160 The one SU 2 instanton solution centered at z \u03bc can be written explicitly

# 160 the one su 2 instanton solution centered at z μ

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The one SU (2) instanton solution centered at z μ can be written explicitly in com- ponents as A cl = 2 g η aμν ( x - z ) ν ( x - z ) 2 + R 2 , (10.55) where the symbol η aμν is anti-symmetric in μ, ν , defined as η aij = aij , i, j = 1 , 2 , 3 , η ai 4 = - η a 4 i = δ ai . (10.56) Note that the one anti-instanton solution would be obtained by replacing η aμν with ¯ η aμν , defined by ¯ η aij = η aij , ¯ η ai 4 = - η ai 4 . The Euclidean action S E is related to the Lorentzian one by S = iS E . The one instanton solution has Euclidean action S cl E = 8 π 2 /g 2 . In background field gauge, the action for the quantum fluctuations is S E = S cl E + Z d 4 x 1 2 ( D μ A 0 ) 2 + g abc F μν a A 0 A 0 + ( D μ Φ) * ( D μ Φ) + ¯ ψγ μ D μ ψ + ( D μ ¯ η a )( D μ η a ) + · · · (10.57) where we have exhibited only quadratic terms in the quantum fluctuations, which is all we need to the one-loop computation. Note that the term proportional to ( D μ A 0 μ ) 2 from expanding the Yang-Mills Lagrangian is canceled by the gauge fixing term. As before, we will write M A , M Φ , M ψ and M η for the kinetic operators on A 0 , Φ, ψ , and ( η, ¯ η ) respectively. The result of the one-loop Gaussian integral is then e - 8 π 2 /g 2 Z d (zero modes) det 0 M ψ det 0 M η p det 0 M A det 0 M Φ , (10.58) where we have separated out the integration over zero modes of A 0 , Φ , ψ, η in the instanton background: these modes are annihilated by the respective kinetic operator M ; det 0 is the determinant taken over modes of nonzero eigenvalues only, defined using a suitable (gauge invariant) regulator. Explicitly, the kinetic operators are M A A 0 = - D 2 A 0 - 2 g abc F bμν A 0 , M ψ = γ μ D μ , M 2 ψ = D 2 - i 2 γ μν F aμν T a , M Φ = M η = - D 2 . (10.59) Here T a are the generators of the SU (2) gauge group. Let us first calculate D 2 in the 161
background of an instanton with z = 0 and R = 1, D 2 = ( μ - igT a A cl )( μ - igT b A cl,μ b ) = - 2 iT a η aμν { x ν r 2 + 1 , ∂ μ } - 4 T a T b η aμν η b μ ρ x ν x ρ ( r 2 + 1) 2 = + 4 r 2 + 1 T a η aμν ( - ix ν μ ) - 4 r 2 ( r 2 + 1) 2 T 2 , (10.60) where r 2 = x μ x μ , T 2 = T a T a . Note that the SO (4) rotation generator - ix [ μ ν ] can be split into the SU (2) × SU (2) generators L a 1 = - i 2 η aμν x μ ν , L a 2 = - i 2 η aμν x μ ν . (10.61) They obey L a 1 L a 1 = L a 2 L a 2 L 2 = - 1 8 ( x μ ν - x ν μ ) 2 . (10.62) Writing the Laplacian in terms of L 2 also, we have D 2 = 1 r 3 r r 3 ∂r - 4 L 2 r 2 - 8 r 2 + 1 T a L a 1 - 4 r 2 ( r 2 + 1) 2 T 2 . (10.63) This expression can be generalized to the kinetic operator acting on fields of nonzero spin, M = - 1 r 3 r r 3 ∂r + 4 L 2 r 2 + 8 r 2 + 1 T a L a 1 + 4 r 2 ( r 2 + 1) 2 T 2 + 16 ( r 2 + 1) 2 T a S a 1 , (10.64) where S a 1 together with S a 2 are the SU (2) × SU (2) generators associated with the intrinsic spin of the field. The fermion ψ transforms in the representation ( 1 2 , 0) (0 , 1 2 ) of the SU (2) × SU (2) SO (4) rotation (Euclidean version of Lorentz) group, whereas A 0 μ transforms in the representation ( 1 2 , 1 2 ). Explicitly, acting on the spinor, S 1 and S 2 are S a 1 = - i 8 η aμν γ μν , S a 2 = i 8 η aμν γ μν , S 2 1 = 3 4 1 - γ 5 2 , S 2 2 = 3 4 1 + γ 5 2 .

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