Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties

Batyrev, Victor V.

Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties

Victor V. Batyrev
(
- Essen U.
)

Oct, 1993

44 pages

Published in:

J.Alg.Geom. 3 (1994) 493-545

e-Print:

alg-geom/9310003 [math.AG]

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Abstract: (arXiv)

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 polynomials

f

with a fixed

n

-dimensional Newton polyhedron

\Delta

in

n

-dimensional algebraic torus

{\bf T} =({\bf C}^*)^n

. If the family

{\cal F}(\Delta)

defined by a Newton polyhedron

\Delta

consists of

(n-1)

-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron

\Delta^*

in the dual space defines another family

{\cal F}(\Delta^*)

of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau

3

-folds. Our method allows to construct many new examples of Calabi-Yau

3

-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families

{\cal F}(\Delta)

and

{\cal F}(\Delta^*)

.

Note:

43 pages, Latex

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