Every flat vector bundle on a manifold comes from a representation of $\pi_1$, by the equivalence between flat vector bundles and local systems.

However, I've seen it mentioned that non-flat vector bundles on a manifold correspond to representations of a central extension $\widehat{\pi}_1$ of $\pi_1$. Is this correct, and what does the action of the central element correspond to geometrically?

Example: In the Atiyah-Bott Yang Mills paper, they mention that a rank $n$ vector bundle with curvature $\kappa$ on a Riemann surface $X$ corresponds to a representation $\rho$ of $\widehat{\pi}_1(X)$ with $$\prod [\rho(a_i),\rho(b_i)]\ = \ \exp(2\pi i \kappa/n)$$ where $a_i,b_i$ are the standard generators of $\pi_1(X)$.



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