N = 2 superconformal algebra

In mathematical physics, the N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and 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, L_n, J_n, for n an integer, and odd elements G+
r, G−
r, where $r\in {\Bbb Z}$ (for the Ramond basis) or $r\in {1\over 2}+{\Bbb Z}$ (for the Neveu–Schwarz basis) defined by the following relations:^[1]

c is in the center

\displaystyle{[L_m,L_n]=(m-n)L_{m+n} +{c\over 12} (m^3-m) \delta_{m+n,0}}

\displaystyle{[L_m,\,J_n]=-nJ_{m+n}}

\displaystyle{[J_m,J_n]={c\over 3} m\delta_{m+n,0}}

\displaystyle{\{G_r^+,G_s^-\}=L_{r+s} +{1\over 2}(r-s)J_{r+s} +{c\over 6} (r^2-{1\over 4}) \delta_{r+s,0} }

\displaystyle{\{G_r^+,G_s^+\}=0=\{G_r^-,G_s^-\}}

\displaystyle{[L_m,G_r^{\pm}]=({m\over 2}-r) G^\pm_{r+m}}

\displaystyle{[J_m,G_r^\pm]= \pm G_{m+r}^\pm}

If $r,s\in {\Bbb Z}$ in these relations, this yields the N = 2 Ramond algebra; while if $r,s\in {1\over 2}+{\Bbb Z}$ 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:

\displaystyle{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 $\alpha$ of Schwimmer & Seiberg (1987):

\alpha(L_n)=L_n +{1\over 2} J_n + {c\over 24}\delta_{n,0}

\alpha(J_n)=J_n +{c\over 6}\delta_{n,0}

\alpha(G_r^\pm)=G_{r\pm {1\over 2}}^\pm

with inverse:

\alpha^{-1}(L_n)=L_n -{1\over 2} J_n + {c\over 24}\delta_{n,0}

\alpha^{-1}(J_n)=J_n -{c\over 6}\delta_{n,0}

\alpha^{-1}(G_r^\pm)=G_{r\mp {1\over 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 $\beta$ , of period two, is given by

\displaystyle{\beta(L_m)=L_m},

\beta(J_m)=-J_m-{c\over 3} \delta_{m,0},

\beta(G_r^\pm)=G_r^\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

{\Bbb Z}\rtimes {\Bbb Z}_2

Twisted operators ${\mathcal L}_n=L_n+ {1\over 2} (n+1)J_n$ were introduced by Eguchi & Yang (1990) and satisfy:

[{\mathcal L}_m,{\mathcal L}_n]=(m-n){\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

\displaystyle{[{\mathcal L}_m,J_n] =-nJ_{m+n} +{c\over 6} (m^2+m)\delta_{m+n,0}}

\displaystyle{\{G_r^+,G_s^-\} =2{\mathcal L}_{r+s}-2sJ_{r+s} +{c\over 3} (m^2+m) \delta_{m+n,0}}

Constructions

Free field construction

Green, Schwarz & Witten (1988) give a construction using two commuting real bosonic fields $(a_n)$ , $(b_n)$

\displaystyle{[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_{-n},\,\,\,\, b_n^*=b_{-n}

and a complex fermionic field $(e_r)$

\displaystyle{\{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 systems

L_n = \sum_m : a_{-m+n} a_m : + \sum_m : b_{-m+n} b_m : + \sum_r (r+{n\over 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} - ib_{r+m}) \cdot e^*_{m}

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

SU(2) supersymmetric coset construction

Di Vecchia et al. (1986) gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of 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 $\ell$ 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},

\displaystyle{[H_m,E_n]=2E_{m+n},}

\displaystyle{[H_m,F_n]=-2F_{m+n},}

the supersymmetric generators are defined by

\displaystyle{G^+_r=(\ell/2+ 1)^{-1/2} \sum E_{-m}\cdot e_{m+r},\,\,\, G^-_r=(\ell/2 +1 )^{-1/2} \sum F_{r+m}\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_{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

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 $G$ .^[3]

Notes

↑ Green, Schwarz & Witten 1998a, pp. 240–241
↑ Wassermann 2010
↑ Wassermann 2010

References

Ademollo, M.; Brink, L.; D'Adda, A.; D'Auria, R.; Napolitano, E.; Sciuto, S.; Giudice, E. Del; Vecchia, P. Di; Ferrara, S.; Gliozzi, F.; Musto, R.; Pettorino, R. (1976), "Supersymmetric strings and colour confinement", Physics Letters B 62 (1): 105–110, Bibcode:1976PhLB...62..105A, doi:10.1016/0370-2693(76)90061-7
Boucher, W.; Freidan, D,; Kent, A. (1986), "Determinant formulae and unitarity for the N = 2 superconformal algebras in two dimensions or exact results on string compactification", Phys. Lett. B 172: 316–322, Bibcode:1986PhLB..172..316B, doi:10.1016/0370-2693(86)90260-1
Di Vecchia, P.; Petersen, J. L.; Yu,, M.; Zheng, H. B. (1986), "Explicit construction of unitary representations of the N = 2 superconformal algebra", Phys. Lett. B 174: 280–284, Bibcode:1986PhLB..174..280D, doi:10.1016/0370-2693(86)91099-3
Eguchi, Tohru; Yang, Sung-Kil (1990), "N = 2 superconformal models as topological field theories", Modern Phys. Lett. A 5: 1693–1701
Goddard, P.; Kent, A.; Olive, D. (1986), "Unitary representations of the Virasoro and super-Virasoro algebras", Comm. Math. Phys. 103: 105–119, Bibcode:1986CMaPh.103..105G, doi:10.1007/bf01464283
Green, Michael B.; Schwarz, John H.; Witten, Edward (1988a), Superstring theory, Volume 1: Introduction, Cambridge University Press, ISBN 0-521-35752-7
Green, Michael B.; Schwarz, John H.; Witten, Edward (1988b), Superstring theory, Volume 2: Loop amplitudes, anomalies and phenomenology, Cambridge University Press, ISBN 0-521-35753-5
Kazama, Yoichi; Suzuki, Hisao (1989), "New N = 2 superconformal field theories and superstring compactification", Nuclear Phys. B 321: 232–268, Bibcode:1989NuPhB.321..232K, doi:10.1016/0550-3213(89)90250-2
Schwimmer, A.; Seiberg, N. (1987), "Comments on the N = 2, 3, 4 superconformal algebras in two dimensions", Phys. Lett. B 184: 191–196, Bibcode:1987PhLB..184..191S, doi:10.1016/0370-2693(87)90566-1
Voisin, Claire (1999), Mirror symmetry, SMF/AMS texts and monographs 1, American Mathematical Society, ISBN 0-8218-1947-X
Wassermann, A. J. (2010) [1998]. "Lecture notes on Kac-Moody and Virasoro algebras". arXiv:1004.1287.
West, Peter C. (1990), Introduction to supersymmetry and supergravity (2nd ed.), World Scientific, pp. 337–8, ISBN 981-02-0099-4

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