
In
algebraic and
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
which has certain properties, such as
Ricci flatness, yielding applications in
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
. Particularly in
superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.
'Superstring theory' is a shorthand for supersymmetric string t ...
, the extra dimensions of
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of
mirror symmetry. Their name was coined by , after , who first conjectured that compact complex manifolds of Kähler type with vanishing first
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...
always admit Ricci-flat Kähler metrics, and , who proved the
Calabi conjecture.
Calabi–Yau manifolds are
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
s that are generalizations of
K3 surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field ...
s in any number of
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on ...
s (i.e. any even number of real
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
s). They were originally defined as compact
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...
s with a vanishing first
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...
and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used.
Definitions
The definition was motivated by the work of
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...
, who proved
Eugenio Calabi's conjecture that any compact
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...
with vanishing first Chern class also admits a (typically different) Kähler metric with vanishing Ricci tensor.
There are many other definitions of a Calabi–Yau manifold used by different authors, some inequivalent. This section summarizes some of the more common definitions and the relations between them.
A Calabi–Yau
-fold or Calabi–Yau manifold of (complex) dimension
is sometimes defined as a compact
-dimensional Kähler manifold
satisfying one of the following equivalent conditions:
* The
canonical line bundle of
is trivial.
*
has a holomorphic
-form that vanishes nowhere.
* The
structure group
In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
of the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
of
can be reduced from
, the
unitary group
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semi ...
, to
, the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...
, in a manner that is compatible with the complex structure.
*
has a Kähler metric with global
holonomy contained in
.
These conditions imply that the first integral Chern class
of
vanishes. Nevertheless, the converse is not true. The simplest examples where this happens are
hyperelliptic surfaces, finite quotients of a complex torus of complex dimension 2, which have vanishing first integral Chern class but non-trivial canonical bundle.
For a compact complex
-dimensional manifold
that admits Kähler metrics, the following conditions are equivalent to each other, but are weaker than the conditions above, though they are sometimes used as the definition of a Calabi–Yau manifold:
*
has vanishing first real Chern class.
*
has a Kähler metric with vanishing Ricci curvature.
*
has a Kähler metric with local
holonomy contained in
.
* A positive power of the
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 nth exterior power of the cotangent bundle \Omega on V.
Over the complex numbers, it is ...
of
is trivial.
*
has a finite cover that has trivial canonical bundle.
*
has a finite cover that is a product of a torus and a
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
manifold with trivial canonical bundle.
If a compact Kähler manifold is simply connected, then the weak definition above is equivalent to the stronger definition.
Enriques surfaces give examples of complex manifolds that have Ricci-flat metrics, but their canonical bundles are not trivial, so they are Calabi–Yau manifolds according to the second but not the first definition above. On the other hand, their double covers are Calabi–Yau manifolds for both definitions (in fact, K3 surfaces).
By far the hardest part of proving the equivalences between the various properties above is proving the existence of Ricci-flat metrics. This follows from Yau's proof of the
Calabi conjecture, which implies that a compact Kähler manifold with a vanishing first real Chern class has a Kähler metric in the same class with vanishing Ricci curvature. (The class of a Kähler metric is the cohomology class of its associated 2-form.) Calabi showed such a metric is unique.
There are many other inequivalent definitions of Calabi–Yau manifolds that are sometimes used, which differ in the following ways (among others):
* The first Chern class may vanish as an integral class or as a real class.
* Most definitions assert that Calabi–Yau manifolds are compact, but some allow them to be non-compact. In the generalization to non-compact manifolds, the difference
must vanish asymptotically. Here,
is the Kähler form associated with the Kähler metric,
.
* Some definitions put restrictions on the
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
of a Calabi–Yau manifold, such as demanding that it be finite or trivial. Any Calabi–Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi–Yau manifold.
* Some definitions require that the holonomy be exactly equal to
rather than a subgroup of it, which implies that the
Hodge number
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
s
vanish for
. Abelian surfaces have a Ricci flat metric with holonomy strictly smaller than
(in fact trivial) so are not Calabi–Yau manifolds according to such definitions.
* Most definitions assume that a Calabi–Yau manifold has a Riemannian metric, but some treat them as complex manifolds without a metric.
* Most definitions assume the manifold is non-singular, but some allow mild singularities. While the Chern class fails to be well-defined for singular Calabi–Yau's, the canonical bundle and canonical class may still be defined if all the singularities are
Gorenstein, and so may be used to extend the definition of a smooth Calabi–Yau manifold to a possibly singular Calabi–Yau variety.
Examples
The fundamental fact is that any smooth
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
embedded in a
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
is a Kähler manifold, because there is a natural
Fubini–Study metric on a projective space which one can restrict to the algebraic variety. By definition, if
is the Kähler metric on the algebraic variety
and the canonical bundle
is trivial, then
is Calabi–Yau. Moreover, there is unique Kähler metric
on
such that
, a fact which was conjectured by
Eugenio Calabi and proved by
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...
(see
Calabi conjecture).
Calabi–Yau algebraic curves
In one complex dimension, the only compact examples are
tori, which form a one-parameter family. The Ricci-flat metric on a torus is actually a
flat metric, so that the
holonomy is the trivial group SU(1). A one-dimensional Calabi–Yau manifold is a complex
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
, and in particular,
algebraic.
CY algebraic surfaces
In two complex dimensions, the
K3 surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field ...
s furnish the only compact simply connected Calabi–Yau manifolds. These can be constructed as quartic surfaces in
, such as the complex algebraic variety defined by the vanishing locus of
for
Other examples can be constructed as elliptic fibrations, as quotients of abelian surfaces, or as
complete intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there s ...
s.
Non simply-connected examples are given by
abelian surfaces, which are real four tori
equipped with a complex manifold structure.
Enriques surfaces and
hyperelliptic surfaces have first Chern class that vanishes as an element of the real cohomology group, but not as an element of the integral cohomology group, so Yau's theorem about the existence of a Ricci-flat metric still applies to them but they are sometimes not considered to be Calabi–Yau manifolds. Abelian surfaces are sometimes excluded from the classification of being Calabi–Yau, as their holonomy (again the trivial group) is a
proper subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
of SU(2), instead of being isomorphic to SU(2). However, the
Enriques surface subset do not conform entirely to the SU(2) subgroup in the
String theory landscape
In string theory, the string theory landscape (or landscape of vacua) is the collection of possible false vacua,The number of metastable vacua is not known exactly, but commonly quoted estimates are of the order 10500. See M. Douglas, "The stat ...
.
CY threefolds
In three complex dimensions, classification of the possible Calabi–Yau manifolds is an open problem, although Yau suspects that there is a finite number of families (albeit a much bigger number than his estimate from 20 years ago). In turn, it has also been conjectured by
Miles Reid that the number of topological types of Calabi–Yau 3-folds is infinite, and that they can all be transformed continuously ( through certain mild singularizations such as
conifold In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux ...
s) one into another—much as
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s can. One example of a three-dimensional Calabi–Yau manifold is a non-singular
quintic threefold in
CP4, which is the
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
consisting of all of the zeros of a homogeneous quintic
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
in the homogeneous coordinates of the CP
4. Another example is a smooth model of the
Barth–Nieto quintic. Some discrete quotients of the quintic by various
actions are also Calabi–Yau and have received a lot of attention in the literature. One of these is related to the original quintic by
mirror symmetry.
For every positive integer
, the
zero set
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or eq ...
, in the homogeneous coordinates of the complex projective space CP
''n''+1, of a non-singular homogeneous degree ''
'' polynomial in ''
'' variables is a compact Calabi–Yau ''
''-fold. The case ''
'' describes an elliptic curve, while for ''
'' one obtains a K3 surface.
More generally, Calabi–Yau varieties/orbifolds can be found as weighted complete intersections in a
weighted projective space. The main tool for finding such spaces is the
adjunction formula.
All
hyper-Kähler manifolds are Calabi–Yau manifolds.
Constructed from algebraic curves
For an algebraic curve
a quasi-projective Calabi-Yau threefold can be constructed as the total space
where
. For the canonical projection
we can find the relative tangent bundle
is
using the relative tangent sequence
and observing the only tangent vectors in the fiber which are not in the pre-image of
are canonically associated with the fibers of the vector bundle. Using this, we can use the relative cotangent sequence
together with the properties of wedge powers that
and
giving the triviality of
.
Constructed from algebraic surfaces
Using a similar argument as for curves, the total space
of the canonical sheaf
for an algebraic surface
forms a Calabi-Yau threefold. A simple example is
over projective space.
Applications in superstring theory
Calabi–Yau manifolds are important in
superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.
'Superstring theory' is a shorthand for supersymmetric string t ...
. Essentially, Calabi–Yau manifolds are shapes that satisfy the requirement of space for the six "unseen" spatial dimensions of string theory, which may be smaller than our currently observable lengths as they have not yet been detected. A popular alternative known as
large extra dimension
In particle physics and string theory (M-theory), the Arkani-Hamed, Dimopoulos, Dvali model (ADD), also known as the model with large extra dimensions (LED), is a model framework that attempts to solve the hierarchy problem (''Why is the force of ...
s, which often occurs in
braneworld models, is that the Calabi–Yau is large but we are confined to a small subset on which it intersects a
D-brane
In string theory, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named.
D-branes are typically classified by their spatial dimensi ...
. Further extensions into higher dimensions are currently being explored with additional ramifications for
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
.
In the most conventional superstring models, ten conjectural dimensions in
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
are supposed to come as four of which we are aware, carrying some kind of
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in Postnikov systems or obstruction theory.
In this article, all ma ...
with fiber dimension six.
Compactification on Calabi–Yau ''n''-folds are important because they leave some of the original
supersymmetry
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
unbroken. More precisely, in the absence of
fluxes, compactification on a Calabi–Yau 3-fold (real dimension 6) leaves one quarter of the original supersymmetry unbroken if the
holonomy is the full SU(3).
More generally, a flux-free compactification on an ''n''-manifold with holonomy SU(''n'') leaves 2
1−''n'' of the original supersymmetry unbroken, corresponding to 2
6−''n'' supercharges in a compactification of
type IIA supergravity or 2
5−''n'' supercharges in a compactification of type I. When fluxes are included the supersymmetry condition instead implies that the compactification manifold be a
generalized Calabi–Yau, a notion introduced by . These models are known as
flux compactifications.
F-theory
In theoretical physics, F-theory is a branch of string theory developed by Iranian-American physicist Cumrun Vafa. The new vacua described by F-theory were discovered by Vafa and allowed string theorists to construct new realistic vacua — in ...
compactifications on various Calabi–Yau four-folds provide physicists with a method to find a large number of classical solutions in the so-called
string theory landscape
In string theory, the string theory landscape (or landscape of vacua) is the collection of possible false vacua,The number of metastable vacua is not known exactly, but commonly quoted estimates are of the order 10500. See M. Douglas, "The stat ...
.
Connected with each hole in the Calabi–Yau space is a group of low-energy string vibrational patterns. Since string theory states that our familiar elementary particles correspond to low-energy string vibrations, the presence of multiple holes causes the string patterns to fall into multiple groups, or
families
Family (from ) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictability, structure, and safety as ...
. Although the following statement has been simplified, it conveys the logic of the argument: if the Calabi–Yau has three holes, then three families of vibrational patterns and thus three families of particles will be observed experimentally.
Logically, since strings vibrate through all the dimensions, the shape of the curled-up ones will affect their vibrations and thus the properties of the elementary particles observed. For example,
Andrew Strominger
Andrew Eben Strominger (; born 1955) is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his ...
and
Edward Witten
Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...
have shown that the masses of particles depend on the manner of the intersection of the various holes in a Calabi–Yau. In other words, the positions of the holes relative to one another and to the substance of the Calabi–Yau space was found by Strominger and Witten to affect the masses of particles in a certain way. This is true of all particle properties.
Calabi-Yau algebra
A Calabi–Yau algebra was introduced by
Victor Ginzburg to transport the geometry of a Calabi–Yau manifold to
noncommutative algebraic geometry
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geo ...
.
See also
*
Quintic threefold
*
G2 manifold
References
*
*
*
*
*
*
*
*
Further reading
*
*
*
*
*
*
*
*
*
* (similar to )
External links
Calabi–Yau Homepageis an interactive reference which describes many examples and classes of Calabi–Yau manifolds and also the physical theories in which they appear.
Spinning Calabi–Yau Space video.*
Calabi–Yau Space' by Andrew J. Hanson with additional contributions by Jeff Bryant,
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...
.
*
Beginner articles
An overview of Calabi-Yau Elliptic fibrations*Lectures on the
Calabi-Yau Landscape
*
Fibrations in CICY Threefolds - (
complete intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there s ...
Calabi-Yau)
{{DEFAULTSORT:Calabi-Yau manifold
Algebraic geometry
Differential geometry
Mathematical physics
String theory
Complex manifolds