Functor category of quantum field theories?

Question

According to the prescription of functorial quantum field theory, a quantum field theory can be viewed as a monoidal functor from some monoidal category of $n$-cobordisms to some monoidal category of vector spaces (typically a category of Hilbert spaces).

Now, what does the functor category of QFTs look like? More precisely, I am having trouble even conceptualizing what "natural transformation between QFTs" would be. I am also interested in the specific case of the category of CFTs (a CFT here would be a monoidal functor from a conformal cobordism category).

Answer 1

The question of what "natural transformation of QFTs" should be is a somewhat subtle one. The issue is most apparent if you work with TQFTs, but it doesn't completely go away if you work with dynamical theories.

Suppose, for example, that you have two $n$D TQFTs $\cal{V},\cal{W}$ and an $(n-1)$D closed manifold $\Sigma$, and let $\bar{\Sigma}$ be a copy of $\Sigma$ in which the direction of time has been reversed. Then you can evaluate your TQFTs to produce vector spaces $\cal{V}(\Sigma), \cal{W}(\Sigma)$. Recall that these are finite-dimensional vector spaces because of the existence of "elbow" cobordisms $\eta_\Sigma : \emptyset \to \Sigma \sqcup \bar{\Sigma}$ and $\epsilon_\Sigma : \bar{\Sigma} \sqcup \Sigma \to \emptyset$. If you apply $\cal V$ to these, you build an isomorphism $\cal V(\bar\Sigma) \cong V(\Sigma)^*$, the linear dual.

Now consider a natural transformation of monoidal functors $f : \cal{V} \to \cal{W}$. Evaluated on $\Sigma$, you would see a linear map $f(\Sigma) : \cal{V}(\Sigma) \to \cal{W}(\Sigma)$, and evaluated on $\bar\Sigma$ would would see a linear map $f(\bar\Sigma) : \cal{V}(\bar \Sigma) \to \cal{W}(\bar \Sigma)$. Symmetric monoidality of $f$ tells you furthermore that $$ f(\Sigma \sqcup \bar\Sigma) = f(\Sigma) \otimes f(\bar\Sigma) : \cal V(\Sigma)\otimes \cal V(\bar\Sigma) \to \cal W(\Sigma)\otimes \cal W(\bar\Sigma).$$

Now, naturality of $f$ tells you that any time you have a cobordism, you should get an equation. So, for example, $f(\epsilon_\Sigma)$ is the equation $$ (*)\quad \cal W(\epsilon_\Sigma) \circ (f(\Sigma) \otimes f(\bar\Sigma)) = \cal V(\epsilon_\Sigma). $$ Finally, note that $f(\bar\Sigma) : \cal V(\Sigma)^* \cong \cal V(\bar\Sigma) \to \cal W(\bar\Sigma) \cong \cal W(\Sigma)^*$ has an "adjoint" map $f(\bar\Sigma)^* : \cal W(\Sigma) \to \cal{V}(\Sigma)$.

After unpacking, the above translations then tell you that: $$ f(\bar\Sigma)^* \circ f(\Sigma) : \cal{V}(\Sigma) \to \cal{V}(\Sigma)$$ is the identity, and similarly on the other side.

As a result, the natural transformation $f$ assigns isomorphisms to all $(n-1)$-manifolds. Thus it is a natural isomorphism $\mathcal{V} \cong \mathcal{W}$.

Note that actually we used nothing about "vector spaces". The problem would any time you have some amount of duality data in your source category. In particular, the problem does not go away if you allow your target to be an even-higher category. If the target is a higher category, then "naturality" would impose not an equality in (*) but an isomorphism, but you would still end up producing an isomorphism $f(\bar\Sigma)^* \circ f(\Sigma) \cong \mathrm{id}_{\cal V(\Sigma)}$.

Nontopologically you might not have elbow cobordisms of "zero length", but you will have some of positive length, and using them still lets you get molified versions of the equation $f(\bar\Sigma)^* \circ f(\Sigma) = \mathrm{id}_{\cal V(\Sigma)}$ where "$f(\bar\Sigma)^*$" gets replaced by a length-depended object, and $\mathrm{id}_{\cal V(\Sigma)}$ gets replaced by the evolution operator for that amount of time.

All together, you see that in order to have a really good (i.e. nontrivial) notion of "natural transformation of QFTs", you really need a version of natural transformations in which "naturality" is imposed only by a morphism, and not an isomorphism.

The bicategory theorists invented such a notion decades ago: there are "lax natural transformations" and "oplax natural transformations". In a lax natural transformation $f : \cal V \to \cal W$ of functors, if you have a 1-morphism $\alpha : X \to Y$, then you get a 2-morphism $f(\alpha) : {\cal W}(\alpha) \circ f(X) \Rightarrow f(Y) \circ {\cal V}(\alpha)$. In an oplax natural transformation, the 2-morphism goes the other direction. Note that "lax" and "oplax" are the two resolutions of a slight conflict: should $f(\alpha)$ go "in the $f$-direction" (i.e. with a $\cal V$ in the domain and a $\cal W$ in the codomain) or "in the $\alpha$ direction" (i.e. with an $X$ in the domain and a $Y$ in the codomain)?

This same conflict propagates to higher categories, and the combinatorics was pretty fun to work out. Scheimbauer and I did so in our paper (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories.