[Paper Review] Double Groupoids, Orbifolds, and the Symplectic Category

Santiago Cañez|arXiv (Cornell University)|May 13, 2011

Homotopy and Cohomology in Algebraic Topology28 references3 citations

TL;DR

This paper introduces symplectic hopfoids—structures in the symplectic category that generalize groupoids for symplectic geometry—and demonstrates their role in constructing symplectic orbifolds via canonical relations. It proves that the core of a symplectic double groupoid is a symplectic groupoid via symplectic reduction, and conjectures that all symplectic orbifolds arise from such constructions, offering a new framework for understanding symplectic stacks and their quantization.

ABSTRACT

Motivated by an attempt to better understand the notion of a symplectic stack, we introduce the notion of a symplectic hopfoid, which should be thought of as the analog of a groupoid in the so-called symplectic category. After reviewing some foundational material on canonical relations and this category, we show that symplectic hopfoids provide a characterization of symplectic double groupoids in these terms. Then, we show how such structures may be used to produce examples of symplectic orbifolds, and conjecture that all symplectic orbifolds arise via a similar construction. The symplectic structures on the orbifolds produced arise naturally from the use of canonical relations. The characterization of symplectic double groupoids mentioned above is made possible by an observation which provides various ways of realizing the core of a symplectic double groupoid as a symplectic quotient of the total space, and includes as a special case a result of Zakrzewski concerning Hopf algebra objects in the symplectic category. This point of view also leads to a new proof that the core of a symplectic double groupoid itself inherits the structure of a symplectic groupoid. Similar constructions work more generally for any double Lie groupoid---producing what we call a Lie hopfoid---and we describe the sense in which a version of the "cotangent functor" relates such hopfoid structures.

Motivation & Objective

• To develop a framework for symplectic stacks using the symplectic category and canonical relations.
• To characterize symplectic double groupoids through symplectic hopfoids as a generalization of groupoids in the symplectic category.
• To show that symplectic orbifolds can be constructed from symplectic hopfoids, conjecturing all such orbifolds arise this way.
• To explore the quantization of symplectic hopfoids as a path toward geometric quantization of symplectic stacks.
• To relate Lie hopfoids in the symplectic category to the cotangent functor and generalized Hopf algebroids.

Proposed method

• Introduces the symplectic category using canonical relations between symplectic manifolds as morphisms.
• Defines symplectic hopfoids as Hopf algebra objects in the symplectic category, generalizing symplectic groupoids.
• Uses symplectic reduction to realize the core of a symplectic double groupoid as a symplectic quotient of the total space.
• Applies the construction to Lie groupoids, showing that the cotangent lift $T^*G \rightrightarrows T^*M$ forms a symplectic hopfoid.
• Demonstrates that quantization of such hopfoids yields quantum groupoid structures via geometric quantization of canonical relations.
• Proposes a quantization procedure using $L^2$-spaces and integration maps, leading to Hopf algebroid-like structures.

Experimental results

Research questions

• RQ1 How can symplectic stacks be characterized using structures in the symplectic category?
• RQ2 What is the role of symplectic hopfoids in constructing symplectic orbifolds?
• RQ3 How does the core of a symplectic double groupoid inherit a symplectic groupoid structure?
• RQ4 Can all symplectic orbifolds be realized as quotients arising from symplectic hopfoids?
• RQ5 What algebraic structure arises from the geometric quantization of a symplectic hopfoid?

Key findings

• The core of a symplectic double groupoid is a symplectic groupoid, proven via symplectic reduction of the total space.
• Symplectic hopfoids provide a characterization of symplectic double groupoids in terms of canonical relations and symplectic quotients.
• The construction yields symplectic orbifolds naturally from symplectic hopfoids, supporting the conjecture that all symplectic orbifolds arise this way.
• Quantization of the cotangent lift $T^*G \rightrightarrows T^*M$ produces a Hopf algebroid structure on $L^2(G) \rightrightarrows L^2(M)$ via integration and convolution maps.
• The quantization process suggests that symplectic hopfoids quantize to quantum groupoid structures, with potential applications to symplectic and cotangent stacks.
• The framework generalizes to Lie hopfoids, with a proposed link to the cotangent functor via canonical relations.