N = 2 superconformal algebra explained

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, L_n, J_n, for n an integer, and odd elements G, G, where

r\in { Z} (for the Ramond basis) or $r\backslash in\; +$ (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 ^{- \}} _{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 ^{+ \}} _{s} = 0 = \{G ^{- \}} _{s} [L _{m,G} ^{\pm} ^{r} ] = \left( {m\over 2 } - r \right) G ^{\pm} _{r + m} [J _{m,G} ^{\pm] =} ^{r} \pm G ^{\pm} _{m + r}

If

r,s\in { Z} in these relations, this yields theN = 2 Ramond algebra; while if $r,s\backslash in\; +$ 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 ^{* = L} _{- n} , J ^{* = J} _{- m} , (G ^{\pm)} _{r} ^{\mp,} ^{- r} c ^{*=c}}

Properties

• The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism

\alpha of : $$\backslash alpha(L\_n)=L\_n\; +\; J\_n\; +\; \backslash delta\_$$ $$\backslash alpha(J\_n)=J\_n\; +\backslash delta\_$$ $$\backslash alpha(G\_r^\backslash pm)=G\_^\backslash pm$$ with inverse: $$\backslash alpha^(L\_n)=L\_n\; -\; J\_n\; +\; \backslash delta\_$$ $$\backslash alpha^(J\_n)=J\_n\; -\backslash delta\_$$ $$\backslash alpha^(G\_r^\backslash pm)=G\_^\backslash pm$$

• In the N = 2 Ramond algebra, the zero mode operators

L _{0} , J _{0} , G ^{\pm} _{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 ^{\pm} _{0} 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 $$\backslash beta(L\_m)\; =\; L\_m,$$ $$\backslash beta(J\_m)=-J\_m-\; \backslash delta\_,$$ $$\backslash beta(G\_r^\backslash pm)=G\_r^\backslash mp$$ 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 $\_n=L\_n+\; (n+1)J\_n$ were introduced by and satisfy: $$[\{\backslash mathcal\; L\}\_m,\{\backslash mathcal\; L\}\_n]\; =\; (m-n)\; \_$$ 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 $$[\{\backslash mathcal\; L\}\_m,J\_n]\; =\; -nJ\_\; +\; \backslash left(m^2\; +\; m\; \backslash right)\; \backslash delta\_$$ $$\backslash \; =\; 2\_-2sJ\_\; +\; \backslash left(m^2+m\backslash right)\; \backslash delta\_$$

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_,\,\,\,\, b_n^*=b_(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 systemsL _{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 fermionsJ _{n} = \sum _{r} : e ^{*e} _{n + r} :

and the two supersymmetric operators

G ^{\pm} _{r} byG ^{+} _{r = \sum} (a _{-m} + i b _{-m} ) ⋅ e _{r+m} , G ^{-= \sum} _{r} (a _{r+m} - ib _{r+m} ) ⋅ 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 byG ^{+} _{r} = (\ell/ 2+ 1 ) ^{-1/2} \sum E _{-m} ⋅ e _{m+r} , G ^{-} _{r} = (\ell/ 2 +1 ) ^{-1/2} \sum F _{r+m} ⋅ 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 ^{*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

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 .

See also

• Virasoro algebra
• Super Virasoro algebra
• Coset construction
• Type IIB string theory

References

• • • • • • • • • • • News: Wassermann . A. J.. Antony Wassermann. Lecture notes on Kac-Moody and Virasoro algebras . 1998 . 1004.1287 . 2010 .