From: Aaron Bergman Subject: Re: K-theory (In reply to "Elementary QFT queries") Date: 27 Oct 2000 10:05:38 GMT Newsgroups: sci.physics.research Summary: [missing] In article , Magnus Jacobsson wrote: > > HOW IS (NON-COMMUTATIVE?) K-THEORY USED IN QFT:S? > > Anyone care to explain? I warn you though: I am pretty > ignorant about the practical details of QFT:s. Try a > user-friendly approach. Or rather mathematician-friendly. I don't know about QFT, but I can describe how it appears in string theory. The first thing to know is that string theory contains nonperturbative objects called D-branes which are simply submanifolds of spacetime on which a string can end -- in other words, boundary states of the worldsheet CFT. Now, for a long time, it was thought that D-brane charges were classified by homology. In IIB, in particular, even dimensional homology. (Why? Because they couple to RR fields which apparently lived in de-Rham cohomology with integer quantization conditions.) Anyways, there's a neat way one can think of D-branes. If one takes a stack of D-branes and anti-D-branes, they should be able to annihilate. This is manifested in the fact that if one computes the spectrum of strings that connect the branes, there is at tachyon. For D odd (an even dimensional brane), this is a complex field with a U(1) symmetry. Sen conjectured that the form of the potential for this field is basically just a mexican hat potential. Now, one can try to construct a codimension 1 soliton (a kink). This, however, can slip around the potential, so is unstable. This is an example of a non-BPS brane. We can imagine instead a swirl in the tachyon field that wraps around the bottom of the potential. This has a winding number and is a codimension two soliton. Most importantly, it's stable. So, the Sen's conjecture is that when the tachyon condenses from this field configuration, one is left with a D-2 brane. So, where does the K-theory come in? Well, just starting with any D-brane, it has a Chan-Paton bundle associated with it which for now is just a vector bundle. We can nucleate any number of D- and anti-D-branes in pairs. So, there is an equivalence relation on bundles (E,E') ~ (E+V,E'+V). Looks familiar already, huh? Now, given the construction above with a bit of a generalization (which is essentially the Atiyah-Bott-Shapiro construction), we can construct any lower D-brane from a whole bunch of 9 and anti-9 branes. So, the moral of all this is that in type IIB string theory, D-branes along with their associated CP bundles are associated to elements in K^0(M) where M is spacetime. Even better, we have the Chern character ch: K^0(M) --> H^{2n}(M,R) so we recover our old description. There are still a few problems, though. For one, there are torsion issues. Another important fact is that the ABS construction only works for submanifolds with a Spin^c structure. Why can't these manifolds become D-branes? It turns out that when one looks at strings on a D-brane, fermions in the action give rise to a Pfaffian with an undetermined sign. In fact, the sign of the Pfaffian is an element of a Z_2 bundle over the loop space of the D-brane. The string partition function is nonanomalous only if this line bundle is trivial. One can compute its Chern class and one obtains that, if there's no NSNS background, the line bundle is trivial if W_3(N) = 0, or a Spin^c structure exists on the submanifold N. K-theory is vindicated. Torsion is still interesting, though. As I'm sure a bunch of people have already noticed, D-brane charges were in homology and not in cohomology which is what K-theory gave us. Ick. There have been a bunch of recent papers suggesting that instead of living in K^0(M) = K_0(C(M)), maybe they live in the K-homology instead. The only difference between the two is the torsion again. I don't know if anyone knows what the answer is here. A few more things. It turns out that one can conjecture that RR fields which were thought to live in even dimensional cohomology instead live in K^{-1}(M) = K^0(SM) (integrate around the circle to get back the form description, IIRC). Now, one has an interesting time quantizing these things. One can also ask what one does in the case of a H = dB_{NSNS} background. In the H is torsion case, one can define a twisted K-theory that the things live in. In the nontorsion case, it is conjectured that the relevant K-theory might be the K-theory of the algebra of sections of the Dixmier-Douady bundle with three-form equal to H. Lastly, what about IIA string theory. In this case, one flips the zeros and ones of the K-theory. Now, one might ask if the circle of the suspension of the manifold is related to the extra circle that shows up in decompactifying IIA to M-theory. As far as I know, this connection has never been shown. Lastly, in noncommutative geometry, it turns out that there is a class of solitons that come from projection operators. This sounds a lot like algebraic K-theory and that's discussed in the Harvey and Moore paper below. Now for the references: In fact, the first relation to K-theory wasn't related to any of the above. It was seen by Moore and Minasian and it had to do with anomaly inflow and other icky stuff: hep-th/9710230 Witten wrote the paper that espouses the view above: hep-th/9810188 The calculation for IIA is in: hep-th/9812135 The anomaly I talked about is in Freed and Witten: hep-th/9907189 RR field and K-theory is in Moore and Witten: hep-th/9912279 Ideas about noncommutative geometry K-homology are in Harvey and Moore: hep-th/0009030 and references therein. Nontrivial torsion H is in Kapustin: hep-th/9909089 General H is in Bouwknegt and Mathai hep-th/0002023 The relation to M-theory is in hep-th/0005090 And finally, an overview by Witten is in hep-th/0007175 Aaron -- Aaron Bergman ============================================================================== From: helling@x4u2.desy.de (Robert C. Helling) Subject: Re: K-theory Date: 27 Oct 2000 10:13:50 GMT Newsgroups: sci.physics.research In article , Magnus Jacobsson wrote: >K-theory should also be unable to distinguish spaces which >are homotopy equivalent, so that K^i(X) and K^i(Y) are isomorphic in >that case. Here I wanted to reply that this is not the case but I wasn't carefully reading and read homology for homotopy. In fact, the information given by K-theory about X is similar to the information given by (Co)Homology: Both are finitely generated abelien groups that means they are of the form direct sum of some copies of Z and some compies of various Z_k the latter being called the torsion. The non-torsion part is usually easily computed (well, hmm) and there is a theorem that this part is the same for both Cohomology and K-theory. Possible differences can only appear in the torsion part and it is quite difficult to find examples where this is the case. This was for example done for some Calabi-Yau spaces in hep-th/0005103. >Iīm afraid I canīt tell you how it is used in QFT, but I wouldnīt be >surprised to find that what they really use there is this latter >non-commutative K-theory. It would be interesting to know...hmm... >...AHA! _I_ can also post a question, canīt I? > >So here it is: > >HOW IS (NON-COMMUTATIVE?) K-THEORY USED IN QFT:S? > >Anyone care to explain? I warn you though: I am pretty ignorant about >the practical details of QFT:s. Try a user-friendly approach. >Or rather mathematician-friendly. I cannot give an example in field theory but K-theory is right now a hot topic for string theory (once again after BigEd has written a paper about this): There the idea is as follows. You might have heart about Dp-branes, that are (p+1) dimensional submanifolds of space-time on which strings can end. It turns out, those carry gauge fields that are sections in some bundle on the Dp-brane. Furthermore, in string-theory, there lurk around a couple of k-form fields in space-time. And a Dp-brane is charged under the k+1-form (electrically, that means, the action is given by the integral of the form over the brane) and magnetically under the (7-p) form. Away from the branes, the forms are closed. As usual, you can determine the number of branes by performing a Gaussian integral. As charges are preserved, this is an integral of motion and supposed to characterize a configuration of branes and fields (among other invariants). Even more, it is hoped the for any snapshot of brane, field, and string dynamics, the possible groundstate in which the system will settle is determined by the charges. As can be seen from what we said above, the charges come from the cohomology groups. Thus, the cohomologies are used to characterize states. One thing I am not going to explain here is that N coinciding branes carry a U(N) gauge field on them. For trivial topology, cohomology and K-theory is just Z, the rank of the vector bundles. Here, this rank and therefore K-theory can directly be identified with the number N of branes. What I have said so far is not the full story. There are also anti-Dp-branes. They are the anti"particles" of the Dp-branes and have negative unit charge. Furthermore, they can annihilate with the Dp-branes. M anti-Dp-branes have U(M) gauge fields of their own. Now, imagine a situation with N Dp's and M anti-Dp's. In total we have U(N) + U(M) bundles. The charges being (N,M). Now, as I said, it is possible for a pair to annihilate. After that we are left with U(N-1) + U(M-1) bundles. As both situations are related by dynamics, they are supposed to be described by the same invariant charges. So, as you might have guessed already, only N-M is invariant. But even more: As far as invariants are concerned we should identify the U(N)+U(M) bundle with the U(N-1)+U(M-1) bundle. But this is nothing but the Grothediek construction you explained in your post. This all is very suggestive that in fact, K-theory gives the correct invariants to characterize situations in string theory. There are more complicated hints: I have (implicitly) only talked about trivial bundles. But bundles might be topologically differen. In that case, the annihilation cannot take place directly. This is taken care of by K-theory. Furthermore, I have been a bit lax about the possible form fields and branes. In fact, there are two (for the purpose of this construction) different string theories: type IIA and type IIB. In IIA, there are only k-forms for odd k. Therefore, only Dp-branes of even p are stable. This coincides with K-theory, that is only non-trivial for the correct dimensionalities. And one could study strings in more complicated space-times for which K-theory has torsion. There, one would expect a maximal number of certain kinds of D-branes. If you try to add one more, they all annihilate to the vacuum. A behaviour like this has been observed in specific examples. Details of all I have said can be found in Witten's hep-th/0007175. One thing I forgot: String theorists also talk a lot about strings moving in some spaces that special back-grounds so the effective theory is given by a theory in non-commutative spaces. There, one naturally expects non-commutative (algebraic) K-theory to play the role of (topological) K-theory I explained above. Robert -- .oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO Robert C. Helling Albert Einstein Institute Potsdam Max Planck Institute For Gravitational Physics and 2nd Institute for Theoretical Physics DESY / University of Hamburg Email helling@x4u2.desy.de Fon +49 40 8998 4706 ============================================================================== From: Volker Braun Subject: Re: K-theory (In reply to "Elementary QFT queries") Date: 28 Oct 2000 15:51:30 GMT Newsgroups: sci.physics.research There seems to be a connection between CFT (especially the D-brane charges in WZW models) and the twisted K theory proposed by Bouwknegt & Mathai, which happens to be what i looked into recently. So what are D-branes is WZW models? I dont know, but here's the upshot of the explanation the other Volker (*) gave me recently. The D-branes are orbits of points where the Lie group (the Target space) acts on itself via conjugation. (*) Volker Schomerus, Stefan Fredenhagen, Thomas Quella So lets take SU(2)=S^3, there are 0-dimensional branes (the orbits of the center = +1, -1 = North and South pole) and 2-dimensional branes (generic orbits). What other ingredients do we need? Since we are interested in twisted K theory, lets allow for a non-zero cohomology class of H in H^3(S^3,Z)=Z. Now without H the 2-branes can just have any radius, but with H they must enclose a quantum of flux, which leads to quantized radii. Finally the CFT people belive that they can transform n 0-dimensional branes to a single 2-dimensional brane with n 'quanta' of radius. Transform means that this is a physically smooth process, but of course we dont understand the complete dynamics. But the SU(2) we started with has finite size, if we stick enough 0-branes at the North pole then this is equivalent to a single brane sitting at the South pole. So somehow CFT predicts torsion charges for the D-branes! And the torsion group here is just Z_k, where k is the integer that labels H's cohomology class. Lets see what twisted K theory yields! So write down the bundle of compact operators over S^3 which (as an C^* algebra) has the Dixmier-Douady class H, and calculate its C^*-algebra K-theory. Just kidding, of course we want to get the result without any real work (there are some loopholes it the free lunch theorem it seems). It turns out the mathematicians (**) have a spectral sequence that converges to this twisted K theory (i'll call it K_H in the following). This sequence has precisely the same E_2 term as the Atiyah-Hirzebruch spectral sequence that converges towards the ordinary K-theory (this somehow follows from the fact that the differences between K and K_H are something 3-dimensional, but the AHSS only requires K^*(point)) (**) Jonathan Rosenberg: Continuous-Trace Algebras from the Bundle Theoretic Point of View. So here is E_2^(p,q) = H^p(S^3, K^q(point)): q=4 | Z 0 0 Z q=3 | 0 0 0 0 q=2 | Z 0 0 Z q=1 | 0 0 0 0 q=0 | Z 0 0 Z -----+---------------- | p=0 p=1 p=2 p=3 So the only interesting differential is d_3 from p=0 to p=3. But Rosenberg determined it to be cup-product with H, i.e. multiplication with k! So we find K_H^0(S^3)=0 and K_H^1(S^3)=Z_k, which fits nicely to the CFT result. Now if this convinced you that twisted K-theory indeed describes physics then i have a bridge i'd like to sell to you. At the very least one should do the analog calculations for more than one group, but this turns out to be not so easy (but there is work in progress). Then one should go on and build out of these CFT's one that corresponds to the compactification of string theory at a special point in the moduli space and then examine the compactification space. I'll leave this as an exercise. Volker PS: Let me point out that K_H is _NOT_ a cohomology theory on topological spaces, so the notation K_H or K(-,H) is awkward at the least. However by definition it is a good homology theory on the associated Algebra. Suspension of the algebra and the space just does not commute because of the twisting. So if you are a pessimist you might say its nonsense, but if you are an optimist this is yet another sign of quantum geometry.