12 a lichnerowicz type formula follows immediately d

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a Lichnerowicz type formula follows immediately : D ( s ) 2 = ( L ( s ) tot ) 2 - ( s 2 - 1 / 4) 1 . Using (2.14), the eigenvalues of the Dirac operator are found to be : D ( s ) ψ ( ± ) ( s ) ,j,m = ± ( j + 1 / 2) 2 - s 2 ψ ( ± ) ( s ) ,j,m , with ψ (+) ( s ) ,j,m = 1 2 Y s - 1 / 2 j,m i 2 Y s +1 / 2 j,m , ψ ( - ) ( s ) ,j,m = γ 3 ψ (+) ( s ) ,j,m = 1 2 Y s - 1 / 2 j,m - i 2 Y s +1 / 2 j,m . In particular it follows that, for Dirac spinors i.e. s = 0, and only in this case, there are no zero eigenvalues. When s = 0, zero is an eigenvalue, 2 | s | times degenerate with eigenspinors Y s - 1 / 2 s - 1 / 2 ,m 0 for positive values of s , 0 Y s +1 / 2 - s - 1 / 2 ,m if s is negative. 13
3 The projective modules over A Through the Gel’fand-Na˘ ımark construction, the topology of M = S 2 ×{ a,b } is encoded in the complex C * -algebra of continuous complex-valued functions on M = S 2 × { a,b } . However, in order to get a fruitful use of a differential structure, we have to restrict it to its dense subalgebra of smooth functions. This proviso made, let { f,g, · · ·} denote elements of A = C ( M ) and let the value of f at a point 9 { x,α } ∈ M , be written as f α ( x ). The vectors of the free right A -module of rank two, identified with A 2 , are of the form X = i =1 , 2 E i f i , where f i ∈ A and { E i ; i = 1 , 2 } is a basis of A 2 . Let Ω ( A ) = k =0 Ω ( k ) ( A ) denote the universal differential envelope of A . Elements of Ω ( k ) ( A ) can be realised, see e.g. [6], as functions on the Cartesian product of ( k + 1) copies of M , vanishing on neighbouring diagonals. For example, F Ω (2) ( A ) is realised as the complex-valued function on M × M × M , F αβγ ( x,y,z ), such that F ααγ ( x,x,z ) = 0 and F αββ ( x,y,y ) = 0. The product in Ω ( A ) is obtained by concatenation, e.g. if F Ω (1) ( A ) and G Ω (2) ( A ) then their product F · G Ω (3) ( A ) is represented by ( F · G ) αβγδ ( x,y,z,t ) = F αβ ( x,y ) G βγδ ( y,z,t ). Ω ( A ) is a differential algebra with differential d acting on f ∈ A = Ω (0) ( A ) as ( d f ) αβ ( x,y ) = f β ( y ) - f α ( x ). On F Ω (1) ( A ) it acts as ( d F ) αβγ ( x,y,z ) = F βγ ( y,z ) - F αγ ( x,z ) + F αβ ( x,y ) and so on for Ω ( k ) ( A ). The involution, defined in A by ( f ) α ( x ) = f α ( x ) * , extends to Ω ( k ) ( A ) as ( F ) αβ ··· ( x,y, · · · ) = F ··· βα ( · · · ,y,x ) * . 10 A (universal) connection on A 2 is given, in the basis { E i ; i = 1 , 2 } , by an Ω (1) ( A )-valued 2 × 2 matrix (( ω )) i k . It acts on X = E i f i as : free ( X ) = E i A d f i + (( ω )) i k f k . (3.1) For X = E i f i and Y = E i g i , let h ( X,Y ) = i,j ( f i ) δ ı j g j = i ( f i ) g i be the standard hermitian product in A 2 with values in A . It extends to a Ω ( A )-valued function on A 2 A Ω ( A ) × A 2 A Ω ( A ) by h ( X F,Y G ) = F h ( X,Y ) G . 9 In this section points of S 2 are denoted by x, y, · · · , while α, β, · · · will assume values in the two-point space { a, b } . 10 Note that d ( f ) = - ( d f ) , f ∈ A and, more generally, if F Ω ( k ) ( A ), then d ( F ) = ( - 1) k +1 ( d F ) .

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