A simple model of 4d-TQFT

TL;DR

This paper introduces a simple 4-dimensional topological quantum field theory (4d-TQFT) model associated with complex roots of unity, exploring its properties and conjectured finite value behavior on closed 4-manifolds.

Contribution

It constructs a new simple 4d-TQFT model based on Delta complexes and formulates a conjecture about its finite-valued behavior on closed 4-manifolds.

Findings

Defined a 4d-TQFT model $M__$ for complex roots of unity.

Conjectured the model's key quantity takes finitely many values on closed 4-manifolds.

Provided specific values for standard 4-manifolds like $S^4$, $S^2 imes S^2$, and $ P^2$.

Abstract

We show that, associated with any complex root of unity $\omega$, there exists a particularly simple 4d-TQFT model $M_\omega$ defined on the cobordism category of Delta complexes. For an oriented closed 4-manifold $X$ of Euler characteristic $\chi(X)$, it is conjectured that the quantity $N^{3\chi(X)/2}M_\omega(X)$, where $N$ is the order of $\omega$, takes only finitely many values as a function of $\omega$. In particular, it is equal to 1 for $S^4$, $\left(3+(-1)^{N}\right)/2$ for $S^2\times S^2$, and $N^{-1/2}\sum_{k=1}^N\omega^{k^2}$ for $\mathbb{C} P^2$.