BRST Cohomology as Derived Invariants

BRST cohomology records gauge-invariant information only after the underlying complex has been shown to resolve the equations of motion and gauge directions. In that setting, ghost number zero represents on-shell gauge-invariant observables, other ghost numbers organize symmetries, deformations, and possible obstructions, and quasi-isomorphic resolutions give the same answer. A formal nilpotent operator on an arbitrarily enlarged field space is not enough, and none of these cohomology groups alone constructs a positive physical Hilbert space.

Required background. Elliptic gauge complexes and Gribov obstructions separate local homological reduction from global gauge fixing; BRST cohomology and physical observables supplies the physical quotient; and chains, homology, and exactness supplies quasi-isomorphisms and contracting homotopies.

Helpful background. Graded algebra and Berezin calculus fixes parity signs for ghosts.

Cohomology of a resolved gauge system

Let $(\mathcal C,s)$ be a graded-commutative cochain algebra with $s^2=0$ and $\operatorname{gh}s=1$. Its cohomology is

$$H^g(s,\mathcal C)= \frac{\ker(s:\mathcal C^g\to\mathcal C^{g+1})} {\operatorname{im}(s:\mathcal C^{g-1}\to\mathcal C^g)}.$$

A representative is not itself an observable: two representatives differing by $sK$ define the same class. For a regular irreducible gauge theory with a complete Koszul–Tate resolution, the antifield-number filtration first restricts functions to the stationary surface and then takes invariants of the gauge action. Consequently $H^0(s)$ is the algebra of gauge-invariant on-shell functions in the declared class of local or multilocal functionals. This conclusion depends on the chosen class: local jet functions, compactly supported functionals, and global observables can have different cohomology.

Auxiliary choices may be added without changing cohomology when they form contractible pairs. If $su=v$, $sv=0$, and the differential of every other generator is independent of $u,v$, define a counting operator $N$ and a homotopy $h$ satisfying $sh+hs=N$. Every closed term with positive $N$-degree is then exact. This proves, for example, that a correctly paired antighost and Nakanishi–Lautrup field do not change BRST cohomology Barnich, Brandt, and Henneaux 2000, §2.7, pp. 18–20. It does not prove invariance after adding an unpaired variable or replacing the complex by one without a quasi-isomorphism.

More generally, a multiplicative quasi-isomorphism $f:(\mathcal C,s)\to(\mathcal C',s')$ induces $H(f):H(s)\to H(s')$. Calling BRST cohomology a derived invariant means invariance under such controlled replacements, not under every gauge-fixing prescription. Analytic domains, support conditions, and topology on completed spaces must also be preserved when infinite-dimensional field spaces are used.

Free Maxwell cohomology

For free Maxwell theory on a contractible region,

$$sA_\mu=\partial_\mu c, \qquad sc=0, \qquad sF_{\mu\nu}=0.$$

Thus polynomials in $F$ and its derivatives give ghost-number-zero classes, subject to the Bianchi identity, equations of motion when the Koszul–Tate part is included, and integrations by parts for integrated local functionals. The potential $A_\mu$ is not closed, and a pure gauge deformation is exact in the resolved complex. A Wilson loop is gauge invariant but is nonlocal; it is not captured by a cohomology calculation restricted to polynomial local jets. This is the simplest warning against conflating local and global observable algebras.

At ghost number one, $cP(F,\partial F,\ldots)$ is closed in the Abelian model. Whether it is a nontrivial class depends on the coefficient space, form degree, and whether one works modulo the spacetime differential $d$. In the relative local cohomology $H^{g,4}(s\mid d)$, ghost number zero contains candidate consistent deformations and ghost number one contains candidate anomalies. For several free vector fields, the deformation that changes the gauge algebra is represented by the Yang–Mills cubic cocycle; extending it beyond first order imposes the Jacobi identity. That sharper relative calculation belongs to the next page, so the absolute group $H^1(s)$ should not be advertised as an anomaly classification by itself.

This Maxwell computation is the exact first application developed physically on the BRST cohomology page: $H^0$ retains field-strength observables, while the appropriate ghost-number-one or relative sector identifies candidate deformations and obstructions. The check $sF=0$ follows because commuting derivatives annihilate $\partial_\mu\partial_\nu c-\partial_\nu\partial_\mu c$.

Failure test and interpretation

Adjoin a variable $u$ with $su=0$ but no partner $v=su$. Then every polynomial in $u$ multiplies old cohomology classes, so the cohomology changes. Likewise, a gauge-fixed complex obtained by deleting zero modes or boundary sectors without a quasi-isomorphism can lose genuine classes. The strongest surviving claim is only that the two complexes each have a nilpotent differential; their observable cohomologies need not agree.

Even when $H^0(s)$ is correct, positivity requires an involution, a state, a null-space quotient, and completion. The quartet mechanism can establish positivity in controlled perturbative settings, but formal BRST cohomology does not imply a nonperturbative Hilbert-space construction.

Exercises

Prove that a contractible pair does not change cohomology.

Solution

Decompose a closed element $a$ into eigencomponents of $N$. For an $N$-eigencomponent $a_n$ with $n>0$, $a_n=n^{-1}(sh+hs)a_n=s(n^{-1}ha_n)$ because $sa_n=0$. Only the $N=0$ component can represent cohomology, and it is independent of the pair.

Check that $F_{\mu\nu}F^{\mu\nu}$ is BRST closed in the Abelian theory.

Solution

Since $sF_{\mu\nu}=\partial_\mu\partial_\nu c-\partial_\nu\partial_\mu c=0$, the graded Leibniz rule gives $s(F_{\mu\nu}F^{\mu\nu})=0$. It is not written as an $s$-variation within the polynomial local complex of nonnegative ghost number, so it represents a gauge-invariant class there.

References

• Barnich, Glenn, Friedemann Brandt, and Marc Henneaux. “Local BRST Cohomology in Gauge Theories.” Physics Reports 338 (2000): 439–569. DOI; Open PDF.
• Henneaux, Marc, and Claudio Teitelboim. Quantization of Gauge Systems. Princeton, NJ: Princeton University Press, 1992. Publisher.