# A mirror theorem for toric complete intersections

@article{Givental1997AMT, title={A mirror theorem for toric complete intersections}, author={Alexander Givental}, journal={arXiv: Algebraic Geometry}, year={1997} }

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions. Revision 03.03.97: we correct an error in Introduction.

#### 401 Citations

THEOREM FOR HOMOGENEOUS SPACES

- 1999

We formulated a mirror-free approach to the mirror conjecture, namely, quantum hyperplane section conjecture, and proved it in the case of nonnegative complete intersections in homogeneous manifolds.… Expand

Quantum hyperplane section theorem for homogeneous spaces

- Mathematics
- 1997

Author(s): Kim, Bumsig | Abstract: We formulated a mirror-free approach to the mirror conjecture, namely, quantum hyperplane section conjecture, and proved it in the case of nonnegative complete… Expand

Homological Mirror Symmetry for Toric Varieties

- 2009

Given a smooth projective toric variety X, we construct an A∞ category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent… Expand

Shift operators and toric mirror theorem

- Mathematics
- 2014

We give a new proof of Givental's mirror theorem for toric manifolds using shift operators of equivariant parameters. The proof is almost tautological: it gives an A-model construction of the… Expand

A mirror construction for the big equivariant quantum cohomology of toric manifolds

- Mathematics
- 2015

We identify a certain universal Landau–Ginzburg model as a mirror of the big equivariant quantum cohomology of a (not necessarily compact or semipositive) toric manifold. The mirror map and the… Expand

Stokes Matrix for the Quantum Cohomology of Cubic Surfaces

- Mathematics
- 2005

We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of smooth cubic… Expand

Pseudoholomorphic Curves and Mirror Symmetry

- Mathematics
- 2006

This survey article was written for Prof. Alan Weinstein’s Symplectic Geometry (Math 242) course at UC Berkeley in Fall 2005. We review dierent constructions arising from the theory of… Expand

TORIC FIBRATIONS AND MIRROR SYMMETRY

- 2007

The relation between the quantum D-modules of a smooth variety X and a toric bundle is studied here. We describe the relation completely when X is a semi-ample complete intersection in a toric… Expand

Wall crossing for symplectic vortices and quantum cohomology

- Mathematics
- 2002

We derive a wall crossing formula for the symplectic vortex invariants of toric manifolds. As an application, we give a proof of Batyrev's formula for the quantum cohomology of a monotone toric… Expand

Equivariant mirrors and the Virasoro conjecture for flag manifolds

- Mathematics
- 2002

We found an explicit description of all $GL(n,\RR)$-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro… Expand

#### References

SHOWING 1-10 OF 22 REFERENCES

Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds

- 1996

In this paper, we first give an intersection theory for moduli problems for nonlinear elliptic operators with certain precompact space of solutions in differential geometry. Then we apply the theory… Expand

Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture

- Mathematics
- 1996

A generalization of the mirror conjecture is proven for the manifolds of complete flags in C^n.

A symplectic fixed point theorem for toric manifolds

- Mathematics
- 1995

In this paper, by a toric manifold we mean a non-singular symplectic quotient M = ℂ n //T k of the standard symplectic space by a linear torus action. Such a toric manifold is in fact a complex… Expand

Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

- Mathematics
- 1996

We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of… Expand

Gromov-Witten classes, quantum cohomology, and enumerative geometry

- Physics, Mathematics
- 1994

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic… Expand

Enumeration of Rational Curves Via Torus Actions

- Physics, Mathematics
- 1995

This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry.

Homological geometry I. Projective hypersurfaces

- Mathematics
- 1995

Introduction Consider a generic quintic hypersurfaceX inCP . It is an example of Calabi– Yau 3-folds. It follows from Riemann–Roch formula, that rational curves on a generic Calabi–Yau 3-fold should… Expand

Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties

- Mathematics, Physics
- 1993

We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV1,...,Vr in a toric… Expand

Gravitational Quantum Cohomology

- Physics
- 1997

We discuss how the theory of quantum cohomology may be generalized to "gravitational quantum cohomology" by studying topological σ models coupled to two-dimensional gravity. We first consider σ… Expand

Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties

- Physics, Mathematics
- 1995

Abstract We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety V or a… Expand