## January 4, 2007

### 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 $C$

(1)$\{ (a \stackrel{f}{\to} b), (c \stackrel{g}{\to} d), \cdots \} \,.$

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:

(2)$\{ (a \stackrel{f}{\to} b), (b \stackrel{g}{\to} c) \} \to \{ (a \stackrel{g\circ f}{\to} c), \}$

and

(3)$\{ (a \stackrel{f}{\to} b), (b \stackrel{g}{\to} c), (d \stackrel{h}{\to} e) \} \to \{ (a \stackrel{g\circ f}{\to} c), (d \stackrel{h}{\to} e) \}$

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

A monoidal functor

(4)$C' \to V$

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

(5)$\{ (a \to b) \}$

an object

(6)$\mathrm{Hom}(a,b) \in \mathrm{Obj}(C) \,,$

to sets of morphisms

(7)$\{ (a \to b), (c \to d) \}$

the corresponding tensor product object

(8)$\mathrm{Hom}(a,b)\otimes \mathrm{Hom}(c,d)$

and to morphisms

(9)$\{ (a \to b), (b \to c) \} \to \{ (a \to c) \}$

the corresponding composition morphism

(10)$\mathrm{Hom}(a,b)\otimes \mathrm{Hom}(b,c) \stackrel{\circ}{\to} \mathrm{Hom}(a,c) \,.$

$\mathrm{Hom}(a,b)$ 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:

(11)$\mathbf{f} : 1 \to \mathrm{Hom}(a,b)$

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

(12)$\mathbf{f_{a,b}} : 1 \to \mathrm{Hom}(a,b)$

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

(13)$\left( 1 \stackrel{f_{a,b} \otimes f_{b,c}}{\to} \mathrm{Hom}(a,b) \otimes \mathrm{Hom}(b,c) \stackrel{\circ}{\to} \mathrm{Hom}(a,c) \right) = \left( 1 \stackrel{f_{a,c}}{\to} \mathrm{Hom}(a,c) \right) \,.$

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

(14)$C_V : C' \to V$

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

(15)$1 : C' \to V$

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

(16)$1 \to C_V$

from the tensor unit functor into the given functor.

Seeing this amounts to nothing but writing down the naturality condition

(17)$\array{ 1 &\stackrel{\mathrm{Id}}{\to}& 1 \\ f_{a,b}\otimes f_{b,c} \downarrow\;\;\;\;\;\;\;\;\; && \;\; \downarrow f_{a,c} \\ \mathrm{Hom}(a,b)\otimes \mathrm{Hom}(b,c) &\stackrel{\circ}{\to}& \mathrm{Hom}(a,c) } \,.$

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

(18)$\rho : C \to \mathrm{Vect} \,.$

If this functor assigns the vector space

(19)$\rho_a$

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

(20)$\mathrm{Hom}(\rho_a,\rho_b) \simeq (\rho_a)^* \otimes \rho_b := \mathrm{Hom}(a,b)$

to

(21)$a\to b \,.$

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

(22)$\rho_{a,b} : \mathbb{C} \to \mathrm{Hom}(a,b)$

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.

Posted at January 4, 2007 7:43 PM UTC

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