[Paper Review] Noncommutative geometry and compactifications of the moduli space of curves

Alastair Hamilton|ArXiv.org|Oct 25, 2007

Algebraic Geometry and Number Theory12 references4 citations

TL;DR

This paper establishes a complete algebraic description of the homology of Kontsevich's compactification of the moduli space of curves using a two-parameter family of differential graded Lie algebras. By equipping the space of noncommutative 0-forms with a Lie bialgebra structure derived from curve pinching, the authors deform Kontsevich's noncommutative symplectic geometry and show that the Chevalley-Eilenberg homology of this deformation precisely recovers the homology of the compactified moduli space.

ABSTRACT

In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Witten's conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers.

Motivation & Objective

• To provide an algebraic characterization of the homology of Kontsevich's compactification of the moduli space of curves.
• To extend Kontsevich's noncommutative symplectic geometry by introducing a Lie bialgebra structure on noncommutative 0-forms.
• To construct a two-parameter family of differential graded Lie algebras as a deformation of the original noncommutative symplectic framework.
• To prove that the Chevalley-Eilenberg homology of this family recovers the homology of the compactified moduli space.
• To establish a correspondence between stable ribbon graph complexes and the homology of the compactified moduli space via algebraic invariants.

Proposed method

• Utilize a Lie cobracket on the space of noncommutative 0-forms to define a Lie bialgebra structure, modeling the topological operation of pinching simple closed curves on Riemann surfaces.
• Construct a two-parameter family of differential graded Lie algebras by deforming the original noncommutative symplectic Lie algebra using the Lie bialgebra structure.
• Apply the Chevalley-Eilenberg complex to compute the homology of the deformed differential graded Lie algebra over ℚ.
• Establish a canonical isomorphism between the homology of the compactified moduli space and the stable homology of the constructed family of differential graded Lie algebras.
• Use explicit tensor constructions associated to stable ribbon graphs to define inverse maps and verify bijectivity of the homology correspondence.
• Verify that the differential in the Chevalley-Eilenberg complex corresponds to edge contractions in ribbon graphs, with contributions from the Lie bracket, Lie cobracket, and deformed bracket on 0-forms.

Experimental results

Research questions

• RQ1 Can the homology of Kontsevich's compactification of the moduli space of curves be fully described using algebraic structures?
• RQ2 How does the Lie bialgebra structure on noncommutative 0-forms encode the topological operation of curve pinching in the compactification?
• RQ3 What is the role of the two-parameter deformation in generalizing Kontsevich's noncommutative symplectic geometry to include compactified moduli spaces?
• RQ4 Is there a canonical isomorphism between the homology of the compactified moduli space and the Chevalley-Eilenberg homology of a deformed differential graded Lie algebra?
• RQ5 Can the stable ribbon graph complex be reconstructed algebraically from the invariants of the deformed Lie algebra?

Key findings

• The Chevalley-Eilenberg homology of the two-parameter family of differential graded Lie algebras is isomorphic to the homology of Kontsevich's compactification of the moduli space of curves.
• The isomorphism is realized via a canonical map that sends tensors associated to stable ribbon graphs to homology classes, with an explicit inverse constructed using coinvariants of the Lie algebra action.
• The differential in the Chevalley-Eilenberg complex corresponds exactly to the three types of edge contractions in the stable ribbon graph complex: from the Lie bracket on the deformed algebra, the Lie bracket on 0-forms, and the Lie cobracket on 0-forms.
• The map between the homology of the compactified moduli space and the homology of the deformed differential graded Lie algebra is a bijective map of Hopf algebras.
• The construction provides a purely algebraic method to generate homology classes on the compactified moduli space via exponentiation in the Maurer-Cartan moduli space of the deformed Lie algebra.
• The deformation parameters γ and ν allow for a continuous family of compactifications, with the original moduli space and its Deligne-Mumford compactification arising as special cases when parameters are set to zero.