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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 VV be a symmetric monoidal category and CC be any category.

Form a new category CC' whose objects are sets of morphisms of CC

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

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

(2){(afb),(bgc)}{(agfc),} \{ (a \stackrel{f}{\to} b), (b \stackrel{g}{\to} c) \} \to \{ (a \stackrel{g\circ f}{\to} c), \}

and

(3){(afb),(bgc),(dhe)}{(agfc),(dhe)} \{ (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)CV C' \to V

is like a VV-enrichment of CC.

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

it assigns to each morphism

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

an object

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

to sets of morphisms

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

the corresponding tensor product object

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

and to morphisms

(9){(ab),(bc)}{(ac)} \{ (a \to b), (b \to c) \} \to \{ (a \to c) \}

the corresponding composition morphism

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

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

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

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

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

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

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

This is a choice of morphism

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

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

(13)(1f a,bf b,cHom(a,b)Hom(b,c)Hom(a,c))=(1f a,cHom(a,c)). \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 VV enriched category is itself already a functor

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

and since the “bare” VV-enriched category CC itself is the tensor unit in the category of all such functors

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

sending everything to Id 1Mor(V)\mathrm{Id}_1 \in \mathrm{Mor}(V) this means the above is a natural transformation:

(16)1C V 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)1 Id 1 f a,bf b,c f a,c Hom(a,b)Hom(b,c) Hom(a,c). \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 VV is closed. Think of V=VectV = \mathrm{Vect}. Think of CC as a being cobordism category.

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

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

If this functor assigns the vector space

(19)ρ a \rho_a

to the object aa of CC, then it assigns an element of

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

to

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

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

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

of an object of V=VectV = \mathrm{Vect}, functoriality of ρ\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 (ρ a,b\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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