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A Derived Legendrian Category for Shifted Contact Stacks

This paper constructs the derived Legendrian category $\mathcal{F}_{c}(X)$ for $n$ -shifted contact derived Artin stacks and the associated $(\infty,2)$ -category of Legendrian correspondences using equivariant descent, establishing their embedding into an AKSZ-defined span category and applying these frameworks to define derived Legendrian surgery and analyze moduli theory.

Original authors: Efe żzbudak, Kadri żlker Berktav

Published 2026-05-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Efe żzbudak, Kadri żlker Berktav

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to map out a vast, complex landscape of mathematical shapes. For a long time, mathematicians have been very good at mapping a specific type of terrain called "Symplectic Geometry." Think of this terrain as a perfectly balanced, frictionless dance floor where every move has a precise, symmetrical partner. In this world, mathematicians use "Lagrangian correspondences" (which you can think of as bridges or walkways) to travel between different points on the dance floor. They have built a giant, organized library (a category) to keep track of these bridges and how you can combine them.

However, the authors of this paper, Efe İzbudak and Kadri İlker Berktav, noticed that some important mathematical landscapes don't fit this perfect "dance floor" model. In these places, the rules are slightly different: the symmetry is "twisted" or "shifted" by a specific factor (like a group of dancers rotating slightly as they move). The authors call these "Shifted Contact Stacks."

If the Symplectic world is a dance floor, the Contact world is like a sloped, spinning slide. The rules of movement here are different. You can't just use the old "bridges" (Lagrangian correspondences) because they don't work on a slide. You need a new kind of connector.

The Big Idea: Building a New Library

The main goal of this paper is to build a new, specialized library for these "sliding" landscapes. They call this the Derived Legendrian Category.

Here is how they built it, using simple analogies:

1. The "Shadow" Trick (Symplectification)
The authors realized that even though the "slide" (the Contact stack) looks different, it is actually just a shadow of a higher, more complex "dance floor" (the Symplectic stack) that is spinning.

The Analogy: Imagine a spinning carousel (the Symplectic world) with a light shining down on it. The shadow cast on the ground is the "slide" (the Contact world).
The Method: To understand the rules of the slide, the authors didn't try to study the slide directly. Instead, they went up to the carousel, studied the spinning dancers, and then "projected" those rules down to the ground. They used a mathematical process called equivariant descent (which is like carefully translating the rules of the spinning carousel so they make sense for the shadow on the ground).

2. The New Connectors: Legendrian Morphisms
In the old library, you connected points with "Lagrangian bridges." In this new library, they connect points with "Legendrian morphisms."

The Analogy: If a Lagrangian bridge is a flat walkway, a Legendrian morphism is like a tightrope that must always move in a specific direction relative to the slope of the slide. It's a very specific, constrained path that fits perfectly on the "slide" terrain.

3. The New Library Structure (The (∞, 2)-Category)
The authors didn't just make a list of paths; they built a full, multi-layered system (an (∞, 2)-category).

Objects: The "slides" themselves (the Contact stacks).
1-Morphisms: The "tightropes" (Legendrian correspondences) connecting them.
2-Morphisms: The ways you can stretch, bend, or transform one tightrope into another (Legendrian spans).
The Magic: They proved that if you take the complex, spinning "dance floor" library (which they already knew how to build) and project it down using their "shadow" method, you get a perfectly valid, working library for the "slides."

What Can You Do With This Library?

The paper shows that this new library isn't just theoretical; it helps solve specific puzzles in mathematics:

The "Surgery" Analogy: Just as a surgeon can cut and rejoin tissue, mathematicians can perform "surgery" on these shapes. The authors showed that if you have a "topological cobordism" (think of it as a tube connecting two shapes), you can use their library to cut and paste these mathematical shapes together in a way that respects the "slide" rules. This is called Derived Legendrian Surgery.
The "Convolution" Analogy: They showed that if you have two functions (like two different recipes), you can combine them using the rules of this library to create a new, complex result. This is useful for studying "discriminant loci" (which are like the "danger zones" or "critical points" in a mathematical landscape).
The "Nilpotent Cone" Application: They applied this to a famous mathematical shape called the "Nilpotent Cone" (related to how matrices behave). They showed that the "self-connections" of this shape in their new library form a special kind of algebra (an $A_\infty$ -algebra) that helps mathematicians understand how this shape can be deformed or changed.

Summary

In short, the authors took a complex, spinning mathematical world (Symplectic geometry) and figured out how to translate its rules to a "slanted" world (Contact geometry). They built a new, rigorous dictionary and rulebook (the Derived Legendrian Category) that allows mathematicians to travel between these slanted shapes, combine them, and perform "surgery" on them, all while keeping the mathematical laws consistent. They did this by looking at the "spinning shadows" of these shapes and carefully translating the rules from the spin down to the ground.

Technical Summary: A Derived Legendrian Category for Shifted Contact Stacks

Problem Statement
In derived algebraic geometry, the geometric relations between derived moduli spaces are typically formalized using the $(\infty, 2)$ -category of shifted symplectic stacks (denoted $\text{Lag}_n$ or $\text{Symp}_n$ ), where morphisms are Lagrangian correspondences. However, certain moduli problems—such as those involving Frobenius actions on $\ell$ -adic local systems, projective Higgs bundles, and Lie 2-groups—do not yield shifted symplectic quotients. Instead, these quotients possess structures that are "fixed up to a character," resulting in what the authors term shifted contact structures.

The existing framework of Lagrangian correspondences is insufficient for these "odd-dimensional" contact settings. The paper addresses the need for a corresponding categorical foundation where Lagrangian correspondences are replaced by Legendrian correspondences, and where the composition of morphisms respects the specific equivariant nature of contact geometry.

Methodology
The authors construct the derived Legendrian category $\mathcal{F}_c(X)$ for an $n$ -shifted contact derived Artin stack $X$ by leveraging the relationship between contact and symplectic geometry via derived symplectification.

Derived Symplectification: An $n$ -shifted contact structure on a stack $X$ is encoded as a weight 1 $\mathbb{G}_m$ -equivariant $n$ -shifted symplectic structure on a principal $\mathbb{G}_m$ -bundle $\tilde{X}$ (the symplectification). This relies on the result that $X \simeq [\tilde{X}/\mathbb{G}_m]$ .
Legendrian Morphisms: Objects in the new category are defined as Legendrian morphisms $f: L \to X$ . These are characterized by their lifts $\tilde{f}: \tilde{L} \to \tilde{X}$ being Lagrangian morphisms in the symplectification $\tilde{X}$ , equivariant with respect to the $\mathbb{G}_m$ -action.
Morphism Spaces via Descent: The morphism spaces $\text{Hom}_{\mathcal{F}_c(X)}(L_1, L_2)$ are constructed by taking the derived intersection of the Lagrangian lifts $\tilde{L}_1$ and $\tilde{L}_2$ within $\tilde{X}$ . This intersection inherits a $\mathbb{G}_m$ -equivariant $(n-1)$ -shifted symplectic structure. The actual morphism space in the contact category is obtained by equivariant descent (taking the stack quotient) of this intersection, yielding an $(n-1)$ -shifted contact stack.
Higher Categorical Structure: Composition and higher morphisms are defined using the $(\infty, 2)$ -category of Lagrangian spans in the symplectification $\tilde{X}$ . The authors demonstrate that the $\mathbb{G}_m$ -equivariance of these spans allows them to descend to the contact base $X$ , preserving the coherence conditions required for an $(\infty, 2)$ -category.
AKSZ Construction: The authors utilize the AKSZ construction to map topological cobordisms to Lagrangian spans in the symplectification, which then descend to define derived Legendrian surgery in the contact category.

Key Contributions and Results

Construction of $\mathcal{F}_c(X)$ : The paper establishes the existence of a well-defined derived Legendrian category $\mathcal{F}_c(X)$ $F_{c} (X)$ for any $n$ $n$ -shifted contact derived Artin stack $X$ $X$ .
- Objects: Legendrian morphisms $f: L \to X$ .
- 1-Morphisms: Derived stacks obtained as the descent of derived intersections of equivariant Lagrangian lifts.
- 2-Morphisms and Composition: Derived from the $(\infty, 2)$ -category of Lagrangian spans via equivariant descent.
Global Category $\text{Leg}_n$ : The authors promote the collection of all $n$ -shifted contact derived stacks into a global $(\infty, 2)$ -category, $\text{Leg}_n$ , where objects are contact stacks, 1-morphisms are Legendrian correspondences, and 2-morphisms are Legendrian spans.
Derived Legendrian Surgery: Using the AKSZ construction, the paper defines a symmetric monoidal $\infty$ -functor $Z_c: \text{Bord}_d \to \text{Leg}_{k-d+1}$ . This functor maps oriented topological cobordisms to Legendrian correspondences, effectively defining derived Legendrian surgery as a categorical operation.

Applications to Moduli Theory
The paper demonstrates the utility of this framework by applying it to specific derived moduli spaces:

Derived Discriminant Loci: For a smooth scheme $L$ , the derived Legendrian category of the 1-jet space $J^1(L)$ encodes the derived discriminant locus $\Delta_{\text{loc}}(f)$ . The composition in this category yields a derived convolution map, providing a geometric analogue to the composition of microlocal kernels.
Nilpotent Cone Deformation: For the derived moduli stack of projective Higgs bundles $\mathcal{P}\text{Higgs}_G(C)$ , the projectivized derived nilpotent cone $\mathbb{P}N$ is an object in $\mathcal{F}_c(\mathcal{P}\text{Higgs}_G(C))$ . The categorical composition equips the homotopy pullback $\mathbb{P}N \times^h_{\mathcal{P}\text{Higgs}_G(C)} \mathbb{P}N$ with an $A_\infty$ -algebra structure, controlling the non-commutative deformation theory of the nilpotent cone.
Topological Field Theories: The functoriality of the construction links topological cobordisms to the algebraic structure of contact stacks, offering a framework for understanding derived Legendrian surgery.

Significance
The paper claims to provide the necessary categorical foundation for studying moduli problems that naturally arise in contact geometry rather than symplectic geometry. By formalizing the descent of Lagrangian correspondences from symplectifications, the authors enable the application of higher categorical tools (spans, composition, surgery) to "odd-dimensional" derived moduli spaces. This bridges the gap between derived symplectic geometry and the specific geometric constraints found in problems involving $\ell$ -adic local systems and Higgs bundles, allowing for the definition of non-commutative deformation algebras and convolution structures in these contexts.

The Big Idea: Building a New Library

What Can You Do With This Library?

Summary

Technical Summary: A Derived Legendrian Category for Shifted Contact Stacks

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