Relative Langlands duality for $\mathfrak{osp}(2n + 1|2n)$

This paper establishes an $S$-duality converse to prior work by proving that the $S$-dual of the action of $\text{SO}(2n+1)\times \text{Sp}(2n)$ on their tautological representations is the symplectic mirabolic space $\text{Sp}(2n)\times\text{Sp}(2n)$ acting on $T^* \text{Sp}(2n)$ and its tautological representations, while also formulating a corresponding global conjecture for the categorical theta-correspondence.

The Gist

Here is an explanation of the paper "Relative Langlands Duality for $osp(2n+1|2n)$" using simple language and creative analogies.

The Big Picture: A Cosmic Mirror Game

Imagine the universe of mathematics is filled with different "worlds" or "languages." For a long time, mathematicians have been trying to find a universal translator between these worlds. This is the Langlands Program, often described as a "Grand Unified Theory" of math.

This specific paper is about a special kind of translation called S-duality. Think of S-duality as a magical mirror. If you look at a complex object in one world (let's call it World A) and hold it up to the mirror, you see a completely different object in World B. The magic is that even though they look totally different, they contain the exact same information. If you understand the rules of World A, the mirror tells you the rules of World B instantly.

The authors (Braverman, Finkelberg, Kazhdan, and Travkin) are proving that a specific, very complicated object in World A has a perfect, matching partner in World B.

The Characters in the Story

To understand the paper, we need to meet the two main characters (the objects) being compared:

1. The "Spinning Top" (World A):

   • What it is: Imagine a giant, multi-dimensional spinning top made of symplectic geometry (a fancy type of math that deals with motion and energy).
   • The Setup: It involves a group called $SO(2n+1)$ (like a sphere with specific symmetries) and a group called $Sp(2n)$ (like a hyper-elastic sheet). They are interacting with a space that looks like a giant, infinite grid of numbers.
   • The Vibe: This object is "twisted." In math terms, it has an anomaly. Imagine trying to build a house where the floor is slightly warped; you can still live there, but you have to wear special shoes (a "metaplectic" structure) to walk on it without falling.

2. The "Double Mirror" (World B):

   • What it is: This is the "S-dual" partner. It looks like a symplectic mirabolic space.
   • The Analogy: If the first object was a spinning top, this one is like a double-mirror system. It involves two copies of the $Sp(2n)$ group interacting with a "cotangent space" (which is like a map showing both position and momentum).
   • The Twist: Because of the anomaly in the first world, the second world also has a "metaplectic" twist. It's like looking at a reflection in a funhouse mirror that slightly distorts the image, but in a very precise, predictable way.

The Main Discovery: The "Perfect Match"

The paper proves a Converse Claim.

• Previous Work: In a previous paper ([BFT]), the authors showed that if you start with the "Double Mirror" (World B) and look in the mirror, you see the "Spinning Top" (World A).
• This Paper: They prove the reverse! If you start with the "Spinning Top" (World A) and look in the mirror, you see the "Double Mirror" (World B).

Why is this hard?
Usually, these mirrors are easy to use. But here, the objects are "anomalous" (twisted). It's like trying to translate a language where the grammar rules change depending on the time of day. The authors had to invent a new dictionary (a new mathematical framework) to show that the translation works perfectly in both directions.

The "Global Conjecture": The Cosmic Map

The paper doesn't just stop at the local mirror game. They also propose a Global Conjecture.

• The Analogy: Imagine the local mirror game happens in a small room. The global conjecture is about what happens if you take that room and wrap it around the entire Earth.
• The Goal: They want to describe how these mathematical "periods" (repeating patterns) relate to L-functions.
• What are L-functions? Think of them as the "DNA" of numbers. They encode deep secrets about prime numbers and geometry.
• The Prediction: The authors predict that if you take a specific mathematical "sheaf" (a bundle of information) from the twisted world and translate it to the dual world, you get a specific object that acts like a Clifford Algebra.
• The "Categorification" Metaphor: In simple terms, they are saying: "The value of this complex number function at a specific point is actually a whole universe of shapes and symmetries." It's like saying the number "5" isn't just a number, but a whole library of books that, when you read them all together, tell you the story of 5.

The "How" (The Secret Sauce)

How did they prove this? They used a few clever tricks:

1. The Hecke Action: Imagine you have a set of Lego blocks. The "Hecke action" is a way of snapping blocks together. The authors showed that the way you snap blocks together in World A is exactly the same as the way you snap them in World B, even though the blocks look different.
2. The "Pseudo-Slice": They found a specific "slice" of the mathematical space (like cutting a cake) where the complex geometry becomes simple enough to analyze. They proved that if you understand this slice, you understand the whole cake.
3. The "De-equivariantized" Algebra: They stripped away the complex symmetries (the "noise") to look at the core algebraic structure (the "signal"). They found that the core structure of World A is identical to the core structure of World B.

Summary for the General Audience

This paper is a victory in the quest to understand the deep connections between different areas of mathematics.

• The Problem: We have two very complex, "twisted" mathematical worlds. We suspect they are mirrors of each other, but the twist makes it hard to prove.
• The Solution: The authors built a bridge. They proved that these two worlds are indeed perfect mirrors. If you know the rules of one, you automatically know the rules of the other.
• The Impact: This helps mathematicians solve problems in one world by translating them into the other, where they might be easier to solve. It also suggests a deep, unified structure underlying the universe of numbers and shapes, linking local geometry to global patterns (like the distribution of prime numbers).

In short: They found the missing link in a cosmic mirror game, proving that two twisted, complex mathematical worlds are actually two sides of the same coin.

Technical Summary

Here is a detailed technical summary of the paper "Relative Langlands Duality for $osp(2n + 1|2n)$" by Alexander Braverman, Michael Finkelberg, David Kazhdan, and Roman Travkin.

1. Problem Statement and Context

The paper addresses a specific instance of Relative Langlands Duality (also known as S-duality in the context of 3d $\mathcal{N}=4$ supersymmetric gauge theories). This framework, developed notably in the work of Ben-Zvi, Sakellaridis, and Venkatesh ([BZSV]), posits a duality between certain symplectic varieties equipped with group actions and their "dual" hyper-spherical varieties.

• The Specific Case: The authors investigate the duality involving the Lie superalgebra $osp(2n+1|2n)$.
• The Setup:
  • Let $V = \mathbb{C}^{2n}$ be a symplectic vector space and $V' = \mathbb{C}^{2n+1}$ be a vector space with a non-degenerate symmetric pairing.
  • Let $M = V' \otimes V$. This space carries a natural symplectic form.
  • The group $G = SO(V') \times Sp(V)$ acts on $M$.
• The Conjecture: The paper aims to prove the converse of a result established in their previous work ([BFT]).
  • Previous Result ([BFT]): The S-dual of the symplectic variety $T^*Sp(2n) \times \mathbb{C}^{2n}_-$ (with a specific metaplectic twist) is the space $M$ with the action of $SO(2n+1) \times Sp(2n)$.
  • Current Goal: Prove that the S-dual of $M$ (with the action of $SO(2n+1) \times Sp(2n)$) is the symplectic variety $T^*Sp(2n) \times \mathbb{C}^{2n}_-$ (with the action of $Sp(2n) \times Sp(2n)$).
  • Anomaly: A crucial technical detail is the presence of an "anomaly." The second factor of the dual group, $Sp(2n)$, corresponds to its metaplectic dual (a twisted form) due to the square root of the determinant line bundle involved in the construction.

2. Methodology

The authors employ a sophisticated blend of geometric representation theory, derived categories, and invariant theory. The core strategy is to establish an equivalence of categories between the "spectral" side (algebraic geometry of the dual group) and the "geometric" side (D-modules/Weyl modules).

A. Categorical Framework

• Spectral Side: They construct a category of perfect dg-modules over a specific dg-superalgebra $A^\bullet$.
  • $A^\bullet = \mathbb{C}[Sp(V)] \otimes \text{Sym}^\bullet(\mathfrak{sp}(V)[-2]) \otimes \text{Sym}^\bullet(\Pi(V)[-1])$.
  • This algebra encodes the functions on the dual variety and the structure of the Langlands dual group.
• Geometric Side: They consider the category $W\text{-mod}_{SO(V')^O \times Sp(V)^O, lc}$, which consists of locally compact, equivariant objects in the category of discrete modules over the Weyl algebra $W$ associated with the space $M_K = M \otimes \mathbb{C}((t))$.
  • This category is a renormalized version of D-modules on the affine Grassmannian twisted by a line bundle.

B. The Main Equivalence

The central theorem asserts an equivalence of triangulated categories:
$$\Phi: D_{Sp(V) \times Sp(V)}^{\text{perf}}(A^\bullet) \xrightarrow{\sim} W\text{-mod}_{SO(V')^O \times Sp(V)^O, lc}$$
This equivalence must commute with the convolution action of the spherical Hecke category.

C. Proof Strategy

The proof proceeds through several key steps:

1. Identification of Hecke Actions: The authors show that the convolution actions of the Hecke categories for $SO(2n+1)$ and $Sp(2n)$ on the unit object $E_0$ (the delta-function at the origin) coincide. This relies on analyzing the geometry of minimal orbits in the affine Grassmannian and identifying the resulting D-modules as Goresky-MacPherson sheaves on a specific cone.
2. Generation by Hecke Action: They prove that the category of interest is generated by the action of the Hecke category on the unit object. This reduces the problem to computing the Ext-algebra of the unit object.
3. De-equivariantization: To compute the Ext-algebra $E^\bullet = \text{Ext}^\bullet(E_0, E_0)$, they "de-equivariantize" the category by restricting the group action to a Levi subgroup $GL(L) \subset Sp(V)$. This simplifies the geometry to a space of differential operators on a lattice.
4. Commutativity and Isomorphism:
   • They prove the Ext-algebra $E^\bullet$ is commutative and formal (quasi-isomorphic to its cohomology).
   • They construct a homomorphism $\psi: G^\bullet \to E^\bullet$ from the algebraic model $G^\bullet$ (derived from the spectral side) to the geometric Ext-algebra.
   • Invariant Theory Argument: The core of the proof involves showing $\psi$ is an isomorphism. They utilize a "pseudo-slice" $\Sigma \subset \mathfrak{sp}(V) \oplus V$ (a Kostant slice crossed with a vector space) and analyze the map from the spectrum of the Ext-algebra to the quotient stack $(\mathfrak{sp}(V) \oplus V) / Sp(V)$.
   • By demonstrating that the map is an isomorphism on a dense open set and using properties of the Weierstrass section and the ramification of the quotient map, they conclude that the algebras are identical.

3. Key Contributions and Results

1. Proof of the Converse Duality: The paper rigorously establishes the S-duality for the pair $(SO(2n+1) \times Sp(2n), M)$ and $(Sp(2n) \times Sp(2n), T^*Sp(2n) \times \mathbb{C}^{2n}_-)$. This confirms the involutive nature of the relative Langlands duality in this specific, anomalous setting.
2. Handling Metaplectic Anomalies: The work provides a concrete realization of how "anomalies" (twisting by square roots of determinant bundles) manifest in the categorical duality. It explicitly shows that the dual of the second $Sp(2n)$ factor is its metaplectic cover, a subtle point often overlooked in non-anomalous cases.
3. Global Conjecture (Theta Correspondence): The authors formulate a global conjecture relating to automorphic forms.
   • They define a sheaf $\Theta$ on the moduli stack of bundles $\text{Bun}_{Sp(2n)}^\omega \times \text{Bun}_{SO(2n+1)}$.
   • They conjecture that the geometric Langlands functor applied to the pullback of the theta-sheaf yields a specific sheaf on the stack of local systems $\text{LocSys}_{Sp(2n)}$.
   • Corollary: They derive a relation involving the Clifford algebra: $\Psi^* \circ \Psi(A_E) \simeq A_E \otimes \text{Cliff}(H^1_{dR}(C, E))$. This suggests a categorification of the value of the L-function at $s=1/2$, linking period sheaves to L-functions.

4. Significance

• Advancement of Relative Langlands Program: This paper is a critical step in the program initiated by [BZSV], moving from the "cotangent" case (where duality is well-understood) to more complex, non-cotangent symplectic varieties with anomalies.
• Super-Group Duality: The title references $osp(2n+1|2n)$, highlighting that the duality is rooted in the representation theory of Lie superalgebras, even though the final statement is phrased in terms of classical groups and their metaplectic covers.
• Bridge between Local and Global: By formulating the global conjecture, the authors connect the local categorical equivalences (proven in the paper) to the global geometric Langlands program, specifically the study of automorphic periods and L-functions.
• Technical Innovation: The use of pseudo-slices and the detailed analysis of the Ext-algebra structure in the presence of metaplectic twists provide new tools for handling similar problems in geometric representation theory.

In summary, the paper successfully proves a deep duality conjecture involving metaplectic groups and provides a concrete categorical framework for understanding the relationship between symplectic representations and their Langlands duals, extending the scope of the Langlands program to include anomalous cases.