N = 2 superconformal algebra

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Short description: 2D supersymmetric generalization to the conformal algebra


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

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 [math]\displaystyle{ r\in {\mathbb Z} }[/math] (for the Ramond basis) or [math]\displaystyle{ r\in {1\over 2}+{\mathbb Z} }[/math] (for the Neveu–Schwarz basis) defined by the following relations:[1]

c is in the center
[math]\displaystyle{ [L_m,L_n] = \left(m-n\right) L_{m+n} + {c\over 12} \left(m^3-m\right) \delta_{m+n,0} }[/math]
[math]\displaystyle{ [L_m,\,J_n]=-nJ_{m+n} }[/math]
[math]\displaystyle{ [J_m,J_n] = {c\over 3} m\delta_{m+n,0} }[/math]
[math]\displaystyle{ \{G_r^+,G_s^-\} = L_{r+s} + {1\over 2} \left(r-s\right) J_{r+s} + {c\over 6} \left(r^2-{1\over 4}\right) \delta_{r+s,0} }[/math]
[math]\displaystyle{ \{G_r^+,G_s^+\} = 0 = \{G_r^-,G_s^-\} }[/math]
[math]\displaystyle{ [L_m,G_r^{\pm}] = \left( {m\over 2}-r \right) G^\pm_{r+m} }[/math]
[math]\displaystyle{ [J_m,G_r^\pm]= \pm G_{m+r}^\pm }[/math]

If [math]\displaystyle{ r,s\in {\mathbb Z} }[/math] in these relations, this yields the N = 2 Ramond algebra; while if [math]\displaystyle{ r,s\in {1\over 2}+{\mathbb Z} }[/math] are half-integers, it gives the N = 2 Neveu–Schwarz algebra. The operators [math]\displaystyle{ L_n }[/math] generate a Lie subalgebra isomorphic to the Virasoro algebra. Together with the operators [math]\displaystyle{ G_r=G_r^+ + G_r^- }[/math], they generate a Lie superalgebra isomorphic to the super Virasoro algebra, giving the Ramond algebra if [math]\displaystyle{ r,s }[/math] are integers and the Neveu–Schwarz algebra otherwise. When represented as operators on a complex inner product space, [math]\displaystyle{ c }[/math] 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:

[math]\displaystyle{ {L_n^*=L_{-n}, \,\, J_m^*=J_{-m}, \,\,(G_r^\pm)^*=G_{-r}^\mp, \,\,c^*=c} }[/math]

Properties

  • The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism [math]\displaystyle{ \alpha }[/math] of (Schwimmer Seiberg): [math]\displaystyle{ \alpha(L_n)=L_n +{1\over 2} J_n + {c\over 24}\delta_{n,0} }[/math] [math]\displaystyle{ \alpha(J_n)=J_n +{c\over 6}\delta_{n,0} }[/math] [math]\displaystyle{ \alpha(G_r^\pm)=G_{r\pm {1\over 2}}^\pm }[/math] with inverse: [math]\displaystyle{ \alpha^{-1}(L_n)=L_n -{1\over 2} J_n + {c\over 24}\delta_{n,0} }[/math] [math]\displaystyle{ \alpha^{-1}(J_n)=J_n -{c\over 6}\delta_{n,0} }[/math] [math]\displaystyle{ \alpha^{-1}(G_r^\pm)=G_{r\mp {1\over 2}}^\pm }[/math]
  • In the N = 2 Ramond algebra, the zero mode operators [math]\displaystyle{ L_0 }[/math], [math]\displaystyle{ J_0 }[/math], [math]\displaystyle{ G_0^\pm }[/math] and the constants form a five-dimensional Lie superalgebra. They satisfy the same relations as the fundamental operators in Kähler geometry, with [math]\displaystyle{ L_0 }[/math] corresponding to the Laplacian, [math]\displaystyle{ J_0 }[/math] the degree operator, and [math]\displaystyle{ G_0^\pm }[/math] the [math]\displaystyle{ \partial }[/math] and [math]\displaystyle{ \overline{\partial} }[/math] operators.
  • Even integer powers of the spectral shift give automorphisms of the N = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism [math]\displaystyle{ \beta }[/math], of period two, is given by [math]\displaystyle{ \beta(L_m) = L_m , }[/math] [math]\displaystyle{ \beta(J_m)=-J_m-{c\over 3} \delta_{m,0}, }[/math] [math]\displaystyle{ \beta(G_r^\pm)=G_r^\mp }[/math] In terms of Kähler operators, [math]\displaystyle{ \beta }[/math] corresponds to conjugating the complex structure. Since [math]\displaystyle{ \beta\alpha \beta^{-1}=\alpha^{-1} }[/math], the automorphisms [math]\displaystyle{ \alpha^2 }[/math] and [math]\displaystyle{ \beta }[/math] generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group [math]\displaystyle{ {\Z}\rtimes {\Z}_2 }[/math].
  • Twisted operators [math]\displaystyle{ {\mathcal L}_n=L_n+ {1\over 2} (n+1)J_n }[/math] were introduced by (Eguchi Yang) and satisfy: [math]\displaystyle{ [{\mathcal L}_m,{\mathcal L}_n] = (m-n) {\mathcal L}_{m+n} }[/math] so that these operators satisfy the Virasoro relation with central charge 0. The constant [math]\displaystyle{ c }[/math] still appears in the relations for [math]\displaystyle{ J_m }[/math] and the modified relations [math]\displaystyle{ [{\mathcal L}_m,J_n] = -nJ_{m+n} + {c \over 6} \left(m^2 + m \right) \delta_{m+n,0} }[/math] [math]\displaystyle{ \{G_r^+,G_s^-\} = 2{\mathcal L}_{r+s}-2sJ_{r+s} + {c\over 3} \left(m^2+m\right) \delta_{m+n,0} }[/math]

Constructions

Free field construction

(Green Schwarz) give a construction using two commuting real bosonic fields [math]\displaystyle{ (a_n) }[/math], [math]\displaystyle{ (b_n) }[/math]

[math]\displaystyle{ {[a_m,a_n]={m\over 2}\delta_{m+n,0},\,\,\,\, [b_m,b_n]={m\over 2}\delta_{m+n,0}}, \,\,\,\, a_n^*=a_{-n},\,\,\,\, b_n^*=b_{-n} }[/math]

and a complex fermionic field [math]\displaystyle{ (e_r) }[/math]

[math]\displaystyle{ \{e_r,e^*_s\}=\delta_{r,s},\,\,\,\, \{e_r,e_s\}=0. }[/math]

[math]\displaystyle{ L_n }[/math] is defined to the sum of the Virasoro operators naturally associated with each of the three systems

[math]\displaystyle{ L_n = \sum_m : a_{-m+n} a_m  : + \sum_m : b_{-m+n} b_m : + \sum_r \left(r+{n\over 2}\right): e^*_{r}e_{n+r} : }[/math]

where normal ordering has been used for bosons and fermions.

The current operator [math]\displaystyle{ J_n }[/math] is defined by the standard construction from fermions

[math]\displaystyle{ J_n = \sum_r : e_r^*e_{n+r} : }[/math]

and the two supersymmetric operators [math]\displaystyle{ G_r^\pm }[/math] by

[math]\displaystyle{ G^+_r=\sum (a_{-m} + i b_{-m}) \cdot e_{r+m},\,\,\,\, G_r^-=\sum (a_{r+m} - ib_{r+m}) \cdot e^*_{m} }[/math]

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

SU(2) supersymmetric coset construction

(Di Vecchia Petersen) gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of (Goddard Kent) 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 [math]\displaystyle{ \ell }[/math] with basis [math]\displaystyle{ E_n,F_n,H_n }[/math] satisfying

[math]\displaystyle{ [H_m,H_n]=2m\ell\delta_{n+m,0}, }[/math]
[math]\displaystyle{ [E_m,F_n]=H_{m+n}+m \ell\delta_{m+n,0}, }[/math]
[math]\displaystyle{ [H_m,E_n]=2E_{m+n}, }[/math]
[math]\displaystyle{ [H_m,F_n]=-2F_{m+n}, }[/math]

the supersymmetric generators are defined by

[math]\displaystyle{ G^+_r = (\ell/2+ 1)^{-1/2} \sum E_{-m} \cdot e_{m+r}, \,\,\, G^-_r = (\ell/2 +1 )^{-1/2} \sum F_{r+m}\cdot e_m^*. }[/math]

This yields the N=2 superconformal algebra with

[math]\displaystyle{ c=3\ell/(\ell+2) . }[/math]

The algebra commutes with the bosonic operators

[math]\displaystyle{ X_n=H_n - 2 \sum_r : e_r^*e_{n+r} :. }[/math]

The space of physical states consists of eigenvectors of [math]\displaystyle{ X_0 }[/math] simultaneously annihilated by the [math]\displaystyle{ X_n }[/math]'s for positive [math]\displaystyle{ n }[/math] and the supercharge operator

[math]\displaystyle{ Q=G_{1/2}^+ + G_{-1/2}^- }[/math] (Neveu–Schwarz)
[math]\displaystyle{ Q=G_0^+ +G_0^-. }[/math] (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

(Kazama Suzuki) generalized the SU(2) coset construction to any pair consisting of a simple compact Lie group [math]\displaystyle{ G }[/math] and a closed subgroup [math]\displaystyle{ H }[/math] of maximal rank, i.e. containing a maximal torus [math]\displaystyle{ T }[/math] of [math]\displaystyle{ G }[/math], with the additional condition that the dimension of the centre of [math]\displaystyle{ H }[/math] is non-zero. In this case the compact Hermitian symmetric space [math]\displaystyle{ G/H }[/math] is a Kähler manifold, for example when [math]\displaystyle{ H=T }[/math]. 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 [math]\displaystyle{ G }[/math].[2]

See also

Notes

  1. ↑ Green, Schwarz & Witten 1998a, pp. 240–241
  2. ↑ 2.0 2.1 Wassermann 2010

References



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