2D Topological Quantum Field Theories, Frobenius Structures, and Higher Algebra
An oriented TQFT is a symmetric monoidal functor Z ∶ Cobₙ → Vectₖ . Equivalently, this is a rule which assigns to each oriented closed (n − 1)-manifold M a vector space Z(M) and to each n-cobordism B ∶ M → N a linear map Z(M) → Z(N), satisfying certain conditions. The classification for 2D oriented...
| Autor: | |
|---|---|
| Formato: | tesis de maestría |
| Fecha de publicación: | 2024 |
| País: | España |
| Recursos: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/105943 |
| Acesso em linha: | https://hdl.handle.net/20.500.14352/105943 |
| Access Level: | acceso abierto |
| Palavra-chave: | TQFTs Frobenius algebras Higher category theory (∞, n)-categories Cobordisms Symmetric monoidal categories Morita contexts Topología Álgebra 1210 Topología 1201.03 Teoría de Categorías |
| Resumo: | An oriented TQFT is a symmetric monoidal functor Z ∶ Cobₙ → Vectₖ . Equivalently, this is a rule which assigns to each oriented closed (n − 1)-manifold M a vector space Z(M) and to each n-cobordism B ∶ M → N a linear map Z(M) → Z(N), satisfying certain conditions. The classification for 2D oriented TQFTs is a classical “folklore” result; they are equivalent to commutative Frobenius algebras — k-algebras A equipped with a linear functional ε ∶ A → k whose kernel contains no non-trivial ideals. This extends to a categorical equivalence Fun^⊗(Cob_2, Vectₖ) ≃ cFrobₖ. A framed extended TQFT is a symmetric monoidal functor Z ∶ Bordᶠʳₙ → C, where here Bordᶠʳₙ and C are (∞, n)-categories. The Cobordism Hypothesis conjectures that framed extended TQFTs are determined by their image at a single point, which must be a fully dualizable object in the target category C. More precisely, the (∞, 0)-category of framed extended TQFTs is equivalent to the core ∞-groupoid of the subcategory of C spanned by its fully dualizable objects: Fun^⊗(Bordᶠʳₙ, C) ≃ Core(Cᶠᵈ). The (∞, n)-category of framed cobordisms Bordᶠʳₙ carries an O(n)-action, which by the previous statement determines a canonical O(n)-action on Core(Cᶠᵈ) for each (∞, n)-category C. This allows stating a Cobordism Hypothesis for G-structured manifolds: G-structured extended TQFTs are equivalent to the homotopy fixed points of a certain G-action on Core(Cᶠᵈ). Notably, and up to homotopy, an orientation is the same as an SO(n)-structure. When specializing to 2D oriented TQFTs and choosing as target the bicategory Alg₂ of algebras, bimodules and intertwiners, these SO(2)-homotopy fixed points correspond to separable symmetric Frobenius algebras. Hence, by taking loops, we recover a particular case of the classical correspondence between unextended 2D oriented TQFTs and commutative Frobenius algebras. |
|---|