Zhengwei Liu: Alterfold theory for TQFT and topological orders in all dimensions #ICBS2024

We first discuss the 2+1 alterfold theory from planar algebras to 2+1 alterfold TQFT. Both Tureav-Viro TQFT and Reshetikhin-Tiraev TQFT can be naturally embedded in the alterfold TQFT through the blow-up procedure. Then we introduce the notion of a Dn algebra equipped with an piecewise-linear isotopic invariant partition function, satisfying reflection positivity and finiteness. Moreover, we construct an n+1 alterfold TQFT from it for all dimensions n. The construction can be iterated for n up to infinity and we obtain a non-trivial TQFT in all dimensions. The alterfold theory also provides a mathematical foundation to study topological orders in all dimensions. The alterfold theory is a fruitful picture language so that what you see is what you get! Various concepts in different topics can be represented naturally in this unified framework. Many remarkable results can be extensively generalized and proved in a conceptual and intuitive way.