Corpus ID: 16958948

Quantum Cohomology Rings of Toric Manifolds

@article{Batyrev1993QuantumCR,
  title={Quantum Cohomology Rings of Toric Manifolds},
  author={V. Batyrev},
  journal={arXiv: Algebraic Geometry},
  year={1993}
}
  • V. Batyrev
  • Published 1993
  • Mathematics
  • arXiv: Algebraic Geometry
We compute the quantum cohomology ring $H^*_{\varphi}({\bf P}, {\bf C})$ of an arbitrary $d$-dimensional smooth projective toric manifold ${\bf P}_{\Sigma}$ associated with a fan $\Sigma$. The multiplicative structure of $H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ depends on the choice of an element $avarphi$ in the ordinary cohomology group $H^2({\bf P}_{\Sigma}, {\bf C})$. There are many properties of the quantum cohomology rings $H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ which are supposed to… Expand
Quantum cohomology for a class of non-Fano toric varieties
The main aim of this paper is to give a description for the structure of the (small) quantum cohomology ring of the toric variety $X={\bb P}(\oplus _{i=1}^r{\cal O}_{{\bb P}^1}(a_i))$ withExpand
On Landau-Ginzburg Systems and $\mathcal{D}^b(X)$ of projective bundles
Let $X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^s} \oplus \bigoplus_{i=1}^r \mathcal{O}_{\mathbb{P}^s}(a_i))$ be a Fano projective bundle over $\mathbb{P}^s$ and denote by $Crit(X) \subsetExpand
Quantum cohomology of partial flag manifolds $$F_{n_1 } \ldots _{n_k }$$
AbstractWe compute the quantum cohomology rings of the partial flag manifolds $$F_{n_1 } \ldots _{n_k } = U(n)/(U(n_1 ) \times \cdots \times U(n_k ))$$ . The inductive computation uses the idea ofExpand
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomialsExpand
On Landau-Ginzburg systems and $\mathcal{D}^b(X)$ of various toric Fano manifolds with small picard group
For a toric Fano manifold $X$ denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. Examples of toric Fano manifolds withExpand
On Landau-Ginzburg systems, Quivers and Monodromy
Let $X$ be a toric Fano manifold and denote by $Crit(f_X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the corresponding Landau-Ginzburg system of equations. For toric Del-Pezzo surfaces andExpand
Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds
In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove theirExpand
Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds
Let $X$ be a compact toric K\"ahler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined openExpand
Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras
A version of mirror symmetry predicts a ring isomorphism between quantum cohomology of a symplectic manifold and Jacobian algebra of the Landau-Ginzburg mirror, and for toric manifoldsExpand
Tropical counting from asymptotic analysis on Maurer-Cartan equations
Let $X = X_\Sigma$ be a toric surface and $(\check{X}, W)$ be its Landau-Ginzburg (LG) mirror where $W$ is the Hori-Vafa potential. We apply asymptotic analysis to study the extended deformationExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 26 REFERENCES
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomialsExpand
Quantum algebraic geometry of superstring compactifications
We investigate the algebrao-geometric structure which is inherent in 2-dimensional conformally invariant quantum field theories with N=2 supersymmetry, and its relation to the Calabi-Yau manifoldsExpand
The Topology of Torus Actions on Symplectic Manifolds
This is an extended second edition of "The Topology of Torus Actions on Symplectic Manifolds" published in this series in 1991. The material and references have been updated. Symplectic manifolds andExpand
The homogeneous coordinate ring of a toric variety
This paper will introduce the homogeneous coordinate ring S of a toric variety X. The ring S is a polynomial ring with one variable for each one-dimensional cone in the fan ∆ determining X, and S hasExpand
Multiple mirror manifolds and topology change in string theory
Abstract We use mirror symmetry to establish the first concrete arena of spacetime topology change in string theory. In particular, we establish that the quantum theories based on certain nonlinearExpand
Construction and couplings of mirror manifolds
Abstract We present an analysis of the conjectured existence of Calabi-Yau “mirror manifolds” for the case where the starting manifold is Y 4,5 . We construct mirror pairs with equal but oppositeExpand
Decomposition of Toric Morphisms
(0.1) This paper applies the ideas of Mori theory [4] to toric varieties. Let X be a projective tonic variety (over any field) constructed from a simplicial fan F. The cone of effective 1-cyclesExpand
Topological Mirrors and Quantum Rings
Aspects of duality and mirror symmetry in string theory are discussed. We emphasize, through examples, the importance of loop spaces for a deeper understanding of the geometrical origin of dualitiesExpand
...
1
2
3
...