Type of Document Dissertation Author Gui, Bin Author's Email Address binguimath@gmail.com URN etd-06272018-151541 Title A unitary tensor product theory for unitary vertex operator algebra modules Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Vaughan Jones Committee Chair Akram Aldroubi Committee Member Dietmar Bisch Committee Member Jesse Peterson Committee Member Thomas Weiler Committee Member Keywords

- algebraic quantum field theory
- tensor category
- conformal field theory
- Vertex operator algebra
- unitary modular tensor category
Date of Defense 2018-03-29 Availability unrestricted AbstractLet V be a unitary vertex operator algebra (VOA) satisfying the following conditions: (1) V is of CFT type. (2) Every N-gradable weak V -module is completely reducible. (3) V is C2-cofinite. Let Rep(V)be the category of unitary V -modules, and let C be a subcategory of Rep(V) whose objects are closed under taking tensor product. Then C is a ribbon fusion category. For any objects Wi; Wj of C, we define a sesquilinear form on thetensor product Wi bWj. We show that if these sesquilinear forms are positive definite (i.e.,

when they are inner products), then the ribbon category C is unitary. We show that if the unitary V -modules and a generating set of intertwining operators in C satisfy certain energy bounds, then these sesquilinear forms are positive definite. Our result can be applied to the modular tensor categories associated to unitary Virasoro VOAs, and unitary affine VOAs

of type An; Dn; G2, and more.

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