I have a question about a statement in Witten's paper 'Analytic Continuation of Chern-Simons Theory' (https://arxiv.org/abs/1001.2933). On page 66, below equation 4.13, he discusses a Lie algebra-valued field $\phi_t$ which is covariantly constant with respect to a gauge field, $A_{\mu}$ on a 4-manifold $I \times M$ ($I$ is the interval), and in addition commutes with a number of other Lie algebra valued fields, i.e.,

$$ D_{\mu}\phi_t=\partial_{\mu}\phi_t+[A_{\mu},\phi_t]=0,~~[\phi_{\mu},\phi_t]=0~~(\textrm{for } \mu\neq t). $$ Witten then claims that when the Dirichlet boundary condition $\phi_t=0|_M$ is obeyed, the unique solution to these equations is $$\phi_t=0$$

How does one show this?

It seems to me that $[\phi_{\mu},\phi_t]=0$ implies that $$\phi_t\in \frak{t},$$ where $\frak{t}$ is the commutative subalgebra of the Lie algebra. Then, the remaining equation becomes $$\partial_{\mu}\phi_t=0,$$ which implies that $\phi_t$ is a constant, and therefore $\phi_t=0$ everywhere due to the boundary condition. However, this does not seem like the most general way of solving these equations.


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