Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 067, 23 pages      arXiv:2212.01695      https://doi.org/10.3842/SIGMA.2023.067

Real Slices of ${\rm SL}(r,\mathbb{C})$-Opers

Indranil Biswas a, Sebastian Heller b and Laura P. Schaposnik c
a) Department of Mathematics, Shiv Nadar University, NH91, Tehsil Dadri, Greater Noida, Uttar Pradesh 201314, India
b) Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, P.R. China
c) Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St, Chicago, IL 60607, USA

Received April 12, 2023, in final form September 05, 2023; Published online September 16, 2023

Abstract
Through the action of an anti-holomorphic involution $\sigma$ (a real structure) on a Riemann surface $X$, we consider the induced actions on ${\rm SL}(r,\mathbb{C})$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of ${\rm SL}(r,\mathbb{C})$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.

Key words: opers; real structure; differential operator; anti-holomorphic involution; real slice.

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