The n-Category Café

you once promised to talk and open a discussion about his article “Branes and Quantization”.

True.

Did you forget that?

No, I didn’t, but I didn’t quite find the time. What an awful feeling.

You are right to pester me about this, really. I promise to write an ($n$Lab)-entry on the article and discuss is here by, let me see, end of July, would that still be okay?

I mean, as long as Edward Witten doesn’t create an $n$Lab entry on his paper himself, before I do.

Dear Zoran,

very good points, indeed. Yes, I agree.

Of course the aim is in this direction. Eventually the goal is to enhance the finitized setup that one experiments with to get some basic concepts right to a more refined setup. Finite groupoids should be generalized to smooth $\infty$-stacks, eventually, categories of vector spaces by suitable $(\infty,2)$-categories for instance, and in that context then one wants to see analysis, operator algebra, etc appear.

I think right this moment the genral idea is to explore which kind of abstract nonsense works most smoothly in the finitized context, so as to be sure about which structure then to internalize in a more sophisticated setup.

The “path integral”-kind of quantization description at our $\sigma$-models is set up for $\infty$-stacks. I have had a bit of discussion with Johan Alm about how the Kan extension prescription relates to the pull-push prescription used there.

I had an idea for how they are related. Johan said he has proved that this idea works. But I didn’t undertstand his proof! :-) Now I think we are still trying to sort this out behind the scenes.

Maybe it would be best if we just presented the problem here to collect the esteemed expert advice from our readers. But Johan will have to decide this when he comes back online.

I’m no expert on Kan extensions, but I think you’ve got the right intuition: left and right Kan extensions are close relatives of coproducts and products, and these tend to agree in contexts with a ‘linear algebra’ flavor. Since Johan Alm is forming Kan extensions that take values in Vect, we shouldn’t be shocked that the left and right Kan extensions agree.

Indeed I often say (to myself) that the special thing about quantum mechanics is that it’s a context where left and right adjoints agree. In category theory we have to very carefully distinguish between the left and right adjoints:

$$hom(L x,y) \cong hom(x,R y)$$

because the homset $hom(-,-)$ is not symmetric in its two arguments. But in quantum mechanics we talk about adjoint operators without worrying about the difference between ‘left’ and ‘right’ adjoints:

$$\langle T^* x, y \rangle = \langle x, T y \rangle$$

but also

$$\langle T x, y \rangle = \langle T^* x, y \rangle$$

The reason is that here the inner product $\langle -, - \rangle$ is symmetric in its two arguments. Well, almost — that’s the mystery of complex conjugation!

Anyway, the ‘almost-symmetry’ of the inner product in a Hilbert space is, I believe, a close relative of the almost-symmetry of the homset in a 2-Hilbert space, like $Hilb$, or even a 2-vector space, like $Vect$.

There is, in short, something about quantum mechanics or linear algebra that doesn’t care much about who is the ‘input’ and who is the ‘output’. Matrices have transposes!

And this is why I’m not surprised that the left and right Kan extensions agree in Johan’s paper.

Is it like or related to the product and coproduct agreeing, for finite sets of vector spaces?

Yes, since this is a special case of a Kan extension.

Limits and colimits are precisely Kan extensions down to a point.

Product and coproduct are precisely limits and colimits over categories without nontrivial morphisms.

Indeed, notice that in Johan’s description, the Kan extension assigns to each object nothing but the direct sum of the vector spaces “sitting above it”.

Well, or almost. If the configuration space groupoid has no nontrivial morphisms, then it’s just this direct sum. In general it is a the image of a projector inside this direct sum.

This gives me a chance to highlight the following very pleasing aspect of Johan’s result:

take his first example, quantization of particles that trace out paths of a certain parameter time.

There may be paths that “take no time to traverse” (their image under the projection along the functor along which we Kan extent is an identity morphism).

The formalism spits out the prescription: project onto the kernel of the operations induced by these 0-time paths. And do this by group averaging over them.

This means: the formalism regards paths of time length 0 automatically as internal gauge transformations and automatically projects onto the “physical states”: those that are invariant under translation along any 0-time path, i.e. those that are gauge invariant.

And it does so by group averaging over the gauge group. This is a famous technique used in gauge theory. Here it all drops out all by itself by just turning the Kan extension crank.

what does it “mean” in general when the left and right Kan extensions agree?

When we restrict Kan extensions to the case of products and coproducts, then this means that we have a category with [[biproducts]], which in turn means that we are at least pretty close to an additive or abelian category (namely that we are enriched over abelian monoids, at least).

It appears that Alm is doing a lot of Laplace Transform like stuff.

Could you expand on that? I am not sure I see which part of Alm’s considerations you are thinking of.

I wrote:

The simple idea/observation I have typed into this pdf

Relation Kan extension/pull-push ?

It’s really, brief, two pages nominally, but the idea itself is barely a page.

The observation there is that with this slight modification, the pull-push quantization happens to coincide with the Kan extension over points.

The big question is: what’s the relation over paths?

I am not sure yet about this. There are some obvious guesses here,

One of these guesses is clearly: the Kan extension $L_g f$

$$\array{ A &&\stackrel{f}{\to}& Vect \\ \downarrow^g && \nearrow_{L_g f} \\ B }$$

of a functor $f$ is computed as the colimit over its transgression to the category $\Gamma(g)$ of weak sections

$$\Gamma(g) = \left\{ \array{ && A \\ & {}^\phi \nearrow & \downarrow^g \\ B &\stackrel{Id}{\to}& B } \right\}$$

of $g$

(all diagrams here with implicit 2-cells filled in)

as

$$Lan_g f \simeq colim ( \Gamma(g) \to [B,A] \stackrel{[B,f]}{\to} [B, Vect])$$

not only over objects, but also over moprhisms.

Now Johan Alm suggests a proof for this, for the situation we are looking at here.

You can find it at the end of the new version here.

I am feeling a bit weird now: if this is right (looks right to me, but elementary as the steps are, it’s a bit intricate) then one would wonder if this is not known in basic Cat theory.

What’s going on?

Is there any fundamental new insight on this quantization scheme or is this just some fancy mathematics with only the hope for something new?

Is there any fundamental new insight on this quantization scheme or is this just some fancy mathematics with only the hope for something new?

Hope or not, the point is to notice that there is an open problem here that is waiting to be solved, this way or, if not this way, then some way:

given some differential $n$-cocycle on some space, what is the extended $n$-dimensional QFT $n$-functor that quantizes it?

- for instance: what is the 3-functor corresponding to Chern-Simons theory? And more importantly: how does it arise from a systematic process from the Chern-Simons actions functional (instead of by guessing and plausibility checking as done these days).

- If that sounds as if it were too obvious a problem, maybe it is helpful to ask the next question: what is the 7-functor corresponding to 7-dimensional Chern-Simons theory? And more importantly: how does it arise from a systematic process from the 7-dimensional Chern-Simons action functional?

You see, recently lots of mathematicians talk about TQFT and QFT, because there has been much progress on formalizing the right axioms that a full QFT should satisfy, so mathematicians can now happily study the space of all solutions to these axioms. For instance for pure TFT it turns out that the “landscape” of TFTs with values in some $V$ is precisely the space of “fully dualizable objects in $V$. Whatever that means.

So that’s fantastic. The landscape of all TQFTs has been fully identified. Why is the random physicist from the street only vaguely aware and even less excited about this fact:

simple: because we have at the moment only a highly insufficient understanding of how these axiomatically characterized TQFTs correspond to the physics they are supposed to describe: its not known which “action functional” or similar gives rise to them. It’s not known which problem they actually solve, which role they play in this world.

The missing link is the formalization of quantization in general and of $\sigma$-model quantization in particular. We don’t just want to say: an $n$-d QFT is an $n$-functor on $Bord_n^S$. We want to say: this physical setup has associated with it this quantum $n$-functor.

Ideally and eventually we’d want to be able to say: this string background defined by such and such topology, metric, and flux and whatnot, gives rise to this 2d CFT, if at all. Currently this is done essentially by educated wild guessing.

So, how do you know that you are quantizing anything?

That’s exactly the point I was trying to make:

given just an abstract TQFT given by a representation of $Bord_n$, you don’t a priori know what it is the quantization of, if anything.

And one central problem that currently prevents us from knowing so is that we have no systematic theory for what it even means precisely to quantize classical background field data to a (T)QFT.

The proposal that this thread is about is a proposal for formalizing what quantization of classical background field data to an extended (T)QFT should mean.

Namely: represent the classical background field data by a $n$-functor on field configurations itself and then push that forward from field configurations to the bordisms that they are field configurations on.