More On Geometric Langlands (a Grand Unified Theory of Math?)

After mentioning in the last posting that Witten is giving talks in Berkeley and Cambridge this week, I found out about various recent developments in Geometric Langlands, some of which Witten presumably will be talking about.

Edward Frenkel has put a draft version of his new book Langlands Correspondence for Loop Groups on his web-site. In the introduction he describes the Langlands Program as “a kind of Grand Unified Theory of Mathematics”, initially linking number theory and representation theory, now expanding into relations with geometry and quantum field theory. The book is nearly 400 pages long, and to be published by Cambridge University Press. Frenkel also notes that recent developments in geometric Langlands have focused on extending the story from the case of flat connections on a Riemann surface to connections with ramification (i.e. certain point singularities are allowed). He has a new paper out on the arXiv about this, entitled Ramifications of the geometric Langlands program, and he writes that:

in a forthcoming paper [by Gukov and Witten] the geometric Langlands correspondence with tame ramification is studied from the point of view of dimensional reduction of four-dimensional supersymmetric Yang-Mills theory.

The title of the forthcoming Gukov-Witten paper is supposedly “Gauge theory, ramification, and the geometric Langlands program.”

Presumably people attending Witten’s talks in Berkeley and Cambridge will get to hear about this new story for the ramified case. For the rest of us, on his web-site David Ben-Zvi has notes from talks this summer by Witten at Luminy where he describes some of this. Ben-Zvi also has an announcement of a series of lectures on geometric Langlands that he’ll be giving at Oxford next April. The summary of the lectures says that he’ll “describe upcoming work of Gukov and Witten which brings together geometric Langlands and link homology theory.” Link homology theory is also known as Khovanov homology, and I wrote about this two years ago here, advertising Atiyah’s speculation that there may be a 4d TQFT story going on, something I always have found very intriguing. Ben-Zvi has recently lectured on Khovanov homology at Austin, and began his lecture by saying that this material relates “themes in 21st century representation theory” to 4d TQFT. He goes on to cover some of the ideas about 4d TQFT and “categorification” that I was very impressed by when I heard about them from a talk by Igor Frenkel a few months ago (described here).

At first I thought Ed Frenkel’s claim that geometric Langlands was going to give a Grand Unified Theory of mathematics was completely over the top, but seeing how some of these very different and fascinating relations between new kinds of mathematics and quantum field theory seem to be coming together, I’m more and more willing to believe that investigating them will come to dominate mathematical physics in the coming years.

Update: Slides from Witten’s Berkeley lectures are here. And many thanks to David Ben-Zvi for the informative comments!

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50 Responses to More On Geometric Langlands (a Grand Unified Theory of Math?)

  1. A.J. says:

    Hi Peter,

    Witten has only delivered one lecture so far, and it was devoted to reviewing background material: mostly S-duality and a few words about topological twisting, all of which can be found in the Kapustin-Witten paper.

  2. Peter Woit says:

    Thanks A.J.!
    It would be great if you could keep us informed about the rest of the lectures…

  3. SFB says:

    It sounds like they are doing interesting math, but leaving physics to the LQG crowd.

  4. atrings says:

    I agree with SFB for the “interesting math “,but not for the”LQG crowd”.

  5. Richard says:

    “At first I thought Ed Frenkel’s claim that geometric Langlands was going to give a Grand Unified Theory of mathematics was completely over the top, but seeing how some of these very different and fascinating relations between new kinds of mathematics and quantum field theory seem to be coming together, I’m more and more willing to believe that investigating them will come to dominate mathematical physics in the coming years.”

    Perhaps a domination of mathematical physics, but the claim of a grand unification of mathematics is in fact way over the top unless you believe that mathematics is nothing but mathematical physics. It probably all depends on your own personal values, biases, points of view, and even whom you believe owns mathematics. Recall Lubos’ wild claim that someday mathematics will be completely subsumed by string theory?

  6. onymous says:

    I expect many of the people who have been working on geometric Langlands for years would be kind of shocked to be called mathematical physicists, Richard. Do they all instantly become mathematical physicists just because Witten got interested in what they’re doing?

  7. Richard says:

    Onymous – I don’t believe I said that.

  8. onymous says:

    Apologies, I misread Peter’s original statement — didn’t notice that he specifically singled out mathematical physics — and so misinterpreted your “…unless you believe that mathematics is nothing but mathematical physics” as an implication that geometric Langlands is mathematical physics. Never mind.

  9. David Ben-Zvi says:

    Hi and thanks for the references! (all notes on my page should be taken with many grains of salt..)
    I should point out that the preprint by Gukov and Witten doesn’t actually talk at all about link homology, so my talk description was perhaps premature, but a connection between
    geometric Langlands and some kind of link homology is to be expected following their ideas (cf Gukov’s Strings talk).
    Cautis and Kamnitzer also have very interesting work in progress on such a relation.

    After all, geometric Langlands is a very general categorification
    program in representation theory, so one would expect it to
    relate to the kinds of categorifications that give rise to Khovanov homology. There just aren’t too many fundamental structures associated with a semisimple Lie group, and they all connect..

    Of course it’s a joke to speak of geometric Langlands as
    a grand unified theory… but the Langlands duality is certainly
    among the broadest themes in math, a kind of nonabelian
    generalization of the Fourier transform, and it’s extremely exciting
    that we can view it in the geometric setting as electric-magnetic duality in four dimensional gauge theories!

  10. relativist says:

    David Ben-Zvi says

    “it’s extremely exciting that we can view it in the geometric setting as electric-magnetic duality in four dimensional gauge theories!”

    Can you expand on that? Sounds very interesting.

  11. urs says:

    it’s extremely exciting that we can view it in the geometric setting as electric-magnetic duality in four dimensional gauge theories!

    Can you expand on that? Sounds very interesting.

    This is the insight of the Kapustin-Witten paper.

    You can find a summary here.

  12. urs says:

    a kind of nonabelian generalization of the Fourier transform

    Is it a nonabelian generalization, or isn’t it rather a categorification of the Fourier transform?

    It seemed to me that much of Langlands can be nicely understood as taking place in categorified linear algebra. I have made remarks on how the Hecke operator looks like a 2-linear map for instance here.

  13. urs says:

    It would be great if you could keep us informed about the rest of the lectures…

    If anyone feels like reporting on interesting lectures online, we have a guest account for that over on the n-Café.

    For instance we had David Roberts guest-reporting from a lecture by Brian Wang here, similar to the many guest reports we had # at the string coffee table.

  14. David Ben-Zvi says:

    Urs:

    “Is it a nonabelian generalization, or isn’t it rather a categorification of the Fourier transform?”

    well it’s both.. the main difficulty is the nonabelian nature
    rather than the categorification, and that is where Langlands tells
    us what to do (in the geometric or classical, noncategorified setting).
    Categorifications of the Fourier transform have been
    used for almost 30 years I think (starting with the Fourier-Deligne
    transform, see eg Laumon’s first ICM), and the geometric Langlands
    program suggests that one can extend this to nonabelian settings
    (G-bundles on curves).

    By the way maybe this is an excuse to air one of my pet
    peeves, the use of the term “Fourier-Mukai”
    to refer to any functor between derived categories given
    by an integral kernel.. I would be surprised if an analyst
    referred to any map on function spaces given by
    integration against a kernel (or any matrix) as a Fourier transform, and the same should hold in the categorified setting — in some
    precise sense (due to Toen and which I’m badly paraphrasing)
    all functors between derived categories are given by integral kernels!
    “Honest” Fourier-Mukai transforms should have additional
    structure and properties (for example taking convolution
    to tensor product). Similarly not any duality is a T-duality!

  15. urs says:

    well it’s both.. [...] Categorifications of the Fourier transform have been used [...] and the geometric Langlands program suggests that one can extend this to nonabelian settings [...]

    Great, thanks! That’s what I was hoping some expert would say. Probably I just talked to the wrong experts so far!

    Because each time I’d ask a question along the lines

    “isn’t an eigenbrane just a categorified eigenvector in some 2-vector space”

    the answer I’d get would be something like

    “no, 2-vector space only appear after we categorify Langlands itself, like Kapranov discussed.”

    http://golem.ph.utexas.edu/category/2006/10/quantization_and_cohomology_we_1.html#c005444

  16. Peter Orland says:

    I don’t know much category theory, but I thought that the
    non-Abelian generalization of the Fourier transform is the character expansion (or Plancherel transform in the non-compact case) for functions on non-Abelian groups. Aside from a character formula, that is the simplest generalization.

    Obviously, I am missing the point and something deeper is meant. Can anyone explain this to a dumb theoretical physicist?

  17. Peter Orland says:

    I just wanted to add that the sort of examples I mentioned don’t help much with non-Abelian duality in classical or quantum field theory. To perform a duality transformation, a zero-curvature condition is Fourier transformed and the parameter integrated over is the dual field. This only really works in the Abelian case. There are non-Abelian generalizations of duality done this way, but they are rather messy, and not obviously useful.

  18. A.J. says:

    Well, Witten finished his lectures, but ran out of time to say much of anything about ramification. There’s just too much information to be covered in (somewhat less than) 3 hours. Most of what he said is pretty well covered in David Ben-Zvi’s notes, and in Urs’s posts on the subject, or in the Kapustin-Witten paper for that matter.

    We did get scans of his notes, so perhaps those will be available online one of these days.

  19. urs says:

    Can anyone explain this to a dumb theoretical physicist?

    One way to get an intuition for what is going on with these Hecke operators and similar transformations is to consider the drastically oversimplified baby toy example situation where the underlying spaces are in fact just – finite sets.

    A vector bundle over a finite set is then just an array of finitely many vector spaces.

    Think of that as a vector whose entries are vector spaces. Such a beast is known as a (Kapranov-Voevodsky) 2-vector.

    The categorification involved here is that which takes the monoid of complex numbers and replaces it by the monoidal category of complex vector spaces.

    So we can imagine doing linear algebra with these vectors whose entries are vector spaces by replacing sums of complex numbers by direct sums of vector spaces and products of complex numbers by tensor products of vector spaces.

    In particular, let X any Y be two finite sets and consider a vector bundle L over X x Y .

    By the above, this is now like a |X| x |Y| matrix with entries being vector spaces. Using the above dictionary, we can define the categorified matrix product of L with a 2-vector over Y, simply by using the ordinary prescription for matrix multiplication but replacing sums of numbers by direct sums of vector spaces and products of numbers by tensor products of vector spaces.

    One can convince onself, that this categorified action of a 2-matrix on a 2-vector can equivalently be reformulated in a more arrow-theoretic way as follows:

    We have projections p1 and p2 from X x Y to X and to Y, respectively. This makes X x Y into a span

    http://golem.ph.utexas.edu/category/2006/10/klein_2geometry_vi.html#c005232

    Given a 2-vector V -> X over X, we may pull it back along p1 to X x Y, tensor the result componentwise with L and push the result of that back along p2.

    This operation produces precisely the naive categorified matrix product that I mentioned above.

    But the nice thing is that this pullback-tensor-pushforward along a “correspondence” like X x Y generalizes to vastly more interesting situations.

    There is an entire zoo of well-known operations of this kind. The Fourier-Moukai transformation is one example. The Hecke transformation that appears in geometric Langlans is another.

    In the above sense, all of these operations can be understodd as linear maps on 2-vector spaces.

    A description of what I just said, including some helpful diagrams and links to further material can be found here:

    Fourier-Mukai, T-Duality and other linear 2-Maps.

  20. urs says:

    Concerning the abelian vs. nonabelian categorified Fourier transform:

    there is something called the “classical limit” of geometric Langlands, as decribed for instance here:

    Pantev on Langlands, II

    The Hecke operation in geometric Langlands is a generalization of the categorified Fourier transformation: is a “2-linear map” in the sense of my comment above

    http://www.math.columbia.edu/~woit/wordpress/?p=492#comment-19258

    such that it coincides with the Fourier-Moukai transformation in this “classical limit”.

    In other words, the Hecke operation is a deformation of the Fourier-Moukai transformation.

  21. Bert Schroer says:

    I never understood what is the relation of elliptic cohomology (not that I don’t know what it presents mathematically since I have followed the area with an ever increasing distance since the days of the Atiyah-Singer index theorem) with particle physics except that Witten has generated a certain enthusiasmus with some particle physicists. Since I have learned to make a distiction between physics and what (some) physicists are doing and since this blog (as Peter’s book) is primarily about the present state of particle physics I think it is a legitimate question to ask about its relation to particle physics. If this is not permitted then this will be my last contribution to this blog.

  22. Anon says:

    To Peter Orland:

    You are correct about the Plancheral theorem. But that tells you that if you know the irreducible representations, and their dimensions/characters, you know how to decompose functions. It doesnt tell you what the characters are.

    In the first instance Langlands is a parameterization of irreducible reps, and a determination of their character; roughly they are in bijection with conjugacy classes in another group.

    The categorification nonsense is an elaboration of this, to say *all* information you can extract comes from this dual group.

  23. Peter Orland says:

    Urs and Anon,

    Thanks for the responses. I understand that a character formula
    of some sort is need to make Plancheral meaningful. What I worry
    about is that even with such a character formula, there isn’t
    enough for non-Abelian electromagnetic duality. In fact, I am skeptical a USEFUL duality for pure Yang-Mills theorists exists.

    To carry out a duality transformation, the Bianchi identity needs
    to be imposed by integrating over a new field (in 3+1 dimensions, this field is a one-form). Then we would like to integrate out the
    orginal gauge field to obtain a action in this new field. Doing this
    in practice is tough. There are tricks for doing it with certain character formulas, but the dual theory is a mess, since the dual
    fields are discretely valued (o.k. on the lattice, but without a good
    continuum interpretation).

    Are these new techniques are somehow better? If so, it would be very interesting.

  24. urs says:

    In the first instance Langlands is a parameterization of irreducible reps, and a determination of their character; roughly they are in bijection with conjugacy classes in another group.

    That’s the original “algebraic” Langlands thing.

    The categorification nonsense is an elaboration of this, to say *all*

    I think the categorification nonsense comes in when you pass from the original to the geometric Langlands correspondence.

    In the original Langlands setup, the Hecke operator is an ordinary linear map, acting on a space of modular forms.

    In the geometric version of the theory, it becomes the Hecke operator that acts on derived coherent sheaves on some moduli space. And that guy is no longer an ordinary linear map. But it is a categorified linear map, if you like (and also if you don’t like it).

    In particular, in a special limit it is nothing but a certain categorification of the Fourier transformation.

  25. urs says:

    I never understood what is the relation of elliptic cohomology [...] with particle physics

    Elliptic cohomology is not about particle physics. It is about string physics.

    Elliptic cohomology is to strings like particles are to K-cohomology #.

    But what is the direct relation of elliptic cohomology to geometric Langlands, that made you bring this up here?

  26. Bert Schroer says:

    Interesting, so after all elliptic cohomology isn’t about particle physics it is rather about ST. That’s precisely what I expected.

  27. Bert Schroer says:

    Urs
    Interesting, so after all elliptic cohomology isn’t about particle physics it is rather about ST. That’s precisely what I expected. I guess I got into the Langland’s column by accident, but without this accident I probably would not have received such a precise answer.

  28. urs says:

    Interesting, so after all elliptic cohomology isn’t about particle physics it is rather about ST.

    Yes, check out the table at the beginning of the introduction of those notes.

    Generalized cohomology theories are labelled by something called their “chromatic filtration”.

    The idea is that a cohomology theory of chromatic level p comes from the physics of “p-particles” – otherwise known as (p-1)-branes.

    K-cohomology has filtration 1. It corresponds to 1-particles (0-branes). Ordinary points, that is.

    Elliptic cohomology has filtration 2. It corresponds to 2-particles, otherwise known as 1-branes or strings.

    Ordinary (singular) cohomology has filtration level 0. There is a precise sense in which it corresponds to 0-particles (or (-1) branes).

    I expect this table is open ended. But I have never seen anything about cohomology theories of chromatic filtration larger than 2.

  29. urs says:

    I am skeptical a USEFUL duality for pure Yang-Mills theorists exists.

    It is a famous conjecture that 4-dimensional Yang-Mills theory has a duality called S-duality.

    Yang-Mills theories (in a given dimension, for a fixed number of supercharges) are parameterized by a complex number

    tau ,

    the coupling constant, and a Lie group

    G,

    the gauge group.

    For N=4 supersymmetric Yang-Mills, there is conjectured to be an isomorphism between Yang-Mills theory for

    (tau,G)

    and that for

    (-1/tau , G^L) .

    -1/tau is, roughly, the inverted coupling constant (therefore: “weak-strong coupling duality”) and G^L is the Lie group that is Langlands dual to G.

    See the first few paragraphs of this, for instance.

    That this is indeed an isomorphism of field theories is not a theorem, but it is supported by enough evidence that makes everybody assume it is indeed true. This is the S-duality conjecture.

    Since the Langlands dual group appears in this conjecture, it has long been speculated that there is indeed a relation between S-duality and the Langlands program. But until recently nobody could really substantiate this.

    The achievement of the Kapustin-Witten work is to show that for the special case that the 4-dimensional Yang-Mills theory is suitably compactified down to two dimensions, the S-dualiy conjecture for Yang-Mills theory is essentially equivalent to the geometric Langlands conjecture.

    All the ingredients of geometric Langlands, like those moduli spaces of bundles and the derived coherent sheaves on them, can be understood in terms of field configurations and boundary conditions of compactified N=4 super Yang-Mills theory.

    Notice that this amounts to further support for the S-duality conjecture, because it increases the number of people that truest the S-duality conjecture by those mathematicians that trust the geometric Langland conjecture.

    But it might also be noteworthy that this suggests that the geometric Langlands duality is only a tiny aspect of a much bigger story – since it is (apparently) just the special case of S-duality applied to a very specific compactification of Yang-Mills theory only.

  30. Peter Orland says:

    Urs,

    Yes, I know about the S-duality conjecture (I would much more interested in a similar conjecture about pure Yang-Mills than N=2 or N=4 Yang-Mills. Theories with adjoint matter are very different from those we know about in nature).

    Though a conjecture is nice, to really prove it operator equivalences are needed. The procedure I discussed before, character expansions of the Bianchi identity, etc., is the first step to find such equivalences. In Abelian theories, this is how Kramers-Wannier duality works. There are some non-Abelian constructions due to Sharachandra and Anishetty, they haven’t proved useful yet.

  31. A.J. says:

    Urs,

    Their’s a mild caveat to be added to your statement that

    All the ingredients of geometric Langlands, like those moduli spaces of bundles and the derived coherent sheaves on them, can be understood in terms of field configurations and boundary conditions of compactified N=4 super Yang-Mills theory.

    The geometric Langlands correspondence is stated in terms of D-modules on the moduli stack of not-necessarily stable G-bundles. Kapustin & Witten’s work doesn’t quite give full information about the moduli stack, but only its semi-stable locus. As I far as I can tell, the relation between N=4 SYM and the Langlands correspondence for D-modules on the full stack hasn’t been completely spelled out.

  32. ks says:

    Question to Urs. First of all thanks a lot for all your explanations. You work on cool stuff anyway ( though it is a little over my head at this time ). Do I understand your research program correctly when I assume that You try to link the standard model and ST in purely algebraic terms by means of higher category theory? Hence when changing the algebraic setting they do not look much different but are connected through certain higher morphisms?

  33. Bert Schroer says:

    Peter Orland
    Conceptual realism demands to separate Kramers-Wannier duality (and its structural extension the order-disorder issue) from speculative ideas. The o-d duality is a local quantum physical phenomenon which has no known analog in higher dimensions. Whereas o-d is a phenomenon which has a solid operator algebraic intrinsic understanding (if you want I can provide you with recent literature) there is nothing like this for the S conjecture.
    By now Wikipedia has more material on wild conjectures than about genuine results. There is the danger that we may be fooled to our own simulacrums and metaphors in particular that conjectures solidify because they comes from somebody with a high status in the community or because they have been hanging around for a long time so that several generations have stepped on them.

  34. urs says:

    wild conjectures

    S-duality is certainly a conjecture, but hardly a wild conjecture.

    I mean, that’s the point: S-duality is apparently as wild as geometric Langlands.

  35. urs says:

    Kapustin & Witten’s work doesn’t quite give full information about the moduli stack, but only its semi-stable locus.

    Right, thanks. There are probably a couple of such technicalities. I am not working on this stuff, so it’s hard to keep them all in mind.

    So what about that “classical limit” in which, apparently, geometric Langlands is only proven so far. Does compactified SYM exactly coincide with the geometric Langlands data in that limit?

  36. David says:

    A couple of comments: Kapustin-Witten’s theory does (as far as I
    understand) cover the full stack of bundles, not just the semistable locus. The sigma-model/mirror symmetry description
    fails outside the semistable locus, but they emphasize
    in the paper that the gauge theory sees the entire stack
    of bundles — I think the problem is us geometers
    have only been able in the past really to process
    the classical aspects of the theory (solns of the equations of motion
    etc) but quantum gauge theory is a lot smarter than we
    are (speaking for myself at least). As far as I know they
    can’t completely say what S-duality predicts off the semistable locus, but the important point is it does actually apply there.

    The classical limit of Langlands is only proven
    generically, missing the hardest locus — it’s a beautiful
    result and one of the best in the subject, but saying classical
    geometric Langlands is understood is on the same
    level as saying you understand (noncompact) Lie groups
    when you understand their diagonalizable elements —
    the hardest part involved unipotents..

    Also I’m not sure I would think of Hecke operators as Fourier transforms – the Hecke operators are the symmetries of moduli
    of bundles (and sheaves on them), while the Fourier-Mukai
    type transforms relate G and G^ the dual group.

    One sense (of many) in which geometric Langlands is a nonabelian
    categorified generalization of the Fourier transform is that while Plancherel helps you decompose spaces of functions on a group,
    geometric Langlands type results help you decompose the
    CATEGORY of all representations of a group —
    since these categories are not semisimple
    there’s a big difference between listing irreducibles and their characters and actually describing the structure of general
    representations. (Geometric Langlands ideas
    can be used to study for example the category
    of Harish-Chandra modules for a real semisimple Lie group).

  37. Peter Orland says:

    Bert,

    I cannot understand your explanation especially well. In my attempt to translate your statement into simple language, I conclude you mean more conjectures than solid statements dominate our field. I don’t need to be reminded of this, since I have
    seen it all over the literature for the last decade or so.

    I was asking if the experts on Langlands believe a useful concrete electric-magnetic duality transformation can be constructed from non-Abelian Fourier transforms (character expansions). I suspect the answer is no, since no one gave me a simple “yes”.

  38. I’m probably mixing algebraic number theory with analytic number theory but is there a relationship between elliptic cohomology and elliptic Mobius transformations?

  39. urs says:

    Also I’m not sure I would think of Hecke operators as Fourier transforms – the Hecke operators are the symmetries of moduli
    of bundles

    Oh, sorry, I misspoke if I said that. The Langlands correspondence is analogous to the Fourier transform, exchanging skyscraper sheaves (analogous to delta-functions) with Hecke-eigensheaves (analogous to plane waves). So, in this analogy, the Hecke operator is like a categorified derivative.

  40. Bert Schroer says:

    I am afraid the sad truth is the answer is “no”. It is better to live in quantum reality than to become complacent with a Disney version of it.
    I was not trying to explain anything in technical terms but only pointing to the obvious observation that Kramers-Wannier on a microscopic level (achieved by Leo Kadanoff) was quantum from the beginning whereas the Seiberg Witten duality is from a physical Disney dreamland which precisely of this is so useful to a large part of mathematics. The kind of mathematics for which it had no use is the operator-algebraic mathematical setting of QT which dates back to von Neumann and has been enriched by the locality principle in AQFT. By the way the manner Kadanoff has extracted (noncommutative) operator commutation relations for the (what we nowadays call) the Ising primary fields from the Euclidean lattice setting (via a partially guessed properties of the transfer matrix formalism) had my deep admiration; the Leitmotiv of all my work with Swieca in the early 70s was related to adapt ate Kadanoff’s order/disorder ideas to the continuous setting of QFT; in many cases we even succeeded to read this back into a continuous functional integrals setting by using an Aharonov-Bohm analog language. Later, when I was working with Rehren on an algebraic approach to chiral conformal QFT I remembered those Kadanoff ideas and we found a completely explicit operator version of an “exchange algebra” for the conformal Ising field theory from which it was possible to compute its n-point Wightman functions. A historical review can be found in
    http://br.arxiv.org/abs/hep-th/0504206
    but thinking about this now, I should have written much more about Leo Kadanoff’s contributions; he really deserved a Nobel prize together with Wilson.
    In those days we also convinced ourselves that this order-disorder idea has no electric-magnetic counterpart in the full QFT setting.

  41. Peter Orland says:

    Bert,

    I also worked extensively on duality. Like you, I concluded that
    there is no simple operator equivalence between a non-Abelian
    gauge theory and its dual. But there are intriguing exceptions
    of systems with non-Abelian systems which do have duality
    transformations and disorder operators. In my Ph.D. thesis I found lattice systems with permutation-group $S_{N}$ symmetry which have nontrivial duals. But I will spare people here from a list of more publications on the subject.

    Regards,
    Peter (O.)

  42. Thomas Larsson says:

    Conceptual realism demands to separate Kramers-Wannier duality (and its structural extension the order-disorder issue) from speculative ideas. The o-d duality is a local quantum physical phenomenon which has no known analog in higher dimensions.

    ???

    The 3D Ising model on a cubic lattice is Kramers-Wannier dual to Ising gauge theory on the same lattice. Why is this not o-d duality in higher dimensions?

  43. urs says:

    I concluded that there is no simple operator equivalence between a non-Abelian gauge theory and its dual.

    Is this saying that you consider the S-duality conjecture to be in fact false?

    If so, I’d be interested in the details of the assumptions that go into this.

    I recall that Bert Schroer was (similarly ?) claiming that the AdS/CFT duality conjecture (in the sense of Maldacena) is false, and that the correct duality statement was along the lines of Rehren’s work.

    In that case I got the impression that two rather different concepts were being compared, and that in fact Rehren’s work had little relation to the setup considered by Maldacena et al. Compare for instance Jacques Distler’s account.

    The crucial difference in this case is that Rehren’s work was based on a fixed and precise axiom set, while Maldacena’s work uses notions of quantum field theory that have not been axiomatized yet.

    For people like Bert Schroer this is reason enough to completely reject all QFT that does not fit into the AQFT axioms. For other people, in contrast, the restrictive applicability of the AQFT axioms is reason enough to reject those.

    To some extent it is a matter of taste concerning which role of rigour you find useful in physics research. I can easily tolerate both these standpoints. But I would like to know in each case which one is assumed by which participant.

  44. woit says:

    Urs,

    You keep ignoring the fact that Peter Orland is asking about pure YM theory, not N=4 SYM. There’s a beautiful story about duality in non-supersymmetric abelian gauge theories, and many people (including Peter) have tried hard to generalize this to the non-abelian case. I gather that he’s trying to understand whether geometric Langlands gives any insight into that problem, and as far as I can tell, the answer is just no.

  45. Peter Orland says:

    Urs,

    Sorry that I am giving long-winded answers to your questions. I am
    mainly interested in advancing methods in asymptotically-free field theories and in constructions which could eventually facilitate calculations. I try to learn other stuff, because I can’t predict what I may need to know in the future. But I am more interested in theoretical, rather than mathematical physics (as people abuse use the term nowadays, to study mathematical techniques, rather than to prove theorems).

    I believe (after some years of trying to show the contrary) there is no USEFUL version of Kramers-Wannier duality which is true for PURE non-Abelian gauge theories. There are non-Abelian dualities for some special $S_N$-invariant systems, which I mentioned above (there is also non-Abelian Bosonization in two dimensions).

    The general problem for duality in non-Abelian theories is constructing dual fields with local commutation or anti-commutation relations. Supersymmetric or other theories with adoint matter have some sort of charge-monopole duality – but such theories are effectively Abelian. These theories are interesting in their own right, but to my way of thinking, they are not as important as Yang-Mills theories coupled only to fundamental (not adjoint) Fermion color charges, or pure Yang-Mills theories.

    There are other notions of duality in QCD. The ‘t Hooft loop is the disorder operator. Unfortunately, there is probably no useful local dual-field-theory formulation for which it is the order parameter.

  46. urs says:

    You keep ignoring the fact that Peter Orland is asking about pure YM theory, not N=4 SYM.

    In as far as I am ignoring anything, it is not on purpose. I’d be glad to be enlightened.

    Maybe I found Peter Orland’s statement

    I concluded that there is no simple operator equivalence between a non-Abelian gauge theory and its dual.

    #

    seemed to refer to arbitrary gauge theories.

    I gather that he’s trying to understand whether geometric Langlands gives any insight into that problem, and as far as I can tell, the answer is just no.

    Hm, maybe here is the source of the misunderstanding. Kapustin-Witten show that geometric Langlands does give insight into the type of duality present in N=4 SYM. So in far as this is different to other types of duality, geometric Langlands apparently does not apply to these.

    Supersymmetric or other theories with adoint matter have some sort of charge-monopole duality – but such theories are effectively Abelian.

    Could you expand on what you mean by “effectively abelian” here? Thanks!

  47. Peter Orland says:

    Urs,

    By “effectively Abelian”, I mean that that the magnetic-monopole charge is well-defined and quantized. In QCD or pure Yang-Mills, there is no precise definition of magnetic-monopole charge.

    In the Georgi-Glashow model (an the related deformation of N=2 supersymmetric gauge theory) a Higgs field breaks the gauge group
    down to the Cartan subgroup. Thus there are Abelian monopoles,
    with quantized charge, etc. These theories have a confined phase for sufficiently small monopole mass, which goes back to Polyakov’s observations in the 70’s. Duality for such theories is not so different from those of Abelian Wilson lattice gauge theories. They are, however, quite different from QCD.

    Now there is an old result made by many people (Fradkin, Shenker, Rabinovici and others) that there is little difference between a Higgs field in a gauge theory and a scalar field in that gauge theory without a Higgs potential. The basic point is that the operator creating a massive vector Boson in the Higgs theory looks just like the operator creating a “meson” built from scalars in the confined phase. From this point of view, any theory with scalar matter
    is not so different from a Higgs theory. In particular, it is possible
    to define magnetic charge, no matter what the scalar potential happens to be. So in such theories charge-monopole duality is a sensible concept.

    The reason why the possibility of duality for Yang-Mills theories is interesting is because it could yield insight into the confinement
    phase. Some sort of magnetic condensation occurs, producing confinement and a mass gap, as simulations show, but we want
    to know why.

  48. anonymous says:

    Off-topic mathematical physics fun: Andre LeClair is claiming there’s a physical system, which, on physical grounds, suggests the Riemann hypothesis is true. Are there any experts around to comment on whether it’s plausible?

    http://www.arxiv.org/abs/math-ph/0611043

  49. relativist says:

    For those like me who don’t know much about the Langlands programme but would like to, a useful account is an older one by Frenkel: `Lectures on the Langlands Program and conformal field theory’, at
    http://www.arxiv.org/PS_cache/hep-th/pdf/0512/0512172.pdf

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