# Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it.

Given a (compact) Riemann surface $$\Sigma$$, a $$SL(n,\mathbb{C})$$-oper is a rank $$n$$ holomorphic vector bundle $$E\rightarrow \Sigma$$, s.th. $$\det(E)\cong \mathcal{O}$$ is trivial, together with a holomorphic connection $$\nabla:E\rightarrow E\otimes K_\Sigma$$ ( $$K_\Sigma$$ being the canonical bundle of $$\Sigma$$) satisfying the following list of axioms:

1) There is a flitration by holomorphic subbundles $$0=F^0\subset F^1\subset \cdots \subset F^n=E$$.

2) The connection is Griffiths transverse: $$\nabla(F^i)\subset F^{i+1}\otimes K_\Sigma$$ .

3) The connection is non-degenerate in the following sense: The induced map $$F^i/F^{i-1}\rightarrow F^{i+1}/F^i\otimes K_\Sigma$$ is an isomorphism.

By non-degeneracy $$E$$ is in fact completely specified by $$F^1$$ and for $$SL(n,\mathbb{C})$$ it has to hold $$F^1\cong K^{\frac{n-1}{2}}_\Sigma$$. Having this, an $$SL(n, \mathbb{C})$$ is equivalent to an $$n$$-th order differential operator $$P:\Omega^{\frac{-n+1}{2}}\rightarrow \Omega^{\frac{n-1}{2}}$$, where $$\Omega$$ is the sheaf of holomorphic sections of $$K_\Sigma$$.

Q1 Is the converse true? I.e. given a holomorphic line bundle $$L\rightarrow \Sigma$$ with sheaf of holomorphic sections $$\mathcal{L}$$ and an n-th order holomorphic differential operator $$P:\mathcal{L}\rightarrow \mathcal{L}$$, can I construct from this an $$SL(n,\mathbb{C})$$ oper?

I think to answer to this is yes, but I cannot track a reference for this.

From the discussion I have in my previous question, it seems that to any rank $$n$$ vector bundle $$H$$ of degree zero with a holomorphic connection one gets at least locally a $$n$$-th order holomorphic differential operator defining its flat sections, due to the existence of a local cyclic section.

Q2 As I think the answer to the first question is yes, the local picture cannot glue in general I suppose. So the question is, do the local cyclic sections glue to a global cyclic section if and only if the bundle $$H$$ has an oper structure?

• I think the answer to your question should be contained, or at least intimated, in these notes of Richard Wentworth: arxiv.org/abs/1402.4203. – Andy Sanders May 29 '20 at 17:39
• See also section 2 of the notes arxiv.org/pdf/math/0501398.pdf of Beilinson-Drinfeld where they also basically answer your question. – Andy Sanders May 29 '20 at 17:41
• @AndySanders, nice, thanks for the references! The beginning of section 2 in the paper by Beilinson-Drinfeld directly answers the first question with the additional assumption that the differential operator has nowhere vanishing principal symbol. So representing the vector bundle locally on a coordinate open as $\mathcal{D}_1(U)/ \mathcal{D}_1P(U)$ the differential operators $P(U)$ only glue to a differential operator with non vanishing principal symbol if the bundle has in fact an oper structure. – mtraube May 29 '20 at 18:39
• Indeed. I'm glad I could say something useful. – Andy Sanders May 29 '20 at 18:41