Higher Central Charges as Obstructions to Conformal Boundary Conditions and Lattice Realizations in 2D Rational CFTs

Kernkonzepte

In 2D rational conformal field theories (CFTs), higher central charges, beyond the chiral central charge, act as obstructions to the existence of energy-conserving boundary conditions and lattice realizations.

Zusammenfassung

Bibliographic Information:

Liu, R., & Ye, W. (2024). Higher obstructions to conformal boundary conditions and lattice realizations. Journal of High Energy Physics, 2024. [Preprint]. arXiv:2411.11757v1

Research Objective:

This research paper investigates the long-held belief that the vanishing of chiral central charges in a 2D conformal field theory (CFT) guarantees the existence of conformal boundary conditions. The authors aim to identify potential higher obstructions beyond chiral central charges and explore their implications for lattice realizations of these CFTs.

Methodology:

The authors utilize the framework of topological quantum field theories (TQFTs) and their correspondence with rational CFTs. They employ the concept of "folding trick" to relate conformal boundary conditions of 2D CFTs to topological boundary conditions of their corresponding 3D TQFTs. They further leverage the notion of higher central charges, defined through anyon data in TQFTs, to analyze obstructions to topological boundaries.

Key Findings:

• The authors demonstrate that the existence of a conformal boundary condition in a 2D rational CFT implies the existence of a topological boundary condition in its corresponding 3D bulk TQFT.
• They establish that higher central charges, beyond the chiral central charge, can obstruct the existence of topological boundary conditions in 3D TQFTs.
• Consequently, they prove that non-trivial higher central charges in a 2D rational CFT prevent the existence of conformal boundary conditions and, therefore, lattice realizations.

Main Conclusions:

The research concludes that vanishing chiral central charge alone is insufficient to guarantee conformal boundary conditions in 2D rational CFTs. Higher central charges emerge as additional, and previously overlooked, obstructions. This finding has significant implications for the study of lattice regularizations of CFTs, suggesting that a CFT with vanishing chiral central charge may still lack a lattice model representation due to these higher obstructions.

Significance:

This study provides a deeper understanding of the relationship between 2D rational CFTs, their boundary conditions, and their potential lattice realizations. The identification of higher central charges as obstructions challenges previous assumptions and opens new avenues for investigating the realizability of CFTs in lattice models.

Limitations and Future Research:

The research primarily focuses on 2D rational CFTs. Exploring whether similar higher obstructions exist in irrational CFTs or more general 2D quantum field theories remains an open question. Further investigation into the properties and computation of higher central charges, particularly their extension to non-rational CFTs, is crucial for advancing this field. Additionally, exploring the role of global symmetries and their interplay with higher central charges in the context of boundary conditions and lattice realizations presents a promising direction for future research.

Statistiken

κ = π/6 Tc−, where κ represents thermal Hall conductance, T is temperature, and c− is the chiral central charge.
ξ1 := (Πa d2aθa)/|Πa d2aθa| = exp(2πic−/8), where ξ1 represents the first higher central charge, a labels anyon types, da is the quantum dimension of anyon a, and θa is the topological spin of anyon a.
ξn := (Πa d2aθna)/|Πa d2aθna|, where ξn represents the nth higher central charge.
For the abelian Chern-Simons theory U(1)2 × U(1)−4, ξ3 = −1 while the Frobenius-Schur exponent (NFS) is 8.

Zitate

"An important consequence of gravitational anomalies is that they are obstructions to an energy-conserving boundary condition of the field theory."
"In the present paper, we answer both of these questions negatively. Our argument is based on earlier results obtained in Ref. [9] as well as a newly developed notion called higher central charges [15–19] and Witt equivalence of chiral CFTs [10]."
"In summary, higher central charges can serve as an obstruction to energy-conserving boundary conditions and lattice regularizations beyond gravitational anomaly."
"Our corollary 2 can be viewed as a special case of higher version of Bloch theorem [41–44]."
"Our results indicate, even with the U(1)-breaking perturbations, the perturbed chiral Luttinger liquid still does not admit a lattice realization."

Wichtige Erkenntnisse aus

Higher obstructions to conformal boundary conditions and lattice realizations

by Ruizhi Liu, ... um arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11757.pdf

Tiefere Fragen

How do these findings concerning higher central charges in 2D CFTs impact our understanding of the AdS/CFT correspondence and its implications for quantum gravity?

This is a very insightful question that delves into the potential interplay between seemingly disparate areas of theoretical physics. Here's a breakdown of the implications:
1. Constraints on Holographic Duals: The AdS/CFT correspondence posits a profound duality between conformal field theories (CFTs) in d dimensions and quantum gravity theories in (d+1)-dimensional anti-de Sitter (AdS) spacetime. The findings of the paper, particularly Corollary 1, suggest that the existence of a well-defined, consistent gravitational dual in AdS3 might be more restrictive than previously thought. If a 2D CFT possesses nontrivial higher central charges, implying the absence of conformal boundary conditions, it raises questions about the nature of its holographic dual in AdS3. It suggests that the corresponding gravity theory might be inherently non-local or ill-defined at the boundary, posing challenges for our understanding of quantum gravity in such scenarios.
2. Boundary Conditions and Black Hole Physics: In the AdS/CFT context, different boundary conditions imposed on the CFT side are often interpreted as different states within the gravitational theory. For instance, thermal states in the CFT correspond to black hole solutions in AdS. The absence of conformal boundary conditions, as dictated by nontrivial higher central charges, might imply constraints on the types of black hole solutions or other gravitational configurations that can be realized holographically.
3. New Probes of Quantum Gravity: Higher central charges, if they can be generalized to a broader class of CFTs, could potentially serve as novel probes of quantum gravity via the AdS/CFT dictionary. Their nontriviality might signal the presence of subtle quantum gravitational effects or degrees of freedom that are not captured by conventional observables like the central charge.
4. Beyond Rational CFTs: A significant challenge lies in extending these findings beyond the realm of rational CFTs. The AdS/CFT correspondence encompasses a vast landscape of CFTs, many of which are not rational. Developing a framework to understand higher obstructions and their gravitational interpretations for more general CFTs is crucial for a complete picture.

Could there be alternative approaches, beyond considering higher central charges, to circumvent these obstructions and potentially enable lattice realizations for a broader class of 2D CFTs?

This is an open question with no definitive answers yet. However, here are some potential avenues for exploration:
1. Modified Lattice Structures: The paper focuses on lattice models with a tensor-factorized Hilbert space and strictly local interactions. Relaxing these assumptions might offer a way forward. For instance, exploring lattices with non-local couplings, tensor network representations with entanglement beyond nearest neighbors, or higher-form symmetries could potentially accommodate CFTs with nontrivial higher central charges.
2. Enlarging Symmetry Considerations: The paper primarily considers global symmetries. Incorporating more intricate symmetry structures, such as non-invertible symmetries or higher-form symmetries, into the lattice regularization procedure might provide additional flexibility and potentially circumvent the obstructions.
3. Non-Unitary Lattice Models: The standard paradigm for lattice realizations often assumes unitarity. Exploring non-unitary lattice models, while challenging, might offer a broader framework that could accommodate CFTs previously deemed unrealizable.
4. Approximation Methods: Instead of seeking exact lattice realizations, developing sophisticated approximation methods, such as tensor network renormalization or other numerical techniques, could provide valuable insights into the low-energy physics of CFTs with nontrivial higher central charges.
5. Beyond the Lattice: It's conceivable that the lattice paradigm, while powerful, might not be the most natural framework for realizing certain CFTs. Exploring alternative regularization schemes, such as those based on quantum field theory techniques or string theory, could offer new perspectives.

If we view the constraints imposed by higher central charges as a form of 'frustration,' can we draw parallels to other frustrated systems in condensed matter physics and leverage those insights to deepen our understanding of CFTs?

This is a thought-provoking analogy that could potentially lead to fruitful cross-fertilization of ideas between CFTs and condensed matter physics. Here's a breakdown of the parallels and potential insights:
1. Frustration and Obstructed Ground States: In condensed matter physics, frustration often arises when local interactions between constituents of a system (e.g., spins in a magnet) cannot be simultaneously satisfied, leading to a highly degenerate ground state or the obstruction of simple ordered phases. Similarly, nontrivial higher central charges in 2D CFTs can be viewed as a form of "topological frustration." The CFT would "prefer" to have a conformal boundary condition (from the perspective of vanishing chiral central charge), but the higher central charges act as constraints, obstructing the realization of such a boundary.
2. Exotic Order and Topological Phases: Frustrated systems in condensed matter physics are renowned for hosting exotic phases of matter, such as spin liquids, which exhibit topological order and long-range entanglement. The "topological frustration" induced by higher central charges might similarly give rise to novel phases or behaviors in CFTs that are not fully understood.
3. Emergent Gauge Fields and Duality: A common theme in frustrated systems is the emergence of gauge fields and duality transformations that provide alternative descriptions of the underlying physics. Exploring dualities in the context of CFTs with nontrivial higher central charges could offer new insights into their properties and potential connections to topological phases.
4. Numerical Methods and Insights: Condensed matter physics has a rich toolbox of numerical methods, such as Monte Carlo simulations and density matrix renormalization group (DMRG) techniques, for studying frustrated systems. Adapting and applying these methods to CFTs with higher central charge constraints could provide valuable numerical insights into their behavior.