There’s an interesting development in the math-physics overlap, with a significant number of physicists getting interested in the theory of automorphic forms, often motivated by the problem of computing string scattering amplitudes. This has led to a group of them writing up a very long and quite good expository treatment of Eisenstein series and automorphic representations, which recently appeared on the arXiv. The emphasis is not on the physics applications (which an introduction explains come about when one is dealing with systems with discrete symmetries like the modular group or higher dimensional generalizations), but on the calculational details of the mathematics. There are quite a few expositions of this material in the mathematics literature but many (mathematicians included), may find the detailed treatment here very helpful.
Another aspect of this area is some overlap with the interesting of mathematicians studying Eisenstein series in the context of Kac-Moody groups. There’s a conference this week bringing together mathematicians and physicists around this topic.
Turning to recent developments in the Langlands correspondence itself, which relates automorphic forms to Galois representations, when I discussed David Nadler’s talk at the Breakthrough Prize symposium (the video is available here), I forgot to mention one thing he talked about that was new to me, the Fargues-Fontaine curve. Nadler explained that Fargues has recently conjectured that the local Langlands correspondence can be understood in terms of ideas from the geometric Langlands correspondence, using the Fargues-Fontaine curve. For more about this from Fargues himself, see materials at his website, which include lecture notes, links to videos of talks at the IAS and MSRI, and this recent survey article. Also informative is some explanation from David Ben-Zvi at MathOverflow.
In April there will be a workshop in Oberwolfach on geometric Langlands that will include this topic, for details of the planned discussions, see here.
Fargues was here today at Columbia, and gave a talk on “p-adic twistors”. Nothing much about Langlands, this was about the question of what the analog is for the Fargues-Fontaine curve when you take the real numbers as your field (the use of “twistors” is that of Simpson’s, see here, not the common use in physics, which is quite different).
I won’t display my extremely limited understanding of this subject by trying to provide my own explanations here. A big problem is that this is mainly about the p-adic Langlands correspondence, something I’ve never been able to understand much about. After making a renewed attempt the last few days, I at least started to get some idea of what are the biggest problematic holes in my knowledge of the math background. Interestingly, it seems many if not most of them have Tate’s name attached (Hodge-Tate, Lubin-Tate, etc, etc…). One pleasant discovery I made is that there are now some excellent expository pieces on this material available, often courtesy of some talented graduate students. One wonderful source I ran into is Alex Youcis’s blog Hard Arithmetic, which has given me some hope that with his help I might soon make a little progress on learning more about this kind of mathematics. I don’t know what’s in the water at Berkeley, but something there keeps producing high-quality blogging by mathematics students, another example is here.
“what’s in the water at Berkeley”
what is this “water” that you speak of ?
zzz,
Maybe that’s it, with the drought they’re drinking something else that would explain the phenomenon…
I can tell you that the bottom of the reservoir does not taste good right now.
Can I ask a naive question here, though in the spirit of the blog? As a mathematician working on and around the Langlands program for over a decade now, I’m sometimes asked about applications to physics. I can mumble “conformal field theory” and “partition function” and “BPS state”. I even talked to a physicist asking about automorphic forms on exceptional groups a few times. But I don’t really know enough to know where this fits in physics.
So really… what’s the shortest path from an honest *theorem/conjecture* in the Langlands program (geometric Langlands is fine) to an honest *physical observation/experiment*? Analogies don’t count. Also… just because “conformal field theory” is full of physics words/motivations… mentioning it doesn’t count on its own.
Is the motivation of physicists to understand the Langlands program limited to the string theorists?
Marty,
No, the motivation to understand the Langlands program is not limited to string theorists (I’m at least one counterexample…)
Maybe one should first differentiate between what mathematicians often mean when they say the “Langlands program” (relating arithmetic questions about Galois groups to automorphic representations) and just one side of that (the automorphic side). One example of quantum field theory that relates both sides is the Witten et al. stuff dealing with the geometric Langlands case, in terms of a mirror symmetry, with its origin in the modular invariance of a twisted N=4 supersymmetric 4d qft (with that explained in terms of a superconformally invariant 6d qft).
As far as I know, the connections to string theory are just on the automorphic side, with one source the modular invariance of string theory amplitudes, explaining why you expect to get automorphic forms.
My own interest (and I think this has also motivated a lot of other work) is in the analogy between automorphic representations and conformal field theory on a Riemann surface, which Witten first pointed out back in 1988 (in “Quantum field theory, Grassmanians, and algebraic curves”). As I’ve learned more and more about the function field and arithmetic cases, the way automorphic representations appear there seems to me to indicate a profound unification between those subjects and those parts of QFT which we are able to understand purely in terms of representation theory. I think there’s a lot more to discover there, but, maybe that’s just me…
Besides the later connection to 4d SSYM, the geometric Langlands stuff from the beginning involved a crucial use of conformal field theory (see for example Frenkel’s 1994 “Affine algebras, Langlands duality and the Bethe ansatz”). As far as I know, the connection between this appearance of QFT in geometric Langlands and the later 4d stuff remains poorly understood.
I keep seeing all sorts of interesting suggested ideas and analogies in this area, the above doesn’t do justice to the range of such ideas. Conformal field theory plays a central role and I would describe that as 2d qft, not string theory (which to me means you try to integrate over the moduli of curves), but that’s an argument over words that is best discouraged (I’ll delete any efforts to carry it on here).
Marty – from my point of view it seems like the flow of information is predominantly in the other direction, from physics to the Langlands program, though perhaps the great breakthrough was the realization that [geometric] Langlands duality and electric-magnetic duality are both part of the same big picture. The geometric Langlands program is amazingly well explained from the perspective of physics — it fits perfectly into the structure of four dimensional gauge theory and even better into that of six dimensional conformal field theory (the relations with 2d CFT are AFAIK subsumed inside these bigger structures). This picture is subtle enough that it appears to know all of the bells and whistles and complications people have discovered in geometric Langlands — at least if you ask it the right questions. Further it suggests a richer structure behind geometric Langlands than one would have imagined before.
From the point of view of physics geometric Langlands allows one to probe electric-magnetic duality on various kinds of defects in certain highly supersymmetric quantum field theories. Some of the structures in the math remain somewhat esoteric from a physics perspective, but others don’t — for example, the appearance of W-algebras in the work of Feigin-Frenkel and Beilinson-Drinfeld (the link to CFT that Peter mentions) is directly tied to one of the hottest topics in gauge theory in recent years, the AGT conjecture. W-algebras themselves are an aspect of the theory of Whittaker vectors, a fundamental theme throughout the Langlands program, which in physicists’ hands becomes something very geometric about “cigars” (making 2 of your 6 dimensions look like a cylinder capped off with a disc at one end). The relation of geometric Langlands with this 6d CFT more broadly puts in right at the heart of many questions in supersymmetric gauge theories, since maybe half of the so-called N=2 SUSY gauge theories (the subject of Seiberg-Witten theory etc) come from compactifications of this one theory, so that one finds the same structures scattered throughout the subject.
By far the clearest introduction to the FF-curve is the guest blog post that Jared Weinstein gave on Persiflage blog—link below
https://galoisrepresentations.wordpress.com/2013/07/23/the-fundamental-curve-of-p-adic-hodge-theory-or-how-to-un-tilt-a-tilted-field/
Thanks David and sdf,
I had forgotten to include a link to that Jared Weinstein blog entry, which is a great example of an expository blog article. There’s a second part to it at
https://galoisrepresentations.wordpress.com/2013/08/12/the-fundamental-curve-of-p-adic-hodge-theory-part-ii/