# Bosonic topological orders and unitary fully dualizable fully extended TQFT

I would like to ask if the following statement can be true: bosonic topological orders in $n$-dimensional space-time 1-to-1 correspond to unitary fully dualizable fully extended TQFT in $n$-dimensions.

Bosonic topological order has been defined in physics using generalized local unitary equivalence in arXiv:1406.5090 Roughly, it corresponds to unitary TQFT, with boundaries, and the boundaries can have boundaries, etc.

• – Arun Debray Jun 25 '18 at 20:32
• As far as I know, this is an open question. One obstacle is that given a quantum system with a Hamiltonian, it's not clear how to define the partition function on manifolds which aren't mapping tori. It's also possible to write down nonisomorphic TQFTs such that the dimensions of the state spaces as well as the partition functions of all mapping tori agree, and this seems to be the data that we know how to extract from a lattice Hamiltonian theory. (I describe an example in this comment). (continued...) – Arun Debray Jun 25 '18 at 20:39
• Unfortunately, extended TQFT doesn't help, as far as I know: it's possible to fully extend both TQFTs, where the target $n$-category is $n$-vector spaces, but for any closed manifold of dimension less than $n$, the data assigned to that manifold by these two theories is the same. (The cobordism hypothesis says that we get extra structure on $Z(\mathrm{pt})$, etc., and this extra structure distinguishes the theories, but above dimension 3 I think it's not explicitly known what that structure is, so this would be hard to use.) – Arun Debray Jun 25 '18 at 20:46
• Thanks for the link. I added an answer to that question. – Xiao-Gang Wen Jun 25 '18 at 20:51
• The issue of mapping torus seems related to this question mathoverflow.net/questions/165830 – Xiao-Gang Wen Jun 25 '18 at 21:02