nLab spinning string

Idea

The spinning string is a variant of the string in direct analogy (but one worldvolume dimension higher) to how the spinning particle is a variant of the particle.

(Which means that “spinning” here is not in the sense of “rotating”, but in the sense of the kind of “intrinsic” spin as exhibited by fermion particles. A spinning string can rotate in addition to its intrinsic spin.)

And just as as the spinning particle automatically exhibits local worldline supersymmetry, the spinning string automatically exhibits local worldsheet supersymmetry. For that reason it is today almost exclusively known as the superstring .

In fact, the very notion of supersymmetry was realized (in parallel to its independent discovery in Russia) when people noticed that the spinning string sigma-model – also called the Neveu-Schwarz-Ramond model (NSR model) – automatically exhibits this extra graded symmetry.

When supplemented with an operation called the GSO projection, the second quantization of the spinning string – its effective target spacetime theory – also becomes a locally supersymmetric theory: spacetime supergravity.

There are other formulations of the superstring which are thought to be equivalent in some sense. Notably the Green-Schwarz superstring is a sigma-model which is not supersymmetric on the worldsheet but instead takes values in a genuine spacetime supermanifold.

Related concepts

• picture number, picture changing operator

• sigma model, brane

  • relativistic particle, spinning particle, superparticle

  • string, spinning string , superstring

  • Goddard-Thorn no-ghost theorem, GSO projection

    string theory FAQ – Does string theory predict supersymmetry?

  • membrane

• string theory

manifest supersymmetry for brane sigma-models:

manifest worldvolume supersymmetry	manifest target+worldvolume supersymmetry	manifest target space supersymmetry
NSR action functional	superembedding approach	Green-Schwarz action functional

graphics grabbed from FSS19c

References

General

The original articles that formulated the NSR model:

• André Neveu, John Schwarz, Factorizable dual model of pions, Nucl. Phys. B31, 86 (1971) (doi:10.1016/0550-3213(71)90448-2)

• Pierre Ramond, Dual theory for Free Fermions, Phys Rev. D3 2415 (1971) (doi:10.1103/PhysRevD.3.2415)

  (introducing the Dirac-Ramond operator)

• Daniel Friedan, Emil Martinec, Stephen Shenker, Conformal invariance, supersymmetry and string theory, Nucl. Phys., B271 (1986) 93 (doi:10.1016/S0550-3213(86)80006-2)

Original articles that realize the local worldsheet supersymmetry of the spinning string (via the supersymmetric Polyakov action):

• Stanley Deser, Bruno Zumino, A complete action for the spinning string, Physics Letters B 65 4 (1976) 369-373 [doi:10.1016/0370-2693(76)90245-8pdf]

• Lars Brink, Paolo Di Vecchia, Paul Howe, A locally supersymmetric and reparametrization invariant action for the spinning string , Physics Letters B 65 Issue 5, 20 (1976) 471-474 [doi:10.1016/0370-2693(76)90445-7]

Further on the spinning string’s worldsheet theory as $D=2$ supergravity formulated in superspace:

• S. James Gates Jr., Liang Lu, Robert N. Oerter: Simplified $SU(2)$ spinning string superspace supergravity, Physics Letters B 218 1 (1989) 33-38 [doi:10.1016/0370-2693(89)90470-X, spire:263912]

Definition of the NSR string in RR-field backgrounds is subtle (in contrast to the classical Green-Schwarz superstring), discussion is in:

• David Berenstein, Robert Leigh, Superstring Perturbation Theory and Ramond-Ramond Backgrounds, Phys. Rev. D 60 106002 (1999) [arXiv:hep-th/9904104]

See also

• Nathan Benjamin, Ying-Hsuan Lin, Lessons from the Ramond sector (arXiv:2005.02394)

Relation to the Berkovits superstring:

• Nathan Berkovits, Manifest Spacetime Supersymmetry and the Superstring (arXiv:2106.04448)

GSO projection and spacetime supersymmetry

The GSO projection and the conjecture that this makes also the target space theory of the spinning string locally supersymmetric is due to

• F. Gliozzi, Joël Scherk, D. I. Olive, Supergravity and its spinor dual model, Phys. Letters B65, 282 (1976)

• F. Gliozzi, Joël Scherk, D. I. Olive, Supersymmetry, supergravity theories and the dual spinor model , Nucl. Phys, B122, 253 (1977)

The demonstration of this fact is due to

• Michael Green, John Schwarz,

  Supersymmetrical dual string theory , Nucl. Phys. B181 (1981) 502;

  Supersymmetrical string theories , Phys. Lett. 109B (1982) 444.

Superconformal invariance of the spinning string was discussed in

• Paul Howe, Super Weyl transformations in two dimensions J. Phys. 12 (1979) 393.

A decent account is in

• Imre Majer, Superstrings (pdf, pdf)

A review of the history of these developments is in

• John Schwarz, String theory origins of supersymmetry, Nucl. Phys. Proc. Suppl. 101 (2001) 54-61 (arXiv:hep-th/0011078)

An exposition of some technical aspects is in the wiki

• The Tangent Bundle, RNS superstring

Reviews and survays are also in any text on string theory. See the references there for more detail.

BRST description

Discussion of the BRST complex of the NSR string:

• José Figueroa-O'Farrill, Takashi Kimura, The BRST cohomology of the NSR string: vanishing and “-ghost’‘ theorems, Comm. Math. Phys. 124 1 (1989) 105-132. [euclid:cmp/1104179078]