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In
mathematical physics Mathematical physics refers to the development of 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 the developme ...
, a super Virasoro algebra is an
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
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 the ...
(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, the ...
. 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 t ...
: the Ramond algebra (named after
Pierre Ramond Pierre Ramond (; born 31 January 1943) is distinguished professor of physics at University of Florida in Gainesville, Florida. He initiated the development of superstring theory. Academic career Ramond completed his BSEE from Newark College of ...
) and the Neveu–Schwarz algebra (named after
André Neveu André Neveu (; born 28 August 1946) is a French physicist working on string theory and quantum field theory who coinvented the Neveu–Schwarz algebra and the Gross–Neveu model. Biography Neveu studied in Paris at the École Normale Supér ...
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 s ...
). 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 L, or l, is the twelfth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''el'' (pronounced ), plural ''els''. History Lamedh ...
= ( 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 L, or l, is the twelfth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''el'' (pronounced ), plural ''els''. History Lamedh ...
= \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. Presenta ...
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 representation In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplic ...
s 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 (1984). It was proven by Peter Goddard, Adrian Kent and David Olive (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 (1986). The construction produce ...
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 bos ...
s on the
closed string In physics, a string is a physical entity postulated in string theory and related subjects. Unlike elementary particles, which are zero-dimensional or point-like by definition, strings are one-dimensional extended entities. Researchers often h ...
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), 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 bos ...
, 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 bos ...
at the edges of the open string.


See also

* ''N'' = 2 superconformal algebra * NS–NS sector * Ramond–Ramond sector *
Superconformal algebra In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superco ...


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