[Paper Review] On the groupoid of transformations of rigid structures on surfaces

Louis Funar, Rùazvan Gelca|ArXiv.org|Jul 5, 1999

Geometric and Algebraic Topology29 references21 citations

TL;DR

This paper provides a rigorous proof that the Moore-Seiberg equations fully present the 2-groupoid of transformations of rigid structures on surfaces, establishing a universal algebraic framework for 3D topological quantum field theories (TQFTs) with corners. Using Cerf theory and Hatcher-Thurston-style techniques, it constructs a canonical representation of this 2-groupoid from maximal TQFTs, generalizing mapping class group representations and linking to Grothendieck's Teichmüller tower via duality groupoids and DAP decompositions.

ABSTRACT

We prove that the groupoid of transformations of rigid structures on surfaces has a finite presentation as a 2-groupoid establishing a result first conjectured by G.Moore and N.Seiberg. An alternative proof was given by B.Bakalov and A.Kirillov Jr. We present some applications to TQFTs. This is also related to recent work on the Grothendieck-Teichmuller groupoid by P.Lochak, A.Hatcher and L.Schneps.

Motivation & Objective

• To rigorously prove that the Moore-Seiberg equations fully present the 2-groupoid of transformations of rigid structures on surfaces.
• To establish a canonical representation of this 2-groupoid from maximal 3D TQFTs with corners.
• To generalize mapping class group representations by embedding them into a universal 2-groupoid structure.
• To clarify the relationship between the duality groupoid and Grothendieck’s Teichmüller tower via rigid structures and DAP decompositions.

Proposed method

• Applies Cerf theory and handlebody decompositions to analyze isotopy classes of curves and their transformations on surfaces.
• Constructs a groupoid of markings (maximal systems of non-isotopic simple closed curves) and derives its presentation.
• Extends the groupoid to overmarkings (curves cutting surfaces into disks, annuli, and pairs of pants) via combinatorial moves.
• Uses Walker’s framework to lift the structure to rigid structures (DAP decompositions with twist data), proving the Moore-Seiberg equations are sufficient.
• Defines a graphical calculus using trivalent graphs with labeled edges to associate vector spaces and conformal blocks.
• Establishes a local, canonical isomorphism between TQFT conformal blocks and tensor products over labeled graphs, yielding a representation of the duality groupoid.

Experimental results

Research questions

• RQ1 Are the Moore-Seiberg equations a complete presentation of the 2-groupoid of transformations of rigid structures on surfaces?
• RQ2 How can a 3D TQFT with corners give rise to a canonical representation of the duality groupoid?
• RQ3 What is the relationship between the duality groupoid and the mapping class group in the context of TQFT?
• RQ4 How do rigid structures (DAP decompositions) and their transformations relate to the Teichmüller tower and Grothendieck’s program?
• RQ5 Can the representation of the duality groupoid be constructed uniformly across all surfaces using TQFT axioms and local gluing?

Key findings

• The Moore-Seiberg equations are proven to be a complete presentation of the 2-groupoid of transformations of rigid structures on surfaces.
• A canonical representation of the duality groupoid is constructed from any maximal TQFT with corners, generalizing mapping class group representations.
• The conformal blocks of a unitary, cyclic TQFT with unique vacuum decompose into primary blocks $W^{i}_{jk}$, indexed by labels on trivalent graphs.
• The isomorphism $\Phi([\sigma,\sigma']) = \Phi(\sigma')^{-1}\Phi(\sigma)$ defines a local, functorial representation of the duality groupoid on tensor products of vector spaces.
• The duality groupoid serves as a universal object containing all mapping class groups of surfaces, analogous to Grothendieck’s Teichmüller tower.
• For non-cyclic TQFTs like the one in [18], a representation can be constructed on an extended groupoid by adding auxiliary boundary structures.