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.
Highlights
Introduction to Differential Cohomology5.1 Hermitian Line Bundles with Covariant Derivative 5.2 Trivializations 5.3 Multiplication 5.4 Integration 5.5 Generalized Cohomology; Tate Twists
’t Hooft anomalies lead to powerful constraints on the dynamics and phases of quantum field theory (QFT)
In the presence of general background fields A for this global symmetry group the instanton number of the dynamical Up1q gauge group can fractionalize
Summary
’t Hooft anomalies lead to powerful constraints on the dynamics and phases of quantum field theory (QFT). They control the properties of boundaries, extended excitations like strings and domain walls, and various defects. ’t Hooft anomalies do not signal an inconsistency of the theory Instead, they show that some contact terms cannot satisfy the Ward identities of global symmetries. They show that some contact terms cannot satisfy the Ward identities of global symmetries They are an obstruction to coupling the system to classical background gauge fields for these symmetries. The generalized ’t Hooft anomalies are an obstruction to making the coupling constants and the various gauge fields spacetime dependent. As with the ordinary ’t Hooft anomalies, we use these generalized anomalies to constrain the phase diagram of the theory as a function of its parameters and to learn about defects constructed by position-dependent coupling constants
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