תובנה - Scientific Computing - # Anomaly Cancellation in Quantum Field Theory

Cancelling Mod-2 Anomalies in 8d Sp(n) Gauge Theory Using the Green-Schwarz Mechanism with Bµν: A Detailed Analysis

Q: Could there be alternative mechanisms, beyond introducing topological degrees of freedom or modifying the Bianchi identity, that might effectively cancel the Witten anomaly in four-dimensional SU(2) and Sp(n) gauge theories?

While the context focuses on canceling anomalies by introducing topological degrees of freedom or modifying the Bianchi identity, alternative mechanisms for anomaly cancellation might exist, particularly for the Witten anomaly in four-dimensional SU(2) and Sp(n) gauge theories. Here are some speculative possibilities: Non-Lagrangian Theories: The Witten anomaly poses a challenge within the framework of conventional Lagrangian quantum field theories. It's conceivable that non-Lagrangian theories, which might not have a well-defined Lagrangian description, could evade or resolve the Witten anomaly in ways not fully understood within the standard framework. Higher-Form Symmetries: Recent developments in understanding higher-form symmetries and their interplay with anomalies might offer new avenues for anomaly cancellation. It's possible that gauging or activating specific higher-form symmetries could lead to a cancellation of the Witten anomaly. Duality Webs: Exploring duality webs connecting different quantum field theories might reveal dual descriptions where the Witten anomaly manifests differently or is absent altogether. For instance, a theory with a Witten anomaly might be dual to a theory with a different type of anomaly that can be canceled by known mechanisms. Boundary Conditions and Domain Walls: In condensed matter physics, anomalies often manifest at the boundaries of systems. It's conceivable that carefully chosen boundary conditions or the presence of domain walls with appropriate properties could lead to an effective cancellation of the Witten anomaly within a larger system. Challenges and Open Questions: Concrete Realizations: Most of these alternative mechanisms are speculative and lack concrete realizations in four-dimensional SU(2) and Sp(n) gauge theories. Theoretical Framework: A comprehensive theoretical framework incorporating these alternative mechanisms into the study of anomalies is still under development.

מושגי ליבה

While the notorious Witten anomaly of 4d SU(2) and Sp(n) gauge theories remains uncanceled by introducing an antisymmetric tensor field H3, the mod-2 anomalies in 8d N=1 Sp(n) gauge theory can be completely canceled by employing a specific configuration of the H3 field and leveraging the Green-Schwarz mechanism.

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arxiv.org

Saito, S., & Tachikawa, Y. (2024). Cancelling mod-2 anomalies by Green-Schwarz mechanism with Bµν. arXiv:2411.09223v1 [hep-th].

This research paper investigates the potential of the Green-Schwarz mechanism, involving an antisymmetric tensor field Bµν, to cancel mod-2 anomalies in specific gauge theories, focusing on SU(2) and Sp(n) gauge theories in four and eight dimensions.

תובנות מפתח מזוקקות מ:

Cancelling mod-2 anomalies by Green-Schwarz mechanism with $B_{\mu\nu}$

by Shota Saito,... ב- arxiv.org 11-15-2024

https://arxiv.org/pdf/2411.09223.pdf

$Cancelling mod-2 anomalies by Green-Schwarz mechanism with $B_{\mu\nu}$$

שאלות מעמיקות

Can this approach be generalized to investigate anomaly cancellation in other types of quantum field theories beyond gauge theories, such as those with different symmetry groups or supersymmetry?

Yes, the approach outlined in the context can be generalized to investigate anomaly cancellation in quantum field theories beyond gauge theories. The core principle relies on the mathematical framework of bordism groups, which can be adapted to accommodate various symmetries and spacetime structures.
Generalization to Different Symmetry Groups:

Beyond Lie Groups: The approach is not limited to Lie groups like SU(n) or Sp(n). It can be applied to theories with discrete symmetry groups or even more exotic symmetries. The key is to identify the classifying space associated with the symmetry group in question.
Global Symmetries: The method can be extended to study anomalies associated with global symmetries. In such cases, one would consider the bordism groups of the classifying space of the global symmetry group.
Incorporating Supersymmetry:

Supergravity Theories: The context already touches upon the case of 8d N=1 supergravity. The same principles can be applied to other supergravity theories in various dimensions. The relevant bordism groups would involve structures like Spin^c or String^c structures, depending on the specific supergravity theory.
Supersymmetric Gauge Theories:  Anomalies in supersymmetric gauge theories can be analyzed by considering the bordism groups of the classifying space of the gauge group, but now equipped with additional structures dictated by supersymmetry. For instance, in theories with extended supersymmetry, one might need to consider generalized cohomology theories like K-theory or elliptic cohomology.
Key Points for Generalization:

Identifying the Classifying Space: The first step is to determine the classifying space associated with the symmetries of the theory.
Choosing the Appropriate Bordism Theory: Depending on the spacetime structure and symmetries, one needs to select the appropriate bordism theory, such as Spin, String, Pin, or their twisted versions.
Computing the Bordism Groups: Calculating the relevant bordism groups can be challenging but provides crucial information about potential anomalies.

Could there be alternative mechanisms, beyond introducing topological degrees of freedom or modifying the Bianchi identity, that might effectively cancel the Witten anomaly in four-dimensional SU(2) and Sp(n) gauge theories?

While the context focuses on canceling anomalies by introducing topological degrees of freedom or modifying the Bianchi identity, alternative mechanisms for anomaly cancellation might exist, particularly for the Witten anomaly in four-dimensional SU(2) and Sp(n) gauge theories. Here are some speculative possibilities:

Non-Lagrangian Theories: The Witten anomaly poses a challenge within the framework of conventional Lagrangian quantum field theories. It's conceivable that non-Lagrangian theories, which might not have a well-defined Lagrangian description, could evade or resolve the Witten anomaly in ways not fully understood within the standard framework.
Higher-Form Symmetries: Recent developments in understanding higher-form symmetries and their interplay with anomalies might offer new avenues for anomaly cancellation. It's possible that gauging or activating specific higher-form symmetries could lead to a cancellation of the Witten anomaly.
Duality Webs:  Exploring duality webs connecting different quantum field theories might reveal dual descriptions where the Witten anomaly manifests differently or is absent altogether. For instance, a theory with a Witten anomaly might be dual to a theory with a different type of anomaly that can be canceled by known mechanisms.
Boundary Conditions and Domain Walls: In condensed matter physics, anomalies often manifest at the boundaries of systems. It's conceivable that carefully chosen boundary conditions or the presence of domain walls with appropriate properties could lead to an effective cancellation of the Witten anomaly within a larger system.
Challenges and Open Questions:

Concrete Realizations:  Most of these alternative mechanisms are speculative and lack concrete realizations in four-dimensional SU(2) and Sp(n) gauge theories.
Theoretical Framework: A comprehensive theoretical framework incorporating these alternative mechanisms into the study of anomalies is still under development.

What are the implications of the successful anomaly cancellation in the 8d Sp(8) theory for the phenomenological viability of string theory as a unified theory of fundamental forces?

The successful cancellation of anomalies in the 8d Sp(8) theory, arising from the heterotic string compactification on T² without vector structure, has significant implications for the phenomenological viability of string theory:

Consistency of String Theory: Anomaly cancellation is a fundamental requirement for the consistency of any quantum field theory, including string theory. The fact that this particular 8d theory, which emerges naturally from string theory, exhibits anomaly cancellation provides strong evidence for the internal consistency of string theory itself.
Existence of Realistic Vacua: String theory, in its critical dimension (10 dimensions), is believed to have a vast landscape of possible vacuum solutions. The successful anomaly cancellation in the 8d Sp(8) theory suggests the existence of at least one consistent vacuum within this landscape. While this specific vacuum might not directly correspond to our observed universe, it provides hope that other realistic vacua with desirable phenomenological features might exist.
Constraints on Model Building: Anomaly cancellation imposes stringent constraints on the allowed matter content and interactions in string theory models. The specific mechanism of anomaly cancellation in the 8d Sp(8) theory, involving a subtle interplay between gravitational and gauge anomalies, provides valuable insights and constraints for constructing more realistic string theory models that aim to reproduce the observed particle physics and cosmology.
Connection to Lower Dimensions: The 8d Sp(8) theory can be further compactified to lower dimensions, potentially leading to four-dimensional theories that resemble the Standard Model of particle physics. The anomaly cancellation in the 8d theory serves as a starting point for ensuring the consistency of these lower-dimensional theories.
Further Research and Open Questions:

Exploring the Landscape:  While the 8d Sp(8) theory provides one example of a consistent vacuum, it's crucial to explore the vast string theory landscape to search for other vacua that better match our observed universe.
Connecting to Phenomenology:  Bridging the gap between the abstract mathematical structures of string theory and the observed phenomenology of particle physics and cosmology remains a significant challenge.
Understanding the Role of Supersymmetry: The 8d Sp(8) theory is supersymmetric. Investigating the role of supersymmetry in anomaly cancellation and its implications for phenomenology is an active area of research.