N = 2 Superconformal Algebra
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In
mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, the 2D ''N'' = 2 superconformal algebra is an infinite-dimensional
Lie superalgebra In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a \Z/2\Z grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. The notion of \Z/2\Z gra ...
, related to
supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
, that occurs in
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
and
two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal fi ...
. It has important applications in
mirror symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...
. It was introduced by 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 In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For exampl ...
. The ''N'' = 2 superconformal algebra is the Lie superalgebra with basis of even elements ''c'', ''L''''n'', ''J''''n'', for ''n'' an
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
, and odd elements ''G'', ''G'', where r\in (for the Ramond basis) or r\in + (for the Neveu–Schwarz basis) defined by the following relations: ::''c'' is in the center :: _m,L_n= \left(m-n\right) L_ + \left(m^3-m\right) \delta_ :: _m,\,J_n-nJ_ :: _m,J_n= m\delta_ ::\ = L_ + \left(r-s\right) J_ + \left(r^2-\right) \delta_ ::\ = 0 = \ :: _m,G_r^= \left( -r \right) G^\pm_ :: _m,G_r^\pm \pm G_^\pm If r,s\in in these relations, this yields the ''N'' = 2 Ramond algebra; while if r,s\in + are half-integers, it gives the ''N'' = 2 Neveu–Schwarz algebra. The operators L_n generate a Lie subalgebra isomorphic to the
Virasoro algebra In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel ...
. Together with the operators G_r=G_r^+ + G_r^-, they generate a Lie superalgebra isomorphic to the
super Virasoro algebra In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance in superstring theory: the Ramond algebra (n ...
, 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: :


Properties

*The ''N'' = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism \alpha of : \alpha(L_n)=L_n + J_n + \delta_ \alpha(J_n)=J_n +\delta_ \alpha(G_r^\pm)=G_^\pm with inverse: \alpha^(L_n)=L_n - J_n + \delta_ \alpha^(J_n)=J_n -\delta_ \alpha^(G_r^\pm)=G_^\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 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- \delta_, \beta(G_r^\pm)=G_r^\mp In terms of Kähler operators, \beta corresponds to conjugating the complex structure. Since \beta\alpha \beta^=\alpha^, the automorphisms \alpha^2 and \beta generate a group of automorphisms of the ''N'' = 2 superconformal algebra isomorphic to the
infinite dihedral group In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p''1''m'' ...
\rtimes _2. *Twisted operators _n=L_n+ (n+1)J_n were introduced by and satisfy: m,_n= (m-n) _ so that these operators satisfy the Virasoro relation with
central charge In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other element ...
0. The constant c still appears in the relations for J_m and the modified relations m,J_n= -nJ_ + \left(m^2 + m \right) \delta_ \ = 2_-2sJ_ + \left(m^2+m\right) \delta_


Constructions


Free field construction

give a construction using two commuting real bosonic fields (a_n), (b_n) : , \,\,\,\, a_n^*=a_,\,\,\,\, b_n^*=b_ and a complex
fermionic field In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of ...
(e_r) : \=\delta_,\,\,\,\, \=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_ a_m : + \sum_m : b_ b_m : + \sum_r \left(r+\right): e^*_e_ : 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_ : and the two supersymmetric operators G_r^\pm by : G^+_r=\sum (a_ + i b_) \cdot e_,\,\,\,\, G_r^-=\sum (a_ - ib_) \cdot e^*_ This yields an ''N'' = 2 Neveu–Schwarz algebra with ''c'' = 3.


SU(2) supersymmetric coset construction

gave a coset construction of the ''N'' = 2 superconformal algebras, generalizing the coset constructions of for the discrete series representations of the Virasoro and super Virasoro algebra. Given a representation of the affine Kac–Moody algebra of
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...
at level \ell with basis E_n,F_n,H_n satisfying : _m,H_n2m\ell\delta_, : _m,F_nH_+m \ell\delta_, : _m,E_n2E_, : _m,F_n-2F_, the supersymmetric generators are defined by : G^+_r = (\ell/2+ 1)^ \sum E_ \cdot e_, \,\,\, G^-_r = (\ell/2 +1 )^ \sum F_\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_ :. The space of
physical state In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. Different states are distinguished by the ways the component parti ...
s consists of
eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
s of X_0 simultaneously annihilated by the X_n's for positive n and the supercharge operator :Q=G_^+ + G_^- (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

generalized the SU(2) coset construction to any pair consisting of a simple
compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact space, compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are ...
G and a closed subgroup H of maximal rank, i.e. containing a
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
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 In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian ...
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 In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel ...
*
Super Virasoro algebra In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance in superstring theory: the Ramond algebra (n ...
* Coset construction * Type IIB string theory


Notes


References

* * * * * * * * * * * * {{DEFAULTSORT:N 2 Superconformal Algebra String theory Conformal field theory Lie algebras Representation theory Supersymmetry