Graded pseudo-traces for strongly interlocked modules for a vertex operator algebra and applications

We define the notion of {\it strongly interlocked} for indecomposable generalized modules for a vertex operator algebra, and show that the notion of graded pseudo-trace is well defined for modules which satisfy this property in certain settings. We prove that in these settings the graded pseudo-trace is a symmetric linear operator that satisfies the logarithmic derivative property. As an application, we prove that all the indecomposable reducible generalized modules for the rank one Heisenberg (one free boson) vertex operator algebras are strongly interlocked, independent of the choice of conformal vector and have well-defined graded pseudo-traces. We also completely characterize which indecomposable reducible generalized modules for the universal Virasoro vertex operator algebras induced from the level zero Zhu algebra are strongly interlocked. In particular, we prove that the universal Virasoro vertex operator algebra with central charge $c$ has modules induced from the level zero Zhu algebra with conformal weight $h$ that are strongly interlocked if and only if either $(c,h)$ is outside the extended Kac table, or the central charge is either $c = 1$ or $25$, the conformal weight satisfies a certain property, and the level zero Zhu algebra module being induced is determined by a Jordan block of size less than a certain specified parameter. We prove that all these modules for the universal Virasoro vertex operator algebra that are strongly interlocked have well-defined graded pseudo-traces. We give several examples of graded pseudo-traces for these Heisenberg and Virasoro strongly interlocked modules.