Let $G$ be a Lie group. Assume that we have an Ad-invariant bilinear symmetric form $$\langle-,-\rangle : \mathfrak{g} \times \mathfrak{g} \to \mathbb{C}.$$ Given a smooth manifold $X,$ we let $\mathcal{M}_X$ denote the moduli space of flat connections on principal $G$-bundles, modulo gauge equivalence. One can then show that the Chern-Simons action determines, for each closed surface $Y,$ a hermitian line bundle $$\mathcal{L}_Y \rightarrow \mathcal{M}_Y$$ with a connection $\theta.$ Further, for each compact oriented $3$-manifold $X,$ we get a parallel section $$s:\mathcal{M}_X \rightarrow r^*_X \mathcal{L}_{\partial X}$$ of the pullback bundle by the restriction $\mathcal{M}_X \to \mathcal{M}_{\partial X}.$

I have read that the fact that $s: \mathcal{M}_X \to r_X^* \mathcal{L}_{\partial X}$is a parallel section implies that the variation of the Chern-Simons action along a path of connections can be computed through parallel transport. This should be done by integrating the connection $\theta$ along the restriction of the path to the boundary. My questions are:

  1. What does "variation of the Chern-Simons action" along a path of connections mean in this context?
  2. Why can it be computed as claimed?

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