The Local Gromov–Witten Theory of $${\mathbb{C}\mathbb{P}^1}$$ and Integrable Hierarchies

  title={The Local Gromov–Witten Theory of \$\$\{\mathbb\{C\}\mathbb\{P\}^1\}\$\$ and Integrable Hierarchies},
  author={Andrea Brini},
  journal={Communications in Mathematical Physics},
  • A. Brini
  • Published 3 February 2010
  • Mathematics
  • Communications in Mathematical Physics
In this paper we begin the study of the relationship between the local Gromov–Witten theory of Calabi–Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large number of cases, the Hamiltonian dispersionless hierarchies that govern the full-descendent genus zero theory. Our main tool is the application of Dubrovin’s formalism, based on associativity equations, to the known results on the genus zero theory from local… 

Vortex Partition Functions, Wall Crossing and Equivariant Gromov–Witten Invariants

In this paper we identify the problem of equivariant vortex counting in a (2,2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov–Witten

Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve

The stringy instanton partition function

A bstractWe perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A1 singularity. This is accomplished via supersymmetric localization on the

Rational reductions of the 2D-Toda hierarchy and mirror symmetry

We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a

Open Topological Strings and Integrable Hierarchies: Remodeling the A-Model

We set up, purely in A-model terms, a novel formalism for the global solution of the open and closed topological A-model on toric Calabi-Yau threefolds. The starting point is to build on recent

On the integrable hierarchy for the resolved conifold

  • M. AlimA. Saha
  • Mathematics
    Bulletin of the London Mathematical Society
  • 2022
We provide a direct proof of a conjecture of Brini relating the Gromov–Witten (GW) theory of the resolved conifold to the Ablowitz–Ladik (AL) integrable hierarchy at the level of primaries. In doing

Frobenius structures on double Hurwitz spaces

We construct Frobenius structures of "dual type" on the moduli space of ramified coverings of $\mathbb{P}^1$ with given ramification type over two points, generalizing a construction of Dubrovin. A

A modified melting crystal model and the Ablowitz–Ladik hierarchy

This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum

Old and New Reductions of Dispersionless Toda Hierarchy

This paper is focused on geometric aspects of two particular types of finite- variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of \Landau{Ginzburg



Gromov--Witten Theory of CP^1 and Integrable Hierarchies

The ancestor Gromov--Witten invariants of a compact {\Kahler} manifold $X$ can be organized in a generating function called the total ancestor potential of $X$. In this paper, we construct Hirota

Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations

We compute, with symplectic field theory (SFT) techniques, the Gromov-Witten theory of $${\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}}$$, i.e., the complex projective line with a orbifold points. A

The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants

The theory of genus zero Gromov-Witten invariants associates to a compact symplectic manifold X a Frobenius manifold H (also known as the small phase space of X) whose underlying flat manifold is the

Topological Strings and (Almost) Modular Forms

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural

Integrable systems and holomorphic curves

In this paper we attempt a self-contained approach to infinite dimensional Hamiltonian systems appearing from holomorphic curve counting in Gromov-Witten theory. It consists of two parts. The first

Extended Seiberg-Witten theory and integrable hierarchy

The prepotential of the effective = 2 super-Yang-Mills theory, perturbed in the ultraviolet by the descendents ?d4??tr ??k+1 of the single-trace chiral operators, is shown to be a particular

Hirota quadratic equations for the extended Toda hierarchy

The Extended Toda Hierarchy (shortly ETH) was introduce by E. Getzler \cite{Ge} and independently by Y. Zhang \cite{Z} in order to describe an integrable hierarchy which governs the Gromov--Witten

The local Gromov-Witten theory of curves

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evalu- ation of basic integrals in the local Gromov-Witten theory of P

Hodge integrals and Gromov-Witten theory

Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of these

Quantum Riemann–Roch, Lefschetz and Serre

Given a holomorphic vector bundle E over a compact Kahler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f :Σ → X with the