Super Virasoro algebra

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In superstring theory, the fermionic fields on the closed string may be either periodic or anti-periodic on the circle around the string. States in the Neveu-Schwarz sector (named for A. Neveu and John Henry Schwarz) admit one option, while those in the Ramond sector (named for Pierre Ramond) admit the other. There are two extensions of the Virasoro algebra to a super Virasoro algebra corresponding to these two sectors, called the Neveu-Schwarz algebra and the Ramond algebra

Note that for a fermionic field, the periodicity depends on the coordinates by which the worldsheet is described. 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 plane, the opposite is correct.

The Neveu-Schwarz sector and Ramond sector are also defined in the open string, according to the boundary conditions of the fermionic field at the edges of the open string.

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[edit] The Neveu-Schwarz algebra

The Neveu-Schwarz algebra is a Lie superalgebra spanned by even elements Lm for integers m and odd elements Gm for m half an odd integer, together a central element D, and these satisfy the relations

[Lm,Ln] = (m − n)Lm+n + D m(m2 − 1)δm+n/8
[Lm,Gn] = (m/2 − n)Gm+n
[Gm,Gn] = GmGn + GnGm = 2Lm+n + D (m2 − 1/4)δm+n/2

[edit] The Ramond algebra

The Ramond algebra is a Lie superalgebra spanned by even elements Lm and odd elements Fm for integers m, together with a central element D, and these satisfy the relations

[Lm,Ln] = (m − n)Lm+n + D m3δm+n/8
[Lm,Fn] = (m/2 −n)Fm+n
[Fm,Fn] = FmFn + FnFm = 2Lm+n + D m2δm+n/2

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[edit] References