Ivancevic_Applied-Diff-Geom

472 Applied Differential Geometry: A Modern IntroductionThe expressions for the curvature tensor now readR (1)µνρ σ =− i 2 θκλ ((∂ κ R (0)µνρ τ )(∂ λ g τγ )g γσ − (∂ κ Γ (0)−Γ (0)νρ β )(Γ (0)µβµτ σ (∂ λ g βγ )g γτ + ∂ µ ((∂ λ g βγ )g γσ ) + (∂ λ Γ (0) σ ))+(∂ κ Γ (0)µρ β )(Γ (0)νβµβτ (∂ λ g τγ )g γσ − Γ (0)ντ σ (∂ λ g βγ )g γττ (∂ λ g τγ )g γσ+∂ ν [(∂ λ g βγ )g γσ ] + (∂ λ Γ (0) σ νβ))), (3.293)R (2)µνρ σ =∂ ν Γ (2)µρ σ + Γ (2)νρ γ Γ (0)µγ σ + Γ (0)νρ γ Γ (2)µγ σ+ i 2 θαβ [(∂ α Γ (1)νρ γ )(∂ β Γ (0)µγ σ ) + (∂ α Γ (0)νρ γ )(∂ β Γ (1)µγ σ )]− 1 8 θα1β 1θ α2β 2(∂ α1 ∂ α2 Γ (0)νρ γ )(∂ β1 ∂ β2 Γ (0)µγ σ ) − (µ ↔ ν), (3.294)where the second order is given implicitly in terms of the Christoffel symbols.The deformed Einstein–Hilbert action is given byS EH = 1 ∫d 4 x det ⋆ e a µ ⋆ R + c.c.2= 1 ∫d 4 x det ⋆ e a µ ⋆ (R +2¯R) = 1 ∫d 4 x det ⋆ e a µ (R +2¯R)∫= S (0)EH + d 4 x (dete a µ )R (2) + (det ⋆ e a µ ) (2) R (0) , (3.295)where we used that the integral together with the Moyal product (by partialintegration) has the property∫∫ ∫d 4 x f ⋆ g = d 4 x fg = d 4 x g ⋆ f.In (3.295) det ⋆ e µ a is the ⋆−determinantdet ⋆ e µ a = 1 4! εµ 1···µ 4 εa1···a 4e µ1a 1⋆e µ2a 2⋆e µ3a 3⋆e µ4a 4= dete µ a +(det ⋆ e µ a ) (2) +. . . ,where(det ⋆ ) (2) = − 1 18 4! θα1β 1θ α2β 2ε µ 1 ...µ 4 a εa1...a 4[(∂ α1 ∂ α2 e 1 aµ1 )(∂ β1 ∂ β2 e 2 aµ2 )e 3 aµ3 e 4µ4 +a∂ α1 ∂ α2 (e 1 aµ1 e 2 aµ2 )(∂ β1 ∂ β2 e 3 aµ3 )e 4 aµ4 + ∂ α1 ∂ α2 (e 1 aµ1 e 2 aµ2 e 3 aµ3 )(∂ β1 ∂ β2 e 4µ4 )].The odd orders of θ vanish in (3.295) but the even orders of θ give nontrivialcontributions.