Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A construction of surface skein TQFTs and their extension to 4-dimensional 2-handlebodies

Leon J. Goertz

TL;DR

The paper constructs a surface skein TQFT AFK_A from a commutative Frobenius algebra A, producing surface skein categories and 3D skein modules that glue via bimodule structures. It then extends this framework to certain 4-dimensional manifolds by Walker-style state-sum methods, identifying when extensions are possible through the perfectness of pairings and introducing a Kirby color to model 4D 2-handle attachments. A detailed analysis is carried out for the α-deformed Bar-Natan theory A_α, including the explicit computation of the Kirby color ω_{2n} and demonstrating extendability to 4D 2-handles (but not all 3-handles). The work also relates the resulting invariants to known 4D theories (e.g., Dijkgraaf–Witten) in specific cases and provides concrete computations for representative 4D handlebodies, highlighting the role of idempotents in disk categories and the interplay between diagrammatic and algebraic constructions. Overall, the paper advances a higher-dimensional skein-theoretic perspective on TQFTs and offers computable invariants for a class of 4D 2-handlebodies grounded in Frobenius-algebra data.

Abstract

For a commutative Frobenius algebra $A$, we construct a $(2,3,3+\varepsilon)$-dimensional TQFT $\mathsf{AFK}_A$ that assigns to a 3-manifold a skein module of embedded $A$-decorated surfaces. These surface skein modules have been first defined by Asaeda--Frohman and Kaiser using skein relations that generalize the combinatorics of Bar-Natan's dotted cobordisms. For 3-manifolds with boundary, we show that surface skein modules carry an action by a certain surface skein category associated with the boundary, which yields a gluing formalism. Our main result concerns a partial extension of $\mathsf{AFK}_A$ to dimension 4, which uses an inductive state-sum construction following Walker. As an example, the equivariant version of Lee's deformation of dotted cobordisms yields a TQFT that extends to 4-dimensional 2-handlebodies but not 3-handlebodies. Finally, we characterize the attachment of 4-dimensional 2-handles by means of a certain Kirby color and use it to compute the invariants of 4-dimensional 2-handlebodies in examples.

A construction of surface skein TQFTs and their extension to 4-dimensional 2-handlebodies

TL;DR

The paper constructs a surface skein TQFT AFK_A from a commutative Frobenius algebra A, producing surface skein categories and 3D skein modules that glue via bimodule structures. It then extends this framework to certain 4-dimensional manifolds by Walker-style state-sum methods, identifying when extensions are possible through the perfectness of pairings and introducing a Kirby color to model 4D 2-handle attachments. A detailed analysis is carried out for the α-deformed Bar-Natan theory A_α, including the explicit computation of the Kirby color ω_{2n} and demonstrating extendability to 4D 2-handles (but not all 3-handles). The work also relates the resulting invariants to known 4D theories (e.g., Dijkgraaf–Witten) in specific cases and provides concrete computations for representative 4D handlebodies, highlighting the role of idempotents in disk categories and the interplay between diagrammatic and algebraic constructions. Overall, the paper advances a higher-dimensional skein-theoretic perspective on TQFTs and offers computable invariants for a class of 4D 2-handlebodies grounded in Frobenius-algebra data.

Abstract

For a commutative Frobenius algebra , we construct a -dimensional TQFT that assigns to a 3-manifold a skein module of embedded -decorated surfaces. These surface skein modules have been first defined by Asaeda--Frohman and Kaiser using skein relations that generalize the combinatorics of Bar-Natan's dotted cobordisms. For 3-manifolds with boundary, we show that surface skein modules carry an action by a certain surface skein category associated with the boundary, which yields a gluing formalism. Our main result concerns a partial extension of to dimension 4, which uses an inductive state-sum construction following Walker. As an example, the equivariant version of Lee's deformation of dotted cobordisms yields a TQFT that extends to 4-dimensional 2-handlebodies but not 3-handlebodies. Finally, we characterize the attachment of 4-dimensional 2-handles by means of a certain Kirby color and use it to compute the invariants of 4-dimensional 2-handlebodies in examples.

Paper Structure

This paper contains 19 sections, 57 theorems, 201 equations, 1 table.

Table of Contents

  1. Introduction
  2. Surface skein theory as a TQFT
  3. Extension of surface skein theory to 4-dimensional manifolds
  4. A Kirby color for 4-dimensional 2-handles
  5. Surface skein theory from commutative Frobenius algebras
  6. Frobenius algebras
  7. Surface skein modules and categories
  8. The surface skein TQFT
  9. Extension to 4-dimensional handles
  10. General construction
  11. Extending rank 1 surface skein theory
  12. The skein module of the solid torus
  13. Strong separability and perfectness
  14. The Kirby color for 2-handles
  15. Computation of the Kirby color
  16. ...and 4 more sections

Key Result

Theorem A

For a commutative Frobenius algebra $A$, surface skein theory assembles into a $(2,3,3+\varepsilon)$-dimensional TQFT, that is, a functor of symmetric monoidal bicategories

Theorems & Definitions (175)

  • Example
  • Theorem A: Construction \ref{['con:theTQFT']}
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem B: Corollary \ref{['cor:allAExtendToOneH']}
  • Example : Theorem \ref{['thm:rkOneTheory']} and Proposition \ref{['prop:rkOneAndDW']}
  • Proposition : Proposition \ref{['prop:Ext2handlesSeparable']}
  • Example : Example \ref{['exm:BNnoExtension']}
  • Theorem C
  • ...and 165 more