The space can be written as C[x1,x2,...]vλ, where vλ is a distinguished ground-state vector. The module category is not semisimple, since one may induce a representation of the abelian Lie algebra where b0 acts by a nontrivial Jordan block. For the rank n free boson, one has an irreducible module Vλ for each vector λ in complex n-dimensional space. Each vector b ∈ Cn yields the operator b0, and the Fock space Vλ is distinguished by the property that each such b0 acts as scalar multiplication by the inner product (b, λ). 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((z1/N)), with the following monodromy condition: if u ∈ V satisfies σ u = exp(2πik/N)u, then un = 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 ramified Galois cover. In the conformal field theory literature, twisted modules are called twisted sectors, and are intimately connected with string theory on orbifolds. Vertex operator superalgebras By allowing the underlying vector space to be a superspace (i.e., a Z/2Z-graded vector space ) one can define a vertex superalgebra by the same data as a vertex algebra, with 1 in V+ and T an even operator. The axioms are essentially the same, but one must incorporate suitable signs into the locality axiom, or one of the equivalent formulations. That is, if a and b are homogeneous, one compares Y(a,z)Y(b,w) with εY(b,w)Y(a,z), where ε is –1 if both a and b are odd and 1 otherwise. If in addition there is a Virasoro element ω in the even part of V2, and the usual grading restrictions are satisfied, then V is called a vertex operator superalgebra. One of the simplest examples is the vertex operator superalgebra generated by a single free fermion ψ. As a Virasoro representation, it has central charge 1/2, and decomposes as a direct sum of Ising modules of lowest weight 0 and 1/2. One may also describe it as a spin representation of the Clifford algebra on the quadratic space t1/2C[t,t−1](dt)1/2 with residue pairing. The vertex operator superalgebra is holomorphic, in the sense that all modules are direct sums of itself, i.e., the module category is