Notes for AKT-140307/0:41:01: Difference between revisions
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'''Proposition: '''$CS(A^g) = CS(A)$ |
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$$Tr(A^g \wedge d A^g + A^g \wedge A^g \wedge A^g) |
$$CS(A^g)=\int_\mathbb{R^3} Tr(A^g \wedge d A^g + \frac23 A^g \wedge A^g \wedge A^g)$$ |
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$$Tr(g^{-1} A \wedge d A g + g^{-1} A g \wedge d g^{-1} \wedge g^{-1} A g + g^{-1} d g \wedge g^{-1} d A g - g^{-1} d g \wedge g^{-1} A \wedge d g +$$ |
$$Tr(g^{-1} A \wedge d A g + g^{-1} A g \wedge d g^{-1} \wedge g^{-1} A g + g^{-1} d g \wedge g^{-1} d A g - g^{-1} d g \wedge g^{-1} A \wedge d g +$$ |
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$$g^{-1} d g \wedge d g^{-1} \wedge A g - g^{-1} A \wedge A \wedge d g + g^{-1} A g \wedge d g^{-1} \wedge d g + g^{-1} d g \wedge d g^{-1} \wedge d g) +$$ |
$$g^{-1} d g \wedge d g^{-1} \wedge A g - g^{-1} A \wedge A \wedge d g + g^{-1} A g \wedge d g^{-1} \wedge d g + g^{-1} d g \wedge d g^{-1} \wedge d g) +$$ |
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Revision as of 01:28, 26 July 2018
Proposition: $CS(A^g) = CS(A)$ $$CS(A^g)=\int_\mathbb{R^3} Tr(A^g \wedge d A^g + \frac23 A^g \wedge A^g \wedge A^g)$$ $$Tr(A^g \wedge d A^g + \frac23 A^g \wedge A^g \wedge A^g) =$$ $$Tr(g^{-1} A \wedge d A g + g^{-1} A g \wedge d g^{-1} \wedge g^{-1} A g + g^{-1} d g \wedge g^{-1} d A g - g^{-1} d g \wedge g^{-1} A \wedge d g +$$ $$g^{-1} d g \wedge d g^{-1} \wedge A g - g^{-1} A \wedge A \wedge d g + g^{-1} A g \wedge d g^{-1} \wedge d g + g^{-1} d g \wedge d g^{-1} \wedge d g) +$$ $$\frac23 Tr( g^{-1} A \wedge A \wedge A g + g^{-1} A \wedge A \wedge d g + g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} A g + g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} d g +$$ $$g^{-1} d g \wedge g^{-1} A \wedge A g + g^{-1} d g \wedge g^{-1} A g \wedge g^{-1} d g + g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} A g + g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g) $$
Now $0 = d g^{-1} g = d g g^{-1} + g d g^{-1}$
So $d g g^{-1} = - g d g^{-1}$
Applying this to the fifth and seventh terms of the equation above yields $$ g^{-1} d g \wedge d g^{-1} \wedge A g = g^{-1} d g \wedge d g^{-1} g \wedge g^{-1} A g = - g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} A g$$ and $$g^{-1} A g \wedge d g^{-1} \wedge d g = g^{-1} A g \wedge d g^{-1} g \wedge g^{-1} d g = - g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} d g $$
Combining this with the fact that the trace is invariant under cyclic permutations show that the
$$Tr(g^{-1} A \wedge d A g - 2 g^{-1} A \wedge A \wedge d g - 2 g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} d g +g^{-1} d g \wedge g^{-1} d A g - g^{-1} d g \wedge g^{-1} A \wedge d g + g^{-1} d g \wedge d g^{-1} \wedge d g) +$$ $$ \frac23 Tr( g^{-1} A \wedge A \wedge A g + 3 g^{-1} A \wedge A \wedge d g + 3 g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} d g +g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g) =$$ $$Tr(g^{-1} A \wedge d A g + g^{-1} d g \wedge g^{-1} d A g - g^{-1} d g \wedge g^{-1} A \wedge d g+ g^{-1} d g \wedge d g^{-1} \wedge d g) + \frac23 Tr( g^{-1} A \wedge A \wedge A g + g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g)$$ Now deal with the extra terms $$Tr(g^{-1} d g \wedge g^{-1} d A g ) = Tr(g^{-1} d g \wedge d(g^{-1} A) g - g^{-1} d g \wedge d g^{-1} \wedge A g) = Tr(g^{-1} d g \wedge d(g^{-1} A) g + g^{-1} d g \wedge g^{-1} A \wedge d g)$$ Finally $$Tr(g^{-1} d g \wedge d(g^{-1} A) g) = Tr(d g \wedge d(g^{-1} A)) = Tr(d (gd(g^{-1} A))) = d Tr(g d(g^{-1} A))$$ Substituting this into equation one and rearranging gives. $$Tr(g^{-1} A \wedge d A g + g^{-1} A \wedge A \wedge A g + \frac13 g^{-1} d g \wedge d g^{-1} \wedge d g) + d Tr(g d(g^{-1} A))$$ A similar argument shows that $$Tr(g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g) = -Tr(g^{-1} d g \wedge d g^{-1} \wedge d g) = dTr( $$ to be completed
