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

c is in the center

${\displaystyle \displaystyle {[L_{m},L_{n}]=(m-n)L_{m+n}+{c \over 12}(m^{3}-m)\delta _{m+n,0}}}$

${\displaystyle \displaystyle {[L_{m},\,J_{n}]=-nJ_{m+n}}}$

${\displaystyle \displaystyle {[J_{m},J_{n}]={c \over 3}m\delta _{m+n,0}}}$

${\displaystyle \displaystyle {\{G_{r}^{+},G_{s}^{-}\}=L_{r+s}+{1 \over 2}(r-s)J_{r+s}+{c \over 6}(r^{2}-{1 \over 4})\delta _{r+s,0}}}$

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

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

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

If ${\displaystyle r,s\in {\mathbb {Z} }}$ in these relations, this yields the N = 2 Ramond algebra; while if ${\displaystyle r,s\in {1 \over 2}+{\mathbb {Z} }}$ are half-integers, it gives the N = 2 Neveu–Schwarz algebra. The operators ${\displaystyle L_{n}}$ generate a Lie subalgebra isomorphic to the Virasoro algebra. Together with the operators ${\displaystyle G_{r}=G_{r}^{+}+G_{r}^{-}}$, they generate a Lie superalgebra isomorphic to the super Virasoro algebra, giving the Ramond algebra if ${\displaystyle r,s}$ are integers and the Neveu–Schwarz algebra otherwise. When represented as operators on a complex inner product space, ${\displaystyle c}$ 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:

${\displaystyle \displaystyle {L_{n}^{*}=L_{-n},\,\,J_{m}^{*}=J_{-m},\,\,(G_{r}^{\pm })^{*}=G_{-r}^{\mp },\,\,c^{*}=c}}$

Properties

• The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism ${\displaystyle \alpha }$ of Schwimmer & Seiberg (1987):

${\displaystyle \alpha (L_{n})=L_{n}+{1 \over 2}J_{n}+{c \over 24}\delta _{n,0}}$

${\displaystyle \alpha (J_{n})=J_{n}+{c \over 6}\delta _{n,0}}$

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

with inverse:

${\displaystyle \alpha ^{-1}(L_{n})=L_{n}-{1 \over 2}J_{n}+{c \over 24}\delta _{n,0}}$

${\displaystyle \alpha ^{-1}(J_{n})=J_{n}-{c \over 6}\delta _{n,0}}$

${\displaystyle \alpha ^{-1}(G_{r}^{\pm })=G_{r\mp {1 \over 2}}^{\pm }}$

• In the N = 2 Ramond algebra, the zero mode operators ${\displaystyle L_{0}}$, ${\displaystyle J_{0}}$, ${\displaystyle G_{0}^{\pm }}$ and the constants form a five-dimensional Lie superalgebra. They satisfy the same relations as the fundamental operators in Kähler geometry, with ${\displaystyle L_{0}}$ corresponding to the Laplacian, ${\displaystyle J_{0}}$ the degree operator, and ${\displaystyle G_{0}^{\pm }}$ the ${\displaystyle \partial }$ and ${\displaystyle {\overline {\partial }}}$ operators.
• Even integer powers of the spectral shift give automorphisms of the N = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism ${\displaystyle \beta }$, of period two, is given by

${\displaystyle \displaystyle {\beta (L_{m})=L_{m}},}$

${\displaystyle \beta (J_{m})=-J_{m}-{c \over 3}\delta _{m,0},}$

${\displaystyle \beta (G_{r}^{\pm })=G_{r}^{\mp }}$

In terms of Kähler operators, ${\displaystyle \beta }$ corresponds to conjugating the complex structure. Since ${\displaystyle \beta \alpha \beta ^{-1}=\alpha ^{-1}}$, the automorphisms ${\displaystyle \alpha ^{2}}$ and ${\displaystyle \beta }$ generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group ${\displaystyle {\mathbb {Z} }\rtimes {\mathbb {Z} }_{2}}$.

• Twisted operators ${\displaystyle {\mathcal {L}}_{n}=L_{n}+{1 \over 2}(n+1)J_{n}}$ were introduced by Eguchi & Yang (1990) and satisfy:

${\displaystyle [{\mathcal {L}}_{m},{\mathcal {L}}_{n}]=(m-n){\mathcal {L}}_{m+n}}$

so that these operators satisfy the Virasoro relation with central charge 0. The constant ${\displaystyle c}$ still appears in the relations for ${\displaystyle J_{m}}$ and the modified relations

${\displaystyle \displaystyle {[{\mathcal {L}}_{m},J_{n}]=-nJ_{m+n}+{c \over 6}(m^{2}+m)\delta _{m+n,0}}}$

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

Constructions

Free field construction

Green, Schwarz & Witten (1988) harvtxt error: no target: CITEREFGreenSchwarzWitten1988 (help) give a construction using two commuting real bosonic fields ${\displaystyle (a_{n})}$, ${\displaystyle (b_{n})}$

${\displaystyle \displaystyle {[a_{m},a_{n}]={m \over 2}\delta _{m+n,0},\,\,\,\,[b_{m},b_{n}]={m \over 2}\delta _{m+n,0}},\,\,\,\,a_{n}^{*}=a_{-n},\,\,\,\,b_{n}^{*}=b_{-n}}$

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

${\displaystyle \displaystyle {\{e_{r},e_{s}^{*}\}=\delta _{r,s},\,\,\,\,\{e_{r},e_{s}\}=0.}}$

${\displaystyle L_{n}}$ is defined to the sum of the Virasoro operators naturally associated with each of the three systems

${\displaystyle L_{n}=\sum _{m}:a_{-m+n}a_{m}:+\sum _{m}:b_{-m+n}b_{m}:+\sum _{r}(r+{n \over 2}):e_{r}^{*}e_{n+r}:}$

where normal ordering has been used for bosons and fermions.

The current operator ${\displaystyle J_{n}}$ is defined by the standard construction from fermions

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

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

${\displaystyle G_{r}^{+}=\sum (a_{-m}+ib_{-m})\cdot e_{r+m},\,\,\,\,G_{r}^{-}=\sum (a_{r+m}-ib_{r+m})\cdot e_{m}^{*}}$

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 ${\displaystyle \ell }$ with basis ${\displaystyle E_{n},F_{n},H_{n}}$ satisfying

${\displaystyle [H_{m},H_{n}]=2m\ell \delta _{n+m,0},}$

${\displaystyle [E_{m},F_{n}]=H_{m+n}+m\ell \delta _{m+n,0},}$

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

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

the supersymmetric generators are defined by

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

This yields the N=2 superconformal algebra with

${\displaystyle c=3\ell /(\ell +2)}$.

The algebra commutes with the bosonic operators

${\displaystyle X_{n}=H_{n}-2\sum _{r}:e_{r}^{*}e_{n+r}:.}$

The space of physical states consists of eigenvectors of ${\displaystyle X_{0}}$ simultaneously annihilated by the ${\displaystyle X_{n}}$'s for positive ${\displaystyle n}$ and the supercharge operator

${\displaystyle Q=G_{1/2}^{+}+G_{-1/2}^{-}}$ (Neveu–Schwarz)

${\displaystyle Q=G_{0}^{+}+G_{0}^{-}.}$ (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 ${\displaystyle G}$ and a closed subgroup ${\displaystyle H}$ of maximal rank, i.e. containing a maximal torus ${\displaystyle T}$ of ${\displaystyle G}$, with the additional condition that the dimension of the centre of ${\displaystyle H}$ is non-zero. In this case the compact Hermitian symmetric space ${\displaystyle G/H}$ is a Kähler manifold, for example when ${\displaystyle H=T}$. 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 ${\displaystyle G}$.[3]

See also

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

Notes

1. Green, Schwarz & Witten 1998a, pp. 240–241 harvnb error: no target: CITEREFGreenSchwarzWitten1998a (help)
2. Wassermann 2010
3. 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 (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, 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, 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