Symmetry Topological Field Theory for Flavor Symmetry

Abstract: In this Letter, we demonstrate that the Symmetry Topological Field Theory (SymTFT) associated to a Quantum Field Theory (QFT) with continuous non-abelian $G$-flavor symmetry is a $BF$-theory with gauge group $G$. We show that gauging $G$-symmetry with a flat connection yields a theory with global symmetry characterized by exchanging the conjugate variables in the quantization of $BF$-theory. We construct the extended operators that generate the $G$-flavor symmetry and the $(d-2)$-form $\text{Rep}(G)$-symmetry of the gauged QFT. We demonstrate that $BF$-theory arises as the theory characterizing $G$-flavor symmetry of a QFT in the AdS/CFT setup. 't Hooft anomalies of the $G$-flavor symmetry are realized as extra terms in the action.

Symmetry Topological Field Theory for Flavor Symmetry

The paper titled "Symmetry Topological Field Theory for Flavor Symmetry" explores the intricate relationship between symmetry topological field theories (SymTFT), quantum field theories (QFT), and flavor symmetry, particularly in the context of non-abelian groups. The authors establish that a SymTFT associated with a QFT possessing continuous non-abelian ( G )-flavor symmetry can be effectively described by a ( BF )-theory, where the gauge group corresponds to ( G ).

Core Concepts

Symmetry Topological Field Theories (SymTFTs): These are theoretical constructs designed to systematically capture and study the symmetries of QFTs, especially when such symmetries are generalized and non-conventional. SymTFTs encapsulate the action of symmetries in a higher-dimension space, often ( d+1 ) dimensions, compared to the original QFT in ( d ) dimensions.

( BF )-Theory: A topological field theory that is pivotal in characterizing the flavor symmetries associated with QFTs. In this context, ( BF )-theory serves as a theoretical framework through which the flavor symmetry of a QFT manifests in the presence of topological boundaries.

Main Contributions

1. Association of SymTFT and ( BF )-Theory: The paper provides a detailed argument demonstrating that the SymTFT for a QFT with ( G )-flavor symmetry is specifically a ( BF )-theory. This association is substantiated through the canonical quantization over time-slices and identification with spaces of partition functions coupled to background flat ( G )-gauge fields.

2. Non-abelian Flavor Symmetry Action: The construction of extended operators, such as untraced Wilson lines and gauge-invariant operators, is used to generate ( G )-flavor symmetry actions on local QFT operators. Such constructions yield equivariant actions under ( G ), even when certain operators are not topological due to the non-abelian nature of the symmetry.

3. Analogy with Holographic Setups: The discussion extends to how ( BF )-theory can be derived as the SymTFT of boundary QFTs in an AdS/CFT framework. By comparing the roles of boundary currents and gauge fields, the paper aligns ( BF )-theory with long-range gauge symmetry present in holography.

Implications and Future Directions

Theoretical Implications: The identification of ( BF )-theories with symmetry topological perspectives provides a new layer of understanding for flavor symmetries in high-energy physics, particularly quantum chromodynamics, where ( SU(N) ) symmetries play a crucial role.

Practical Implications: Understanding these symmetries through the lens of SymTFTs could enhance computational techniques in particle physics and inform the development of more refined predictive models for particle interactions.

Anomalies and Extensions: The paper touches upon incorporating 't Hooft anomalies through additional terms in the ( BF )-action, suggesting future research avenues in anomaly resolutions and symmetry protection in quantum fields.

Conclusion

This examination of symmetry topological field theories and their relation to flavor symmetries via ( BF )-theories enriches the theoretical understanding of higher-dimensional symmetries in quantum field theories. By resolving the challenges of non-topological operator actions and demonstrating holographic analogies, the study opens new pathways for exploring generalized symmetries and their implications in the landscape of theoretical physics. Further investigations might delve into more complex anomaly structures and explore broader applications across different physical theories.