# Gromov - Witten invariants and integrable hierarchies of topological type

@article{Dubrovin2013GromovW,
title={Gromov - Witten invariants and integrable hierarchies of topological type},
author={Boris Dubrovin},
journal={arXiv: Mathematical Physics},
year={2013}
}
• B. Dubrovin
• Published 3 December 2013
• Mathematics
• arXiv: Mathematical Physics
We outline two approaches to the construction of integrable hierarchies associated with the theory of Gromov - Witten invariants of smooth projective varieties. We argue that a comparison of these two approaches yields nontrivial constraints on Chern numbers of varieties with semisimple quantum cohomology.
Double Ramification Cycles and Integrable Hierarchies
In this paper we present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. We conjecture that in the semisimple case our hierarchy is related to the
On Gromov–Witten invariants of P
• Mathematics
• 2018
We propose a conjectural explicit formula of generating series of a new type for Gromov–Witten invariants of P of all degrees in full genera.
On Emergent Geometry of the Gromov-Witten Theory of Quintic Calabi-Yau Threefold
We carry out the explicit computations that are used to write down the integrable hierarchy associated with the quintic Calabi-Yau threefold. We also do the calculations for the geometric structures
On Gromov–Witten invariants of $\mathbb{P}^1$
• Mathematics
Mathematical Research Letters
• 2019
We propose a conjectural explicit formula of generating series of a new type for Gromov--Witten invariants of $\mathbb{P}^1$ of all degrees in full genera.
GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back
. Dubrovin establishes the relationship between the GUE partition function and the partition function of Gromov–Witten invariants of the complex projective line. In this paper, we give a direct proof
Degree zero Gromov--Witten invariants for smooth curves
. For a smooth projective curve, we derive a closed formula for the generating series of its Gromov–Witten invariants in genus g and degree zero. It is known that the calculation of these invariants
On the integrable hierarchy for the resolved conifold
• Mathematics
Bulletin of the London Mathematical Society
• 2022
. We provide a direct proof of a conjecture of Brini relating the Gromov– Witten theory of the resolved conifold to the Ablowitz–Ladik integrable hierarchy at the level of primaries. In doing so, we
Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality
• Mathematics
• 2021
For each of the simple Lie algebras g = Al, Dl or E6, we show that the all-genera one-point FJRW invariants of g-type, after multiplication by suitable products of Pochhammer symbols, are the
Classical Hurwitz numbers and related combinatorics
• Mathematics
• 2016
In 1891 Hurwitz [30] studied the number Hg,d of genus g ≥ 0 and degree d ≥ 1 coverings of the Riemann sphere with 2g + 2d− 2 fixed branch points and in particular found a closed formula for Hg,d for

## References

SHOWING 1-10 OF 58 REFERENCES
An algebro-geometric proof of Witten's conjecture
• Mathematics
• 2007
We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating
Intersection theory on the moduli space of curves and the matrix airy function
We show that two natural approaches to quantum gravity coincide. This identity is nontrivial and relies on the equivalence of each approach to KdV equations. We also investigate related mathematical
Geometry and analytic theory of Frobenius manifolds
Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of
ON ALMOST DUALITY FOR FROBENIUS MANIFOLDS
We present a universal construction of almost duality for Frobenius man- ifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We
Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation
• Mathematics
• 1998
Abstract:We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a
Derived categories of toric varieties III
We prove that the derived McKay correspondence holds for the cases of finite abelian groups and subgroups of $$\mathrm{GL}(2,\mathbf {C})$$GL(2,C). We also prove that K-equivalent toric birational
On Betti numbers and Chern classes of varieties with trivial odd cohomology groups
It was noticed in a very recent preprint of T. Eguchi, K. Hori, and Ch.-Sh. Xiong (hep-th/9703086) that a curious identity between Betti numbers and Chern classes holds for many examples of Fano
Frobenius manifolds and Virasoro constraints
• Mathematics
• 1998
Abstract. For an arbitrary Frobenius manifold, a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. In