Note 4
Dirac Operator
From 3-dimensional Spinor Group to 4-dimensional Spin Manifold
1 <Clifford algebra>
n has inner product.
Orthonormal basis of the inner product space e1, …, en
Algebra generated from e1, …, en has next relations.
eiej = -ejei (i 
j )
(ei )2 = -1 (i = 1, …, n ) (1)
The algebra is called n-dimensional Clifford algebra, expressed by Cln.
Cln has vector space generated from ei1…eik against i1 < …<ik
0 ≤ k ≤ n. When k = 0, ei1∧…∧eik = 1.
2 <Dirac operator>
Differential operator defined over open set of n
D = γ1 
+ … +γn 
γiγj = -γjγi (i j )
(γi )2 = -1 (i = 1, …, n )
D becomes Dirac operator.
3 <Representation space>
Cln is presentation space of Cln , for Cln’s vector space generated from ei1…eik against i1 < …<ik
4 <Exterior algebra>
At (1), now relation is changed to (ei )2 = 0 (i = 1, …, n ) (2)
New elation is called exterior algebra, abbreviated to ex.
Exterior algebra’s product is expressed by wedge product ∧.
Vector space of exterior algebra is generated from
ei1∧…∧eik . (3)
0 ≤ k ≤ n. When k = 0, ei1∧…∧eik = 1.
Now 0 ≤ k ≤ n. When k = 0, ei1∧…∧eik = 1.
Vector subspace generated from (3) against fixed k is expressed by ∧k.
5 <Differential form>
Basis of n is expressed by dx1, …, dxn.
∧k valued function on ex is expressed by the next,
α = 
αi1,...,αk dxi1, …, dxin. (4)
(4 ) is called k-dimensional differential form.
6 <exterior differentiation operator, associated operator>
All of k-dimensional differential forms is expressed by Ωk .
Next operators are given against Ωk .
Exterior differentiation operator d : Ωk →Ωk+1
Associated operator d* : Ωk →Ωk-1
7 <Spinor group>
Rotation group of 3-dimensional Euclid space SO ( 3 )
SO ( 3 ) is homeomorphic with 3-dimensional sphere S3 that is called spinor group.
n-dimensional spinor group is expressed by Spin ( n ).
Spinor group has two 2-dimensional complex expression S±.
S+ is called plus 2-dimensional spinor.
S- is called minus 2-dimensional spinor.
8 <Spinor representation>
By S+ and S- , S is expressed to the next.
S = S+ ⊕ S-
9 <Riemann manifold>
Euclid space 2l
Dirac operator is expressed to the next by generating element er .
D = 
er
(5)
When Euclid space is lifted to oriented Riemann manifold, the condition of 2-dimensional Stiefel Whitney class is defined .
The condition is the next.
w2(TM ) = 0
TM is tangent bundle.
w is vector bundle ξ’s base space B’s Z2 coefficient’s cohomology group’s element.
wi (ξ) ∈ Hi ( B ; Z2 ) i = 0, 1, 2, …
10 <Spin Riemann manifold >
2l-dimensional spin Riemann manifold M
Dirac operator D
Spinor field that satisfies Ds = 0 is called harmonic spinor.
Space given by harmonic spinor H
From S = S+ + S-
Decomposition H = H+ ⊕ H-
dim H+ - dim H- becomes topological invariant.
The invariant is called index D.
11 <Seiberg-Witten equation>
Oriented compact4-dimensional spin manifold M
Complex linear bundle over M L
U ( 1 ) connection A of L is fixed.
Plus spinor bundle S+
Section of S+ ⊗ L ∅
Seiberg-Witten equation is defined by the next.
DA∅ = 0, FA+ = [ ∅ ∅- ]+ (6)
Here
DA∅ = 0 Dirac equation
FA+ = 
2+
0 +2+ = S+ ⊗ S+
0 0-dimensional differential form
2+ Self-dual 2-dimensional differential form
