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Unitarity of the Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, II

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Abstract

This is the second part in a two-part series of papers constructing a unitary structure for the modular tensor category associated to a unitary rational vertex operator algebra (VOA). We define, for a unitary rational vertex operator algebra V, a non-degenerate sesquilinear form \(\Lambda \) on each vector space of intertwining operators. We give two sets of criteria for the positivity of \(\Lambda \), both concerning the energy bounds condition of vertex operators and intertwining operators. These criteria can be applied to many familiar examples, including unitary Virasoro VOAs, unitary affine VOAs of type A, D, and more. Having shown that \(\Lambda \) is an inner product, we prove that \(\Lambda \) induces a unitary structure on the modular tensor category of V.

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Notes

  1. The primitive form of \(\Lambda \) appeared in [Was98]. See the discussion in Remark 6.4.

  2. By proposition 3.4, if an intertwining operator is energy bounded, then so is its adjoint intertwining operator. Therefore, it suffices to require that \({\mathcal {F}}\cup \overline{{\mathcal {F}}}\), instead of \({\mathcal {F}}\), generates the monoidal category of V, where \(\overline{{\mathcal {F}}}=\{W_{{\overline{i}}}:i\in {\mathcal {F}} \}\). Moreover, we are also interested in the general case where \({\mathcal {F}}\cup \overline{{\mathcal {F}}}\) generates, not the whole tensor category of V, but a smaller tensor subcategory. In this case, this tensor subcategory might not be modular, but only a ribbon fusion category. In Sect. 5.3, the statement of conditions A and B will take care of this general case.

  3. This theorem is also proved in [CWX]. We would like to thank Sebastiano Carpi for letting us know this fact.

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Acknowledgements

This research is supported by NSF Grant DMS-1362138. I am grateful to my advisor, professor Vaughan Jones, for his consistent support throughout this project.

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  1. Department of Mathematics, Vanderbilt University, Nashville, USA

    Bin Gui

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Communicated by Y. Kawahigashi

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Gui, B. Unitarity of the Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, II. Commun. Math. Phys. 372, 893–950 (2019). https://doi.org/10.1007/s00220-019-03534-0

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