Functional Integral Construction of Topological Quantum Field Theory

TL;DR

This paper develops a lattice-based approach to topological quantum field theories across all dimensions, introducing a new construction called alterfold TQFT that unifies various existing models and enables explicit calculations of invariants.

Contribution

It introduces the unitary n+1 alterfold TQFT from n-dimensional lattice models, unifying multiple constructions and providing explicit models for 4-manifold invariants.

Findings

Constructed a non-invertible 3+1 alterfold TQFT from a linear functional.

Derived the local quantum symmetry as a unitary spherical 3-category with explicit 20j-symbols.

Enabled explicit computation of scalar invariants of 2-knots in 4-manifolds.

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.