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'', ''L''_''n'', ''J''_''n'', for ''n'' an integer, and odd elements ''G'', ''G'', where $r\in$ (for the Ramond basis) or $r\in +$ (for the Neveu–Schwarz basis) defined by the following relations:

::''c'' is in the center
:: _m,L_n= \left(m-n\right) L_ + \left(m^3-m\right) \delta_
:: _m,\,J_n-nJ_
:: _m,J_n= m\delta_
::$\ = L_ + \left(r-s\right) J_ + \left(r^2-\right) \delta_$
::$\ = 0 = \$
:: _m,G_r^= \left( -r \right) G^\pm_
:: _m,G_r^\pm \pm G_^\pm

If $r,s\in$ in these relations, this yields the
''N'' = 2 Ramond algebra; while if $r,s\in +$ 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:

:$$

Properties

*The ''N'' = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism $\alpha$ of : $\alpha(L_n)=L_n + J_n + \delta_$ $\alpha(J_n)=J_n +\delta_$ $\alpha(G_r^\pm)=G_^\pm$ with inverse: $\alpha^(L_n)=L_n - J_n + \delta_$ $\alpha^(J_n)=J_n -\delta_$ $\alpha^(G_r^\pm)=G_^\pm$
*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 $\overline$ 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 $\beta(L_m) = L_m ,$ $\beta(J_m)=-J_m- \delta_,$ $\beta(G_r^\pm)=G_r^\mp$ In terms of Kähler operators, $\beta$ corresponds to conjugating the complex structure. Since $\beta\alpha \beta^=\alpha^$, the automorphisms $\alpha^2$ and $\beta$ generate a group of automorphisms of the ''N'' = 2 superconformal algebra isomorphic to the infinite dihedral group $\rtimes _2$.
*Twisted operators $_n=L_n+ (n+1)J_n$ were introduced by and satisfy: m,_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 m,J_n= -nJ_ + \left(m^2 + m \right) \delta_ $\ = 2_-2sJ_ + \left(m^2+m\right) \delta_$

Constructions

Free field construction

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

:$, \,\,\,\, a_n^*=a_,\,\,\,\, b_n^*=b_$

and a complex fermionic field $(e_r)$

:$\=\delta_,\,\,\,\, \=0.$

$L_n$ is defined to the sum of the Virasoro operators naturally associated with each of the three systems

:$L_n = \sum_m : a_ a_m : + \sum_m : b_ b_m : + \sum_r \left(r+\right): e^*_e_ :$

where normal ordering has been used for bosons and fermions.

The current operator $J_n$ is defined by the standard construction from fermions

:$J_n = \sum_r : e_r^*e_ :$

and the two supersymmetric operators $G_r^\pm$ by

:$G^+_r=\sum (a_ + i b_) \cdot e_,\,\,\,\, G_r^-=\sum (a_ - ib_) \cdot e^*_$

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
: _m,H_n2m\ell\delta_,
: _m,F_nH_+m \ell\delta_,
: _m,E_n2E_,
: _m,F_n-2F_,
the supersymmetric generators are defined by
:$G^+_r = (\ell/2+ 1)^ \sum E_ \cdot e_, \,\,\, G^-_r = (\ell/2 +1 )^ \sum F_\cdot e_m^*.$
This yields the N=2 superconformal algebra with
:$c=3\ell/(\ell+2) .$
The algebra commutes with the bosonic operators
:$X_n=H_n - 2 \sum_r : e_r^*e_ :.$
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
:$Q=G_^+ + G_^-$ (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 $G$.

See also

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

Notes

References

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{{DEFAULTSORT:N 2 Superconformal Algebra
String theory
Conformal field theory
Lie algebras
Representation theory
Supersymmetry