Abstract

Jones introduced planar algebras in 1999, which has been generalized to surface algebras in 2019 by Liu. In this talk, we introduce an isotopy invariant partition function BIMSA for a/b-alternating colored 3-manifolds M with defects of surfaces S, lines L and vertices V, where L and V are labelled by objects and morphisms in a spherical fusion category C. BIMSA conceptually unifies various notions of C, such as the Drinfeld center Z©, Turaev-Viro TQFT as a-colored subtheory, Reshetikhin-Turaev TQFT as b-colored subtheory, higher Frobenius-Schur indicators, etc. BIMSA also provides fruitful computational tools, so that several remarkable results could be easily verified, such as TV©(M)=RT(Z©)(M). While considering the a-b duality as a Fourier duality, we recover various inequalities in Quantum Fourier Analysis, such as the positivity of Liu-Palcoux-Wu invariants etc. This is joint work with Shuang Ming, Yilong Wang and Jinsong Wu.