Background Fields and Gauging
A background gauge field probes how a theory responds to symmetry data; making that field dynamical constructs a new theory. This chapter develops sources, bundles, spurions, and counterterms before explaining what a consistent gauging operation must specify.
Enter this chapter
Section titled “Enter this chapter”Begin with Current Sources and Generating Functionals to generate symmetry responses by differentiation.
Helpful background. The Generating Functional and Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supply source differentiation and geometric connections.
Topics
Section titled “Topics”- Current Sources and Generating Functionals — Organizes current insertions and connected response.
- Coupling to Background Gauge Fields and Bundles — Uses connections and bundles as nondynamical global probes.
- Spurions, Local Counterterms, and Symmetry Response — Distinguishes covariance of couplings from symmetry at fixed couplings.
- Background Fields versus Dynamical Gauging — Separates evaluating a theory on a background from summing over backgrounds.
- Gauging Continuous and Finite Symmetries — Specifies dynamical integrals or finite sums and their consistency conditions.
- Residual, Quotient, and Emergent Dual Symmetries — Determines which transformations survive gauging and which new symmetries can emerge.
How the ideas fit together
Section titled “How the ideas fit together”A background partition function retains its dependence on a chosen connection and bundle. Gauging replaces that choice by a weighted sum or integral modulo gauge equivalence, so anomaly freedom, allowed sectors, and counterterms become part of defining the result. Finite gauging already exhibits this distinction; see Bhardwaj and Tachikawa 2018, § 2.1.
The gauging diagram places the consistency checks before the sum. Assigning a transformation to a fixed coupling, called spurionic covariance, instead compares members of a family; it need not supply a conserved charge in any one member.
Review the chapter
Section titled “Review the chapter”Why is replacing an ordinary derivative by a covariant derivative insufficient to specify a gauged quantum theory?
Solution
It defines a local coupling but leaves unanswered which bundles are summed over, what action and measure weight them, which transformations are quotiented, and whether anomalies obstruct the construction. These choices can distinguish theories with the same local covariant derivative.
Where to go next
Section titled “Where to go next”Continue to Gauge Structure, Global Form, and Observables to identify the global data and physical probes of the new theory.