Relative quantum field theory (1212.1692v3)

Abstract: We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions.

Summary of "Relative Quantum Field Theory"

The paper "Relative Quantum Field Theory" by Daniel S. Freed and Constantin Teleman provides a comprehensive exploration into the concept of relative quantum field theories (RQFTs), elaborating on their structural framework, examples, and implications in both gauge theories and string theories. The foundational goal is to elucidate the notion of RQFTs, particularly in contexts such as gauge theory involving algebraic structures rather than group structures and their peculiarities when reduced in dimensionality.

Framework and Definitions

Relative Quantum Field Theory: The authors introduce the concept of RQFTs as a modification of typical quantum field theories where the theory is associated with another, usually in a higher dimension. The relative theory provides a boundary condition or transition between different theories. Formally, a relative quantum field theory $F$ is defined as a homomorphism $F: 1 \rightarrow \alpha$, where $\alpha$ is an extended quantum field theory in one higher dimension. This definition inverts the standard approach by considering anomalies and their resolution through higher-dimensional fields.

Examples and Interpretation

Relative $\sigma$-Models: The authors explore relative $\sigma$-models, where fields $p: \text{Map}(X, M) \to B X$ relate mappings to a manifold $M$ and classifying space $BX$. Here, integration over fibers of these mappings connects classical $\sigma$-models to their relative counterparts. The theory $\alpha$ is topological, while the theory $F$ incorporates geometric data like metrics.

Relative Gauge Theories: The paper extends to gauge theories based on central extensions with finite groups $\pi$. The relative structure is articulated through fibrations and fields forming groupoids, aiming to resolve puzzles in dimensional reductions observed in string theory contexts. The article cites examples like supersymmetric gauge theories where dimensional reductions reveal deeper group structure anomalies resolved by relative theories.

Dimensional Reduction and Implications

The reduction of these theories in lower dimensions, specifically the six-dimensional $(0,2)$-superconformal theories in string theory, is examined. The authors propose that dimensional reductions often involve theories transforming through layers of anomalies, necessitating relative formulations. These insights enable exploration into anomalies not computable through standard quantum field theoretic means, providing intuitive yet rigorous frameworks for understanding symmetries and dualities, such as Langlands duals in four-dimensional reductions.

Practical and Theoretical Implications

The paper explores how relative quantum field theories impact contemporary understanding of gauge anomalies, modular tensor categories, and the structure of quantum field theories. By extending the notion of quantum theories through simplicial sets and category theory, there is potential for computing and understanding HFQFTs in terms of geometric topology.

Speculation on Future Developments

The conceptual pathway laid expands the toolbox for theoretical physicists tackling anomalies in highly symmetric spaces, offering resolution pathways by leveraging higher-dimensional structures. These relative structures serve as potentially transformative lenses through which complex systems in quantum mechanics and field theory might be re-evaluated. Future development may include exploring the quantization paths for self-dual fields, a crucial point hinted at within this paper.

In conclusion, "Relative Quantum Field Theory" offers foundational insights into the role and formulation of relative theories in quantum mechanics and their practical implications. By codifying them within existing theoretical constructs, researchers can reinterpret challenging anomalies, leverage mathematical structures, and explore new dimensions of quantum field theories.