A **state sum model** is a smooth invariant defined on smooth triangulated, or PL manifolds, by summing a local partition function over labels attached to the elements of the triangulation.


Typical examples in 4d are the [Crane-Yetter invariant](https://ncatlab.org/nlab/show/Crane-Yetter+model) on ribbon fusion categories and [Cui's invariant](https://www.ems-ph.org/journals/show_abstract.php?issn=1663-487X&vol=10&iss=4&rank=1) based on $G$-crossed braided spherical fusion categories, but there are also others like the Kashaev invariant.


A **chain mail invariant** is defined on [handle decompositions](https://en.wikipedia.org/wiki/Handle_decomposition), and assigns data to the handles and uses a [Kirby diagram](https://en.wikipedia.org/wiki/Kirby_calculus) or similar to calculate the invariant, typically by interpreting the diagram in a graphical calculus of a category.


State sum models are often chain mail invariants (see Roberts "Refined state-sum invariants of 3- and 4-manifolds", <https://arxiv.org/abs/1810.05833>, <https://doi.org/10.1007/s00220-017-3012-9>), and chain mail invariants are always state sum models since a triangulation gives rise to a handle decomposition.


An **extended TQFT** is a (higher) functor from the fully extended bordism category to some category of higher vector spaces. State sum models are expected to give extended TQFTs, and in low dimensions, this relationship can be made precise (<https://repositories.lib.utexas.edu/bitstream/handle/2152/ETD-UT-2011-05-3139/DAVIDOVICH-DISSERTATION.pdf>).


In 4d, there are TQFTs defined via path integrals such as the Seiberg-Witten and Donaldson invariants, which can detect exotic smooth structures. Since these theories are defined fully locally, one would expect them to be extended TQFTs, or even state sums, or even chain mails. But I'm not aware of an explicit, rigorous construction. (In fact, 4d state sum models are often homotopy invariants.)


*Are there rigorously defined ETQFTs, state sums or chain mail theories that detect smooth structures?*



Are there 4d state sum models, extended TQFTs or chain mail invariant that detect smooth structures?
[This MO answer](https://mathoverflow.net/a/362517/184) by Arun Debray gives an example in the unoriented case where two specific homeomorphic manifolds can be distinguished by a specific TFT of this kind.


In general all these constructions produce "semisimple" TFTs and it has been shown by David Reutter that in the oriented case such TFTs cannot detect smooth structures. See his paper [Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure](https://arxiv.org/abs/2001.02288).


It is possible that some (extended) TFT with a more exotic target category could do the trick. (In fact the identity functor gives a trivial example that does detect all smooth structures, but obviously we want something less tautological).


