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Taufunctions on Hurwitz spaces
 Math. Phys. Anal. Geom
, 2003
"... Abstract. We construct a flat holomorphic line bundle over a connected component of the Hurwitz space of branched coverings of the Riemann sphere P 1. A flat holomorphic connection defining the bundle is described in terms of the invariant Wirtinger projective connection on the branched covering cor ..."
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Abstract. We construct a flat holomorphic line bundle over a connected component of the Hurwitz space of branched coverings of the Riemann sphere P 1. A flat holomorphic connection defining the bundle is described in terms of the invariant Wirtinger projective connection on the branched covering corresponding to a given meromorphic function on a Riemann surface of genus g. In genera 0 and 1 we construct a nowhere vanishing holomorphic horizontal section of this bundle (the “Wirtinger taufunction”). In higher genus we compute the modulus square of the Wirtinger taufunction.
On G−function of Frobenius manifolds related to Hurwitz spaces
 IMRN (2004), no
"... Abstract. The semisimple Frobenius manifolds related to the Hurwitz spaces Hg,N(k1,...,kl) are considered. We show that the corresponding isomonodromic taufunction τI coincides with (−1/2)power of the Bergmann taufunction which was introduced in a recent work by the authors [8]. This enables us to ..."
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Abstract. The semisimple Frobenius manifolds related to the Hurwitz spaces Hg,N(k1,...,kl) are considered. We show that the corresponding isomonodromic taufunction τI coincides with (−1/2)power of the Bergmann taufunction which was introduced in a recent work by the authors [8]. This enables us to calculate explicitly the Gfunction of Frobenius manifolds related to the Hurwitz spaces H0,N(k1,...,kl) and H1,N(k1,...,kl). As simple consequences we get formulas for the Gfunctions of the Frobenius manifolds C N / ˜ W k (AN−1) and C × C N−1 × {ℑz> 0}/J(AN−1), where ˜ W k (AN−1) is an extended affine Weyl group and J(AN−1) is a Jacobi group, in particular, proving the conjecture of [13]. In case of Frobenius manifolds related to Hurwitz spaces Hg,N(k1,...,kl) with g ≥ 2 we obtain formulas for τI  2 which allows to compute the real part of the Gfunction.
Isomonodromic taufunction of Hurwitz Frobenius manifolds and its applications
 Int. Math. Res. Not. (2006), Art. ID
"... Abstract. In this work we find the isomonodromic (JimboMiwa) taufunction corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss several applications of this result. First, we get an explicit expression for the Gfunction (solution of Getzler’s equation) of the Hurwitz Frobeni ..."
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Abstract. In this work we find the isomonodromic (JimboMiwa) taufunction corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss several applications of this result. First, we get an explicit expression for the Gfunction (solution of Getzler’s equation) of the Hurwitz Frobenius manifolds. Second, in terms of this taufunction we compute the genus one correction to the free energy of hermitian twomatrix model. Third, we find the JimboMiwa taufunction of an arbitrary RiemannHilbert problem with quasipermutation monodromy matrices. Finally, we get a new expression (analog of genus one RaySinger formula) for the determinant of Laplace operator in the Poincaré metric on Riemann surfaces of an arbitrary genus.
Bergmann taufunction on Hurwitz spaces and its applications
"... Abstract. The main result of this work is a computation of the Bergmann taufunction on Hurwitz spaces in any genus. This allows to get an explicit formula for the Gfunction of Frobenius manifolds associated to arbitrary Hurwitz spaces, get a new expression for determinant of Laplace operator in Po ..."
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Abstract. The main result of this work is a computation of the Bergmann taufunction on Hurwitz spaces in any genus. This allows to get an explicit formula for the Gfunction of Frobenius manifolds associated to arbitrary Hurwitz spaces, get a new expression for determinant of Laplace operator in Poincaré metric on Riemann surfaces of arbitrary genus, and compute JimboMiwa taufunction of an arbitrary RiemannHilbert problem with quasipermutation monodromies. 1
On the isomonodromic taufunction for the Hurwitz spaces of branched coverings of genus zero and one
 Mathematical Research Letters
"... The Hurwitz space Hg,N is the space of equivalence classes of pairs (L, π), where L is ..."
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The Hurwitz space Hg,N is the space of equivalence classes of pairs (L, π), where L is
On the genus Two Free Energies for Semisimple Frobenius
 Manifold, Russian Journal of Mathematical Physics
"... We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the socalled ”genus two Gfunction”. Conjecturally the genus two Gfunction vanishes for a series of import ..."
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We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the socalled ”genus two Gfunction”. Conjecturally the genus two Gfunction vanishes for a series of important examples of Frobenius manifolds associated with simple singularities as well as for P1orbifolds with positive Euler characteristics. We explain the reasons for such Conjecture and prove it in
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, 2008
"... On the isomonodromic taufunction for the Hurwitz spaces of branched coverings of genus zero and one ..."
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On the isomonodromic taufunction for the Hurwitz spaces of branched coverings of genus zero and one
A Note on Symmetries of WDVV Equations
, 2008
"... We investigate symmetries of WittenDijkgraafE.VerlindeH.Verlinde (WDVV) equations proposed by Dubrovin from bihamiltonian point of view. These symmetries can be viewed as canonical Miura transformations between genuszero bihamiltonian systems of hydrodynamic type. In particular, we show that t ..."
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We investigate symmetries of WittenDijkgraafE.VerlindeH.Verlinde (WDVV) equations proposed by Dubrovin from bihamiltonian point of view. These symmetries can be viewed as canonical Miura transformations between genuszero bihamiltonian systems of hydrodynamic type. In particular, we show that the moduli space of twoprimary models under symmetries of WDVV can be characterized by the polytropic exponent h. Furthermore, we also discuss the transformation properties of free energy at genusone level.