Neveu-Schwarz sector
Encyclopedia
In mathematical physics
, a super Virasoro algebra is an extension of the Virasoro algebra
to a Lie superalgebra
. There are two extensions with particular importance in superstring theory
: the Ramond algebra (named after Pierre Ramond
) and the Neveu–Schwarz algebra (named after Andre Neveu
and John Henry Schwarz). Both algebras have N=1 supersymmetry
and an even part given by the Virasoro algebra. They describe the symmetries of a superstring in two different sectors, called the Ramond sector and the Neveu–Schwarz sector.
where δi is zero unless i = 0, in which case it is 1.
The odd part of the algebra has basis Gr, where r is either an integer (the Ramond case), or half an odd integer (the Neveu–Schwarz case). In both cases, C is central in the superalgebra, and the additional graded brackets are given by
Note that this last bracket is an anticommutator, not a commutator, because both generators are odd.
The unitary highest weight representations of these algebras have a classification analogous to that for the Virasoro algebra, with a continuum of representations together with an infinite discrete series. The existence of these discrete series was conjectured by Daniel Friedan
, Zongan Qiu, and Stephen Shenker
(1984). It was proven by Peter Goddard, Adrian Kent and David Olive
(1986), using a supersymmetric generalisation of the coset construction
or GKO construction.
s on the closed string may be either periodic or anti-periodic on the circle around the string. States in the "Ramond sector" admit one option, described by the Ramond algebra, while those in the "Neveu–Schwarz sector" admit the other, described by the Neveu–Schwarz algebra.
For a fermionic field
, the periodicity depends on the choice of coordinates on the worldsheet
. In the w-frame, in which the worldsheet of a single string state is described as a long cylinder, states in the Neveu–Schwarz sector are anti-periodic and states in the Ramond sector are periodic. In the z-frame, in which the worldsheet of a single string state is described as an infinite punctured plane, the opposite is true.
The Neveu–Schwarz sector and Ramond sector are also defined in the open string and depend on the boundary conditions of the fermionic field
at the edges of the open string.
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, a super Virasoro algebra is an extension of the Virasoro algebra
Virasoro algebra
In mathematics, the Virasoro algebra is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in conformal field theory and string theory....
to a Lie superalgebra
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
. There are two extensions with particular importance in 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...
: the Ramond algebra (named after Pierre Ramond
Pierre Ramond
Pierre Ramond is a Distinguished Professor of Physics at University of Florida in Gainesville, Florida...
) and the Neveu–Schwarz algebra (named after Andre Neveu
André Neveu
André Neveu is a French physicist working on string theory and quantum field theory who coinvented the Neveu-Schwarz algebra and the Gross-Neveu model.Neveu studied in Paris at the École normale supérieure...
and John Henry Schwarz). Both algebras have N=1 supersymmetry
Supersymmetry algebra
In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras, and hence are Lie superalgebras...
and an even part given by the Virasoro algebra. They describe the symmetries of a superstring in two different sectors, called the Ramond sector and the Neveu–Schwarz sector.
The N = 1 super Virasoro algebras
There are two minimal extensions of the Virasoro algebra with N = 1 supersymmetry: the Ramond algebra and the Neveu–Schwarz algebra. They are both Lie superalgebras whose even part is the Virasoro algebra: this Lie algebra has a basis consisting of a central element C and generators Lm (for integer m) satisfying- [Lm,Ln] = (m − n)Lm+n + m(m2 − 1)δm+n C/12,
where δi is zero unless i = 0, in which case it is 1.
The odd part of the algebra has basis Gr, where r is either an integer (the Ramond case), or half an odd integer (the Neveu–Schwarz case). In both cases, C is central in the superalgebra, and the additional graded brackets are given by
- [Lm,Gr] = (m/2 − r)Gm+r,
- {Gr,Gs} = 2Lr+s + (r2 − 1/4)δr+s C/3.
Note that this last bracket is an anticommutator, not a commutator, because both generators are odd.
The unitary highest weight representations of these algebras have a classification analogous to that for the Virasoro algebra, with a continuum of representations together with an infinite discrete series. The existence of these discrete series was conjectured by Daniel Friedan
Daniel Friedan
Daniel Harry Friedan is an American theoretical physicist and is one of two sons of the feminist author and activist Betty Friedan. He earned his Ph.D...
, Zongan Qiu, and Stephen Shenker
Stephen Shenker
Stephen Hart Shenker is an American theoretical physicist who works on string theory. He is a professor at Stanford University and former director of the Stanford Institute for Theoretical Physics. His brother Scott Shenker is a computer scientist...
(1984). It was proven by Peter Goddard, Adrian Kent and David Olive
David Olive
David Olive CBE FLSW FRS, is a British theoretical physicist. Olive made fundamental contributions to the string theory and duality theory. He was Professor of physics at Imperial College, London...
(1986), using a supersymmetric generalisation of the coset construction
Coset construction
In mathematics, the coset construction is a method of constructing unitary highest weight representations of the Virasoro algebra, introduced by Peter Goddard, Adrian Kent and David Olive...
or GKO construction.
Application to superstring theory
In superstring theory, the fermionic fieldFermionic field
In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi-Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields....
s on the closed string may be either periodic or anti-periodic on the circle around the string. States in the "Ramond sector" admit one option, described by the Ramond algebra, while those in the "Neveu–Schwarz sector" admit the other, described by the Neveu–Schwarz algebra.
For a fermionic field
Fermionic field
In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi-Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields....
, the periodicity depends on the choice of coordinates on the worldsheet
Worldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
. In the w-frame, in which the worldsheet of a single string state is described as a long cylinder, states in the Neveu–Schwarz sector are anti-periodic and states in the Ramond sector are periodic. In the z-frame, in which the worldsheet of a single string state is described as an infinite punctured plane, the opposite is true.
The Neveu–Schwarz sector and Ramond sector are also defined in the open string and depend on the boundary conditions of the fermionic field
Fermionic field
In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi-Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields....
at the edges of the open string.
See also
- N = 2 superconformal algebraN = 2 superconformal algebraIn mathematical physics, the N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and conformal field theory. It has important applications in mirror symmetry...
- Superconformal algebraSuperconformal algebraIn theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. It generates the superconformal group in some cases .In two dimensions, the superconformal algebra is infinite-dimensional...