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For a gauge theory of a gauge group $G$ (better here: a simply-connected and compact $G$; but if you wish, you can comment more general cases), we view the spacetime as the base manifold and the principal bundle is the principal $G$-bundle. The gauge field is locally the 1-form connection taking values in the Lie algebra of $G$.

For example, we may consider a Yang-Mills (YM) gauge theory in 4 dim.

If we regard another new gauge field, instead, as a Cech 1-cocycle with values in the center Z$(G)$, I propose that here, we can consider the symmetry transformation acts by multiplying each transition function defining a $G$-bundle by the corresponding value of the Cech 1-cocycle. The YM Lagrangian depends only on the curvature of the gauge field, is not affected by this transformation, hence this new gauge field as a Cech 1-cocycle is a symmetry transformation on the YM Lagrangian as well as the partition function.

    1. Does the above description have a precise formulation in mathematics? How?
    1. Is there a similar statement for Chern-Simons gauge theory in 3 dim?
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