N = 2 superconformal algebra

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{{su|p=+|b=r}}, G{{su|p=−|b=r}}, where $r\backslash in\; \{\backslash mathbb\; Z\}$ (for the Ramond basis) or $r\backslash in\; \{1\backslash over\; 2\}+\{\backslash mathbb\; Z\}$ (for the Neveu–Schwarz basis) defined by the following relations:{{harvnb|Green|Schwarz|Witten|1988a|pp=240–241}}

::c is in the center

::$[L\_m,L\_n]\; =\; \backslash left(m-n\backslash right)\; L\_\{m+n\}\; +\; \{c\backslash over\; 12\}\; \backslash left(m^3-m\backslash right)\; \backslash delta\_\{m+n,0\}$

::$[L\_m,\backslash ,J\_n]=-nJ\_\{m+n\}$

::$[J\_m,J\_n]\; =\; \{c\backslash over\; 3\}\; m\backslash delta\_\{m+n,0\}$

::$\backslash \{G\_r^+,G\_s^-\backslash \}\; =\; L\_\{r+s\}\; +\; \{1\backslash over\; 2\}\; \backslash left(r-s\backslash right)\; J\_\{r+s\}\; +\; \{c\backslash over\; 6\}\; \backslash left(r^2-\{1\backslash over\; 4\}\backslash right)\; \backslash delta\_\{r+s,0\}$

::$\backslash \{G\_r^+,G\_s^+\backslash \}\; =\; 0\; =\; \backslash \{G\_r^-,G\_s^-\backslash \}$

::$[L\_m,G\_r^\{\backslash pm\}]\; =\; \backslash left(\; \{m\backslash over\; 2\}-r\; \backslash right)\; G^\backslash pm\_\{r+m\}$

::$[J\_m,G\_r^\backslash pm]=\; \backslash pm\; G\_\{m+r\}^\backslash pm$

If $r,s\backslash in\; \{\backslash mathbb\; Z\}$ in these relations, this yields the

N = 2 Ramond algebra; while if $r,s\backslash in\; \{1\backslash over\; 2\}+\{\backslash mathbb\; Z\}$ 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:

:$\{L\_n^*=L\_\{-n\},\; \backslash ,\backslash ,\; J\_m^*=J\_\{-m\},\; \backslash ,\backslash ,(G\_r^\backslash pm)^*=G\_\{-r\}^\backslash mp,\; \backslash ,\backslash ,c^*=c\}$

• The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism $\backslash alpha$ of {{harvtxt|Schwimmer|Seiberg|1987}}: $$\backslash alpha(L\_n)=L\_n\; +\{1\backslash over\; 2\}\; J\_n\; +\; \{c\backslash over\; 24\}\backslash delta\_\{n,0\}$$ $$\backslash alpha(J\_n)=J\_n\; +\{c\backslash over\; 6\}\backslash delta\_\{n,0\}$$ $$\backslash alpha(G\_r^\backslash pm)=G\_\{r\backslash pm\; \{1\backslash over\; 2\}\}^\backslash pm$$ with inverse: $$\backslash alpha^\{-1\}(L\_n)=L\_n\; -\{1\backslash over\; 2\}\; J\_n\; +\; \{c\backslash over\; 24\}\backslash delta\_\{n,0\}$$ $$\backslash alpha^\{-1\}(J\_n)=J\_n\; -\{c\backslash over\; 6\}\backslash delta\_\{n,0\}$$ $$\backslash alpha^\{-1\}(G\_r^\backslash pm)=G\_\{r\backslash mp\; \{1\backslash over\; 2\}\}^\backslash pm$$
• In the N = 2 Ramond algebra, the zero mode operators $L\_0$, $J\_0$, $G\_0^\backslash pm$ 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 $G\_0^\backslash pm$ the $\backslash partial$ and $\backslash overline\{\backslash partial\}$ operators.
• Even integer powers of the spectral shift give automorphisms of the N = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism $\backslash beta$, of period two, is given by $$\backslash beta(L\_m)\; =\; L\_m\; ,$$ $$\backslash beta(J\_m)=-J\_m-\{c\backslash over\; 3\}\; \backslash delta\_\{m,0\},$$ $$\backslash beta(G\_r^\backslash pm)=G\_r^\backslash mp$$ In terms of Kähler operators, $\backslash beta$ corresponds to conjugating the complex structure. Since $\backslash beta\backslash alpha\; \backslash beta^\{-1\}=\backslash alpha^\{-1\}$, the automorphisms $\backslash alpha^2$ and $\backslash beta$ generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group $\{\backslash Z\}\backslash rtimes\; \{\backslash Z\}\_2$.
• Twisted operators $\{\backslash mathcal\; L\}\_n=L\_n+\; \{1\backslash over\; 2\}\; (n+1)J\_n$ were introduced by {{harvtxt|Eguchi|Yang|1990}} and satisfy: $$[\{\backslash mathcal\; L\}\_m,\{\backslash mathcal\; L\}\_n]\; =\; (m-n)\; \{\backslash mathcal\; L\}\_\{m+n\}$$ 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 $$[\{\backslash mathcal\; L\}\_m,J\_n]\; =\; -nJ\_\{m+n\}\; +\; \{c\; \backslash over\; 6\}\; \backslash left(m^2\; +\; m\; \backslash right)\; \backslash delta\_\{m+n,0\}$$ $$\backslash \{G\_r^+,G\_s^-\backslash \}\; =\; 2\{\backslash mathcal\; L\}\_\{r+s\}-2sJ\_\{r+s\}\; +\; \{c\backslash over\; 3\}\; \backslash left(m^2+m\backslash right)\; \backslash delta\_\{m+n,0\}$$

{{harvs|last1=Green|last2=Schwarz|last3=Witten|year=1988a|year2=1988b|txt}} give a construction using two commuting real bosonic fields $(a\_n)$, $(b\_n)$

:$\{[a\_m,a\_n]=\{m\backslash over\; 2\}\backslash delta\_\{m+n,0\},\backslash ,\backslash ,\backslash ,\backslash ,\; [b\_m,b\_n]=\{m\backslash over\; 2\}\backslash delta\_\{m+n,0\}\},$

\,\,\,\, a_n^*=a_{-n},\,\,\,\, b_n^*=b_{-n}

and a complex fermionic field $(e\_r)$

:$\backslash \{e\_r,e^*\_s\backslash \}=\backslash delta\_\{r,s\},\backslash ,\backslash ,\backslash ,\backslash ,\; \backslash \{e\_r,e\_s\backslash \}=0.$

$L\_n$ is defined to the sum of the Virasoro operators naturally associated with each of the three systems

:$L\_n\; =\; \backslash sum\_m\; :\; a\_\{-m+n\}\; a\_m\; :\; +\; \backslash sum\_m\; :\; b\_\{-m+n\}\; b\_m\; :\; +\; \backslash sum\_r\; \backslash left(r+\{n\backslash over\; 2\}\backslash right):\; e^*\_\{r\}e\_\{n+r\}\; :$

where normal ordering has been used for bosons and fermions.

The current operator $J\_n$ is defined by the standard construction from fermions

:$J\_n\; =\; \backslash sum\_r\; :\; e\_r^*e\_\{n+r\}\; :$

and the two supersymmetric operators $G\_r^\backslash pm$ by

:$G^+\_r=\backslash sum\; (a\_\{-m\}\; +\; i\; b\_\{-m\})\; \backslash cdot\; e\_\{r+m\},\backslash ,\backslash ,\backslash ,\backslash ,\; G\_r^-=\backslash sum\; (a\_\{r+m\}\; -\; ib\_\{r+m\})\; \backslash cdot\; e^*\_\{m\}$

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

{{harvtxt|Di Vecchia|Petersen|Yu|Zheng|1986}} gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of {{harvtxt|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 $\backslash ell$ with basis $E\_n,F\_n,H\_n$ satisfying

:$[H\_m,H\_n]=2m\backslash ell\backslash delta\_\{n+m,0\},$

:$[E\_m,F\_n]=H\_\{m+n\}+m\; \backslash ell\backslash delta\_\{m+n,0\},$

:$[H\_m,E\_n]=2E\_\{m+n\},$

:$[H\_m,F\_n]=-2F\_\{m+n\},$

the supersymmetric generators are defined by

:$G^+\_r\; =\; (\backslash ell/2+\; 1)^\{-1/2\}\; \backslash sum\; E\_\{-m\}\; \backslash cdot\; e\_\{m+r\},\; \backslash ,\backslash ,\backslash ,\; G^-\_r\; =\; (\backslash ell/2\; +1\; )^\{-1/2\}\; \backslash sum\; F\_\{r+m\}\backslash cdot\; e\_m^*.$

This yields the N=2 superconformal algebra with

:$c=3\backslash ell/(\backslash ell+2)\; .$

The algebra commutes with the bosonic operators

:$X\_n=H\_n\; -\; 2\; \backslash sum\_r\; :\; e\_r^*e\_\{n+r\}\; :.$

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.{{harvnb|Wassermann|2010}}

{{harvtxt|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 $G$.

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

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{{DEFAULTSORT:N 2 Superconformal Algebra}}

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