Kapranov's analogies

Question

I just wonder about Kapranov's "Analogies between Langlands Correspondence and topological QFT". I would like to read a more detailed exposition and how one turns that analogy into concrete conjectures. Do there exist texts from courses or seminars on that? Has this book by İkeda already been published and reviewed?

Answer 1

There are notes from more recent (2000) lectures of Kapranov on the subject on my webpage.

Despite this, as far as I know, there isn't any convincing argument at this point that there should be a general higher dimensional Langlands theory. What we learn from physicists is that one can expect deep dualities for algebraic surfaces and threefolds, but probably not in general. And we also learn that the shape of these dualities won't be one you are likely to guess from analogies with curves and with higher dimensional class field theory, but for which string theory is a great guide. One way to put this is that we now (post 2005) understand that geometric Langlands is an aspect of four-dimensional topological field theory (more specifically, maximally supersymmetric gauge theory) — so one ought to look for higher dimensional analogs of this, and they are very very few and very special.

Perhaps the main issue in making direct generalizations (that Kapranov addresses) is the great complexity in describing Hecke algebras in higher dimensions — in fact this is one of the most exciting areas of current research, under the name Hall Algebras (cf work of Joyce, Kontsevich-Soibelman and others). It turns out that Hall algebras of 3-dimensional (Calabi-Yau) varieties have a remarkably rich representation theoretic structure, which (upon dimensional reduction to surfaces or curves) encapsulates much of classical representation theory and geometric Langlands etc.

In any case the most exciting thing around as far as I'm concerned is a 6-dimensional supersymmetric field theory (which lacks a good name — it's called the (0,2) theory or the M5 brane theory) which seems to enfold everything we know in all of the above examples. It is a conformal field theory, and "explains" Langlands duality simply as the SL2(Z) conformal symmetry of tori (when one considers it on R^4 times a torus). One can expect this theory to lead to an interesting Langlands-like program for algebraic surfaces, which includes in particular the interesting recent work in this direction of Braverman-Finkelberg (arXiv:0711.2083 and 0908.3390).

I should mention also that since Kapranov's paper there has been a lot of work, by Kazhdan, Braverman, Gaitsgory and Kapranov in particular, on representation theory for higher dimensional local fields — it's just that there isn't yet a clear Langlands type picture for this. Again one should expect something special to happen for surfaces, of which we have many glimpses — in particular the beautiful theory of Cherednik's double affine Hecke algebra and its Fourier transform, which is I think accepted as a clear hint at a Langlands picture for surfaces. But in any case I strongly believe that the current evidence indicates we should be learning from the physicists, reading papers by Gaiotto etc, to figure out what should happen for surfaces rather than just believing that we should try to extrapolate from what we know for curves.