Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules

A unitary and strongly rational vertex operator algebra (VOA) \({\mathbb {V}}\) is called strongly unitary if all irreducible \({\mathbb {V}}\)-modules are unitarizable. A strongly unitary VOA \({\mathbb {V}}\) is called completely unitary if for each unitary \({\mathbb {V}}\)-modules \({\mathbb {W}}_1,{\mathbb {W}}_2\) the canonical non-degenerate Hermitian form on the fusion product \({\mathbb {W}}_1\boxtimes {\mathbb {W}}_2\) is positive. It is known that if \({\mathbb {V}}\) is completely unitary, then the modular category \(\textrm{Mod}^\textrm{u}({\mathbb {V}})\) of unitary \({\mathbb {V}}\)-modules is unitary (Gui in Commun Math Phys 372(3):893–950, 2019), and all simple VOA extensions of \({\mathbb {V}}\) are automatically unitary and moreover completely unitary (Gui in Int Math Res Not 2022(10):7550–7614, 2022; Carpi et al. in Commun Math Phys 1–44, 2023). In this paper, we give a geometric characterization of the positivity of the Hermitian product on \({\mathbb {W}}_1\boxtimes {\mathbb {W}}_2\), which helps us prove that the positivity is always true when \({\mathbb {W}}_1\boxtimes {\mathbb {W}}_2\) is an irreducible and unitarizable \({\mathbb {V}}\)-module. We give several applications: (1) We show that if \({\mathbb {V}}\) is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group G, and if \({\mathbb {V}}^G\) is strongly unitary, then \({\mathbb {V}}^G\) is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if \({\mathbb {V}}\) is unitary and strongly rational, and if \({\mathbb {U}}\) is a simple current extension which is unitarizable as a \({\mathbb {V}}\)-module, then \({\mathbb {U}}\) is a unitary VOA.