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| {{DISPLAYTITLE:''N'' = 2 superconformal algebra}}
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| 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 (string theory)|mirror symmetry]]. It was introduced by {{harvs|txt | last1=Ademollo | first1=M. | last2=Brink | first2=L. | last3=D'Adda | first3=A. | last4=D'Auria | first4=R. | last5=Napolitano | first5=E. | last6=Sciuto | first6=S. | last7=Giudice | first7=E. Del | last8=Vecchia | first8=P. Di | last9=Ferrara | first9=S. | last10=Gliozzi | first10=F. | last11=Musto | first11=R. | last12=Pettorino | first12=R. | title=Supersymmetric strings and colour confinement | doi=10.1016/0370-2693(76)90061-7 | year=1976 | journal=Physics Letters B | volume=62 | issue=1 | pages=105–110}} as a gauge algebra of the U(1) fermionic string.
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| ==Definition==
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| 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.
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| The '''''N'' = 2 superconformal algebra''' is the Lie superalgebra with basis of even elements ''c'', ''L''<sub>''n''</sub>, ''J''<sub>''n''</sub>, for ''n'' an integer, and odd elements ''G''{{su|p=+|b=''r''}}, ''G''{{su|p=−|b=''r''}}, where <math>r\in {\Bbb Z}</math> (for the Ramond basis) or <math>r\in {1\over 2}+{\Bbb Z}</math> (for the Neveu–Schwarz basis) defined by the following relations:<ref>{{harvnb|Green|Schwarz|Witten|1998a|pp=240–241}}</ref>
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| ::''c'' is in the center
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| ::<math>\displaystyle{[L_m,L_n]=(m-n)L_{m+n} +{c\over 12} (m^3-m) \delta_{m+n,0}}</math>
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| ::<math>\displaystyle{[L_m,\,J_n]=-nJ_{m+n}}</math>
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| ::<math>\displaystyle{[J_m,J_n]={c\over 3} m\delta_{m+n,0}}</math>
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| ::<math>\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} }</math>
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| ::<math>\displaystyle{\{G_r^+,G_s^+\}=0=\{G_r^-,G_s^-\}}</math>
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| ::<math>\displaystyle{[L_m,G_r^{\pm}]=({m\over 2}-r) G^\pm_{r+m}}</math>
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| ::<math>\displaystyle{[J_m,G_r^\pm]= \pm G_{m+r}^\pm}</math>
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| If <math>r,s\in {\Bbb Z}</math> in these relations, this yields the
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| '''''N'' = 2 Ramond algebra'''; while if <math>r,s\in {1\over 2}+{\Bbb Z}</math> are
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| half-integers, it gives the '''''N'' = 2 Neveu–Schwarz algebra'''. The operators <math>L_n</math> generate a Lie subalgebra isomorphic to the [[Virasoro algebra]]. Together with the operators <math>G_r=G_r^+ + G_r^-</math>, they generate a Lie superalgebra isomorphic to the [[super Virasoro algebra]],
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| giving the Ramond algebra if <math>r,s</math> are integers and the Neveu–Schwarz algebra otherwise. When represented as operators on a [[inner product space|complex inner product space]], <math>c</math> 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:
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| :<math>\displaystyle{L_n^*=L_{-n}, \,\, J_m^*=J_{-m}, \,\,(G_r^\pm)^*=G_{-r}^\mp, \,\,c^*=c}</math>
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| ==Properties==
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| *The ''N'' = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism <math>\alpha</math> of {{harvtxt|Schwimmer|Seiberg|1987}}:
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| ::<math>\alpha(L_n)=L_n +{1\over 2} J_n + {c\over 24}\delta_{n,0}</math>
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| ::<math>\alpha(J_n)=J_n +{c\over 6}\delta_{n,0}</math>
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| ::<math>\alpha(G_r^\pm)=G_{r\pm {1\over 2}}^\pm</math>
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| :with inverse:
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| ::<math>\alpha^{-1}(L_n)=L_n -{1\over 2} J_n + {c\over 24}\delta_{n,0}</math>
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| ::<math>\alpha^{-1}(J_n)=J_n -{c\over 6}\delta_{n,0}</math>
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| ::<math>\alpha^{-1}(G_r^\pm)=G_{r\mp {1\over 2}}^\pm</math>
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| *In the ''N'' = 2 Ramond algebra, the zero mode operators <math>L_0</math>, <math>J_0</math>, <math>G_0^\pm</math> and the constants form a five-dimensional Lie superalgebra. They satisfy the same relations as the fundamental operators in [[Kähler manifold|Kähler geometry]], with <math>L_0</math> corresponding to the Laplacian, <math>J_0</math> the degree operator, and <math>G_0^\pm</math> the <math>\partial</math> and <math>\overline{\partial}</math> operators.
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| * Even integer powers of the spectral shift give automorphisms of the ''N'' = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism <math>\beta</math>, of period two, is given by
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| ::<math>\displaystyle{\beta(L_m)=L_m},</math>
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| ::<math>\beta(J_m)=-J_m-{c\over 3} \delta_{m,0},</math>
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| ::<math>\beta(G_r^\pm)=G_r^\mp</math>
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| :In terms of Kähler operators, <math>\beta</math> corresponds to conjugating the complex structure. Since <math>\beta\alpha \beta^{-1}=\alpha^{-1}</math>, the automorphisms <math>\alpha^2</math> and <math>\beta</math> generate a group of automorphisms of the ''N'' = 2 superconformal algebra isomorphic to the [[infinite dihedral group]] <math>{\Bbb Z}\rtimes {\Bbb Z}_2</math>.
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| * Twisted operators <math>{\mathcal L}_n=L_n+ {1\over 2} (n+1)J_n</math> were introduced by {{harvtxt|Eguchi|Yang|1990}} and satisfy:
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| ::<math>[{\mathcal L}_m,{\mathcal L}_n]=(m-n){\mathcal L}_{m+n}</math>
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| :so that these operators satisfy the Virasoro relation with central charge 0. The constant <math>c</math> still appears in the relations for <math>J_m</math> and the modified relations
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| ::<math>\displaystyle{[{\mathcal L}_m,J_n] =-nJ_{m+n} +{c\over 6} (m^2+m)\delta_{m+n,0}}</math>
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| ::<math>\displaystyle{\{G_r^+,G_s^-\} =2{\mathcal L}_{r+s}-2sJ_{r+s} +{c\over 3} (m^2+m) \delta_{m+n,0}}</math>
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| ==Constructions==
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| ===Free field construction===
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| {{harvtxt|Green|Schwarz|Witten|1988}} give a construction using two commuting real [[bosonic field]]s <math>(a_n)</math>, <math>(b_n)</math>
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| :<math> \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}</math>
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| and a complex [[fermionic field]] <math>(e_r)</math>
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| :<math> \displaystyle{\{e_r,e^*_s\}=\delta_{r,s},\,\,\,\, \{e_r,e_s\}=0.}</math>
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| <math>L_n</math> is defined to the sum of the Virasoro operators naturally associated with each of the three systems
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| :<math>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} :</math>
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| where [[normal ordering]] has been used for bosons and fermions.
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| The current operator <math> J_n</math> is defined by the standard construction from fermions
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| :<math>J_n = \sum_r : e_r^*e_{n+r} : </math>
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| and the two supersymmetric operators <math> G_r^\pm</math> by
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| :<math> G^+_r=\sum (a_{-m} + i b_{-m}) \cdot e_{r+m},\,\,\,\, G_r^-=\sum (a_{r+m} - ib_{r+m}) \cdot e^*_{m}</math>
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| This yields an ''N'' = 2 Neveu–Schwarz algebra with ''c'' = 3.
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| ===SU(2) supersymmetric coset construction===
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| {{harvtxt|Di Vecchia|Petersen|Yu|Zheng|1986}} gave a coset construction of the ''N'' = 2 superconformal algebras, generalizing the [[coset construction]]s of {{harvtxt|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 <math>\ell</math> with basis <math>E_n,F_n,H_n</math> satisfying
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| :<math>[H_m,H_n]=2m\ell\delta_{n+m,0},</math>
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| :<math>[E_m,F_n]=H_{m+n}+m \ell\delta_{m+n,0},</math>
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| :<math> \displaystyle{[H_m,E_n]=2E_{m+n},}</math>
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| :<math>\displaystyle{[H_m,F_n]=-2F_{m+n},}</math>
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| the supersymmetric generators are defined by
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| :<math> \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^*.}</math>
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| This yields the N=2 superconformal algebra with
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| :<math>c=3\ell/(\ell+2)</math>.
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| The algebra commutes with the bosonic operators
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| :<math>X_n=H_n - 2 \sum_r : e_r^*e_{n+r} :.</math>
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| The space of [[physical state]]s consists of [[eigenvector]]s of <math>X_0</math> simultaneously annihilated by the <math>X_n</math>'s for positive <math>n</math> and the supercharge operator
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| :<math>Q=G_{1/2}^+ + G_{-1/2}^-</math> (Neveu–Schwarz)
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| :<math>Q=G_0^+ +G_0^-.</math> (Ramond)
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| 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]].<ref>{{harvnb|Wassermann|2010}}</ref>
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| ===Kazama–Suzuki supersymmetric coset construction===
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| {{harvtxt|Kazama|Suzuki|1989}} generalized the SU(2) coset construction to any pair consisting of a simple [[compact Lie group]] <math>G</math> and a closed subgroup <math>H</math> of maximal rank, i.e. containing a [[maximal torus]] <math>T</math> of <math>G</math>, with the additional condition that
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| the dimension of the centre of <math>H</math> is non-zero. In this case the compact [[Hermitian symmetric space]] <math>G/H</math> is a Kähler manifold, for example when <math>H=T</math>. 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 <math>G</math>.<ref>{{harvnb|Wassermann|2010}}</ref>
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| ==See also==
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| *[[Virasoro algebra]]
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| *[[Super Virasoro algebra]]
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| *[[Coset construction]]
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| *[[Type IIB string theory]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation | last1=Ademollo | first1=M. | last2=Brink | first2=L. | last3=D'Adda | first3=A. | last4=D'Auria | first4=R. | last5=Napolitano | first5=E. | last6=Sciuto | first6=S. | last7=Giudice | first7=E. Del | last8=Vecchia | first8=P. Di | last9=Ferrara | first9=S. | last10=Gliozzi | first10=F. | last11=Musto | first11=R. | last12=Pettorino | first12=R. | title=Supersymmetric strings and colour confinement | doi=10.1016/0370-2693(76)90061-7 | year=1976 | journal=Physics Letters B | volume=62 | issue=1 | pages=105–110}}
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| *{{citation|first=W.|last=Boucher|first2=D,|last2=Freidan|authorlink2=Daniel Friedan|first3=A.|last3=Kent|title=Determinant formulae and unitarity for the ''N'' = 2 superconformal algebras in two dimensions or exact results on string compactification|journal=Phys. Lett. B|volume=172|year=1986|pages=316–322| doi = 10.1016/0370-2693(86)90260-1 }}
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| *{{citation|last=Di Vecchia|first= P.|last2 =Petersen|first2=J. L.|last3=Yu,|first3= M.|last4=Zheng|first4= H. B.|title=Explicit construction of unitary representations of the ''N'' = 2 superconformal algebra|journal=Phys. Lett. B |volume=174 |year=1986|pages=280–284}}
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| *{{citation|last=Eguchi|first=Tohru|last2= Yang|first2=Sung-Kil|title=''N'' = 2 superconformal models as topological field theories|
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| journal=Modern Phys. Lett. A|volume= 5 |year=1990|pages=1693–1701}}
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| *{{citation|first=P.|last= Goddard|authorlink=Peter Goddard (physicist)|first2= A.|last2= Kent|first3=D.|last3= Olive|authorlink3=David Olive|url =http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104114626 |title=Unitary representations of the Virasoro and super-Virasoro algebras|journal= Comm. Math. Phys. |volume= 103|year=1986|pages=105–119}}
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| *{{citation|first=Michael B.|last=Green|authorlink=Michael B. Green|first2=John H.|last2=Schwarz|authorlink2=John Henry Schwarz|first3=Edward|last3=Witten|
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| authorlink3=Edward Witten|title=Superstring theory, Volume 1: Introduction|publisher=Cambridge University Press|year=1988a|isbn=0-521-35752-7}}
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| *{{citation|first=Michael B.|last=Green|authorlink=Michael B. Green|first2=John H.|last2=Schwarz|authorlink2=John Henry Schwarz|first3=Edward|last3=Witten|
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| authorlink3=Edward Witten|title=Superstring theory, Volume 2: Loop amplitudes, anomalies and phenomenology|publisher=Cambridge University Press|year=1988b|
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| isbn=0-521-35753-5}}
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| *{{citation|last=Kazama|first= Yoichi|last2= Suzuki|first2= Hisao|title=New ''N'' = 2 superconformal field theories and superstring compactification|journal=Nuclear Phys. B |volume=321 |year=1989|pages= 232–268|doi=10.1016/0550-3213(89)90250-2}}
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| *{{citation|last=Schwimmer|first= A.|last2= Seiberg|first2= N.|authorlink2=Nathan Seiberg|title=Comments on the ''N'' = 2, 3, 4 superconformal algebras in two dimensions|journal=Phys. Lett. B |volume=184|year=1987|pages=191–196}}
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| *{{citation|first=Claire|last=Voisin|authorlink=Claire Voisin|title=Mirror symmetry|series=SMF/AMS texts and monographs|volume=1|year=1999|publisher=American Mathematical Society|isbn=0-8218-1947-X}}
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| *{{Citation | last1=Wassermann | first1=A. J. | title=Lecture notes on Kac-Moody and Virasoro algebras | origyear=1998 | arxiv=1004.1287 | year=2010}}
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| *{{citation|title=Introduction to supersymmetry and supergravity|first=Peter C.|last=West|edition=2nd|publisher=World Scientific|year=1990|isbn=981-02-0099-4|pages=337–8}}
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| {{DEFAULTSORT:N 2 Superconformal Algebra}}
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| [[Category:String theory]]
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| [[Category:Conformal field theory]]
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| [[Category:Lie algebras]]
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| [[Category:Representation theory]]
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| [[Category:Supersymmetry]]
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