Pure spinor
Encyclopedia
In a field of mathematics
known as representation theory
pure spinors (or simple spinors) are spinor
representations of the special orthogonal group that are annihilated by the largest possible subspace
of the Clifford algebra
. They were introduced by Élie Cartan
in the 1930s to classify complex structures. Pure spinors were introduced into the realm of theoretical physics, and elevated in their importance in the study of spin geometry more generally, by Roger Penrose
in the 1960s, where they became among the basic objects of study in twistor theory
.
vector space
C2n with even complex dimension
2n and a quadratic form
Q, which maps a vector v to complex number Q(v). The Clifford algebra
Cliff2n is the ring
generated by products of vectors in C2n subject to the relation
Spinor
s are modules
of the Clifford algebra, and so in particular there is an action of C2n on the space of spinors. The subset of C2n that annihilates a given spinor ψ is a complex subspace Cm. If ψ is nonzero then m is less than or equal to n. If m is equal to n then ψ is said to be a pure spinor.
Not all spinors are pure. In general pure spinors may be separated from impure spinors via a series of quadratic equation
s called pure spinor constraint
s. However in 6 or less real dimensions all spinors are pure. In 8 dimensions there is, projectively, a single pure spinor constraint. In 10 dimensions, the case relevant for superstring theory
, there are 10 constraints
where Γμ are the gamma matrices, which represent the vectors C2n that generate the Clifford algebra. In general there are
constraints.
. In the year 2000 Nathan Berkovits, professor at Instituto de Fisica Teorica in São Paulo-Brazil introduced the pure spinor formalism in his paper Super-Poincare covariant quantization of the superstring. This formalism is the only known quantization
of the superstring which is manifestly covariant with respect to both spacetime
and worldsheet supersymmetry
. In 2002 Nigel Hitchin
introduced generalized Calabi–Yau manifolds in his paper Generalized Calabi–Yau manifolds, where the generalized complex structure
is defined by a pure spinor. These spaces describe the geometries of flux compactifications in string theory.
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...
known as representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
pure spinors (or simple spinors) are spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
representations of the special orthogonal group that are annihilated by the largest possible subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
of the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
. They were introduced by Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
in the 1930s to classify complex structures. Pure spinors were introduced into the realm of theoretical physics, and elevated in their importance in the study of spin geometry more generally, by Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
in the 1960s, where they became among the basic objects of study in twistor theory
Twistor theory
In theoretical and mathematical physics, twistor theory maps the geometric objects of conventional 3+1 space-time into geometric objects in a 4 dimensional space with metric signature...
.
Definition
Consider a complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
C2n with even complex dimension
Complex dimension
In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...
2n and a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
Q, which maps a vector v to complex number Q(v). The Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
Cliff2n is the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
generated by products of vectors in C2n subject to the relation
Spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s are modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
of the Clifford algebra, and so in particular there is an action of C2n on the space of spinors. The subset of C2n that annihilates a given spinor ψ is a complex subspace Cm. If ψ is nonzero then m is less than or equal to n. If m is equal to n then ψ is said to be a pure spinor.
The set of pure spinors
Every pure spinor is annihilated by a half-dimensional subspace of C2n. Conversely given a half-dimensional subspace it is possible to determine the pure spinor that it annihilates up to multiplication by a complex number. Pure spinors defined up to complex multiplication are called projective pure spinors. The space of projective pure spinors is the homogeneous spaceHomogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
- SO(2n)/U(n).
Not all spinors are pure. In general pure spinors may be separated from impure spinors via a series of quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
s called pure spinor constraint
Constraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...
s. However in 6 or less real dimensions all spinors are pure. In 8 dimensions there is, projectively, a single pure spinor constraint. In 10 dimensions, the case relevant for superstring theory
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
, there are 10 constraints
where Γμ are the gamma matrices, which represent the vectors C2n that generate the Clifford algebra. In general there are
constraints.
Pure spinors in string theory
Recently pure spinors have attracted attention in string theoryString 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...
. In the year 2000 Nathan Berkovits, professor at Instituto de Fisica Teorica in São Paulo-Brazil introduced the pure spinor formalism in his paper Super-Poincare covariant quantization of the superstring. This formalism is the only known quantization
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of the superstring which is manifestly covariant with respect to both spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
and worldsheet supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
. In 2002 Nigel Hitchin
Nigel Hitchin
Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...
introduced generalized Calabi–Yau manifolds in his paper Generalized Calabi–Yau manifolds, where the generalized complex structure
Generalized complex structure
In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure...
is defined by a pure spinor. These spaces describe the geometries of flux compactifications in string theory.