Moduli spaces of vector bundles on a curve and opers

Abstract

Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \,2$$ , and let $$\xi $$ be a holomorphic line bundle on X with $$\xi ^{\otimes 2}\,=\, {\mathcal O}_X$$ . Fix a theta characteristic $${\mathbb {L}}$$ on X. Let $${\mathcal M}_X(r,\xi )$$ be the moduli space of stable vector bundles E on X of rank r such that $$\bigwedge ^r E\,=\, \xi $$ and $$H^0(X,\, E\otimes {\mathbb L})\,=\, 0$$ . Consider the quotient of $${\mathcal M}_X(r,\xi )$$ by the involution given by $$E\, \longmapsto \, E^*$$ . We construct an algebraic morphism from this quotient to the moduli space of $$\textrm{SL}(r,{\mathbb C})$$ opers on X. Since $$\dim {\mathcal M}_X(r,\xi )$$ coincides with the dimension of the moduli space of $$\textrm{SL}(r,{\mathbb C})$$ opers, it is natural to ask about the injectivity and surjectivity of this map.