Parabolic opers and differential operators
Résumé
Parabolic ${\rm SL}(r,\mathbb C)$--opers were defined and investigated in \cite{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 ${\rm SL}(r,\mathbb C)$--opers on a Riemann
surface $X$ with given singular divisor $S \,\subset\, X$ and with fixed parabolic weights satisfying
the condition that all parabolic weights at any $x_i\, \in\, S$ 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 ${\rm SL}(r,\mathbb C)$--connection.