Classification of isomonodromy problems on elliptic curves

This paper describes isomonodromy problems in terms of flat $G$-bundles over punctured elliptic curves $\Sigma_\tau$ and connections with regular singularities at marked points. The bundles are classified by their characteristic classes, which are elements of the second cohomology group $H^2(\Sigma_\tau,{\mathscr Z}(G))$, where ${\mathscr Z}(G)$ is the centre of $G$. For any complex simple Lie group $G$ and any characteristic class the moduli space of flat connections is defined, and for them the monodromy-preserving deformation equations are given in Hamiltonian form together with the corresponding Lax representation. In particular, they include the Painlevé VI equation, its multicomponent generalizations, and the elliptic Schlesinger equations. The general construction is described for punctured complex curves of arbitrary genus. The Drinfeld–Simpson (double coset) description of the moduli space of Higgs bundles is generalized to the case of the space of flat connections. This local description makes it possible to establish the Symplectic Hecke Correspondence for a wide class of monodromy-preserving problems classified by the characteristic classes of the underlying bundles. In particular, the Painlevé VI equation can be described in terms of $\operatorname{SL}(2,{\mathbb C})$-bundles. Since ${\mathscr Z}(\operatorname{SL}(2,{\mathbb C}))={\mathbb Z}_2$, the Painlevé VI equation has two representations related by the Hecke transformation: 1) as the well-known elliptic form of the Painlevé VI equation (for trivial bundles); 2) as the non-autonomous Zhukovsky–Volterra gyrostat (for non-trivial bundles).
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