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 r ∈ Z {\displaystyle r\in {\mathbb {Z} }} (for the Ramond basis) or r ∈ 1 2 + Z {\textstyle 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 ] = ( m − n ) L m + n + c 12 ( m 3 − m ) δ m + n , 0 {\displaystyle [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 ] = − n J m + n {\displaystyle [L_{m},\,J_{n}]=-nJ_{m+n}}
[ J m , J n ] = c 3 m δ m + n , 0 {\displaystyle [J_{m},J_{n}]={c \over 3}m\delta _{m+n,0}}
{ G r + , G s − } = L r + s + 1 2 ( r − s ) J r + s + c 6 ( r 2 − 1 4 ) δ r + s , 0 {\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}}
{ G r + , G s + } = 0 = { G r − , G s − } {\displaystyle \{G_{r}^{+},G_{s}^{+}\}=0=\{G_{r}^{-},G_{s}^{-}\}}
[ L m , G r ± ] = ( m 2 − r ) G r + m ± {\displaystyle [L_{m},G_{r}^{\pm }]=\left({m \over 2}-r\right)G_{r+m}^{\pm }}
[ J m , G r ± ] = ± G m + r ± {\displaystyle [J_{m},G_{r}^{\pm }]=\pm G_{m+r}^{\pm }}
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