# Picard-Fuchs equations and mirror maps for hypersurfaces

@article{Morrison1991PicardFuchsEA, title={Picard-Fuchs equations and mirror maps for hypersurfaces}, author={David R. Morrison}, journal={arXiv: High Energy Physics - Theory}, year={1991} }

We describe a strategy for computing Yukawa couplings and the mirror map, based on the Picard-Fuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes in the case of quintic hypersurfaces.) We then explain a technique of Griffiths which can be used to compute the Picard-Fuchs equations of hypersurfaces. Finally, we carry out the computation for four specific examples (including quintic hypersurfaces, previously done by Candelas et al.). This… Expand

#### 124 Citations

Monodromy of Inhomogeneous Picard-Fuchs Equations

- Mathematics, Physics
- 2013

We study low-degree curves on one-parameter Calabi-Yau hypersurfaces, and their contribution to the space-time superpotential in a superstring compactification with D-branes. We identify all lines… Expand

On the mixed-twist construction and monodromy of associated Picard-Fuchs systems

- Mathematics
- 2021

We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard-rank ρ ≥ 16. Upon identifying a particular Jacobian elliptic fibration on its… Expand

Picard-Fuchs equations of special one-parameter families of invertible polynomials

- Mathematics
- 2011

The thesis deals with calculating the Picard-Fuchs equation of special one-parameter families of invertible polynomials. In particular, for an invertible polynomial g(x1, . . . , xn) we consider the… Expand

Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds

- Mathematics
- 2006

Abstract In this paper we are concerned with the monodromy of Picard-Fuchs differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the… Expand

Nonabelian mirrors and Gromov-Witten invariants

- Physics, Mathematics
- 2020

We propose Picard-Fuchs equations for periods of nonabelian mirrors in this paper. The number of parameters in our Picard-Fuchs equations is the rank of the gauge group of the nonabelian GLSM, which… Expand

Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

- Physics, Mathematics
- 1995

Abstract We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton corrected Yukawa couplings and the topological one-loop partition function to the case of complete… Expand

Monodromy of Picard-Fuchs dierential equations for Calabi-Yau threefolds

- Mathematics
- 2008

In this paper we are concerned with the monodromy of Picard-Fuchs dif- ferential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric… Expand

Considerations of one modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kahler potentials and mirror maps

- Physics
- 1993

Abstract We consider Calabi-Yau compactifications with one Kahler modulus. Following the method of Candelas et al. we use the mirror hypotheses to solve the quantum theory exactly in dependence of… Expand

Picard-Fuchs Equations and Special Geometry

- Physics
- 1993

We investigate the system of holomorphic differential identities implied by special Kahlerian geometry of four-dimensional N=2 supergravity. For superstring compactifications on Calabi-Yau threefolds… Expand

Rational curves on Calabi-Yau manifolds: verifying predictions of mirror symmetry

- Mathematics, Physics
- 1993

Mirror symmetry, a phenomenon in superstring theory, has recently been used to give tentative calculations of several numbers in algebraic geometry. In this paper, the numbers of lines and conics on… Expand

#### References

SHOWING 1-10 OF 42 REFERENCES

Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians

- Mathematics
- 1992

We give a mathematical account of a recent string theory calcula- tion which predicts the number of rational curves on the generic quintic three- fold. Our account involves the interpretation of… Expand

Picard-Fuchs equations and the moduli space of superconformal field theories

- Physics
- 1991

Abstract We derive simple techniques which allow us to relate Picard-Fuchs differential equations for the periods of holomorphic p -forms on certain complex manifolds, to their moduli space and its… Expand

On the Periods of Certain Rational Integrals: II

- Mathematics
- 1969

In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show how… Expand

Differential equations for periods and flat coordinates in two-dimensional topological matter theories

- Physics
- 1991

Abstract We consider two-dimensional topological Landau-Ginzburg models. In order to obtain the free energy of these models, and to determine the Kahler potential for the marginal perturbations, one… Expand

Topological field theory and rational curves

- Mathematics, Physics
- 1993

We analyze the quantum field theory corresponding to a string propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear σ-model and… Expand

Calabi-Yau manifolds in weighted P4

- Physics
- 1990

Abstract It has recently been recognized that the relation between exactly solvable conformal field theory compactifications of the Heterotic String and Calabi-Yau manifolds necessarily involves the… Expand

Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems

- Mathematics
- 1970

0. Introduction 229 Par t I. Summary of main results 231 1. The geometric situation giving rise to variation of Hodge structure. . . . 231 2. Data given by the variation of Hodge structure 232 3.… Expand

Duality in {Calabi-Yau} Moduli Space

- Physics
- 1990

We describe a duality in the moduli space of string vacua which pairs topologically distinct Calabi-Yau manifolds and shows that the yield isomorphic conformal theories. At the level of the… Expand

On the zeta function of a hypersurface

- Mathematics
- 1962

Abstract : This article is concerned with the further development of the methods of p-adic analysis used in an earlier article to study the zeta function of an algebraic variety defined over a finite… Expand

Intersection form for quasi-homogeneous singularities

- Mathematics
- 1977

We consider quasi-homogeneous polynomials with an isolated singular point at the origin. We calculate the mixed Hodge structure of the cohomology of the Milnor fiber and give a proof for a conjecture… Expand