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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