@article{0690436906114003a4f4f6d373f3c3ef,

title = "A Map Between Moduli Spaces of Connections",

abstract = "We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to lift such connections to an elliptic curve. The transformation is as follows: given an elliptic curve C with elliptic quotient π:C→P1, and the logarithmic connection (E,∇) on P1, we may pullback the connection to the elliptic curve to obtain a new connection (π∗E,π∗∇) on C. After suitable birational modifications we bring the connection to a particular normal form. The whole transformation is equivariant with respect to bundle automorphisms and therefore defines a map between the corresponding moduli spaces of connections. The aim of this paper is to describe the moduli spaces involved and compute explicit expressions for the above map in the case where the target space is the moduli space of rank 2 logarithmic connections on an elliptic curve C with two simple poles and trivial determinant.",

author = "Frank Loray and Valente Ram{\'i}rez",

note = "Funding Information: Most of the present work was carried out while the second author was a postdoc at IRMAR. He would like to thank the IRMAR and the Universit{\'e}de Rennes for hosting him during this period. We would like to thank Thiago Fassarella and N{\'e}stor Fernandez Vargas for many valuable discussions on this topic. We also thank Nicolas Tholozan who helped us to understand the action of Φtop on the symplectic 2-form on the monodromy side. We{\textquoteright}re also thankful to the anonymous referees for providing many suggestions to improve the content and clarity of the text. F.L. acknowledges the support of CNRS and the project Foliage ANR-16-CE40-0008. V.R. was supported by the grants PAPIIT IN-106217, CONACYT 219722, and the PRESTIGE postdoc program (coordinated by Campus France and co-financed under the Marie Curie Actions - COFUND of the FP7). He also acknowledges the support of the Centre Henri Lebesgue ANR-11-LABX-0020-01. Publisher Copyright: {\textcopyright} 2020, Institute of Mathematics. All rights reserved.",

year = "2020",

month = dec,

day = "2",

doi = "10.3842/SIGMA.2020.125",

language = "English",

volume = "16",

pages = "1--42",

journal = "Symmetry, integrability and geometry : methods and applications (SIGMA)",

issn = "1815-0659",

publisher = "Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine",

}