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February 7, 2012

Good News

Posted by John Baez

I’m in Sydney talking to Ross Street and other category theorists at Macquarie University. Once I’d expressed worries about the fate of Australian category theory now that Street has retired. I’m happy to say that we can put those worries to rest.

First of all, while Street has formally retired, he still comes to work every day and is very active. Second, Steve Lack has moved from Sydney University to Macquarie, so he can now have graduate students. On top of that, he’s gotten a 4-year grant that’ll let him spend most of his time on research! Richard Garner is here too, and has a 5-year research-only postdoc. Also in the math department are Michael Batanin, Alexei Davydov and Mark Weber. And if that weren’t enough, Dominic Verity has joined Mike Johnson over in Macquarie’s computing department.

So, it’s a great place to go if you want to talk to category theorists! I apologize to anyone whom I left out.

And on a more personal note…

… Jeffrey Morton has gotten a 2-year postdoc position in Christoph Schweigert’s group in Hamburg! As you may recall, this is Urs Schreiber’s old home. So, we can expect to see more interplay between string theory, TQFTs, and the study of categorified quantum theory using spans.

And there’s even more good news. Jeffrey Morton and Jamie Vicary are on the brink of releasing a paper that sheds new light on the quantum harmonic oscillator! They’ve both written about ways to categorify the harmonic oscillator, and now they’ve joined forces and figured out how to understand Khovanov’s categorified Heisenberg algebra using combinatorics!

You may recall how groupoidification lets us understand the canonical commutation relations

aa *a *a=1 a a^* - a^* a = 1

in a very simple way, by making precise the fact that ‘there’s one more way to put a ball in a box and then take one out than to take one out and then put one in’. Khovanov’s categorified Heisenberg algebra has some relations whose meaning is far less obvious. They’re usually drawn using string diagrams, and they’re pretty, but what do they say about reality?

Jeffrey and Jamie have figured it out! The math is nontrivial—it involves bicategories, spans of groupoids, and the representation theory of symmetric groups, all working together. But the meaning of Khovanov’s relations can be understood in terms of balls in a box! This will, I think, add a whole new layer of depth to our understanding of a very basic item in physics: the quantum harmonic oscillator.

So, stay tuned.

Posted at February 7, 2012 12:03 AM UTC

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Re: Good News

They’ve both written about ways to categorify the harmonic oscillator, and now they’ve joined forces and figured out how to understand Khovanov’s categorified Heisenberg algebra using combinatorics!

Does this help in understanding what a categorified quantum harmonic oscillator might have to do with physics (apart from its decategorification being the ordinary quantum harmonic oscillator)?

Posted by: Urs Schreiber on February 7, 2012 10:12 AM | Permalink | Reply to this

Re: Good News

I think it will, though I don’t know how.

You may remember David Corfield wondering about the divide between ‘Baez–Dolan categorification’ and ‘Khovanov categorification’. The divide was never very large mathematically, but the two schools had different emphases.

The Khovanov school has made great progress categorifying the favorite algebraic structures of mathematical physicists (quantum groups, the Heisenberg algebra, etc.) and relating this categorification process to ‘dimension-boosting’ in field theory (see for example Witten’s March 28th 2012 talk). The Baez–Dolan school, if I dare call it that, was focused on reducing these algebraic structures to their underlying combinatorics: for example, understanding the annihilation and creation operators in terms of finite sets equipped with extra stuff.

What Jamie Vicary and Jeffrey Morton have done is to explicitly reconcile and unify these two approaches, though so far only in the special case of the categorified Heisenberg algebra. That seems like a step forwards, though I don’t know where it’ll lead.

Posted by: John Baez on February 7, 2012 10:41 AM | Permalink | Reply to this

Re: Good News

…wondering about the divide between ‘Baez–Dolan categorification’ and ‘Khovanov categorification’.

That was a while ago! For the record, that was here and here.

Posted by: David Corfield on February 7, 2012 2:48 PM | Permalink | Reply to this

Re: Good News

see for example Witten’s March 28th 2012 talk

So that talk seems to be going to be about the relation between QFT and Khovanov homology, which we talked about for instance at 4d QFT for Khovanov Homology.

Right, so this is what makes me ask: can one somehow see what role specifically the categorified harmonic oscillator might play in this relation?

How does even the categorified Heisenberg algebra come into the story of Khovanov homology? In the paper that you pointed to, I can’t see any relation to Khovanov homology being made.

Maybe some expert on that “school” is around and can explain?

Posted by: Urs Schreiber on February 7, 2012 12:28 PM | Permalink | Reply to this

Re: Good News

Jeffrey runs a blog, and the last post links to a talk about this work with Jamie – Groupoidification and Khovanov’s Categorification of the Heisenberg Algebra.

Posted by: David Corfield on February 7, 2012 3:19 PM | Permalink | Reply to this

Re: Good News

the abstract of Witten’s talk is the same as the abstract of http://arxiv.org/abs/1108.3103

Posted by: phx on February 8, 2012 11:09 AM | Permalink | Reply to this

Re: Good News

This is a good question, which I’ve thought about while we’ve been working on this project. I can’t immediately say for sure that there is a close relation with physics, but I can suggest what occurs to me.

Basically, what we’ve shown is that Khovanov’s categorified Heisenberg algebra has, as a concrete model, a slightly souped-up version of the groupoidification based on finite sets. From my point of view, what’s interesting about it is that the souping-up - from a setting in a monoidal category to a setting in a monoidal 2-category - has the same form as the souping up of a TQFT into an extended TQFT. In fact, it involves exactly the same extension of the span category Span 1(Gpd)Span_1(Gpd) to a 2-category Span 2(Gpd)Span_2(Gpd) that I talk about for the ETQFT construction that gives the Dijkgraaf-Witten model.

In that case, the physical meaning of raising the categorical dimension is to talk about the local structure of a (topological) quantum field theory. So, e.g. Hilbert spaces in a TQFT are interpreted as slightly degenerate 2-linear maps between 2-Hilbert spaces. What it means physically is that the 2-Hilbert spaces describe boundaries between regions, and composing the functors describes how to glue together a Hilbert space for a composite system. So my guess is that categorifying the QHO will give something most naturally understood in terms of composite systems build out of oscillator-like components.

So, maybe this would make more sense if I summarize what we did:

In the un-souped picture, Fock space is represented by the groupoid FinSet 0FinSet_0 whose objects are finite sets and whose morphisms are bijections. The paper John linked to shows how this gets taken to the polynomial algebra [[z]]\mathbb{C}[[z]] which we think of as Fock space where the raising operator is multiplication by zz and the lowering operator is the derivative z\partial_z. These are represented by spans from FinSet 0FinSet_0 to itself, where one leg is the identity, and the other is the map that takes a set SS to S{}S \sqcup \{ \star \}. This is “putting a ball into a box” in the combinatorial language John mentioned. In that language, “states” for the system are combinatorial “stuff types”, and the raising and lowering operators are interpreted as acting on them through operations that amount to moving “balls” (elements) in and out of “boxes” (the underlying sets of the stuff types). Then, e.g. the commutation relation just expresses the fact that you have one extra way to remove a ball from a box if you first put an extra ball in.

The un-souped picture stops here, because it lives in a monoidal (1-)category Span 1(Gpd)Span_1(Gpd). The souped-up picture uses the 2-category Span 2(Gpd)Span_2(Gpd), which has spans of span maps as its 2-morphisms. (Actually, isomorphism classes of these things, the way Span 1(Gpd)Span_1(Gpd) has iso-classes of spans as 1-morphisms).

Then one thing we find is that, when these 2-morphisms are available, it’s exactly true that the raising and lowering operators are adjoints - actually, formal ambidextrous adjoints in the 2-category Span 2(Gpd)Span_2(Gpd). One can easily write down the units and counits for the two adjunctions. (This is all a special case of something that works for any span and its “dual” span with the orientation of source and target reversed).

This is the most essential part of how the connection to Khovanov’s picture works. It uses a diagram calculus that makes sense in any situation where there are ambidextrous adjoints (as described e.g. in this paper by Aaron Lauda). The caps and cups in this calculus denote the units and counits for the adjunctions. There are a few other basic ingredients (crossings), but these can all be represented as 2-morphisms in the span 2-category.

So the diagram calculus, which defines a free monoidal category in terms of some generating diagrams and relations, has a model in this combinatorial picture. (Though it’s better for us to think of it as a 2-category with one object, so Khovanov’s objects are our morphisms). Upward- and downward-pointing strands denote raising/lowering operators (i.e. moving “balls” in and out of “boxes”), and a general object is some sequence (e.g. “put two balls in, then take one out, then put one back in”, etc). The diagrams denote “process movies”, where we change the process by applying the units, counits, crossings, etc. These are specific 2-spans, and have fairly straightforward interpretations in the combinatorics (e.g. one of the units means “GO FROM [do nothing] TO [put ball X in, then remove it again]”, and so on).

Then simple combinatorial arguments show that the spans used in the groupoidified QHO and the associated units and counits from their adjunctions do in fact satisfy all the relations Khovanov asks for. So they form a model of the theory described by the diagram calculus.

There are some other details, like where to find the symmetrizers and antisymmetrizers in that calculus. The short version is they’re not in Span 2(Gpd)Span_2(Gpd), but they can be found in 2Vect2Vect (or, really, 2Hilb2Hilb), after you map Span 2(Gpd)Span_2(Gpd) into it by “2-linearization”, which is this thing that restricts to groupoidification in a special case.

The 2-vector space in question is the category of representations of FinSet 0FinSet_0 - or, if you like, the product of all the representation categories of the symmetric groups, which amounts to the same thing. The categorified Heisenberg algebra acts on this category in a way that’s fairly straightforward to describe explicitly.

Notice that if you want to get back the groupoidified QHO from the 2-linear picture, you have to think of the groupoids and spans which make it up as actually being spans and 2-spans, in the hom-category Hom(1,1)Hom(1,1), where 11 is the terminal groupoid. (There’s some stuff to say about this, but I’ll leave it for now.) When you do this, since the groupoid 11 has only one - trivial - irreducible representation, you don’t see anything but the trivial (“bosonic”) representations of the symmetric group, though in the souped up version they are there.

Posted by: Jeffrey Morton on February 8, 2012 2:13 PM | Permalink | Reply to this

Re: Good News

Basically, what we’ve shown is that Khovanov’s categorified Heisenberg algebra has, as a concrete model, a slightly souped-up version of the groupoidification based on finite sets. […] same form as the souping up of a TQFT into an extended TQFT.

I see, thanks. That’s a very useful statement. So your QHO is the extension/localization down to the point of Khovanov’s Heisenberg algebra. I had missed that last time that I heard you talk. (And, to be frank, looking back at these slides I still find it hard to see where you say this. Seems to be an important statement that deserves to be highlighted up front.)

Okay, that nicely clarifies the relation between your approach and Khovanov’s. But still: what is the interpretation of either, physically?

With your picture it is becoming clear that one is describing ingredients of a “classical finite extended TQFT”. That’s fine, I am happy to know this in isolation. But I’d be happier if I now knew some relation or application of this to something else.

If I see correctly, then in his article Khovanov states as motivation for his categorified Heisenberg algebra only (?) that Frenkel suggested to him to work that out. Your construction now already gives me much more motivation. But it feels like there should be more here.

The trouble is: Witten’s work shows that everything Khovanov homology is to be thought of as categorified Chern-Simons theory. But there is no quantum harmonic oscillator anywhere in ordinary Chern-Simons, so it’s hard to see what a categorification of it should now appear.

So I’d tend to guess it’s not related to the Witten-Khovanov story after all. To what might it be related?

Posted by: Urs Schreiber on February 8, 2012 8:31 PM | Permalink | Reply to this

Re: Good News

That interpretation is just my best guess at how this categorification should be physically interpreted, and it’s based on a tentative analogy. Jamie and I have discussed it a few times, but I’ve never managed to expand the analogy into a convincing physical story, which is why I don’t want to over-sell that guess right now.

For one thing, talking about extending the QHO “down to a point” is surely misleading, since the QHO is defined only at a point in the first place. I don’t think the TQFT analogy can be pushed further than the guess that there’s some notion of composite systems, which can be expressed using the bicategory Span 2(Gpd)Span_2(Gpd). But whatever sort of composition the spans represent, can’t be literally come from gluing cobordisms together by taking some kind of groupoid-structured moduli spaces, as in the Dijkgraaf-Witten situation. It makes no sense to me to speak of increasing the “codimension” of a theory which is already defined at a point…

In fact, the structure of the cobordism category defines a particular bunch of composite spans which one has to work with - representing the pair-of-pants, etc. Then all the “composite systems” are (groupoids of connections on) surfaces or other manifolds built up from the generators of the cobordism category.

The kind of composites that naturally show up here don’t come about that way. One thing to say here is that this framework has a new ingredient, which is the “Fock Monad”, which Jamie introduced in this paper. In that case, he’s abstracting how the QHO gets built from Hilb to a more general dagger-monoidal category setting. The monad FF takes an object XX (say a Hilbert space) and gives a “Fock space object” F(X)F(X) which is a free commutative monoid object on XX. The categorified QHO is based on a 2-categorical analog - it exists for Span 2(Gpd)Span_2(Gpd), and also for 2Vect2Vect. The raising and lowering operators come from the bialgebra structure that defines, and they’re the generators we build composites from. They don’t look like, e.g. “pair of pants” spans - for one thing, the source and target are the same. The composites have something to do with Feynman diagrams, but not obviously with surfaces or manifolds. Perhaps there’s a nice story here, but it’s not obvious.

(I’d also mention that we only did this for the Heisenberg algebra essentially because it’s the first example that this Fock monad wants to give you - namely, the groupoid you feed into the 2-category version of FF is just X=1X=1, the trivial groupoid. The free symmetric monoidal category on it has objects that are nn-fold products of its only object, and morphisms that are permutations. That is, F(1)FinSet 0F(1) \simeq FinSet_0. But of course, you can put other groupoids XX into FF. You’d get a more complicated diagram calculus and a different categorified algebra out. The story here is: the individual particles in a harmonic oscillator don’t have any interesting internal states.)

Posted by: Jeffrey Morton on February 9, 2012 12:16 AM | Permalink | Reply to this

Re: Good News

You can get Khovanov homology starting from a categorified quantum group, namely categorified SL q(2)\mathrm{SL}_q(2). Quantum groups show up as symmetries, not of the ordinary quantum harmonic oscillator, but of the qq-deformed version. So, the relation between Khovanov homology and what Jamie and Jeffrey have done might become clearer when someone categorifies the qq-deformed harmonic oscillator.

One way to do that might be to replace the category of finite sets by the category of finite-dimensional vector spaces over 𝔽 q\mathbb{F}_q. The idea is summarized here, but there’s been a lot of work on this by many people from many angles. I think it’ll be good to combine that work with Jeffrey and Jamie’s new ideas.

However, connecting the categorified harmonic oscillator to Khovanov homology may not be the fastest way to figure out what the categorified harmonic oscillator is good for! I only mentioned Khovanov homology because that’s a case where categorifying a simple algebraic gadget, namely a quantum group, gives rise to math that people have already succeeded in connecting to ‘traditional physics’. When we go ahead and categorify all known algebraic gadgets, a picture should emerge that includes Khovanov homology, qq-deformed harmonic oscillators, ordinary harmonic oscillators, and lot of other stuff. This will be fun, but it may be slow. It may be faster, though painful, to simply sit and think about the categorified harmonic oscillator and what it might be good for.

Posted by: John Baez on February 9, 2012 12:59 AM | Permalink | Reply to this

Re: Good News

Hi Jeffrey,

I should find the time to sit down and work through your notes in detail. Unfortunately, currently I can’t even spend a second on that. But maybe I can make you give me a crisp and precise statement.

Currently I have this: first you said

what we’ve shown is that Khovanov’s categorified Heisenberg algebra has […] a slightly souped-up version […] what’s interesting about it is that the souping-up […] has the same form as the souping up of a TQFT into an extended TQFT.

But then

[…] talking about extending the QHO “down to a point” is surely misleading,

In these vague terms these two quotes seem to contradict each other. I guess it’s because I am interpreting your vague description incorrectly.

Is there a way to give a concise formal definition-theorem style statement of the relation between Khovanov’s categorified Heisenberg algebra, your construction and the notion of extended TQFT?

Posted by: Urs Schreiber on February 14, 2012 10:58 AM | Permalink | Reply to this

Re: Good News

Urs:

I’m not sure I can give a form of the analogy with extended TQFT which is precise enough to satisfy you, since I can’t give one which satisfies me. However, I can make it more precise than the version you quoted. To interpret that statement the way I wanted it, note that I have in mind those TQFT which come from groupoidification via gauge theory, since that was the case which motivated me to give a definition of the bicategory Span 2(Gpd)Span_2(Gpd) in the first place.

So, to recap what I’m comparing it to: TQFT’s are monoidal functors Z:nCobVectZ : nCob \rightarrow Vect, and the defining feature of the special class I’m thinking of is that they factor through Span(Gpd)Span(Gpd). That is, picking a gauge group GG there’s A:nCobSpan(Gpd)A : nCob \rightarrow Span(Gpd), which assigns to any manifold its groupoid of flat GG-connections. That’s actually a contravariant functor A G:TopGpdA_G : Top \rightarrow Gpd, but since any cobordism is a cospan, it extends to the AA I described. Then degroupoidification, D:Span(Gpd)VectD : Span(Gpd) \rightarrow Vect turns that into a TQFT via Z=DA GZ = D \circ A_G. (Let’s say GG is finite, to avoid extra complications about topological or Lie groupoids, and generalizations of groupoidification).

Then the same A GA_G gives an extended TQFT as a (monoidal) 2-functor Z:nCob 22VectZ : nCob_2 \rightarrow 2Vect which factors through Span 2(Gpd)Span_2(Gpd) (the 2-category with iso. classes of spans of span maps as 2-morphisms), and the 2-linearization 2-functor Λ:Span 2(Gpd)2Vect\Lambda : Span_2(Gpd) \rightarrow 2Vect, since A GA_G extends to A:nCob 2Span 2(Gpd)A : nCob_2 \rightarrow Span_2(Gpd). As we’ve discussed before, this is an example of the Freed-Hopkins-Lurie-Teleman approach to extended TQFT. Since degroupoidification is a special case of 2-linearization (i.e. Λ\Lambda looks like DD when restricted to Hom(1,1)Hom(1,1)), the usual TQFT picture shows up in the case of closed manifolds, where the codimension-2 (object) manifolds happen to be empty.

Now, as for the QHO, there’s a similar sort of effect. The groupoidification of the (single-variable) Heisenberg algebra uses a representation in Span(Gpd)Span(Gpd). Rather than A G(M)A_G(M) for manifolds MM, the important object is FinSet 0FinSet_0, and rather than A G(S)A_G(S) for cobordisms SS, the important morphisms are the groupoidified creation/annihilation operators. Call all that H 1H_1 so that D(H 1)=HD(H_1) = H, the Heisenberg algebra acting on Fock space. Now the analogy is that DD can still be seen as Λ\Lambda acting on Hom(1,1)Hom(1,1), and to “extend” the QHO means to consider a case where these groupoids and spans are not suspended between 11 and 11, but rather have some different “boundary” objects.

I put “boundary” in quotation marks because this is where the analogy is imprecise. In the case of a TQFT, taking nontrivial objects here means considering manifolds MM with nonempty boundary BB, so the object is the groupoid A G(B)A_G(B), and we have a span like 1A G(M)A G(B)1 \leftarrow A_G(M) \rightarrow A_G(B), or maybe some variation if we split B=B 1B 2B = B_1 \cup B_2. But this principle - moving from Hom(1,1)Hom(1,1) to some nontrivial hom-space - is the basis for stealing the word “extended” for this higher generalization of the QHO. I say that it’s misleading to think of extending “down to a point” because the groupoids in question don’t come from topological spaces as they do in the TQFT example. (Also, even in the analogous case, we’d only be extending to codimension 2).

The boundary object which we use in the setup that reproduces Khovanov’s monoidal category is FinSet 0FinSet_0: we go from Hom(1,1)Hom(1,1) to Hom(FinSet 0,FinSet 0)Hom(FinSet_0,FinSet_0). I don’t see any way to relate this to a topological space per se, but like the groupoid of connections, it classifies configurations for some system (namely one whose configurations are “sets of bosons”). Then there’s a way to look at Khovanov’s setup in terms of a factorization. First, there’s an injection I:HSpan 2(Gpd)I : H \rightarrow Span_2(Gpd), where HH is Khovanov’s diagram category (that’s the closest we get here to a category of cobordisms and the map AA: in each case we get the image in Span 2(Gpd)Span_2(Gpd) of a free monoidal category on some generators and relations). Then applying Λ:Span 2(Gpd)2Vect\Lambda : Span_2(Gpd) \rightarrow 2Vect we get the full “extended” QHO.

(This last step is needed to get the Karoubi envelope of the diagram category, which Khovanov uses, since the hom-categories in 2Vect2Vect are abelian.)

Now, the original groupoidified QHO lives inside this, if we restrict down to Hom(1,1)Hom(1,1) (which is always possible, since everything has a unique functor into 11, so any hom-category in Span 2(Gpd)Span_2(Gpd) has a projection to Hom(1,1)Hom(1,1)). Now, what Khovanov’s picture categorifies (hence the Span 2(Gpd)Span_2(Gpd) picture) is the multi-variable Heisenberg algebra, whereas the groupoidified QHO (the projection on Hom(1,1)Hom(1,1)) recovers only the one-variable form). There’s some slightly complicated story here, which is not quite clear to me, but certainly picking a “boundary object” in FinSet 0FinSet_0 (analogous to picking a connection on the boundary in the TQFT case) is closely related to the process of choosing a particular one of the raising/lowering operators a na_n in the multivariable Heisenberg algebra. So the multivariable form is an “extended” version of the single-variable form, in some sense analogous to “extended” TQFT.

Posted by: Jeffrey Morton on February 19, 2012 7:35 PM | Permalink | Reply to this

Re: Good News

Hi Jeffrey,

thanks for the reply. Now I believe I see where our misunderstanding is. It seems to be related to this sentence here:

I say that it’s misleading to think of extending “down to a point” because the groupoids in question don’t come from topological spaces as they do in the TQFT example.

Unless, of course, I am mistaken, but my impression is that here you are conflating “being an extension to a point” with “being a sigma-model”.

Because, the fact that A G(M)A_G(M) “comes from a space” means that there is a target space BG\mathbf{B}G (a discretly smooth groupoid) which represents the configuration spaces of fields (is the universal moduli stack for field configurations) in that

A G(X) SmoothGrpd(X,BG disc)Grpd(Π(X),BG) \begin{aligned} A_G(X) &\simeq Smooth \infty Grpd(X, \mathbf{B}G_{disc}) \simeq \infty Grpd(\Pi(X), B G) \end{aligned}

for all XX.

But “extension to the point” can be considered independently of whether or not the fields have such a target space / moduli stack description.

Namely, under an “extension of the categorified quantum harmonic oscillator to the point”, I would understand any bicategory 𝒞\mathcal{C} with all duals, such that its looping Ω𝒞\Omega \mathcal{C} on the tensor unit object is the category underlying the categoried quantum harmonic operator.

Once we have that, it would be a second step to ask in which ways this may “come from spaces”, namely to ask for symmetric monoidal 2-functors

Bord 2𝒞. Bord_2 \to \mathcal{C} \,.

Do you see what I mean?

From what you say here, my impression is that the extension to the point in this sense you have indeed considered:

Now the analogy is that DD can still be seen as Λ\Lambda acting on Hom(1,1)Hom(1,1), and to “extend” the QHO means to consider a case where these groupoids and spans are not suspended between 1 and 1, but rather have some different “boundary” objects.

You continue with:

[this] is the basis for stealing the word “extended” for this higher generalization of the QHO.

but, by what I said above, I don’t think this is “stealing”. Instead, it’s what I think this should be called!

Posted by: Urs Schreiber on February 20, 2012 10:09 AM | Permalink | Reply to this

Re: Good News

Hi Urs:

Okay, yes I agree, we had a mismatch of terminology. I was (and still am) unhappy with “extension to a point”, but your usage is clear enough now. The main reason I don’t like it is because to me it suggests that the groupoids of fields come from a sigma-model, in which case it’s fairly clear to me how to determine which groupoids should appear at the next stage when we raise the codimension of the manifold. I’m (somewhat) less wary about using “extended” in the case of the QHO, but even there, it’s less clear exactly how we “delooped”, if you like. We reused the groupoid FinSet 0FinSet_0 for various reasons, but none of them clearly correspond to what we do in the sigma-model when we increase the codimension. I just find that using “extension to a point” suggests something more systematic than I know how to do in this case.

In particular, when we first started looking at the QHO in Span 2(Gpd)Span_2(Gpd), before we discovered the correspondence with Khovanov, I didn’t think of it as a “delooping”, to get a new level of objects, but rather in terms of filling in 2-morphisms between the spans given in the groupoidification. (This is a slightly slippery thing about Span 2(Gpd)Span_2(Gpd), namely that Span 1(Gpd)Span_1(Gpd) is both a homotopy category, and a hom-category, of it. Our choice of how to “extend” the QHO implicitly used this fact.)

Anyway, the most important generating 2-morphisms we introduce are just the units and counits which make it precise that the creation and annihilation operators are “adjoints” (as they are in Hilbert space in the other sense of “adjoint”). So it still seems as if this point of view is relevant. These are the cups and caps of Khovanov’s diagrams.

On the other hand, when we look at Hom(1,1)Hom(1,1), or as you say, loop on the monoidal unit, then we do get the groupoidification of the harmonic oscillator by taking those spans whose middle object is FinSet 0FinSet_0, and composites of particular 2-morphisms. (Namely, one of the units and its dual, the counit from the other adjunction). Specifically, this gives the creation operator for the groupoidified QHO an interpretation as a “process movie” which does something like the following: “take [create-nn-times-then-annihilate-nn-times] to [create-(n+1)(n+1)-times-then-annihilate-(n+1)(n+1)-times], using a unit”. The annihilation operator does the reverse.

(This gets back to the slippery thing about Span 2(Gpd)Span_2(Gpd). Take a functor ff, and turn it into a span as usual, by letting the other leg of the span be the identity. This is adjoint to f f^{\dagger}, where the orientation is reversed. It so happens that the unit of one side of the adjunction just looks exactly like the span ff, and the counit of the other side looks exactly like f f^{\dagger}, if you ignore the source and target objects and look only at the span in the middle of the 2-morphism. So the same span appears wearing a “1-morphism hat” and a “2-morphism hat”. The way we get the groupodified QHO back out depends entirely on this ambiguity, if you start the way we actually did. Maybe there’s a different way to the same answer, and a natural reason to get FinSet 0FinSet_0 as the groupoid for “the point” in the sense you were using the term.)

But there are other 1-morphisms in Hom(1,1)Hom(1,1), which degroupoidify to give different Hilbert spaces, not the usual Fock space. They are groupoids of various possible classes of Feynman diagram. The 2-morphisms between them also come from the units and counits, which are “creation” and “annihilation” 2-morphisms in the sense that they take one class of Feynman diagrams, and either insert or delete a “creation/annihilation pair”.

So one thing I would do to try to understand the physical significance of this “extended QHO” (if any) is to answer the qusteion “what do these Hilbert spaces and maps represent?”, keeping in mind that there’s a special case which is supposed to be just the usual QHO. (I can’t do this at the moment, but it strikes me as probably pretty straightforward, since one can churn out completely explicit descriptions of them quite mechanically.) Whatever they are, they’re particular sectors of the “extended QHO”.

Posted by: Jeffrey Morton on February 20, 2012 5:07 PM | Permalink | Reply to this

Re: Good News

What is the relation between the two categorified quantum harmonic oscillators discussed here
and the Fock functors in the humongous book Monoidal Functors, Species and Hopf Algebras by Aguiar and Mahajan?

Posted by: Maarten Bergvelt on February 15, 2012 7:30 PM | Permalink | Reply to this

Re: Good News

Aguiar and Mahajan are considering Joyal’s species and various generalizations of those. I consider species, or ‘structure types’, to be version 1.0 of ‘stuff types’. A stuff type is a groupoid over the groupoid of finite sets:

F:XFinSet 0F : X \to FinSet_0

It’s a structure type if the functor FF forgets only structure, i.e. it’s faithful.

You can degroupoidify a stuff type and get a vector in Fock space, or more precisely a formal power series in one variable.

Similarly, a stuff operator, or span of groupoids with FinSet 0FinSet_0 appearing at both feet of the span, degroupoidifies to give an operator on Fock space. The annihilation and creation operators are examples.

Morton and Vicary extend this framework by working with a bicategory consisting of:

• groupoids

• spans of groupoids

• (equivalence classes of) spans of spans of groupoids

The annihilation and creation operators are morphisms in here. Suitably ‘linearizing’ this setup gives, among other things, Khovanov’s categorified Heisenberg algebra.

Posted by: John Baez on February 16, 2012 4:06 AM | Permalink | Reply to this

Re: Good News

Thanks a lot, John.

I am confused (of course!). Species are just a special type of stuff
types, right, so I was expecting that the Aguiar and Mahajan
construction would be a special case of the degroupoidification
construction.

But it seems that they get, by applying a Fock functor, from the
choice of a species not a vector in Fock space, but a specific,
interesting, Hopf algebra, like the Hopf algebra of symmetric
functions, or the symmetric algebra of a vector space. So they are
doing something different from what you guys have been doing.

So I guess I am asking whether there is a stuff type analog of the
Aguiar and Mahajan Fock functors.

Hope this makes sense.

Posted by: Maarten Bergvelt on February 16, 2012 5:49 PM | Permalink | Reply to this

Re: Good News

I need to think about this a bit more, sorry… I started thinking about it and got distracted! I don’t think what they’re doing is so very different from what we’re doing, but I forget the relation and need to look at their book to remember it.

Posted by: John Baez on February 28, 2012 5:17 AM | Permalink | Reply to this

Re: Good News

Well, I think their setup is more general: they seem to get creation and annihilation operators for not just the quantum harmonic operator, but for also other “Fock spaces”, like the algebra of symmetric functions. I have never seen quantum mechanical system associated to symmetric functions, so that sounds interesting.

Also they emphasize that the quantum harmonic oscillator Fock space is not just a Hilbert space, but in fact a Hopf algebra.The Hopf algebra structure is not made explicit in the groupoidification approach, but it is hidden in the normal ordering. You can define a State Field correspondence, that associates to a state f(z)f(z) the field Y(f)=:f(a+a *):Y(f)=:f(a+a^*):. The action of Y(f)Y(f) has a nice expression using the coproduct:

Y(f)g(z)=fgB(f,g),Y(f)g(z)= f'g' B(f'', g''),

where the coproduct of ff is fff'\otimes f'' and B is the inner product. In the groupoidification scheme the inner product is categorified, but I haven’t seen the coproduct with you guys. Presumably it is easy to define?

It would be interesting to see how the whole span of groupoids yoga applies to the more general Fock spaces from Aguiar and Majahan.

Posted by: Maarten Bergvelt on February 29, 2012 12:49 AM | Permalink | Reply to this

Re: Good News

Hi Maarten. I think that bosonic Fock space is just the algebra of symmetric functions!

In the linear algebraic world, the most general way I know about of getting Fock spaces is by applying the free commutative monoid monad to an object in a symmetric monoidal category. This includes things like antisymmetric Fock space. But if you know something that doesn’t fit this mould, that would be interesting.

All these “linear algebra” Fock spaces come equipped with a Hopf algebra structure. In the groupoidified case, however, it’s not clear that it survives - although you do certainly at least get a bialgebra.

Posted by: Jamie Vicary on March 30, 2012 2:38 PM | Permalink | Reply to this

Heisenberg Lie n-algebra

Here is an observation, concerning categorified Heisenberg algebras and their relation to physics.

For (X,ω)(X, \omega) a symplectic vector space, the ordinary Heisenberg Lie algebra is the sub-Lie algebra of the Poisson bracket Lie algebra on the constant and the linear functions

𝔥𝔢𝔦𝔰(X,ω)𝔭𝔬𝔦𝔰𝔰𝔬𝔫(X,ω). \mathfrak{heis}(X, \omega) \hookrightarrow \mathfrak{poisson}(X, \omega) \,.

Now, a higher analog of the notion of Poisson bracket Lie algebra to one of Poisson bracket Lie n-algebra for all nn has been given by Chris Rogers.

In particular, let (X,ω)(X, \omega) be an n-plectic vector space, equivalently regarded as an n-plectic manifold with constant nn-plectic form. Then it makes sense to ask whether an element in the Poisson bracket Lie nn-algebra 𝔭𝔬𝔦𝔰𝔰𝔬𝔫(X,ω)\mathfrak{poisson}(X, \omega) is constant or linear with respect to translation along XX.

One easily checks that the sub-complex on the constant and linear elements is a sub-Lie nn-algebra

𝔥𝔢𝔦𝔰(X,ω)𝔭𝔬𝔦𝔰𝔰𝔬𝔫(X,ω). \mathfrak{heis}(X, \omega) \hookrightarrow \mathfrak{poisson}(X, \omega) \,.

It seems natural to call this the Heisenberg Lie nn-algebra of (X,ω)(X, \omega).

Its relation to nn-dimensional quantum field theory is fairly clear: it does provide something like a localization of the Heisenberg Lie algebra of a QFT “to the point”.

So this gives a higher analog of Heisenberg Lie algebras. And it does so as just a small part of some much bigger structure, that of Poisson bracket Lie nn-algebras, which Chris has provided a fair bit of good theory and examples for, and which we have recently seen to have an origin in a very general abstract construction in higher topos theory.

It is of course quite different in nature to the categorified Heisenberg algebra of Khovanov (or at least so it seems): the Rogers-Heisenberg Lie nn-algebra reproduces the ordinary Heisenberg Lie algebra not by decategorification in the sense of passing to K-classes, but by transgression to loop space (up to technical details).

Just an observation. Just happened to be thinking about this during off hours lately.

Posted by: Urs Schreiber on March 5, 2012 4:24 PM | Permalink | Reply to this

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