Defects in skein theory and TQFT

Abstract: Given a 3-manifold $M$ with a network of line and point defects in its boundary, we define the skein module of this configuration, generalizing the well-studied case of 3-manifolds which only admit point defects in the boundary. We prove that when all defects are labelled by semisimple data, our skein module is isomorphic to the state space of $\partial M$ in the defect version of the Reshetikhin-Turaev TQFT constructed by Carqueville-Runkel-Schaumann. Our defect skein modules follow naturally by globalizing the graphical calculus of module categories and functors thereof, and generalize the possible defect data considered in the defect TQFT beyond the semisimple case.

• The paper presents a novel framework that establishes an explicit isomorphism between defect skein modules and TQFT state spaces.
• It employs categorical and diagrammatic techniques using ribbon graphs, module categories, and Frobenius algebra labels to encode defects.
• Explicit computations on 3-manifolds validate the framework, extending TQFT constructions to both semisimple and non-semisimple cases.

Defect Skein Modules and State Spaces in TQFT

Introduction and Motivation

The paper "Defects in skein theory and TQFT" (2606.07432) develops a framework for skein modules associated to 3-manifolds with a network of line and point defects on the boundary, generalizing standard skein-theoretic constructions linked to Reshetikhin-Turaev (RT) Topological Quantum Field Theories (TQFTs). The authors introduce a machinery that comprehensively incorporates not just point, but also line defects labeled by module categories and their functors, operating within the categorical language of ribbon categories. This approach links the diagrammatic/graphical calculi intrinsic to skein theory with the higher-categorical and module-theoretic structures underlying defect TQFTs, especially as formulated by Carqueville–Runkel–Schaumann (CRS).

A salient feature is the treatment of both semisimple and non-semisimple defect data, which extends previous work that was either restricted to semisimple settings or considered only basic defect structures. The main theoretical result is an explicit isomorphism between the defect skein module, built diagrammatically from ribbon graphs respecting the defect stratification, and the state space of the associated boundary in the defect RT TQFT as constructed in the CRS framework.

Algebraic, Categorical, and Topological Preliminaries

The construction operates in the setting of modular fusion categories $\mathcal{C}$ but accommodates general ribbon categories at the level of algebraic definitions. The ambient categorical context is that of $Rex$, the $(2,1)$-category of finitely cocomplete $\mathbf{k}$-linear categories, right exact functors, and natural transformations. Module categories, Deligne-Kelly tensor products, and module functors are used as labels for defects, reflecting modern higher-categorical perspectives.

Defects are modeled as stratifications: 1-strata (line defects) are locally labeled by module categories, 0-strata (point defects) by functors or multimodules, generalizing the labeling of ordinary open-closed TQFT. Important structural roles are played by $\Delta$-separable symmetric Frobenius algebras (for surface/line defect labeling) and their associated module categories. The theory is developed within a precise smooth stratified topology framework, leveraging regularity, orientation, and framing conventions, ensuring compatibility of graphical calculus with the stratified structure.

Defect Skein Modules: Construction and Relations

Defect skein modules $\mathrm{Sk}(M)$ generalize ordinary $\mathcal{C}$-skein modules by encoding the local algebraic data of defects in the boundary stratification:

• Generators: Embedded, framed ribbon graphs in $M$ colored according to the defect labeling (module categories/objects of $\mathcal{C}$ functors at vertices, etc.).
• Locality and Relations: Relations (skein relations) are imposed locally, in embedded balls/cubes respecting the defect stratification. These relations model the corresponding Hom spaces, enforcing the graphical calculus of ribbon categories, module categories, and functorial data.

The approach systematically handles all local moves—interior (pure $\mathcal{C}$), line defect (module category), point defect (functor/multimodule)—and establishes that all relations required are realizable by local moves in appropriately embedded neighborhoods. The formal combinatorics of the construction ensure that state spaces for surfaces with arbitrary line and point defect networks are well-defined.

Correspondence with Defect TQFT State Spaces

The critical result proven is the isomorphism:

$Rex$0

where the latter is the defect RT state space for the boundary surface, as in CRS [CRS18LineSurfaceDefects]. The proof is constructive, explicitly giving the correspondence between skein-theoretic objects (defect-colored ribbon graphs) and TQFT states (morphisms in the appropriately colimit-coned categorical settings arising from triangulations and CRS projectors).

The argument systematically aligns local relations and isotopy classes in the skein module with the idempotent/projector-based quotient defining the defect RT state space. Key technical tools include:

• Handle Attachments and Surgery: Skein equivalence under handle sliders mirrors the TQFT construction via surgeries and cobordisms.
• Functoriality and Factorization: The graphical calculus globalizes to factorization homology, matching the colimit constructions underlying extended TQFT.
• Zipping and Subduction: Processes for moving coupons and propagating module actions along defects connect the linear algebra of module categories with diagrammatics.

When all defect data are semisimple (i.e., labels are from fusion module categories and functors), the correspondence is tautological (recovering standard CRS), but the framework extends strictly further, allowing for non-semisimple module categories and functors.

Examples and Computational Implications

The paper provides explicit, nontrivial computations for the defect skein modules of standard 3-manifolds (solid tori with networked defects, handlebodies, spheres with defects). These computations utilize the full machinery and showcase the reduction of seemingly complicated situations to explicit linear algebra, in terms of coends and relative monoidal Hom spaces.

An illustrative case is the computation of the defect skein module for a solid torus with a single line and point defect, showing its identification with

$Rex$1

where $Rex$2 is the Frobenius algebra and $Rex$3 is the bimodule labeling the defect.

These explicit computations clarify the effect of defect handle slides (the equivalence under 2-handle moves), encode the nontrivial interplay of local module-theoretic structure with global topological manipulations, and provide a template for calculational approaches in non-semisimple settings.

Higher-Categorical and Factorization Homology Connections

The framework aligns defect skein modules with factorization homology as a colimit over stratifications decorated by module categories and related categorical data. This placement in higher algebraic settings, such as braided tensor Morita 2-categories, suggests promising routes for systematic generalizations (non-semisimple TQFT, stratified factorization homology, higher defect types).

The defect skein module construction corresponds to the global sections of the (factorization) algebra governing the higher-categorical structure. There are clear connections to Crane–Yetter-type invariants and extended boundary theories, especially through the perspective of anomaly inflow and bulk-boundary correspondence.

Future Directions

The paper lays the groundwork for the following research avenues:

• Non-Semisimple TQFTs: Systematic construction of defect state spaces and categorified invariants for non-semisimple modular data, guided by skein-theoretic methods [DGG+22, CGHP23, CGP23].
• Higher Defect Networks: Stratified settings incorporating defects of all admissible codimensions, controlled by diagrammatic calculus and categorical traces, aligned with the language of Factorization Homology and Morita theory.
• Algorithmic Computability: The explicit nature of the local-to-global graphical calculus paves the way for algorithmic approaches to TQFT state spaces with defects.
• Extended/Derived Structures: Connecting with extended TQFTs, blob homology, and higher trace frameworks to study gauge and symmetry-defect interplay.

Conclusion

The paper achieves an overview between diagrammatic skein theory and the algebraic-categorical machinery of defect TQFT, generating a robust, expressive formalism for state spaces in the presence of arbitrary defect networks on 3-manifold boundaries. The explicit isomorphism between defect skein modules and TQFT state spaces not only validates the graphical approach but rigorously anchors defect modifications in both semisimple and non-semisimple contexts. The framework presented constitutes a technically powerful tool for ongoing work in quantum topology, higher categories, and topological phases of matter.

---

References

• "Defects in skein theory and TQFT" (2606.07432)
• Carqueville, Runkel, Schaumann, "Orbifolds of n-dimensional defect TQFTs" [CRS19OrbifoldsNdimensionalDefect]
• [DGG+22] D. Gaiotto, et al., "3-Dimensional TQFTs from nonsemisimple modular tensor categories".
• [CGHP23] C. Galindo, et al., "Skein categories for 3+1 TQFTs from nonsemisimple categories".

(Paper citations and further technical details are accessible in the full text (2606.07432).)