2D HQFTs and Frobenius $(\mathcal{G},\mathcal{V})$-categories (2501.10113v1)

Abstract: Homotopy Quantum Field Theories as variants of Topological Quantum Field Theories are described by functors from some cobordism category, enriched with homotopical data, to a symmetric monoidal category $\mathcal{V}$. A new notion of HQFTs is introduced using target pairs of spaces $(X,Y)$ acounting for basepoints being sent to points in $Y$. Such $(X,Y)$-HQFTs are classified in dimension 1 by dualizable representations of $\mathcal{G}:=\Pi_1(X,Y)$, the relative fundamental groupoid. For dimension 2, the notion of crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories is introduced, generalizing crossed Frobenius $G$-algebras, where $G$ is only a group. After stating generalities of these multi-object generalizations, a classification theorem of 2-dimensional $(X,Y)$-HQFTs via crossed loop Frobenius $(\mathcal{G},\mathcal{V})$-categories is proven.

• The paper introduces and uses crossed loop Frobenius (G,V)-categories to classify two-dimensional Homotopy Quantum Field Theories (HQFTs), building upon existing algebraic frameworks.
• The classification of 2D HQFTs hinges on categorizing them via dualizable representations of the relative fundamental groupoid $\Pi_1(X,Y)$, which encodes essential homotopical data.
• Developing crossed loop Frobenius categories offers a refined algebraic toolkit for modeling HQFTs and opens avenues for future research in higher dimensions and connections to related fields.

Analysis of "2D HQFTs and Frobenius $(G,V)$-categories"

The paper explores the sophisticated terrain of two-dimensional Homotopy Quantum Field Theories (HQFTs), focusing on their classification through the lens of Frobenius $(G,V)$-categories. By building upon the foundational structures of Topological Quantum Field Theories (TQFTs), the author introduces an enriched framework that integrates homotopical data, thereby extending the algebraic tools available for the paper of HQFTs.

Homotopy Quantum Field Theories are described as functors from a cobordism category, endowed with homotopical data, to a symmetric monoidal category $V$. The paper's essential contribution lies in introducing and classifying these theories using what is termed crossed loop Frobenius $(G,V)$-categories—an innovation expanding the algebraic models associated with HQFTs.

At the heart of this work is the classification of 2D HQFTs. This hinges on the categorization of these theories via dualizable representations of the relative fundamental groupoid of a space pair $(X,Y)$. The groupoid $\Pi_1(X,Y)$, formulated from the fundamental groupoid relative to a subspace, plays a pivotal role as it encapsulates the homotopical data that distinguishes HQFTs from their TQFT counterparts.

Key results include explicit classification theorems for both one-dimensional $(X,Y)$-HQFTs and two-dimensional variants. For dimension one, the paper confirms that such HQFTs are fully classified by dualizable representations of $\Pi_1(X,Y)$. For the more complex two-dimensional HQFTs, the classification employs the newly introduced crossed loop Frobenius $(G,V)$-categories, extending Turaev's earlier work on $G$-algebras by generalizing to the groupoid context.

The formalism of crossed loop Frobenius $(G,V)$-categories leverages the enriched structure of groupoids, providing a comprehensive algebraic framework that encompasses not only classical cobordism structures but also incorporates loop space data. This approach unifies the representation of both closed and open surfaces in HQFTs, drawing connections with the paper of categories enriched over symmetric monoidal categories.

Implications of these findings are manifold. Practically, the development of crossed loop Frobenius categories offers a nuanced algebraic toolkit for modeling HQFTs that can be potentially applied to diverse problems in algebraic topology and quantum algebra. Theoretically, this work opens avenues for future research, particularly in extending HQFT classifications to higher dimensions and other algebraic settings.

This paper presents a concrete step towards understanding the algebraic underpinnings of HQFTs, providing a robust foundation for future theoretical developments. The intricacies inherent in intertwining homotopical data with quantum field theories are deftly handled, suggesting further exploration into the interactions between topology, category theory, and quantum physics. Future work may involve exploring the potential extensions to wider classes of categories or establishing connections with quantum computing models, leveraging these rich algebraic structures.