## August 12, 2005

### K-theory for dummies, I

#### Posted by urs I feel like talking about K-theory, K-homology, and the like. Here I start leisurely with a bird’s eye overview of some central ideas.

So this are the main concepts, informally:

Pick some topological space $X$. Consider all complex vector bundles over X. Including their homomorphisms these form a category $\mathrm{Vect}\left(X\right)$.

There is sort of a semi-ring structure on that category in that vector bundles can be added, using the direct sum, and multiplied, using the direct product. When we take equivalence classes in $\mathrm{Vect}\left(X\right)$ the result is in fact an ordinary semi-ring. There is a standard way, called the Grothendieck group completion, to throw in additive inverses such as to obtain an honest ring. This ring is denoted ${K}_{0}\left(X\right)$ and is called the K-theory of $X$.

Instead of $X$, one can consider the suspension $\mathrm{SX}$ of $X$. This is sort of a ‘sphere’ which at every latitude looks like $X$, if you wish. More precisely, it is the space obtained by taking the product of $X$ with the unit interval and identifying all points of $X$ attached to $0\in \left[0,1\right]$ and all those attached to $1\in \left[0,1\right]$:

(1)$SX:=\frac{X×\left[0,1\right]}{\left(X×\left\{0\right\}\right)\cup \left(X×\left\{1\right\}\right)}$

The $n$-th iteration of taking the suspension is written ${S}^{n}X$ and one defines

(2)${K}_{n}\left(X\right):={K}_{0}\left({S}^{n}X\right)\phantom{\rule{thinmathspace}{0ex}}.$

Bott periodicity says that ${K}_{n}\left(X\right)$ is isomorphic to ${K}_{n+2}\left(X\right)$:

(3)${K}_{n}\left(X\right)\simeq {K}_{n+2}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$

(In the case of K-theory for real vector bundles we’d have periodicity 8 instead of 2).

Since K-theory is something algebraic, it is natural to do away with topological spaces and replace them by their algebraic equivalent, ${C}^{*}$-algebras.

Given the ${C}^{*}$-algebra $A=C\left(X\right)=\left\{f:X\to ℂ\right\}$ of continuous complex-valued functions on $X$, a vector bundle over $X$ can equivalently be characterized as a projection in ${M}_{n}\left(A\right)$, i.e. as an $n×n$-matrix $P$ with values in $A$ that satisfies $P={P}^{*}={P}^{2}$.

So an alternative way to define ${K}_{0}$, but now generalized to arbitrary ${C}^{*}$-algebras $A$, is as the abelian group ${K}_{0}\left(A\right)$ which has one generator $\left[p\right]$ for every projector in ${M}_{n}\left(A\right)$ for all $n$ subject to the identification of projections which can be continuously connected and to the relation

(4)$\left[p\right]+\left[q\right]=\left[p\oplus q\right]\phantom{\rule{thinmathspace}{0ex}}.$

This formulation makes it easy to define sort of a dual K-theory.

For every ${C}^{*}$-algebra $A$ there is are dual ${C}^{*}$-algebras ${D}_{\rho }\left(A\right)$ defined as the commutant-up-to-compact-operators of any rep of $A$. So given $A$, choose any rep $\rho :A\to B\left(H\right)$ of $A$ in terms of bounded operators on some seperable Hilbert space $H$. Then

(5)${D}_{\rho }\left(A\right):=\left\{T\in B\left(H\right):\left[T,\rho \left(a\right)\right]\sim 0\phantom{\rule{thinmathspace}{0ex}},\forall a\in A\right\}\phantom{\rule{thinmathspace}{0ex}},$

where ${T}_{1}\sim {T}_{2}$ means that ${T}_{1}-{T}_{2}\in K\left(H\right)$ is a compact operator.

It may happen that $A$ didn’t have a unit. Let the result of making it unital by adjoining a unit be denoted by $\stackrel{˜}{A}$.

The K-theory of $D\left(\stackrel{˜}{A}\right):={D}_{\rho }\left(\stackrel{˜}{A}\right)$ (for any $\rho$) is now called K-homology. One writes

(6)${K}^{p}\left(A\right):={K}_{1-p}\left(D\left(\stackrel{˜}{A}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

The position of the indices here denotes the covariant or contravariant nature of these maps, when these are regarded as functors from the category of ${C}^{*}$-algebras to that of abelian groups.

The reason for calling the map $A↦{K}^{p}\left(A\right)$ a homology is that in the case where $A=C\left(X\right)$ is the ${C}^{*}$-algebra of functions on $X$, this map can be shown to define what is called a generalized homology theory.

This is any theory which associates a list of groups to any topological space such that a couple of crucial properties copied from ordinary (simplicial, say) homology, are satisfied.

So one might wonder what the cohomology theory corresponding to ${K}^{p}$ might be. Turns out that this is nothing but the K-theory that we started with (which is another way to understand the position of the indices):

(7)${K}^{p}\left(A\right)\simeq \mathrm{Hom}\left({K}_{p}\left(A\right),ℤ\right)\phantom{\rule{thinmathspace}{0ex}}.$

But at the level of the above definitions this duality is neither straightforward to see nor really nice. It becomes much neater once a cool result is established which says that ${K}^{p}\left(C\left(X\right)\right)$ is nothing but equivalence classes of generalized Dirac operators on rank-$p$ vector bundles over $X$, loosely speaking.

More precisely, this works as follows:

A Fredholm operator is a bounded operator $F$ which admits a notion of index:

(8)$\mathrm{Index}\left(F\right):=\mathrm{Dim}\left(\mathrm{Kernel}\left(F\right)\right)-\mathrm{Dim}\left(\mathrm{Cokernel}\left(F\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

A Fredholm module is like a spectral triple with a Fredholm operator instead of a Dirac operator, namely its a triple $\left(H,\rho \left(A\right),F\right)$ of a Hilbert space $H$ on which $F$ and the ${C}^{*}$-algebra $A$ is represented by $\rho :A\to B\left(H\right)$ and such that $F$ satisfies three technical conditions.

If in addition a Clifford algebra ${C}_{p}$ is represented on $H$ which commutes with $F$ this is called a $p$-multigraded Fredholm module.

Now, Kasparov went ahead and defined the Kasparov K-homology group ${\mathrm{KK}}^{-p}\left(A\right)$ to be the abelian group of equivalence classes of Fredholm modules.

More precisely, ${\mathrm{KK}}^{-p}\left(A\right)$ is the abelian group generated by $p$-multigraded Fredholm modules $\left(H,\rho \left(A\right),F\right)$ up to unitary equivalence, up to modules which can be continuously connected, and up to the relation

(9)$\left[a\right]+\left[b\right]=\left[a\oplus b\right]\phantom{\rule{thinmathspace}{0ex}},$

where $a,b$ are Fredholm modules, $\left[a\right],\left[b\right]$ their images in ${\mathrm{KK}}^{-1}\left(X\right)$ and $\oplus$ denotes the rather obvious direct sum of such modules.

So, this definition looks rather similar in spirit to the one above. And indeed, it turns out that Kasparov’s groups are isomorphic to the ordinary K-homology groups like this:

(10)${\mathrm{KK}}^{-p}\left(A\right)\simeq {K}^{p}\left(A\right)\phantom{\rule{thinmathspace}{0ex}}.$

This is nice, among other things because it gives a nice interpretation for the pairing between K-homology and K-theory:

First of all, any Dirac operator on $X$ gives rise to a Fredholm module (that’s pretty much the motivation for the definition of Fredholm module, I guess). More generally, every symmetric elliptic first-order differential operator $D$ on some vector bundle over $X$ gives a Fredholm module and hence an element in ${\mathrm{KK}}^{-\cdot }\left(C\left(X\right)\right)$ and hence an element in K-homology ${K}^{\cdot }\left(C\left(X\right)\right)$.

Now suppose in addition an element in K-theory ${K}_{\cdot }\left(C\left(X\right)\right)$ itself is given, i.e. the class of a further vector bundle $V$ over $X$. We can in natural way get the operator ${D}_{V}$ obtained by tensoring with $V$ and then compute it’s index. This gives a map

(11)$\begin{array}{ccc}{K}^{p}\left(C\left(X\right)\right)×{K}_{p}\left(C\left(X\right)\right)& \to & ℤ\\ \left(\left[D\right],\left[V\right]\right)& ↦& \mathrm{Index}\left({D}_{V}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

So to coin a slogan summarizing this situation:

K-theory consists of classes of vector bundles, K-homology consists of classes of Dirac operators and under the index map Dirac operators are dual to vector spaces.

Before quitting I should admit that most of what I said here comes from the very nice textbook

N. Higson & J. Roe
Analytic K-Homology
Oxford Univ. Press (2000)

Posted at August 12, 2005 7:14 PM UTC

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### Re: K-theory for dummies, I

So, I went through this brief period in graduate school trying to find a use in string theory for E-theory. Perhaps now I can pass along this responsibility to you :).

Posted by: Aaron Bergman on August 13, 2005 7:41 AM | Permalink | Reply to this

### E-theory

Many thanks, I’ll certainly have a look at this!

(Hm, let’s see, now I know of the existence of E-theory, F-theory, K-theory, M-theory, Z-theory. Did I forget anything? Does ${\varphi }^{4}$-theory count? ;-)

Posted by: Urs on August 13, 2005 6:36 PM | Permalink | Reply to this

### Re: K-theory for dummies, I

BTW, Aaron (or anyone else), I was wondering: What if we do something like K-theory for vector bundles with connection? Shouldn’t that be related to branes with gauge field VEVs turned on? Or if not, why not?

Posted by: Urs on August 14, 2005 8:24 PM | Permalink | Reply to this

### Re: K-theory for dummies, I

What if we do something like K-theory for vector bundles with connection?

I have taklked to Ulrich Bunke and Mathai Varghese about this over lunch. Unless I have misunderstood something the answer was that I should look at ‘smooth K-theory’. But I need to check the references they pointed me to which was,

Bunke&Schlick: ‘Smooth K-theory’,

Bismut, Gillet& Soule

as well as Hopkins and Singer.

But I gotta run now…

Posted by: Urs on August 15, 2005 12:38 PM | Permalink | Reply to this

### Re: K-theory for dummies, I

Hi Urs,

I think Bunke & Schick’s “A Model for Smooth K-Theory” is still in preparation. Apparently, Bunke spoke about it at the Oberwolfach workshop of Jun 12-18, 2005.

However, Hopkins & Singer’s paper is available at arXiv:math.AT/0211216.

Bismut, Gillet & Soule seemed to have worked on the GRR formula in Arakelov theory, which appears to be very out of left field with regards to smooth K-theory. Is it another work in preparation?

Posted by: Rongmin on August 16, 2005 4:40 AM | Permalink | Reply to this

### Re: K-theory for dummies, I

Thanks for these hints!

Is it another work in preparation?

Maybe not, I’ll check.

Posted by: Urs on August 16, 2005 7:19 AM | Permalink | Reply to this

### Re: K-theory for dummies, I

Thanks for these hints!

No worries, Urs. While searching for Bunke & Schick I happened to find something interesting about those reports. ;) Anyway, do let me know about the paper by Bismut, Gillet & Soule if you have more info.

Cheers.

Posted by: Rongmin on August 17, 2005 10:05 AM | Permalink | Reply to this

### Re: K-theory for dummies, I

Thanks, Urs. I have a large pile of NCG lecture notes to go through. But I haven’t changed my mind about the categorification program!

Posted by: Kea on August 14, 2005 11:35 PM | Permalink | Reply to this

### Re: K-theory for dummies, I

But I haven’t changed my mind about the categorification program!

Wait, what precisely are you referring to, here?

Posted by: Urs on August 15, 2005 7:08 AM | Permalink | Reply to this

### Re: K-theory for dummies, I

Maybe the view that fundamental theories need also to include inbuilt quantum logic hence topos (or higher topos) theory. I don’t follow physicsforums or any like that, so I only know what I’ve been told.

Please correct me if I’m wrong.

Posted by: David Roberts on August 15, 2005 7:25 AM | Permalink | Reply to this

### Re: Logic

Hi David. You listened! Wow. Hope all is well back in SA.

Posted by: Kea on August 16, 2005 2:52 AM | Permalink | Reply to this

### Re: Logic

Ok. I still need to catch up on topos theory. Is there some systematic procedure (like internalization or so) which would allow me to move any construction from the ordinary topos Set to any other?

Like, does it make sense to ask what a bundle in a ‘quantum logic topos’ would look like, for instance, or anything similar?

Posted by: Urs on August 16, 2005 7:24 AM | Permalink | Reply to this

### Re: Logic

Hi Urs. I guess the easiest answer is that these aren’t really the right questions to ask. To begin with there is no one topos in which one can do quantum logic. The logic may be non-classical in a topos but it is always distributive. Hence the need to look at ‘higher’ structures. As for ‘transferring’ constructions, that would sort of defeat the purpose of thinking internally, which is to say one is living in some given logical universe from beginning to end (whatever that means). I highly recommend the papers of Lawvere.

Posted by: Kea on August 16, 2005 10:05 PM | Permalink | Reply to this

### Re: Logic

these aren’t really the right questions to ask.

[…]

I highly recommend the papers of Lawvere.

All right, thanks. I am probably still stuck in my native topos.

But in case you feel like talking about it a little more, maybe you could help me by for instance pointing me to your favorite example where some problem has been solved by switching to some exotic topos and doing something in there.

Posted by: Urs on August 16, 2005 10:43 PM | Permalink | Reply to this

### Re: Logic

Hi Kea,

let me make another attempt. You wrote:

As for ‘transferring’ constructions, that would sort of defeat the purpose of thinking internally, which is to say one is living in some given logical universe from beginning to end (whatever that means). I highly recommend the papers of Lawvere.

I haven’t had the chance to read Lawvere yet, but I did talk to somebody who knows a little about these issues.

This person seemed to be telling me that the idea is to first ‘externalize’ a given concept, by formulating it abstractly without any reference to $\mathrm{Set}$, and then internalizing the result into some topos other that $\mathrm{Set}$.

So for instance we could say that a bare monoid externally looks like an object $O$ together with an arrow

(1)$O×O\stackrel{m}{\to }O\phantom{\rule{thinmathspace}{0ex}}.$

(Plus maybe an associativity property, if you like.)

So, an ordinary monoid is a monoid object $O$ in $\mathrm{Set}$. Now given any other topos $T$, shouldn’t I be able to have a monoid object in $T$? Is this sort of process not what you have in mind?

Posted by: Urs on August 19, 2005 7:15 PM | Permalink | Reply to this

### Re: Logic

“So, an ordinary monoid is a monoid object O in Set. Now given any other topos T, shouldn’t I be able to have a monoid object in T? Is this sort of process not what you have in mind?”

Hi Urs

Yes, of course, one can talk about monoid
(or ‘something else’) objects in T. Perhaps we’re just talking semantics. It was the word ‘transferring’ that seemed wrong: as if you had started in Set rather than ‘externally’.

Posted by: Kea on August 20, 2005 4:48 AM | Permalink | Reply to this

### Re: Logic

Ok, thanks.

Now I am a bit puzzled by the following: suppose we formulated quantum mechanics externally. I guess that should be possible. Assume $Q$ is our definition of an externalized QM-object.

Now, what if you were to internalize $Q$ in some quantum logic topos $T$?

Posted by: Urs on August 20, 2005 7:05 AM | Permalink | Reply to this

### Re: Logic

“suppose we formulated quantum mechanics externally”

Ah! Well I guess this is where things get tricky. I’m not going to say what a ‘quantum topos’ is, but there is a topos analogue to this, outlined in detail in McLarty’s book. That is, the arrows of interest are thought of as ‘interpretations’ of propositions, so there are different levels of reality lurking even in the 1-categorical setting.

“Assume Q is our definition of an externalized QM-object”

So what would you like Q to be?

Posted by: Kea on August 21, 2005 12:13 AM | Permalink | Reply to this

### Re: Logic

So what would you like $Q$ to be?

An externalized Hilbert space $ℋ$ together with a choice of of externalized linear operator $ℋ\stackrel{H}{\to }ℋ$.

In order to do that, we’d first need to externalize the concept of ‘field’, I guess. Maybe as a warmup it would be sufficient to concentrate on rings, or even just semi-rings. These would be objects $R$ with arrows

(1)$R×R\stackrel{\oplus }{\to }R$
(2)$R×R\stackrel{\otimes }{\to }R$

satisfying a distributivity law. Then $ℋ$ should be a module of that, i.e. something like a left-$R$-space, hence an object $ℋ$ with an arrow

(3)$R×ℋ\stackrel{l}{\to }ℋ$

respecting $\otimes$ and $\oplus$ in the obvious way. The inner product would be an arrow

(4)$ℋ×ℋ\to R\phantom{\rule{thinmathspace}{0ex}}.$

A linear operator would be any arrow

(5)$ℋ\to ℋ$

which is compatible with the $R$-action.

(Here by ‘compatible’ I mean the existence of some obvious commuting diagrams.)

Posted by: Urs on August 22, 2005 1:39 PM | Permalink | Reply to this

### Re: Logic

“…we’d first need to externalize the concept of field, I guess…”

In other words we want some axioms for which Hilbert spaces are a nice model and with respect to which there is a special object R. Exactly! This is the sort of thing I’m working on.

Have you heard much about *-autonomous categories and linear logic and stuff like that? Quite a big branch of Comp Sci these days.

Posted by: Kea on August 22, 2005 11:55 PM | Permalink | Reply to this

### Re: Logic

Exactly! This is the sort of thing I’m working on.

Cool! I am glad we managed to arrive at that point of mutual understanding. In my 2-NCG-ramblings I was thinking about how to externalize QM and then internalize it in something like $\mathrm{Cat}$ (hoping to capture string physics this way), while you are apparently thinking about externalizing QM and then internalizing it in some quantum-topos.

Just for the record I’d note that it is in principle possible to combine both steps. Let $T$ be your favorite quantum topos. Then we can look at categories internalized in $T$. They live in the 2-category $T\mathrm{Cat}$ with $T$-internal functors being morphisms and $T$-internal natural transformations being 2-morphisms. Now let $Q$ be a ‘QM-object’ internalized in $T\mathrm{Cat}$ and you’d have QM ‘categorified’ both in my (John Baez’s, really) sense as well as in your sense.

Have you heard much about $*$-autonomous categories and linear logic and stuff like that?

No, unfortunately not. I you can point me to a readable introductory text that explains why these are relevant for what we are talking about, I’d be grateful.

Posted by: Urs on August 23, 2005 10:12 AM | Permalink | Reply to this

### Re: Logic

“…and you’d have QM categorified both in my (John Baez’s, really) sense as well as in your sense”

Yes, I guess so. Wonderful. The reason I prefer my method is that I don’t think String Theory is the right way to look at the physics.

“If you can point me to a readable introductory text…”

Unfortunately no. The definition of such categories is easy enough: you can look it up, but it’s not of any direct interest to this discussion. The difficulty in understanding the point of it, though, is a matter of familiarity with logic, which is a whole other branch of maths in itself and I haven’t managed to find any easy introductions to linear logic. However, there are some good basic texts in logic which cover most of the jargon. If anyone has any good references, please let us know.

Posted by: Kea on August 23, 2005 11:11 PM | Permalink | Reply to this

### Re: Logic

The reason I prefer my method is that I don’t think String Theory is the right way to look at the physics.

Ok, fine. Let me just note that you don’t need to be interested in strings to be interested in internalizing things in $\mathrm{Cat}$. Even thought I keep rambling on about it, it is only rather recently that connections between Baezian categorification and string ideas have emerged. In Oberwolfach Hendryk Pfeiffer has told me about some cool applications of categorification to the study of the relation between TFTs in various dimensions. (There was an amazing transparancy at the very end of his talk, but I cannot find it on his website. It’s not quite this one.) It’s certainly a method which is interesting in its own right.

The definition of such categories is easy enough: you can look it up, but it’s not of any direct interest to this discussion.

Oh, ok. You were just testing me. :-)

I thought you were trying to pave the way for telling me more about what you have in mind concerning your notion of ‘categorification’ of physics.

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$I haven’t managed to find an easy

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$introduction to linear logic.

If it is of no relevance to our discussion, never mind. I happen to have plenty of other things that I should read eventually. :-)

On the other hand, if you feel like telling us more about your stuff, I’d be happy to see you posting here.

Posted by: Urs on August 24, 2005 7:47 PM | Permalink | Reply to this

### Re: Logic

How is Hendryk? I haven’t seen him since mid 2003 and I often wonder how he is.

“…if you feel like telling us more about your stuff, I’d be happy to see you posting here.”

On a String Theory blog? Cool. When I’ve finished with what’s keeping me busy, I’ll take you up on that.

Posted by: Kea on August 24, 2005 11:11 PM | Permalink | Reply to this

### Re: Logic

Adelaide is fine, except I used all of yesterday trying to prove a lemma when I’m supposed to be giving a talk on Fri. eek.

Your package is still in tansit (well, maybe it arrived today, but I haven’t gone in to uni).

As far as listening to your views on Q grav - I think I’m hooked. We _really_ need new experiments. Not that I actually care anymore about physics - I’m only doing mathematical gauge theory stuff as that’s what my supervisors do. Ah, n-stacks and higher topoi.

Actually that reminds me (crossing to another thread) - since we shoud soon have a concept of bisite we can then think about Grothendieck bitopoi on a bicat. Hmmm. Gotta be patient, I s’pose.

D

Posted by: David Roberts on August 17, 2005 3:11 AM | Permalink | Reply to this
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