### FFRS on Uniqueness of CFT: Sewing as Natural Transformation

#### Posted by Urs Schreiber

A comment on “*sewing*” in 2-dimensional quantum field theory, and on its description in terms of natural transformations as used in the recent FFRS paper discussed here.

What do quantum field theorists mean by *sewing*?

Let $V$ be a symmetric monoidal category and $C$ be any category.

Form a new category $C'$ whose objects are sets of morphisms of $C$

Morphisms of $C'$ are “sewings”, namely choices of composable morphisms from a given collection of morphisms. The following examples of typical morphisms say it all:

and

This category naturally has a symmetric monoidal structure, simply induced from the disjoint union of sets.

A monoidal functor

is like a $V$-enrichment of $C$.

Rather: if $C$ is codiscrete, i.e. if it has precisely one morphism between any ordered pair of objects, then such a functor is precisely a $V$-enrichment of $C$:

it assigns to each morphism

an object

to sets of morphisms

the corresponding tensor product object

and to morphisms

the corresponding composition morphism

$\mathrm{Hom}(a,b)$ is like the set of morphisms from $a$ to $b$. Only that it is not a set in general, but an object of $V$.

A particular morphism from $a$ to $b$ is hence an “element” of $\mathrm{Hom}(a,b)$. That is, a morphism from the tensor unit into the Hom-object:

in $\mathrm{Mor}(V)$.

Obviously, we may regard $C$ itself as trivially $V$-enriched, in that we assume all Hom-objects to be the tensor unit in $V$.

Consider then a functor from this “bare” $C$ into a $V$-enrichment of $C$ which is the identity on objects.

This is a choice of morphism

for each pair of objects $a,b$ such that composition is respected, i.e. such that

In words: it’s a collection of morphisms, one for each source and target object, that is closed under composition.

But since our $V$ enriched category is itself already a functor

and since the “bare” $V$-enriched category $C$ itself is the tensor unit in the category of all such functors

sending everything to $\mathrm{Id}_1 \in \mathrm{Mor}(V)$ this means the above is a natural transformation:

from the tensor unit functor into the given functor.

Seeing this amounts to nothing but writing down the naturality condition

As FFRS notice, this condition expresses precisely the structure of *sewing constraints* encountered in the study of representations of cobordism categories.

Why?

Assume furthermore that $V$ is closed. Think of $V = \mathrm{Vect}$. Think of $C$ as a being cobordism category.

A representation of $C$ in $\mathrm{Vect}$ is a functor

If this functor assigns the vector space

to the object $a$ of $C$, then it assigns an element of

to

From the point of view that the linear map assigned by $\rho$ to a morphism $a \to b$ in $C$ is itself an element

of an object of $V = \mathrm{Vect}$, **functoriality of $\rho$ is a sewing constraint** in the above sense.

At first sight, this might look like nothing more but a game with words and concepts. What’s the point?

The point is - if you like - *a kind of holography* encountered in 2-dimensional conformal field theory.

Namely, it turns out that the linear maps ($\rho_{a,b}$ in the above notation) assigned to cobordisms by a 2-dimensional conformal quantum field theory which come to us a priori as *morphisms* (“correlators”), are usefully thought of as elements of a vector space assigned by a *three*-dimensional quantum field theory to the given 2-cobordism (or rather its “complex double”, which is closed).

What were morphisms for 2-dimensional QFT now become objects for 3-dimensional QFT.

That’s why the “sewing” perspective on representations of 2-dimensional cobordisms categories is so useful: it allows to bring the 3-dimensional perspective into the game.