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We define the notion of strongly interlocked for indecomposable generalizedmodules for a vertex operator algebra, and show that the notion of gradedpseudo-trace is well defined for modules which satisfy this property. We provethat the graded pseudo-trace is a symmetric linear operator that satisfies thelogarithmic derivative property. As an application, we prove that all theindecomposable reducible generalized modules for the rank one Heisenberg (onefree boson) vertex operator algebras are strongly interlocked, independent ofthe choice of conformal vector and thus have well-defined graded pseudo-traces.We also completely characterize which indecomposable reducible generalizedmodules for the universal Virasoro vertex operator algebras induced from thelevel zero Zhu algebra are strongly interlocked. In particular, we prove thatthe universal Virasoro vertex operator algebra with central charge c hasmodules induced from the level zero Zhu algebra with conformal weight h thatare strongly interlocked if and only if either (c,h) is outside the extendedKac table, or the central charge is either c = 1 or 25, the conformal weightsatisfies a certain property, and the level zero Zhu algebra module beinginduced is determined by a Jordan block of size less than a certain specifiedparameter. We give several examples of graded pseudo-traces for Heisenberg andVirasoro strongly interlocked modules.

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