Orientable 2-Dimensional Cobordisms Are Classified by TQFTs, Enabling Advances in Khovanov Homology

Topological quantum field theories, mathematical frameworks describing physical systems where properties remain unchanged under continuous deformations, form the basis for understanding diverse phenomena in physics and mathematics. Leon J. Goertz and Paul Wedrich investigate a specific class of these theories, focusing on those applicable to two-dimensional surfaces, and present a new framework for classifying topological quantum field theories designed for orientable cobordisms, structures defining the boundaries of these surfaces. This work addresses a gap between fully oriented and unoriented theories, motivated by applications in areas like knot theory and the study of three-dimensional manifolds, where surfaces are frequently treated as unoriented despite underlying theoretical structures requiring orientation. By establishing a classification for orientable cobordisms, the researchers provide a crucial intermediate step towards a more complete understanding of topological quantum field theories and their applications in complex mathematical and physical models.

Dimensional topological quantum field theories (TQFTs) for orientable cobordisms represent a central focus of this work, approached with rigorous mathematical precision. Motivation stems from skein-theoretic models of surfaces embedded in three-manifolds and Khovanov homology, where surfaces are often treated as unoriented despite the associated two-dimensional TQFTs not necessarily requiring full unorientation. The research introduces Cob2 to denote the symmetric monoidal category of oriented two-dimensional cobordisms between oriented closed one-manifolds, and UCob2 to represent the symmetric monoidal category of unoriented two-dimensional cobordisms between unoriented closed one-manifolds. Consequently, the category OCob2 is defined as a non-full subcategory of UCob2, encompassing all objects and specifically those morphisms that preserve orientation.

Orientable Cobordisms, Frobenius Algebras, and TQFTs

Scientists investigate the composition of orientable cobordisms, recognizing that even the Klein bottle, though unorientable, can be constructed from orientable components. The set of orientable cobordisms is not closed under composition, but its closure is OCob2. The goal is to provide a complete description of OCob2, positioning it between Cob2 and UCob2, and to fully classify symmetric monoidal functors originating from it, which are essential for defining orientable TQFTs. The research demonstrates that, as symmetric monoidal categories, Cob2 is freely generated by a commutative Frobenius algebra, OCob2 by a commutative Frobenius algebra with involution, and UCob2 by a commutative Frobenius algebra with involution and a compatible Möbius morphism.

These characterizations are recalled for both oriented and unoriented cases, and then used to deduce the intermediate case. Let V be a symmetric monoidal category serving as the target for these TQFTs. A corollary establishes that evaluating functors at the circle S1 creates an equivalence between functors from OCob2 and the category of involutive Frobenius algebras, providing a section for the forgetful functor. To provide a presentation for OCob2, scientists define key morphisms including multiplication (m), unit (u), diagonal (Δ), evaluation (ε), involution (φ), and a punctured Klein bottle (K). Proposition 10 proves that the orientable cobordism category OCob2 is generated as a PROP by these morphisms, subject to specific relations. Proposition 11 demonstrates that a symmetric monoidal functor from OCob2 to Cob2 is uniquely and well-defined by assigning the circle S1 to itself and the morphisms (m, u, Δ, ε, φ) to their counterparts in Cob2, providing a section for the forgetful functor.

Orientable Cobordisms Categorised, Bridging Theory Gaps

This work establishes a refined classification of two-dimensional topological field theories (TQFTs), bridging the gap between fully oriented and unoriented theories. Scientists developed an intermediate framework, categorizing TQFTs defined on orientable cobordisms, which are surfaces that consistently maintain a defined “inside” and “outside”. This new classification provides a more nuanced understanding of how these theories behave and expands the possibilities for their application. The team rigorously defined the category OCob2, encompassing cobordisms constructed from orientable surfaces, and demonstrated its position between the established categories of fully oriented (Cob2) and fully unoriented (UCob2) cobordisms.

They proved that OCob2 is generated by a specific set of morphisms, mathematical building blocks representing fundamental surface transformations, including multiplication (m), unit (u), comultiplication (∆), and counit (ε). These morphisms satisfy precise relationships ensuring a consistent algebraic structure, specifically a unital, associative, and commutative algebra. Crucially, the research demonstrates a direct correspondence between functors from OCob2 and involutive Frobenius algebras. A corollary establishes that evaluating these functors at the circle S1 creates an equivalence between functors from OCob2 and the category of involutive Frobenius algebras, meaning these two mathematical structures are essentially interchangeable.

This connection allows scientists to classify orientable TQFTs using the well-understood properties of these algebras. Furthermore, the team constructed a diagram illustrating the relationships between TQFTs originating from Cob2, OCob2, and UCob2, demonstrating how oriented TQFTs can be extended to OCob2 by simply choosing the identity as the involution. This work justifies the use of commutative Frobenius algebras in constructing skein theories for three-manifolds, particularly in the Asaeda, Frohman, Kaiser TQFT and Bar-Natan’s description of Khovanov homology, where the Frobenius algebra H*(CP1) is commonly used. The research confirms that this algebra defines a TQFT on OCob2, which is sufficient for these applications.

Orientable Cobordisms and Simplified TQFT Classification

This work establishes a framework for classifying two-dimensional topological quantum field theories (TQFTs) that applies specifically to orientable cobordisms. Building upon existing classifications for fully unoriented TQFTs, this research identifies a simplified structure applicable when dealing with orientable surfaces. This intermediate approach proves valuable because many models, such as those arising from skein theory and Khovanov homology, often treat surfaces as unoriented even while the underlying TQFT may not be fully unoriented. The researchers demonstrate that this framework accurately captures the essential properties of these TQFTs, providing a well-defined and faithful section for a specific forgetful functor. The authors acknowledge that this work focuses on orientable cobordisms and does not directly address fully unoriented cases, representing a natural progression in the understanding of TQFT classifications.