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Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ where $\Omega$ is the symplectic form on $\mathcal{A}$. In that setup we can express how the complexified action changes the norm.

In the survey "Yang-Mills Connections and Einstein-Hermitian Metrics" of Itoh and Nakajima at the bottom of page 453, the authors quote a proposition in [BGS] that identifies explicitly the dependency of the "Ray-Singer" metric (on the determinant line bundle of the cohomology) on the original metric. They then relate it to the effect on the norm of the complexified action on the line bundle.

This relates two different actions of the complex Gauge group: The first is the complexified action on the line bundle, and we can measure how this action makes the norm of an element of the line bundle vary. The second is the action on the space of Hermitian metrics on the original vector bundle: This action induces an action on the space of Hermitian metrics on the determinant line bundle. We can then measure for a fixed element how its norm varies when we modify the Hermitian metric.

One can find a similar thing in Donaldson's "INFINITE DETERMINANTS, STABLE BUNDLES AND CURVATURE" page 238 corollary 16.

I do not understand why these two actions should be a priori related: As a matter of fact after computing the derivatives along a one-parameter subgroup and using the moment map one can see that they are actually related but in the aforementioned articles it seems that we do not need this computation to obtain the result.

Does anyone have an idea about that? Thanks.

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