N = 2 superconformal algebra

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In mathematical physics, the 2D N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and two-dimensional conformal field theory. It has important applications in mirror symmetry. It was introduced by M. Ademollo, L. Brink, and A. D'Adda et al. (1976) as a gauge algebra of the U(1) fermionic string.

Definition

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There are two slightly different ways to describe the N = 2 superconformal algebra, called the N = 2 Ramond algebra and the N = 2 Neveu–Schwarz algebra, which are isomorphic (see below) but differ in the choice of standard basis. The N = 2 superconformal algebra is the Lie superalgebra with basis of even elements c, Ln, Jn, for n an integer, and odd elements G+
r
, G
r
, where   (for the Ramond basis) or   (for the Neveu–Schwarz basis) defined by the following relations:[1]

c is in the center
 
 
 
 
 
 
 

If   in these relations, this yields the N = 2 Ramond algebra; while if   are half-integers, it gives the N = 2 Neveu–Schwarz algebra. The operators   generate a Lie subalgebra isomorphic to the Virasoro algebra. Together with the operators  , they generate a Lie superalgebra isomorphic to the super Virasoro algebra, giving the Ramond algebra if   are integers and the Neveu–Schwarz algebra otherwise. When represented as operators on a complex inner product space,   is taken to act as multiplication by a real scalar, denoted by the same letter and called the central charge, and the adjoint structure is as follows:

 

Properties

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  • The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism   of Schwimmer & Seiberg (1987):       with inverse:      
  • In the N = 2 Ramond algebra, the zero mode operators  ,  ,   and the constants form a five-dimensional Lie superalgebra. They satisfy the same relations as the fundamental operators in Kähler geometry, with   corresponding to the Laplacian,   the degree operator, and   the   and   operators.
  • Even integer powers of the spectral shift give automorphisms of the N = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism  , of period two, is given by       In terms of Kähler operators,   corresponds to conjugating the complex structure. Since  , the automorphisms   and   generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group  .
  • Twisted operators   were introduced by Eguchi & Yang (1990) and satisfy:   so that these operators satisfy the Virasoro relation with central charge 0. The constant   still appears in the relations for   and the modified relations    

Constructions

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Free field construction

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Green, Schwarz, and Witten (1988a, 1988b) give a construction using two commuting real bosonic fields  ,  

 

and a complex fermionic field  

 

  is defined to the sum of the Virasoro operators naturally associated with each of the three systems

 

where normal ordering has been used for bosons and fermions.

The current operator   is defined by the standard construction from fermions

 

and the two supersymmetric operators   by

 

This yields an N = 2 Neveu–Schwarz algebra with c = 3.

SU(2) supersymmetric coset construction

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Di Vecchia et al. (1986) gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of Goddard, Kent & Olive (1986) for the discrete series representations of the Virasoro and super Virasoro algebra. Given a representation of the affine Kac–Moody algebra of SU(2) at level   with basis   satisfying

 
 
 
 

the supersymmetric generators are defined by

 

This yields the N=2 superconformal algebra with

 

The algebra commutes with the bosonic operators

 

The space of physical states consists of eigenvectors of   simultaneously annihilated by the  's for positive   and the supercharge operator

  (Neveu–Schwarz)
  (Ramond)

The supercharge operator commutes with the action of the affine Weyl group and the physical states lie in a single orbit of this group, a fact which implies the Weyl-Kac character formula.[2]

Kazama–Suzuki supersymmetric coset construction

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Kazama & Suzuki (1989) generalized the SU(2) coset construction to any pair consisting of a simple compact Lie group   and a closed subgroup   of maximal rank, i.e. containing a maximal torus   of  , with the additional condition that the dimension of the centre of   is non-zero. In this case the compact Hermitian symmetric space   is a Kähler manifold, for example when  . The physical states lie in a single orbit of the affine Weyl group, which again implies the Weyl–Kac character formula for the affine Kac–Moody algebra of  .[2]

See also

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Notes

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References

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  • Ademollo, M.; Brink, L.; D'Adda, A.; D'Auria, R.; Napolitano, E.; Sciuto, S.; Giudice, E. Del; Vecchia, P. Di; Ferrara, S.; Gliozzi, F.; Musto, R.; Pettorino, R. (1976), "Supersymmetric strings and colour confinement", Physics Letters B, 62 (1): 105–110, Bibcode:1976PhLB...62..105A, doi:10.1016/0370-2693(76)90061-7
  • Boucher, W.; Friedan, D; Kent, A. (1986), "Determinant formulae and unitarity for the N = 2 superconformal algebras in two dimensions or exact results on string compactification", Phys. Lett. B, 172 (3–4): 316–322, Bibcode:1986PhLB..172..316B, doi:10.1016/0370-2693(86)90260-1
  • Di Vecchia, P.; Petersen, J. L.; Yu, M.; Zheng, H. B. (1986), "Explicit construction of unitary representations of the N = 2 superconformal algebra", Phys. Lett. B, 174 (3): 280–284, Bibcode:1986PhLB..174..280D, doi:10.1016/0370-2693(86)91099-3
  • Eguchi, Tohru; Yang, Sung-Kil (1990), "N = 2 superconformal models as topological field theories", Mod. Phys. Lett. A, 5 (21): 1693–1701, Bibcode:1990MPLA....5.1693E, doi:10.1142/S0217732390001943
  • Goddard, P.; Kent, A.; Olive, D. (1986), "Unitary representations of the Virasoro and super-Virasoro algebras", Comm. Math. Phys., 103 (1): 105–119, Bibcode:1986CMaPh.103..105G, doi:10.1007/bf01464283, S2CID 91181508
  • Green, Michael B.; Schwarz, John H.; Witten, Edward (1988a), Superstring theory, Volume 1: Introduction, Cambridge University Press, ISBN 0-521-35752-7
  • Green, Michael B.; Schwarz, John H.; Witten, Edward (1988b), Superstring theory, Volume 2: Loop amplitudes, anomalies and phenomenology, Cambridge University Press, Bibcode:1987cup..bookR....G, ISBN 0-521-35753-5
  • Kazama, Yoichi; Suzuki, Hisao (1989), "New N = 2 superconformal field theories and superstring compactification", Nuclear Physics B, 321 (1): 232–268, Bibcode:1989NuPhB.321..232K, doi:10.1016/0550-3213(89)90250-2
  • Schwimmer, A.; Seiberg, N. (1987), "Comments on the N = 2, 3, 4 superconformal algebras in two dimensions", Phys. Lett. B, 184 (2–3): 191–196, Bibcode:1987PhLB..184..191S, doi:10.1016/0370-2693(87)90566-1
  • Voisin, Claire (1999), Mirror symmetry, SMF/AMS texts and monographs, vol. 1, American Mathematical Society, ISBN 0-8218-1947-X
  • Wassermann, A. J. (2010) [1998]. "Lecture notes on Kac-Moody and Virasoro algebras". arXiv:1004.1287.
  • West, Peter C. (1990), Introduction to supersymmetry and supergravity (2nd ed.), World Scientific, pp. 337–8, ISBN 981-02-0099-4