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

Clay Cordova, Daniel S. Freed, Ho Tat Lam, Nathan Seiberg

TL;DR

The paper extends the concept of ’t Hooft anomalies to the space of coupling constants and spacetime-dependent backgrounds, showing that generalized anomalies constrain phase structures and defect dynamics across dimensions. It develops a unifying anomaly inflow framework using invertible field theories in one higher dimension and differential cohomology, linking parameter-space phenomena to interfaces and RG flows. Through pedagogical examples—from a particle on a circle to various fermionic and gauge theories in 2d, 3d, and 4d—it derives concrete anomaly data and dynamical consequences, including level crossings, gapless edge modes, and nontrivial IR behavior. The work provides a robust mathematical toolkit (differential cohomology, Quillen superconnections, and invertible theories) and lays groundwork for applying these ideas to four-dimensional gauge theories in a companion paper.

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 Poincaré symmetry) to background gauge fields (and a metric for the Poincaré 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.

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

TL;DR

The paper extends the concept of ’t Hooft anomalies to the space of coupling constants and spacetime-dependent backgrounds, showing that generalized anomalies constrain phase structures and defect dynamics across dimensions. It develops a unifying anomaly inflow framework using invertible field theories in one higher dimension and differential cohomology, linking parameter-space phenomena to interfaces and RG flows. Through pedagogical examples—from a particle on a circle to various fermionic and gauge theories in 2d, 3d, and 4d—it derives concrete anomaly data and dynamical consequences, including level crossings, gapless edge modes, and nontrivial IR behavior. The work provides a robust mathematical toolkit (differential cohomology, Quillen superconnections, and invertible theories) and lays groundwork for applying these ideas to four-dimensional gauge theories in a companion paper.

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 Poincaré symmetry) to background gauge fields (and a metric for the Poincaré 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.

Paper Structure

This paper contains 42 sections, 171 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction and Summary
  2. Anomalies and Symmetries
  3. Anomalies in Parameter Space: Defects
  4. Anomalies in Parameter Space: Families of QFTs
  5. Synthesis via Anomaly Inflow
  6. An Intuitive Interpretation in Terms of $-1$-Form Symmetries
  7. Another Synthesis
  8. Examples and Summary
  9. A Particle on a Circle
  10. Spacetime Dependent Coupling $\theta$
  11. Coupling to Background $U(1)$ Gauge Fields
  12. The Anomaly
  13. Dynamical Consequences: Level Crossing
  14. Massive Fermions
  15. Fermion Quantum Mechanics
  16. ...and 27 more sections

Figures (1)

  • Figure 1: Interfaces defined by spatially varying coupling constant $\theta(x)$. In (a), the variation is smooth and the resulting interface dynamics are universal. In (b), the variation is abrupt. The resulting worldvolume dynamics is not universal and can be modified by coupling to degrees of freedom on the interface (schematically denoted $\mathcal{O}(x)$ above. As we will discuss, for certain special choices of $\mathcal{O}(x)$, an abrupt interface can be made completely transparent.

Theorems & Definitions (37)

  • Remark 1.11
  • Remark 1.12
  • Remark 1.13
  • Remark 1.14
  • Example 1.15
  • Remark 1.20
  • Remark 1.21
  • Remark 1.23
  • Example 1.24
  • Remark 1.25
  • ...and 27 more