Clifford Algebra Note 4 Dirac Operator From 3-dimensional Spinor Group to 4-dimensional Spin Manifold

  

Clifford Algebra

 

Note 4

Dirac Operator

From 3-dimensional Spinor Group to 4-dimensional Spin Manifold

 

TANAKA Akio

 

1 <Clifford algebra>

ⁿ has inner product.

Orthonormal basis of the inner product space     e₁, …, e_n

Algebra generated from e₁, …, e_n has next relations.

e_ie_j = -e_je_i  (i  j )

(e_i )² = -1   (i = 1, …, n )       (1)

The algebra is called n-dimensional Clifford algebra, expressed by Cl_n.

Cl_n has vector space generated from e_i1…e_ik  against i₁ < …<i_k

0 ≤ k ≤ n. When k = 0, e_i1∧…∧e_ik = 1.

2 <Dirac operator>

Differential operator defined over open set of ⁿ

D = γ₁  + … +γ_n  

γ_iγ_j = -γ_jγ_i  (i  j )

(γ_i )² = -1   (i = 1, …, n )

D becomes Dirac operator.

3 <Representation space>

Cl_n is presentation space of Cl_n , for  Cl_n’s vector space generated from e_i1…e_ik  against i₁ < …<i_k

4 <Exterior algebra>

At (1), now relation is changed to (e_i )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

e_i1∧…∧e_ik .     (3)

0 ≤ k ≤ n. When k = 0, e_i1∧…∧e_ik = 1.

Now 0 ≤ k ≤ n. When k = 0, e_i1∧…∧e_ik = 1.

Vector subspace generated from (3) against fixed k is expressed by ∧^k.

5 <Differential form>

Basis of ⁿ is expressed by dx₁, …, dx_n.

∧^k valued function on ex is expressed by the next,

α = αi₁,...,α_k dx_i1, …, dx_in.    (4)

(4 ) is called k-dimensional differential form.

6 <exterior differentiation operator, associated operator>

All of k-dimensional differential forms is expressed by Ω^k .

Next operators are given against Ω^k .

Exterior differentiation operator     d : Ω^k →Ω^k+1

Associated operator     d* : Ω^k →Ω^k-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 e_r .

D = e_r      (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.

w_i (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.

D_A∅ = 0,  F_A⁺ = [ ∅ ∅^- ]⁺      (6)

Here

D_A∅ = 0   Dirac equation

F_A⁺ = ²₊　　　　　　

⁰ +²₊ = S⁺ ⊗ S⁺

⁰   0-dimensional differential form

²₊  Self-dual 2-dimensional differential form  

 

Tokyo January 20, 2008

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