Conifold
Encyclopedia
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 compactifications of string theory
, the base is usually a five-dimension
al real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.
Conifolds are important objects in string theory
: Brian Greene
explains the physics
of conifolds in Chapter 13 of his book The Elegant Universe - including the fact that the space can tear near the cone, and its topology
can change. This possibility was first noticed by and employed by to prove that conifolds provide a connection between all (then) known Calabi-Yau compactifications in string theory; this partially supports a conjecture by whereby conifolds connect all possible Calabi-Yau complex 3-dimensional spaces.
A well-known example of a conifold is obtained as a deformation limit of a quintic - i.e. a quintic hypersurface in the projective space
. The space 
has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equations
in terms of homogeneous coordinates
on 
, for any fixed complex 
, has complex dimension three. This family of quintic hypersurfaces is the most famous example of Calabi-Yau manifold
s. If the complex structure parameter
is chosen to become equal to one, the manifold described above becomes singular since the derivative
s of the quintic polynomial
in the equation vanish when all coordinates
are equal or their ratio
s are certain fifth roots of unity. The neighbourhood of this singular point looks like a cone
whose base is topologically
just
.
In the context of string theory
, the geometrically singular conifolds can be shown to lead to completely smooth physics of strings. The divergences are "smeared out" by D3-branes wrapped on the shrinking three-sphere in Type IIB string theory and by D2-branes wrapped on the shrinking two-sphere in Type IIA string theory, as originally pointed out by . As shown by , this provides the string-theoretic description of the topology
-change via the conifold transition originally described by , who also invented the term "conifold" and the diagram
for the purpose. The two topologically distinct ways of smoothing a conifold are thus shown to involve replacing the singular vertex (node) by either a 3-sphere (by way of deforming the complex structure) or a 2-sphere (by way of a "small resolution"). It is believed that nearly all Calabi-Yau manifolds can be connected via these "critical transitions", resonating with Reid's conjecture.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, a conifold is a generalization of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. Unlike manifolds, conifolds can contain conical singularities i.e. points whose neighbourhoods look like cones
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
over a certain base. In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, in particular in flux compactifications of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, the base is usually a five-dimension
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
al real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.
Conifolds are important objects in string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
: Brian Greene
Brian Greene
Brian Greene is an American theoretical physicist and string theorist. He has been a professor at Columbia University since 1996. Greene has worked on mirror symmetry, relating two different Calabi-Yau manifolds...
explains the physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
of conifolds in Chapter 13 of his book The Elegant Universe - including the fact that the space can tear near the cone, and its topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
can change. This possibility was first noticed by and employed by to prove that conifolds provide a connection between all (then) known Calabi-Yau compactifications in string theory; this partially supports a conjecture by whereby conifolds connect all possible Calabi-Yau complex 3-dimensional spaces.
A well-known example of a conifold is obtained as a deformation limit of a quintic - i.e. a quintic hypersurface in the projective space
. The space has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equationsin terms of homogeneous coordinates
on , for any fixed complex , has complex dimension three. This family of quintic hypersurfaces is the most famous example of Calabi-Yau manifoldCalabi-Yau manifold
A Calabi-Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as in theoretical physics...
s. If the complex structure parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
is chosen to become equal to one, the manifold described above becomes singular since the derivativeDerivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s of the quintic polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
in the equation vanish when all coordinates
are equal or their ratioRatio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s are certain fifth roots of unity. The neighbourhood of this singular point looks like a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
whose base is topologically
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
just
.In the context of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, the geometrically singular conifolds can be shown to lead to completely smooth physics of strings. The divergences are "smeared out" by D3-branes wrapped on the shrinking three-sphere in Type IIB string theory and by D2-branes wrapped on the shrinking two-sphere in Type IIA string theory, as originally pointed out by . As shown by , this provides the string-theoretic description of the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
-change via the conifold transition originally described by , who also invented the term "conifold" and the diagram
for the purpose. The two topologically distinct ways of smoothing a conifold are thus shown to involve replacing the singular vertex (node) by either a 3-sphere (by way of deforming the complex structure) or a 2-sphere (by way of a "small resolution"). It is believed that nearly all Calabi-Yau manifolds can be connected via these "critical transitions", resonating with Reid's conjecture.