BRST–BV Quantization and Anomalies

BRST–BV quantization replaces a singular gauge quotient with a graded complex. The chapter follows its classical resolution, quantum master equation, and anomaly obstructions while separating formal consistency from existence of a physical quantum theory.

Enter this chapter

Begin with Koszul–Tate resolution and the BRST bicomplex; exactness assumptions matter before cohomology can be interpreted as observables. Helpful background. The BRST differential and gauge-fixed complex provide the field-theory notation. The BV master equation and gauge fixing provide the odd-symplectic construction. What is an anomaly? distinguishes a quantum obstruction from ordinary explicit symmetry breaking.

Topics

1. Koszul–Tate Resolution and the BRST Bicomplex — Resolves equations of motion and gauge identities under regularity assumptions.
2. BRST Cohomology as Derived Invariants — Interprets invariant information through a cochain complex.
3. Local BRST Cohomology, Consistent Deformations, and Currents — Classifies local currents, deformations, and possible obstructions.
4. The BV Complex and Classical Master Equation — Encodes gauge structure in the classical BV action.
5. Gauge-Fixing Fermions, Canonical Transformations, and Observables — Explains when gauge-fixing changes preserve the relevant observables.
6. Quantum Master Equation and Anomaly Obstructions — Locates the order-by-order obstruction to the quantum identity.
7. Local Anomaly Descent and Wess–Zumino Consistency — Derives consistency conditions for local anomaly representatives.
8. Global Anomalies, Determinant Lines, and Eta Invariants — Detects global phases invisible to local descent.
9. Anomaly Cancellation and Positive-Hilbert-Space Existence — Separates anomaly cancellation from positivity and nonperturbative existence.
10. Perturbative Gauge QFT: Constructions and Scope — States the scope of formal perturbative gauge constructions.
11. Chiral Gauge Perturbative Construction: Open Obligations — Identifies the additional obligations in chiral gauge constructions.

How the ideas fit together

Resolution makes classical cohomology meaningful; the master equation organizes the higher gauge structure Barnich, Brandt, and Henneaux 2000, §§ 5–7 and 9–11 (PDF); Gomis, París, and Samuel 1995, §§ 4–6 (PDF). At quantum order, a nontrivial local obstruction blocks an allowed counterterm. Global anomaly phases require another test, expressed through determinant lines and reduced eta invariants Dai and Freed 1994, Theorems 1.9 and 5.3 (PDF). Cancellation of both does not supply convergence or a positive continuum representation.

Hypotheses and licensed conclusions

Domains, hypotheses, licensed results, excluded upgrades, and failure tests for BRST–BV quantization

Object and domain	Required hypotheses	Licensed result	Excluded converse or upgrade	Adversarial check
Stationary ideal and reducible gauge generators	Regular equations; complete Noether identities and reducibility tower; Koszul–Tate acyclicity in positive antifield number	A resolution of on-shell functions and a controlled BRST bicomplex	Nilpotence alone does not prove exactness or identify the intended quotient	Introduce a dependent equation or missed reducibility relation and exhibit surviving positive-antifield-number homology
Proper classical BV action on the odd cotangent space	Nondegenerate odd symplectic pairing; complete gauge generators; properness; $(S,S)=0$ with consistent degree and sign conventions	Nilpotent BV differential and a derived critical-locus model	A classical master action supplies neither a measure nor a quantum theory	Drop an antifield term encoding a field-dependent bracket and calculate $s^2$
Gauge-fixed BV integral or perturbative functional	Regular gauge fermion; Lagrangian slice avoiding singular strata; compatible measure or renormalization; vanishing boundary terms	Equality of BRST-closed observables under admissible canonical changes	Off-shell Green functions need not agree, and no global Gribov-free slice follows	Move the slice through a Faddeev–Popov zero mode or a boundary contribution
Renormalized quantum master identity	Local causal renormalization; consistency of the breaking; trivial obstruction in the relevant local BRST cohomology; admissible finite counterterms	Order-by-order restoration of the master identity and formal gauge-invariant observables	Formal solvability does not imply convergence, a global adiabatic limit, or positivity	Choose a nontrivial ghost-number-one cocycle and try to remove it with an exact counterterm
Local chiral anomaly in an even-dimensional theory	Specified representation, chirality, bundle, trace normalization, local functional class, and descent representative	Wess–Zumino-consistent anomaly class and its consistent/covariant-current relation	Vanishing de Rham anomaly polynomial does not exclude torsion or large-transformation anomalies	Use one $SU(2)$ Weyl doublet: the cubic local test vanishes while the Pfaffian changes sign
Determinant or Pfaffian line over background-field space	Specified spin structure and representation; family of Fredholm operators; control through zero modes; trivial holonomy on every relevant loop or bordism class	A globally consistent fermion phase for the tested parameter space	Triviality on one loop or component does not prove global triviality or interacting existence	Build the mapping torus of a large transformation and evaluate its eta phase or mod-two index
Candidate anomaly-free chiral gauge theory	All local and global cancellations; local regulator; all-order identities; cutoff and volume limits; locality; positive representation; nontrivial dynamics	A physical continuum theory only when every analytic and representation hypothesis is separately proved	Charge sums plus a formal BRST charge are not sufficient for a positive Hilbert space	Demand the missing uniform estimates and positivity theorem after the anomaly arithmetic succeeds

Structured table data (JSON).

Review the chapter

Can a vanishing local anomaly polynomial leave a global anomaly?

Solution

Yes. Vanishing local curvature can leave a flat determinant or Pfaffian line with nontrivial holonomy around a loop in field space. The local test does not establish the global trivialization needed for a consistent fermionic phase.

Where to go next

Continue to BV–BFV boundaries and gluing, where surface terms modify the master-equation structure.

References

• Barnich, Glenn, Friedemann Brandt, and Marc Henneaux. “Local BRST Cohomology in Gauge Theories.” Physics Reports 338 (2000): 439–569. DOI; Open PDF.
• Dai, Xianzhe, and Daniel S. Freed. “Eta-Invariants and Determinant Lines.” Journal of Mathematical Physics 35 (1994): 5155–5194. DOI; Open PDF.
• Gomis, Joaquim, Jordi París, and Stuart Samuel. “Antibracket, Antifields and Gauge-Theory Quantization.” Physics Reports 259 (1995): 1–145. DOI; Open PDF.