Editing Vertex operator algebra (section)

==Modules==
Much like ordinary rings, vertex algebras admit a notion of module, or representation.  Modules play an important role in conformal field theory, where they are often called sectors. A standard assumption in the physics literature is that the full [[Hilbert space]] of a conformal field theory decomposes into a sum of tensor products of left-moving and right-moving sectors:

:<math>\mathcal{H} \cong \bigoplus_{i \in I} M_i \otimes \overline{M_i}</math>

That is, a conformal field theory has a vertex operator algebra of left-moving chiral symmetries, a vertex operator algebra of right-moving chiral symmetries, and the sectors moving in a given direction are modules for the corresponding vertex operator algebra.

=== Definition ===
Given a vertex algebra ''V'' with multiplication ''Y'', a ''V''-module is a vector space ''M'' equipped with an action ''Y''<sup>M</sup>: ''V'' ⊗ ''M'' → ''M''((''z'')), satisfying the following conditions:

: (Identity) ''Y''<sup>M</sup>(1,z) = Id<sub>M</sub>
: (Associativity, or Jacobi identity) For any ''u'', ''v'' ∈ ''V'', ''w'' ∈ ''M'', there is an element

:<math>X(u,v,w;z,x) \in M[[z,x]][z^{-1}, x^{-1}, (z-x)^{-1}]</math>

such that ''Y''<sup>M</sup>(''u'',''z'')''Y''<sup>M</sup>(''v'',''x'')''w'' and ''Y''<sup>M</sup>(''Y''(''u'',''z''–''x'')''v'',''x'')''w''
are the corresponding expansions of <math>X(u,v,w;z,x)</math> in ''M''((''z''))((''x'')) and ''M''((''x''))((''z''–''x'')).
Equivalently, the following "[[Jacobi identity]]" holds:

:<math>z^{-1}\delta\left(\frac{y-x}{z}\right)Y^M(u,x)Y^M(v,y)w - z^{-1}\delta\left(\frac{-y+x}{z}\right)Y^M(v,y)Y^M(u,x)w = y^{-1}\delta\left(\frac{x+z}{y}\right)Y^M(Y(u,z)v,y)w.</math>

The modules of a vertex algebra form an [[abelian category]].  When working with vertex operator algebras, the previous definition is sometimes given the name ''weak <math>V</math>-module'', and genuine ''V''-modules must respect the conformal structure given by the conformal vector <math>\omega</math>. More precisely, they are required to satisfy the additional condition that ''L''<sub>0</sub> acts semisimply with finite-dimensional eigenspaces and eigenvalues bounded below in each coset of '''Z'''.  Work of Huang, Lepowsky, Miyamoto, and Zhang{{citation needed|date=January 2023}} has shown at various levels of generality that modules of a vertex operator algebra admit a fusion tensor product operation, and form a [[braided tensor category]].

When the [[category (mathematics)|category]] of ''V''-modules is semisimple with finitely many irreducible objects, the vertex operator algebra ''V'' is called rational.  Rational vertex operator algebras satisfying an additional finiteness hypothesis (known as Zhu's ''C''<sub>2</sub>-cofiniteness condition) are known to be particularly well-behaved, and are called ''regular''.  For example, Zhu's 1996 modular invariance theorem asserts that the characters of modules of a regular VOA form a vector-valued representation of <math>\mathrm{SL}(2, \mathbb{Z})</math>.  In particular, if a VOA is ''holomorphic'', that is, its representation category is equivalent to that of vector spaces, then its partition function is <math>\mathrm{SL}(2, \mathbb{Z})</math>-invariant up to a constant.  Huang showed that the category of modules of a regular VOA is a [[modular tensor category]], and its fusion rules satisfy the [[Verlinde formula]].

=== Heisenberg algebra modules ===

Modules of the Heisenberg algebra can be constructed as [[Fock space]]s <math>\pi_\lambda</math> for <math>\lambda \in \mathbb{C}</math> which are induced representations of the [[Heisenberg Lie algebra]], given by a vacuum vector <math>v_\lambda</math> satisfying <math>b_nv_\lambda = 0</math> for <math>n > 0</math>, <math>b_0v_\lambda = 0</math>, and being acted on freely by the negative modes <math>b_{-n}</math> for <math>n>0</math>.  The space can be written as <math>\mathbb{C}[b_{-1}, b_{-2}, \cdots]v_\lambda</math>. Every irreducible, <math>\mathbb{Z}</math>-graded Heisenberg algebra module with gradation bounded below is of this form.

These are used to construct lattice vertex algebras, which as vector spaces are direct sums of Heisenberg modules, when the image of <math>Y</math> is extended appropriately to module elements.

The module category is not semisimple, since one may induce a representation of the abelian Lie algebra where ''b''<sub>0</sub> acts by a nontrivial [[Jordan block]].  For the rank ''n'' free boson, one has an irreducible module ''V''<sub>λ</sub> for each vector λ in complex ''n''-dimensional space.  Each vector ''b'' ∈ '''C'''<sup>n</sup> yields the operator ''b''<sub>0</sub>, and the Fock space ''V''<sub>λ</sub> is distinguished by the property that each such ''b''<sub>0</sub> acts as scalar multiplication by the inner product (''b'', λ).

=== Twisted modules ===

Unlike ordinary rings, vertex algebras admit a notion of twisted module attached to an automorphism.  For an automorphism σ of order ''N'', the action has the form ''V'' ⊗ ''M'' → ''M''((''z''<sup>1/N</sup>)), with the following [[monodromy]] condition: if ''u'' ∈ ''V'' satisfies σ ''u'' = exp(2π''ik''/''N'')''u'', then ''u''<sub>n</sub> = 0 unless ''n'' satisfies ''n''+''k''/''N'' ∈ '''Z''' (there is some disagreement about signs among specialists).  Geometrically, twisted modules can be attached to branch points on an algebraic curve with a [[Ramification (mathematics)|ramified]] [[Galois cover]].  In the conformal field theory literature, twisted modules are called [[twisted sector]]s, and are intimately connected with string theory on [[orbifold]]s.