Anomalies in the Space of Coupling Constants and Their Dynamical Applications I (1905.09315v3)

Abstract: It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincar\'{e} symmetry) to background gauge fields (and a metric for the Poincar\'{e} symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects 't Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of 't Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary 't Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized 't Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to this perspective. Also, we use Quillen's superconnections to derive the anomaly for a free spinor field with variable mass. In a companion paper we will study four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.

• The paper presents a generalized framework where 't Hooft anomalies extend to coupling constants, constraining the dynamics of quantum field theories.
• It utilizes differential cohomology and higher-dimensional anomaly inflow to explain phase transitions and defect dynamics in various models.
• Concrete examples in gauge theories and quantum mechanics illustrate how anomaly-induced constraints impact vacuum structures and parameter evolution.

Anomalies in the Space of Coupling Constants and Their Dynamical Applications I

The paper, authored by Clay Córdova, Daniel S. Freed, Ho Tat Lam, and Nathan Seiberg, presents an exploration of 't Hooft anomalies generalized to the space of coupling constants within quantum field theories (QFTs). The investigation reveals novel insights into the role that parameter-dependent anomalies play in dictating the dynamics and phases of these theories. Building from commonly understood 't Hooft anomalies, which constrain symmetries by coupling terms to background gauge fields and analyzing path integrals for gauge invariance, this work extends the framework to treat coupling constants as spacetime-dependent background fields, revealing a richer tapestry of physical phenomena.

Overview of Key Concepts and Results

• Generalized 't Hooft Anomalies: Traditional 't Hooft anomalies arise in theories where the partition function isn't invariant under gauge transformations due to the introduction of background fields for global symmetries. The paper extends these anomalies to include dynamical coupling constants, interpreting them through generalized differential cohomology.
• Anomaly Inflow and Higher Dimensions: By introducing higher-dimensional manifolds, the anomalies in the space of parameters can manifest as boundary effects along lower dimensions. The 'inflow' from these anomalies plays a critical role in maintaining the consistency of the theory across varying parameter spaces.
• Application to Various Theoretical Constructs: The authors provide concrete examples using quantum mechanical systems like a particle on a circle, fermions across different dimensions, and 2D gauge theories. These examples concretize how anomalies lead to observable physical consequences such as phase transitions, defect dynamics, and more.
• Theoretical Constructs Using Differential Cohomology: The paper utilizes the framework of differential cohomology to rigorously define the interaction of background fields and parameter spaces, offering a mathematical synthesis of anomalies as geometric and topological phenomena.

Implications and Future Directions

1. Constraint on Parameter Evolution: The work articulates the unlawful nature of continuous parameter space evolutions that disregard anomaly-induced constraints. In practical terms, theories cannot be trivially gapped across all parameter limits due to anomalies, indicating inevitable phase transitions or disparate vacuum structures at certain critical parameter values.
2. Defect Dynamics and Interface Anomalies: The insights into spacetime-varying parameters unveil how boundary states must align with generalized anomalies to preserve consistency. This finding provides critical insights into interface and defect theories within higher-dimensional QFTs.
3. Extension Beyond Traditional Symmetries: Interestingly, the framework provides a potential extension towards new symmetry concepts, potentially recasting what have traditionally been seen as isolated or minute parameter changes into broader, symmetry-related phenomena.
4. Future Refinements and Applications: While already rich in insight, the framework established invites further hypothesis testing and theoretical refinement, especially concerning parameter spaces in more complex gauge theories and their physical ramifications in both theoretical and applied contexts.

In conclusion, the paper illustrates a profound connection between anomalous behavior in parameter spaces and the rich tapestry of quantum field dynamic behavior. The rigorous mathematical underpinning utilising differential cohomology provides a robust framework for future explorations, including potential applications in both condensed matter and high-energy theoretical physics domains. The expansion of 't Hooft anomaly concepts to parameter spaces not only enriches the theoretical landscape but also suggests new questions and lines of inquiry to pursue in understanding the fundamental nature of field theories.