Local BRST Cohomology, Consistent Deformations, and Currents

Local BRST cohomology classifies local densities only after quotienting both BRST-exact changes and total derivatives. In spacetime dimension $n$, the relative group $H^{g,n}(s\mid d)$ contains first-order interactions at ghost number zero, anomaly candidates at ghost number one, and conserved-current information at negative ghost number, subject to locality, regularity, and the chosen field complex. The method identifies candidates and obstructions; it does not guarantee convergence, positivity, or a global gauge fixing.

Required background. The Koszul–Tate resolution and BRST bicomplex supply the antifield filtration, while BRST cohomology as derived invariants distinguishes representatives from classes.

Helpful background. Wess–Zumino consistency and descent anticipates the ghost-number-one sector, and quantum currents and improvements explain why total derivatives and trivial currents must be removed.

The relative complex of local forms

Let $\Omega^{p,g}_{\mathrm{loc}}$ be local horizontal $p$-forms of ghost number $g$, built from fields, ghosts, antifields, and finitely many jets. The BRST differential $s$ and horizontal exterior derivative $d=dx^\mu\partial_\mu$ obey

$$s^2=d^2=0, \qquad sd+ds=0.$$

An $n$-form $a^{g,n}$ is a relative cocycle when

$$sa^{g,n}+da^{g+1,n-1}=0.$$

Two cocycles define the same class if $a^{g,n}\sim a^{g,n}+sm^{g-1,n}+dn^{g,n-1}$. After integration over a boundaryless spacetime, or with support and boundary conditions that kill the surface term, the $d$-exact change does not affect the functional. Without that support or boundary hypothesis, the quotient is not licensed.

Filtering $s=\delta+\gamma+\cdots$ by antifield number gives a practical algorithm. First use Koszul–Tate acyclicity to eliminate positive-antifield cycles that are not tied to genuine Noether data. Next compute longitudinal cohomology in gauge-covariant variables. Finally solve the descent equations upward or downward, checking at every step whether a lift is obstructed. The general descent mechanism and its hypotheses are developed in Barnich, Brandt, and Henneaux 2000, §§9.1–9.6, pp. 70–78.

The grading gives the interpretation only in a specified complex. For local $n$-forms, $H^{0,n}(s\mid d)$ contains consistent first-order deformations and invariant counterterms; $H^{1,n}(s\mid d)$ contains consistent local anomaly candidates; and $H^{-1,n}(s\mid d)$ is related to nontrivial global symmetries and conserved currents. A class can disappear when nonlocal counterterms are allowed, but permitting them changes the problem rather than proving the local anomaly trivial.

Yang–Mills as a deformation of free vectors

Take $N$ free Abelian potentials $A^a_\mu$ with ghosts $c^a$. Write the master action as a formal deformation

$$S=S_0+gS_1+g^2S_2+\cdots .$$

The classical master equation gives, order by order,

$$(S_0,S_0)=0, \qquad s_0S_1=0, \qquad \frac12(S_1,S_1)+s_0S_2=0,$$

where $s_0=(S_0,\,\cdot\,)$. A representative of the non-Abelian first-order class is, up to correlated sign and normalization conventions,

$$S_1=\int d^4x\left[ -\frac12 f_{abc}F^{a\mu\nu}A^b_\mu A^c_\nu +A^{*\mu}_a f^a{}_{bc}A^b_\mu c^c -\frac12c^*_a f^a{}_{bc}c^b c^c \right].$$

The antifield-independent term is the cubic vertex; the term linear in $A^*$ deforms the gauge transformation; the $c^*$ term deforms the gauge algebra. The first-order equation requires the appropriate antisymmetries of $f_{abc}$. At second order, $(S_1,S_1)/2$ is $s_0$-exact only when

$$f^a{}_{e[b}f^e{}_{cd]}=0,$$

the Jacobi identity. Barnich, Brandt, and Henneaux derive the uniqueness of this algebra-deforming cubic vertex for free vectors and locate the second-order obstruction in local BRST cohomology Barnich, Brandt, and Henneaux 2000, §13.3, pp. 127–129. This is the precise first application handed to the physical BV master-equation treatment.

An independent check is to assign $(\operatorname{gh},\text{form degree})=(0,4)$ to every integrand term: $\operatorname{gh}A^*= -1$, $\operatorname{gh}c=1$, and $\operatorname{gh}c^*=-2$. A second check sets $f=0$, recovering the free theory. Neither check replaces the Jacobi obstruction calculation.

Equivalently, the physical master-equation construction receives not merely a cubic vertex but the correlated deformation of transformations and algebra. Omitting either antifield term can make the antifield-independent vertex look acceptable while destroying the master equation.

Trivial classes and failed extensions

If $S_1=s_0B$ modulo a total derivative, the corresponding interaction is generated by a local field or canonical redefinition and is not a new coupling in this classification. Conversely, finding a nontrivial first-order class is not enough: failure of the second-order equation means that no local $S_2$ completes that proposed deformation with the same field content. Choosing antisymmetric constants that violate Jacobi is the adversarial example. The cubic vertex still solves the linearized problem, but the nonlinear gauge theory does not exist as the claimed deformation.

The analysis is local in jet space. Boundary charges, Wilson operators, bundle topology, and global anomalies need different complexes. Power counting may further restrict allowed representatives, but the cohomological calculation itself does not establish ultraviolet renormalizability.

As an independent consistency check, integrate the cocycle on a compactly supported test configuration. Replacing $a$ by $a+sm+dn$ changes the functional by an $s$-exact term because the integral of $dn$ vanishes. If support reaches a boundary, the same calculation produces a boundary integral and the claimed equivalence fails unless boundary data are added.

Exercises

Check the ghost number of the three terms in $S_1$.

Solution

The cubic field term has ghost number zero. The $A^*Ac$ term has $-1+0+1=0$, and the $c^*cc$ term has $-2+1+1=0$. Thus the deformed master action retains ghost number zero.

Why is stopping at $s_0S_1=0$ insufficient?

Solution

That equation tests only the coefficient of $g$. At order $g^2$, the class of $(S_1,S_1)/2$ must vanish in $H^{1,4}(s_0\mid d)$. For the vector deformation its nontrivial part is proportional to the Jacobiator, so arbitrary antisymmetric constants need not extend.

References

• Barnich, Glenn, Friedemann Brandt, and Marc Henneaux. “Local BRST Cohomology in Gauge Theories.” Physics Reports 338 (2000): 439–569. DOI; Open PDF.
• Barnich, Glenn, and Marc Henneaux. “Consistent Couplings between Fields with a Gauge Freedom and Deformations of the Master Equation.” Physics Letters B 311 (1993): 123–129. DOI; Open PDF.