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?
Kapranov’s Analogies – Understanding Kapranov’s Analogies in Arithmetic Geometry
arithmetic-geometrybig-picturelanglands-conjectures
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The analogy doesn't quite give a number theoretic version of the Poincare conjecture. See Sikora, "Analogies between group actions on 3-manifolds and number fields" (arXiv:0107210): the author states the Poincare conjecture as "S3 is the only closed 3-manifold with no unbranched covers." The analogous statement in number theory is that Q is the only number field with no unramified extensions, and indeed he points out that there are a few known counterexamples, such as the imaginary quadratic fields with class number 1.
The paper also has a nice but short summary of the so-called "MKR dictionary" relating 3-manifolds to number fields in section 2. Morishita's expository article on the subject, arXiv:0904.3399, has more to say about what knot complements, meridians and longitudes, knot groups, etc. are, but I don't think there's an explanation of what knot surgery would be and so I'm not sure how Kirby calculus fits into the picture.
Edit: An article by B. Morin on Sikora's dictionary (and how it relates to Lichtenbaum's cohomology, p. 28): "he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology."
This is a "big-picture" question, but allow me to illustrate some recent progress by taking a small example close to my heart.
Let us adjoin to the field $\mathbb{Q}_p$ a primitive $l$-th root of $1$, where $p$ and $l$ are primes, to get the extension $K|\mathbb{Q}_p$. We notice that this extension is unramified if $l\neq p$ but ramified if $l=p$. When we adjoin all the $l$-power roots of $1$, we get the $l$-adic cyclotomic character $\chi_l:\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)\to\mathbb{Q}_l^\times$ which is unramified if $l\neq p$ but ramified if $l=p$. But we cannot just say that $\chi_p$ is ramified and be done with it. We have to somehow express the fact that $\chi_p$ is a natural and a "nice" character, not an arbitrary character $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)\to\mathbb{Q}_p^\times$, of which there are very many because the topologies on the groups $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$, $\mathbb{Q}_p^\times$ are somehow "compatible".
The fact that $\chi_p$ is a "nice" character is expressed by saying that it is crystalline. In general, we can talk of crystalline representions of $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$ on finite-dimensional spaces over $\mathbb{Q}_p$; the actual definition is in terms of a certain ring $\mathbf{B}_{\text{cris}}$, constructed by Fontaine, which can be understood in terms of crystalline cohomology.
My illustrative example is about the $l$-adic criterion for an abelian variety $A$ over $\mathbb{Q}_p$ to have good reduction. For $l\neq p$, this can be found in a paper by Serre and Tate in the Annals, and it is called the Néron-Ogg-Shafarevich criterion. It says that $A$ has good reduction if and only if the representation of $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$ on the $l$-adic Tate module $V_l(A)$ is unramified.
What happens when $l=p$ ? It is too much to expect that $V_p(A)$ be an unramified representation when $A$ has good reduction; we have seen that even $\chi_p$ is not unramified. What Fontaine proved is that the $p$-adic representation $V_p(A)$ is crystalline (if $A$ has good reduction). To complete the analogy with the case $l\neq p$, Coleman and Iovita proved in a paper in Duke that, conversely, if the representation $V_p(A)$ is crystalline, then the abelian variety $A$ has good reduction.
I hope you find this enticing.
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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.