FQFT and cohomology
is determined uniquely by an -functor
on the strict -category obtained from by discarding the n-morphisms. (Of course, not every such -functor determines such a transformation; the missing condition is “naturality” at the top level.)
Analogous statements hold for general (weak) n-categories, although they are more complicated to formulate; see below.
We spell out explicitly the -functorial nature of transformation for low values of .
Saying that such a function extends to a functor :
is equivalent to saying that these components form a natural transformation. Since there are no nontrivial 2-morphisms in —in other words, the forgetful functor is faithful—such an extension to a functor is necessarily unique.
So we may regard the component function of as a 0-functor
A pseudonatural transformation between (strict, say, for ease of of notation) 2-functors between (strict, for simplicity) 2-categories is in components a 1-functor that functorially assigns pseudonaturality squares:
We may regard this as a 1-functor
from the underlying 1-category of to the arrow category of , whose objects are morphisms in , whose morphisms are squares in , and whose composition is pasting of such squares (see double category for details).
Again, saying that this 1-functor extends to a 2-functor from to the arrow 2-category of says precisely that these components form a pseudonatural transformation, and any such extension is unique when it exists since the forgetful 2-functor is locally faithful.
The pseudonaturality condition on , which is componentwise the equation
and the fact that there are only identity 3-morphisms in implies that this already uniquely extends to a 2-functor
where on the right we have the 2-category whose objects are morphisms in , whose morphisms are squares in and whose 2-morphisms are cylinders bounded by these squares.
For and two strict -functors, transformations which are in components given by -functors
are entirely specified by their underlying -functors
For weak -categories analogous statements hold, but may have a less straightforward formulation. What is always true is that the transformation is specified by its values on -morphisms (and below) and will be functorial in a weak sense on these, but these -morphisms and below will usually not form an -category themselves, since they will compose coherently only up to -morphisms.
One way to bring the general case into the above simple form is to invoke models by semi-strict ∞-categories. By Simpson's conjecture, every ∞-category has a model in which everything is strict except possibly the identities and their unitalness coherence laws. This means that if is such a semistrict model of an -category, then is an -semicategory and the transformation
is an -functor on that. (By naturalness we have that is guaranteed also to respect the weak identities in in some way, but that way is not so easy to formalize.)
More generally, for any algebraic notion of weak -category, there is a corresponding algebraic “-dimensional” structure containing only the operations on -dimensional cells and below in the given notion of weak -category. This is not in general a notion of weak -category, but it may suffice to formulate the above principle precisely. If the starting notion of -category had strict associativity and interchange, then the resulting -dimensional structure will be a notion of -semicategory.
It follows that with two such -dimensional QFTs, a transformation does look in components itself like an QFT – which is twisted by and in some sense (see below) – , but in dimension .
More specifically, if is a symmetric monoidal (∞,n)-category with tensor unit there is the trivial FQFT given by the constant -functor .
One can see in examples that the transformations
encode boundary conditions on cobordisms with boundary for the theory . Conversely, this means that one discovers on the boundary of the -dimensional QFT the -dimensional QFT . Or rather, this is the case if instead of natural transformations one uses canonical transformations: those component maps that are required to be natural only with respect to the invertible -morphisms in .
For the case of and 2-dimensional cobordisms without any extra structure, a detailed version of these statements are given in (Schommer-Pries). For and the holographic relation between Reshetikhin–Turaev model and rational 2d CFT in FFRS-formalism some remarks are in (Schreiber).
In the study of quantum field theory and string theory such kinds of relations between -dimensional QFTs and -dimensional QFTs on their boundary have been called the holographic principle . The degree to which this principle has been formalized and the degree to which this formalization has been verified varies greatly. Examples include
This is probably the oldest known holographic relation between QFTs.
In the 2-d QFT called the Poisson sigma-model with target the Poisson Lie algebroid coming from a Poisson manifold the boundary 3-point function computes the deformation quantization of the classical system described by that Poisson manifold.
(This was in fact used implicitly by Maxim Kontsevich to solve deformation quantization. The relation to the Poisson sigma-model was made explicit by Cattaneo and Felder.)
The boundary theory of the 2-dimensional A-model QFT on a target space that is a complexification of a classical phase space is the 1-dimensional QFT (= quantum mechanics) that is the geometric quantization of this phase space.
This is described at quantization via the A-model.
The AdS/CFT correspondence conjectures specifically a holographic relation between quantum gravity on a 10-dimensional asymptotically anti-de Sitter spacetime and a superconformal Yang-Mills theory on its asymptotic boundary. More generally, this specific construction is supposed to apply, with variations, in many different dimensions.
See also higher category theory and physics.
from slide 65 on.
A formally comparatively well understood case of QFT holography is the relation between 3-dimensional Chern-Simons theory and the 2-dimensional WZW-model. This is formalized by the Reshetikhin–Turaev model on the 3-dimensional side and the Fuchs-Runkel-Schweigert construction on the 2-dimensional side.
Remarks on how the relation between Reshitikhin-Turaev and FSR seem to have an interpretation in terms transformations between 3-functors are at
There is it discussed how the basic string diagram that in FSR formalism encodes a field insertion on, possibly, a defect line and encodes the disk amplittudes of the CFT is the string diagram Poincaré-dual to the cylinder in a 3-category of 3-vector spaces.
For references on the holographic principle in QFT, see there.