Ramond Boundary Condition
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In
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, a super Virasoro algebra is an extension of the
Virasoro algebra In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string t ...
(named after Miguel Ángel Virasoro) to a
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. In most of these theories, th ...
. There are two extensions with particular importance 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 th ...
: the Ramond algebra (named after Pierre Ramond) and the Neveu–Schwarz algebra (named after André Neveu and
John Henry Schwarz John Henry Schwarz (; born November 22, 1941) is an American theoretical physicist. Along with Yoichiro Nambu, Holger Bech Nielsen, Joël Scherk, Gabriele Veneziano, Michael Green, and Leonard Susskind, he is regarded as one of the founders of st ...
). 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.


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 ''L''''m'' (for integer ''m'') satisfying L_m , L_n = ( m - n ) L_ + \frac m ( m^2 - 1 ) \delta_ where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 ...
. The odd part of the algebra has basis G_r, 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 L_m , G_r = \left( \frac - r \right) G_ \ = 2 L_ + \frac \left( r^2 - \frac \right) \delta_ Note that this last bracket is an
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
, not a commutator, because both generators are odd. The Ramond algebra has a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Present ...
in terms of 2 generators and 5 conditions; and the Neveu—Schwarz algebra has a presentation in terms of 2 generators and 9 conditions.


Representations

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 Harry Friedan (born October 3, 1948) is an American theoretical physicist and one of three children of the feminist author and activist Betty Friedan. He is a professor at Rutgers University. Biography Education and career Friedan earned h ...
, Zongan Qiu, and
Stephen Shenker Stephen Hart Shenker (born 1953) 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 co ...
(1984). It was proven by Peter Goddard, Adrian Kent and
David Olive David Ian Olive ( ; 16 April 1937 – 7 November 2012) was a British theoretical physicist. Olive made fundamental contributions to string theory and duality theory, he is particularly known for his work on the GSO projection and Montonen– ...
(1986), using a supersymmetric generalisation of the
coset construction In mathematics, the coset construction (or GKO construction) is a method of constructing unitary highest weight representations of the Virasoro algebra, introduced by Peter Goddard, Adrian Kent and David Olive David Ian Olive ( ; 16 April ...
or GKO construction.


Application to superstring theory

In superstring theory, the
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 bo ...
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 (periodic conditions are referred to as Ramond
boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
), described by the Ramond algebra, while those in the "Neveu–Schwarz sector" admit the other (anti-periodic conditions are referred to as Neveu–Schwarz boundary conditions), described by the Neveu–Schwarz algebra. For a
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 bo ...
, the periodicity depends on the choice of coordinates on the
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 as a direct generalization of the world line concept for a point particle in special a ...
. 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 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 bo ...
at the edges of the open string.


See also

* ''N'' = 2 superconformal algebra * NS–NS sector * Ramond–Ramond sector * Superconformal algebra


Notes


References

* * * * * *{{cite journal , authors=Mezincescu, L.; Nepomechie, I.; Zachos, C. K. , year=1989 , title=(Super)conformal algebra on the (super)torus , journal=Nuclear Physics B , volume=315 , issue=1 , page=43 , doi=10.1016/0550-3213(89)90448-3, bibcode = 1989NuPhB.315...43M Theoretical physics String theory Lie algebras Conformal field theory Boundary conditions