Lagrangian structures on the derived moduli of constructible sheaves

This paper establishes that the moduli of $\mathcal{D}(k)$-valued constructible sheaves and perverse sheaves on a compact oriented manifold with a conically smooth stratification are $(2-n)$-shifted Lagrangian, a result derived from constructing a relative left $n$-Calabi--Yau structure via lax gluing of categorical cubes and identifying symplectic leaves for perverse sheaves with prescribed monodromy.

The Gist

Imagine you are an architect trying to understand the shape of a very strange, crumpled piece of paper. This isn't just any paper; it's a mathematical object called a stratified space. Think of it like a layered cake, but the layers aren't flat. Some layers are smooth sheets, some are sharp edges, some are points where everything meets, and some are "cones" that taper off to a point.

The paper you are asking about, written by Merlin Christ and Enrico Lampetti, is a guidebook for finding hidden symmetry and balance (mathematicians call this "Lagrangian structures") inside these crumpled, layered shapes.

Here is the story of their discovery, broken down into simple concepts.

1. The Problem: The "Crumpled" Map

In the world of math, we often study "sheaves." Imagine a sheaf as a map of local weather patterns on a landscape.

• On a smooth hill, the weather is predictable and changes slowly.
• But on our "crumpled" landscape (the stratified space), the weather changes drastically at the edges, the corners, and the sharp points.

Mathematicians want to know: "If I look at all the possible weather maps on this crumpled shape, is there a hidden order? Is there a way to measure the 'distance' between two different maps?"

2. The Key Ingredient: The "Calabi-Yau" Compass

To find this order, the authors use a special tool called a Calabi-Yau structure.

• The Metaphor: Think of a Calabi-Yau structure as a perfectly balanced compass. In a normal room, a compass might spin wildly. But in a Calabi-Yau world, the compass always points true North, no matter how you turn the room.
• The Twist: Usually, this compass works on smooth, round shapes (like a sphere). But this paper asks: Can we build a compass that works on our crumpled, layered cake?

3. The Solution: The "Lego Cube" Strategy

The authors realized that you can't build a compass for the whole crumpled cake at once. It's too messy. Instead, they used a strategy they call "Cubical Gluing."

• The Analogy: Imagine you are building a giant 3D puzzle (a cube).
  • You start with small, smooth Lego blocks (these represent the smooth parts of the cake).
  • You know exactly how to put a compass on each smooth block.
  • Now, you have to snap these blocks together to form the crumpled shape.
  • The authors discovered a special "Lax Glue" (a flexible, slightly stretchy glue). This glue allows them to snap the blocks together without breaking the compasses on the individual blocks.

By using this "Lax Glue," they proved that if every small piece has a compass, the entire giant puzzle inherits a compass too.

4. The Big Discovery: The "Shifted" Balance

Once they built the compass for the whole shape, they found something amazing.

• The Result: The collection of all possible weather maps (the "moduli space") on this crumpled shape isn't just a random mess. It has a hidden symplectic structure.
• The Metaphor: Imagine a dance floor.
  • A Symplectic structure is like the rules of a dance that ensure every dancer has a partner and no one bumps into anyone. It's a perfect, balanced flow.
  • The authors found that for a shape with dimension $n$, this dance floor is "shifted" by $2-n$.
  • What does "shifted" mean? Think of it like a dance floor that exists in a different dimension or time. If you are in a 3D world ($n=3$), the dance floor is "shifted" by -1. It's a bit abstract, but it means the rules of the dance are slightly different from the usual ones, yet they still work perfectly.

5. The Special Case: The "Knot" and the "Ring"

The paper gets even more specific. Imagine you have a smooth ball (like a beach ball) and you poke a hole in it, or you tie a knot inside it.

• The Setup: You have a smooth 3D space with a knot inside it.
• The Discovery: If you look at the "weather maps" that behave in a specific way around that knot (called "perverse sheaves"), you find a special symplectic leaf.
• The Metaphor: Imagine the dance floor has a VIP section. If you fix the "monodromy" (which is just a fancy word for "how the weather spins as you walk around the knot"), you enter this VIP section. Inside this section, the dance rules are perfectly balanced (symplectic).

Why Does This Matter?

You might ask, "Who cares about crumpled paper and dance floors?"

1. Physics: These mathematical structures are deeply connected to quantum physics and string theory. The "Calabi-Yau" namesake comes from shapes used to describe extra dimensions in the universe.
2. New Math: This paper gives us a new way to understand shapes that aren't smooth. It tells us that even the most jagged, broken, or layered shapes have a hidden, perfect mathematical harmony.
3. Future Applications: The authors mention that these structures help build "BPS Lie algebras," which are tools used to count and organize complex mathematical objects, much like how a librarian organizes a massive library.

Summary

In short, Christ and Lampetti took a messy, layered mathematical shape. They showed that by breaking it down into smooth pieces and gluing them together with a special "flexible glue," the whole shape inherits a hidden, perfect balance (a Lagrangian structure). This balance allows mathematicians to study these complex shapes using the powerful tools of symplectic geometry, opening the door to understanding everything from knots in 3D space to the fundamental laws of physics.

Technical Summary

Here is a detailed technical summary of the paper "Lagrangian structures on the derived moduli of constructible sheaves" by Merlin Christ and Enrico Lampetti.

1. Problem Statement

The paper addresses the problem of understanding the symplectic and Lagrangian geometry of moduli spaces associated with constructible sheaves on stratified spaces.

• Context: Previous work by Brav–Dyckerhoff established that the moduli of local systems (locally constant sheaves) on a smooth, oriented manifold $X$ of dimension $n$ carries a $(2-n)$-shifted symplectic structure. If $X$ has a boundary, the restriction map to the boundary is Lagrangian.
• Gap: These results were limited to smooth manifolds and local systems. The authors seek to generalize this to constructible sheaves on conically smooth stratified spaces (which include singular spaces like algebraic varieties with stratifications) and to perverse sheaves (a specific subcategory of constructible sheaves defined by perversity functions).
• Goal: To construct a unified framework that endows the derived moduli stacks of constructible and perverse sheaves with shifted Lagrangian and Poisson structures, generalizing the smooth case to singular, stratified settings.

2. Methodology

The authors employ a sophisticated blend of higher category theory, derived algebraic geometry, and stratified topology.

A. Categorical Cubes and Calabi–Yau Structures

Instead of working directly with moduli stacks, the authors work at the level of the underlying stable $\infty$-categories.

• Categorical Cubes: They define a categorical $n$-cube as a functor $C_*: I^n \to \text{LinCat}_k$ (where $I = \{0 \to 1\}$).
• Cubical Calabi–Yau Structures: They utilize the notion of cubical Calabi–Yau structures (introduced in [CDW23]). A cubical $m$-Calabi–Yau structure on a cube involves a class in total negative cyclic homology that induces relative Calabi–Yau structures on the "tip" functor and recursive Calabi–Yau structures on the boundary faces.
• Gluing via Directed Pushouts: A central technical innovation is the use of directed pushouts (partially lax pushouts in $(\infty, 2)$-categories). The authors prove that relative Calabi–Yau structures can be glued along these directed pushouts (Proposition 2.17). This allows them to construct global structures by gluing local ones.

B. Stratified Topology and Unzipping

To apply the categorical machinery to stratified spaces, they use the unzipping procedure for conically smooth stratified spaces (from [AFT17]):

• Unzipping: A conically smooth stratified space $X$ of depth $n$ can be resolved into a sequence of smooth manifolds with corners, denoted $\text{Unzip}^{[k,n]}(X)$.
• Tubular Neighborhoods: The stratification is analyzed via tubular neighborhoods of strata. The category of constructible sheaves on $X$ is reconstructed as an iterated directed pushout of categories of local systems on these tubular neighborhoods and their boundaries (links).
• The Categorical Cube: They associate a specific categorical $(n+1)$-cube, denoted $\text{Cons}_P(X)_*$, to the stratified space. The "tip" of this cube is the category of constructible sheaves $\text{Cons}_P(X)$, and the other vertices correspond to local systems on the strata and their links.

C. Derived Moduli of Objects

They utilize the theory of derived moduli of objects (Toën–Vaquié, Brav–Dyckerhoff):

• If a $k$-linear stable $\infty$-category $\mathcal{C}$ admits a left $d$-Calabi–Yau structure, its moduli stack $\mathcal{M}_{\mathcal{C}}$ carries a $(2-d)$-shifted symplectic structure.
• If a functor $F: \mathcal{C} \to \mathcal{D}$ admits a relative left $d$-Calabi–Yau structure, the induced map of moduli stacks $\mathcal{M}_{\mathcal{D}} \to \mathcal{M}_{\mathcal{C}}$ is a $(2-d)$-shifted Lagrangian morphism.

3. Key Contributions and Results

Main Theorem: Cubical Calabi–Yau Structure on Stratified Spaces

Theorem 3.51 & Corollary 3.52:
For a compact oriented manifold $X$ of dimension $d$ with a conically smooth stratification $P$ of depth $n$:

1. The categorical $(n+1)$-cube $\text{Cons}_P(X)_*$ admits a canonical left $d$-Calabi–Yau structure.
2. This structure is constructed by iteratively gluing Calabi–Yau structures on the unzipped manifolds (which are smooth manifolds with corners) using the lax gluing property of directed pushouts.
3. Consequently, the functor from the colimit of the cube (excluding the tip) to the tip:
   $$\text{colim}_{I^{n+1} \setminus \{(1,\dots,1)\}} \text{Cons}_P(X)_* \longrightarrow \text{Cons}_P(X)$$
   inherits a relative left $d$-Calabi–Yau structure.

Lagrangian Structures on Moduli Stacks

Theorem 1 (Corollary 4.20):
The relative Calabi–Yau structure induces $(2-d)$-shifted Lagrangian morphisms between derived stacks:
$$\text{Cons}_P(X) \longrightarrow \mathcal{M}_{\text{colim} \dots}$$
$${}^p\text{Perv}_P(X) \longrightarrow \mathcal{M}_{\text{colim} \dots}$$
where ${}^p\text{Perv}_P(X)$ is the derived stack of perverse sheaves with a fixed perversity function $p$.

Poisson Structures

Corollary 4.22:
Since the moduli stacks admit Lagrangian morphisms to a symplectic target (or are Lagrangian in a relative sense), they naturally carry $(2-d)$-shifted Poisson structures. This generalizes the known Poisson structures on moduli of local systems on Riemann surfaces to higher-dimensional stratified spaces.

Symplectic Leaves and Monodromy

Theorem 2 (Theorem 4.38):
The authors specialize to the case where the stratification is defined by a smooth submanifold $D \subset X$ of codimension 2.

• They identify symplectic leaves of the moduli of perverse sheaves by fixing the monodromy around the circle bundle $\partial D \to D$ (the link of $D$).
• For prescribed monodromy data $\lambda = \{\lambda_1, \dots, \lambda_m\}$, the derived stack ${}^p\text{Perv}^{r,\lambda}_D(X)$ carries a $(2-d)$-shifted symplectic structure.
• Example: If $X$ is a 3-manifold and $D$ is a knot, the moduli of perverse sheaves with fixed monodromy around the knot carries a $(-1)$-shifted symplectic structure.

4. Significance and Implications

1. Unification of Smooth and Singular Cases: The paper provides a rigorous framework that unifies the symplectic geometry of smooth manifolds (local systems) with the geometry of singular spaces (constructible/perverse sheaves). It shows that the "shifted symplectic" nature is intrinsic to the topology of the stratification, not just the smoothness of the space.
2. New Class of Examples: It generates a vast new class of examples of shifted Lagrangian morphisms and symplectic stacks arising from topology and algebraic geometry (e.g., moduli of sheaves on algebraic varieties with stratifications).
3. Connection to BPS Algebras: The authors highlight the relevance of these results to BPS Lie algebras and cohomological Hall algebras (CoHA). Recent work suggests that symmetric derived stacks with $0$- or$(-1)$-shifted symplectic structures allow for the construction of BPS algebras. The identification of symplectic leaves for perverse sheaves (Theorem 2) provides the necessary geometric input to study the cohomology of these moduli spaces via CoHA.
4. Technical Advancement: The development of lax gluing for cubical Calabi–Yau structures is a significant methodological contribution to higher category theory, enabling the construction of global structures from local data in a way that was previously difficult for higher-dimensional cubes.

In summary, Christ and Lampetti successfully extend the theory of shifted symplectic and Lagrangian structures from smooth manifolds to the rich world of stratified spaces, providing a powerful tool for studying the geometry of constructible and perverse sheaves and their applications in mathematical physics.