QFT of Charged n-Particle: Extended Worldvolumes

August 2, 2007

QFT of Charged n-Particle: Extended Worldvolumes

Posted by Urs Schreiber

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For a while I had lost my License to Quantize. But now I got it back. So I resume talking about the Quantum theory of the Charged $n$ -Particle (I, II, III, IV, V, VI, VII, VIII, IX).

Recall, this is supposed to be the study of the situation where we

- have an $n$ -particle ( $n = 1$ : a particle, $n = 2$ : a string, $n = 3$ : a membrane) modeled by an $n$ -category $par$ (its parameter space) which is essentially an $(n - 1)$ -category

${par}_{1} = {•}$ for the pointlike particle

${par}_{2} = {•_{1} \to •_{2}}$ for the line-like string

${par}_{3} = {\begin{matrix} ↗ ↘ \\ •_{1} & ⇓ & •_{2} \\ ↘ ↗ \end{matrix}}$ for the disk-shaped membrane

- propagating (i.e. being mapped into) a target space $tar$ (for instance $tar = P_{n} (X)$ the path $n$ -groupoid of a spacetime $X$ for a sigma-model, or $tar = Σ G_{(n)}$ an $n$ -group for a gauge theory )

- and coupled there to a background field modeled by an $n$ -bundle with connection, thought of as the corresponding parallel transport $n$ -functor $tra : tar \to n Vect$

and we want to consider the quantization of this setup, which is supposed to be itself an $n$ -functor, namely the Segal-like extended worldvolume QFT $q (tra) : n Cob \supset par \times Interval \to n Vect$ which assigns to points an $n$ -vector space of states, to 1-dimensional cobordisms a morphisms of $n$ -vector spaces, and so on.

While in the original Segal picture an $n$ -dimensional QFT is thought of as a mere 1-functor which sends $n$ -dimensional cobordisms going between their $(n - 1)$ -dimensional boundaries to linear maps between vector spaces associated to these boundaries, we are here modelling our $(n - 1)$ -dimensional boundaries – namely what we call “parameter space” – by $(n - 1)$ -categories, thus assigning data not just to the top two dimensions, but all the way down to points.

This is idea is usually called an extended functorial QFT. Some people refer to it as “localized” QFT. I am not sure exactly what the best term would actually be. But in need of a term, I shall here refer to this kind of $n$ -categorical refinement of functorial QFT as local refinement.

But – and this is what I am going to talk about here – there is actually also a different, alternative way to extend a Segal-like QFT functor to an $n$ -functor discussed in the literature:

instead of allowing the original $(n - 1)$ -dimensional objects to decompose into lower-dimensional pieces and thus regarding them as $(n - 1)$ -morphisms instead of objects, one sometime leaves the objects the way they are and instead throws in more morphisms between the existing cobordisms. For instance one may want not to divide out homeomorphisms of $n$ -dimensional cobordisms in an $n$ -dimensional topological QFT, but instead throw in one 2-morphism for each such homeomorphism. This concept I shall here refer to as global refinement.

(See the literature mentioned here.)

One may wonder what to make of this parallel existence of two different notions of extended QFT. Here I want to point out that

Local refinement is adjoint to global refinement.

In a sense that can be made precise in the context of the $n$ -particle. It crucially involves Gray tensor products of higher categories.

One nice way to think about this turns out to be this:

The worldvolume (worldline for the particle, worldsheet for the string, etc.) of the $n$ -particle arises from the Gray tensor product of the particle with the timeline.

I thought that’s a cute statement which deserves to be put in bold here. Below I give the precise statement that this is referring to.

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Depending on what one is looking at, there are different notions of “passage of time” in different QFTs: in ordinary non-relativistic quantum mechanics of a point particle, time flows along the real numbers. In a topological QFT time would not really flow at all.

In the context of the $n$ -particle, we model this by considering a 1-category $Interval$ whose morphisms represent time steps. So for the non-relativistic particle we’d take $Interval = Σ ℝ,$ the real numbers under addition, regarded as a 1-object category.

For a topological QFT we might take $Interval$ to be the walking (co)-monad: the category with a single object and an endomorphism on it which is such that its composite with itself may be contracted back to just the morphism itself, and the other way around, such that everything is nicely associative.

$n = 1$ : Worldlines

For a 1-particle of shape $par$ the worldlines it traces out should hence be the morphisms in the category $worldvol : = par \times Interval .$ So if the particle is just a point $par = {•}$ , its “worldvolumes” are just Riemannian line segments, hence morphisms $• \overset{t}{\to} •$ in $Interval$ . But, for instance, we might take the 1-particle for instance to really be a collection of two particles, setting $par = {•_{1} •_{2}}$ . Then its worldvolumes would consist either of time intervals for one or for the other of the other particle. I mention this just to demonstrate the idea.

$n = 2$ : Worldsheets

So far so trivial. The situation becomes somewhat more interesting as we pass to the 2-particle, modeled by a $(2 - 1 = 1)$ -category of the form ${•_{1} \to •_{2}}$ . Assume $Interval = Σ ℝ$ , for definiteness.

Then we might be tempted to simply copy the construction for the 1-particle from above and say that $worldvol : = {•_{1} \to •_{2}} \times Interval$ is the worldvolume category of that string.

But does that make sense? Not if we take the ordinary cartesian tensor product of categories here: then ${•_{1} \to •_{2}} \times Interval$ would just contain a bunch of 1-morphisms and would in no way model the strip swept out by the string as it propagates along.

But then, we shouldn’t do that in the first place! It’s a violation of the dao. The 2-particle lives in the context of $2 Cat$ – even though $par$ and $Interval$ might both be degenerate in that no nontrivial 2-morphisms are present. But actually, as soon as we become more serious about all this, these will be 2-categories proper (for instance as we pass to the super $n$ -particle). Certainly the walking monad is a 2-category, for instance.

As soon as we follow the dao again by properly working in 2 $Cat$ , abstract nonsense alone turns out to produce the proper model for the worldsheet of the string for us:

as you all know, the “right” tensor product on $2 Cat$ to use is not the naive cartesian product $\times$ , which is the ordinary cartesian product on the sets of morphisms. That’s because this would fail to be adjoint to the internal $Hom$ of $2 Cat$ (the 2-category of 2-functors between two given 2-categories).

Rather, one defines a tensor product $\otimes : 2 Cat \times 2 Cat \to 2 Cat$ such that it is the adjoint functor to the internal Hom, i.e. such that for all 2-categories $A$ , $B$ and $C$ we have an equivalence $Hom (A \otimes B, C) ≃ Hom (A, Hom (B, C))$ of 2-categories.

If you don’t know how $\otimes$ is defined (I didn’t before we once talked about it here) it is actually easy and instructive to work it out in the simple cases which we are interested in here, where $A = Interval = Σ ℝ$ and $B = par = {•_{1} \to •_{2}}$ and $C$ is something ( $C = 2 Vect$ in our context, but that’s not relevant at all for what I am talking about here).

An object in $Hom (par, C)$ is just a morphism $V_{1} \overset{f}{\to} V_{2}$ in $C$ (which we would think of as a 2-space of state $V_{i}$ over each endpoint of the string, together with a 2-linear map between them associated to the bulk of the string). Now, since we are in $2 Cat$ , a morphism in $Hom (par, C)$ is, being a pseudonatural transformation of 2-functors on 1-categories, a square $\begin{matrix} V_{1} & \overset{f}{\to} & V_{2} \\ ↓ & ⇓^{U (t)} & ↓^{t} \\ V'_{1} & \overset{f'}{\to} & V'_{2} \end{matrix}$ in $C$ !

We already see intuitively that if the top and bottom horizontal lines of this square are supposed to be the spaces of 2-states of a string, then the square connecting them ought to be the propagation map associated to a pice of worldsheet along which our string travelled.

And indeed, that’s what demanding an equivalence $Hom (Interval \otimes par, C) ≃ Hom (Interval, Hom (par, C))$ implies: the 2-category which we are supposed to address as $Interval \otimes par$ has to be such that a 2-functors from here to $C$ are given by squares in $C$ as above. But that means that $Interval \otimes par$ has to be the abstract such square of length $t$ ! $Interval \otimes par = {\begin{matrix} •_{1} & \to & •_{2} \\ ↓ & ⇓^{\sim} & ↓^{t} \\ •_{1} & \to & •_{2} \end{matrix} ∣ t \in ℝ} .$

(Physicists will have noticed that all I am talking here are the free propagators, where the $n$ -particle just travels along its way, without interacting with other $n$ -particles.)

Indeed, that’s how the Gray tensor product works: it says that the product $C \otimes D$ of two 2-categories is such that what used to be a commuting square $\begin{matrix} (c_{1}, d_{1}) & \overset{g}{\to} & (c_{1}, d_{2}) \\ ↓^{f} & ↓^{f} \\ (c_{2}, d_{1}) & \overset{g}{\to} & (c_{2}, d_{2}) \end{matrix}$ in $C \times D$ for any two morphisms $c_{1} \overset{f}{\to} c_{2}$ and $d_{1} \overset{g}{\to} d_{2}$ in $C$ and $D$ respectively, is replaced in $C \otimes D$ by a mere 2-isomorphism $\begin{matrix} (c_{1}, d_{1}) & \overset{g}{\to} & (c_{1}, d_{2}) \\ ↓^{f} & ⇓_{μ (f, g)}^{\sim} & ↓^{f} \\ (c_{2}, d_{1}) & \overset{g}{\to} & (c_{2}, d_{2}) \end{matrix} .$

Here we see that these 2-morphisms added by using the Gray tensor product instead of the ordinary cartesian product provide us with the worldsheet of the string!

$n = 3$ : Membrane Worldvolumes

I don’t recall having seen the analogue of the Gray tensor product discussed for 3-categories (can anyone point me to some literature?) but from going throught the same kind of simple argument as before, for the simple parameter space categories which we are dealing with, it is clear how the pattern continues.

For instance for $par = {\begin{matrix} ↗ ↘ \\ •_{1} & ⇓ & •_{2} \\ ↘ ↗ \end{matrix}}$ the disk-shaped membrane, I think one finds, with $\otimes$ denoting the tensor product in $3 Cat$ which is adjoint to the internal Hom there, that $worldvol : = {\begin{matrix} ↗ ↘ \\ • & ⇓ & • \\ ↘ ↗ \end{matrix}} \otimes Interval$ is the 3-category whose 3-morphisms are filled cylinders over this disk.

And so on.

Finally: locally and globally extended QFT

With that said, the statement that local and global extension of QFT are mutually adjoint concepts is now most immediate:

The propagation $n$ -functor of our $n$ -particle $q (tra) : worldvol \to n Vect$ is a locally extended QFT: it assigns data to points up to $n$ -dimensional pieces of worldvolume.

And it is an object in $q (tra) \in Hom (worldvol, n Vect) .$ But since we have, by our definition now, $worldvol = Interval \otimes par$ we have in fact $q (tra) \in Hom (Interval \otimes par, n Vect) .$ This means that under the equivalence $Hom (Interval \otimes par, n Vect) ≃ Hom (Interval, Hom (par, n Vect))$ our locally refined extended QFT $q (tra)$ maps to an $n$ -functor $\tilde{q} (tra) : Interval \to n {Vect}^{par} .$

But consider what this is like:

Assume, just for simplicity, again that $Interval = Σ ℝ$ . Then, first, $\tilde{q} (tra)$ looks very much like the naive Segal-like functor describing an $n$ -dimensional QFT in the Schrödinger picture: it assignes some kind of space of states to the single point, and a propagator $U (t)$ to the time interval of length $t$ .

But we need to remember that $\tilde{q} (tra)$ is really to be regarded as an $n$ -functor: this implies that there may be morphisms between such assignments of propagators $U (t) \to U' (t)$ .

I’d have trouble drawing all the pictures here which make this quite manifest, but if it isn’t quite clear already grab an envelope in your vicinity and picture this situation: it is indeed the one of globally extended QFT: there is a manifest assignment of data only to $(n - 1)$ -dimensional copies of our $n$ -particle (a refined notion of space of state) and to the $n$ -dimensional worldvolume swept out by it – but then there are higher morphisms relating the top-level assignments to each other.

This becomes more pronounced as we use more sophisticated models for $Interval$ .

On the other hand, by the general abstract nonsense at work here, the globally refined extended QFT $\tilde{q} (tra)$ encodes precisely the same information as the locally refined extended QFT $q (tra)$ which we started with. When you write it out, you’ll see that the globally extended QFT, which seems to be missing information on the level of lower dimensional cobordisms, actually has its own refined version of “spaces of states” which it assigns to the entire $(n - 1)$ -dimensional $n$ -particle. This keeps all the information about what is going on in lower-dimensions.

Well, it’s all a game with abstract nonsense of course. But it is actually useful. I expect to talk about what this is useful for another time. Enough for now.

Posted at August 2, 2007 5:32 PM UTC

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31 Comments & 7 Trackbacks

Re: QFT of Charged n-Particle: Extended Worldvolumes

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Just to emphasize:

I don’t recall having seen the analogue of the Gray tensor product discussed for 3-categories (can anyone point me to some literature?)

I would also be very interested in any information regarding this, so does someone know? /Jens

Posted by: Jens on August 3, 2007 3:29 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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I think what you are looking for is discussed in chapter 3 of Sjoerd Crans’s thesis, where he constructs a Gray tensor product of strict $ω$ -categories. In outline, it works something like this: he constructs a functor

Γ \to ω - Cat

from the monoidal category of combinatorial cubes to $ω$ -categories, similar in spirit to how Ross Street constructs the free $ω$ -category on a parity complex (Crans uses Michael Johnson’s pasting schemes instead). By the usual Kan trick, this functor induces a functor to cubical sets,

ω - Cat \to {Set}^{Γ^{op}},

and this functor is full and faithful. Via Day convolution, there is a monoidal biclosed structure on cubical sets, and this restricts to a monoidal biclosed structure on $ω$ -Cat: the desired Gray tensor product.

(Oh wait: did you mean strict 3-categories?)

Posted by: Todd Trimble on August 3, 2007 9:01 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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I think what you are looking for is discussed in chapter 3 of Sjoerd Crans’s thesis,

Thanks! Great. I’ll have a look at it when I am less tired.

But, do you know, is my idea correct: does the Gray-like tensor product $C \otimes D$ of 3-categories $C$ and $D$ involve throwing in cylindrical 3-morphisms for each pair of 2-morphism from one and 1-morphism from the other 3-category?

Oh wait: did you mean strict 3-categories?

I will want to know this for 3-categories which are Gray categories. But I’d already be happy to know the corresponding statement for just strict 3-categories.

Posted by: Urs Schreiber on August 3, 2007 10:36 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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Does the Gray-like tensor product C⊗D of 3-categories C and D involve throwing in cylindrical 3-morphisms for each pair of 2-morphism from one and 1-morphism from the other 3-category?

Yes, that sounds right. Sjoerd has material on the tensor product of two globes (or ‘globs’), which generalizes this idea.

I will want to know this for 3-categories which are Gray categories.

In that case, you might prefer this other paper of Sjoerd’s.

Posted by: Todd Trimble on August 4, 2007 12:25 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

In that case, you might prefer this other paper of Sjoerds.

Excellent recommendation. When you see this, Urs, you will start to appreciate what raising and lowering operators are really about. (Just a thought from a real not-a-real physicist).

Posted by: Kea on August 4, 2007 12:45 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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When you see this, Urs, you will start to appreciate what raising and lowering operators are really about.

Says the student to his Zen-master:

Can you give me maybe just a tiny little further hint as to what this koan might be about?

;-)

So, I have seen the paper. Not read every word of it, but I think I grok the idea.

Also, I know what raising and lowering operators are, and, from John Baez’s observations and Jeffrey Morton’s work, what they might really be.

But if asked what this has to do with Gray tensor products, I wouldn’t quite know what to say.

Let’s see: maybe you are thinking along the following lines:

suppose we fix a 1-category called $K$ , then, using Crans’s $ω$ -categorical tensor product, we get from it an $ω$ -functor $a_{K}^{†} : n Cat \to (n + 1) Cat$ acting as $a_{K}^{†} : C_{n} \mapsto K \otimes C_{n} .$

Hm.

Hm. Hm.

Hm.

What if we form the adjoint $ω$ -functor $a : (n + 1) Cat \to n Cat$ ?

Hm. Hm. Interesting thought, I must say.

You know, from my perspective, this would not really be “raising and lowering” operators:

rather, this would be dimensional reduction and dimensional oxidation!

Say this (i.e. the 2-category associated to this 2-graph) is the membrane

${par}_{3} = {\begin{matrix} ↗ ↘ \\ •_{1} & ⇓ & •_{2} \\ ↘ ↗ \end{matrix}}$

and this is the string ${par}_{2} = {•_{1} \to •_{2}} .$ Then we are suppsed to get the string from the membrane by dimensional reduction of its worldvoume ${worldvol}_{{par}_{3}} : = {par}_{3} \otimes Σ ℝ,$ So maybe we should act with the adjoint of the dimensional oxidation operation ${par}_{3} \mapsto a_{{par}_{2}} ({par}_{3}) .$ If it exists. Hm.

Okay, for the experts: in Sjoerd Crans’s setup, does the functor $a_{k}$ adjoint to $a_{K}^{†}$ , as defined above exist for, say, $K = {par}_{2}$ , as above?

What is the result of acting with this adjoint functor on ${par}_{3}$ ? Can anyone tell?

Would maybe be quite interesting for applications.

Posted by: Urs Schreiber on August 4, 2007 10:05 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

In that case, you might prefer this other paper of Sjoerd’s.

Great, thank you!

Posted by: Jens on August 4, 2007 9:20 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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Does the Gray-like tensor product C⊗D of 3-categories C and D involve throwing in cylindrical 3-morphisms for each pair of 2-morphism from one and 1-morphism from the other 3-category?

Yes, that sounds right. Sjoerd has material on the tensor product of two globes (or ‘globs’), which generalizes this idea.

Thanks. Very useful. I see precisely the cylinder that I was referring to, which I like to visualize like this

appear on top of p. 11 of Sjoerd Crans’s text.

By the way, when looking at Sjoerd Crans’s On combinatorial models for higher dimensional homotopies I realized that something pretty interesting is going on here, in general. (The following will be totally obvious to you, but I had not really thought about it quite this way before).

Naively, there are two different heuristic ways to think of what the tensor product of an $n$ -category $C_{n}$ with an $m$ -category $D_{m}$ should be:

if we regard an $n$ -category as a mere collection of sets with extra structure on them, we would just expect the cartesian product on sets and hence that $C_{n} \otimes D_{m}$ is a $\max (n, m)$ -category.

But if we seriously think than an $n$ -category is a model for something of “dimension $n$ ” (like an $n$ -dimensional space), we would tend to expect that $C_{n} \otimes C_{m}$ is an $n \cdot m$ -category!

And in fact, that’s what Crans, working with $ω$ -categories from the get go, does obtain.

So, his cubical tensor product of two 2-categories $C_{2}$ and $D_{2}$ , as he explains somewhere, indeed is not a 2-category, but a 4-category.

He explains how from this 4-category we obtain precisely the familiar 2-category which is the Gray tensor product $C_{2} \otimes_{Gray} D_{2}$ by dividing out 3- and 4-morphisms of his 4-category.

That’s interesting, since this means that as a result we get something like a mixture of the two naive heuristic expectations mentioned above: on 1-morphisms the tensor product really acts as if it were taking the product of “1-dimensional intervals”, hence producing 2-dimensional worldsheets, er, I mean, 2-dimensional cubes, while on 2-morphisms (top level) it acts essentially like the cartesian product on sets.

It may be obvious to you all. But I find this quite charming.

Posted by: Urs Schreiber on August 4, 2007 9:35 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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I agree: it is cool that the true Gray tensor product of two 2-categories is a 4-category! But I thought the categorical dimension of the tensor product was $m + n$ , since the product of an $m$ -cube and an $n$ -cube is an $m + n$ -cube. (Also, $m \cdot n$ doesn’t seem right because what if $n = 0$ ?)

Posted by: Todd Trimble on August 4, 2007 1:26 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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It’s really no fair: the product of an $n$ -dimensional thing and an $m$ -dimensional thing has dimension given by the sum $n + m$ , not the product $n \times m$ . There’s something logarithmic about dimension. The case $n = m = 2$ is just an extra twist the math gods threw in to confuse us.

Here’s a fun little puzzle. We all know the great thing about ‘two twos’, namely

$2 + 2 = 4$ $2 \times 2 = 4$ $2^{2} = 4$

But, these identities are just the first three of an infinite sequence! What are the rest?

Posted by: John Baez on August 5, 2007 8:25 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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The wizard came by and posed a little riddle:

But, these identities are just the first three of an infinite sequence! What are the rest?

Here is what I think the answer is:

define $op : ℕ \times ℕ \times ℕ \to ℝ$ recursively as follows:

$op (0, n, x) : = {\underset{︸}{x + x + \dots + x}}_{n arguments}$ and $op (k + 1, n, x) : = {\underset{︸}{op (k, x, \dots op (k, x, op (k, x, x))}}_{x appears n times} .$

For instance $op (2,3,2) = op (1,3,3) = op (0,3, op (0,3,3)) = 3^{2} = 3 \cdot 3 = 3 + 3 + 3 .$

Then $op (k,2,2) = 4$ for all $k \in ℕ$ .

I also have a question for the wizard:

why is it we see $op (k, \cdot, \cdot)$ for $0 \leq k \leq 2$ all around us (addition, multiplication, taking powers), while the rest does not seem to appear “naturally” (of course one can force it) either in math or in the natural sciences?

My hunch is that it is related to the way arithmetic on Church numerals works in Lambda-Calculus. But I am not sure if that is the “exaplanation” of this phenomenon, or just another aspect of this phenomenon.

Posted by: Urs Schreiber on August 6, 2007 9:55 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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John wrote:

Here’s a fun little puzzle. We all know the great thing about ‘two twos’, namely $2 + 2 = 4$ $2 \times 2 = 4$ $2^{2} = 4$ But, these identities are just the first three of an infinite sequence! What are the rest?

I wrote John a little email on this a week ago; he just got back and asked me if I would post it. I’ll quote it below; it may be substantially the same as Urs’s solution, but afterwards I’ll toss in some buzzwords and trivia as well.

Heh, here’s one solution to your 2 + 2 = 2 x 2 = … puzzle. Well, it’s just a formalization of Knuth arrow notation, where $2^{2} = 2 ↑ ↑ 2 = 2 ↑ ↑ ↑ 2 = . . .$ . Define a sequence of functions $f_{0}, f_{1}, f_{2}, . . . : ℕ_{+} \times ℕ_{+} \to ℕ_{+}$ ( $ℕ_{+}$ = positive integers) by $f_{0} (m, n) = m + n$ $f_{k + 1} (1, n) = n$ $f_{k + 1} (m + 1, n) = f_{k} (f_{k + 1} (m, n), n) .$ This is just the idea that “ $f_{k + 1}$ -tion is repeated $f_{k}$ -tion”, as in “multiplication is repeated addition”, “exponentiation is repeated multiplication”, and so on. By induction, $f_{1} (m, n) = m n,$ $f_{2} (m, n) = n^{m},$ and $f_{k + 1} (2, n) = f_{k} (f_{k + 1} (1, n), n) = f_{k} (n, n)$ whence $f_{k + 1} (2, 2) = f_{k} (2, 2) = 4$ by induction.

Writing all this reminds me of notation systems for writing really big numbers. The Knuth arrow notation can be used to define the moderately fast-growing Ackermann sequence $1^{1}, 2 ↑ ↑ 2, 3 ↑ ↑ ↑ 3, . . ., n ↑ . . . ↑ n$ ( $n$ Knuth arrows), …; see here for a definition of the Ackerman function $A (m, n)$ . In other words, the Ackermann sequence is defined by

A (n) = f_{n + 1} (n, n) .

It grows faster than any primitive recursive function. As noted above $A (2)$ is just 4, but $A (3)$ is an exponential stack of 3’s of length $3^{3^{3}} = 3^{27} = 7625597484987$ . Just try to get an idea of how big $A (4)$ is! (See also Conway-Guy, The Book of Numbers, pp. 60-61.)

But that’s nuthin’. Much, much faster growing functions can be defined! If you mentally get off on this sort of thing, you probably already know about Conway’s chained arrow notation, and how large countable ordinals can be used to define obscenely rapidly growing sequences. And then of course there’s the busy beaver function, which grows faster than any recursive function. See also Scott Aaronson’s engaging article here, for more on the busy beaver.

Some other bits of trivia: the largest number used in a mathematical proof was for a long time the Skewes number, on the order of $10^{10^{10^{34}}}$ . According to Conway and Guy, it has to do with the fact that Gauss’s estimate for $π (x)$ (the number of primes less than $x$ ),

Li (x) = \int_{2}^{x} 1 / \ln (x) dx,

is occasionally better than Riemann’s more refined guess

R (x) = \sum_{n \geq 1} \frac{μ (n)}{n} Li (x^{1 / n})

where $μ$ denotes the Möbius function. J.E. Littlewood proved this happens for infinitely many $x$ , and Skewes showed it happened for some $x$ less than the Skewes number. Subsequently it has been shown to occur for some $x$ less than $1.4 \times 10^{316}$ , so Skewes’s number is now something of a quaint artifact from a bygone era.

But anyway, the Skewes number is just triflingly, pathetically puny compared with the current record-holder for largest number used in a proof, namely Graham’s number. In Knuth arrow notation, it’s defined to be $G = g_{64}$ where $g_{1}$ is defined to be $3 ↑ ↑ ↑ ↑ 3$ , and $g_{n}$ is defined to be $3 ↑ . . . ↑ 3$ , using $g_{n - 1}$ Knuth arrows. Holy smokes!

(This number comes up in a problem of Ramsey theory; it’s the smallest known upper bound for a number whose actual value was thought for a long time to be just 6! Well, now it’s known it must be at least 11. The gap is narrowing…)

Finally, check out this description of a big number duel held earlier this year at MIT.

Posted by: Todd Trimble on August 12, 2007 7:41 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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I did mean $n + m$ , of course. Was a typo, sorry. As I said, what is so nice here is that an $n$ -category really behaves like an $n$ -dimensional space under this tensor product.

I find that in A tensor product for Gray-categories Sjoerd Crans emphasizes this striking property very nicely.

The difference between the cartesian product and Gray’s tensor product […] is not just the difference between a commuting square and a square commuting up to a 2-arrow. I.e., this difference is not due to some Main Principle of Category Theory, that in any category it is unnatural and undesirable to speak about equality oft wo objects [22, p. 179].

[…]

No, the conceptual difference lies in the treatment of dimension. The cartesian product of 2-categories, and of $ω$ -categories, is basically set-theoretical: $C \times D$ has as basic ingredient pairs $(x, y)$ of dimension $p$ for $x \in C_{p}$ and $y \in D_{p}$ , and functoriality then gives, more generally, pairs $(x, y)$ of dimension $\max (p, q)$ .

The tensor product of 2-categories, and of $ω$ -categories, is basically topological:

$C \otimes D$ has as basic ingredient expressions $c \otimes d$ of dimension $p + q$ for $c \in C_{p}$ and $d \in D_{q}$ ,

That’s pretty cool. I expect that, for instance, this will finally allow to say what the wedge product of differential forms really is.

Unfortunately, while I can follow this particular paper of Crans’s easily, in the other one I have trouble following from the point on where coends are being used.

By the way, is it know how far we’d get with just Crans’s $(ω - Cat)_{\otimes}$ -categories? I.e. is there any chance that, like every tricategory is equivalent to a Gray category, every tetracategory is equivalent to a $(3 - Cat)_{\otimes}$ -category?

I guess that not, but is it known?

Posted by: Urs Schreiber on August 5, 2007 11:42 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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Urs, I think a $(Gray - Cat)_{\otimes}$ -category is what Crans would also call a 4d-tas. (The plural is teisi: the word ‘tas’ is Welsh and means either ‘stack’ or ‘file’. About 10 or 12 years ago, Street defined structures he called ‘ $n$ -files’, as a tentative proposal for semi-strict $n$ -categories, and I thought I once heard that Crans’s work on teisi was originally motivated by a desire to take up Street’s idea seriously.) Crans has thought long and hard about $n$ -teisi and I believe has proved a lot about them; perhaps he can be induced to join this discussion.

According to the abstract of this paper of Crans, 4-teisi with one object and one arrow are close to, but not exactly the same as, braided monoidal 2-categories. My understanding is also that $n$ -teisi have not yet been rigorously defined for higher $n$ , but that the intuition behind them led Crans independently to the notion of sylleptic monoidal and symmetric monoidal 2-categories, which he then proved equivalent to the definitions of Day-Street. Your best bet for online references to teisi may be the paper linked above, but there’s also this.

I think I may be able to get more precise information on the status of teisi with respect to coherence of tetracategories; I’ve heard a few rumors. Let me try to get back to you on this.

Posted by: Todd Trimble on August 6, 2007 1:55 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

You know, from my perspective, this would not really be raising and lowering operators: rather, this would be dimensional reduction and dimensional oxidation!

Yes, but I don’t see how I can agree with that given that (to me at least) the Crans paper never had anything to do with string/membrane theory. When I came across it several years ago I was thinking about Mach’s principle and general relativity and raising/lowering operators in a point-particle QFT (brane diagrams get forced upon one anyway by twistor geometry but never in the context of ordinary spacetime) and I had never spent one day of my life thinking about string theory.

Given how far you have come with the higher categories, don’t you think its time you questioned the physical foundations of this stuff? How are you going to use it to calculate gluon scattering amplitudes for the LHC?

Posted by: Kea on August 4, 2007 11:19 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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but I don’t see how I can agree with that given that (to me at least) the Crans paper never had anything to do with string/membrane theory.

In that case, I don’t see how we came to be talking to each other, since it doesn’t mention the raising and lowering operators you were mentioning either! ;-)

Given how far you have come with the higher categories, don’t you think its time you questioned the physical foundations of this stuff? How are you going to use it to calculate gluon scattering amplitudes for the LHC?

Well, first I finish the theory of the charged quantum $n$ -particle #, which will allow me to make sense of $n$ -particles propagating on arbitrary target categories. Then I find that target space category such that the Soibelman limit # of the $n$ -algebra of observables # of the super # 2-particle # propagating on it yields the noncommutative algebra which Connes found # to correspond to the KK-compactification which yields the standard model.

But that’s just my evil top secret plan. Please don’t mention it to anyone! (I am a little bit behind schedule, too. Maybe the LHC will have to start without me…)

Posted by: Urs Schreiber on August 6, 2007 10:12 AM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

and this functor is full and faithful.

Sorry, I shouldn’t have said the cubical nerve is full.

Posted by: Todd Trimble on August 4, 2007 2:16 AM | Permalink | Reply to this

Read the post QFT of Charged n-Particle: Towards 2-Functorial CFT
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Tracked: August 6, 2007 11:48 PM

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Tracked: August 7, 2007 11:15 PM

Re: QFT of Charged n-Particle: Extended Worldvolumes

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I would like to restate the question at the end of one of my comment above. It looks like an important question to me but might be missed in the speculative discussion it appeared in:

In the $(ω Cat)_{\otimes}$ -CATegory of all $ω$ -categories which Sjoerd Crans discusses, consider the $ω$ -functor $2_{1} \otimes \cdot : ω Cat \to ω Cat$ given by tensoring $C \mapsto 2_{1} \otimes C$ with the 1-category that Sjoerd Crans calls $2_{1} : = {•_{1} \to •_{2}} .$

Question: Does this $ω$ -functor have an adjoint?

If so, how does it act on simple $ω$ -categories like the $2_{n}$ , for instance?

Posted by: Urs Schreiber on August 8, 2007 11:14 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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Oh yes, you did ask that! (Sorry, I’m a little slow…)

Sjoerd proves that $ω$ -Cat with its tensor product is monoidal biclosed, so yes, on general grounds the $ω$ -functor $2_{1} \otimes -$ has a right adjoint. Let me call it $2_{1} \Rightarrow -$ . I think it’s just what you’d expect: the 0-cells of $2_{1} \Rightarrow D$ are 1-cells of $D$ ; the 1-cells of $2_{1} \Rightarrow D$ are 2-cell squares in $D$ ; the 2-cells of $2_{1} \Rightarrow D$ are 3-dimensional cylinders in $D$ , and so on: the $n$ -cells of $2_{1} \Rightarrow D$ are cylinders of shape $2_{n} \otimes 2_{1}$ in $D$ .

(Or were you asking something different?)

Posted by: Todd Trimble on August 9, 2007 3:18 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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Obviously the last line of my previous post before the parenthetical question should be “of shape $2_{1} \otimes 2_{n}$ in $D$ ”.

Posted by: Todd Trimble on August 9, 2007 3:23 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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the $n$ -cells of $2_{1} \Rightarrow D$ are cylinders of shape $2_{1} \otimes 2_{n}$ in $D$ .

Ah, I see. Very nice.

(Or were you asking something different?)

No, that was precisely what I was asking. And the answer is also like what I would have intuitively expected.

I just wasn’t sure I could see if this is indeed adjoint to $2_{1} \otimes \cdot$ .

Right, so this actually matches nicely with what Crans mentions in the introduction of A tensor product for Gray-categories on p. 3, where he turns the reader’s attention to the fact that a transformation of $k$ -functors $C \to D$ is a morphism $C \otimes 2_{k - 1} \to D$ (or rather its component map is, I would say).

But a transformation of $k$ -functors should really take values in cylinders over $(k - 1)$ -dimensional bases in $D$ . By what you just said about the adjoint of $2_{k - 1} \otimes \cdot$ , this follows formally from $Hom (C \otimes 2_{k - 1}, D) ≃ Hom (C, 2_{k - 1} \Rightarrow D) .$

That’s all very nice (in as far as I got it right). It rather neatly provides the right formal language for “holography”, it seems. I am enchanted. Thanks for telling me about this stuff!

Posted by: Urs Schreiber on August 9, 2007 4:06 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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a transformation of $k$ -functors $C \to D$ is a morphism $C \otimes 2_{k - 1} \to D$ (or rather its component map is, I would say)

I am wondering if in this formalism of Crans’ one can make a similarly nice statement concerning the following issue which I think I am facing:

suppose, for definiteness, that $\begin{matrix} F (a) & \overset{F (f)}{\to} & F (b) \\ ^{η (a)} ↓ & ^{η (f)} ⇓ & ^{η (b)} ↓ \\ G (a) & \overset{G (f)}{\to} & G (b) \end{matrix}$ is the component of a transformation $η : F \to G$ of 2-functors $F$ and $G$ .

I seem to be faced with an application where $F$ and $G$ always coincide on objects, and where it is required to think of the collection of all such transformations as decomposed into

a) those for which $η$ is trivial over objects, i.e. $η (a) = {Id}_{F (a)} = {Id}_{G (a)}$

b) those for which $η$ is trivial over morphisms, i.e. $η (f) = {Id}_{η (b) \circ F (f)} = {Id}_{G (f) \circ η (a)}$

c) the rest.

So in case b), $η$ would just be a natural trasformation, i.e just natural at the level of 1-morphisms, wheras is a) $η$ would be just a natural transformation at the level of 2-morphisms (in that only bigons are involved, no squares).

In my application here, it seems I want to talk about categories of 2-functors where I restrict the morphisms to be either of type a) or of type b).

Probably I am not saying this well. And that’s in fact my question: can anyone help me see more clearly what I might be looking for here?

Posted by: Urs Schreiber on August 20, 2007 7:27 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: Extended Worldvolumes

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I’ll make another attempt to ask this question:

Denote, for the purpose of this comment, by $2 {Cat}_{nat}$ the strict 2-category whose objects are strict 2-cats and whose morphisms are natural transformations (meaning they have components which are squares filled with identities 2-cells),

and denote by $2 {Cat}_{pseudo}$ the usual Gray-category where we allow the transformations to be pseudo.

The former sits inside the latter $2 {Cat}_{nat} ↪ 2 {Cat}_{pseudo} .$ In my previous comment I think I was essentially asking about the cokernel of this inclusion, I guess, and if there is anything useful to be said about it.

Posted by: Urs Schreiber on August 20, 2007 7:47 PM | Permalink | Reply to this

Read the post That Shift in Dimension
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Excerpt: What makes the Kontsevich-Cattaneo-Felder theorem tick? How can it be that an n-dimensional quantum field theory is encoded in an (n+1)-dimensional one?
Tracked: August 25, 2007 2:11 AM

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Tracked: September 3, 2007 1:50 PM

Re: QFT of Charged n-Particle: Extended Worldvolumes

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Hi Todd! Can you hear me?!

I am hoping that maybe you might find some time to help me a little more with learning about Sjoerd Crans’ and related results on the tensor product on $ω$ -Cat.

I am in Trondheim visiting Nils Baas, and we are trying to figure out how his notion of concordance of 2-bundles is the same as the notion one obtains by looking at the 2-bundles as their classifying 2-functors and then at (pseudo)-natural transformations of these.

I suggested that it must boil down to the closedness statement, for $C$ and $D$ two 2-categories, that

$Hom (C \otimes 2_{1}, D) ≃ Hom (C, Hom (2_{1}, D)) .$

Here the left side would be about “concordance” (“homotopy”): two 2-functors $F_{1}, F_{2} : C \to D$ are “concordant” if there is a 2-functor $\hat{F} : C \otimes 2_{1} \to D$ restricting to $F_{1}$ on the “left boundary” and to $F_{2}$ on the “right boundary” of the “interval” $2_{1} : = {• \to \circ} .$

On the other hand, the right side would reformulate the same situation as a (pseudo or lax) natural transformation, namely as

a) a 1-functor on 1-morphisms of $C$ with values in squares of $D$

b) such that the naturality tin cans with $F_{1}$ and $F_{2}$ filling top and bottom 2-commute.

My idea is that by hitting everything in sight here with some notion of realizations of nerves, we see that this tells us that concordance classes of 2-bundles are precisely the same as “natural” classes of the same 2-bu

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August 2, 2007