Classification of irreducible modules for the vertex operator algebra \(M(1)^+\)
The authors classify the irreducible modules for the fixed point vertex operator subalgebra \(M(1)^+\) of the vertex operator algebra \(M(1)\) associated to the Heisenberg algebra with central charge \(1\) under the \(-1\) automorphism \(\theta\). The classification result says that any irreducible \(M(1)^+\) module is isomorphic to either a submodule of a \(M(1)\)-module or a submodule of a \(\theta\)-twisted \(M(1)\)-module. The proof is based on results in [\textit{C. Dong} and \textit{R. L. Griess jun.}, J. Algebra 208, 262-275 (1998; Zbl 0918.17023)], in particular on the fact that \(M(1)^+\) is generated by the Virasoro element \(\omega\) and a singular vector \(J\) of weight \(4\) for the Virasoro algebra. This enables the authors to explicitly determine the structure of Zhu's algebra \(A(M(1)^+)\) as a commutative algebra over \(\mathbb C\) with two variables, and then use Zhu's algebra to classify the irreducible modules.
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