Hamiltonian actions on 0-shifted cosymplectic groupoids

This paper introduces 0-shifted cosymplectic structures on differentiable stacks to develop a Hamiltonian action theory that includes a reduction procedure, a Kirwan convexity theorem, and examples of Morse-Bott Lie groupoid morphisms.

The Gist

Here is an explanation of the paper "Hamiltonian Actions on 0-Shifted Cosymplectic Groupoids" using simple language and creative analogies.

The Big Picture: A New Way to Map Time and Space

Imagine you are trying to describe the movement of a complex machine, like a clock with thousands of gears, or the flow of traffic in a busy city. In physics and math, we usually use Symplectic Geometry to map these movements. Think of Symplectic Geometry as a perfect, rigid grid that works great for systems where energy is conserved and time is just a background setting.

But what if time is the main character? What if the system changes depending on when you look at it? This is where Cosymplectic Geometry comes in. It's like adding a "time dimension" to your map. It allows us to study systems that evolve over time, like a pendulum slowing down or a planet orbiting a star that is also moving.

This paper introduces a brand new, super-advanced version of this map called 0-Shifted Cosymplectic Groupoids.

The Core Concepts, Explained with Analogies

1. The "Fuzzy" Map (Pre-cosymplectic Manifolds)

In the old days, mathematicians wanted their maps to be perfect: every point had a unique direction and speed. But in the real world, things get messy. Sometimes, different points in space behave exactly the same way, or they get "stuck" in a loop.

The authors start with a "fuzzy" map called a Pre-cosymplectic manifold.

• Analogy: Imagine a river. In some parts, the water flows fast and straight. In other parts, there are whirlpools or calm pools where the water doesn't move forward at all.
• The Problem: If you try to draw a single arrow for every drop of water, you get stuck in the whirlpools.
• The Solution: Instead of trying to map every single drop, the authors group the drops that behave the same way into "leaves" (like pages in a book). They study the shape of the whole "book" (the foliation) rather than just the individual pages.

2. The "Groupoid" (The Traveler's Passport)

To handle these "fuzzy" maps and the loops/whirlpools, the authors use a tool called a Lie Groupoid.

• Analogy: Think of a Groupoid as a massive, magical passport system for a city.
  • A normal map just shows you where you are.
  • A Groupoid passport shows you where you are, where you can go, and how you got there. It records every possible path between two points.
  • If you get stuck in a whirlpool (a loop), the passport records that you went in a circle and came back to the same spot. This allows mathematicians to "smooth out" the wrinkles in the map without losing the history of the movement.

3. The "0-Shifted" Twist

The term "0-shifted" sounds scary, but it's just a technical way of saying: "We are looking at the map from the perspective of the whole group of travelers, not just one person."

• Analogy: Imagine you are looking at a dance floor.
  • Standard view: You look at one dancer.
  • 0-Shifted view: You look at the entire dance floor as a single, shifting organism. You see how the patterns of the whole group relate to each other. This view is "Morita invariant," which means it doesn't matter if you zoom in or out; the essential shape of the dance remains the same.

4. The "Moment Map" (The GPS Tracker)

The paper talks about Hamiltonian Actions and Moment Maps.

• Analogy: Imagine a symphony orchestra. The conductor (the Lie Group) tells the musicians how to play. The Moment Map is like a GPS tracker that shows exactly where the music is "going" in terms of energy and shape.
• In this paper, the authors show how to build a GPS tracker for these complex, time-dependent, "fuzzy" dance floors. They prove that even when the system is messy, the GPS tracker still works and points to a specific, predictable shape.

The Main Discoveries (The "What Did They Do?" Section)

The authors didn't just invent a new map; they proved three major things about how it works:

1. The Reduction Procedure (Simplifying the Chaos)

• The Idea: Sometimes a system is too complicated to study. You want to remove the parts that are just "spinning in circles" (symmetries) to see the core shape.
• The Result: They created a recipe to "peel away" the extra layers of the groupoid. If you take a complex, time-dependent system and remove the repetitive loops, you are left with a cleaner, simpler system (a "Cosymplectic Stack") that is easier to understand.
• Real-world vibe: It's like taking a tangled ball of yarn, finding the loose ends, and pulling them out until you have a neat, straight line.

2. The Kirwan Convexity Theorem (The Shape of the Future)

• The Idea: If you have a system with a lot of symmetry (like a spinning top), the possible states it can be in form a specific shape.
• The Result: They proved that the "map" of all possible states is always a convex polyhedron (a shape like a diamond or a cube, but in higher dimensions).
• Analogy: Imagine you are filling a bucket with water. No matter how you pour it, the water will always settle into a flat, predictable surface. This theorem says that for these complex time-dependent systems, the "surface" of all possible outcomes is always a nice, solid, geometric shape.

3. Morse-Bott Morphisms (The Smooth Hills)

• The Idea: Mathematicians love to find the "peaks" and "valleys" in a landscape (critical points).
• The Result: They showed that the paths defined by their new GPS tracker (the Moment Map) are "Morse-Bott."
• Analogy: Imagine walking on a mountain range. Usually, you might trip on a single sharp peak. But here, the "peaks" are actually smooth, flat plateaus or gentle rolling hills. This makes the system much more stable and predictable. It means the system doesn't have sudden, chaotic jumps; it flows smoothly over these flat tops.

Why Does This Matter?

This paper is like upgrading the operating system for a computer that simulates the universe.

• Old System: Could only handle static, perfect grids (Symplectic).
• New System: Can handle time, loops, and messy, real-world data (0-Shifted Cosymplectic Groupoids).

The "Delzant Classification" Bonus:
The authors mention that this new math could help classify "Toric Cosymplectic Stacks."

• Translation: They are creating a library catalog for a specific type of complex, time-dependent shape. Just as a librarian organizes books by genre and author, these mathematicians are organizing these complex shapes by their geometric "fingerprints" (convex polytopes).

Summary

In short, Daniel L´opez-Garcia and Fabricio Valencia have built a new mathematical toolkit. This toolkit allows us to:

1. Map systems that change over time and get "stuck" in loops.
2. Simplify these messy systems by peeling away the repetitive parts.
3. Prove that the remaining shapes are always predictable, solid, and smooth.

It's a bridge between the messy reality of time-dependent physics and the clean, beautiful world of geometric shapes.

Technical Summary

Here is a detailed technical summary of the paper "Hamiltonian Actions on 0-Shifted Cosymplectic Groupoids" by Daniel López-García and Fabricio Valencia.

1. Problem Statement and Motivation

The paper addresses the geometric challenges associated with time-dependent Hamiltonian systems, which are difficult to model using standard symplectic geometry. While cosymplectic geometry (involving a closed 2-form $\omega$ and a non-vanishing closed 1-form $\eta$) offers a framework for such systems, many physical scenarios involve pairs $(\omega, \eta)$ that do not satisfy the strict non-degeneracy conditions required for a full cosymplectic structure. Instead, these pairs define precosymplectic structures, where the intersection of the kernels $\ker(\omega) \cap \ker(\eta)$ forms a regular distribution, leading to a singular leaf space.

The central problem is to develop a global geometric framework that:

1. Integrates these singular foliations associated with precosymplectic structures.
2. Generalizes the theory of Hamiltonian actions and moment maps to this setting.
3. Extends these concepts to the realm of differentiable stacks and Lie 2-groups, allowing for a robust reduction procedure and convexity theorems analogous to the symplectic case.

2. Methodology

The authors employ a multi-layered approach combining differential geometry, foliation theory, and higher category theory:

• From Manifolds to Groupoids: They begin by analyzing precosymplectic manifolds $(M, \omega, \eta)$. They utilize the Lichnerowicz map $\flat: TM \to T^*M$ defined by $\flat(v) = \iota_v \omega + \eta(v)\eta$. They establish that for precosymplectic manifolds, $\ker(\flat) = \ker(\omega) \cap \ker(\eta)$, which defines a regular foliation $\mathcal{F}_\flat$.
• Integration via Lie Groupoids: To handle the singular quotient space $M/\mathcal{F}_\flat$, the authors integrate the associated Lie algebroid to a Lie groupoid $G \rightrightarrows M$. They introduce the concept of a 0-shifted cosymplectic structure on $G$, defined by a pair of closed basic forms $(\omega, \eta)$ such that the induced Lichnerowicz map is a quasi-isomorphism between the tangent complex of the groupoid and the cotangent complex.
• Stacky Framework: They prove that this structure is Morita invariant, allowing the definition of 0-shifted cosymplectic structures on differentiable stacks $[M/G]$.
• Lie 2-Group Actions: The theory is extended to actions of foliation Lie 2-groups ($K_1 \rightrightarrows K_0$) on these groupoids. The authors adapt the theory of moment maps to this higher categorical setting, defining a moment map morphism $\mu: G \to (\mathfrak{k}/\mathfrak{n})^*$.
• Reduction and Convexity: They apply techniques from symplectic geometry (specifically referencing works on 0-shifted symplectic structures) to construct a Marsden–Weinstein–Meyer reduction and prove a Kirwan-type convexity theorem for these new structures.

3. Key Contributions

A. Definition of 0-Shifted Cosymplectic Groupoids

The paper introduces the notion of a 0-shifted cosymplectic structure on a Lie groupoid $G \rightrightarrows M$. This is characterized by:

• A pair of closed basic forms $(\omega, \eta)$.
• The condition that the Lichnerowicz map $\flat$ induces a quasi-isomorphism on the tangent/cotangent complexes.
• This definition generalizes the standard cosymplectic structure (where the groupoid is a manifold) and the precosymplectic structure (where the groupoid integrates the characteristic foliation).

B. Hamiltonian Actions and Moment Map Morphisms

The authors define Hamiltonian actions of Lie 2-groups on 0-shifted cosymplectic groupoids. Key features include:

• The moment map is a Lie groupoid morphism $\mu: G \to (\mathfrak{k}/\mathfrak{n})^*$, where $\mathfrak{n}$ is the ideal of the Lie algebra $\mathfrak{k}$ acting trivially on the leaf space.
• The map satisfies equivariance conditions with respect to the coadjoint action of the Lie 2-group.
• This framework unifies the study of Hamiltonian actions on precosymplectic manifolds and their global stacky counterparts.

C. Reduction Procedure

A reduction procedure is established for 0-shifted cosymplectic groupoids. If $0$ is a regular value of the moment map:

• The zero fiber $\mu^{-1}(0)$ inherits a Lie 2-group action.
• A quotient groupoid is constructed (analogous to the symplectic quotient), resulting in a new 0-shifted cosymplectic groupoid.
• Corollary 4.4 highlights that even for ordinary Lie group actions on cosymplectic manifolds, this procedure yields cosymplectic stacks (or orbifolds/manifolds under specific freeness/properness conditions), providing new insights into classical reduction.

D. Convexity and Morse-Bott Properties

• Kirwan Convexity Theorem: The authors prove that for a compact Lie 2-group action, the image of the moment map intersected with a Weyl chamber (the "moment body") is a closed convex polyhedron.
• Morse-Bott Morphisms: They demonstrate that the component functions of the moment map morphism are Morse-Bott Lie groupoid morphisms. Specifically, the critical point sets are saturated, and the indices of non-degenerate critical submanifolds are even.

4. Key Results

1. Lemma 2.1: Establishes that for a precosymplectic manifold, the kernel of the Lichnerowicz map is exactly $\ker(\omega) \cap \ker(\eta)$, ensuring the associated distribution is regular.
2. Proposition 3.2 (Morita Invariance): Proves that the property of being a 0-shifted cosymplectic structure is invariant under Morita equivalence, validating the definition on differentiable stacks.
3. Proposition 4.3 (Reduction): Constructs the reduced 0-shifted cosymplectic groupoid $(K \times_N R_1 \rightrightarrows R_0, \omega_{red}, \eta_{red})$ from a Hamiltonian action, proving the existence of a principal bundle structure and the preservation of the geometric structure.
4. Proposition 4.5 (Convexity): Confirms that the moment body $\Delta(G)$ is a closed convex polyhedron under clean actions of compact Lie 2-groups.
5. Proposition 4.6 (Morse-Bott): Shows that component functions of the moment map are Morse-Bott with even indices, generalizing classical results to the stacky setting.

5. Significance and Impact

• Unification of Geometric Theories: The paper successfully bridges the gap between cosymplectic geometry (time-dependent systems), presymplectic geometry (singular systems), and higher geometry (stacks and Lie 2-groups).
• Globalization of Singular Systems: By using Lie groupoids and stacks, the authors provide a rigorous global framework for systems where the leaf space is singular, a common issue in time-dependent Hamiltonian mechanics that standard manifold theory cannot easily resolve.
• New Classification Tools: The results pave the way for a Delzant-type classification for "toric cosymplectic stacks," linking combinatorial data (convex polytopes) to geometric objects in this generalized setting.
• Extension of Classical Theorems: The work demonstrates that fundamental theorems of symplectic geometry (Kirwan convexity, Marsden-Weinstein reduction, Morse-Bott properties) have robust analogues in the 0-shifted cosymplectic context, suggesting a deep structural parallel between symplectic and cosymplectic geometries when viewed through the lens of higher stacks.

In summary, this paper provides a foundational theory for Hamiltonian dynamics on singular spaces by introducing 0-shifted cosymplectic groupoids, offering powerful tools for reduction, classification, and the analysis of time-dependent systems in a global, stack-theoretic framework.