Before I turn to the main topic of this posting, a lecture by Jacob Lurie, I’d like to point to something else involving him, a comment and posting at Mathematics Without Apologies, a blog you should be following anyway. On the topic of the usefulness of “proof assistants”, I liked Lurie’s point that a major problem with this is:
Working in a formal system, more or less by definition, means that you can’t ignore steps which are routine and focus attention on the ones that contain the fundamental ideas.
But if you want to discuss this, it should be over there, the topic of this posting is something very different.
Last week I noticed that Lurie had given a talk at Harvard on “Categorifying Fourier Theory”, which is available here. I enjoyed watching it, ending up quite intrigued by the abstract picture he was painting, but rather discouraged by the lack of any example that would give insight into what it might be useful for. Neglecting to mention the example that explains why an abstract theorem is useful is unfortunately all-too-common practice among mathematicians. Perhaps in this case with Dick Gross, Jean-Pierre Serre and John Tate in the front row, he felt it unnecessary. Luckily though, he gave the same talk recently in Arizona, and there (in the question session) did give a fascinating motivating example.
His starting point was the Fourier transform, which one can generalize to any abelian group G, and think of as identifying complex functions on G with complex functions on the dual (or character) group G^. The standard Fourier transform is the case G=G^=R, Fourier series are the case G=U(1), G^=Z. He then went on to discuss two levels of abstraction, or categorification of this. The first identifies a representation of G on a vector space V with a function from G^ to vector spaces (the isotypic decomposition of the representation). The second identifies representations of G on categories with representations of G^ on categories.
It was this equivalence of representations on categories that was his main result, for which in Arizona he gave the example of G the group of invertible Laurent series. The idea is that this group can be identified with its dual group G^ (in some sense as algebraic groups), using the Weil symbol (for a definition and context, see here). Lurie’s claim that was new to me was that the equivalence in this case is essentially the GL(1) version of the general local geometric Langlands conjecture, which is supposed to be an equivalence of two representations on categories, for more general (non-abelian) groups G.
At least for me, understanding of some sophisticated mathematical phenomenon really starts when I understand the simplest example of the phenomenon. For the number field case of Langlands theory, my initial efforts to understand the subject didn’t lead anywhere until I realized that maybe it was best to first think about the local version, which was a statement about representation theory that I could make some sense of. I was hopeful that thinking about the simplest case of that, the abelian case, would give great insight, found though that the Abelian case is already quite non-trivial (local class field theory). For the geometric Langlands case, I found that the discussion of the local version in Edward Frenkel’s book was very helpful, but I always wondered about the abelian case. Now I’m hopeful that the abelian story is something that although I’ve never seen it, is well-understood, and that a helpful reader will point me to a reference.
Another reason for being interested in this particular topic is that it has some connection to the relationship between Langlands theory and QFT that first got me interested in all of this. Back in 1987 Witten wrote some fascinating papers giving an abstract formulation of free fermion theories on Riemann surfaces (see here and here) with tantalizing connections to what later became geometric Langlands. In this work the group of invertible Laurent series and the Weil symbol play a central role. There was also later work by Takhtajan on this, see here and here. I wonder why the most recent version of the last reference deletes the material on the multiplicative group case, which is the one Lurie mentions.
The abelian case of geometric Langlands is geometric class field theory. Serre’s text on algebraic groups and class fields is the standard reference. But there are many expositions! See http://mathoverflow.net/questions/73054/a-reference-for-geometric-class-field-theory for links.
Thanks Marty,
What I’m curious about is something that should be much simpler, the local case, and just for the group GL(1,C). Lurie’s argument seems to indicate that this case follows from his very general, very simple construction. Also, he’s claiming an equivalence of group actions on categories, and I’d like to know more about what that means here (as opposed to just a usual Langlands correspondence of representations).
Peter,
I’ve never understood what this langlands QFT connection is supposed to have anything to do with new physics. Is this purely a mathematical interest, or is there potential for a new stimulating insight into fundamental physics?
Justin,
Personally I believe the connections between qft and the representation theory point of view on number theory (e.g. Langlands theory) indicate a deep unity that if better understood may lead to new insights in fundamental physics, for more, see
ww.math.columbia.edu/~woit/mathphys.pdf
But that’s a far off goal, and the question I’m asking here is a purely mathematical one, I was just pointing out the tantalizing connection to QFT.
Thanks for the links Peter! I haven’t yet watched the videos, which sound great, but here’s some perspective on the story. First of all the local geometric Langlands conjecture is (roughly speaking) an equivalence of two-categories, one associated to G and one associated to the dual group G^. This is completely natural from the physics — a four-dimensional topological field theory (such as twisted N=4 super Yang Mills) should attach a 2-category to a closed 1-manifold, in this case the circle, so that an equivalence of field theories (S-duality) gives an equivalence of 2-categories. Roughly speaking, following Frenkel-Gaitsgory, one side is the 2-category of categorical representations of the loop group, and the other is the 2-category of sheaves of categories over the stack of Langlands parameters — very roughly, modules over the tensor category of quasicoherent sheaves on the space of Langlands paramters. In the abelian case, geometric Langlands is captured by Cartier duality (algebraic Pontrjagin duality, i.e. Fourier transform) — this goes back at least to Laumon’s work on Fourier transforms and that of Rothstein. There they work in the global setting, using the self-duality of the Jacobian, but this has a well-known local origin, the Contou-Carrere Cartier self-duality of invertible Laurent series (or of its quotient by invertible Taylor series, i.e., the abelian affine Grassmannian). This is what Jacob is generalizing I believe, though I need to find out what the precise statement is. Sadly as with most things in this subject it’s hard to find introductory references.
In any case let’s work out the GL_1 case in the simpler Betti form (the Betti Geometric Langlands conjecture is a new modified form of the usual de Rham one which, unlike its predecessor, really is a TFT, an algebraic model of Kapustin-Witten’s – to appear). On both sides we’re not dealing with any old two-categories, but with monoidal categories (the 2-categories being the collection of their module categories). On the automorphic side this is the “categorical group algebra” of the group K^* of Laurent series — i.e. the monoidal category built so that its module categories are simply categories with an action of the group. This means (in the Betti version) local systems on K^* with convolution (in the de Rham case this is replaced with D-modules). But K^* splits as a product of two contractible pieces (the infinite dimensional vector space of monic Taylor series and its dual, the formal group of K/O) which don’t affect the category, and then Z (counting degree of leading term) and C^* (the coefficient of the leading term). So we need to describe local systems on Z x C^*, with convolution. Sheaves on Z are the same as representations of the dual multiplicative group C^*, ie quasicoherent sheaves on pt/C^*. The Mellin transform (multiplicative Fourier transform) identifies local systems on C^* (ie reps of pi_1 = Z) with quasicoherent sheaves on the dual torus C^* (labeling monodromies). So we have an equivalence of monoidal categories betweek Loc(K^*) with convolution and QCoh(C^* x pt/C^*) with tensor product, hence of their 2-categories of module categories. Now note the latter is C^*/C^*= C^* local systems on the punctured disc — i.e., the space of Langlands parameters (it’s fun to work this out for general tori to not get confused between the group and its dual). This is the (Betti) local geometric class field theory (the de Rham version involves the stack of rank one connections on the punctured disc, which is more painful).
As an illustration of Betti vs de Rham by the way it’s useful to look at how we can think of sheaves on C^*. In the Betti version, local systems=reps of pi_1=modules for group algebra of pi_1=quasicoherent sheaves on the dual C^*. In the de Rham case, D-modules on C^*= modules for C[z,z^-1] adjoined z d/dz = (by Mellin transform) modules for C[s] (s corresponding to z d/dz) adjoined t, t^-1 (corresponding to z) which acts on s as shift by integers. In other words we have difference modules on the dual C, which are the same as Z-equivariant quasicoherent sheaves on C. ANALYTICALLY these would be the same as sheaves on the quotient C/Z= C^*, but algebraically the two categories are very different. However, they both contain the same “basic objects of interest”, the “eigensheaves” (rank one local systems with various monodromies) and things finitely constructed out of them (finite rank local systems=flat connections). Thus the two theories are different ways to “integrate” the same basic objects. The claim is the kind of objects seen by Kapustin-Witten are closer to the former than the latter.
Thanks David!
That’s very helpful. It would be great if someone would write this sort of thing up in detail. I’m still quite curious how one might understand the picture you’re explaining in terms of Lurie’s “categorification” of Pontryagin duality.
Peter wrote: ” I’m still quite curious how one might understand the picture you’re explaining in terms of Lurie’s “categorification” of Pontryagin duality.”
I can’t tell what level of understanding you’re at and what you’re wondering about, so I’ll assume this remark means Ben-Zvi’s story is not screaming out categorification to you, as it is to me.
The point is that sheaves are like functions, but functions on a space form a set, while sheaves form a category. It’s good to draw up a big “analogy chart” with two columns, at left listing things you can do with the set of complex-valued functions on a space (or abelian group), at right the corresponding things you can do with the category of sheaves over that space. Most fundamentally, you can add and multiply functions on a space, and you can convolve functions on a group. All these operations have analogues for sheaves! Think about the analogue of the “group algebra” over in the right column. The fun starts when you get to the Fourier transform.
John,
Thanks for the further comments elucidating Ben-Zvi. I do see the general “categorification” going on here, I’m afraid I wasn’t very clear about what I still don’t see. It’s really about this specific example, how the identification given by the Weil pairing and the general story about group actions on categories told by Lurie ends up giving the equivalence between the things David describes. I suppose I just need to think carefully about the hints he provides…
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