N = 2 superconformal algebra

Template:Short description

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 Template:Harvs 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 GTemplate:Su, GTemplate:Su, where $r \in ℤ$ (for the Ramond basis) or $r \in \frac{1}{2} + ℤ$ (for the Neveu–Schwarz basis) defined by the following relations:^[1]

c is in the center

[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{c}{12} (m^{3} - m) δ_{m + n, 0}

[L_{m}, J_{n}] = - n J_{m + n}

[J_{m}, J_{n}] = \frac{c}{3} m δ_{m + n, 0}

{G_{r}^{+}, G_{s}^{-}} = L_{r + s} + \frac{1}{2} (r - s) J_{r + s} + \frac{c}{6} (r^{2} - \frac{1}{4}) δ_{r + s, 0}

{G_{r}^{+}, G_{s}^{+}} = 0 = {G_{r}^{-}, G_{s}^{-}}

[L_{m}, G_{r}^{\pm}] = (\frac{m}{2} - r) G_{r + m}^{\pm}

[J_{m}, G_{r}^{\pm}] = \pm G_{m + r}^{\pm}

If $r, s \in ℤ$ in these relations, this yields the N = 2 Ramond algebra; while if $r, s \in \frac{1}{2} + ℤ$ 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 $α$ of Template:Harvtxt: $α (L_{n}) = L_{n} + \frac{1}{2} J_{n} + \frac{c}{24} δ_{n, 0}$ $α (J_{n}) = J_{n} + \frac{c}{6} δ_{n, 0}$ $α (G_{r}^{\pm}) = G_{r \pm \frac{1}{2}}^{\pm}$ with inverse: $α^{- 1} (L_{n}) = L_{n} - \frac{1}{2} J_{n} + \frac{c}{24} δ_{n, 0}$ $α^{- 1} (J_{n}) = J_{n} - \frac{c}{6} δ_{n, 0}$ $α^{- 1} (G_{r}^{\pm}) = G_{r \mp \frac{1}{2}}^{\pm}$
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 $β$ , of period two, is given by $β (L_{m}) = L_{m},$ $β (J_{m}) = - J_{m} - \frac{c}{3} δ_{m, 0},$ $β (G_{r}^{\pm}) = G_{r}^{\mp}$ In terms of Kähler operators, $β$ corresponds to conjugating the complex structure. Since $β α β^{- 1} = α^{- 1}$ , the automorphisms $α^{2}$ and $β$ generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group $ℤ ⋊ ℤ_{2}$ .
Twisted operators $ℒ_{n} = L_{n} + \frac{1}{2} (n + 1) J_{n}$ were introduced by Template:Harvtxt and satisfy: $[ℒ_{m}, ℒ_{n}] = (m - n) ℒ_{m + n}$ 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 $[ℒ_{m}, J_{n}] = - n J_{m + n} + \frac{c}{6} (m^{2} + m) δ_{m + n, 0}$ ${G_{r}^{+}, G_{s}^{-}} = 2 ℒ_{r + s} - 2 s J_{r + s} + \frac{c}{3} (m^{2} + m) δ_{m + n, 0}$

Constructions

Free field construction

Template:Harvs give a construction using two commuting real bosonic fields $(a_{n})$ , $(b_{n})$

[a_{m}, a_{n}] = \frac{m}{2} δ_{m + n, 0}, [b_{m}, b_{n}] = \frac{m}{2} δ_{m + n, 0}, a_{n}^{*} = a_{- n}, b_{n}^{*} = b_{- n}

and a complex fermionic field $(e_{r})$

{e_{r}, e_{s}^{*}} = δ_{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} (r + \frac{n}{2}) : 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} - i b_{r + m}) \cdot e_{m}^{*}

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

SU(2) supersymmetric coset construction

Template:Harvtxt gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of Template:Harvtxt 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 $ℓ$ with basis $E_{n}, F_{n}, H_{n}$ satisfying

[H_{m}, H_{n}] = 2 m ℓ δ_{n + m, 0},

[E_{m}, F_{n}] = H_{m + n} + m ℓ δ_{m + n, 0},

[H_{m}, E_{n}] = 2 E_{m + n},

[H_{m}, F_{n}] = - 2 F_{m + n},

the supersymmetric generators are defined by

G_{r}^{+} = (ℓ / 2 + 1)^{- 1 / 2} \sum E_{- m} \cdot e_{m + r}, G_{r}^{-} = (ℓ / 2 + 1)^{- 1 / 2} \sum F_{r + m} \cdot e_{m}^{*} .

This yields the N=2 superconformal algebra with

c = 3 ℓ / (ℓ + 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.^[2]

Kazama–Suzuki supersymmetric coset construction

Template:Harvtxt 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 $G$ .^[2]

Notes

Template:Reflist

References

↑ Template:Harvnb
↑ ^2.0 ^2.1 Template:Harvnb

[1] Template:Harvnb

[Wassermann2010-2] 2.0 ^2.1 Template:Harvnb

[1]

[2]

N = 2 superconformal algebra

Contents

Definition

Properties

Constructions

Free field construction

SU(2) supersymmetric coset construction

Kazama–Suzuki supersymmetric coset construction

See also

Notes

References

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N = 2 superconformal algebra

Definition

Properties

Constructions

Free field construction

SU(2) supersymmetric coset construction

Kazama–Suzuki supersymmetric coset construction

See also

Notes

References

Navigation menu

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