Download Morse Theory is a part pf differential geometry, concerned with

Morse Theory is a part of differential geometry, concerned with Riemannian (i.e.
smooth and compact) n-dimensional manifolds M , and differentiable maps. We study
smooth real-valued functions f : M  , and importantly, their critical points.
A point c  M , with c  (c1 ,..., cn ) is a critical point of the function f if
f
( x1 ,..., xn )  0 , for all i  1,..., n.
xi
x c
A critical point is called non-degenerate if and only if the Hessian is non-singular. i.e.
2 f
xi x j
 0 , for all i, j  1,..., n.
x c
A smooth function f is labelled a Morse function if and only if has a finite number
of non-degenerate critical points.
The Morse Lemma states (from Topology and Geometry for Physicists, Nash and
Sen):
If c  M is a non-degenerate critical point of a smooth, real-valued function f  M ,
where M is smooth and compact of dimension n, then there exists a local coordinate
system ( x1 ,..., xn ) in a neighbourhood of c  M such that
f  f (c)  ( x1 ) 2 
 ( xk ) 2  ( xk 1 ) 2 
 ( xn ) 2
holds throughout the neighbourhood. k denotes the number of negative terms present,
is said to be the index of f at c.
??PROOF OF LEMMA??
Exploiting properties of supersymmetric quantum mechanics and ideas from quantum
mechanical perturbation theory, one can establish a very surprising connection with
Morse Theory.
The Morse Inequalities describe a relationship between critical points of a Morse
function, M p ( f ) , and the Betti numbers of a Riemannian manifold, B p ( M ).
M p ( f ) is defined as the number of critical points of f , with index p.
To define B p ( M ) , we first need some requirements/definitions. Let M be the
Riemannian manifold and let  p ( M ) be the space of smooth p-forms of M with
0  p  n. Let d :  p   p 1 denote exterior differentiation. We define the p-th Betti
number of M , B p ( M ), as the dimension of the p-th (de Rham) cohomology
H p (M , )
 ker d n1 
Bp ( M ) : dim H p ( M , )  dim 

 Im dn 
Another equally useful definition of the Betti number is that B p is equal to the
number of zero eigenvalues of the Laplacian H, acting on p-forms in M.
The fundamental result of Hodge Theory is that the (cohomology) space H p is
isomorphic to the space of p-forms annihilated by the Hamiltonian H.
[Quoted from Schrödinger Operators, Cycon, Froese, Kirsch and Simon]
ker dn1
 , as required.
This statement implies B p  dim H p  dim 

Im
d
n

Hodge Theory is also a useful tool to obtain the definition of the inner product on
forms. We will also define the Hodge operator, which will be used in the proof of the
Morse Inequalities.
If we choose a Riemannian metric on the manifold M, denoted gm , this implies that
for all m  M , we have an inner product gm (, ) on the tangent space T  ( M ). For
smooth vector fields on the manifold, the inner product is also a smooth function.
The usual inner product on M is denoted ,  . We want to define this inner product on
the space of p-forms, for each form and also the Hamiltonian operator. If we choose
an orthonormal basis for T  ( M ) given by 1 ,...,n  , these elements can be used to
form a basis on the space of p-forms ( M ) . This basis is given by
i1 i2 
i p
1  i1  i2 
 ip  n
Choosing our desired basis to be orthonormal, we obtain the inner product on pforms. This inner product is independent on the choice of orthonormal basis on the
tangent space. We can then deduce this inner product holds on all tangent space to the
manifold.
We can now introduce the Hodge star operator  , acting on forms. The Hodge
operator is a linear operator mapping p-forms to (n- p)-forms. i.e.
 :  p (M )  n p ( M ) . Using our inner product on p-forms, we can define this
operator. Given   p ( M ) , there exists a unique (n-p)-form  n p (M ) , such
that for every  n p (M ) ,
 ,      ,
holds.   1   n is the volume n-form, with 1 ,...,n an orthonormal frame of
the tangent space. The Hodge star operator has the follow useful properties:
i.         
ii.   p (M ),  n p (M )   ,   (1) p ( n p )  , 
iii.   p (M )  ( )  (1) p ( n p ) 
We can now state the Morse Theorem:
Let M be a Riemannian n-dimensional manifold, and f : M 
M. Then,
(i). For all p  0,1,..., n,
Bp (M )  M p ( f )
(ii). For all
k
 (1)
k p
p 0
k  0,1,..., n,
we
have
the
weak
a Morse function of
Morse
Inequalities
we have the string Morse Inequalities
k
Bp (M )   (1)k  p M p ( f )
p 0
We know we can derive the Betti numbers of M on a supersymmetric quantum
mechanical system by finding the number of zero eigenvalues of H acting on p-forms.
It is very difficult to compute the spectrum of H. it was Witten’s observation that if
we deform the system with respect to t  and a Morse function f, the Betti numbers
remain t-independent but the spectrum of H t is much simpler to derive.
This deformation is the first step in proving the Morse Inequalities using the methods
of Witten. The exterior derivatives d , d * and the Laplacian H    dd *  d *d are
replaced by
dt  etf detf
dt*  etf d *etf
H t : dt dt*  dt*dt
Qt  d t  d t
where t 
and f is a Morse function.
 
The Betti number for the deformed system is given by Bp (t )  dim H dpt . dt differs
tf
from d only by conjugation with e . Thus
ker(dt )  e  tf ker(d ) and
Im(dt )  e tf Im(d ).
 etf ker(d ) 
This implies Bp (t )  dim  tf

 e Im(d ) 
 ker(d ) 
 dim 
  Bp
 Im(d ) 
We have proved that the Betti number of the deformed Laplacian is t-independent and
so equal to the usual Betti number B p . As B p  B p (t ), we can also deduce that the
number of zero eigenvalues of H t is equal to the number of zero eigenvalues at t  0.
This is important due to the fact that the limit of H t as t goes to infinity simplifies.
The study of H t as t goes to infinity shows us that we can place upper bounds on B p ,
in terms of the critical points of the Morse function f. This is the method Witten uses
to reproduce the Morse Inequalities.
We need to find an expression for H t which exposes the role of the critical points of
f.
Let xi i  1,..., n be a coordinate system in some neighbourhood of the manifold M.
Define a i* acting on p-forms by
ai*  dxi   , where   H
a i is the adjoint of a i* . Thus,
a i dx j1 
p
 dx p   (1) k g ijk dx j1 
j
 dx jk 1  dx ik  dx jk 1 
 dx
jp
k 1
This is the same as
a i  (1) np  n 1   dxi   
where  is the Hodge star operator.
g ij is the metric on the tangent space T*. a i and a i* are the zero-th-order operators,
reminiscent of fermionic creation-annihilation operators given respectively by exterior
and interior multiplication. We can deduce the following properties:
a , a   g
i
i*
ij
, and
dxi , dx j  g ij
If we choose a flat orthonormal space for this coordinate system, then g ij   ij . This
will be an important requirement during the following calculations.
We can write dt , dt* and H t in terms of a i , a i* and the original exterior derivatives
and Laplacian. Now, for all p-forms   H:
dt  e  tf d (etf  )
 etf etf d  etf etf tdf  
 d  tdf  
n
But df  
i 1
f
dxi
i
x
So d  tdf    d  t 
i
f i
dx  
xi
f


  d  t  i a i*  
i x


This implies dt  d  t  fi ai* , where f i 
i
f
.
x i
Similarly, dt*  etf d * (e  tf  )
 d *  tdf  
 d *  t  ( fi )dxi  
i
Thus, dt*  d *  t  fi ai .
i
Now we can use these new definitions to find a new expression for
H t  dt dt*  dt*dt  dt , dt* .




 
dt , dt*   d  t  f i ai*  ,  d *  t  f j a j  



i

 
j

 
   t 2  fi f j ai* , a j  tA ,
i, j
Where A is another 0th-order operator, given by
A  A1  A1* , with


A1  d ,  f j a j  .
 j

Now we use the requirement that the coordinate system is on a flat orthonormal space.
i.e. g ij   ij and a i* , a j    ij .
So, H t    t 2  fi f j ij  tA
ij
   t (f )2  tA
2
2
 f 
as  fi f j    i    f  .
i, j
i  x 
2
ij
we want to derive a more useful expression for the operator A. So, let d   a i* i ,
i

where  i  i .
x


Thus, A1  d ,  f j a j 
 j



   a i * i ,  f j a j 
j
 i

i*
j
 a , f j a  i  ai* i , f j a j 
i, j
  fi  i   a i*a j f ij .
i
i, j
This implies A1 is equal to the following expression
A1    i ( fi )   f ji a ja i
i
ij
This gives us a new expression for A, as shown below:
A  A1  A1   fi  i   aia j fij   fi  i    fi    f ji a ja i
2
i
ij
i
i
ij
 2 fij a a    fi  , (since f ij is symmetric)
i
2
j
ij
i
  fij  2a a   ij 
i
j
ij
But we know  ij  a i , a j  . So the final expression for A is
A   fij ai , a j 
ij
So we have found a new expression for H t with respect to zero-order operators:
H t    t 2  f   t  fij a i* , a j 
2
i, j
2 f
. This expression of H t has a direct connection with the critical
xi x j
points of f, as desired.
where fij 
Since f is a Morse function, i.e. a function with a finite number of non-degenerate
2
critical points, this implies that  f  vanishes at a finite number of points. Let
c 
(a)
k
a 1
be the critical points of f with index equal to a. Therefore, by the Morse
Lemma, we can write f in such a way with coordinate system ( x1 ,..., xn ) as
f  f (c(a) )  ( x1)2 
 ( xind (a) )2  ( xind (a)1)2 
 ( xn )2
in a neighbourhood of each c ( a ) on M.
If we require dx1 ,..., dx n to be orthonormal, we obtain a metric in a neighbourhood of
each critical point c ( a ) . Using partition of unity, we can ‘patch’ together such metrics
of each critical point to obtain a metric on the whole of the manifold, M.
We can write H t in terms of local coordinates in a neighbourhood of the critical point
c(a) .
 f f
H t    t 2   ij i j
 x x
2 f

ai* , a j 

t

i
j 

x

x

We get, (for H t in terms of the local coordinates of a critical point):
H t( a )
   4t
2
i
xi2
 2t
ind ( a )

i 1
n
n
i 1
i 1
   4t 2  xi2  2t  i ai* , ai 
if i  1,..., ind (a )
1,
with i  
 1, if i  ind (a)  1,..., n
 acts on p-forms as follows:
n

 ai* , ai   2t
 ai* , ai 




i ind ( a ) 1
 2 f i1
dx 
2
i 1 xi
n
 dxip )  
( fdxi1 
 dxip
2
This implies that  is equivalent to 2 .
xi
Therefore, H t( a ) can be rewritten as
 2

    2  4t 2 xi2  2ti ai* , ai   .
i 1  xi

n
H
(a)
t
n
n
i 1
i 1
Define K ( a ) as    4 xi2  A( a ) , with A( a )  2 i  ai* , ai  .
To derive the eigenvalues of H t , we must first find the eigenvalues of K ( a ) (which
are easier to derive). The eigenvalues of H t will then be given in terms of the
spectrum of  a K ( a ) , for all a  1,..., ind (a) .
On all p-forms dxi1 
n
 dx p ,    4 xi2 acts the same way. i.e. the expression acts
i
i 1
n
as a scalar operator. This implies the eigenvalues of    4 xi2 are the harmonic
i 1
n
oscillator eigenvalues, given by 2 1  2ni  , ni  0,1,
i 1
Each corresponding independent eigenform is of the type
 dx i 
1
 dx p , 1  i1  i2 
i
 i p  n,
where  is the harmonic oscillator eigenfunction. Now,
a , a fdx 


i*
i
i1


 fdxi1   dxi p , if i  i ,..., i
p
1
 dx  
i
i
 fdx 1   dx p , if i  i1,..., i p

ip
n


This implies A( a )  2 i  ai* , ai  acts diagonally on the eigenforms mentioned
i 1
earlier.
i.e. A(a) dx 1 
i
 dxip
n
 2 i ai* , ai  dxi1 
 dx p
i
i 1
 n
i1
 2 
i dx 
  i n1
2  dxi1 
 i 1 i


 dxi p , if i i1,..., i p 

 dxi p , if i  i1,..., i p 
n
 2 i i dxi1 
 dx p .
i
i 1
where
i


1, if i  i ,..., i
p
1
 
1, if i  i1,..., i p



are the eigenvalues of A( a ) .
Therefore the spectrum of K ( a ) is given by


n
 ( K ( a ) )  2   (1  2ni )  i i 
 i 1

Where  denotes the spectrum.
n
We want to see when the eigenvalues are equal to zero. Now,

i 1
i i
 n, with
equality if and only if the index of a is equal to p, and i  sign( i ) , (implying
 1, if i  1,..., p
.
1, if i  p  1,..., n
i  
With this information, we can see that K ( a ) has a zero eigenvalues only when ni  0,
for all i. Therefore we know


0, if p  ind (a)
dim ker K ( a )  p ( M )  

1, if p  ind (a)
So we see there is one zero eigenvalue for each K ( a ) , only when the index of the
critical value is equal to p. We are required to ‘patch’ together each K ( a ) to describe a
metric over all critical points of f on the manifold.
Take  K ( a ) to be this metric. The spectrum is then given by
a


  K (a) 
  K (a) 
a
a
We can now deduce that
 ker  K ( a )

a
dim 
(a)   M p ,

Im  K 
a


where m p is the number of critical points c ( a ) with index equal to p.
We now want to find an approximation for dimker  Ht  with respect to the
calculations we have just made. A theorem from Schrödinger Operators, (Cycon,
Froese, Kirsch, Simon), states
Let Ht and  K ( a ) be as above. Let Enp (t ) denote the nth eigenvalues of H t acting
a
on p-forms, counting multiplicity. Denote enp as the nth eigenvalue of  K ( a ) acting on
a
p-forms, counting multiplicity. Then, for fixed n and large t, H t has atleast n
eigenvalues and
lim
t 
Enp (t )
 enp .
t
As t goes to infinity, the number of zero eigenvalues of H t must be less than or equal
to the number of zero eigenvalues of  K ( a ) . I.e.
a
 ker  K ( a )

ker H t


a
dim 
 dim 

(a) 
Im H t 

Im  K 

a


but the left hand expression is equal to the Betti number B p ( M ) , and the second
expression is equal to M p ( f ) . This implies
Bp (M )  M p ( f )
This proves the weak Morse Inequalities.
Now the strong Morse Inequalities will be proved. We know the first M p eigenvalues
of  a K ( a ) are zero. This implies eMp p 1 is the first non-zero eigenvalues of the
operator. The corresponding eigenvalue of H t is given by
lim
t 
EMp p 1 (t )
t

 eMp p 1 , where Enp (t ) grows like t for all n  M p  1. Of the

eigenvalues E1p (t ),..., EMp p (t ) , we know the first B p are zero, due to the weak Morse
Inequalities. This leaves the infinitesimally small collection of low-lying eigenvalues
E
p
B p 1

(t ),..., EMp p (t ) . One can deduce that these low-lying eigenvalues come in even-
odd pairs. I.e. for all Enp (t ) with p odd, there exists Enp' ' (t ) with p’ even. Both with
B p 1  n  M p .
To prove the Strong Morse Inequalities, we must first define Rtp as the ( M p  B p ) dimensional space of low-lying eigenvalues of H t . Recall, Qt : dt  dt , acting on
( M ) and Qt2  H t . Qt acts on p-forms as follows (by definition of the exterior
derivative acting on forms):
Qt :  p ( M )   p 1 ( M )   p 1 ( M )
Qt acts on Rtp the same way as on the space of forms.
Qt is one-to-one on np 0 Rtp   Rt0  Rt1 
 Rtn  . I.e. if Qt : Rtp  Rtp 1  Rtp 1 and
Qt : Rtq  Rtq 1  Rtq 1 , for some p and q, but the resulting space or equivalent, this
implies p=q. This implies that if p is even, Qt takes the even space to the space of odd
eigenvalues. Also, if p is odd, Qt takes the odd space to the space of even
eigenvalues. So,
2j
2 j 1
p even
p 0
p odd
p 1
2 j 1
2j
p odd
p 1
p even
p 0
Qt :  Rtp   Rtp
Qt :  Rtp   Rtp
This implies the dimension of the spaces on the right hand side is greater than that of
the dimensions of the left hand side spaces. Therefore,
 M 0  B0  
 M1  B1  
  M 2 j  B2 j    M 1  B1  
  M 2 j 1  B2 j 1    M 0  B0  
which are the strong Morse Inequalities.
[1800]
  M 2 j 1  B2 j 1 
  M 2 j  B2 j  ,