In mathematics, mirror symmetry is a conjectural relationship between certain
Calabi–Yau manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...
s and a constructed "mirror manifold". The conjecture allows one to relate the number of
rational curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s on a Calabi-Yau manifold (encoded as
Gromov–Witten invariant
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic man ...
s) to integrals from a family of varieties (encoded as
period integrals on a
variation of Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
s). In short, this means there is a relation between the number of genus
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s of degree
on a Calabi-Yau variety
and integrals on a dual variety
. These relations were original discovered by
Candelas
The candela ( or ; symbol: cd) is the unit of luminous intensity in the International System of Units (SI). It measures luminous power per unit solid angle emitted by a light source in a particular direction. Luminous intensity is analogous to ...
,
de la Ossa, Green, and Parkes
in a paper studying a generic
quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathem ...
in
as the variety
and a construction
from the quintic
Dwork family In algebraic geometry, a Dwork family is a one-parameter family of hypersurfaces depending on an integer ''n'', studied by Bernard Dwork. Originally considered by Dwork in the context of local zeta-functions, such families have been shown to have re ...
giving
. Shortly after,
Sheldon Katz wrote a summary paper outlining part of their construction and conjectures what the rigorous mathematical interpretation could be.
Constructing the mirror of a quintic threefold
Originally, the construction of mirror manifolds was discovered through an ad-hoc procedure. Essentially, to a generic
quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathem ...
there should be associated a one-parameter family of
Calabi-Yau manifolds
which has multiple singularities. After
blowing up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with th ...
these
singularities, they are resolved and a new Calabi-Yau manifold
was constructed. which had a flipped Hodge diamond. In particular, there are isomorphisms
but most importantly, there is an isomorphism
where the string theory (the ''A-model'' of
) for states in
is interchanged with the string theory (the ''B-model'' of
) having states in
. The string theory in the A-model only depended upon the Kahler or symplectic structure on
while the B-model only depends upon the complex structure on
. Here we outline the original construction of mirror manifolds, and consider the string-theoretic background and conjecture with the mirror manifolds in a later section of this article.
Complex moduli
Recall that a generic
quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathem ...
in
is defined by a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
of degree
. This polynomial is equivalently described as a global section of the
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
.
Notice the vector space of global sections has dimension
but there are two equivalences of these polynomials. First, polynomials under scaling by the
algebraic torus In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher ...
(non-zero scalers of the base field) given equivalent spaces. Second, projective equivalence is given by the
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
group of
,
which is
dimensional. This gives a
dimensional parameter space
since
, which can be constructed using
Geometric invariant theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas ...
. The set
corresponds to the equivalence classes of polynomials which define smooth Calabi-Yau quintic threefolds in
, giving a
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
of Calabi-Yau quintics. Now, using
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Al ...
and the fact each Calabi-Yau manifold has trivial
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
, the space of
deformations has an isomorphism
with the
part of the
Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
on
. Using the
Lefschetz hyperplane theorem In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the ...
the only non-trivial cohomology group is
since the others are isomorphic to
. Using the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
and the
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle o ...
, which is the
top Chern class, the dimension of this group is
. This is because
Using the
Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
we can find the dimensions of each of the components. First, because
is Calabi-Yau,
so
giving the Hodge numbers
, hence
giving the dimension of the moduli space of Calabi-Yau manifolds. Because of the
Bogomolev-Tian-Todorov theorem, all such deformations are unobstructed, so the smooth space
is in fact the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
of quintic threefolds. The whole point of this construction is to show how the complex parameters in this moduli space are converted into
Kähler parameters of the mirror manifold.
Mirror manifold
There is a distinguished family of Calabi-Yau manifolds
called the
Dwork family In algebraic geometry, a Dwork family is a one-parameter family of hypersurfaces depending on an integer ''n'', studied by Bernard Dwork. Originally considered by Dwork in the context of local zeta-functions, such families have been shown to have re ...
. It is the
projective family
over the complex plane
. Now, notice there is only a single dimension of complex deformations of this family, coming from
having varying values. This is important because the Hodge diamond of the mirror manifold
has
Anyway, the family
has symmetry group
acting by