— DRAFT — IASSNS-HEP-97/72

Lecture 7

IV. Faddeev–Popov Ghosts — BRST Quantization

Eric D’Hoker

It is standard practice to recast the Faddeev-Popov determinant, which arose from fixing Diff(Σ) gauge invariance, in terms of ghost fields. (See also Faddeev’s lectures.) One introduces Grassmann-valued ghost fields (c, c̄) = (c^{z}(dz)^{−1}, cz̄(dz̄)^{−1}) and anti-ghost fields (b, b̄) = (b_{zz}(dz)^{2}, b_{z̄z̄}(dz̄)^{2}) transforming as the gauge transformation vector fields (v, v̄) and as the metric deformations (δg_{zz}, δg_{z̄z̄}), respectively. It is convenient to study such b-c systems for general weight first, since we shall make use of such systems as well: c ∈ K^{1−n}; b ∈ K^{n}. For now n is an integer, but later n will also be allowed to be a half-integer.

A. Determinants and b-c systems

We therefore generalize by considering a system of a pair of Grassmann-valued fields (c, c̄) = (c(dz)^{1−n}, c̄(dz̄)^{1−n}) and (b, b̄) = (b(dz)^{n}, b̄(dz̄)^{n}), so that the above case is n = 2. In this notation, b and c are independent whereas b̄, c̄ are their complex conjugates. The action is first order in derivatives:

S_{n}(b, c) = \frac{1}{2π} ∫_{Σ} dμ_{g} (b∇_{(1−n)}^{z}c + b̄∇z̄ ; (n−1)c̄) .  (4.1)

We have the following Theorem:

\begin{aligned}Z_{(n)}^{−}(g)| & det ϕ_{j}(z_{k})|^{2} | det ψ_{a}(w_{b})|^{2} \\ & = ∫ D(bb̄)D(cc̄)e^{−S_{n}(b,c)} ∏_{j=1}^{N_{(n)}^{−}} b(z_{j})b̄(z_{j}) ∏_{a=1}^{N_{(n−1)}^{−}} c(w_{a})c̄(w_{a}) , \end{aligned}  (4.2)

where the measures D(bb̄) and D(cc̄) are obtained from the L^{2} norms on K^{n}, K̄^{n}, K^{(1−n)} and K̄^{(1−n)}; ϕ_{j} ∈ Ker ∇_{(n)}^{z}, ψ_{a} ∈ Ker ∇_{(1−n)}^{z} and N_{(n)}^{±} = dim Ker Δ_{(n)}^{±}. Note that for n ≥ 2 there are no ψ_{α}’s whereas for h = 0 there are no ϕ_{j}’s.

IV.1

We begin the proof by recalling a finite-dimensional Grassmann integral, over variables b_{i}, c_{i}, i = 1, …, N, such that {b_{i}, b_{j}} = {c_{i}, c_{j}} = {b_{i}, c_{j}} = 0. We use the standard Grassmann integration rules:

∫ dc_{i} c_{j} = δ_{ij}  ∫ db_{i} b_{j} = δ_{ij}  (4.3)

and ∫ dc_{i} 1 = ∫ db_{i} 1 = 0. Then, for any matrix M_{ij}, we have

∏_{i=1}^{N} ∫ db_{i}dc_{i} e^{b_{i}M_{ij}c_{j}} = det M .  (4.4)

To prove the Theorem, we begin by expanding b and c in an orthonormal basis of eigenfunctions of Δ_{(n)}^{−} and Δ_{(1−n)}^{−} as follows. Let Ψ_{α} be an orthonormal basis of eigenfunctions of Δ_{(1−n)}^{−} corresponding to the non-zero eigenvalues λ_{α}^{2}, and let ψ_{a} be an orthonormal basis of zero modes of Δ_{(1−n)}^{−}. We have

\{ \begin{matrix}Δ_{(1−n)}^{−}Ψ_{α} = λ_{α}^{2}Ψ_{α} \\ Δ_{(1−n)}^{−}ψ_{a} = 0 , \end{matrix}  (4.5)

and we define Φ_{α} by

\{ \begin{matrix}\sqrt{2} ∇_{(1−n)}^{z}Ψ_{α} = λ_{α}Φ_{α} \\ −\sqrt{2} ∇_{z}^{(1−n)}Φ_{α} = λ_{α}Ψ_{α} , \end{matrix}  (4.6)

so that Φ_{α} provides an orthonormal basis of eigenfunctions of Δ(−n)^{+} for non-zero eigenvalues. Let ϕ_{j} be an orthonormal basis for Ker ∇_{z}^{(−n)}. As a result, (g_{zz̄})^{n}Φ_{α}^{∗} together with the zero modes (g_{zz̄})^{n}ϕ_{j}^{∗} provides an orthonormal basis of eigenfunctions for Δ_{(n)}^{−}. Thus, we have the decompositions

\begin{aligned}c & = ∑_{α} c_{α}Ψ_{α} + ∑_{a} c_{a}ψ_{a} \\ b & = ∑_{α} b_{α}(g_{zz̄})^{n}Φ_{α}^{∗} + ∑_{j} b_{j}(g_{zz̄})^{n}ϕ_{j}^{∗} . \end{aligned}  (4.7)

The L^{2} norm then induces the following measures:

\{ \begin{matrix}D(bb̄) = ∏_{α} db_{α} db̄_{α} ∏_{j} d b_{j} d b̄_{j} \\ D(cc̄) = ∏_{α} dc_{α} dc̄_{α} ∏_{a} dc_{a} dc̄_{a} \end{matrix}  (4.8)

IV.2 and it is straightforward to evaluate the action:

\begin{aligned}S_{n}(b, c) & = \frac{1}{2π} ∑_{α,β} b_{α}c_{β} ∫_{Σ} dμ_{g}(g_{zz̄})^{n}Φ_{α}^{∗}∇_{(1−n)}^{z}Ψ_{β} + c.c. \\ & = \frac{1}{2π\sqrt{2}} ∑_{α,β} b_{α}c_{β}λ_{β} ∫_{Σ} dμ_{g}(g_{zz̄})^{n}Φ_{α}^{∗}Φ_{β} + c.c \\ & = \frac{1}{2π\sqrt{2}} ∑_{α}(b_{α}c_{α}λ_{α} + b̄_{α}c̄_{α}λ_{α}) . \end{aligned}  (4.9)

Notice that the zero mode coefficients b_{j}, c_{a} do not enter the action. Modulo a constant (\frac{1}{2π\sqrt{2}} )^{ζ(0)} coming from the ζ-function regularization,

∫ ∏_{α} db_{α}db̄_{α}dc_{α}dc̄_{α} e^{−S_{n}(b,c)} = ∏_{α} λ_{α}^{2} = Det′Δ_{(1−n)}^{−} .  (4.10)

The remaining finite product of factors can only involve the zero modes, since the number of factors of b and c precisely equals the number of zero-mode integrations.

\begin{aligned}∫ ∏_{a} dc_{a}dc̄_{a} & ∏_{j} db_{j}db̄_{j} ∏_{j=1}^{N_{(n)}^{−}} b(z_{j})b̄(z_{j}) ∏_{a=1}^{N_{(1−n)}^{−}} c(w_{a})c̄(w_{a}) \\ & = | det ϕ_{j}(z_{k})|^{2} | det ψ_{a}(w_{b})|^{2} . \end{aligned}  (4.11)

Recall that our bases ϕ_{j}, ψ_{a} had been chosen orthonormal. In an arbitrary basis, the result would be

\frac{| det ϕ_{j}(z_{k})|^{2}}{det(ϕ_{j}, ϕ_{k})} \frac{| det(ψ_{a}(w_{b}))|^{2}}{det(ψ_{a}, ψ_{b})} .  (4.12)

Putting all together, we obtain the result in the Theorem by using the definition of the determinant combination Z_{(n)}^{−}(g).

Notice that the functional integral without b or c insertions,

∫ D(bb̄)D(cc̄)e^{−S_{n}(b,c)} ,

vanishes because of the existence of zero modes which do not enter the action at all.

IV.3 B. Ghost representation of the Faddeev-Popov determinant

For bosonic string theory, n = 2 is the suitable weight, and we have the following general form for the amplitudes

A = ∑_{h=0}^{∞} e^{−χΦ_{0}} ∫_{ℳ_{h}} dm_{j}dm̄_{j} ∫ D(bb̄)D(cc̄)e^{−S_{2}(b,c)} ∏_{j} |(μ_{j}, b)|^{2} ∫_{Map(Σ;M)} Dx 𝒪 e^{−S_{m}(x)}

(4.13)

where

𝒪 = \{ \begin{matrix}• & ∏_{a=N−2}^{N} c(z_{a})c̄(z_{a})V_{1} … V_{N−3}W (z_{N−2})W (z_{N−1})W (z_{N}) & (N ≥ 3; h = 0) \\ • & 0 & (N = 0, 1, 2; h = 0) \\ • & c(z_{N})c̄(z_{N})V_{1} … V_{N−1}W (z_{N}) & (N ≥ 1; h = 1) \\ • & V_{1} … V_{N} , & (h ≥ 2) \end{matrix}

(4.14)

and S_{2}(b, c) is the action for the ghost fields in the case n = 2. This representation is valid for any matter action S_{m}, and any set of vertex operators V_{i}. The fields c and b are called the Faddeev-Popov ghosts and anti-ghosts.

C. Conformal Field Theory of the b-c system

From the Weyl transformation law of the determinant combinations Z_{(n)}^{−}(g), and the Weyl invariance of det ϕ_{j}(z_{k}) and det ψ_{a}(w_{b}), we know that the b-c system is a conformal field theory with central charge

−2c_{(n)}^{−} = −2(6n^{2} − 6n + 1) .

In particular, we confirm that for the ghost system of the bosonic string (i.e. n = 2) the central charge is c_{ghosts} = −26, cancelling the central charge of the matter part. We now study this conformal field theory for any n ∈ \frac{1}{2}ℤ, assuming that a spin structure has been chosen on Σ. In particular, for n = \frac{1}{2}, we have a Dirac fermion ψ̃_{+} = b, ψ_{+} = c; ψ̃_{−} = b̄, ψ_{−} = c̄ (in which case b, c are in the same space of Dirac spinors), or two Majorana spinors: (ψ_{+}, ψ̃_{+}) and (ψ_{−}, ψ̃_{−}) = (ψ_{+}, ψ̃_{+})^{∗}. (See Witten’s lectures.)

IV.4

From the classical action, we obtain the stress tensor by varying with respect to δg^{zz} and using

δ∇_{(1−n)}^{z} = \frac{1}{2}δg^{zz}∇_{z}^{(1−n)} + \frac{1 − n}{2}∇_{z}(δg^{zz}) .  (4.15)

We find for the classical stress tensor:

\begin{aligned}T^{(bc)}(z) = T_{zz}^{(bc)} & = (1 − n)∇_{z}bc − nb∇_{z}c \\ & = (1 − n)(∂_{z}b − nΓb)c − nb(∂_{z}c − (1 − n)Γc) \\ & = (1 − n)(∂_{z}b)c − nb∂_{z}c . \end{aligned}  (4.16)

Using Weyl rescaling, we may study the system on a surface with locally flat metric ĝ = 2|dz|^{2}. At the same time, covariant derivatives become just Cauchy-Riemann operators: ∇_{(n)}^{z} becomes ¯∂_{(n)} and ∇_{z}^{(n)} becomes ∂_{(n)} (¯∂ and ∂ for short). The action is then

S_{n}(b, c) = \frac{1}{2π} ∫_{Σ} d^{2} z(b¯∂c + b̄∂c̄) .  (4.17)

The field equations are ¯∂c = ¯∂b = 0, ∂c̄ = ∂b̄ = 0. The OPE of b and c is given by the Green function of ¯∂:

b(z)c(w) ∼ \frac{1}{z − w} ∼ c(z)b(w) .  (4.18)

The quantum stress tensor may now be defined by subtracting the OPE singularities from the classical stress tensor. The holomorphic stress tensor is defined by

T^{(bc)}(z) = lim_{w→z} \{(1 − n)∂b(z)c(w) − nb(z)∂c(w) + \frac{1}{(z − w)^{2}} \} .  (4.19)

One immediately verifies

\begin{aligned}T^{(bc)}(z)T^{(bc)}(w) & ∼ & −\frac{(6n^{2} − 6n + 1)}{(z − w)^{4}} & + \frac{2}{(z − w)^{2}} T^{(bc)}(w) + \frac{1}{z − w} ∂_{w}T_{(w)}^{(bc)} \\ T^{(bc)}(z)c(w) & ∼ & \frac{1 − n}{(z − w)^{2}} c(w) & + \frac{1}{z − w} ∂_{w}c(w) \\ T^{(bc)}(z)b(w) & ∼ & \frac{n}{(z − w)^{2}} b(w) & + \frac{1}{(z − w)}∂_{w}b(w) . \end{aligned} (4.20)

(There is no singularity in the expansion of b(z)b(w) or c(z)c(w).) The classical action S_{n} is invariant under U(1)_{bc} × U(1)b̄c̄ phase rotations:

\{ \begin{matrix}c → e^{iθ}c \\ b → e^{−iθ}b \end{matrix}  \{ \begin{matrix}c̄ → e^{−iθ′} c̄ \\ b̄ → e^{iθ′}b̄ \end{matrix}  (4.21)

IV.5 Only the diagonal subgroup with θ = θ′ leaves the measure D(bb̄)D(cc̄) invariant, and results in a quantum symmetry.

The current associated with U(1)_{bc} is given by

j_{z}^{(bc)} = : cb : = lim_{w→z} \{c(w)b(z) − \frac{1}{w − z} \} ,  (4.22)

and we find the OPE’s:

j_{z}^{(bc)}c(w) ∼ −\frac{1}{z − w} c(w) ;  j_{z}^{(bc)}b(w) ∼ \frac{1}{z − w} b(w) .  (4.23)

The current j_{z}^{(bc)} has an anomaly, as can be established from the fact that j_{z}^{(bc)} does not transform as a tensor (for n ≠ \frac{1}{2}):

T^{(c)}(z)j_{w}^{(bc)} ∼ \frac{1 − 2n}{(z − w)^{3}} + \frac{1}{(z − w)^{2}} j_{w}^{(bc)} + \frac{1}{z − w} ∂_{w}j_{w}^{(bc)} .  (4.24)

Thus, under analytic diffeomorphisms z → z′ = f(z), we have

j′^{(bc)}(z′)dz′ = j^{(bc)}(z)dz + (n − \frac{1}{2}) d(ln f′(z))  (4.25)

Note that there is no anomaly for the case of spinors when n = 1/2. Also, on a worldsheet with general metric g, j^{(bc)} is not conserved; instead ∇^{z}j_{z}^{(bc)} = (\frac{1}{2} − n) R_{g}.

D. Bosonization of the b-c system

In Witten’s lectures, a Bose-Fermi correspondence was established between a 2-dimen-sional Dirac fermion and a real scalar boson field. The Dirac fermion corresponds here to the case n = \frac{1}{2}. We will generalize the Bose-Fermi correspondence to the case of any n ∈ \frac{1}{2}ℤ. We begin by noticing that the b-c stress tensor can be expressed in terms of the U(1)_{bc} current only:

T^{(bc)}(z) = \frac{1}{2}j^{(bc)}(z)^{2} + (n − \frac{1}{2}) ∂_{z}j^{(bc)}(z) ,  (4.26)

where the product of operators is always renormalized à la OPE. Using

j^{(bc)}(z)j^{(bc)}(w) ∼ \frac{1}{(z − w)^{2}} ,  (4.27)

IV.6 it is straightforward to verify that T^{(bc)} satisfies all identities listed above.

We now introduce a scalar field ϕ, with action (for general metric on Σ)

S_{Q}(x) = \frac{1}{4π} ∫_{Σ} dμ_{g} \{\frac{1}{2}g^{mn}∂_{m}ϕ∂_{n}ϕ + QR_{g}ϕ\} .  (4.28)

This system was studied extensively in Problem Set #1. Let us consider it only on locally flat metrics. Its central charge is c = 1 + 3Q^{2}, and there is a current ∂_{z}ϕ. Identifying the two systems will require that they have equal central charge:

1 + 3Q^{2} = −12n^{2} + 12n − 2  or  Q = i(2n − 1) .  (4.29)

(The second solution Q = −i(2n − 1) is equivalent upon changing the sign of ϕ.) The identification of the currents gives

\begin{matrix}j^{(bc)}(z) & = i∂_{z}ϕ \\ T^{(bc)}(z) = −\frac{1}{2}(∂_{z} & ϕ)^{2} + \frac{1}{2} Q∂_{z}^{2}ϕ . \end{matrix}  (4.30)

The fields b, c and b̄, c̄ may be identified in terms of the chiral boson fields ϕ_{+}(z),ϕ_{−}(z̄), with ϕ(z, z̄) = ϕ_{+}(z) + ϕ_{−}(z̄) by

\{ \begin{matrix}c(z) = e^{iϕ_{+}(z)} \\ b(z) = e^{−iϕ_{+}(z)} \end{matrix}  \{ \begin{matrix}c̄(z̄) = e^{+iϕ_{−}(z̄)} \\ b̄(z̄) = e^{−iϕ_{−}(z̄)} . \end{matrix}  (4.31)

The U(1)_{bc}×U(1)b̄c̄ transformations now simply shift ϕ_{+} and ϕ_{−} by θ and −θ′, respectively.

It is now easy to see that the number (N_{b}, N_{c}) of b and c operator insertions yielding non-zero correlation functions is constrained by the Riemann-Roch-Atiyah-Singer theorem:

N_{b} − N_{c} − \frac{i}{2}Qχ = 0  or  N_{b} − N_{c} = (\frac{1}{2} − n) χ ,

where χ is the Euler number of Σ.

E. The b-c Fock Space

The full Fock space is obtained as a tensor product of the bc Fock space ℱ^{(bc)} and the b̄c̄ Fock space ℱ^{(b̄c̄)}:

Fock(b, c, b̄, c̄) = ℱ^{(bc)} ⊗ ℱ^{(b̄c̄)} .  (4.32)

IV.7 We proceed to construct ℱ^{(bc)}, and ℱ^{(b̄c̄)} is constructed analogously. To do so, we consider the system on an annulus Σ, centered at z = 0, with flat metric g = 2|dz|^{2}, and we expand the holomorphic fields b(z) and c(z) in a Laurent series:

\begin{aligned}c(z) & = ∑_{r} c_{r} z^{−r−(1−n)} \\ b(z) & = ∑_{r} b_{r} z^{−r−n} \end{aligned}  r ∈ ℤ .  (4.33)

The shifts in the powers of z reflect the conformal weights of the fields. The OPE of b and c yields

{b_{r}, b_{s}} = {c_{r}, c_{s}} = 0  {b_{r}, c_{s}} = δ_{r+s,0} .  (4.34)

The stress tensor (renormalized as in IV.7) yields the Virasoro generators

\begin{aligned}L_{m}^{(bc)} & = ∑_{r∈ℤ}(mn − r)b_{r}c_{m−r}  m ≠ 0 \\ L_{0}^{(bc)} & = ∑_{r=1}^{∞} r(b_{−r}c_{r} + c_{−r}b_{r}) − \frac{1}{2} n(n − 1) . \end{aligned}  (4.35)

The U(1)_{bc} current gives the bc number operator:

U = ∮ \frac{dz}{2πi}j_{z}^{(bc)} = ∑_{r=1}^{∞}(c_{−r}b_{r} − b_{−r}c_{r}) + \frac{1}{2}(c_{0}b_{0} − b_{0}c_{0}) + n − \frac{1}{2} .  (4.36)

(The constant shifts in L_{0}^{(bc)} and U_{0} require great care, and are fundamental in the sequel.) The polarization here is again inherited from Minkowski worldsheet signature:

b_{r}^{†} = b_{−r}  c_{r}^{†} = c_{−r};  r ∈ ℤ .  (4.37)

The ground state of the b-c system is doubly degenerate, since b_{0} and c_{0} commute with L_{0} (in fact they do not enter into the expession for L_{0}), and since b_{0}, c_{0} generate an algebra {b_{0}, c_{0}} = 1, b_{0}^{2} = c_{0}^{2} = 0 of which the ground state is a two-dimensional representation. We define the basis vectors by |↑⟩ and |↓⟩, with

b_{r}, c_{r} |↑⟩ |↓⟩ = 0  r > 0 .  (4.38)

IV.8 We choose the states so that

\begin{aligned}c_{0} |↑⟩ & = 0 \\ b_{0} |↓⟩ & = 0 . \end{aligned}  (4.39)

It follows that

\begin{aligned}b_{0} |↑⟩ & = |↓⟩ & U |↑⟩ & = n |↑⟩ \\ c_{0} |↓⟩ & = |↑⟩ & U |↓⟩ & = (n − 1) |↓⟩ . \end{aligned}  (4.40)

It is natural to choose the following inner products

⟨↓|↑⟩ = ⟨↑|↓⟩ = 1  (4.41)

(Notice that ⟨↑|↑⟩ = ⟨↓|↓⟩ = 0 by their properties under b_{0} and c_{0}.)

The b-c Fock space is naturally obtained by applying the creation operators {b_{−r}, c_{−r}}_{r∈ℕ} to either one of the ground states |↑⟩ or |↓⟩. The inner product thus inherited on ℱ^{(bc)} is not positive definite.

F. BRST Quantization

We now return to string theory, where the relevant b-c system is the one with n = 2, in the notation used above. The combined matter-ghost action is (in a locally flat g)

S = \frac{1}{2π} ∫ d^{2}z (\frac{1}{2}∂x · ¯∂x + b¯∂c + b̄∂c̄) .  (4.42)

The combined stress tensor (we denote the x stress tensor by T^{(x)})

T (z) = T^{(x)}(z) + T^{(bc)}(z)  (4.43)

has zero central charge in D = 26 and is thus a primary field of weight (2, 0). The associated Virasoro generators are

\{ \begin{matrix}L_{m} & = ∑_{r∈ℤ}(2m − r)b_{r}c_{m−r} + \frac{1}{2} ∑_{r∈ℤ} x_{r} · x_{m−r} (m ≠ 0) \\ L_{0} & = \frac{1}{2}p^{2} − 1 + ∑_{r=1}^{∞} r(x_{−r}x_{r} + b_{−r}c_{r} + c_{−r}b_{r}) \end{matrix}  (4.44)

IV.9 The second term (−1) in the second equation arises from the ghost stress tensor on p. IV. 8 for n = 2. The Virasoro algebra of L_{m} reduces to the Witt algebra (i.e. with zero central charge)

[L_{ℓ}, L_{m}] = (ℓ − m)L_{ℓ+m}  (4.45)

Recall the ghost number

U = 1 + c_{0}b_{0} + ∑_{r=1}^{∞}(c_{−r}b_{r} − b_{−r}c_{r})  (4.46)

and recall that the round states |↑⟩ and |↓⟩ have ghost number 2 and 1, respectively. The full Fock space of states of the closed bosonic string, including the ghost states, is

ℱ_{closed} = ( ⨁_{k∈ℝ^{26}} ℱ_{k} ⊗ ℱ̃_{k}) ⊗ ℱ^{(bc)} ⊗ ℱ^{(b̄c̄)}  (4.47)

We now wish to identify, inside ℱ, the subspace of physical states. We shall concentrate on ℱ_{k} ⊗ ℱ^{(bc)}, the treatment of ℱ̃_{k} ⊗ ℱ^{(b̄c̄)} is analogous.

Since the Virasoro algebra of L_{ℓ} generators holds with vanishing central charge, it is natural to impose

L_{ℓ} | ϕ⟩ = 0 ∀ ℓ ∈ ℤ .

However, since the Fock space now also contains ghost states, this condition is insufficient. One would like to supplement it with a condition of “no ghosts”. But from the treatment above, we see that ghost number changes under conformal mappings, so the no ghost condition is not invariant. The proper framework for imposing physical state conditions, when ghosts are present, is BRST quantization. It permits to recast the physical state conditions in terms of cohomology.

We begin by discussing a finite-dimensional case, with finite-dimensional Lie group G and associated Lie algebra 𝒢. Let Λ be a representation of 𝒢, with representation matrices Λ_{i}, i = 1, …, dim G, and let f_{ij}^{k} be the structure constants of 𝒢:

[Λ_{i}, Λ_{j}] = f_{ij}^{k}Λ_{k}  (4.48)

IV.10 One introduces anti-ghosts b_{i} in the adjoint representation of 𝒢 and ghosts c^{i} in the dual to the adjoint representation of 𝒢. (For compact G, these are identified, but in general, it may not be possible to do so.) These satisfy

{b_{i}, c^{j}} = δ_{j}^{j}  {c^{i}, c^{j}} = {b_{i}, b_{j}} = 0 .  (4.49)

The number operator is the analogue of the ghost number

U = c^{i}b_{i}  [U, c^{i}] = c^{i}; [U, b_{i}] = −b_{i}  (4.50)

and is invariant under 𝒢. The Fock space ℱ is determined by choosing a polarization. The natural polarization here is as follows

b_{i} | 0⟩ = 0  ℱ = {c^{i}} | 0⟩ ,  (4.51)

and ℱ contains 2^{dim G} states.

This model may be realized concretely by taking c^{i}, i = 1, …, dim G to be a basis of left-invariant 1-forms on G, and b_{i} to be the contraction with the form c^{i}: b_{i} = ∂/∂c^{i}. The operator U gives the degree of the form and the ground state | 0⟩ corresponds to forms of degree 0, i.e. constants. The Fock space ℱ is the vector space of all left-invariant forms on G: Ω^{inv}(G).

In BRST theory, one introduces the BRST operator (still for our finite-dimensional group G):

Q ≡ c^{i}Λ_{i} − \frac{1}{2} f_{ij}^{k}c^{i}c^{j}b_{k} .  (4.52)

This operator acts on ℱ ⊗Λ, where Λ is the representation vector space of Λ. This operator is nilpotent. To see this, first show that Q^{2} = 0 when Λ_{i} = 0; the remaining terms with Λ present are

Q^{2} = \frac{1}{2}c^{i}c^{j} ([Λ_{i}, Λ_{j}] − f_{ij}^{k}Λ_{k}) ,  (4.53)

which vanishes in view of the structure relations on Λ.

In terms of left-invariant forms on the group G, the operator Q has a natural interpretation as well. This is understood by examining how a differential acts on a left-invariant

IV.11 differential form in Ω^{(p)}(G): if ω ∈ Ω^{(p)}(G) then ω = ω_{i_{1}} … i_{p} c^{i_{1}} ∧ … ∧c^{i_{p}}, with constant ω_{i_{1}...i_{p}}. The differential on c^{i} is given by the Cartan structure equations

dc^{i} + \frac{1}{2} f_{jk}^{i}c^{j} ∧ c^{k} = 0  (4.54)

and thus its action on ω may be recast in a purely algebraic form

dω = −\frac{1}{2} f_{jk}^{i}c^{j}c^{k}b_{i} .  (4.55)

Furthermore, when ω transforms under the representation Λ of 𝒢, then Q is nothing but the covariant derivative acting on ℱ ⊗ Λ, and its square vanishes in view of the flatness of the connection c^{i}.

We now consider states in the Fock space ℱ_{(m)} of degree m = 0, 1, …, dim G. The BRST operator Q has degree 1, and maps Q: ℱ_{(m)} → ℱ_{(m+1)} for m ≤ dim G − 1, and Q: ℱ_{(dim G)} → 0. A state |ψ⟩ is BRST invariant if

Q |ψ⟩ = 0  (i.e. Q-closed)

and BRST trivial if for some |x⟩ ∈ ℱ_{(m−1)}, we have

|ψ⟩ = Q |χ⟩  (i.e. Q-exact)

BRST cohomology is defined by the following equivalence relation: |ψ⟩ ∼ |ψ′⟩ provided there is some |χ⟩ such that

|ψ′⟩ = |ψ⟩ + Q |χ⟩ .

BRST cohomology classes of degree m are denoted by H^{(m)}(Q). The BRST cohomology of degree 0 is easily computed. First, |ψ⟩ ∈ ℱ_{(0)} means that b_{j} |ψ⟩ = 0 for all j, so that the condition of BRST invariance becomes

Q |ψ⟩ = c^{i}Λ_{i} |ψ⟩ = 0  ⇒  Λ_{i} |ψ⟩ = 0

Here, we have used the fact that, since {b_{i}, c^{j}} = δ_{i}^{j}, we have c^{i} |ψ⟩ ≠ 0 for all i, and these vectors are linearly independent. Thus, all BRST invariant states of ghost number 0 are

IV.12 𝒢-invariant. Clearly no states in ℱ_{(0)} are BRST trivial. Vice versa, all 𝒢-invariant states of ghost number 0 are BRST invariant.

In string theory, the BRST formalism is implemented as follows. The Lie aglebra 𝒢 is the infinite-dimensional Virasoro algebra, with structure constants

[L_{i}, L_{j}] = (i − j)L_{i+j} + central; f_{ij}^{k} = (i − j)δ_{i+j}^{k} .  (4.56)

We do not specify the central term, since we shall not infer its implications from the finite-dimensional model. The representation Λ of 𝒢 is given by the matter fields x, with generators L_{i}^{(x)}, and central charge 26. The ghost and anti-ghost modes are

c^{ℓ} = c_{−ℓ}  and  b_{ℓ}  ℓ ∈ ℤ .  (4.57)

The polarization for the string theory BRST is different from that for the finite-dimensional model

b_{ℓ}, c_{ℓ} | ↓⟩ , | ↑⟩ = 0  ℓ > 0 .  (4.58)

The BRST operator is defined by the structure constants f_{ij}^{k}, but the precise ordering prescription, and the effects of the central terms cannot be obtained from the finitedimensional model. It will turn out (see next paragraph) that the correct prescription is given by

Q ≡ ∑_{r∈ℤ}(L_{r}^{(x)} − δ_{r,0})c_{−r} − \frac{1}{2} ∑_{r,s∈ℤ}(r − s) : c_{−r}c_{−s}b_{r+s} :  (4.59)

The correct prescription may be derived most efficiently from the fact that the BRST charge derives from a conserved (in fact analytic) BRST current, which arises as the Noether current of BRST symmetry of the action S. The transformation rules are (they may be derived from the commutation of the operator λQ, for some constant Grassmann parameter λ, with the fields x^{μ}, b, c):

\{ \begin{matrix}δx^{μ} & = λc∂x^{μ} \\ δc & = λc∂c \\ δb & = 2λ(T^{(x)} + T^{(bc)}) . \end{matrix}  (4.60)

IV.13 The resulting BRST current is

\begin{aligned}j^{BRST}(z) & = −\frac{1}{2} c∂x · ∂x + bc∂c + \frac{3}{2} ∂^{2}c \\ & = cT^{(x)} + \frac{1}{2} cT^{(bc)} + \frac{3}{2} ∂^{2}c . \end{aligned}  (4.61)

The last term on the right hand side ensures that j^{BRST} is a (1, 0) form as a quantum operator. The quantum (renormalized) current is defined as usual by OPE singularity subtractions. The BRST charge is then

Q = ∮ \frac{dz}{2πi} j^{BRST}(z) ,  (4.62)

and agrees with the expression of (4.59).

Using the OPE algebra for the b-c system, derived in §C, we find that for D = 26 we have Q^{2} = 0. Conversely, if Q^{2} = 0, then the Virasoro algebra of L_{m} = L_{m}^{(x)} + L_{m}^{gh)} must be satisfied with zero central charge. This is easily established from the fact that

L_{m} = {Q, b_{m}}  ⟹  [L_{m}, Q] = 0  (4.63)

and hence

[L_{m}, L_{n}] = {Q, [L_{m}, b_{n}]} = (m − n){Q, b_{m+n}} = (m − n)L_{m+n} .

The BRST invariance condition Q |ψ⟩ = 0 now reproduces the physical state conditions L_{m} |ψ⟩ = 0, m ∈ ℤ, in the following way. First, we define a map i from ℱ_{k}^{phys} into ℱ_{k}^{phys} ⊗ ℱ^{(bc)}. which maps physical states in ℱ_{k}^{phys} into states of ghost number 1 as follows:

i: ℱ_{k}^{phys} → ℱ_{k} ⊗ ℱ^{(bc)}, |ε, k⟩ → |ε, k, ↓⟩ = |ε, k⟩ ⊗ | ↓⟩  (4.64)

Since b_{m} | ↓⟩ = 0 for all m ∈ ℕ and c_{n} | ↓⟩ = 0 for all n ∈ ℕ^{+}, it follows that if |ϵ, k⟩ is physical, then Q |ε, k, ↓⟩ = 0. Thus, i is really a map into Ker Q

i: ℱ_{k}^{phys} → Ker Q .  (4.65)

Next, let us consider a state |ψ⟩ ∈ ℱ_{k} ⊗ ℱ^{(bc)}, of the form

|ψ⟩ = |ϵ, k⟩ ⊗ | ↓⟩  (4.66)

IV.14 with ghost number 1, and belonging to Ker Q: Q |ψ⟩ = 0. Then, in view of b_{m} | ↓⟩ = 0 for m ∈ ℕ and c_{n} |↓⟩ = 0 for n ∈ ℕ^{+}, we have

\begin{aligned}Q |ψ⟩ & = ∑_{r∈ℤ}(L_{r}^{(x)} − δ_{r,0})c_{−r} |ε, k⟩ ⊗ | ↓⟩ \\ & = \{(L_{0}^{(x)} − 1)c_{0} + ∑_{r=1}^{∞} L_{r}^{(x)}c_{−r}\} |ε, k⟩ ⊗ | ↓⟩ , \end{aligned}  (4.67)

The condition Q |ψ⟩ = 0 implies (L_{0}^{(x)} − 1) |ε, k⟩ = 0 and L_{r}^{(x)} |ε, k⟩ = 0 for r ∈ ℕ^{+}, which are the correct physical state conditions on ℱ_{k}. (The ground state | ↑⟩ would miss the conditions (L_{0}^{(x)} − 1) |ε, k⟩ = 0.)

Which of these states are BRST trivial? Again, from the definition of Q and the fact that b_{m} |ψ⟩ = c_{m} |ψ⟩ = 0 for m ∈ ℕ^{+}, it follows that if |ψ⟩ = Q |λ⟩, with |λ⟩ of ghost number 0 then

|ψ⟩ = ∑_{r=1}^{∞} L_{−r}^{(x)} |λ_{r}⟩  (4.68)

for some states |λ_{r}⟩. But then |λ⟩ is spurious.

In fact, one can show that any state |ψ⟩ in Ker Q, of ghost number 1, is of the form

|ψ⟩ = |ε, k⟩ ⊗ | ↓⟩ + Q |x⟩

where |ε, k⟩ ∈ ℱ_{k}^{phys} and |x⟩ is some state of ghost number 0. Thus, the cohomology of Q of ghost number 1, i.e. space of states of ghost number 1 which are BRST invariant, modulo BRST trivial states, is isomorphic to the physical state space ℱ^{+} defined previously:

ℱ^{+} = ℱ^{phys}/ℱ^{spurious} ∼ H^{(1)}(Q)  (4.69)

IV.15