Super Schlesinger Equations

The equations describing the dependence of the universal connection on the deformation parameters. The universal deformation space of the connections with regular singularities on P ¹ was constructed by L. Schlesinger [1] (see [2] for a modern treatment). Some solutions to the (super) Schlesinger equations (“strict special” ones, with the structure group reduced to the orthogonal one, and supplied with an additional piece of data) correspond to semisimple Frobenius (super)manifolds [3,4].

Let us consider a supermanifold ℂ^n|n with its natural SUSY structure ℂ^n|n spanned by the vector fields , where () are natural coordinates. Let B be the universal covering of ℂⁿ|ⁿ ∖ (), where pairwise distinct integers α and β run over 1,...,n, then the supermanifold B× P ^1|1 has the direct product SUSY structure, and D∼ _α is the inverse image in B× P ^1|1 of the submanifold in ℂ ^n|n× P ^1|1, where (λ,ξ) are natural coordinates on a big cell of P ^1|1. Furthermore, define D∼ _∞ = B× ∞ ⊂ B× P ^1|1, where ∞...