Bicategories for boundary conditions and for surface defects in 3-d TFT (1203.4568v3)

Abstract: We analyze topological boundary conditions and topological surface defects in three-dimensional topological field theories of Reshetikhin-Turaev type based on arbitrary modular tensor categories. Boundary conditions are described by central functors that lift to trivializations in the Witt group of modular tensor categories. The bicategory of boundary conditions can be described through the bicategory of module categories over any such trivialization. A similar description is obtained for topological surface defects. Using string diagrams for bicategories we also establish a precise relation between special symmetric Frobenius algebras and Wilson lines involving special defects. We compare our results with previous work of Kapustin-Saulina and of Kitaev-Kong on boundary conditions and surface defects in abelian Chern-Simons theories and in Turaev-Viro type TFTs, respectively.

• The paper introduces a bicategorical framework that unifies the characterization of boundary conditions and surface defects in 3D TFTs.
• It employs advanced categorical methods, including central functors and modular tensor categories, to connect bulk theories with interface structures.
• The framework leverages Witt equivalence for precise classification, opening pathways for applications in condensed matter phenomena and lattice model interfaces.

Topological Boundary Conditions and Surface Defects in 3D Topological Field Theories

The paper "Bicategories for boundary conditions and for surface defects in 3-d TFT" by Jürgen Fuchs, Christoph Schweigert, and Alessandro Valentino presents an elaborate framework to describe boundary conditions and surface defects in three-dimensional topological field theories (TFTs), particularly those of Reshetikhin-Turaev type. The discussion centers around modular tensor categories and utilizes advanced categorical methods, such as bicategories, central functors, and the Witt group. In what follows, we review the core ideas and implications of this work, which offers significant contributions to the theoretical understanding of 3D TFTs.

Boundary Conditions

Boundary conditions in 3D TFTs refer to the end of a three-dimensional region and are described mathematically by certain categorically structured functors. For a bulk theory characterized by a modular tensor category $\mathcal{C}$, boundary conditions can be encapsulated via a central functor. This functor, denoted as $F_{\text{bulk}\to a}$, maps elements of the bulk category to the monoidal category $\mathcal{W}_a$ of Wilson lines within the boundary. Importantly, $F_{\text{bulk}\to a}$ inherits a structure that allows it to encompass adiabatic analogues of bulk-to-boundary processes. When lifted into the Drinfeld center $\mathcal{Z}(\mathcal{W}_a)$, this functor becomes a braided equivalence. The existence of boundary conditions is intimately tied to the requirement that $\mathcal{C}$ be Witt-trivial, meaning it can be expressed as a Drinfeld center of a fusion category.

Surface Defects

Surface defects, which mediate interfaces between distinct three-dimensional regions, are similarly encoded in terms of central functors and modular tensor categories. Given two such regions, delineated by categories $\mathcal{C}_1$ and $\mathcal{C}_2$, two central functors can map each side’s bulk Wilson lines into the defect category $\mathcal{W}_d$. Again, by forming the Deligne product and considering the constraints from the Drinfeld center, an equivalence becomes evident: $$\mathcal{C}_1 \boxtimes \mathcal{C}_2^\text{rev} \simeq \mathcal{Z}(\mathcal{W}_d)$$.

Witt Group

The necessity for both boundary conditions and surface defects hinges on the Witt equivalence of the relevant modular tensor categories. Specifically, if such structures exist between two regions, then the categories labeling those regions must reside in the same class within the Witt group $\mathfrak{W}$. This equitable relation is particularly striking because it aligns with abstract mathematical processes inherent to modular tensor categories.

Implications and Future Directions

These insights provide profound implications both theoretically and practically. Modifications to TFTs under boundary conditions and defects offer new avenues for exploring condensed matter phenomena, such as phases in quantum Hall systems. Moreover, identifying such conditions is paramount to modeling interfaces in lattice models, extending the scope of the Reshetikhin-Turaev construction to include substructures of lower dimensionality in three-dimensional manifold categorizations.

Conclusion

The paper establishes a robust framework to analyze and describe topological boundary conditions and surface defects in 3D topological field theories, leveraging category theory tools such as central functors and the Witt group. This enhances the understanding of modular tensor categories and their application to 3D TFTs, providing a seminal contribution to the mathematical underpinnings of theoretical physics. Future studies may expand upon these foundations to explore more complex interactions between different categorical structures in higher-dimensional settings or across different fields of physical application.