Categorical Trace and Sections of 2-Transport

November 17, 2006

Categorical Trace and Sections of 2-Transport

Posted by Urs Schreiber

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What is a trace?

In quantum field theory, the answer is: let $\mathrm{QFT} : n\mathrm{Cob} \to \mathrm{Vect}$ be 1-functor describing an $n$ -dimensional QFT. Then the trace of

(1)

\mathrm{QFT}(X \times [0,1])

(2)

\mathrm{QFT}(X \times S^1) \,.

What is a 2-trace?

In an extended QFT we want to refine $\mathrm{QFT}$ to an $n$ -functor #. Do we get an $n$ -trace this way?

Curiously, the right way to think of the extended $\mathrm{QFT}$ is, apparently, to think in terms of spaces of sections of $n$ -bundles #.

I am trying to understand the implications of a general abstract proposal # for what that might actually mean.

As a consistency check, I would like to understand if in the case where the QFT’s target space is a 2-group and the 2-bundle is a 2-representation: does the canonical space of states

(3)

[\mathrm{sect},[\mathrm{conf},[\mathrm{par},\mathrm{phas}]] \ni \mathrm{s} \stackrel{\sim}{\mapsto} qft \in [\mathrm{par},[\mathrm{sect},[\mathrm{conf},\mathrm{phas}]]

know about the notion of 2-character of a 2-representation as discussed by Kapranov and Ganter ?

$MathML-enabled post (click for more details).$

Here $\mathrm{conf}$ denotes the configuration space of the QFT, $\mathrm{phas}$ the space of phases, $\mathrm{par}$ the parameter space and $\mathrm{sect}$ the space of states - as described here.

Clearly, I am taking the liberty here to think more about arrow theory than about the technicalities to be dealt with in concrete implementations. If you are rather interested in the latter, you might enjoy the most recent work by Graeme Segal, the originator of the functorial description of CFT.

What I instead would like to understand here is this:

a general idea for the canonical extended QFT associated to a charged $n$ -particle

To a given $n$ -transport # over target space $P$ , coupled to an $n$ -particle represented by a parameter space $\mathrm{par}$ , we can, in a natural way, associate the space $\mathrm{sect}$ of sections of the $n$ -transport “transgressed” to the $n$ -particle’s configuration space.

(1)

H : \mathrm{sect} \stackrel{\subset}{\to} [\mathrm{conf},[\mathrm{par},\mathrm{phas}]] \,.

But Freed et al. tell us (and moreover Bruce Bartlett and Simon Willerton # tell me, who is greatly indebted to them for lots of good conversation) that we should, roughly, be looking for a map from parameter space $\mathrm{par}$ to a space of sections.

I notice that, quite abstractly, there is a canonical candidate, namely the image of $H$ under what ought to be a canonical equivalence in the world of 2-categories:

(2)

[\mathrm{sect}, [\mathrm{conf},[\mathrm{par},\mathrm{phas}]]] \ni H \stackrel{i}{\mapsto} i(H) \in [\mathrm{par}, [\mathrm{sect},[\mathrm{conf},\mathrm{phas}]]] \,.

To flesh out this abstract nonsense, we need

an application

Take target space to be a strict 2-group

(3)

P = \Sigma(G_2) \,.

Take parameter space to be the open 2-particle

(4)

\mathrm{par} =\{a \to b\}

or the closed one,

(5)

\mathrm{par} = \Sigma(\mathbb{Z}) \,,

coupled to a 2-bundle with 2-connection on target space

(6)

\rho : \Sigma(G_2) \to 2\mathrm{Vect} \,,

which is nothing but a 2-representation of $G_2$ .

For the different but similar setup of lax 2-representations of discrete 2-groups, Kapranov and Ganter have proposed a notion of 2-character. The canonical “space of states”

(7)

i(H) \in [\mathrm{par}, [\mathrm{sect},[\mathrm{conf},\mathrm{phas}]]]

ought to know about such 2-characters.

Does it?

To decide this, we need just follow that tao of abstract nonsense, work out $i(H)$ and check.

I found this to be harder that I had expected. But now I do have something like a partial result, constructing not the full $i(H)$ in the above example but just some restriction of it.

I am not sure if anything but the introduction and the remark at the end is actually readable, but in case anyone cares, some details can be found in these notes:

$\;\;\;$ sections of 2-reps

The result of these considerations (in as far as they happen to be correct) is:

In the above example (open string $a \to b$ propagating on a 2-group and coupled to a 2-rep) the “canonical space of states” $i(H)$ associates collections of 2-morphisms $\array{ & \nearrow \searrow^{\mathrm{Id}} \\ \mathbb{C} &e \Downarrow \;& \mathbb{C} \\ & \searrow \nearrow_{\rho(g)} }$ for all $g \in \mathrm{Mor}_1(\Sigma(G_2))$ to $a$ , similar collections to $b$ and associates to the morphism $a \to b$ a map from one to the other that comes from conjugation with represented 2-group elements.

Actually, one gets something more general than this. But in this special case this structure is indeed rather similar in nature to the proposal by Kapranov and Ganter.

reminder: the 2-character by Kapranov and Ganter #

Kapranov and Ganter considered 2-reps $\rho$ of 2-groups $G_2$ of the form

(8)

G_2 = \mathrm{Disc}(G) = (1 \to G) \,,

which have an ordinary group $G$ as objects and only identity morphisms. But that restriction has no bearing on the notion of 2-trace they invent in order to define 2-characters of such reps.

Let

(9)

\rho : \Sigma(G_2) \to T \,,

be the lax 2-functor representing the 2-group, where $T$ is some 2-category of 2-vector spaces. So

(10)

\rho \;\; : \;\; \array{ & \nearrow \searrow^{g} \\ \bullet &h \Downarrow\;& \bullet \\ & \searrow \nearrow_{g'} } \;\; \mapsto \;\; \array{ & \nearrow \searrow^{\rho(g)} \\ A &\rho(h) \Downarrow\;& A \\ & \searrow \nearrow_{\rho(g')} } \,.

Then, by definition, the Kapranov-Ganter 2-trace of the image under $\rho$ of some $g \in G$ is the Hom-space

(11)

\mathrm{Tr}(\rho(g)) = \mathrm{Hom}_{\mathrm{Hom}_T(A,A)}(\mathrm{Id}_{A}, \rho(g)) \,.

An element in that space $\mathrm{Tr}(\rho(g))$ is hence a 2-morphism $e$ of the form

(12)

\array{ & \nearrow \searrow^{\mathrm{Id}} \\ A &e \Downarrow\;& A \\ & \searrow \nearrow_{\rho(g)} } \,.

Moreover, for any element $h \in G$ we may map that morphism to the one obtained by conjugating with $\rho(h)$ and inserting the unit $\mathrm{Id}_A \Rightarrow \stackrel{\rho(h)}{\to}\stackrel{\rho(h^{-1})}{\to}$

(13)

\array{ \mathrm{Id}_A \\ e' \Downarrow\; \\ \stackrel{\rho(h)}{\to}\stackrel{\rho(g)}{\to}\stackrel{\rho(h^{-1})}{\to} } \;\; := \;\; \array{ \mathrm{Id}_A \\ \Downarrow \\ \stackrel{\rho(h)}{\to}\stackrel{\rho(h^{-1})}{\to} \\ e \Downarrow\; \\ \stackrel{\rho(h)}{\to}\stackrel{\rho(g)}{\to}\stackrel{\rho(h^{-1})}{\to} } \,.

This way, the categorical trace actually becomes a functor on the action groupoid of $G$ acting on itself.

Posted at November 17, 2006 8:08 PM UTC

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transgression

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I was asked to say something about how the way I turn functors on target space into functors on configuration space above is related to the ordinary notion of transgression.

I’ll illustrate this with the example that shows how an abelian gerbe on $X$ transgresses to a line bundle on $L X$ .

Let $X$ be a space and let $P_2(X)$ be 2-paths in $X$ , i.e. objects are points in $X$ , morphisms are paths between points and 2-morphisms are surfaces between paths.

Let $\mathrm{par}$ be some model for the circle. Let’s take

(1)

\mathrm{par} = \Sigma(\mathbb{Z}) \,,

the category with a single object and $\mathbb{Z}$ worth of morphism, with composition corresponding to addition of natural numbers.

Regard this as a 2-category with only identity 2-morphisms.

Now, one finds that the functor 2-category

(2)

[\mathrm{par},P_2(X)]

has

objects being closed paths in $X$ , to be thought of as loops with basepoint

morphisms being cylinders between such based loops, containing the information of the path traced out by thre basepoint

2-morphisms are deformations of the path of this basepoint.

So $[\mathrm{par},P_2(X)]$ is much like the 1-category

(3)

P_1(L X)

whose objects are (based) loops in $X$ and whose morphisms are (based) cylinders in $X$ , only that $[\mathrm{par},P_2(X)]$ also knows how to deal with transformations that move the basepoint.

Now consider on $P_2(X)$ a 2-functor that sends paths to 1-dimensional vector spaces:

(4)

\mathrm{tra} : P_2(X) \to \Sigma(1d\mathrm{Vect}_\mathbb{C})

such that this guy is smooth and locally trivializable in a suitable sense.

I claim that such 2-functors are equivalent to bundle gerbes # with connection #.

(I have shown how every such functor gives rise to a bundle gerbe with connection here. For the converse statement you will have to trust me for the moment.)

So $\mathrm{tra}$ sends each path to a 1-dimensional vector space, sends each surface to a morphism between such vector spaces, and sends composition of paths to tensor product of vector spaces.

Notice that in the case that this 2-functor is trivial in the sense that it sends every path to the canonical 1-d vector space $V_\gamma = \mathbb{C}$ we can regard it as a functor

(5)

\mathrm{tra} : P_2(X) \to \Sigma(\Sigma(U(1))) \,.

If, moreover, that transport were flat, we could think of this as a pseudofunctor on the fundamental groupoid of $X$ , or equivalently as a groupoid 2-cocycle.

But we need not. Here I want to consider the most general line bundle gerbe with connection and hence an arbitrary functor $P_2(X) \to \Sigma(1d\mathrm{Vect}_\mathbb{C})$ .

The standard construction now turns $\mathrm{tra}$ into a 2-functor $\mathrm{tra}_*$ on our “extended” loop space 2-category $[\mathrm{par},P_2(X)]$ :

(6)

\mathrm{tra}_* : [\Sigma(\mathbb{Z}),P_2(X)] \to [\Sigma(\mathbb{Z}), \Sigma(1d\mathrm{Vect})] \,.

Here $\mathrm{tra}_*$ simply takes any object $\ell : \Sigma(\mathbb{Z}) \to P_2(X)$ and sends it to the object

(7)

\Sigma(\mathbb{Z}) \stackrel{\ell}{\to} P_2(X) \stackrel{\mathrm{tra}}{\to} \Sigma(1d\mathrm{Vect})

in $[\Sigma(\mathbb{Z}),\Sigma(1d\mathrm{Vect})]$ . Similarly for cylinders and basepoint trajectory deformations.

We can easily read off what $\mathrm{tra}_*$ does in detail:

it sends every based loop to the 1-dimensional vector space associated with it (to become the fiber of a line bundle on loop space)

it sends every cylinder between loops with given basepoint trajectory to an isomorphism between the vector spaces associated to the loops (to become a parallel transport on the bundle over loop space), but twisted by the vector space associated to the base point trajectory

but furthermore, it associates a morphism between the vector spaces of the basepoint trajectory and a deformed trajectory to every such deformation.

This last bit is a certain gauge transformation that we don’t usually consider when talking about line bundles on loop space.

And in fact, it is only apparently present here. If we have a closer look, these “twists” are actually not there.

It’s probably quite obvious to everybody reading this, but since I already typed that much, I can just as well also spell this out in detail.

So let $\ell_1$ be an incoming based loop and let $\ell_2$ be the outgoing loop of a cylinder $\Sigma$ , whose basepoint runs along the path $\gamma$ . Then $\mathrm{tra}_*(\Sigma)$ is a 2-morphism in $\Sigma(1d\mathrm{Vect})$ of the form

(8)

\array{ \bullet &\stackrel{V_{\ell_1}}{\to}& \bullet \\ V_\gamma\downarrow\; &\;\Downarrow\mathrm{tra}(\Sigma)& \; \downarrow V_\gamma \\ \bullet &\stackrel{V_{\ell_2}}{\to}& \bullet } \,,

i.e. a linear map

(9)

\mathrm{tra}(\Sigma) : V_{\ell_1}\otimes V_\gamma \to V_\gamma \otimes V_{\ell_2} \,.

But all these space are 1-dimensional! Hence this is canonically isomorphic to a plain map

(10)

\mathrm{tra}(\Sigma) : V_{\ell_1} \to V_{\ell_2} \,.

Moreover, the nature of the base point deformations (coming from a modification of pseudonatural transformations) tells us precisely that this morphism is independent of which base line trajectory we choose.

In other words: $\mathrm{tra}_*$ is well defined on isomorphism classes of 1-morphisms in $[\Sigma(\mathbb{Z}),P_2(X)]$ . Hence it descends to a 1-functor

(11)

\tra_* : P_1(L X) \to 1d\mathrm{Vect} \,.

That’s the line bundle with connection on loop space obtained from the line bundle gerbe with connection on target space.

Posted by: urs on November 20, 2006 8:58 PM | Permalink | Reply to this

Re: transgression

One word of warning: The diff geom notion of
transgression (Chern) is the opposite of the usage in alg top (Serre - Borel- H. Cartan)
even though the underlying implementation
is similar.

For a fibration F –> E –> B
Chern: transgression:
part of H^*(B)–> H*-1(F)
Borel-Serre: part of H^*-1(F) –> H*(B)

A special particuarly important case:
\Omega B –> PB –> B

jim

Posted by: jim stasheff on November 21, 2006 2:56 PM | Permalink | Reply to this

Read the post Basic Question on Homs in 2-Cat
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Excerpt: A basic question on internal homs in 2-categories.
Tracked: November 21, 2006 3:53 PM

Re: Categorical Trace and Sections of 2-Transport

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In particular, the tensor unit $1 \in [P, n\mathrm{Vect}]$ is that constant functor I was talking about, which sends everything to the identity on the tensor unit in $n\mathrm{Vect}$ .

so if i want to think of this as something like a bundle it’s the trivial one with fibre $\mathbb{R}$ or $\mathbb{C}$ ??

so its sections are functions on the base $B$ but incorporated in some $n$ -cat version of $C^\infty(B)$ cf. your remarks on comm alg with its $n$ -cat interp

I finally noticed that you consider only bundles with connection that is, path liftings = parallel tranport as being a bundle with connection

so the bundle without connection is some equivalence class that approach would then generalize to fibre spaces

Posted by: jim stasheff on November 21, 2006 4:12 PM | Permalink | Reply to this

bundles and sections

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so if i want to think of this as something like a bundle it’s the trivial one with fibre $\mathbb{R}$ or $\mathbb{C}$ ??

Yes, exactly.

A (smooth) vector bundle with connection on a (smooth) space $X$ is a (smooth) functor

(1)

\mathrm{tra} : P_1(X) \to \mathrm{Vect}

for a suitable category $P_1(X)$ of paths in $X$ .

Using the monoidal structure of $\mathrm{Vect}$ , the functor category

(2)

[P_1(X),\mathrm{Vect}]

becomes monoidal, too.

The tensor unit

(3)

1 : P_1(X) \to \mathrm{Vect}

in $[P_1(X),\mathrm{Vect}]$ then is the functor, which sends every path to the identity morphism on the ground field (let’s say that’s $\mathbb{C}$ ):

(4)

1 : (x \stackrel{\gamma}{\to} y) \; \mapsto \; ( \mathbb{C} \stackrel{\mathrm{Id}}{\to} \mathbb{C} ) \,.

Moreover, notice that a morphism

(5)

e : 1 \to \mathrm{tra}

is a collection of linear maps

(6)

e_x : \mathbb{C} \to V_x = \mathrm{tra}(x)

for all $x \in X$ such that

(7)

\array{ \mathbb{C} &\stackrel{\mathrm{Id}}{\to}& \mathbb{C} \\ e_x \downarrow\; && \; \downarrow e_y \\ V_x &\stackrel{\mathrm{tra}(\gamma)}{\to}& V_y }

commutes, for all $\gamma \in \mathrm{Mor}(P_1(X))$ .

But this says exactly that $e$ is a flat (covariantly constant) section of the vector bundle with connection represented by $\mathrm{tra}$ .

We get all sections, if we suppress the 1-morphism appropriately.

In a formally nice way (which is of course overkill for the simple example at hand), I would say this as follows:

Let

(8)

\mathrm{par} = \{ \bullet \}

be the category with a single object and no nontrivial morphisms.

Denote by

(9)

\mathrm{conf} \subset [\mathrm{par},P_1(X)]

the category whose objects are functors $\mathrm{par} \to P_1(X)$ and whose morphisms are all “thin” natural transformations between these.

For the 1-categorical example discussed here this simply means that there are no nontrivial morphisms in $\mathrm{conf}$ at all and $\mathrm{conf}$ is just the discrete category on the set of functors $\mathrm{par} \to P_1(X)$ - which in turn is nothing but $X$ itself. (All this becomes more interesting as we move to higher categorical dimensions).

So then we can “transgress” # everything from $P_1(X)$ to $\mathrm{conf}$ to get functors

(10)

1_* : \mathrm{conf} \to \mathrm{Vect}

and

(11)

\mathrm{tra}_* : \mathrm{conf} \to \mathrm{Vect} \,.

Now we find that the space of morphisms

(12)

\mathrm{sect} = [1_*,\mathrm{tra}_*]

is indeed precisely the space of sections of the vector bundle described by $\mathrm{tra}$ .

Furthermore, notice that, when everything is smooth, the endomorphism

(13)

f : 1_* \to 1_*

of the functor $1_*$ are just any (smooth) collection of maps

(14)

f_x : \mathbb{C} \to \mathbb{C}

for all $x \in X$ , not having to satisfy any condition in this example. So when everything is smooth we have

(15)

C^\infty(X) = \mathrm{End}(1_*) \,.

Acting with an element of $C^\infty(X)$ of a section of the vector bundle is now nothing but the composition

(16)

\mathrm{End}(1_*)\times \mathrm{Hom}(1_*,\mathrm{tra}_*) \to \mathrm{Hom}(1_*,\mathrm{tra}_*) \,.

Of course, for ordinary vector bundles this way of looking at the situation is complete overkill. But I think that thought of it this way, everything categorifies rather nicely.

I finally noticed that you consider only bundles with connection that is, path liftings = parallel tranport as being a bundle with connection

Yes, that’s right. Sorry if this was not clear.

Yes, in my setup I say a $n$ -bundle with connection is a suitable transport $n$ -functor. That’s it.

When I began thinking about this stuff I was always inclined to try to build total spaces of $n$ -bundles, construct the transport groupoids of these and consider transport functors into these.

This is still an interesting question one can consider. But I found it to be rather fruitful to allow for the freedom of not having to specify the total space.

Of course, the total space of the bundle is encoded in the values of the transport $n$ -functor on all $(n-1)$ -morphisms. For instance for a 1-bundle, we get the collection of all fibers as the collection of all vector spaces $V_x = \mathrm{tra}(x)$ .

so the bundle without connection is some equivalence class that approach would then generalize to fibre spaces

Yes, that would be one way to get the bundle without connection.

Another would be to pull back to “configuration spaces” of the sort discussed above, which “forget” the information about the highest-dimensional morphisms.

Posted by: urs on November 21, 2006 4:52 PM | Permalink | Reply to this

Re: bundles and sections

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Above # I discussed the concept of endomorphisms of the trivial $n$ -transport for the toy example of ordinary line bundles:

So when everything is smooth we have

(1) $C^\infty(X) = \mathrm{End}(1_*)$

Meanwhile, I am beginning to extract some honest theorems in the notes on sections of 2-reps posted above.

A simple one is a nice illustration of this idea of $n$ -monoids of endomorphisms of the trivial transport, but now for the case where we are looking at a closed 2-particle propagating on a 2-group and coupled to a strict 2-rep with values in $\mathrm{Vect}$ -modules.

So let target space be the 2-groupoid

(2)

\Sigma(G_2)

for $G_2$ a strict 2-group, coming from the crossed module

(3)

t : H \to G \,.

The trivial rep

(4)

1 : \Sigma(G_2) \to \mathrm{Bim} \subset {}_{\mathrm{Vect}}\mathrm{Mod}

sends the single object to the $\mathbb{C}$ -algebra $\mathbb{C}$ and every morphism to the identity morphism on that.

What is $\mathrm{End}(1)$ in this case?

That’s just a matter of unwrapping all definitions and reading off the result. The result is:

proposition: The category $\mathrm{End}(1)$ is equivalent, as a monoidal category, to the category of representations of the cokernel $G/\mathrm{Im}(t)$ of $t$ :

(5)

\mathrm{End}(1) \simeq \mathrm{Rep}(G/\mathrm{Im}(t)) \,.

It follows in particular, that the space of sections of a 2-vector bundle on $\Sigma(G_2)$ is a $\mathrm{Rep}(G/\mathrm{Im}(t))$ -module category.

There are more fact of this sort one can extract here, though the others require more concentration.

The most obvious next question is: what is the 2-vector space structure on the space of sections on the configuration space of the closed 2-particle propagating on $\Sigma(G_2)$ ?

So let

(6)

1_* : \mathrm{conf} \to [\mathrm{par},\mathrm{Bim}]

be the trivial 2-vector bundle on configuration space.

For that we have:

proposition: The category $\mathrm{End}(1_*)$ is equivalent, as a monoidal category, to loops in the category of representations of the loop groupoid of $G_2$

(7)

\mathrm{End}(1_*) \simeq \Lambda\mathrm{Rep}(\Lambda G_2) \,.

Here, the loop groupoid of a 2-group is a slight generalization of the loop groupoid of an ordinary group, as used by Simon Willerton #.

It is almost the action groupoid of $G$ on itself, but with morphisms identified that are ismomorphic under certain 2-morphisms in $G_2$ .

Moreover, loops in reps is simply the functor category

(8)

\Lambda\mathrm{Rep}(\Lambda G_2) := [\Sigma(\mathbb{Z}),\mathrm{Rep}(\Lambda G_2)] \,.

So the space of sections on configuration space forms a $\Lambda\mathrm{Rep}(\Lambda G_2)$ -module category.

But - as I tried to motivate in my post above - what we should really be interested in is the 2-functor on parameter space that in encoded in this space of sections on configuration space.

It is getting late today, so I am not completely convinced I can trust myself, but maybe I have shown that the functor we get by the $([A,[B,C]] \to [B,[A,C]])$ -gymnastics # goes

(9)

\mathrm{par} \to {}_{\mathrm{Rep}(\Lambda G_2)}\mathrm{Mod} \,.

That might be considered encouraging. After all, my hope was that when doing all this one categorical level further up, we get 2-functors

(10)

\mathrm{par} \to \mathrm{Bim}(\mathrm{Rep}(\hat \Omega G)) \subset {}_{\mathrm{Rep}(\hat \Omega G)}\mathrm{Mod} \,,

because for those I know how they encode 2D CFT correlators.

Posted by: urs on November 22, 2006 7:51 PM | Permalink | Reply to this

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November 17, 2006