Fix a $ au$-connection DL on a line bundle L over $X imes mathbb{C}$, where X is a compact connected Riemann surface of genus at least three. Let $mathcal{M}_X(D_L)$ denote the moduli space of all semistable $ au$-connections of rank n, where $ngeq 2$, with the property that the induced $ au$-connection on the top exterior product is isomorphic to (L, DL). Let $mathcal{M}_Y(D_M)$ be the similar moduli space for another Riemann surface Y with genus(Y) = genus(X), where DM is a $ au$-connection on a line bundle M over $Y imes mathbb{C}$. We prove that if the variety $mathcal{M}_X(D_L)$ is isomorphic to $mathcal{M}_Y(D_M)$, then X is isomorphic to Y. Let $mathcal{M}^D_X$ denote the moduli space of all rank n flat connections on X. We prove that $mathcal{M}^D_X$ determines X up to finitely many Riemann surfaces. For the very general Riemann surface X, the variety $mathcal{M}^D_X$ determines X.
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