Functional Integral Construction of Topological Quantum Field Theory

• URL: http://arxiv.org/abs/2409.17103v1
• Date: Wed, 25 Sep 2024 17:15:35 GMT
• Title: Functional Integral Construction of Topological Quantum Field Theory
• Authors: Zhengwei Liu,
• Abstract summary: We introduce the unitary $n+1$ alterfold TQFT and construct it from a linear functional on an $n$-dimensional lattice model. A unitary spherical $n$-category is mathematically defined and emerges as the local quantum symmetry of the lattice model. 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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