N = 2 superconformal algebra

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, L_n, J_n, for n an integer, and odd elements G+
r, G−
r, where r ∈ Z (for the Ramond basis) or r ∈ 1 2 + Z (for the Neveu–Schwarz basis) defined by the following relations:

If r , s ∈ Z in these relations, this yields the N = 2 Ramond algebra; while if r , s ∈ 1 2 + Z are half-integers, it gives the N = 2 Neveu–Schwarz algebra. The operators L n generate a Lie subalgebra isomorphic to the Virasoro algebra. Together with the operators G r = G r + + G r − , they generate a Lie superalgebra isomorphic to the super Virasoro algebra, giving the Ramond algebra if r , s are integers and the Neveu–Schwarz algebra otherwise. When represented as operators on a complex inner product space, c 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:

L n ∗ = L − n , J m ∗ = J − m , ( G r ± ) ∗ = G − r ∓ , c ∗ = c

Properties

The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism α of Schwimmer & Seiberg (1987):

with inverse:

In the N = 2 Ramond algebra, the zero mode operators L 0 , J 0 , G 0 ± and the constants form a five-dimensional Lie superalgebra. They satisfy the same relations as the fundamental operators in Kähler geometry, with L 0 corresponding to the Laplacian, J 0 the degree operator, and G 0 ± the ∂ and ∂ ¯ operators.

Even integer powers of the spectral shift give automorphisms of the N = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism β , of period two, is given by

In terms of Kähler operators, β corresponds to conjugating the complex structure. Since β α β − 1 = α − 1 , the automorphisms α 2 and β generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group Z ⋊ Z 2 .

Twisted operators L n = L n + 1 2 ( n + 1 ) J n were introduced by Eguchi & Yang (1990) and satisfy:

so that these operators satisfy the Virasoro relation with central charge 0. The constant c still appears in the relations for J m and the modified relations

Free field construction

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

[ a m , a n ] = m 2 δ m + n , 0 , [ b m , b n ] = m 2 δ m + n , 0 , a n ∗ = a − n , b n ∗ = b − n

and a complex fermionic field ( e r )

{ e r , e s ∗ } = δ r , s , { e r , e s } = 0.

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

L n = ∑ m : a − m + n a m : + ∑ m : b − m + n b m : + ∑ r ( r + n 2 ) : e r ∗ e n + r :

where normal ordering has been used for bosons and fermions.

The current operator J n is defined by the standard construction from fermions

J n = ∑ r : e r ∗ e n + r :

and the two supersymmetric operators G r ± by

G r + = ∑ ( a − m + i b − m ) ⋅ e r + m , G r − = ∑ ( a r + m − i b r + m ) ⋅ e m ∗

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

SU(2) supersymmetric coset construction

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 ℓ with basis E n , F n , H n satisfying

[ H m , H n ] = 2 m ℓ δ n + m , 0 , [ E m , F n ] = H m + n + m ℓ δ m + n , 0 , [ H m , E n ] = 2 E m + n , [ H m , F n ] = − 2 F m + n ,

the supersymmetric generators are defined by

G r + = ( ℓ / 2 + 1 ) − 1 / 2 ∑ E − m ⋅ e m + r , G r − = ( ℓ / 2 + 1 ) − 1 / 2 ∑ F r + m ⋅ e m ∗ .

This yields the N=2 superconformal algebra with

c = 3 ℓ / ( ℓ + 2 ) .

The algebra commutes with the bosonic operators

X n = H n − 2 ∑ r : e r ∗ e n + r : .

The space of physical states consists of eigenvectors of X 0 simultaneously annihilated by the X n 's for positive n and the supercharge operator

Q = G 1 / 2 + + G − 1 / 2 − (Neveu–Schwarz) Q = G 0 + + G 0 − . (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.

Kazama–Suzuki supersymmetric coset construction

Kazama & Suzuki (1989) generalized the SU(2) coset construction to any pair consisting of a simple compact Lie group G and a closed subgroup H of maximal rank, i.e. containing a maximal torus T of G , with the additional condition that the dimension of the centre of H is non-zero. In this case the compact Hermitian symmetric space G / H is a Kähler manifold, for example when H = T . 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 G .