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Heat kernel proof of A-S index theory
Hu xiyu
July 10, 2016

1
1
...
g is the Levi-civita connection on T M ,R = Rg ∈ Ω2 (End(T M ))
...


by chern-weil theory,we know:
1
...

2
...

so we can def Ag = A(M ) ∀g is a metric on M
...
2

dirac bundle

(E, E ) is a dirac bundle
...

F E/S ∈ EndCl(M ) (E) the twisting curvature of E
...
3

supertrace

StrE/S : Endcl(M ) → CM
...
∀w ∈ Ω(M )
...

1

1
...

2π

by chern-weil theory,we know:
1
...

2
...

F E ⊗W/S=F

E/S

⊗ 1W + 1E ⊗ F W
...

+

−

ch(E⊗W )/S , ch(E ⊗W ) = chE/S (E/S)(ch(F W ) − ch(F W ))
...


1
...

M

2
2
...

1
complete Φ(d) with the norm
...

The extension of operator d in H 1 (M ) is called d,assume dc is d restrict on
∞
1
1
∞
C0 (M ),dc extend to H0 (M ) called dc ,H0 (M ) is Co (M ) complete with the
1 (M ) ⊂ H 1 (M )
...
1 , of course H0
1
Sobelev theory tell us, if M is a complete Riemann manifold,then H0 (M ) =
H 1 (M )
...

Φ(δ) = {w ∈ C ∞ 1 − f orm|
2

|w|2 +

|δw|2 < +∞}

By a lemma from Gaffney (Ann
...

the laplace operator(with Direchlet boundary condition or Neumann boundary condition) is ∆D = δdc , ∆N = δ C d, When M is with smooth bound1
ary,then ∆ = ∆N = general∆oprator,assume M is complete (H0 (M ) =
1 (M )),then because d = d, δ
H
C
C = δ,Gaffney proved(Ann
...

As we all known, ∆ is self-adjoint operator,so e−∆t make up a bounded selfadjoint operator semi group,by the self-adjoint operator theory,if dEλ is the
spectrum measure of ∆,then
∞
−∆t

e

dλt dEλ , t > 0

=
0


...
when f ∈ L2 (M ),

∞
i

∆ (e

−∆t

λi e−∆t dEλ (f )
...
2

⊂ C ∞ (M ) is basic on Weyl theory
...
1: assume M is a complete manifold,then there exist a heat kernel
H(x, y, t) ∈ C ∞ (M ×M ×R+ ),and (e−∆t f )(x) = M H(x, z, t−s)H(z, y, s)dz,∀f ∈
L2 (M ),satisfied:
(1)H(x, y, t) = H(y, x, t)
...

t→0+

∂
(3)(∆ − ∂t )H = 0
...


Proof:
(A) first proof, ∀f ∈ L2 (M ), e−∆t f ∈ C ∞ (M × R+ )
...

0

in fact,we have:
∂
−∆t f )
∂t (e
∞

∞

0

0
∞

= lim 1 [ e−λ(t+ ) dEλ (f ) −
→0

A

= lim [

→0 0
A

e−λ −1 −λt
e dEλ (f )

e−λt dEλ (f )]

+
A

e−λ −1
λe−λt dEλ (f )]
λ

−λe−λt dEλ (f ) + O(Ae−At f )

=
0

∞

−→

−λe−λt dEλ (f ) as A → +∞
...

∂
so what have we proved ,in fact we proved in the classical sence, ∂t (e−∆t f )
exist
...

in fact to proof

∂
−∆t f )
∂t (e

∞

=

−λe−λ dEλ (f ) (in weak sence),we need only

0

∞
to proof:∀ψ ∈ C0 (M × R+ ) we have:
∞

∂ψ −∆t
(e
f) = −
∂t

−λe−λt dEλ (f ))

ψ(
0

but because of ( ),this is obvious,and similar we can proof that (in the weak
sence):
∂2
(∆ + 2 )i (e−λt f ) =
∂t

∞

(λ + λ2 )i e−λt dEλ (f )
0

rmk: there is some thing to explain,why ∆(e−λt f ) =

∞

λe−λt dEλ (f )
...
and
we have proved e−λt f ∈ ∞ Φ(Li ),and we know that ∞ Φ(Li ) ⊂ C ∞
...


4

and now we easy to observe that,
if f1 (x, t) = e−λt f ,then:
∂
f1 = −
∂t

∞

λe−λt dEλ (f ) = −∆(e−λt f ) = ∆f1
0


...

Rmk:f1 (x, t) ∈ C ∞ ,so derivatives is in classical sense
...

M

∞
by the decomposition of unity,we can assume f ∈ C0 (M ),and suppf sufficed small
...

and ∀N > 0, lim ◦x P (x, y, t) = O(tN ),and when d(x, y) suffice small,t →
t→0

+0,we have expansion:
P (x, y, t) ∼

exp(−d(x, y)2 /4t)
(4πt)n/2

ai (x, y)ti
...
a(x, y) ∈ C ∞ (M ×
M ), a0 (x, y) = 1,for 0 < < s < t − s,
e−∆ P (x, y, t − ) − e−∆(t− ) P (x, y, )
t−

d
−∆(t−s) P (x, y, s))ds
ds (e

=
t−

=
t−

=

[∆e−∆(t−s)P (x,y,s) + e−∆(t−s) ∂P (x, y, s)]ds
∂s
e−∆(t−s) ◦x P (x, y, s)ds
...


P (x, y, ) = P (x, y, t) −
0

5

assume F (x, y, s) = ◦x P (x, y, s),then
t

∆i e−∆(t−s) (x, y, s)ds
0
t ∞

=

λi e−λ(t−s) dEλ (F (x, y, s))ds

0 0
t ∞

=
s0 0
s0 ∞

+
0 0
t ∞

=
s0 0

λi e−λ(t−s) dEλ (F (x, y, s))ds
λi e−λ(t−s) dEλ (F (x, y, s))ds
λi e−λ(t−s) dEλ (F (x, y, s))ds +

s0

O(sN )ds
...
because:
|e−∆(t−s) ◦x P (x, y, s)| = O(sN )
...
becauseH(x, y, t) =
∞
lim e−∆t P (x, y, ) so ∀f (y) ∈ C0 (M ),we have:
→0

P (x, y, )f (y)dy = e−∆t f (x)
...

on the other hand ,by the definition of H(x, y, t),we can check:
when y is fix and t > 0,H(x, y, t) ∈ L2 (M ), so
e−∆t f (x) =

H(x, y, t)f (y), ∀f ∈ L2 (M )
...


we call H(x, y, t) is the kernel of e−∆t
...

now we proof prop (3),by definition:
H(x, y, t) = lim e−∆t− P (x, y, ), ∀ > 0
...


proof prop (4),by e−∆s e−∆(t−s) = e−∆t and (

) wo know:

H(x, z, t − s)H(z, y, s)dz
...

rmk: for the manifold with bounded,we can also proof the existence of
heat kernel with Dirchlet or Neumann boundary condition
...
Chavel
...
3

quasi fundamental solution of heat equation

it is well known that for Rn , the fundamental solution of heat equation
2
∂
(∆ − ∂t )u = 0 is exp(− r )/(4πt)n/2
...

i≥0

where d(x, y) is the Riemann distance of two point of M
...
, n), r = d(x, y)
...

it is well known that there exist functions ψ(r), φ(r) only depend on r :
√
d log( g) dψ
d2 ψ
∆ψ = 2 + (
)
...

dr dr

r2

n

ψ = (4πt)− 2 e− 4t

φ = φ0 + φ1 t +
...

i=0
...

∂t
∂t
∂t
dr dr

because of
∆ψ −

√
d log g dψ
∂
ψ=
,
∂t
dr
dr
dψ
r
= − ψ
...
N

√
r d log g
dφk k
)φk − r
]t ,
2 dr
dr

problem become to solve the equations:
√
r d log g
dφk
= (k +
)φk = ∆φk−1 , k = 0, 1
...

r
dr
2 dr
which is equivalent to:
d k 1
1
(r g 4 φk ) = rk g ∆φk−1 , k ≥ 1
...

dr
∂dr
solve it:

1

φ0 = g 4
r(x,y)
1
4

φk (x, y) = g r

1

−k

rk−1 (∆φk−1 )g 4 dr
...
and:

(∆ −

n
r2
∂
)uN = (4πt)− 2 e− 4t ∆φN tN
...
t:

1
θ(r) = 1, when|r| ≤
...


8

take:
PN (x, y, t) = θ(r(x, y))uN (x, y, t),
of course:
P n(x, y, t) ∈ C ∞ (M ×M ×R+ ), and lim P (x, y, ) = δx (y), (∆−
→0

∂
)PN = O(tN ),
∂t

this is to say:
PN (x, y, t) is the quasi fundamental solution of heat equation
...


2
...
1:
H(x, y, t) > 0, ∀ t > 0
...

Lemma4
...

∂r
proof is similar to Lemma4
...

T hm4
...
:
assume M is a complete riemann manifold ,Ric ≥ (n − 1)k,∀x ∈ M, r0 > 0,
heat kernel of B(x, r0 ), H(x, y, t) and heat kernel ε(r(x, y), t) of geodesic ball
V (k, r0 ) in space form satisfied:
ε(r(x, y), t) ≤ H(x, y, t)
...


9

proof strategy: basically we use the formula
ui (∆u)i + Ric( u,

1
2 ∆(|

u|2 ) =
i,j

u2 +
ij

u)
...


i

T hm4
...

i≥0

rmk:this thm is well known
...
1

rough outline of heat kernel proof of Atiyahsinger index theorem
proof strategy

start point: Mckean-Singer formula
...

x∈M

from this we know Fredholm operator deformation invariance,in the same
time we need chern-weil theory
...
in the expansion on heat kernel ,we need to proof when t → 0, the limit
exist and find a way to calculate it
...
indentify the limit as t → 0
...


3
...

on compact manifold M , D : Γ(M, E) → Γ(M, E) is a self-adjoint operator
0 D−

...
D+ = D|+ , D− = D|−
...

dimention of superspace E = E + ⊕ E − :
dimE = dimE + − dimE −
...

Lemma:
1
...

2
...

the proof of this two lemma is easy,leave sas exercise
...
3

Mckean-Singer formula

the formula is:
2

ind(D+ ) = Str(e−tD ) =

Str(K(x, y))
...


=
0

proof 1:
we have first eigenvalue estimate on compact manifold:
2

|Str(e−tD − P0 )| ≤ Cvol(M )e−tλ
...

2

on the other hand ,we need to show Stre−tD is independent with t,in fact:
d
2
2
Str(e−tD ) = −Str(D2 e−tD )
...

2
2
d
[ , ] supercommunater =⇒ dt Str(e−tD ) = −Str[D, De−tD ] = 0
...
e
...
=⇒ indD = n+ − n−
...
e
...

what we have proved is:
2

indD = Str(e−tD ) =

3
...


analytic formula of indD+

from the discuss of heat kernel in section 2,we know following result(section
2 only discuss the case of function but use the similar way we can get similar
result on bundle):
n

+∞

ti Ki (x), Ki ∈ Γ(M, End(E))
...


(3)

M
i− n
2

t

∞

ai (D+ D− ) −

i=0

n

ti− 2 ai (D+ D− )
...

take t → 0,the only thing make sense is the series of order
to proof:
ind(D+ ) = a n (D− D+ ) − a n (D+ D− )
...

our strategy is following:
t→0
Step1: proof Str(Kt (x, y)) has a limit as t → 0 i
...

step2:use a rescaling of space,time,clifford bundles ,to find a way that make
us only need to calculate the leader coefficient
...
5

From the McKeanSinger formula to the index theorem

Let M be a compact oriented Riemannian manifold of even dimension n
...
The diagonal kt (x, x)
is a section of End(E) which is iso- morphic to Cl(M )⊗EndCl(M ) (E)
...
Elements of EndCl(M ) (E) are given 0-degree
...

the following theorem hold:
Theorem 1
...
The coefficients ki have degree less or equal to 2i
...
2
...


+

ind(D ) = Str(e

−tD2

∞

)=

t

i− n
2

∞
+

−

i=0

n

ti− 2 ai (D+ D− )
...

Proof: Let e1 ,
...
For any multi-index I ⊂ {1,
...
Then the set {cI } is a basis for
Cl(V )
...

2
q
...
d
so take t → 0 , we get:
n

∞

n

ti− 2 Str(ki (x))
...
Let V be a Euclidean space
...
, ∧en if
e1 ,
...
Furthermore, the supertrace defined above equals:
n

Str(a) = (2i) 2 (T ◦ σ(a))
...
)
proof:
the dimension of Str space is one because of CLn−1 (V ) = [CL(V ), CL(V )]
and it never be empty because there is a natural defined supertrace on
CL(V )
...
for
instance the chirality operator Γ
...

q
...
d
so we know:
∀ section a ⊗ b ∈ Γ(M ⊗ CL(M ) ⊗ EndCL(M ) (E):
n

StrE (a ⊗ b)(x) = (−2i) 2 σn (a(x))StrE/S (b(x))
...

2
2
theorem1 implies then the following theorem for the index of a Dirac operator associated to a Clifford connection which is known as the local index
theorem
...
(Local index theorem) Let M be a compact, oriented evendimensional manifold and let E be a Clifford module with Clifford connection E
...
Then lim Str(kt (x, x))|dx|
t→0

exists and is obtained by taking the n -th form piece of
(2πi)n/2 A(M )ch(E/S)
...

15

However, since the index of a Dirac operator is independent of the Clifford
superconnection used to define it, we get AtiyahSinger index formula for any
Dirac operator
...
(AtiyahSinger Index Theorem) Let M be a compact, oriented,
even-dimensional manifold and let D be a Dirac operator on a Clifford module E
...

M

hence theorem 3 is a consequence of theorem 1
...


3
...
11 we mainly follow Chapter 4 of [BGV] but rearrange the different steps in order to make the proof clearer, at least for us
...
The idea of the proof is to work in normal coordinates x around a
point x0 ∈ M
...
Using the symbol isomorCl(V
phism , we can look at k(t, x) as a section of ∧(V ∗ ) ⊗ E(W )
...
We use Lichnerowicz formula to get the explicit form of the operator
L such that the kernel k(t, x) satisfies the heat equation (∂t + L)k(t, x) = 0
...
In a third step, we define a rescaling of space, time and the Clifford
algebra, introduced by Getzler
...


16

A

Chern-Weil theory

some text in Appendix A

B

Complete proof of theorem 1

some text in Appendix B

References
[1] Lectures on Differential Geometry
...
Shing- Tung Yau (Harvard University)
[2] N
...
Getzler, M
...
Julien
Meyer
...

[4] on the Atiyah-Singer Index Theorem Liviu I
...
Last revision: November 15, 2013
...
Pte
...


17



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