What is the dimension of LQG (Rovelli's TQFT)?

[atyy]

In http://arxiv.org/abs/1010.1939 Rovelli describes his present model as a generalized TQFT. The current quantum groups stuff is also strongly TQFT inspired (as spin foams in general are). This makes me wonder whether LQG should not in fact be "higher dimensional". The reason is that I've often read in Baez's TWF that the boundary theory of a TQFT has local degrees of freedom, even though there are none in the bulk. In which case, shouldn't LQG be a TQFT in 5D with 4D gravity emerging on the boundary?

I suppose the more general idea is that the boundary is a constraint, and so all we need is that LQG should be a TQFT with constraints, which is indeed the idea behind the current spin foam models. Nonetheless, the naive argument would still be that spin foams should come not from a 4D TQFT, but from a 5D one. Does anyone know what the "dimension" of Rovelli's generalized TQFT is, if there is such a thing?

 

[marcus]

In http://arxiv.org/abs/1010.1939 Rovelli describes his present model as a generalized TQFT. The current quantum groups stuff is also strongly TQFT inspired (as spin foams in general are). This makes me wonder whether LQG should not in fact be "higher dimensional". The reason is that I've often read in Baez's TWF that the boundary theory of a TQFT has local degrees of freedom, even though there are none in the bulk. In which case, shouldn't LQG be a TQFT in 5D with 4D gravity emerging on the boundary?

I suppose the more general idea is that the boundary is a constraint, and so all we need is that LQG should be a TQFT with constraints, which is indeed the idea behind the current spin foam models. Nonetheless, the naive argument would still be that spin foams should come not from a 4D TQFT, but from a 5D one. Does anyone know what the "dimension" of Rovelli's generalized TQFT is, if there is such a thing?

http://arxiv.org/abs/1012.4707 is the standard wide-audience review at this point. There is a section on page 14:
"Loop gravity as a generalized TQFT"
Look at second paragraph of right column on page 14.

==quote==
...Therefore loop gravity is essentially a TQFT in the sense of Atiyah, where the cobordism between 3 and 4d manifold is replaced by the cobordism between graphs and foams...
==endquote==

 

[marcus]

In http://arxiv.org/abs/1010.1939 Rovelli describes his present model as a generalized TQFT. ...

Atyy, I looked back at that October paper you pointed to and it said the same thing, on page 2, about dimensionality:

==quote==
TQFT ON MANIFOLDS WITH DEFECTS

Atiyah has provided a compelling definition of a general covariant QFT, by giving axioms for topological quantum field theory (TQFT) [22, 23]. In Atiyah scheme, a 4d TQFT is defined by the cobordisms between 3d manifolds...
The model defined by (4) belongs to a simple generalization of Atiyah’s TQFT, where: (i) boundary Hilbert spaces are not necessarily finite dimensional; (ii) 4d manifolds are replaced by two-complexes; (iii) 3d manifolds are replaced by graphs [24–26]. Graphs bound two-complexes in the same manner in which 3d manifolds bound 4d manifolds.
==endquote==

I'm puzzled by your post because I see no suggestion either place of anything that would elevate the dimensionality to 5D. Is there something I'm missing?

 

[atyy]

I'm puzzled by your post because I see no suggestion either place of anything that would elevate the dimensionality to 5D. Is there something I'm missing?

Well, I think I know what my confusion was. First, since a TQFT doesn't have local degrees of freedom, but its boundary theory does, I had thought that since one wanted a 4D theory with local degrees of freedom, one should start with a 5D TQFT and look at its 4D boundary theory. However, it seems that in the 3D Chern-Simons case, the boundary theory is a CFT, which isn't background independent. So I guess the lesson of AdS/CFT and the 3D TQFT/2D CFT correspondences are the same: a boundary CFT can produce a background independent bulk theory.