Editing Vertex operator algebra (section)

=== 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]].