Relation between cohomology and the BRST operator

Ghostly Lie algebra cohomology

Let $mathfrak{g}$ be our Lie algebra and $V_rho$ a representation space with representation map $rho : mathfrak{g} to mathrm{End}(V_rho)$. $V_rho$ is, by the action through the representation, naturally a $mathfrak{g}$-module (people missing the ring structure in $mathfrak{g}$ - just embed it into the universal enveloping algebra). We define the associated Chevally-Eilenberg complex as the complex of $V_rho$-valued differential forms on $mathfrak{g}$:

$$ dots overset{mathrm{d}}{to} Lambda^{p-1}mathfrak{g}^* otimes V_rho overset{mathrm{d}}{to} Lambda^{p}mathfrak{g}^* otimes V_rhooverset{mathrm{d}}{to} Lambda^{p+1}mathfrak{g}^* otimes V_rhooverset{mathrm{d}}{to}$$

whose cohomology we call the Lie algebra cohomology of $mathfrak{g}$ with coefficients in $V_rho$. Now, the algebraist is disturbed: There's an ugly differential in our complex, spoiling the fun! Let's build an operator expression for it:

Recall that, on $Lambda^p mathfrak{g}^*$, we have two natural operations:

Contraction, which is

$$iota : Lambda^1 mathfrak{g}^* times mathfrak{g} to Lambda^0 mathfrak{g}^*, (omega,G) mapsto omega(G)$$

extended to all $Lambda^pmathfrak{g}^ast$ by setting $iota(omega wedge xi,G) = (omegawedgexi)(G) wedge (-1)^{mathrm{deg}(omega)}omegawedge(xi(G))$ and the wedge product, which is

$$wedge : Lambda^p mathfrak{g}^* times mathfrak{g}^* to Lambda^{p+1} mathfrak{g}^*, (omega,k)mapsto kwedgeomega $$

and these define two operators $iota_G = iota(-,G)$ and $wedge_k = k wedge -$ acting upon $p$-forms.

Now, choose any canonically dual basis of $mathfrak{g}$ resp. $mathfrak{g}^*$, let's call them $T_a$ resp. $S^a$, and write

$$ mathrm{d} = wedge_{S^a}rho(T_a) - frac{1}{2}wedge_{S^a}wedge_{S^b}iota_{[T_a,T_b]}$$

Using it on the basis elements of $Lambda^{p}mathfrak{g}^* otimes V_rho$, we can show by direct computation that this is indeed the differential from the Chevalley-Eilenberg complex, and thus an operator expression for the differential. Defining $c^a := wedge_{S^a}$ as the ghost and $b_a := iota_{T_a}$ as the anti-ghost yields that the Chevalley-Eilenberg differential is indeed the BRST operator

$$ Q = mathrm{d} = c^arho(T_a) - frac{1}{2}f^c_{ab} c^a c^b b_c $$

What does $Q$ compute in physics?

Classically, we apply this approach to symplectic manifolds/phase spaces $mathcal{M}$ that possess a (symplectomorphic) group action by a Lie group $G$, and we construct the equivariant moment map

$$ mu : mathcal{M} to mathfrak{g}^*$$

defined by being equivariant under the coadjoint action of $G$ on $mathfrak{g}^*$ and fulfilling $mathrm{d}(mu(dot{})(g)) = omega(rho(g),dot{})$ with $omega$ as the symplectic form. If the action of $G$ represents a gauge symmetry, we would like to obtain the coisotropic reduction $tilde{mathcal{M}} := mathcal{M}/ G$ containing no redundancies. Define the submanifold $M_0 := mu^{-1}(0)$ and observe that the poisson algebra of functions on $tilde{mathcal{M}}$ fulfills

$$ C^infty(tilde{mathcal{M}}) = H^0(mathfrak{g};C^infty(M_0))$$

since the zeroth cohomology of a Lie algebra with coefficients in a module consists of precisely the elements of the module that are invariant under the group action and because the natural projection $pi : mathcal{M}_0 to tilde{mathcal{M}}$ provides a pullback from functions on the reduction to $mathcal{M}_0$. We don't want to rehash the derivation of the Koszul complex here, suffice it to say that $H^0(mathfrak{g};C^infty(M_0))$ can be computed by looking at the complex

$$ dots to Lambda^2 mathfrak{g} otimes C^infty(mathcal{M}) to Lambda mathfrak{g} otimes C^infty(mathcal{M}) to C^infty(mathcal{M}) to 0$$

and computing $H^0 = C^infty(mathcal{M}_0)$ and $H^p = 0$ otherwise, leading to the projective resolution

$$ dots to Lambda^2 mathfrak{g} otimes C^infty(mathcal{M}) to Lambda mathfrak{g} otimes C^infty(mathcal{M}) to C^infty(mathcal{M}) to C^infty(mathcal{M}_0) to 0 $$

yielding, since the tensor product is left exact, a projective resolution for $Lambda^p mathfrak{g}^* otimes C^infty(mathcal{M}_0)$

This yields a bicomplex $C^{p,q} := Lambda^p mathfrak{g}^* otimes Lambda^q mathfrak{g} otimes C^infty(mathcal{M})$, from which a usual graded complex $mathcal{C}^p$ may be constructed by $mathcal{C}^p := bigoplus_{r + s = p}C^{r,s}$, which is the infamous BRST complex, and which can be written as $mathcal{C}^p = Lambda^p(mathfrak{g} oplus mathfrak{g}^*) otimes C^infty(mathcal{M})$

With some algebraic magic involving the Poisson superalgebra structure of this complex, one can retrace the steps for deriving an explicit from for the differential from the ghostly cohomology for Lie algebras, and obtain that, here,

$$ mathrm{d} = {Q,dot{}}$$

with $Q in mathcal{C}^1$ being the classical BRST operator, and this time the ghosts and antighosts are the images of the generators of $mathfrak{g}$ and $mathfrak{g}^*$ under the natural embedding of these into $Lambda(mathfrak{g} oplus mathfrak{g}^*$).

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A lengthier, but still quick and very readable discussion of this can be found in Josê Figueroa-O'Farrill's lecture notes on "BRST Cohomology".