# 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} }

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.

## 19 Citations

Double Ramification Cycles and Integrable Hierarchies

- Mathematics
- 2015

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

- Mathematics
- 2020

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 $\mathbb{P}^1$

- MathematicsMathematical 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

- Mathematics
- 2022

. 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

- Mathematics
- 2022

. 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

- MathematicsBulletin 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…

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