Abelianisation of Logarithmic sl2\mathfrak{sl}_2-Connections

We prove a functorial correspondence between a category of logarithmic $\mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \Sigma \to X$. The proof is by constructing a pair of inverse functors $\pi^{\text{ab}}, \pi_{\text{ab}}$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.Comment: Comments are always welcome! Version edits: Added a running example (2.5, 2.10, 2.23, 2.25, 2.27, 2.35, 2.39, 2.43, 3.13), Lemma 2.9, and more figures (1, 2, 4, 7, 8, 11). Expanded the discussion after Definition 2.46. Journal reference to follo

Isomonodromic tau-function of Hurwitz Frobenius manifolds and its applications

In this work we find the isomonodromic (Jimbo-Miwa) tau-function corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss several applications of this result. First, we get an explicit expression for the G-function (solution of Getzler's equation) of the Hurwitz Frobenius manifolds. Second, in terms of this tau-function we compute the genus one correction to the free energy of hermitian two-matrix model. Third, we find the Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with quasi-permutation monodromy matrices. Finally, we get a new expression (analog of genus one Ray-Singer formula) for the determinant of Laplace operator in the Poincar\'e metric on Riemann surfaces of an arbitrary genus.Comment: The direct proof of variational formulas on branched coverings is added. The title is modified due to observed coincidence of isomonodromic tau-function of Hurwitz Frobenius manifolds with Bergman tau-function on Hurwitz spaces introduced by the author