432 research outputs found
Intersections in genus 3 and the Boussinesq hierarchy
In this note we prove that the enlarged Witten's conjecture is true in the
case of the Boussinesq hierarchy for correlators in genus 3 with descendants
only at one point
Combinatorics of binomial decompositions of the simplest Hodge integrals
We reduce the calculation of the simplest Hodge integrals to some sums over
decorated trees. Since Hodge integrals are already calculated, this gives a
proof of a rather interesting combinatorial theorem and a new representation of
Bernoulli numbers.Comment: 16 page
Some relations for one-part double Hurwitz numbers
In this very short note we slightly generalize some relations for one-part
double Hurwitz numbers from math.AG/0209282.Comment: 3 page
On double Hurwitz numbers in genus 0
We study double Hurwitz numbers in genus zero counting the number of covers
\CP^1\to\CP^1 with two branching points with a given branching behavior. By
the recent result due to Goulden, Jackson and Vakil, these numbers are
piecewise polynomials in the multiplicities of the preimages of the branching
points. We describe the partition of the parameter space into polynomiality
domains, called chambers, and provide an expression for the difference of two
such polynomials for two neighboring chambers. Besides, we provide an explicit
formula for the polynomial in a certain chamber called totally negative, which
enables us to calculate double Hurwitz numbers in any given chamber as the
polynomial for the totally negative chamber plus the sum of the differences
between the neighboring polynomials along a path connecting the totally
negative chamber with the given one.Comment: 17 pages, 3 figure
On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket
In our recent paper we proved the polynomiality of a Poisson bracket for a
class of infinite-dimensional Hamiltonian systems of PDE's associated to
semi-simple Frobenius structures. In the conformal (homogeneous) case, these
systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the
first Poisson structure of their hierarchy. Our approach was based on a very
involved computation of a deformation formula for the bracket with respect to
the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the
structure of that deformation formula. In particular, we reprove it using a
deformation formula for weak quasi-Miura transformation that relates our
hierarchy of PDE's with its dispersionless limit.Comment: 21 page
On double Hurwitz numbers with completed cycles
In this paper, we collect a number of facts about double Hurwitz numbers,
where the simple branch points are replaced by their more general analogues ---
completed (r+1)-cycles. In particular, we give a geometric interpretation of
these generalised Hurwitz numbers and derive a cut-and-join operator for
completed (r+1)-cycles. We also prove a strong piecewise polynomiality property
in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural
ELSV/GJV-type formula, that is, an expression in terms of some intrinsic
combinatorial constants that might be related to the intersection theory of
some analogues of the moduli space of curves. The structure of these
conjectural "intersection numbers" is discussed in detail.Comment: 31 page
Tautological relations in Hodge field theory
We propose a Hodge field theory construction that captures algebraic
properties of the reduction of Zwiebach invariants to Gromov-Witten invariants.
It generalizes the Barannikov-Kontsevich construction to the case of higher
genera correlators with gravitational descendants.
We prove the main theorem stating that algebraically defined Hodge field
theory correlators satisfy all tautological relations. From this perspective
the statement that Barannikov-Kontsevich construction provides a solution of
the WDVV equation looks as the simplest particular case of our theorem. Also it
generalizes the particular cases of other low-genera tautological relations
proven in our earlier works; we replace the old technical proofs by a novel
conceptual proof.Comment: 35 page
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