Gaitsgory and Raskin Prove Geometric Langlands Conjecture, Advancing Mathematics and Physics

A team of nine mathematicians, led by Dennis Gaitsgory and Sam Raskin, has proven the geometric Langlands conjecture, a decades-long problem originating from Robert Langlands’ work connecting number theory and harmonic analysis. This proof builds upon earlier contributions from Alexander Beilinson, Vladimir Drinfeld, Dima Arinkin, Peter Scholze and Laurent Fargues, and is expected to advance research on local versions of the Langlands conjectures. The achievement also draws parallels to S-duality in quantum field theory, as demonstrated by Edward Witten and Anton Kapustin in 2007, potentially offering new approaches to number-theoretic problems.

Origins and Scope of the Langlands Programme

The Langlands programme began approximately 60 years ago with the work of Robert Langlands, who posited a connection between number theory and harmonic analysis. An earlier success within the programme was the 1995 proof of Fermat’s Last Theorem by Andrew Wiles, demonstrating the potential for resolving long-standing mathematical problems through this framework. The geometric Langlands conjecture, formulated in the 1980s by Vladimir Drinfeld, proposes a correspondence between different mathematical objects, specifically relating to the properties of Riemann surfaces.

The proof of the geometric Langlands conjecture may offer progress on the arithmetic version of the Langlands programme, given the close relationship between the mathematical objects involved. Understanding this correspondence requires recognising that the two mathematical worlds connected by the programme are not separate entities, but rather different facets of a single underlying reality. Establishing a relationship between a surface’s fundamental group – describing how loops can be tied around it – and specific types of sheaves, tools used in algebraic geometry, was central to the proof.

The recent proof builds upon earlier work, with Alexander Beilinson and Vladimir Drinfeld having described how to construct the necessary sheaves in the 1990s. Further refinement of this relationship occurred in 2012 through the contributions of Dima Arinkin and Dennis Gaitsgory, with Gaitsgory subsequently outlining a potential path toward proving the conjecture. The proof is expected to boost research on local versions of the Langlands conjectures, which focus on specific objects within the broader global settings, and may offer opportunities for Langlands programme applications in these areas.

Peter Scholze has played a key role in connecting the local and global programmes, and his work, alongside that of Laurent Fargues, has created a pathway for applying methods from the global geometric Langlands programme to the local arithmetic context. Furthermore, the geometric Langlands programme exhibits surprising connections to theoretical physics, specifically to the concept of S-duality, a symmetry relating electric and magnetic fields in quantum field theory. Edward Witten and Anton Kapustin demonstrated in 2007 that S-duality mirrors the symmetry of the geometric Langlands correspondence, suggesting a deep connection between the mathematical programme and fundamental symmetries in physics. Minhyong Kim is currently investigating ways to rigorously establish these connections, utilising concepts from physics as metaphors for problems in number theory.

One immediate consequence of the proof is its potential to boost research on local versions of the Langlands conjectures, which focus on specific objects within the number theory. These local contexts offer a refined focus for investigation.

Furthermore, the geometric Langlands programme has surprising connections to theoretical physics, particularly to the concept of S-duality – a symmetry in quantum field theory relating electric and magnetic fields. Edward Witten and Anton Kapustin demonstrated in 2007 that S-duality possesses a symmetry mirroring that of the geometric Langlands correspondence, suggesting that the mathematical programme may reflect a deep symmetry within quantum physics.