A Lagrangian Formulation of Symmetry TQFTs for Continuous Symmetries in Quantum Field Theory

Core Concepts

This paper presents a Lagrangian formulation of Symmetry TQFTs (SymTFTs) for continuous symmetries in quantum field theory, focusing on U(1) global symmetries and proposing a framework for continuous non-abelian G(0) symmetries.

Abstract

Bibliographic Information:

Brennan, T. D., & Sun, Z. (2024). A SymTFT for Continuous Symmetries. Journal of High Energy Physics. arXiv:2401.06128v3 [hep-th]

Research Objective:

This paper aims to extend the framework of Symmetry TQFTs (SymTFTs) to incorporate continuous symmetries, specifically focusing on U(1) and proposing a framework for continuous non-abelian G(0) symmetries. This is crucial for a comprehensive understanding of symmetries in quantum field theories (QFTs), including gapless, interacting theories arising from spontaneously broken continuous symmetries.

Methodology:

The authors employ a Lagrangian formulation of SymTFTs. They analyze the U(1) SymTFT by studying its operator content, canonical quantization, and possible gapped boundary conditions. They demonstrate how the SymTFT couples to a QFT and discuss the behavior of the U(1) symmetry and SymTFT operators within the QFT. They further explore how different IR phases of a QFT with U(1) global symmetry are realized in the SymTFT and how to realize different global structures of the U(1)(0) symmetry. Finally, they discuss the SymTFT's application in coupling the QFT to non-flat connections and comment on the dynamical gauging of the U(1) symmetry.

Key Findings:

• The authors successfully formulate a Lagrangian description of the SymTFT for U(1) global symmetries, analogous to the ZN SymTFT described by a BF theory.
• They demonstrate how the U(1) SymTFT can describe symmetry defect operators, gapped boundary conditions (Dirichlet and Neumann), and spontaneous symmetry breaking.
• They illustrate the application of the continuous SymTFT in understanding anomalies, their obstruction of Neumann boundary conditions, and the realization of non-invertible chiral Q/Z symmetry in 4d theories.
• The authors propose a SymTFT for continuous non-abelian G(0) symmetries, drawing parallels with the topologically B-twisted 3d N = 4 G(0) gauge theory.

Main Conclusions:

This paper provides a significant advancement in understanding continuous symmetries within the framework of SymTFTs. The Lagrangian formulation for U(1) symmetries and the proposed framework for G(0) symmetries offer valuable tools for studying a wider range of QFTs and their symmetry structures.

Significance:

This research significantly contributes to the study of symmetries in QFTs by extending the powerful framework of SymTFTs to encompass continuous symmetries. This opens up new avenues for investigating gapless and interacting theories, anomalies, and the interplay between continuous and discrete symmetries.

Limitations and Future Research:

While the paper provides a solid foundation for continuous SymTFTs, further research is needed to fully understand the non-abelian G(0) case, particularly the full spectrum of topological operators and the nature of the G(0) symmetry defect operator in the QFT.

Stats

Quotes

"Symmetry is a powerful tool for studying dynamics in QFT: it provides selection rules, constrains RG flows, and often simplifies analysis."
"The advantage of the SymTFT over the standard construction of SPT phases is that it admits a description of more general global symmetries in QFTs such as non-invertible and categorical symmetries and allows one to capture the topological manipulations via changing the symmetry boundaries."
"In this paper, we will demonstrate how to one can incorporate continuous symmetries into the framework of symmetry TQFTs."

Key Insights Distilled From

A SymTFT for Continuous Symmetries

by T. Daniel Br... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2401.06128.pdf

Deeper Inquiries

How can the framework of continuous SymTFTs be applied to study specific examples of gapless, interacting QFTs arising from spontaneously broken continuous symmetries?

Answer:
The framework of continuous SymTFTs provides a powerful tool for studying gapless, interacting QFTs arising from spontaneously broken continuous symmetries. Here's how:
1. Describing Goldstone Modes:

Effective Action: Spontaneous symmetry breaking (SSB) leads to the emergence of gapless Goldstone modes. The low-energy dynamics of these modes are described by an effective action, often a non-linear sigma model (NLSM), with a target space determined by the symmetry breaking pattern.
SymTFT and NLSM: The SymTFT can be used to constrain the form of this effective action. The target space of the NLSM must be compatible with the topological structure of the SymTFT, particularly its boundary conditions. This provides non-trivial constraints on the low-energy dynamics.
2. 't Hooft Anomalies and Constraints:

Anomaly Matching: 't Hooft anomalies of the original symmetry must be matched in the IR theory, even after SSB. The SymTFT provides a way to track these anomalies and ensure they are appropriately realized in the low-energy description.
Constraints on Phases: Anomalies can obstruct the existence of certain gapped phases or impose relations between different phases. This is reflected in the SymTFT through the absence of certain boundary conditions or the existence of non-trivial maps between them.
3. Interplay with Other Symmetries:

Mixed Anomalies: Continuous symmetries often mix with other global symmetries, leading to mixed anomalies. The SymTFT framework naturally incorporates these mixed anomalies, providing a complete picture of the symmetry structure.
Non-invertible Symmetries: SSB can lead to the emergence of non-invertible symmetries in the IR. The continuous SymTFT framework can be extended to incorporate these non-invertible symmetries, capturing the full complexity of the low-energy theory.
Specific Examples:

O(N) Model: The O(N) model in 3d exhibits SSB to O(N-1). The SymTFT for the U(1) subgroup of O(N) can be used to study the low-energy dynamics of the Goldstone mode, which is described by an NLSM with a target space S^(N-1).
Superfluids and Superconductors: These systems exhibit SSB of a U(1) symmetry. The U(1) SymTFT can be used to study the properties of the gapless phonon mode in superfluids and the constraints on vortex excitations.
In summary, continuous SymTFTs provide a powerful framework for studying gapless, interacting QFTs arising from SSB by:

Constraining the effective action of Goldstone modes.
Tracking 't Hooft anomalies and their realization in the IR.
Capturing the interplay with other global symmetries, including non-invertible ones.

Could there be alternative formulations of SymTFTs for continuous symmetries, potentially using different mathematical frameworks?

Answer:
Yes, it's plausible that alternative formulations of SymTFTs for continuous symmetries exist, potentially employing different mathematical frameworks. Here are some possibilities:
1. Higher-Category Theory:

Motivation: Continuous symmetries are naturally associated with Lie groups, which have a richer structure than finite groups. This suggests that higher-category theory, which generalizes the notion of categories to include higher-dimensional morphisms, might provide a more natural framework.
Potential Advantages: Higher categories could capture the infinitesimal structure of Lie groups more directly, potentially leading to a more elegant and powerful description of continuous SymTFTs.
2. Geometric Quantization:

Motivation: The canonical quantization of the U(1) SymTFT discussed in the context involves geometric quantization of a particle on a circle. This suggests that geometric quantization, which provides a way to construct Hilbert spaces from symplectic manifolds, could play a more prominent role.
Potential Advantages: Geometric quantization might offer a more direct connection to the geometry of the gauge fields and the topology of spacetime, potentially leading to new insights.
3. Non-Commutative Geometry:

Motivation: Non-commutative geometry generalizes the notion of spaces to include non-commutative algebras. This framework has been fruitful in studying gauge theories and quantum gravity.
Potential Advantages: Non-commutative geometry could provide a way to incorporate the non-trivial commutation relations between the operators in the SymTFT more naturally, potentially leading to a more unified description.
4. Generalized Cohomology Theories:

Motivation: Generalized cohomology theories, such as K-theory and bordism, provide powerful tools for studying topological phases of matter.
Potential Advantages: These theories could offer a more refined classification of continuous SymTFTs and their boundary conditions, potentially revealing new connections to other areas of physics and mathematics.
Challenges and Open Questions:

Mathematical Complexity: Exploring these alternative formulations would require significant mathematical development and might involve highly abstract concepts.
Physical Interpretation: It's crucial to ensure that any alternative formulation has a clear and compelling physical interpretation in terms of symmetries and anomalies in QFT.
In conclusion, while the current framework of continuous SymTFTs is promising, exploring alternative formulations using different mathematical frameworks could lead to a deeper understanding of continuous symmetries in QFT and potentially uncover new connections to other areas of physics and mathematics.

How does the understanding of continuous symmetries within SymTFTs contribute to the broader pursuit of unifying quantum field theory and gravity?

Answer:
The understanding of continuous symmetries within the framework of SymTFTs holds significant promise for advancing the quest to unify quantum field theory (QFT) and gravity. Here's how:
1. Holography and Quantum Gravity:

AdS/CFT Correspondence: The AdS/CFT correspondence provides a concrete realization of holography, relating gravitational theories in Anti-de Sitter (AdS) spacetime to conformal field theories (CFTs) on its boundary.
SymTFTs as Boundary Theories: SymTFTs, being topological, are naturally realized as boundary theories of gravitational theories in one higher dimension. This suggests that they could play a crucial role in understanding the holographic dictionary and the emergence of gravity from entanglement in the boundary theory.
2. Emergent Spacetime and Geometry:

Symmetry as a Guiding Principle: One of the key challenges in unifying QFT and gravity is understanding the emergence of spacetime and geometry from more fundamental degrees of freedom. Symmetries, particularly gauge symmetries, are believed to play a central role in this emergence.
SymTFTs and Background Independence: SymTFTs provide a framework for studying symmetries in a background-independent manner, encoding them in terms of topological invariants. This could offer valuable insights into how spacetime geometry could emerge from the structure of symmetries and their anomalies.
3. Beyond Standard Model Physics:

Grand Unified Theories (GUTs): GUTs attempt to unify the fundamental forces of nature by embedding the Standard Model gauge group into a larger symmetry group. These theories often involve continuous symmetries beyond the Standard Model.
SymTFTs and GUTs: SymTFTs could provide a new perspective on GUTs, potentially offering insights into their low-energy structure, the nature of symmetry breaking, and the origin of observed particle masses and mixings.
4. Quantum Information and Entanglement:

Entanglement and Geometry: Recent developments suggest a deep connection between entanglement in QFT and the emergence of spacetime geometry. SymTFTs, with their focus on topological invariants, could provide a natural framework for exploring this connection.
Quantum Error Correction: The topological nature of SymTFTs makes them robust against local perturbations, hinting at potential connections to quantum error correction codes, which are crucial for building fault-tolerant quantum computers. Understanding these connections could have implications for both quantum information theory and quantum gravity.
In summary, the study of continuous symmetries within the framework of SymTFTs offers a promising avenue for progress in unifying QFT and gravity by:

Providing insights into the holographic dictionary and the emergence of gravity.
Offering a background-independent framework for studying symmetries and their role in emergent spacetime.
Shedding light on beyond Standard Model physics and grand unified theories.
Deepening our understanding of the connections between entanglement, geometry, and quantum information.