N = 2 superconformal 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 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$&minus; r$, 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 $$[{\mathcal L}_m,J_n] = -nJ_{m+n} + {c \over 6} \left(m^2 + m \right) \delta_{m+n,0}$$ $$\{G_r^+,G_s^-\} = 2{\mathcal L}_{r+s}-2sJ_{r+s} + {c\over 3} \left(m^2+m\right) \delta_{m+n,0}$$

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 $$G$$.