January 18, 2007

D-Branes from Tin Cans, III: Homs of Homs

Posted by Urs Schreiber In D-Branes from Tin Cans, I: Arrow Theory of Disks I had started talking about the exercise of finding the right arrow theory for $n$-disk holonomy - in the classical as well as in the quantized world.

For $n=1$ this is quite familiar:

Classically, we have a vector bundle $V$ with connection $\nabla$ over a space $X$. A path $x \stackrel{\gamma}{\to} y$ in $X$ is a 1-disk. Picking a section of $V$ over $x$ and over $y$, respectively, i.e. morphisms $e_1(x) : \mathbb{C} \to V_x$ and $\bar e_2(y) : V_y \to \mathbb{C}$ amounts to choosing a boundary insertion on the 1-disk. The disk holonomy of the 1-disk $\gamma$ under the transport of $\nabla$ with boundary insertions $e_1$ and $\bar e_2$ is the number

(1)$\mathbb{C} \stackrel{e_1(x)}{\to} V_x \stackrel{\mathrm{tra}_\nabla(\gamma)}{\to} V_y \stackrel{\bar e_2(y)}{\to} \mathbb{C} \,,$

where $\mathrm{tra}_\nabla$ denotes the parallel transport induced by $\nabla$.

There is a quantum version of this. After quantizing the above setup (for instance by pushing it forward to a point) we obtain a vector transport functor on the abstract worldline of the particle, which sends a worldline of length $t$ to a morphism $H \stackrel{U(t) = \exp(i t \nabla^2)}{\to} H$ in $\mathrm{Hilb}$.

Now we can compute the “quantum 1-disk holonomy”. This is known as the (1-)disk correlator. If $\psi_1$ is a vector in the Hilbert space $H$ and $\bar \psi_2$ is a covector in that space, then the quantum mechanical 1-disk correlator with $\psi_1$ and $\psi_2$ as “boundary insertions” looks like this: $\mathbb{C} \stackrel{\psi}{\to} H \stackrel{\exp(i t \nabla^2)}{\to} H \stackrel{\bar \psi_2}{\to} \mathbb{C} \,,$ You may be more familar with this entity in the equivalent notation $\cdots = \langle \psi_2 | \exp(i t \nabla^2)\psi_1\rangle \,.$

The goal of the exercise now is to reformulate this situation arrow-theoretically in such a way, that

a) it is possible to blindly categorify it

b) and that blindly categorifying it yields all the structure we expect to see for 2-disk correlators of strings (= 2-particles), for 3-disk correlators of membranes (= 3-particles), and so on. Since for higher $n$ this will involve lots of tin can diagrams and since all information about the existence and the nature of D-branes is hopefully contained in these (unless that exercise remains unseccessful), the name of the game is building D-branes from tin cans. Obviously.

When I started doing this exercise, I noticed a curious fact: the arrow-theory for the 2-disk-correlator of the 2-particle looked almost exactly like the arrow-theory for the section of the 3-particle, living over the disk.

This was encouraging, because this relation between states of a 3-dimensional QFT and correlators of a 2-dimensional QFT is precisely one of the things whose better underdstanding motivates going through this exercise here in the first place.

But at that time I did not fully appreciate what was going on. Even though, with hindsight, I could have:

pairing an $n$-section of an $n$-transport functor with an $n$-cosection involves the $\mathrm{Hom}$-$n$-functor, as described in the section 1.2 “A Rosetta Stone: arrow theory of quantum mechanics” in On 2D QFT - from Arrows to Disks. But the result of the pairing is itself an ($n-1$)-transport functor. So we can take its sections, in turn, and pair them. The result is an $(n-2)$-transport functors. And so on.

The general mechanism at work here is quite general and not restricted to the funny context that I am applying it to. I bet somewhere out there some category theorists have long ago thought about what I will briefly describe next:

We are faced with the situation where we have a bunch of $n$-functors, here called $\mathrm{tra}_*$ with a tensor product on them. The tensor unit functor I denote $1_* \,.$

A “section” of $\mathrm{tra}_*$ is a morphism $e_1 : 1_* \to \mathrm{tra}_* \,.$

A “cosection” is a morphism $\bar e_2 : \mathrm{tra}_* \to 1_* \,.$

The natural pairing between them, giving rise to the morphism $(\bar e_2, e_1) := 1_* \stackrel{e_1}{\to} \mathrm{tra}_* \stackrel{\bar e_2}{\to} 1_*$ should be thought of as the image of $\mathrm{Id} \in \mathrm{End}(\mathrm{tra}_*)$ under the Hom-$(n-1)$-functor $\mathrm{Hom} \left( \array{ \mathrm{tra}_* && \mathrm{tra}_* \\ e_1\uparrow\;\; &,& \;\;\downarrow^{\bar e_2} \\ 1_* && 1_* } \right) \; : \; \mathrm{End}(\mathrm{tra}_*) \to \mathrm{End}(1_*) \,.$

But since $\mathrm{End}(1_*)$ is itself monoidal (of course), with tensor unit being $\mathrm{Id}_{1_*}$, we find that we can, in turn, try to consider sections $D_1 : \mathrm{Id}_{1_*} \to (\bar e_2, e_1)$ of the pairing $(\bar e_2, e_1)$ itself. And cosections $\bar D_2 : (\bar e_2, e_1) \to \mathrm{Id}_{1_*} \,.$ And hence the corresponding pairing: If one thinks about it, one sees that, for 2-functors and in components, this diagram expresses the decoration of disks by diagrams of the sort In the center we have the value of $\mathrm{tra}_*$ over the disk $\Sigma$ in question. On top and bottom we see the “open string insertions” given by the sections $e_1$ and $e_2$. The semi-disks on the left and right come from the second order sections $D_1$ and $\bar D_2$. They describe how the endpoints of the string move along the boundary of the disk.

It was exactly this diagram that I considered in Arrow Theory of Disks. But there I added the semi-disks on the left and right more or less “by hand”.

My point here is that we should really be thinking of these D-branes as arising from a “higher order inner product” given by an iterated Hom-construction, in the above sense.

Of course there is then an entire hierarchy of higher order pairings. At level three it looks roughly like this: Here, now, we have first order sections $e$, second order sections $D$ and third-order sections $F$. All of them paired in the indicated way.

It becomes increasingly hard to draw this stuff. In components and for 3-functors, the above globe gives rise to a decoration of disks by diagrams like This is supposed to remind you of the third diagram on p. 10 in hep-th/0512076. More on that later.

Posted at January 18, 2007 7:43 PM UTC

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Re: D-Branes from Tin Cans, III: Homs of Homs

This is interesting… I think I’m beginning to understand it. It worries me that it gets difficult to draw as $n$ increases though.

Its tough to keep up, but some of Urs’s ideas are slowly starting to sink in, and I’m still wrestling with a few of them.

The one point which Urs (and I) seem to stress is that one should think of bundles (with connection) as transport functors (bottom up) rather than as total space things (top down). So, for instance, one thinks of a vector bundle with connection on a space X as a functor $P_1(X) \to Vect$, rather than as a bundle $E \to X$ of vector spaces, equipped with a connection.

I’m still not completely certain of where that takes one at the end of the day… perhaps it leads to complications, and I’d like to know what others think about this philosophy.

For instance, take a presheaf on a category C. One can think of it bottom-up as a functor $F : C^op \rightarrow \Set$. This is Urs’s (and my) way, and it also happens to be the standard way. You can even think of it as a bundle with connection on C. The space of flat sections of the presheaf is then the space $\Gamma(F) = Hom(1, F)$ where $1$ is the trivial presheaf, which sends everything to $Z$.

Alternatively, you can think of it top-down as a discrete fibration $\pi : E \rightarrow C$. A section is then a functor $F : C \to E$ such that $\pi F = \id_C$. Passing between these two pictures is a standard exercise in category theory.

Here’s the thing. When you go to higher dimensions, things get a bit more tricky. Given a 2-category $C$, a presheaf of categories on $C$ is a weak 2-functor $F : C \to Cat$. The space of sections is theen the space of pseduonatural transformations and modifications $[1, F]$, where $1$ is the trivial presheaf of categories.

Note the words ‘pseudo’ appearing above. It shows that things get a bit tricky, perhaps, in the bottom-up approach.

The top-down approach now says that a presheaf of categories on $C$ is a 2-functor $\pi : E \rightarrow C$. A flat section is a weak 2-functor $s : C \to E$ such that $\pi s = \id_C$. Passing between the two pictures is called the Grothendieck construction.

Anyhow, the latter picture is perhaps slightly more simple… or perhaps that’s an illusion, since a section is still pseudo stuff.

Posted by: Bruce Bartlett on January 20, 2007 12:47 AM | Permalink | Reply to this

the 3-dimensional visualization blockade

It worries me that it gets difficult to draw as $n$-increases though.

I think: better a picture difficult to draw than no picture at all.

This is really a general point, which we talked about a while ago ##:

one nice thing about categorical language is that it allows us to conceive involved phenomena in terms of geometry. This is fun, because, as humans, we have a more direct access to geometrical figures than to the equivalent page-long linear strings of symbols.

But this advantage only holds for really low dimensions. From a certain point on, we have to fall back to lower-dimensional shadows of higher dimensional structures. Or even throw all intuition in the wind and fully depend on linear notation for simplicial sets.

On the other hand, so far we are only talking about strings and membranes. That’s still well within our visualization skills. So in as far as the above was difficult to draw, it was so mainly due to my puny typesetting skills.

Posted by: urs on January 20, 2007 4:20 PM | Permalink | Reply to this

Re: the 3-dimensional visualization blockade

From a certain point on, we have to fall back to lower-dimensional shadows of higher dimensional structures. Or even throw all intuition in the wind and fully depend on linear notation for simplicial sets.

In principle, there should be an option in between, allowing us to use intermediate-dimensional pictures (more than linear, but low enough to visualise) that are still perfectly precise and complete (like notation for simplicial sets but unlike shadows). In effect, we should be able to use rigorous pictorial notation for (something around the level of) simplicial bicategories!

Posted by: Toby Bartels on January 21, 2007 4:37 AM | Permalink | Reply to this

From Characteristic Functions to Classifying Maps

Bruce wrote:

The one point which Urs (and I) seem to stress is that one should think of bundles (with connection) as transport functors (bottom up) rather than as total space things (top down). So, for instance, one thinks of a vector bundle with connection on a space X as a functor $P_1(X) \to Vect$, rather than as a bundle $E \to X$ of vector spaces, equipped with a connection.

I’m still not completely certain of where that takes one at the end of the day… perhaps it leads to complications, and I’d like to know what others think about this philosophy.

I doubt it’s always best to think of bundles (with connection) over $X$ as maps from $X$ rather than maps to $X$. But, it’s a very important idea. And, it’s an incredibly flexible idea!

The first time we meet this idea is when learn about the ‘characteristic function’ of a subset. Suppose $X$ is a set. Then we can think of a subset $S \subseteq X$ as a map to $X$, namely the inclusion

$i: S \hookrightarrow X$

or a map from $X$, namely the characteristic function

$\chi: X \to \{0,1\}$

such that $\chi(x) = 1$ for $x \in S$ and $\chi(x) = 0$ for $x \notin S$.

We all know that subsets and characteristic functions convey the same information. But, we don’t all stop to think about what happens when we categorify this idea!

We should think of an inclusion $i: S \hookrightarrow X$ as a ‘bundle of truth values’ over $X$, whose fiber over each point $x \in X$ is the truth value saying whether $x$ is in $S$ or not.

Then we have:

Bundles of truth values $i: S \hookrightarrow X$ over a set $X$ are classified by maps from $X$ to the set of truth values.

If you know that truth values are $-1$-categories, you’ll immediately want to categorify the above result and see what we get for bundles of $0$-categories — that is, sets.

We get this:

Bundles of sets $p: S \to X$ over a set $X$ are classified by maps from $X$ to the category of sets.

Note here we need to think of the set $X$ as a special sort of category — one with only identity morphisms — to make sense of a ‘map from $X$ to the category of sets’. If we do this, the above result is true.

Now we can categorify again:

Bundles of categories $p: C \to X$ over a set $X$ are classified by maps from $X$ to the 2-category of categories.

This is also true — but we get a more interesting truth if we replace $X$ by a category:

Bundles of categories $p: C \to X$ over a category $X$ are classified by maps from $X$ to the 2-category of categories.

This is again true, and it’s what you called the ‘Grothendieck construction’. People usually call a bundle of categories an ‘opfibration’. To get something that makes sense, we need to think of $X$ as a 2-category with just identity morphisms. Also, of course, we should use weak 2-functors (aka ‘pseudofunctors’) as our maps between 2-categories. So, people usually say it this way:

Opfibrations $p: C \to X$ over a category $X$ are classified by pseudofunctors $\chi: X \to Cat$.

People also usually replace $X$ by its opposite category and get this:

Fibrations $p: C \to X$ over a category $X$ are classified by pseudofunctors $\chi: X \to Cat^{op}$.

It’s cool how the humble idea of ‘characteristic function’ turns into this noble-sounding result when we categorify it a couple of times!

But we can go much further. For example, Claudio Hermida has shown something like this:

Fibrations $p: C \to X$ over a 2-category $X$ are classified by pseudo-2-functors $\chi: X \to 2Cat^{op}$.

But even this is just the tip of the iceberg!

In fact, we can do many other things with this ‘classifying’ idea. I explained a big bunch of them in a lecture I gave as a warmup to an introduction to higher gauge theory. I also wrote a whole issue of This Week’s Finds about various variations on this theme: week223.

If you read this stuff, you’ll see that the usual classification of ‘covering spaces’ fits perfectly into the philosophy we’ve been discussing. So does the ‘classifying space’ idea in topology! So does the ‘Postnikov tower’ idea in algebraic topology! So does the ‘Schreier theory’ idea for classifying short exact sequences of groups! So does the ‘smooth anafunctor’ idea for classifying principal bundles with connection! So does the ‘smooth 2-anafunctor’ idea for classifying principal 2-bundles with 2-connection! Also, Danny Stevenson has done a great job studying 2-connections in terms of categorified Schreier theory for Lie 2-algebras!

So, it’s a very fruitful line of thought.

There’s also an important meta-point here: it’s surprisingly useful to take simple ideas and categorify them.

Posted by: John Baez on January 20, 2007 2:15 AM | Permalink | Reply to this

Re: From Characteristic Functions to Classifying Maps

John wrote, about the idea of thinking of $n$-bundles with connection in terms of maps from base space:

But, it’s a very important idea. And, it’s an incredibly flexible idea!

To all the good reasons John mentioned, I could maybe add the following:

As I briefly mentioned in my talk in Toronto # one nice thing about this description is that the formalism answers a lot of questions for you.

The theory of higher bundles with connection involves lots of “twists” that you would not easily guess, but which have to be included.

For instance: when people first thought about morphisms of bundle gerbes, they defined them to be simply morphisms of the corresponding transition bundles. This turned out to be not good enough, in that it did not reproduce the desired classification of bundle gerbes by third integral cohomology.

The right notion of morphism of bundle gerbes is now known as a “stable morphism”. It involves a certain “twist” by a certain bundle.

But if one translates all this into the language of parallel transport 2-functors, there is only one possible notion of morphism - namely a modification of 2-functors - and this automatically produces the right “twists” for you.

Or take the problem of guessing what a connection (“and curving”) on a nonabelian principal bundle gerbe should be. It’s very non-obvious in ordinary terms. Paolo Aschieri and Brano Jurčo correctly guessed the funny twist that one has to introduce in order to understand that. But here, again, everything is automatic when we realize that these connection 1-forms come from pseudonatural transformations of transport 2-functors (as discussed here).

Another good check for a good formalism is John’s advice – also greatly promoted by Toby Bartels – to always check your definitions against the degenerate cases. A good definition is one that applies without further qualifications also to all degenerate cases.

So given a theory of $n$-bundles, you want to check if it smoothly applies to the case $n=0$. What is a 0-bundle?

In the ordinary formulation, with maps going from total space to base space, that becomes difficult, because a 1-bundle is, in this formalism already a 0-functor: namely a projection map $p : P \to X$.

In order to fit 0-bundles into this picture (“bundles of numbers”), one would have to invoke further reasoning.

Not so if we go the other way around:

in the language of parallel transport, a 1-bundle is a 1-functor, from a 1-category of points and paths in base space to a 1-category of fibers and fiber morphisms.

A 0-bundle should hence be nothing but a 0-functor from points in base space to 0-fibers. But that’s nothing but a function on base space.

And that’s precisely what we want.

Posted by: urs on January 20, 2007 4:46 PM | Permalink | Reply to this

Re: From Characteristic Functions to Classifying Maps

Also, Danny Stevenson has done a great job studying 2-connections in terms of categorified Schreier theory for Lie 2-algebras!

To amplify this point (again) I add that what Danny does is essentially nothing but the differential version of the picture of a $n$-bundle with connection as an $n$-functor from $n$-paths to $n$-fibers.

Or, the other way round, this latter way of looking at things is nothing but the integrated version of what Danny considers.

Danny considers splittings of the $n$-Atiyah algebroid sequence

(1)$\mathrm{ad}P \to T P/G \to T X \,,$

whereas the idea of realizing $n$-bundles with connection as parallel transport functors cinsiders splittings of the integrated version of this, the $n$-Atiyah groupoid sequence

(2)$\mathrm{Ad}P \to P \times_G P \to X x X \,.$

I have talked about that in $n$-Transport and Higher Schreier Theory.

Posted by: urs on January 20, 2007 5:03 PM | Permalink | Reply to this

Re: From Characteristic Functions to Classifying Maps

Thanks for these interesting comments. It is really cool how all these things are categorifications of the `subset’ idea! Its also a nice application of -1 categories. Urs’s pictures are actually remarkably good.

I’ve read the anafunctor treatment of this in the slides of talks on higher gauge theory, and it looks very elegant. Here in Toronto Nick Gurski has tried to teach me about profunctors. Could someone explain how profunctors and anafunctors are related, because their underlying ideas looks slightly similar…? I mean, vaguely speaking, they’re similar in that an anafunctor is a kind of span, and a span is a kind of many-to-many function, and profunctors seem to encapsulate a many-to-many idea too.

Posted by: Bruce Bartlett on January 21, 2007 2:38 AM | Permalink | Reply to this

Re: From Characteristic Functions to Classifying Maps

Could someone explain how profunctors and anafunctors are related […]?

I don’t know profunctors very well, but I do know this: Up to equivalence, an anafunctor is the same thing as a representable profunctor. Actually, people will often say that (up to equivalence) a representable profunctor is the same as a simple functor, but they have to use the Axiom of Choice to say this; the equivalence of anafunctors and representable profunctors, in contrast, requires no Choice (and can therefore be internalised into a wider variety of categories, even if you belive Choice in the category of sets).

Actually, representable profunctors are most directly related to saturated anafunctors, which in the context of Lie groupoids (that is internal to the category of smooth manifolds and restricting to the groupoid case) are known as Hilsum–Skandalis maps. But up to equivalence, every anafunctor is saturated (in a way that respects composition and natural transformations).

There may be an unsaturated version of profunctors, too, but this is probably not important.

Posted by: Toby Bartels on January 21, 2007 4:31 AM | Permalink | Reply to this

Anafunctors and Profunctors

If this relation between anafunctors and profunctors has really not been written down in full detail, somebody should do it! We could make it an $n$-category Café project and work it out together here (similar to the way we do Klein 2-geometry here).

It gets all very interesting as we move up the ladder: A bundle-gerbe “is” a 2-anafunctor (“is” means: certainly is with my personal definition of 2-anafunctor, which probably is essentially equivalent to the notion given by Makkai) and there are indications that we want to think of morphisms of bundle gerbes as profunctors between the corresponding groupoids. I need to fully understand this at some point.

Posted by: urs on January 21, 2007 1:02 PM | Permalink | Reply to this

Re: Anafunctors and Profunctors

This sounds like a great idea : an n-category cafe project on the relationship between anafunctors and profunctors.

Is this kind of like doing a Wiki-research-ipedia?

I suggest Toby make the first post, giving us some basic references.

Posted by: Bruce Bartlett on January 21, 2007 8:11 PM | Permalink | Reply to this

Re: Anafunctors and Profunctors

I’m sure that the relationship has been written down, if not published. Certainly Makkai said as much in a reply to John on the categories mailing list some years ago; that’s how I noticed the fact.

But it’s pretty easy to see from the definitions. The reference for anafunctors is of course Makkai’s paper. I don’t know a reference for profunctors, but they are in Wikipedia. Adopting that notation, a profunctor is representable if, given any object of C, the resulting (covariant) functor from D to Set is representable.

Posted by: Toby Bartels on January 22, 2007 2:12 AM | Permalink | Reply to this

Re: Anafunctors and Profunctors

But it’s pretty easy to see from the definitions.

Okay, I’ll try. If I interpret you correctly, this should mean that the anafunctor $\array{ |F| &\stackrel{F_1}{\to}& D \\ p \downarrow \;\; \\ C }$ that we build from a representable profunctor $F : C \to \mathrm{Set}^{D^\mathrm{op}}$ contains in the pre-image $p^{(-1)}(a \stackrel{\gamma}{\to} b)$ all those morphisms in $D$ that represent $F(a \stackrel{\gamma}{\to} b) \,.$ And $F_1$ then simply injects these into $D$.

Is that the idea?

Hm, not sure that I see yet why that would satisfy the required conditions on $p$.(?)

Posted by: urs on January 22, 2007 2:43 PM | Permalink | Reply to this
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