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Higher level twisted Zhu algebras. (English) Zbl 1317.81233

Summary: The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra \(V\), graded by \(\Gamma /\mathbb{Z}\) for some subgroup \(\Gamma\) of \(\mathbb{R}\) containing \(\mathbb{Z}\), and with a Hamiltonian operator \(\operatorname{H}\) having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level \(p\) Zhu algebras \(\operatorname{Zhu}_{p,\; \Gamma}(V)\), and we prove the following theorems: For each \(p\), there is a bijection between the irreducible \(\operatorname{Zhu}_{p,\; \Gamma}(V)\)-modules and the irreducible \(\Gamma\)-twisted positive energy \(V\)-modules, and \(V\) is \((\Gamma, H)\)-rational if and only if all its Zhu algebras \(\operatorname{Zhu}_{p, \Gamma}(V)\) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for \(H\). We provide an explicit description of the level \(p\) Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra \(\operatorname{Vir}^c\) and the universal affine Kac-Moody vertex algebra \(V^k(\mathfrak{g})\) at non-critical level. We also compute the inverse limits of these directed systems of algebras.{
©2011 American Institute of Physics}

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
57R10 Smoothing in differential topology
17B69 Vertex operators; vertex operator algebras and related structures
17B70 Graded Lie (super)algebras
19L50 Twisted \(K\)-theory; differential \(K\)-theory
17B68 Virasoro and related algebras
17B81 Applications of Lie (super)algebras to physics, etc.
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References:

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