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.