Introduction
The early history of mirror symmetry has been told many times; we will only
summarize it briefly here. The story begins with the introduction of Calabi-Yau
compactifications in string theory in 1985 [11]. The idea is that, since superstring
theory requires a ten-dimensional space-time, one reconciles this with the observed
universe by requiring (at least locally) that space-time take the form
R1,3
× X,
where
R1,3
is usual Minkowski space-time and X is a very small six-dimensional
Riemannian manifold. The desire for the theory to preserve the supersymmetry
of superstring theory then leads to the requirement that X have SU(3) holonomy,
i.e., be a Calabi-Yau manifold. Thus string theory entered the realm of algebraic
geometry, as any non-singular projective threefold with trivial canonical bundle
carries a metric with SU(3) holonomy, thanks to Yau’s proof of the Calabi conjecture
[113].
This generated an industry in the string theory community devoted to produc-
ing large lists of examples of Calabi-Yau threefolds and computing their invariants,
the most basic of which are the Hodge numbers h1,1 and h1,2.
In 1989, a rather surprising observation came out of this work. Candelas,
Lynker and Schimmrigk [12] provided a list of Calabi-Yau hypersurfaces in weighted
projective space which exhibited an obvious symmetry: if there was a Calabi-Yau
threefold with Hodge numbers given by a pair
(h1,1,h1,2),
then there was often also
one with Hodge numbers given by the pair
(h1,2,h1,1).
Independently, guided by
certain observations in conformal field theory, Greene and Plesser [36] studied the
quintic threefold and its mirror partner. If we let be the solution set in
P4
of
the equation
x0
5
+ · · · + x4
5
ψx0x1x2x3x4 = 0
for ψ C, then for most ψ, is a non-singular quintic threefold, and as such, has
Hodge numbers
h1,1(Xψ)
= 1,
h1,2(Xψ)
= 101.
On the other hand, the group
G =
{(a0,...,a4)|ai μ5,
4
i=0
ai = 1}
{(a, a, a, a, a)|a μ5}
acts diagonally on P4, via
(x0,...,x4) (a0x0,...,a4x4).
Here μ5 is the group of fifth roots of unity. This action restricts to an action on
Xψ, and the quotient Xψ/G is highly singular. However, these singularities can be
xi
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