Generalized vertex algebras and duality
Now we return to the setting of Chapter 5. The axioms for a generalized vertex operator algebra do not quite apply to the structure of Ω*, the vacuum space (3.29) (for the Heisenberg algebra (ĥ * ) z ) equipped with the relative vertex operators Y*. In fact, the finite-dimensionality axiom (6.4) and the boundedness axiom (6.5) fail to hold in general, since L is not necessarily positive definite. Moreover, the Jacobi identity (5.11) (Theorem 5.1) differs from the Jacobi identity (6.12) by the insertion of the factor c(ā, b). Of course, a suitable group G would also have to be defined. We would like to extend our axioms so as to include this structure. Then the corresponding generalization of the duality arguments of Chapter 7 would provide an alternate duality-based proof of the Jacobi identity (5.11) (recall Remark 7.18), and in particular, of Theorems 8.6.1 and 8.8.23 of [FLM3]. (The arguments in Chapter 7 above, in the Appendix of [FLM3] and in [FHL] are not sufficiently general to give a duality-based proof of Theorem 8.8.23 of [FLM3] — the special case of (5.11) in which h * = 0 and 〈ā, b〉 ? ∈ Z — because the rational lattice in that theorem is not assumed even or integral, and has an alternating form c(·,·); in the proof of Proposition 7.16, the vector v3 is arbitrary.)
Unable to display preview. Download preview PDF.