Parabolic opers and differential operators

TL;DR

This paper studies parabolic SL(r,C)-opers on Riemann surfaces, establishing a canonical correspondence with certain holomorphic differential operators satisfying specific symbol conditions, thus linking geometric structures with differential operators.

Contribution

It introduces a new characterization of parabolic SL(r,C)-opers as differential operators with fixed principal and sub-principal symbols, extending the understanding of opers on curves with parabolic structures.

Findings

Space of opers is identified with affine space of differential operators

Differential operators have principal symbol 1 and zero sub-principal symbol

Logarithmic SL(r,C)-connections are characterized by these operators

Abstract

Parabolic SL(r,C)-opers were defined and investigated in [BDP] in the set-up of vector bundles on curves with a parabolic structure over a divisor. Here we introduce and study holomorphic differential operators between parabolic vector bundles over curves. We consider the parabolic SL(r,C)-opers on a Riemann surface X with given singular divisor S and with fixed parabolic weights satisfying the condition that all parabolic weights at any point $x_i$ in S are integral multiples of $\frac{1}{2N_i+1}$, where $N_i > 1$ are fixed integers. We prove that this space of opers is canonically identified with the affine space of holomorphic differential operators of order r between two natural parabolic line bundles on X (depending only on the divisor S and the weights $N_i$) satisfying the conditions that the principal symbol of the differential operators is the constant function 1 and the sub-principal symbol vanishes identically. The vanishing of the sub-principal symbol ensures that the logarithmic connection on the rank r bundle is actually a logarithmic SL(r, C)-connection.