Although Jarah Evslin is an anarchist, his talk at the postdoc journal club was extremely organized and useful. His topic was
* What is K-theory bad for and what it does not classify?
We immediately figured out a couple of possible answers - such as the hamburgers, and so forth. However, Jarah started and everything suddenly made much more sense. Jarah is a very mathematically skillful guy, but nevertheless, category theory is too hardcore even for him. Nevertheless, he wanted to solve similar tasks as category theory and classify various conserved brane charges and fluxes, and outlined the following sequence of mathematical objects:
These sets are increasingly refined. Everytime you fall one step lower, your space only classifies a subspace of the upper one, and moreover makes some identifications (quotienting). Therefore, every new entry in the list above is a set which is smaller "by two contributions" than the previous one, given a natural definition of "smaller". Actually, every new entry is not quite obtained as a quotient of the subset of the previous one because it can "twist" the previous entry so that Z_2 cubed is replaced by Z_8, as an example below will indicate. Also, every new entry is "more conserved" - its charges (elements) are invariant under a broader class of physical processes than the elements of the previous classifying set.
Jarah has explained many examples in which the neighboring structures differ. For example, one-cycles on a genus g surface may be understood as embeddings. In that case, a D1-brane and a different D1-brane nearby cannot annihilate - this pair is distinguished in the structure called "embeddings". Then you may reduce the set of possible embeddings to homotopy; the first homotopy on the genus g surface is a non-Abelian group. Its abelianization gives you the homology; homology is a smaller group. I won't write anything else about the difference between embeddings and homotopy.
Instead, let's continue with the difference between homotopy and homology. Jarah has discussed various special cases - such as the branes on RP^7 x S^3. Homology gives you Z_2 for H_1, H_3, as well as H_5, and you might think that the natural group is Z_2 cubed - three independent numbers are added modulo 2. However, K-homology gives you a different answer. It shifts the Z_2's relatively to each other, so that Z_2 cubed is actually replaced by Z_8. This means that if you annihilate two 3-branes, instead of nothing you obtain a 1-brane, and so forth. He also defined K-theory, twisted K-theory, and gave several other examples.
We have had many discussions led by Jarah about the ability to refine K-theory in such a way that it is invariant under all dualities or satisfies similarly big constraints. Jarah's S-covariant K-theory is not invariant under T-duality, for example, simply because he classifies possible fluxes of the 3-form H, but he does not classify the possible topologies (geometry) even though these topologies may be T-dual to the H-fluxes. My feeling is that something that would classify all these conserved objects and respected all dualities would have to know - more or less - about the whole landscape, and therefore knowing this ultimately refined K-theory is almost equivalent to knowing the whole "theory of everything".
There are issues about these refined K-homologies being groups or just semigroups or nothing like that. It's intuitively clear that one expects a group structure for objects that can be thought of as small perturbations of a background. For example, K-theory should count all generalized D-branes - everything that contributes to the total energy/action by an amount proportional to 1/g_{string}. At weak coupling, that's much smaller than the tension/action of a nontrivial closed string background (e.g. NS5-branes) that goes like 1/g_{string}^2, and therefore it is natural that we get a structure of an Abelian group as K-theory. Once we add both NS-NS H-fluxes as well as R-R fluxes, it's not shocking to learn that the classifying set won't behave as a group. The condition of the objects being small perturbations of the same background does not hold anymore: similarly, we also don't expect the set of Calabi-Yau topologies to be a naturally definable group because we have no canonical way how to "add" topologies! ;-)
At any rate, Jarah's talk was very interesting. ___________________________________________________________________________ ___ E-mail: l...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/ eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call) Webs: http://schwinger.harvard.edu/~motl/http://motls.blogspot.com/ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^
Alas, I only have a very rudimentary understanding even of the definition of K-theory. About all I know is that it has to do with understanding the products of bundles that live on pairs of D-barD branes.
If anyone feels like giving a 60 ASCII line introductory course on K-theory, I'd appreciate it (as, I am sure, other readers here would, too).
> Although Jarah Evslin is an anarchist, his talk at the postdoc journal > club was extremely organized
Probably if he exports enough entropy in political opinions he can use that to decrease the entropy in his scientific work. ;-)
> nevertheless, category theory is too hardcore even for him.
For those who have missed it, this is referring to a small discussion Lubos and I had on the relation between category theory and physics/string theory. It started with this post of mine
> he wanted to solve similar tasks as category theory
Where I should add that the point that I was arguing for in the above was not directly related to derived categories and related stuff, but more concerned with the idea that the category theoretic approach to nonabelian gerbes indicates that there is a nice parallel
stringification <-> categorification .
Categorification essentially means taking a given mathematical structure and allowing all "point-like" objects in that structure to become "map-like". This way they become sort of 1-dimensional ("morphisms") and acquire "internal structure".
(More precisely, categorification means defining a given concept in terms of arrow diagrams and then *internalizing* these diagrams in the 2-category Cat of all (small) categories. Since the morphisms in Cat are functors, this implies that sets go over in categories, functions into functors and equations between functions in natural isomorphisms between functors.)
This is morally rather similar to how I can regard a point particle with certain properties and then "magnify" it to realize that it is really just the endpoint of a string (on some brane, say), whose "internal structure" (its oscillations) determine the properties of the particle.
This might appear as a rather far-fetched analogy, but, as I try to indicate in the above mentioned posts, it becomes rather precise in the context of categorized fiber bundles which describe strings in nonabelian 2-form backgrounds, as well as when conformal field theory is formulated as point particle dynamics on loop space.
> and classify various > conserved brane charges and fluxes, and outlined the following sequence of > mathematical objects:
Thanks for the article! I agree with everything, for example I agree that its your feeling that finding something S and T invariant would give the whole landscape, although I'm not so optimistic myself. Your blog has finally given me a reference to point people to when they ask me for references to get the basic idea of the K-theory classification.
[Moderator's note: I'm not sure whether my blog is too useful for understanding K-theory. ;-) Thanks anyway. LM]
I've thought a bit more about your question about a Dp wrapping a cycle with nontrivial nonintegral B-flux carrying apparently non-quantized D(p-2)-charge. There are lots of ways to define D(p-2)-charge, but B on a noncontractible cycle doesn't seem to give something that needs to be quantized. The way to check this is to take a D(8-p) brane worldsheet and see how much the action changes as you move it around a cycle linking that D(p-2)-charge. The nontriviality of the cycle supporting the B flux seems to make this impossible without having the D(8-p) cut the Dp-brane, in which case the Dirac argument is harder to follow. This happens because in order to link the D(p-2) its got to link a point on the 2-cycle with the B flux, but the Dp wraps this 2-cycle. So it seems like the Dirac argument that you might think would have quantized this B flux doesn't seem to go through as simply as usual, maybe it doesn't work at all. Thus if you want to define the D(p-2) charge to be a quantized thing then you probably want to not include this contribution.
On the other hand if the 2-cycle is contractible there's no problem, you can link the D(p-2)-brane without intersecting the Dp-brane. But in this case the integral of H on the interior of the 2-cycle exactly cancels this contribution to the D(p-2) charge a la Wati Taylor.
You can do the same thing in M-theory, where its easier to understand, with an M5 linking a 3 cycle with C flux which you might say yields M2 charge. Then you can check to see if the partition function of a second M5 whose 6d worldsheet sweeps out a 7 cycle linking the M2 is well defined. The problem is that the 7-cycle needs to intersect the original M5 and so some M2 charge can be on either side anyway, ie there's not really any Dirac quantization when you cut inside the region where the charge is smeared (this is why M2's can smear in an M5 but not outside).
It's a good question, I still think I need to understand it better.
In article <32bdvfF3cmjg2U1-100...@individual.net>, Urs Schreiber <Urs.Schrei...@uni-essen.de> wrote: > ... If anyone feels like giving a 60 ASCII line introductory course on K-theory, > I'd appreciate it (as, I am sure, other readers here would, too).
There's a lot of K-theories out there, but the easiest is vector bundle K-theory. (This is K^0, to be precise). Look at ordered pairs of vector bundles
(E,F)
with the following operation
(A,B)+(E,F) = (A+E,B+F)
and quotiented by the following equivalence relation
(E,F) ~ (E+G,L+G)
For a vector bundle, we write the class [(E,0)] as [E] and [(E,F)] as [E] - [F]. We also have a multiplication given by the tensor product of bundles making K^0 into a ring..
The Chern character acts as a ring homomorphism from K-theory to cohomology in even degree. If you ignore torsion (ie, tensor with Q), it's an isomorphism. More generally, there's a spectral sequence relating ordinary cohomology to K-theory, the Atiyah-Hirzebruch spectral sequence.
If you decide you like coherent sheaves rather than vector bundles, you can from their Grothendieck group by taking the free group on coherent sheaves quotiented by the relation
0 -> A -> B -> C -> 0 ==> [A] + [C] = [B]
(hoping I have my signs right.) Eventually, you might decide that you'd rather work with the complexes themselves rather than an Abelian group with relations. That road takes you to the derived category.
Another interesting question is what happens in the presence of a H-flux background. If H is torsion, one can look at SU(N)/Z_n vector bundles which are classified by a torsion three form. If H isn't torsion, however, things get funkier and you need to go beyond the scope of this post.
[...]
> > This means that if you annihilate two 3-branes, instead > > of nothing you obtain a 1-brane, and so forth.
> Hm. Can I understand that heuristically somehow?
The tachyon field can have a winding around the vacuum.
On Wed, 15 Dec 2004 13:28:57 -0500, Urs Schreiber wrote: >> * Homotopy
Note: Kahn's spectral sequence computes Homology starting from Homotopy groups.
>> * Homology
Note: the (Poincare-dual of the) Atiyah-Hirzebruch spectral sequence computes K-groups starting from Homology groups.
>> * K-homology >> * S-covariant K-homology
> So what precisely happens in the step from holomogy to K-homology?
Well then a spectral sequence rears its ugly head, removes some degrees of freedom and combines the remaining ones in a different way. There are simple examples where the K-groups are strictly smaller than the homology groups, and Lubos already mentioned an example where (Z_2)^3 gets modified into Z_8.
This is more than just similar to RG flow, although I don't think I understand the full extent. The twisted K-theory of Lie Groups was first "computed" using RG arguments by Fredenhagen & Schomerus. And the vertex operator people have spectral sequences that determine the VOA at the endpoint of RG flow (in very special cases).
>> So what precisely happens in the step from homology to K-homology?
> Well then a spectral sequence rears its ugly head, removes some degrees of > freedom and combines the remaining ones in a different way.
Hm, I am afraid I'll need a more detailed definition of what K-homology really is! How is it defined?
Anyway, thanks to you and Aaron for offering help. Unfortunately I am currently unbelievably busy with other stuff. But when you two are still around after New Year's eve to help me learn some K-theory, I'll be glad to bother you with further questions then.
> >> So what precisely happens in the step from homology to K-homology?
> > Well then a spectral sequence rears its ugly head, removes some degrees of > > freedom and combines the remaining ones in a different way.
> Hm, I am afraid I'll need a more detailed definition of what K-homology > really is! How is it defined?
You don't want to know. Really. Just think of it as dual to K-theory (ie, K-cohomology).
(And, besides, I don't remember the geometric definition.)
It can get worse -- you can generalize this to a bivariant theory, ie, KK-theory or E-theory. Despite some attempts on the former, I don't know of any use for it in physics. You can also generalize in a different direction to get a whole hierarchy of generalized cohomology theories starting with elliptic cohomology. That, of course, is related to string theory. If you don't know already, you'd like it because it ought to be related to 2-bundles.
math.AT/0306027
But I'm talking about things I really don't understand. In particular, there are some papers by Kriz and Sati on these generalized cohomology theories that I definitely don't understand.
> You can also generalize in a different > direction to get a whole hierarchy of generalized cohomology theories > starting with elliptic cohomology. That, of course, is related to string > theory. If you don't know already, you'd like it because it ought to be > related to 2-bundles.
> math.AT/0306027
Thanks for this reference, I was not aware of it yet.
N. Bass, B. Dundas & J. Rognes: Two-Vector Bundles and Forms of Elliptic Cohomology .
Not sure if their construction should be called a 2-bundle, though. As far as I can see from skimming the paper, their 2-vector bundles are obtained by categorifying just the vector space part of an ordinary vector bundle, and that in the sense of Kapranov&Voevodsky, who decreed that a "2-vector space" should be some nth power of the category Vect of ordinary vector spaces.
This is not obviously the same as a 2-bundle (in the sense of math.CT/0410328), where the entire concept of a bundle is categorifed by internalizing it in a smooth sub-2-category of Cat. But certainly it is related, and I would like to understand this relation eventually.
(The difference between Vect^n and vector spaces internalized in Cat is discussed a little in math.AQ/0307263. For instance in Vect^n there is no subtraction functor.)
Can we cook up a 2-group with vector-bundle valued matrices as morphisms and matrices of vector space isomorphisms as 2-morphisms? If yes, then the relation between 2-vector bundles and 2-bundles is easy. I don't see yet what the horizontal product of vector space isomorphism-valued matrices would be, though.
> N. Baas, B. Dundas & J. Rognes: > Two-Vector Bundles and Forms of Elliptic Cohomology .
> Not sure if their construction should be called a 2-bundle, though.
There was some behind-the-scene discussion of this point, which Aaron should be aware of:
The relation between the two approaches is "so far fairly indirect". The 2-K theory resulting from Baas' "2-vector bundles" is apparently not satisfactory as it is restricted to discrete groups. Whether or not elliptic cohomology can be (better) done using 2-vector 2-bundles in the sense of Baez and Bartels is still an open question.
Baas' attempt to improve the situation as well as more discussion on elliptic cohomology, categorified K-theory and so on is here:
