4 1. INDE X THEOR Y
the Operator o f convolution b y the invers e Fourier transfor m o f /. I n particular ,
suppose /(A ) = e~ A2/2. The n
f(D)u{x) = -j=J_ e-^-y?'
2u{v)dy,
in other words, f{D) i s a smoothing Operator with Gaussian kernel. A smoothing
Operator wit h compactl y supporte d kerne l i s a compac t Operator , s o wheneve r
f{D) i s cu t dow n t o a compac t domai n b y multiplicatio n b y a compactl y
supportedp , we get a compact Operator . Tha t is , f(D) i s locally compact .
Let's confin e ou r attentio n fo r th e moment , however , t o th e cas e that M i s a
compact manifold. The n ellipti c regularity has the following simpl e consequence :
the comple x fi£
2
i s the direc t su m o f tw o close d subcomplexe s invarian t unde r
Z), th e comple x W * = kerD o f Harmonie form s (o n whic h th e differentia l d i s
identically zero) , an d it s orthogona l complemen t H
±]
moreover , th e restrictio n
of D t o
H1-
i s invertible, an d this implies that H
1-
i s an aeyclic complex it ha s
trivial cohomology . Thu s w e ge t th e Hodge theorem H
l
DR(M) = W(M). Sinc e
W(M), a s an eigenspace of a generalized Dirac Operator on a compact manifold ,
must b e finit e dimensional , thi s give s u s the analyti c explanatio n tha t w e were
seeking for th e finite-dimensionality o f de Rham cohomology .
Index theor y
The notion of index is abstractly defined i n functional analysi s for all Fredholm
Operators: a Predholm Operato r fro m on e Hubert spac e to anothe r i s a bounde d
Operator which is invertible modul o th e compac t Operators . Suc h a n Operato r T
has kerT an d kerT * finite-dimensional, an d it s index is, by definition, Ind(T ) =
dim ker T dim ker T*. T o relate thi s definitio n t o th e Dira c Operator s tha t w e
have bee n considering , le t u s introduc e th e notio n o f a chopping funetion.
DEFINITION 1.5: A choppin g funetio n is a continuous odd funetion x : R
[—1,1] such that x(t) ± 1 as t —* ±oo .
LEMMA 1.6: Let D be a generalized Dirac Operator on a compact manifold M.
Then x{D) i$
a
Fredholm Operator, and if xi and X2 dre two different chopping
functions, then the corresponding Fredholm Operators xi {D) and X2 (D) differ by
a compact Operator.
PROOF: Sinc e x
1
1 an d xi ~ X2 ar e function s vanishin g a t infinity , th e
corresponding Operator s ar e compac t b y 1.3. D
Since
x{D)ls
self-adjoint , th e index of
x{D)1S
zero . However , Dirac Operator s
on even-dimensiona l manifold s frequentl y com e equippe d wit h a piec e o f extr a
strueture called a grading: thi s is a self-adjoint involution
3
e which anticommute s
An involution i s a n Operato r whos e Squar e i s 1.
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