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 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 $r\in {\mathbb {Z} }$ (for the Ramond basis) or ${\textstyle r\in {1 \over 2}+{\mathbb {Z} }}$ (for the Neveu–Schwarz basis) defined by the following relations:[1]

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

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

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

If $r,s\in {\mathbb {Z} }$ in these relations, this yields the N = 2 Ramond algebra; while if ${\textstyle r,s\in {1 \over 2}+{\mathbb {Z} }}$ 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},\,\,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 $\beta (L_{m})=L_{m},$ $\beta (J_{m})=-J_{m}-{c \over 3}\delta _{m,0},$ $\beta (G_{r}^{\pm })=G_{r}^{\mp }$ 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 ${\mathbb {Z} }\rtimes {\mathbb {Z} }_{2}$ .
Twisted operators ${\textstyle {\mathcal {L}}_{n}=L_{n}+{1 \over 2}(n+1)J_{n}}$ 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 $[{\mathcal {L}}_{m},J_{n}]=-nJ_{m+n}+{c \over 6}\left(m^{2}+m\right)\delta _{m+n,0}$ $\{G_{r}^{+},G_{s}^{-}\}=2{\mathcal {L}}_{r+s}-2sJ_{r+s}+{c \over 3}\left(m^{2}+m\right)\delta _{m+n,0}$

Constructions

Free field construction

Green, Schwarz & Witten (1988) give a construction using two commuting real bosonic fields $(a_{n})$ , $(b_{n})$

{[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 $(e_{r})$

\{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}\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 fermions

J_{n}=\sum _{r}:e_{r}^{*}e_{n+r}:

and the two supersymmetric operators $G_{r}^{\pm }$ by

G_{r}^{+}=\sum (a_{-m}+ib_{-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},

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

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

the supersymmetric generators are defined by

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 $G$ .[2]

Notes

Green, Schwarz & Witten 1998a, pp. 240–241
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.; Friedan, 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 (3–4): 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 (3): 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", Mod. Phys. Lett. A, 5 (21): 1693–1701, Bibcode:1990MPLA....5.1693E, doi:10.1142/S0217732390001943
Goddard, P.; Kent, A.; Olive, D. (1986), "Unitary representations of the Virasoro and super-Virasoro algebras", Comm. Math. Phys., 103 (1): 105–119, Bibcode:1986CMaPh.103..105G, doi:10.1007/bf01464283, S2CID 91181508
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, Bibcode:1987cup..bookR....G, ISBN 0-521-35753-5
Kazama, Yoichi; Suzuki, Hisao (1989), "New N = 2 superconformal field theories and superstring compactification", Nuclear Physics B, 321 (1): 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 (2–3): 191–196, Bibcode:1987PhLB..184..191S, doi:10.1016/0370-2693(87)90566-1
Voisin, Claire (1999), Mirror symmetry, SMF/AMS texts and monographs, vol. 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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[1] Green, Schwarz & Witten 1998a, pp. 240–241

[Wassermann2010-2] Wassermann 2010