N = 2 superconformal algebra

Short description: 2D supersymmetric generalization to the conformal 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 $r\in \mathbb{\mathbb{Z}}$ (for the Ramond basis) or $r\in \frac{1}{2}+\mathbb{\mathbb{Z}}$ (for the Neveu–Schwarz basis) defined by the following relations:

c is in the center

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

$[L_m,J_n]=-n{J}_{m+n}$

$[J_m,J_n]=\frac{c}{3}m{\delta }_{m+n,0}$

$\{G_r^{+},G_s^{-}\}=L_{r+s}+\frac{1}{2}(r-s)J_{r+s}+\frac{c}{6}(r^2-\frac{1}{4}){\delta }_{r+s,0}$

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

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

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

If $r,s\in \mathbb{\mathbb{Z}}$ in these relations, this yields the N = 2 Ramond algebra; while if $r,s\in \frac{1}{2}+\mathbb{\mathbb{Z}}$ 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): $$\alpha (L_n)=L_n+\frac{1}{2}{J}_n+\frac{c}{24}{\delta }_{n,0}$$ $$\alpha (J_n)=J_n+\frac{c}{6}{\delta }_{n,0}$$ $$\alpha (G_r^{\pm })=G_{r\pm \frac{1}{2}}^{\pm }$$ with inverse: $${\alpha }^{-1}(L_n)=L_n-\frac{1}{2}{J}_n+\frac{c}{24}{\delta }_{n,0}$$ $${\alpha }^{-1}(J_n)=J_n-\frac{c}{6}{\delta }_{n,0}$$ $${\alpha }^{-1}(G_r^{\pm })=G_{r\mp \frac{1}{2}}^{\pm }$$
• 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 $\bar{\partial }$ 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 $$\beta (L_m)=L_m,$$ $$\beta (J_m)=-J_m-\frac{c}{3}{\delta }_{m,0},$$ $$\beta (G_r^{\pm })=G_r^{\mp }$$ 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 $\mathbb{\mathbb{Z}}⋊{\mathbb{\mathbb{Z}}}_2$.
• Twisted operators ${{ℒ}}_n=L_n+\frac{1}{2}(n+1)J_n$ were introduced by (Eguchi Yang) 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 $$[{{ℒ}}_m,J_n]=-n{J}_{m+n}+\frac{c}{6}(m^2+m){\delta }_{m+n,0}$$ $$\{G_r^{+},G_s^{-}\}=2{{ℒ}}_{r+s}-2s{J}_{r+s}+\frac{c}{3}(m^2+m){\delta }_{m+n,0}$$

Constructions

Free field construction

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

$[a_m,a_n]=\frac{m}{2}{\delta }_{m+n,0},[b_m,b_n]=\frac{m}{2}{\delta }_{m+n,0},a_n^{*}=a_{-n},b_n^{*}=b_{-n}$

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

$L_n=\sum _m:a_{-m+n}{a}_m:+\sum _m:b_{-m+n}{b}_m:+\sum _r(r+\frac{n}{2}):e_r^{*}{e}_{n+r}:$

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

(Di Vecchia Petersen) gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of (Goddard Kent) 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.

Kazama–Suzuki supersymmetric coset construction

(Kazama Suzuki) 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

References

• Ademollo, M.; Brink, L.; D'Adda, A.; D'Auria, R.; Napolitano, E.; Sciuto, S.; Giudice, E. Del; Vecchia, P. Di et al. (1976), "Supersymmetric strings and colour confinement", Physics Letters B 62 (1): 105–110, doi:10.1016/0370-2693(76)90061-7, Bibcode: 1976PhLB...62..105A, https://cds.cern.ch/record/438898 
• 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, doi:10.1016/0370-2693(86)90260-1, Bibcode: 1986PhLB..172..316B 
• 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, doi:10.1016/0370-2693(86)91099-3, Bibcode: 1986PhLB..174..280D 
• 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, Bibcode: 1990MPLA....5.1693E 
• Goddard, P.; Kent, A.; Olive, D. (1986), "Unitary representations of the Virasoro and super-Virasoro algebras", Comm. Math. Phys. 103 (1): 105–119, doi:10.1007/bf01464283, Bibcode: 1986CMaPh.103..105G, http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104114626 
• 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, Bibcode: 1987cup..bookR....G 
• Kazama, Yoichi; Suzuki, Hisao (1989), "New N = 2 superconformal field theories and superstring compactification", Nuclear Physics B 321 (1): 232–268, doi:10.1016/0550-3213(89)90250-2, Bibcode: 1989NuPhB.321..232K 
• 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, doi:10.1016/0370-2693(87)90566-1, Bibcode: 1987PhLB..184..191S 
• Voisin, Claire (1999), Mirror symmetry, SMF/AMS texts and monographs, 1, American Mathematical Society, ISBN 0-8218-1947-X 
• Wassermann, A. J. (2010). "Lecture notes on Kac-Moody and Virasoro algebras". 
• West, Peter C. (1990), Introduction to supersymmetry and supergravity (2nd ed.), World Scientific, pp. 337–8, ISBN 981-02-0099-4