Editing Vertex operator algebra (section)

== Superconformal structures ==
The Virasoro algebra has some [[supersymmetry|supersymmetric extensions]] that naturally appear in [[superconformal field theory]] and [[superstring theory]].  The ''N''=1, 2, and 4 [[superconformal algebra]]s are of particular importance.

Infinitesimal holomorphic superconformal transformations of a [[supercurve]] (with one even local coordinate ''z'' and ''N'' odd local coordinates θ<sub>1</sub>,...,θ<sub>N</sub>) are generated by the coefficients of a super-stress–energy tensor ''T''(z, θ<sub>1</sub>, ..., θ<sub>N</sub>).

When ''N''=1, ''T'' has odd part given by a Virasoro field ''L''(''z''), and even part given by a field

:<math>G(z) = \sum_n G_n z^{-n-3/2}</math>

subject to commutation relations

* <math>[G_m,L_n] = (m-n/2)G_{m+n}</math>
* <math>[G_m,G_n] = (m-n)L_{m+n} + \delta_{m,-n} \frac{4m^2+1}{12}c</math>

By examining the symmetry of the operator products, one finds that there are two possibilities for the field ''G'': the indices ''n'' are either all integers, yielding the [[Ramond algebra]], or all half-integers, yielding the [[Neveu–Schwarz algebra]].  These algebras have unitary discrete series representations at [[central charge]]

:<math>\hat{c} = \frac{2}{3}c = 1-\frac{8}{m(m+2)} \quad m \geq 3</math>

and unitary representations for all ''c'' greater than 3/2, with lowest weight ''h'' only constrained by ''h''≥ 0 for Neveu–Schwarz and ''h'' ≥ ''c''/24 for Ramond.

An ''N''=1 superconformal vector in a vertex operator algebra ''V'' of central charge ''c'' is an odd element τ ∈ ''V'' of weight 3/2, such that
:<math>Y(\tau,z) = G(z) = \sum_{m \in \mathbb{Z}+1/2} G_n z^{-n-3/2},</math>

''G''<sub>−1/2</sub>τ =  ω, and the coefficients of ''G''(''z'') yield an action of the ''N''=1 Neveu–Schwarz algebra at central charge ''c''.

For ''N''=2 supersymmetry, one obtains even fields ''L''(''z'') and ''J''(''z''), and odd fields ''G''<sup>+</sup>(z) and ''G''<sup>−</sup>(z). The field ''J''(''z'') generates an action of the Heisenberg algebras (described by physicists as a ''U''(1) current). There are both Ramond and Neveu–Schwarz ''N''=2 superconformal algebras, depending on whether the indexing on the ''G'' fields is integral or half-integral.  However, the ''U''(1) current gives rise to a one-parameter family of isomorphic superconformal algebras interpolating between Ramond and Neveu–Schwartz, and this deformation of structure is known as spectral flow.  The unitary representations are given by discrete series with central charge ''c'' = 3-6/''m'' for integers ''m'' at least 3, and a continuum of lowest weights for ''c'' &gt; 3.

An ''N''=2 superconformal structure on a vertex operator algebra is a pair of odd elements τ<sup>+</sup>, τ<sup>−</sup> of weight 3/2, and an even element μ of weight 1 such that τ<sup>±</sup> generate ''G''<sup>±</sup>(z), and μ generates ''J''(''z'').

For ''N''=3 and 4, unitary representations only have central charges in a discrete family, with ''c''=3''k''/2 and 6''k'', respectively, as ''k'' ranges over positive integers.