where D$_{b}$ is the covariant derivative for the connection b and Ad$_{U}$ β = UβU − $^{1}$. The block ( δφ δU δφ δσ ) is a square matrix in the sense that it is a map R 4 → R 4 when we identify R 3 ≃ su (2) and R 4 ≃ C $^{2}$. The maps δb δB and δψ$_{L}$ δψ ′ L are also “square matrices” in the same way. The matrix J is then lower triangular, so that its determinant is just the product of the determinant of these three blocks. The map ( δφ δU δφ δσ ) is linear on functions, the map δb δB is linear on 1 -forms and the map δψ$_{L}$ δψ ′ L is linear on spinors. It is then easy to see that their determinants can be computed as determinants in some finite dimensional vector spaces at each point in space-time: R 4 for ( δφ δU δφ δσ ) , su (2) for δb δB and C 2 for δψ$_{L}$ δψ ′ $_{L}$. The determinant of ϕ ↦→ Uϕ on C 2 is obviously 1 for any U . In the same way, results in linear algebra and Lie algebras show that the determinant of β ↦→ Ad$_{U}$ β is 1 on su (2) . In the basis σ$_{1}$, σ$_{2}$, σ$_{3}$, 1l of su (2) ⊕ R ≃ R 4 and the basis ( 1 $_{0}$) , ( i $_{0}$) , ( 0 $_{1}$) , ( 0 i ) of C 2 ≃ R $^{4}$, the map ( δφ δU δφ δσ ) takes the explicit form of the 4 × 4 matrix    φ$_{2}$$_{y}$ − φ$_{2}$$_{x}$ φ$_{1}$$_{y}$ φ$_{1}$$_{x}$ − φ$_{2}$$_{x}$ − φ$_{2}$$_{y}$ − φ$_{1}$$_{x}$ φ$_{1}$$_{y}$ φ$_{1}$$_{y}$ φ$_{1}$$_{x}$ − φ$_{2}$$_{y}$ φ$_{2}$$_{x}$ − φ$_{1}$$_{x}$ φ$_{1}$$_{y}$ φ$_{2}$$_{x}$ φ$_{2}$$_{y}$   $^{,}$ where φ = ( φ 1 $_{x}$+ iφ 1 y φ 2 $_{x}$+ iφ 2 y ) . The determinant of this matrix is ( φ 2 1 x + φ 2 1 y + φ 2 x + φ 2 $_{y}$) 2 = η $^{4}$. The determinant of J is then η $^{4}$, which shows that the functional integral measures are related by [ dφ ][ db ][ dψ$_{L}$ ] = η $^{4}$[ dσ ][ dU ][ dB ][ $_{dψ}$′ $_{L}$] = η $^{3}$[ dη ][ dU ][ dB ][ $_{dψ}$′ $_{L}$] We would like to thank A. Abada, S. Lazzarini and T. Schücker for stimulating discus- sions. ACKNOWLEDGMENTS ∗ thierry.masson@cpt.univ-mrs.fr 15