ω = d g g^{- 1}, A = g A g^{- 1} - ω .

d ω = ω \land ω

. Direct substitution gives

F = g F g^{- 1},

so trace invariance proves

C_{2} = C_{2}

For the Chern-Simons 3-form, expansion and graded cyclicity give

Y (A) - Y (A) = \frac{1}{8 π ^{2}} d Tr (ω \land g A g^{- 1}) + \frac{1}{24 π ^{2}} Tr (ω^{3}) .

Since cyclicity also gives

Tr (ω \land g A g^{- 1}) = Tr (g^{- 1} d g \land A),

the required two-form may be chosen as

T = \frac{1}{8 π ^{2}} Tr (g^{- 1} d g \land A) .

Thus

Y ⟼ Y + d T + \frac{1}{24 π ^{2}} Tr [(d g g^{- 1})^{3}] .

Solved by gpt-5.6-sol high.

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