $f$ is $G$-invariant, we get $(i_{X_f}\omega)(Y)=df(Y)=0.$
Then $$\omega(X_f,Y)=\tilde{g}(JX_f,Y)=\tilde{g}(JX-Y+JZ,Y)=-\tilde{g}(Y,Y).$$
It follows that $Y=0$.

Now let $f_1,f_2\in C^\infty(T^*M)^G$, at $(x,\xi)\in T^*\mathring{\Sigma},$ we have
$$X_{f_i}=X_i+Z_i,\ i=1,2$$
where $X_i\in T_{(x,\xi)}G(x,\xi)$, $Z_i\in T_{(x,\xi)}T^*\mathring{\Sigma}$.

Let $\hat{f}_1={f_1}|_{T^*\mathring{\Sigma}}$,
then we claim that $X_{\hat{f}_1}=Z_1$.
In fact $i_{X_{\hat{f}_1}}\omega|_{T^*\Sigma}=d{\hat{f}_1}$. $\forall Y\in T_{(x,\xi)}T^*\mathring{\Sigma}$, we have
\begin{eqnarray*}
\tilde{g}\langle X_{\hat{f}_1}, Y\rangle&=&\omega(X_{\hat{f}_1},JY)\\
&=&d{\hat{f}_1}(JY)\\
&=&df_1(JY)\\
&=&\omega(X_{f_1}, JY)\\
&=&\tilde{g}\langle Z_1, Y\rangle.
\end{eqnarray*}

Then at $(x,\xi)$,
\begin{eqnarray*}
\{f_1,f_2\}_1&=&\omega(X_{f_1},X_{f_2})\\
&=&\tilde{g}(JX_1+JZ_1,X_2+Z_2)\\
&=&\tilde{g}(JZ_1,Z_2)\\
&=&\omega|_{T^*\Sigma}(Z_1,Z_2)\\
&=&\{\hat{f}_1,\hat{f}_2\}_2.
\end{eqnarray*}
By continuity, $\{f_1,f_2\}_1=\{\hat{f}_1,\hat{f}_2\}_2$ on
$T^*\Sigma$ everywhere.

{\bf{Proof of Corollary {\ref{coro1}}}:}
As $M$ is a polar $G$-manifold, the inclusion: $\Sigma\slash \Pi \rightarrow M\slash G$ is a homomophism {\cite{PalaisTerng}}. By Chevalley restriction theorem {\cite{PalaisTerng}}, the restriction
$|_{\Sigma}: C^\infty(M)^G\rightarrow C^\infty(\Sigma)^\Pi$ is an isomorphism, which implies that $M/G$ is diffeomophic to $\Sigma/\Pi$.
We now show that $T^*M\sslash G$ is diffeomophic to $T^*\Sigma\sslash \Pi$. By Theorem {\ref{mainthm}}, $C^\infty(T^*M)^G/I^G$ is isomophic to $C^\infty(T^*\Sigma)^\Pi$ as Poisson algebra, where $I^G$ is the ideal of $G$-invariant smooth functions on $X$ vanishing on $u^{-1}(0)$.

It's enough to show that the inclusion $T^*\Sigma\sslash\Pi\rightarrow T^*M\sslash G$ is a homomorphism. By Proposition {\ref{prop1}}, it sufficies to show
$$G\cdot(x,\xi)\cap T^*\Sigma=\Pi\cdot(x,\xi),\ \forall (x,\xi)\in T^*\Sigma.$$

Clearly $\Pi\cdot(x,\xi)\subseteq G\cdot(x,\xi)\cap T^*\Sigma.$
On the other hand, $\forall h_1(x,\xi)\in G(x,\xi)\cap T^*\Sigma$, we have $h_1 x \in G\cdot x\cap \Sigma$ and $h_1\xi^\#\in T_{h_1x}\Sigma$.
By Corollary 4.9 in {\cite{PalaisTerng}}, we get
$$G\cdot x\cap\Sigma= \Pi\cdot x,\ \forall x\in\Sigma.$$
Hence
\begin{eqnarray}{\label{h1h2}}
h_1x=h_2x,\ h_2\in \Pi.
\end{eqnarray}
Then $(h_2^{-1}h_1)x=x$ and so $h_2^{-1}h_1\in G_x$.
Since $(x,\xi)\in T^*\Sigma$, we get $\xi^\#\in T_x(G\cdot x)^{\perp}.$
By Lemma {\ref{lemma3.1}}, the slice representation: $G_x\times T_x(G\cdot x)^{\perp}\rightarrow T_x(G\cdot x)^{\perp}$ is polar with a section $T_x\Sigma$ and generalized Weyl group $\Pi_x$. By Corollary 4.9 in {\cite{PalaisTerng}} again,
$$G_x\cdot\xi^\#\cap T_x\Sigma=\Pi_x\cdot\xi^\#.$$
As $h_2^{-1}h_1\in G_x$, $h_2\in \Pi$, $h_1\xi^\#\in T_{h_1x}\Sigma$, we get $h_2^{-1}h_1\xi^\#\in G_x\cdot\xi^\#\cap T_x\Sigma.$
Then there exists $h_3\in \Pi_x$ such that
$$h_2^{-1}h_1\xi^\#=h_3\xi^\#.$$
Hence $h_1\xi^\#=h_2h_3\xi^\#\in\Pi\cdot\xi^\#.$
Combined with {\ref{h1h2}}, we obtain
$h_1(x,\xi^\#)=(h_2x,h_2h_3\xi^\#)=h_2h_3(x,\xi^\#)\in \Pi\cdot(x,\xi^\#).$
So $G(x,\xi)\cap T^*\Sigma\subseteq \Pi\cdot(x,\xi).$
{\hfill $\square$}\medskip


\section{Statement of results}\label{intro}
\noindent Let us reproduce the setting of \cite{KM} in a slightly more general form. Let $G$ be a
Lie
group
and  $\Gamma$
a lattice in $G$. Denote by $X$ the \hs\ $G/\Gamma$ and by $\mu$ the $G$-invariant probability measure on $X$. In what follows, $\|\cdot\|_p$ will stand for the $L^p$ norm. Fix a basis $\{Y_1,\dots,Y_n\}$
for the Lie algebra
$\frak g$
of
$G$, and, given a smooth
 function $h \in C^\infty(X)$ and $\ell\in{\mathbb{N}}$, define the ``{\sl $L^2$, order $\ell$" Sobolev norm} $\|h \|_{2,\ell}$ of $h $ by
 $$
 \|h \|_{2,\ell} {\, \stackrel{\mathrm{def}}{=}\, } \sum_{|\alpha| \le \ell}\|D^\alpha h \|_2,
 $$
 where

 $\alpha = (\alpha_1,\dots,\alpha_n)$ is a multiindex, $|\alpha| = \sum_{i=1}^n\alpha_i$, and $D^\alpha$ is a differential operator of order $|\alpha|$ which is a monomial in  $Y_1,\dots, Y_n$, namely $D^\alpha = Y_1^{\alpha_1}\cdots Y_n^{\alpha_n}$.
This definition depends on the basis,
however, a change of basis
would only  distort
$ \|h \|_{2,\ell}$
by a bounded factor. We also let
$$C^\infty_2(X) = \{h \in C^\infty(X): \|h \|_{2,\ell} < \infty\text{ for any }\ell = {\mathbb{Z}}_+\}.$$
 Now let $\Delta$ be a {real-valued} function on $X$, and for $z{\in{\mathbb{R}}}$ denote
$$\pd(z) {\, \stackrel{\mathrm{def}}{=}\, }\mu\left(\Delta^{-1}\big([z,\infty)\big)\right).$$
Say that $\Delta$ is {\sl DL\/} (an abbreviation for
``distance-like'') if {there exists $z_0 \in{\mathbb{R}}$   such that  $\pd(z_0) > 0$ and
\begin{itemize}
\item[\rm (a)] $\Delta$ is  uniformly continuous on $\Delta^{-1}\big([z_0,\infty)\big)$; that is,    $\forall\,\varepsilon>0$ there exists a neighborhood $U$ of the identity in ${G}$ such that   for any $x\in X$ with $\Delta(x) \ge z_0$,
$$  g\in U \quad\Longrightarrow \quad|\Delta(x)  - \Delta(gx) | < \varepsilon;$$
\item[\rm (b)] the function $\pd$
does not decrease very fast, more precisely, if
\begin{equation}\label{dl}\tag{DL}
\exists\, c,\delta >
\text{ such that }
{\pd(z ) \ge c  \pd(z - \delta)}
\ \ \ \forall\, z\ge {z_0}.
\end{equation}
\end{itemize}}
The paper \cite{KM} gives several  examples of DL functions on \hs s of semisimple Lie groups. The main goal of that paper was to study statistics of excursions of generic trajectories of   flows on $X$ into sets $\Delta^{-1}\big([z,\infty)\big)
$ for large {enough} $z$. A crucial ingredient of the argument was approximation of characteristic functions of those sets
by smooth functions with uniformly bounded Sobolev norms. However, as was recently observed by Dubi Kelmer and Shucheng Yu,
the argument in the main approximation statement, namely
 \cite[Lemma~4.2]{KM}, contains a mistake. To state a corrected version below,  we
need to weaken the regularity assumption on the smooth functions approximating the sets  $\Delta^{-1}\big([z,\infty)\big)$. Namely, for $\ell\in{\mathbb{Z}}_+$ and $C >
0$, let us say that a  nonnegative function
$h \in C^\infty_2(X) $ is {\sl $(C,\ell)$-regular\/} if
\begin{equation}\label{reg}\tag{REG}
\|h\|_{2,\ell} \le C
\sqrt{\|h\|_1
.
\end{equation}
Note that  the argument of  \cite{KM} used a stronger condition:
\begin{equation}\label{regold}\tag{REG-old}
\|h\|_{2,\ell} \le C  \|h\|_
.
\end{equation}
Equivalently one can replace $\sqrt{\|h\|_1}$ in \eqref{reg} with $ \|h\|_2$, but for technical reasons it is more convenient to use the square root of the $L^1$ norm.
Here is the corrected statement of  \cite[Lemma~4.2]{KM}:

\begin{thm}\label{lem4.2} Let  $\Delta$ be a
DL function on $X$. Then for any $\ell\in{\mathbb{Z}}_+$ there exists  $C > 0$
such
that for every $z {\ge z_0}$ one can find two  $(C,\ell)$-regular nonnegative
functions $h'$ and $h''$ on $X$ such that
\eq{4.2}{
h' \le 1_{\Delta^{-1}([z,\infty))} \le h''\quad \text{and}\quad
\pd(z)
\le \|h'\|_1 \le \|h''\|_1
\le \frac1c
\pd(z),}
with $c$ {and $z_0$} as in \eqref{dl}.
\end{thm}

Fix  a right-invariant
Riemannian metric on $G$ and the corresponding metric `dist' on $X$. For $g\in G$, let us denote by $\|g\|$ the
distance between $g\in G$ and the identity element of $G$. (Note that $\|g\| = \|g^{-1}\|$ due to the right-invariance of the metric.)
Now say  that the $G$-action on  $X$ is {\sl exponentially
mixing}
if there exist   $\lambda ,E > 0$ and $\ell\in{\mathbb{Z}}_+$
such that for any   $\varphi, \psi \in C^\infty_2(X)$  and for any $g\in G$ one has
\begin{equation}\label{em}\tag{EM}
| \langle g\varphi ,\psi \rangle | \le E{e^{ - \lambda \|g\|}\left\| \varphi  \right\|_{2,\ell}} {\left\| \psi  \right\|_{2,\ell}}.\end{equation}
Here $\langle \cdot ,\cdot \rangle$ stands for the inner product in $L^2(X,\mu)$.

One of the main goals of \cite{KM} was, given a sequence $\{f_t: t\in{\mathbb{N}}\}$ of elements of $G$ and a sequence of non-negative functions $\{h_t: t\in{\mathbb{N}}\}$ on $X$ such that $$\sum_{t=1}^\infty \|h_t\|_1
= \infty,$$ compare the growth of $\sum_{t=1}^N  h_t(f_tx) $ for $\mu$-a.e.\ $x\in X$ with the growth of $\sum_{t=1}^N\|h_t\|_1
$ as $N\to\infty$. Results like that usually go by the name `dynamical Borel-Cantelli lemmas', see \cite{CK, HNPV}. In \cite[Proposition 4.1]{KM} such a conclusion was shown to follow from the exponential mixing of the $G$-action on $X$, the {\sl exponential divergence} of $\{f_t\}$, namely the condition
\begin{equation}\label{ed}\tag{ED}
\sup_{t\in{\mathbb{N}}}\sum_{s=1}^\infty e^{-\lambda\|f_sf_t^{-1}\|} < \infty \quad
\forall\,\lambda > 0,
\end{equation}
and the  regularity assumption \eqref{regold} on functions $\{h_t\}$.

\smallskip
In the following theorem we weaken the regularity condition \eqref{regold} to \eqref{reg} and
derive  the same conclusion:

\begin{thm}\label{prop4.1}
Suppose that
the $G$-action on $X$ is exponentially mixing.
Let $\{f_{t} : {t}\in {\mathbb{N}}\}$ be a sequence of elements of $G$ satisfying \eqref{ed},    and let  $ \{h_t : {t}\in {\mathbb{N}}\}$
be a
sequence of non-negative  $(C,\ell)$-regular functions on $X$ such that $\|h_t\|_1 \le 1$ for all $t$,  and $\sum_{t=1}^\infty \|h_t\|_1
= \infty$. Then
$$
\lim_{N\to\infty}\dfrac{\sum_{t=1}^N  h_t(f_tx)}{\sum_{t=1}^N \|h_t\|_1
}
= 1 \quad\text{for
$\mu$-a.e.\
$x\in X$.}
$$
\end{thm}

Using the above theorem in place of   \cite[Proposition 4.1]{KM} and Theorem \ref{lem4.2}
 in place of \cite[Lemma  4.2]{KM}, one can then recover
\cite[Theorem  4.3]{KM}, that is, prove

\begin{thm}\label{thm4.3}
Suppose that
the $G$-action on $X$ is exponentially mixing.
Let $\{f_{t} : {t}\in {\mathbb{N}}\}$ be a sequence of elements of $G$ satisfying \eqref{ed},     let  $\Delta$ be a DL function on $X$, and let {$\{r_{t} : {t}\in {\mathbb{N}}\}\subset [z_0,\infty)$} be
such that
\eq{div}{
{\sum_{{t}=1}^{\infty} \pd(r_{t})} = \infty
.}
Then for some positive
$c \le 1$ and for  almost
all $x\in X$ one has
$$
 c \le \liminf_{N\to\infty}\frac {\#\{ 1 \le {t}\le N\mid \Delta(f_{t}x)
\ge r_{t}\}}  {\sum_{t=1}^{N} \pd\big(r_{t}\big)} \le \limsup_{N\to\infty}\frac {\#\{ 1 \le {t}\le N\mid \Delta(f_{t}x)
\ge r_{t}\}}  {\sum_{t=1}^{N} \pd\big(r_{t}\big)} \le \frac1c.
$$
{Consequently, for any $\{r_{t} \}$ satisfying \equ{div} and almost all $x\in X$ one has $\Delta(f_tx) \ge r_t$ for infinitely many $t\in{\mathbb{N}}$. That is, in the terminology of \cite{KM}, the family of sets
$$\{\Delta^{-1}\big([z,\infty)\big): z\in{\mathbb{R}}\}$$ is Borel-Cantelli for $\{f_{t}\}$.}
\end{thm}

 \bigskip
\noindent { {\bf Acknowledgements}. The authors are grateful to  Dubi Kelmer for bringing their attention to the mistake in \cite[Lemma  4.2]{KM}, and to Shucheng Yu for a suggestion how to correct it. Thanks are also due to Nick Wadleigh for helpful discussions.}

\section{Proofs}\label{proofs}
 Let us state a general form of Young's inequality, whose proof we give for the sake of self-containment of the paper.  Denote by $m$ the Haar measure on $G$ normalized so that the quotient map $G\to X$ locally sends $m$ to $\mu$. For $\psi\in L^1(G, m)$ and $h\in L^1(X, \mu)$, define $\psi\ast h$ by $$(\psi\ast h)(x) {\, \stackrel{\mathrm{def}}{=}\, }  \int_G \psi(g)h(g^{-1}x) \,dm(g).$$

 \begin{lem}\label{young}  Let $\psi\in L^1(G, m)$ and $h\in L^2(X, \mu)$. Then $\|\psi\ast h\|_2\leq \|\psi\|_1  \|h\|_2.$ \end{lem}

 \noindent{\it Proof.} We have
  \begin{equation*}
 \begin{split}|(\psi\ast h)(x)| &\leq \int_G|\psi(g)|^{1/2}  |h(g^{-1}x)|\cdot |\psi(g)|^{1/2} \,dm(g)\\ \text{(by Cauchy-Schwarz)} &\le \left(\int_G |\psi(g)| \,dm(g)\right)^{1/2}  \left( \int_G|h(g^{-1}x)|^2   |\psi(g)| \,d m(g) \right)^{1/2}.\end{split}
 \end{equation*}

  Thus we have $$|(\psi\ast h)(x)|^2 \leq \|\psi\|_1  \int_G |\psi(g)|\cdot  |h(g^{-1}x)|^2  \,d m(g).$$ Integrating over $X$ and using Fubini's Theorem gives
 \begin{equation*}
 \begin{split}
 \|\psi\ast h\|_2^2&\leq \|\psi\|_1  \int_X\int_G|\psi(g)|\cdot |h(g^{-1}x)|^2\,dm(g)\,d\mu(x)\\ &=\|\psi\|_1\int_G|\psi(g)|\int_X|h(g^{-1}x)|^2\,d\mu(x)\,dm(g),
 \end{split}
 \end{equation*}
which, by the $G$-invariance of $\mu$, is the same as $$\|\psi\|_1\int_G|\psi(g)|\,dm(g)\int_X|h(x)|^2\,d\mu(x)= \|\psi\|_1^2 \cdot \|h\|_2^2.\qquad\qquad\qquad \qquad\qquad\qed$$

  \begin{proof}[Proof of Theorem \ref{lem4.2}] We follow the
proof of  \cite[Lemma  4.2]{KM}.
 For  $z\in{\mathbb{R}}$, let us
use the notation $$A(z) {\, \stackrel{\mathrm{def}}{=}\, } \Delta^{-1}\big([z,\infty)\big).$$ Then, for  $\varepsilon  > 0$, let us
denote by $A'(z,\varepsilon )$ the set of all points of $A(z)$ which are not
$\varepsilon $-close to $\partial A(z)$, i.e. $$A'(z,\varepsilon ){\, \stackrel{\mathrm{def}}{=}\, } \{x\in A(z) :
\text{dist}\big(x,\partial A(z)\big) \ge \varepsilon \},$$ and  by $A''(z,\varepsilon )$ the
$\varepsilon $-neighborhood of  $A(z)$, namely
$$A''(z,\varepsilon ){\, \stackrel{\mathrm{def}}{=}\, } \{x\in X : \text{dist}\big(x,
A(z)\big) \le \varepsilon \}.$$
Choose {$z_0$,} $\delta$ and $c$ {as in}  (DL).   Then, using the
uniform continuity of $\Delta$ {on $\Delta^{-1}\big([z_0,\infty)\big)$}, find $\varepsilon  > 0$ such that
$$
|\Delta(x) -
\Delta(y)| < \delta\text{ whenever }{\Delta(x) \ge z_0 \text{ and }}{\rm dist}(x,y) < \varepsilon
$$
It
follows that for all $z{\ge z_0}$,
$$
A(z+\delta) \subset A'(z,\varepsilon )\subset A(z)\subset A''(z,\varepsilon )\subset
A(z-\delta);$$ therefore  one
can apply (DL) to conclude that
\eq{4.4}{
c \mu\big( A(z)\big)\le \mu\big( A'(z,\varepsilon )\big) \le \mu\big(
A''(z,\varepsilon )\big) \le \frac1c\mu\big(A(z)\big)
}

Now  take a non-negative $\psi\in C^\infty(G)$ of $L^1$ norm $1$ such that
supp$\,\psi$ belongs to the ball of radius $\varepsilon /4$ centered in $e\in G$.
 Fix  $z{\ge z_0}$ and consider   functions $h' {\, \stackrel{\mathrm{def}}{=}\, } \psi* 1_{A'(z,\varepsilon /2)}$ and $h'' {\, \stackrel{\mathrm{def}}{=}\, } \psi* 1_{A''(z,\varepsilon /2)}$. Then one clearly has
$$
1_{A'(z,\varepsilon )} \le h' \le 1_{A(z)} \le h'' \le 1_{A''(z,\varepsilon )}
$$
which, together with  \equ{4.4},
 immediately implies \equ{4.2}.  It remains to choose $\ell\in{\mathbb{Z}}_+$ and find $C$ {(independent of $z$)} such that both $h'$ and $h''$ are $(C,\ell)$-regular. Take a multiindex $\alpha$ with $|\alpha| \le \ell$, and write $$\|D^\alpha h'\|_2 =  \|D^\alpha(\psi* 1_{A'(z,\varepsilon /2)})\|_2  = \|D^\alpha(\psi)*
1_{A'(z,\varepsilon /2)}\|_2.$$ Then, by the Young
inequality,
$$
\|D^\alpha h'\|_2 \le \|D^\alpha(\psi)\|_1   \sqrt{\mu\big(A'(z,\varepsilon /2)\big)}  \le
\|D^\alpha(\psi)\|_1  \sqrt {\mu\big(A(z)\big)}  \le
\|D^\alpha(\psi)\|_1\left(\frac{\|h'\|_1}{c}\right)^{1/2}
.
$$
Similarly, $$\|D^\alpha h''\|_2 \le \|D^\alpha(\psi)\|_1 \sqrt{\mu\big(A''(z,\varepsilon /2)\big)} \le \|D^\alpha(\psi)\|_1
\left(\frac{\mu\big(A(z)\big)}{c}\right)^{1/2}\le
\|D^\alpha(\psi)\|_1
\left(\frac{\|h''\|_1}{c}\right)^{1/2};$$
hence, with $C = \frac1{\sqrt{c}}\sum_{|\alpha|\le \ell}\|D^\alpha(\psi)\|_1$, both  $h'$ and  $h''$ are
$(C,\ell)$-regular, and the theorem is proven.
 \end{proof}

  \begin{proof}[Proof of Theorem \ref{prop4.1}] Denote $\int_X h_{t}\,d\mu = \|h_t\|_1$ by $a_t$. Following the argument in  \cite{KM}, our goal is to show that the sequence of functions $\{h_t\circ f_t\}$ satisfies a second-moment condition dating back to the work of Schmidt and
  Sprind\v zuk:
\begin{equation}\label{sp}\tag{SP}
\sup_{1\le M < N}\frac{\int_X\Big(\sum_{{t} = M}^{N}
h_{t}(f_tx) - \sum_{{t} = M}^{N}a_t  \Big)^2\,d\mu}{
\sum_{{t} = M}^{N}a_t} < \infty.
\end{equation}
the conclusion of the theorem will then follow in view of  \cite[Lemma  2.6]{KM}, which is a special case of \cite[Chapter I, Lemma 10]{Sp}.

\smallskip
Take $1 \le M < N$. As in \cite[Remark  2.7]{KM}, one can rewrite the numerator as \linebreak
$\sum_{s,t=M}^N \left(\langle f_t^{-1}h_t, f_s^{-1}h_s\rangle -a_sa_t\right)$, and then estimate it using the exponential mixing of the $G$-action on $X$:
\begin{equation*}\begin{split}
\left|\sum_{s,t=M}^N  \langle f_t^{-1}h_t, f_s^{-1}h_s\rangle -a_sa_t\right| &\le \sum_{s,t=M}^N \left|\langle f_s f_t^{-1}h_t,  h_s\rangle -a_sa_t\right| \\
\text{(with $E,\lambda,\ell$ as in \eqref{em}) } &\le E\sum_{s,t=M}^N  {e^{ - \lambda \|f_s f_t^{-1}\|}\left\| h_t  \right\|_{2,\ell}} {\left\| h_s  \right\|_{2,\ell}}\\
\text{(by the $(C,\ell)$-regularity of $\{h_t\}$) }&\le EC^2 \sum_{s,t=M}^N   {e^{ - \lambda \|f_s f_t^{-1}\|} \sqrt{a_sa_t}}.
\end{split}\end{equation*}
Now, following an observation communicated to us by Shucheng Yu, split the above sum according to the comparison between $a_s$ and $a_t$:
\eq{3sums}{  \sum_{a_s = a_t}  e^{ - \lambda \|f_s f_t^{-1}\|} \sqrt{a_sa_t}   \ +   \sum_{a_s < a_t}  e^{ - \lambda \|f_s f_t^{-1}\|} \sqrt{a_sa_t} \   +     \sum_{a_s > a_t}  e^{ - \lambda \|f_s f_t^{-1}\|} \sqrt{a_sa_t},}
where the values of $s,t$ in the last three sums range between $M$ and $N$.  By symmetry, the last two sums are equal. Thus \equ{3sums} is not greater than
\begin{equation*}\begin{split}
  \sum_{a_s = a_t} e^{ - \lambda \|f_s f_t^{-1}\|} a_t  +   2\sum_{ a_s < a_t}  e^{ - \lambda \|f_s f_t^{-1}\|} a_t &\le 2  \sum_{s,t=M}^N e^{ - \lambda \|f_s f_t^{-1}\|} a_t \\
&\le 2  \sum_{t=M}^N a_t  \sum_{s=M}^Ne^{ - \lambda \|f_s f_t^{-1}\|}
\\
&\le 2  \sum_{t=M}^N a_t \cdot \sup_{t\in{\mathbb{N}}}\sum_{s=1}^\infty e^{ - \lambda \|f_s f_t^{-1}\|}\ ,  \end{split}\end{equation*}
and the proof of \eqref{sp} is finished in view of \eqref{ed}.
 \end{proof}