Functional Integral Construction of Topological Quantum Field Theory

Abstract: We introduce regular stratified piecewise linear manifolds to describe lattices and investigate the lattice model approach to topological quantum field theory in all dimensions. We introduce the unitary $n+1$ alterfold TQFT and construct it from a linear functional on an $n$-dimensional lattice model on an $n$-sphere satisfying three conditions: reflection positivity, homeomorphic invariance and complete finiteness. A unitary spherical $n$-category is mathematically defined and emerges as the local quantum symmetry of the lattice model. The alterfold construction unifies various constructions of $n+1$ TQFT from $n$-dimensional lattice models and $n$-categories. In particular, we construct a non-invertible unitary 3+1 alterfold TQFT from a linear functional and derive its local quantum symmetry as a unitary spherical 3-category of Ising type with explicit 20j-symbols, so that the scalar invariant of 2-knots in piecewise linear 4-manifolds could be computed explicitly.