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Relative Langlands Duality

David Ben-Zvi, Yiannis Sakellaridis, Akshay Venkatesh

TL;DR

This work introduces a relative Langlands duality that assigns to a favorable Hamiltonian G-space M a dual Hamiltonian M for the Langlands dual group, with automorphic periods and spectral L-functions interpreted as quantized data. The authors develop a structural theory of hyperspherical spaces, showing they arise from Whittaker induction from symplectic subspaces, and construct their duals via a splitting of the associated SL2 data and a Whittaker induction procedure. A central aim is to match automorphic and spectral data through conjectural 'L-sheaves' and 'period sheaves,' extending along global and local unramified settings, and to situate the duality within a 4d arithmetic QFT framework inspired by Kapustin–Witten. They formulate and explore precise conjectures on polarization, anomaly, and rational/spectral splits, providing concrete examples and outlining open problems and future directions for a fully-fledged arithmetic-geometric Langlands correspondence. The paper paves a path toward a unified microlocal perspective on periods and L-functions across global, local, geometric, and arithmetic tiers via a duality that mirrors electric-magnetic symmetry in topological field theory.

Abstract

We propose a duality in the relative Langlands program. This duality pairs a Hamiltonian space for a group $G$ with a Hamiltonian space under its dual group $\check{G}$, and recovers at a numerical level the relationship between a period on $G$ and an $L$-function attached to $\check{G}$; it is an arithmetic analog of the electric-magnetic duality of boundary conditions in four-dimensional supersymmetric Yang-Mills theory.

Relative Langlands Duality

TL;DR

This work introduces a relative Langlands duality that assigns to a favorable Hamiltonian G-space M a dual Hamiltonian M for the Langlands dual group, with automorphic periods and spectral L-functions interpreted as quantized data. The authors develop a structural theory of hyperspherical spaces, showing they arise from Whittaker induction from symplectic subspaces, and construct their duals via a splitting of the associated SL2 data and a Whittaker induction procedure. A central aim is to match automorphic and spectral data through conjectural 'L-sheaves' and 'period sheaves,' extending along global and local unramified settings, and to situate the duality within a 4d arithmetic QFT framework inspired by Kapustin–Witten. They formulate and explore precise conjectures on polarization, anomaly, and rational/spectral splits, providing concrete examples and outlining open problems and future directions for a fully-fledged arithmetic-geometric Langlands correspondence. The paper paves a path toward a unified microlocal perspective on periods and L-functions across global, local, geometric, and arithmetic tiers via a duality that mirrors electric-magnetic symmetry in topological field theory.

Abstract

We propose a duality in the relative Langlands program. This duality pairs a Hamiltonian space for a group with a Hamiltonian space under its dual group , and recovers at a numerical level the relationship between a period on and an -function attached to ; it is an arithmetic analog of the electric-magnetic duality of boundary conditions in four-dimensional supersymmetric Yang-Mills theory.
Paper Structure (274 sections, 44 theorems, 650 equations, 2 figures, 5 tables)

This paper contains 274 sections, 44 theorems, 650 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. What is a "period?"
  3. The automorphic side of the Langlands program.
  4. Relative Langlands: Automorphic Side
  5. Relative Langlands: Spectral Side
  6. Matching automorphic and spectral
  7. Electric-Magnetic Duality and Topological Field Theory
  8. Arithmetic Field Theory
  9. Aims and outline of the current paper
  10. Some examples
  11. Some open questions
  12. Recent developments
  13. Acknowledgments
  14. Notation and conventions
  15. The coefficient fields $\mathbb F$ and $k$
  16. ...and 259 more sections

Key Result

Lemma 2.6.1

Given two Weil sheaves $\mathcal{F}, \mathcal{G}$ on an $\mathbb{F}_q$-variety $X$, let $f$ and $\check{g}$ be the trace functions associated to, respectively, $\mathcal{F}$ and $D \mathcal{G}$; then Here, and in what is written below, we will keep denoting by $\mathrm{Hom}(\mathcal{F}, \mathcal{G})$ the (derived) homomorphisms over the base change of schemes and stacks to $\mathbb F = \overline{

Figures (2)

  • Figure 1.1.1: Automorphic states and observables
  • Figure 1.1.2: Spectral states and observables

Theorems & Definitions (271)

  • Remark 1.1.3: Number fields and ramification
  • Remark 1.1.6: Squares and square-roots
  • Remark 1.2.1: Gaiotto-Witten data, Nahm poles and Whittaker induction
  • Remark 1.2.2
  • Remark 1.3.1: The fourth dimension
  • Conjecture 1.3.2: Meta-conjecture
  • Remark 2.3.4
  • Remark 2.5.4: Super-vector spaces and Galois descent
  • Remark 2.5.6
  • Example 2.5.9
  • ...and 261 more