107 research outputs found
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
correction to free energy in hermitian two-matrix model
Using the loop equations we find an explicit expression for genus 1
correction in hermitian two-matrix model in terms of holomorphic objects
associated to spectral curve arising in large N limit. Our result generalises
known expression for in hermitian one-matrix model. We discuss the
relationship between , Bergmann tau-function on Hurwitz spaces, G-function
of Frobenius manifolds and determinant of Laplacian over spectral curve
On the isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one
The isomonodromic tau-function for the Hurwitz spaces of branched coverings
of genus zero and one are constructed explicitly. Such spaces may be equipped
with the structure of a Frobenius manifold and this introduces a flat
coordinate system on the manifold. The isomonodromic tau-function, and in
particular the associated -function, are rewritten in these coordinates and
an interpretation in terms of the caustics (where the multiplication is not
semisimple) is given.Comment: 18 page
Isomonodromic tau function on the space of admissible covers
The isomonodromic tau function of the Fuchsian differential equations
associated to Frobenius structures on Hurwitz spaces can be viewed as a section
of a line bundle on the space of admissible covers. We study the asymptotic
behavior of the tau function near the boundary of this space and compute its
divisor. This yields an explicit formula for the pullback of the Hodge class to
the space of admissible covers in terms of the classes of compactification
divisors.Comment: a few misprints corrected, journal reference adde
Genus one contribution to free energy in hermitian two-matrix model
We compute an the genus 1 correction to free energy of Hermitian two-matrix
model in terms of theta-functions associated to spectral curve arising in large
N limit. We discuss the relationship of this expression to isomonodromic
tau-function, Bergmann tau-function on Hurwitz spaces, G-function of Frobenius
manifolds and determinant of Laplacian in a singular metric over spectral
curve.Comment: 25 pages, detailed version of hep-th/040116
Multiplicative Functions of Numbers Set and Logarithmic Identities. Shannon and factorial logarithmic Identities, Entropy and Coentrop
The multiplicative functions characterizing the finite set of positive numbers are introduced in the work.With their help we find the logarithmic identities which connect logarithm of sum of the set numbers and logarithmsof numbers themselves. One of them (contained in the work of Shannon) interconnects three information functions:information Hartley, entropy and coentropy. Shannon's identity allows better to understand the meaning andrelationship of these collective characteristics of information (as the characteristics of finite sets and as probabilisticcharacteristics). The factorial multiplicative function and the logarithmic factorial identity are formed also frominitial set numbers. That identity connects logarithms of factorials of integer numbers and logarithm of factorial oftheir sum
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