Editing Vertex operator algebra (section)

== Additional constructions ==
* Fixed point subalgebras: Given an action of a symmetry group on a vertex operator algebra, the subalgebra of fixed vectors is also a vertex operator algebra.  In 2013, Miyamoto proved that two important finiteness properties, namely Zhu's condition C<sub>2</sub> and regularity, are preserved when taking fixed points under finite solvable group actions.
* Current extensions: Given a vertex operator algebra and some modules of integral conformal weight, one may under favorable circumstances describe a vertex operator algebra structure on the direct sum.  Lattice vertex algebras are a standard example of this.  Another family of examples are framed VOAs, which start with tensor products of Ising models, and add modules that correspond to suitably even codes.
* Orbifolds: Given a finite cyclic group acting on a holomorphic VOA, it is conjectured that one may construct a second holomorphic VOA by adjoining irreducible twisted modules and taking fixed points under an induced automorphism, as long as those twisted modules have suitable conformal weight.  This is known to be true in special cases, e.g., groups of order at most 3 acting on lattice VOAs.
* The coset construction (due to Goddard, Kent, and Olive): Given a vertex operator algebra ''V'' of central charge ''c'' and a set ''S'' of vectors, one may define the commutant ''C''(''V'',''S'') to be the subspace of vectors ''v'' strictly commute with all fields coming from ''S'', i.e., such that ''Y''(''s'',''z'')''v'' ∈ V<nowiki>[[</nowiki>''z''<nowiki>]]</nowiki> for all ''s'' ∈ ''S''.  This turns out to be a vertex subalgebra, with ''Y'', ''T'', and identity inherited from ''V''.   And if ''S'' is a VOA of central charge ''c''<sub>S</sub>, the commutant is a VOA of central charge ''c''–''c''<sub>S</sub>.  For example, the embedding of ''SU''(2) at level ''k''+1 into the tensor product of two ''SU''(2) algebras at levels ''k'' and 1 yields the Virasoro discrete series with ''p''=''k''+2, ''q''=''k''+3, and this was used to prove their existence in the 1980s.  Again with ''SU''(2), the embedding of level ''k''+2 into the tensor product of level ''k'' and level 2 yields the ''N''=1 superconformal discrete series.
* BRST reduction: For any degree 1 vector ''v'' satisfying ''v''<sub>0</sub><sup>2</sup>=0, the cohomology of this operator has a graded vertex superalgebra structure.  More generally, one may use any weight 1 field whose residue has square zero.  The usual method is to tensor with fermions, as one then has a canonical differential.  An important special case is quantum Drinfeld–Sokolov reduction applied to affine Kac–Moody algebras to obtain affine ''W''-algebras as degree 0 cohomology.  These ''W'' algebras also admit constructions as vertex subalgebras of free bosons given by kernels of screening operators.