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: 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. Given a vertex algebra V with multiplication Y, a V-module is a vector space M equipped with an action YM: V ⊗ M → M((z)), satisfying the following conditions: (Identity) YM(1,z) = IdM (Associativity, or Jacobi identity) For any u, v ∈ V, w ∈ M, there is an element such that YM(u,z)YM(v,x)w and YM(Y(u,z–x)v,x)w are the corresponding expansions of in M((z))((x)) and M((x))((z–x)). Equivalently, the following "Jacobi identity" holds: The modules of a vertex algebra form an abelian category. When working with vertex operator algebras, the previous definition is given the name "weak module", and V-modules are required to satisfy the additional condition that L0 acts semisimply with finite-dimensional eigenspaces and eigenvalues bounded below in each coset of Z. Work of Huang, Lepowsky, Miyamoto, and Zhang 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 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 C2-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 SL2(Z). In particular, if a VOA is holomorphic, i.e., its representation category is equivalent to that of vector spaces, then its partition function is SL2(Z)-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. To connect with our first example, the irreducible modules of the rank 1 free boson are given by Fock spaces Vλ with some fixed momentum λ, i.e., induced representations of the Heisenberg Lie algebra, where the element b0 acts by scalar multiplication by λ.