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First, let me introduce the background of the problem: While studying chapter 4 of the Hitchin's paper "THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE", I encountered an issue. Hitchin ...

This is a technical question on Taubes' paper: Gauge theory over periodic end. on Page 378. Recall that: Let $Y$ be a closed manifold, with $b_1=1$, and $\tilde Y$ be the $\mathbb Z$-covering of $...