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, G, where r\in {\mathbb Z} (for the Ramond basis) or r\in {1\over 2}+{\mathbb Z} (for the Neveu–Schwarz basis) defined by the following relations:

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

If r,s\in {\mathbb Z} in these relations, this yields the N = 2 Ramond algebra; while if r,s\in {1\over 2}+{\mathbb 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^\pm)^=G_{-r}^\mp, ,,c^=c}

Properties

• The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism \alpha of : \alpha(L_n)=L_n +{1\over 2} J_n + {c\over 24}\delta_{n,0} \alpha(J_n)=J_n +{c\over 6}\delta_{n,0} \alpha(G_r^\pm)=G_{r\pm {1\over 2}}^\pm with inverse: \alpha^{-1}(L_n)=L_n -{1\over 2} J_n + {c\over 24}\delta_{n,0} \alpha^{-1}(J_n)=J_n -{c\over 6}\delta_{n,0} \alpha^{-1}(G_r^\pm)=G_{r\mp {1\over 2}}^\pm
• In the N = 2 Ramond algebra, the zero mode operators L_0, J_0, G_0^\pm 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^\pm the \partial and \overline{\partial} operators.
• Even integer powers of the spectral shift give automorphisms of the N = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism \beta, of period two, is given by \beta(L_m) = L_m , \beta(J_m)=-J_m-{c\over 3} \delta_{m,0}, \beta(G_r^\pm)=G_r^\mp In terms of Kähler operators, \beta corresponds to conjugating the complex structure. Since \beta\alpha \beta^{-1}=\alpha^{-1}, the automorphisms \alpha^2 and \beta generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group {\Z}\rtimes {\Z}_2.
• Twisted operators {\mathcal L}n=L_n+ {1\over 2} (n+1)J_n were introduced by and satisfy: [{\mathcal L}m,{\mathcal L}n] = (m-n) {\mathcal L}{m+n} 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 [{\mathcal L}m,J_n] = -nJ{m+n} + {c \over 6} \left(m^2 + m \right) \delta{m+n,0} {G_r^+,G_s^-} = 2{\mathcal L}{r+s}-2sJ_{r+s} + {c\over 3} \left(m^2+m\right) \delta_{m+n,0}

Constructions

Free field construction

give a construction using two commuting real bosonic fields (a_n), (b_n)

: {[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}

and a complex fermionic field (e_r)

: {e_r,e^*s}=\delta{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 = \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} :

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 = \sum_r : e_r^*e_{n+r} :

and the two supersymmetric operators G_r^\pm by

: G^+r=\sum (a{-m} + i b_{-m}) \cdot e_{r+m},,,,, G_r^-=\sum (a_{r+m} - ib_{r+m}) \cdot e^*_{m}

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

SU(2) supersymmetric coset construction

gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of 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 \ell with basis E_n,F_n,H_n satisfying :[H_m,H_n]=2m\ell\delta_{n+m,0}, :[E_m,F_n]=H_{m+n}+m \ell\delta_{m+n,0}, :[H_m,E_n]=2E_{m+n}, :[H_m,F_n]=-2F_{m+n}, the supersymmetric generators are defined by : 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^*. This yields the N=2 superconformal algebra with :c=3\ell/(\ell+2) . The algebra commutes with the bosonic operators :X_n=H_n - 2 \sum_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

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.

Notes

References

• {{citation|last1=Eguchi|first1=Tohru|last2= Yang|first2=Sung-Kil|title=N = 2 superconformal models as topological field theories| journal=Mod. Phys. Lett. A|volume= 5 |year=1990|issue=21|pages=1693–1701|doi=10.1142/S0217732390001943|bibcode=1990MPLA....5.1693E }}
• {{citation|first1=Michael B.|last1=Green|authorlink=Michael B. Green|first2=John H.|last2=Schwarz|authorlink2=John Henry Schwarz|first3=Edward|last3=Witten| authorlink3=Edward Witten|title=Superstring theory, Volume 1: Introduction|publisher=Cambridge University Press|year=1988a|isbn=0-521-35752-7}}
• {{citation|first1=Michael B.|last1=Green|authorlink=Michael B. Green|first2=John H.|last2=Schwarz|authorlink2=John Henry Schwarz|first3=Edward|last3=Witten| authorlink3=Edward Witten|title=Superstring theory, Volume 2: Loop amplitudes, anomalies and phenomenology|publisher=Cambridge University Press|year=1988b| isbn=0-521-35753-5|bibcode=1987cup..bookR....G}}

References

1. {{harvnb. Green. Schwarz. Witten. 1988a
2. {{harvnb. Wassermann. 2010