For several years now, David Ben-Zvi, Yiannis Sakellaridis and Akshay Venkatesh have been working on a project involving a relative version of Langlands duality, which among many other things provides a perspective on L-functions and periods of automorphic forms inspired by the quantum field theory point of view on geometric Langlands. For some talks about this, see quite a few by David Ben-Zvi (for example, talks here, here, here and here, slides here and here), the 2022 ICM contribution from Yiannis Sakellaridis, and Akshay Venkatesh’s lectures at the 2022 Arizona Winter School (videos here and here, slides here and here). Also helpful are notes from Ben-Zvi’s Spring 2021 graduate course (see here and here).
A paper giving details of this work has now appeared, with the daunting length of 451 pages. I’m looking forward to going through it, and learning more about the wide range of ideas involved. A recent post advertised James’s Arthur’s 204 page explanation of the original work of Langlands, and the ongoing progress on the original number field versions of his conjectures. It’s worth noting that while there are many connections to the ideas originating with Langlands, this new work shows that the “Langlands program” has expanded into a striking vision relating different areas of mathematics, with a strong connection to deep ideas about quantization and quantum field theory. The way in which these ideas bring together number theory and quantum field theory provide new evidence for the deep unity of fundamental ideas about mathematics and physics.
Peter,
how should a traditional physicist imagine the progress that the Langlands program will bring or could bring to quantum field theory?
Klaus,
I’m afraid a traditional physicist can happily just ignore the Langlands program. But at a deep level I’d argue that representation theory and quantum theory are closely related, and the Langlands program is a vast generalization of representation theory, seemingly with connections to QFT that could illuminate the relation between QFT and representation theory.
For some notes on the relation between QM and representation theory, see
https://www.math.columbia.edu/~woit/qmnumbertheory.pdf
One aim of those notes is to explain the theta correspondence, and this seems to be a point where there is some relation (that I don’t currently understand…) to the relative Langlands story here.
Peter, thanks for the kind post!
The theory of theta series and the oscillator reps does fit very nicely into the arithmetic QFT metaphor / fairy tale (all the precise math in our paper is in the function field setting).
Let’s think of the Hilbert space of L^2 functions on the line over F a local field (real or p-adic) not just as individual quantum mechanical systems but as the states of an “arithmetic 3d QFT” on the “arithmetic 2-manifold” F. Likewise for quantizations of other symplectic vector spaces than the plane (eg other theta correspondences).
Now we upgrade this quantum mechanics story by considering its symmetries, which come by quantizing groups G of hamiltonian symmetries of the plane. We could take the multiplicative group for the Tate-Iwasawa theory of abelian L-functions or the full SL2 (or metaplectic group) for the theta correspondence.
To each of these groups G we have [on level of metaphor!!] an “arithmetic 4d QFT”, which to a global field and fixed level structure (“arithmetic 3-manifold”) attaches the space of automorphic forms (with level structure coming from a defect along a “link”, the conductor).
The “oscillator” 3d QFT is now upgraded to a boundary theory for this 4d QFT. The state associated to this boundary theory on our arithmetic 3-manifold is the theta-series (so the period of an automorphic form is its pairing with another state – whence integral representations of L-functions). More generally the theta series gives an operator from L2(R) (or an adelic version) to the space of automorphic forms, which can be interpreted as the path integral with a prescribed boundary theory. Finally all this story has an electric-magnetic dual description in geometry of spaces of Galois representations into the dual group of G, which becomes the role of the theta correspondence as realizing cases of Langlands functoriality.
Thanks David!
That’s very helpful in giving me an idea of how this is supposed to work in a context where I have some idea what is going on.