Coset Space Dimensional Reduction

Coset Space Dimensional Reduction

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. . . . 49 4.4

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Lagrangian

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dL(y) · L ⁻¹ (y) = e(y) = e ^a Q _a + e ⁱ Q _i (2.26)

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de ^a = 1

2 f _bc ^a e ^b ∧ e ^c + f _bi ^a e ^b ∧ e ⁱ , de ⁱ = 1

2 f _ab ⁱ e ^a ∧ e ^b + 1

2 f _jk ⁱ e ^j ∧ e ^k (2.28)

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2 y ^β δ _β ^c δ _α ^b f _cb ⁱ + · · · (2.29)

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e ^a (y ^s )Q _a + e ⁱ (y ^s )Q _i = e ^a (y)r ⁻¹ Q _a r + e ⁱ (y)r ⁻¹ Q _i r + dr ⁻¹ · r (2.31)

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r ⁻¹ Q _a r ≡ adj(r)Q _a = D _a ^b (r)Q _b (2.32)

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r ⁻¹ Q _a r = exp( − φ ⁱ Q _i )Q _a exp(φ ⁱ Q _i )

= Q _a + φ ⁱ [Q _a , Q _i ] + · · ·

= Q _a + φ ⁱ f _ai ^b Q _b + · · · (2.33)

D _a ^b (r) = δ _a ^b + φ ⁱ f _ai ^b + · · · (2.34)