Editing Vertex operator algebra (section)

== Vertex operator superalgebras ==
By allowing the underlying vector space to be a superspace (i.e., a '''Z'''/2'''Z'''-graded vector space <math> V=V_+\oplus V_-</math>) one can define a ''vertex superalgebra'' by the same data as a vertex algebra, with 1 in ''V''<sub>+</sub> 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 ε''Y''(''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 ''V''<sub>2</sub>, 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 ''t''<sup>1/2</sup>'''C'''[''t'',''t''<sup>−1</sup>](''dt'')<sup>1/2</sup> 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 equivalent to the category of vector spaces.

The tensor square of the free fermion is called the free charged fermion, and by boson–fermion correspondence, it is isomorphic to the lattice vertex superalgebra attached to the odd lattice '''Z'''.{{sfn|Kac|1998}} This correspondence has been used by Date–Jimbo–Kashiwara-Miwa to construct [[soliton]] solutions to the [[Kadomtsev–Petviashvili equation|KP hierarchy]] of nonlinear PDEs.