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My questions here are focused on $D$-dimensional topological quantum field theories (TQFTs) which are unitary and which have finite dimensional Hilbert space on a closed spatial manifold $M^{d-1}$. Say the partition function $$Z(M^{d-1} \times S^1)=\dim \mathcal{H}$$ is finite. The spacetime $M^{d-1} \times S^1$ can be viewed as a closed spatial manifold $M^{d-1}$ and a compact time.

We know that there are a large classes of 3-dimensional (3d) TQFT defined on a spacetime 3-manifold. Many of them are some version of Chern-Simons TQFT. Some of them can defined on any closed manifold, other TQFTs require to have the spin-manifold to be defined. For example, even for the Chern-Simons thoery, we have the $3d$ spin Chern-Simons theory requiring the spin-manifold.

My question: What are some examples of 4d TQFT with or without requiring spin structure, say the $4d$ spin-manifold? There are a large classes of TQFTs described by twisted discrete gauge theories, say the Dijkgraaf-Witten theory. What are other 4d TQFT with or without requiring spin structure / spin-manifold? Do we have the continuous Lagrangian formulation or the lattice formulations of the theories?

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    $\begingroup$ The spin-statistics theorem asserts that, if you insist that your (T)QFTs be unitary, then you need spin structures iff you have fermions. So super Dijkgraaf–Witten theories, for example, require spin structures. See for example arxiv.org/abs/1505.05856. $\endgroup$
    – Theo Johnson-Freyd
    Commented Dec 19, 2016 at 5:23
  • $\begingroup$ Heegard Floer homology is an example. $\endgroup$
    – Soutrik
    Commented Jan 31, 2017 at 14:41
  • $\begingroup$ Heegard Floer homology is an example for what? $\endgroup$
    – miss-tery
    Commented Feb 1, 2017 at 3:57

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One of the most famous 4d TQFTs is the Crane-Yetter TQFT, or its Hamiltonian lattice formulation, the Walker-Wang model. See my question How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?.

Crane-Yetter is defined for a ribbon fusion (premodular) category $\mathcal{C}$. As far as I know, it is unitary if $\mathcal{C}$ is. The Crane-Yetter theory doesn't require spin structures, although it can be defined as a spin TQFT if your ribbon fusion category has a $\mathbb{Z}_2$-grading. Such a grading can arise when you have a nontrivial twist on the transparent objects (basically a finite supergroup symmetry).

Dijkgraaf-Witten are special cases of Crane-Yetter, as I've shown in my thesis (and super Dijkgraaf-Witten probably as well).

Recently, Crane-Yetter has been generalised from ribbon fusion categories to $G$-crossed braided fusion categories, see this article by Shawn Cui.

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