## October 30, 2007

### Higher Clifford Algebras

#### Posted by John Baez

Lately Urs has been dreaming of categorified Clifford algebras. But he’s not the only one! We should send one of our spies to this talk tomorrow:

• Chris Douglas, Higher Clifford algebras, Topology Seminar, Chicago University, talk in E203 at 4:30 pm, pre-talk in the same room at 3:00, October 30, 2007.

Here’s the abstract:

Real K-Theory is 8-periodic. This periodicity can be seen algebraically from the periodicity of Clifford algebras: Clifford algebras form a 2-category, and in that 2-category, the generator Cl(1) has order 8. The analogous algebraic objects for elliptic cohomology might be called “higher Clifford algebras” and ought to form a 3-category. We introduce a candidate such 3-category whose objects are invertible conformal nets. We show that the generating net, the net of free fermions, will have order at least 24. This is joint work with Arthur Bartels and André Henriques.

Life is getting interesting.

For more on the 8-hour Clifford clock, the 2-category of Clifford algebras, and the super-Brauer group, try these:

Posted at October 30, 2007 12:30 AM UTC

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### Re: Higher Clifford Algebras

higher Clifford algebras […] ought to [be] invertible conformal nets.

Sounds good. That’s probably what WIAEO? suggests.

$spin(n)$ acts on Clifford algebra, $\mathrm{string}(n)$ via $\hat \Omega_k \mathrm{spin}(n)$ on fermions on the circle.

Dirac-Ramond theory also suggests that 2-Clifford is Clifford algebra on the circle.

What I think we’d eventually want is a way to say: under the principle of least resistance under categorification Dirac operators are this-or-that.

Namely “quantized” covariant derivatives. With “quantized” meaning: send Grassmann to Clifford.

We already know that covariant derivatives are, under the principle-of-least-resistance (namely morphisms into the curvature). So it remains to understand what the notion of “quantization” here is.

Posted by: Urs Schreiber on October 30, 2007 1:22 AM | Permalink | Reply to this

### Re: Higher Clifford Algebras

Mike Shulman said he’d spy on this talk and report back to us.

Posted by: John Baez on October 30, 2007 5:03 AM | Permalink | Reply to this

### Re: Higher Clifford Algebras

As usual at Chicago, Chris first gave a pre-talk’ with background material before the main talk. The pretalk was about Clifford algebras, which you all know about (or can read about in TWF). To summarize: we have a sequence of Clifford algebras $Cl_n$ which are generated by $n$ anticommuting square roots of $\pm 1$. The sequence is periodic up to Morita equivalence; $Cl_8$ is $\mathbb{R}(16)$, the algebra of $16\times 16$ real matrices, which is Morita equivalent to $\mathbb{R}$, and from then on it repeats every 8 with extra matrix dimensions thrown in.

By the way, Chris remarked on something which I’ve never thought about before: it’s also true that $Cl_6$ is $\mathbb{R}(8)$, so why don’t we get a period of 6 instead of 8? The answer is that the Clifford algebras are really best thought of as $\mathbb{Z}/2$-graded algebras, and $Cl_6$ is not Morita equivalent to $\mathbb{R}$ as a graded algebra.

All of this has relevance to K-theory, because it turns out that $K^n(X)$ can be represented geometrically by bundles of Clifford modules’ over $X$. Let’s start with $K^0$; we know that elements of $K^0(X)$ are formal differences’ $V-W$ of vector bundles over $X$. We can model the formal difference $V-W$ with an honest geometric object by using the $\mathbb{Z}/2$-graded vector bundle $V\oplus W$, where $V$ is even and $W$ is odd. Such a thing should represent the zero class in K-theory just when $V$ and $W$ are isomorphic; this can be rephrased as saying that there exists an odd operator $e$ on $V\oplus W$ (hence, taking $V$ to $W$ and vice versa) such that $e^2=1$. But this just says that $V\oplus W$ has an action of the first Clifford algebra $Cl_1$.

More generally, Karoubi proved that for any $n$, $K^{-n}(X)$ can be represented by $Cl_n$-module bundles on $X$ modulo those such that the $Cl_n$-action extends to a $Cl_{n+1}$-action. When $n=0$ this is what we had above, since a $Cl_0$-module is just a vector space. This allows us to deduce Bott periodicity for K-groups from the algebraic periodicity (up to Morita equivalence) of Clifford algebras.

K-theory tells us about bundles of C-modules for a Clifford algebra C, so it cares about the category C-mod of C-modules. A vector bundle can be thought of as a functor from the category of paths in X to Vect, and similarly a C-module bundle is a functor to C-mod.

Now, Clifford algebras live naturally as the 0-cells of a symmetric monoidal bicategory CL2, whose 1-cells are bimodules and whose 2-cells are bimodule maps. Internal equivalence in this bicategory gives us the notion of Morita equivalence. We find the category C-mod inside CL2 as the hom-category from C to the unit 1. Moreover, the endomorphism category CL2(1,1) of the unit object is just the category Vect that we originally thought $K^0$ was telling us about.

What we want to do is find a higher’ version of all this that applies to elliptic cohomology. We can no longer get away with using finite-dimensional things, so we have to replace our ordinary vector spaces with Hilbert spaces and our Clifford algebras with von Neumann algebras (algebras that embed into operators on a Hilbert space and are complete in an induced topology).

We also move up one level, so that instead of ordinary bundles, we’re talking about functors defined on the 2-category of points, paths, and surfaces in X. Instead of Vect, these 2-functors should land in the 2-category of von Neumann algebras, bimodules, and maps. Thus, this looks like a 2-dimensional QFT, rather than a 1-dimensional one. Stolz and Teichner had the idea that this should tell us something about $Ell^0(X)$, the 0-degree part of the elliptic cohomology of $X$.

The goal is now to get information about $Ell^n(X)$ for higher $n$. By analogy, we hope to find some object $?_n$ like $Cl_n$, so that elements of $Ell^n(X)$ will be represented by functors into $?_n$-mod. We hope to find some sort of 3-category $??3$ such that $??3(?_n,1)$ is the 2-category $?_n$-mod and $??3(1,1)$ is the 2-category of von Neumann algebras, bimodules, and maps that we started with for $Ell^0$.

The solution proposed by Chris, Arthur Bartels, and Andre Henriques (DBH), is a 3-categorical structure whose objects are conformal nets. A conformal net is a cosheaf of von Neumann algebras on the category of intervals’. The idea is that you have a von Neumann algebra for each subinterval of an interval, with inclusions for inclusions of intervals, and gluing conditions for unions of subintervals. Chris told us that conformal nets have been around for a while in other contexts, with slightly different definitions. There is a conformal net $Ff$ called the net of free fermions which plays the role of $Cl_1$, and the three of them have proven that its order is divisible by 24 (this would be the periodicity, analogous to the 8-periodicity of Clifford algebras). They don’t know what the order actually is yet, but they’re betting on 24 or 48.

The thing I haven’t said much about is the type of 3-categorical structure’ they’re using. It’s not a 3-category or tricategory, but something more like a 3-dimensional version of a double category (or a framed bicategory), provided you interpret that correctly. A double category can be defined to be an internal category object in the 2-category Cat; what they’re looking at is an internal bicategory object’ in the 2-category of symmetric monoidal categories. This notion is something I thought about myself a while ago as a natural generalization of framed bicategories, but never developed much, so it makes me happy to see that the same idea has occurred to other people independently.

It’s easy to define an internal strict $n$-category object in any category with pullbacks: you have objects $C_k$ of $k$-cells for all $0\le k\le n$, source and target maps $C_k\to C_{k-1}$, compositions $C_k\times_{C_\ell} C_k\to C_k$, and so on, satisfying the obvious associativity, unit, and interchange axioms. Given this, it’s not too hard to see what you mean by a pseudo’ such object (which DBH call a coherent $n$-category object’) in a 2-category with pullbacks: you weaken the associativity, unit, and interchange axioms to be up to coherent isomorphism in your 2-category. For instance, a pseudo 1-category object in the 2-category of categories is just a pseudo double category. A pseudo 1-category object in the 2-category of symmetric monoidal categories is a symmetric monoidal pseudo double category, and so on. What I’m calling pseudo $n$-category objects in Cat’ were actually defined by Michael Batanin, who called them monoidal $n$-globular categories’.

The really neat thing, from my point of view, is that they also impose fibrational conditions on the source and target maps, just like in my definition of framed bicategory. As in the case of monoidal framed bicategories, which I mentioned in the appendix to my paper, here fibrational conditions imply that if you look only at the objects of $C_0$, the objects of $C_1$, and the objects and morphisms of $C_2$, you get an ordinary tricategory—and often with a lot less work than would be involved in checking the definition by hand, since here the only coherence involved is 2-categorical’.

I’ve thought some about this myself, including the example (related to one of theirs) of rings, algebras, and bimodules, but didn’t get around to doing much with it. I also didn’t come up with a good name for these things. The best I’ve thought of so far is 3-level category’. This fits the way I tend to think about them, generalizes nicely to $n$-level category’, and, like $n$-fold category’, suggests something $n$-dimensional which isn’t quite the same as an $n$-category.

Actually, DBH end up weakening this notion further, although it’s not clear whether this extra weakness is essential. You can actually define an internal bicategory object in any category with pullbacks, by equipping yourself with maps such as $C_1\times_{C_0} C_1\times_{C_0} C_1 \to C_2$ that picks out an associator for each composable triple. And you can then pseudoify’ that in a 2-category, in an appropriate way. DBH call this a coherent bicategory object’. If what you pseudoify is the notion of bicategory with strict associativity’, they call it a coherent semi-strict bicategory object’. Fibrational conditions on a coherent 2-category object should allow you to lift’ the coherence to make it a coherent bicategory object as well.

Chris has given me permission to link to his precise definitions of coherent semi-strict bicategory object and coherent 2-category object.

Mike

Posted by: Mike on November 1, 2007 9:55 PM | Permalink | Reply to this

### Re: Higher Clifford Algebras

A vector bundle can be thought of as a functor from the category of paths in $X$ to $\mathrm{Vect}$, and similarly a $C$-module bundle is a functor to $C-\mathrm{mod}$.

All right. So now we are even talking about differential K-theory. Since, unless we restrict to constant paths, these functors yield vector bundles equipped with a connection.

Now, Clifford algebras live naturally as the 0-cells of a symmetric monoidal bicategory CL2, whose 1-cells are bimodules and whose 2-cells are bimodule maps. Internal equivalence in this bicategory gives us the notion of Morita equivalence. We find the category C-mod inside CL2 as the hom-category from C to the unit 1. Moreover, the endomorphism category CL2(1,1) of the unit object is just the category Vect that we originally thought $K^0$ was telling us about.

Okay, so let’s put this together with the previous statement. We find that instead of thinking about our vector bundle directly as a functor, we should rather be thinking about it as the component map of a pseudonatural transformation.

For let $1 : P_2(X) \to CL2$ be the tensor unit in the 2-category of 2-functors from 2-paths to CL2, i.e. the one that sends each point to $\mathrm{Cl}_0 \simeq \mathbb{R}$ and all paths to identities. Moreover, let $C : P_2(X) \to CL2$ be the 2-functor that sends everything to the identity on the Clifford algabra $C$. Then a $C$-module bundle (with connection) is a (pseudonatural) transformation $V : 1 \to C \,.$

When we replace $C$ by something less trivial, we obtain twisted K-theory.

Posted by: Urs Schreiber on November 2, 2007 4:14 PM | Permalink | Reply to this

### Re: Higher Clifford Algebras

Viewing C-module bundles as natural transformations $V:1 \rightarrow C$ is exactly right, and is how Stolz and Teichner originally formulated this story, and is our preferred way to think about it. In particular, the idea that a twisting of a theory (eg K-theory) and the degree grading have exactly the same status is a nice consequence of this viewpoint, as you say. One of the key tests that is going to tell us that we have the right notion of Higher Clifford Algebras is when we can recognize $bgl_1 tmf$ (the classifying space for twistings of tmf) from our 3-category.

Posted by: Chris Douglas on November 5, 2007 5:52 PM | Permalink | Reply to this

### Re: Higher Clifford Algebras

is how Stolz and Teichner originally formulated this story

Hm. Last time I talked to them, they were thinking of this in slightly different terms. Unless my memory is failing me, I then pointed out in our elliptic seminar that this is equivalent to having a morphism into the twisting object.

Anyway, it’s not so important who had wich idea first (except that for me it is a bad issue that I am lagging behind so much with writeups).

Anyway, I am pretty fond of this idea of realizing (twisted) $n$-bundles (with connection) as morphism into $(n+1)$-bundles.

I gave a talk on that at the Fields institue last winter, and have been developing it a little further since. One way to look at it on the classical side is as a vast generalization (nonabelian and higher $n$) of Stokes’ theorem, really. If you think about it. It’s that shift in dimenson which one encounters every once in a while.

And then the quantum analog of this statement: $n$-dimensional QFTs as morphisms into $(n+1)$-dimensional ones.

This is what physicsist’s call the holographic principle.

And when thinking in terms of $n$-transport one finds that this is really the holographic principle of $n$-category theory at work.

This principle simply says:

the transformation of two $n$-functors is itself, in components, an $(n-1)$-functor.

Once we equip all our $n$-functors here with extra structure that makes them what I call transport $n$-functors (smoothness, local trivializability), this obvious statement becomes a powerful generating mechanism for interesting phenomena.

There are lots of fun variations on this theme. Like

The covariant derivative of an $n$-connection $\mathrm{tra}$ is a morphism into the corresponding curvature $(n+1)$-transport.

(I think this particular point should eventually be most relevant to what you are considering here: Dirac operators are “quantized” covariant derivatives (Grassmann to Clifford). We know by the above what $n$-covariant derivatives are. So what weneed to figure out is what the $n$-“symbol map” does to them. I am pretty sure the answer involves the 2-Clifford algebras you found, but I need to understand this better.)

Sections, States, Twists and Holography, recalling there some of the aspects that made it into my Toronto notes.

What I am eventually trying to finish is the thing sketched in Towards 2-functorial CFT:

Classical WZW 2-transport is a morphism into CS 3-transport (“trivializing” in a generalized way the Chern-Simons 3-transport which obstructs the lift of a $G$-transport to a $\mathrm{String}(G)$-transport).

After quantization this becomes the statement that 2dWZW CFT 2-transport is a morphism into 3dCS TFT 3-transport.

I am claiming that this last statement is secretly the mechanism behind the FRS construction.

Only problem that I keep making these claims while still not producing the corresponding writeups. There is too much to do.

Anyway, it is good to see you emphasize the importance of morphisms into twisting objects in particular and to see your highly exciting work on Clifford 2-algebras in general. And thanks to Mike Shulman for reporting on it!

Posted by: Urs Schreiber on November 5, 2007 6:33 PM | Permalink | Reply to this

### Re: Higher Clifford Algebras

Only problem that I keep making these claims while still not producing the corresponding writeups.

Maybe at least some of these things can now finally be written up. I arrived in Aarhus last week, and I’m almost done taking care of all the practical stuff associated with relocating. So, I predict that I will start working on this again during the second half of this week, eagerly looking forward to actually doing some research again…

Posted by: Jens on November 5, 2007 7:42 PM | Permalink | Reply to this

### Re: Higher Clifford Algebras

Hi Jens!

Good to hear from you, indeed.

I arrived in Aarhus last week

Ah, that’s good. I was wondering if you are already there.

I am about to leave to Trondheim for a week, where I’ll visit Nils Baas together with Konrad.

There are $x+1$ things I need to do when I get back, but let’s keep an eye on our stuff. We should proceed by writing out the big story further, along the pictures already produced, and then iteratively filling in the details.

Posted by: Urs Schreiber on November 5, 2007 8:05 PM | Permalink | Reply to this

### Re: Higher Clifford Algebras

By the way, in the unlikely event that anyone happens to consider following any of the links I was providing: don’t bother until in a couple of hours. The server is down which hosts the pictures and docs to be found there.

Posted by: Urs Schreiber on November 5, 2007 10:01 PM | Permalink | Reply to this

### Karoubi K-theory

I turned the part on Karoubi’s Clifford-algebraic description of K-theory into an $n$Lab entry:

[[Karoubi K-theory]]

I am wondering if anyone can provide me with more or maybe just other references than collected there so far. I’d be grateful for any pointers.

Posted by: Urs Schreiber on July 14, 2009 9:01 PM | Permalink | Reply to this
Read the post 2-Vectors in Trondheim
Weblog: The n-Category Café
Excerpt: On line 2-bundles.
Tracked: November 5, 2007 9:55 PM

### Re: Higher Clifford Algebras

Clifford modules are those super Lie modules for the super Lie algebra $\mathbb{R}^{1|q}$ on which $\mathbb{R}^{1|0}$ acts as a multiple of the identity.

Another way to say this is that the Clifford algebra $Cl_q$ is obtained from the universal enveloping algebra of the super Lie algebra $\mathbb{R}^{1|q}$ by identifying the generator of $\mathbb{R}^{1|0}$ with the identity in the algebra.

Here $\mathbb{R}^{1|q}$ is the super Lie algebra spanned by a single even element $I$ and $q$ odd elements $\{\theta_i\}$ with the only non-trivial super Lie bracket being

$[\theta_i,\theta_j] = \delta_{ij} \cdot I \,.$

So possibly it would be fruitful to look for super Lie 2-algebra extensions of $\mathbb{R}^{1|q}$.

Posted by: Urs Schreiber on May 30, 2008 6:43 PM | Permalink | Reply to this

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