n
′
M2 =
(P 2
i
i − 2) = 2
na†n ani − 1 = 2(N − 1) ,
′
M
ˆ
2
αmM = −i 1
X
=
a†
2
αP j
nα∆n − ∆†nanα
,
1
n−1
∞
∆
1
i
i
n = i √
m(n − m)a a
m(n + m)a† a
n
2
m
n−m,i −
m
n+m,i
, (8.2.1)
m=1
m=1
where again the i summation is over both a and α, representing modes coming from
both the physical Xa(σ) (with xa identified as the usual spacetime coordinate) and
the ghost modes Xα(σ) (with xα the ghost coordinates of sect. 3.4), as in the mode
expansion (7.1.7).
To understand the relation of the first-quantized BRST quantization [4.4,5] to
that derived from the light cone (and from the OSp(1,1|2)), we show the Sp(2) sym-
metry of the ghost coordinates. We first combine all the ghost oscillators into an
“isospinor” [4.1]:
1
δ
Cα =
C′,
.
(8.2.2)
∂/∂σ
δC
This isospinor directly corresponds (except for lack of zero modes) to ˆ
Xα of the
OSp(1,1|2) formalism from the light cone: We identify
ˆ
Xα = (xα + pασ) + Cα ,
(8.2.3)
and we can thus directly construct objects which are manifestly covariant under the Sp(2) of Mαβ. The periodic inverse derivative in (8.2.2) is defined in terms of the saw-tooth function
1
1
f (σ) =
dσ′ 1 ǫ(σ − σ′) − (σ − σ′) f(σ′) ,
(8.2.4)
∂/∂σ
2
π
where ǫ(σ) = ±1 for ±σ > 0. The product of the derivative with this inverse
derivative, in either order, is simply the projection operator which subtracts out
the zero mode. (For example, C+ is just C minus its zero mode.) Along with P a,
this completes the identification of the nonzero-modes of the two formalisms. We can then rewrite the other relevant operators (8.1.10,12) in terms of Cα:
dσ
p2 + M2 =
(P 2 + Cα′C ′ − 2) ,
2π
a
α
8.2. OSp(1,1|2)
159
dσ
Q
2
′
α = −i 1
+
2
C
Cα′C − 2) ,
2π
α(P a
α
dσ
M
′
αβ = −i 1
.
(8.2.5)
2
C
2π (αCβ)
Again, all definitions include normal ordering. This first-quantized IGL(1) can then be seen to agree with that derived from OSp(1,1|2) in (8.2.1) by expanding in zero-modes.
For the closed string, the OSp(1,1|2) algebra is extended to an IOSp(1,1|2) algebra following the construction of (7.1.17): As for the open-string case, the D-dimensional indices of the light-cone formalism are extended to (D+4)-dimensional indices, but
just the values A = (±, α) are kept for the BRST algebra. To obtain the analog of
the IGL(1) formalism, we perform the transformation (3.4.3a) for both left and right-handed modes. The extension of IGL(1) to GL(1|1) analogous to that of OSp(1,1|2)
to IOSp(1,1|2) uses the subalgebras (Q, J3, p−, p˜c = ∂/∂c) of the IOSp(1,1|2)’s of each set of open-string operators. After dropping the terms containing ∂/∂˜
c, x− drops out,
and we can set p+ = 1 to obtain:
∂
Q
→
− ic1(p 2 + M2) + M+i
+
2
a
Q+ ,
∂c
∂
J3
→
c
+ M3
,
∂c
p
2
−
→
− 1(p + M2) ,
2
a
∂
p˜c
→
.
(8.2.6)
∂c
These generators have the same algebra as N=2 supersymmetry in one dimension,
with Q and p˜c corresponding to the two supersymmetry generators (actually the
complex combination and its complex conjugate), J3 to the O(2) generator which
scales them, and p− the 1D momentum. The closed-string algebra GL(1|1) is then
constructed in analogy to the IOSp(1,1|2), taking sums for the J’s and differences for the p’s.
The application of this algebra to obtaining the gauge-invariant action will be
discussed in chapt. 11.
160
8. BRST QUANTUM MECHANICS
8.3. Lorentz gauge
We will next consider the OSp(1,1|2) algebra which corresponds to first-quantization in the Lorentz gauge [3.7], as obtained by the methods of sect. 3.3 when applied to the Virasoro algebra of constraints.
From (3.3.2), for the OSp(1,1|2) algebra we have
dσ
δ
δ
δ
Qα = −i
Cα( 1P 2 − 1) + 1C(αCβ)′i
− Bi
+ 1 (CαB′ − Cα′B)i
2π
2
2
δCβ
δC
2
α
δB
δ
+ 1Cβ(C ′Cα)′i
.
(8.3.1)
4
β
δB
B is conjugate to the time-components of the gauge field, which in this case means
the components g00 and g01 of the unit-determinant part of the world-sheet metric
(see chapt. 6). This expression can be simplified by the unitary transformation Qα →
UQαU−1 with
δ
ln U = −1 (1CαC
′
.
(8.3.2)
2
2
α) δB
We then have the OSp(1,1|2) (from (3.3.7)):
δ
δ
δ
J−α =
Cα −i1P 2 + i + Cβ′
+ B′
,
2
− B
δCβ
δB
δCα
δ
δ
Jαβ =
C(α
,
J
,
δCβ)
+α = 2
Cα δB
δ
δ
J−+ =
2B
+ Cα
.
(8.3.3)
δB
δCα
A gauge-fixed kinetic operator for string field theory which is invariant under the full OSp(1,1|2) can be derived,
δ
δ
δ
K = 1 J α, J
=
1 P 2
+ B′i
= 1 (p2 + M2) ,
2
−
−α,
i
− 1 + Cα′i
δB
2
δCα
δB
2
(8.3.4)
as the zero-mode of the generators G(σ) from (3.3.10):
δ
G(σ) = −1 J α, J
2
−
−α, δB
δ
δ
′
δ
δ
′
= −i(1P 2
+ Cα
+ B′
+ B
. (8.3.5)
2
− 1) + Cα′ δCα
δCα
δB
δB
The analog in the usual OSp(1,1|2) formalism is
1
α, [J
2]
2 {J −
−α, p+
} = pApA ≡ 2p+p− + pαpα = ✷ − M2 .
(8.3.6)
8.3. Lorentz gauge
161
This differs from the usual light-cone gauge-fixed hamiltonian p−, which is not OSp(1,1|2) invariant. Unlike the ordinary BRST case (but like the light-cone formalism), this
operator is invertible, since it’s of the standard form K = 1p2 +
2
· · ·. This was made
possible by the appearance of an even number (2) of anticommuting zero-modes. In
ordinary BRST (sect. 4.4), the kinetic operator is fermionic: c(✷ − M2) − 2 ∂ M+ is
∂c
not invertible because M+ is not invertible.
As usual, the propagator can be converted into a form amenable to first-quantized
path-integral techniques by first introducing the Schwinger proper-time parameter:
1
∞
=
dτ e−τK
,
(8.3.7)
K
0
where τ is identified with the (Wick-rotated) world-sheet time. At the free level, the analysis of this propagator corresponds to solving the Schrödinger equation or, in
the Heisenberg picture (or classical mechanics of the string), to solving for the time dependence of the coordinates which follow from treating K as the hamiltonian:
[K, Z] = iZ′
,
.
Z − [K, Z] = 0
→
Z = Z(σ + iτ )
(8.3.8)
for Z = P, C, B, δ/δC, δ/δB. Thus in the mode expansion Z = z
†
0+
∞(z e−in(σ+iτ)+
1
n
z
†
nein(σ+iτ )) the positive-energy zn ’s are creation operators while the negative-energy zn’s are annihilation operators. (Remember active vs. passive transformations: In the Schrödinger picture coordinates are constant while states have time dependence e−tH ; in the Heisenberg picture states are constant while coordinates have time dependence etH ( )e−tH .)
When doing string field theory, in order to define real string fields we identify
complex conjugation of the fields as usual with proper-time reversal in the mechanics, which, in order to preserve handedness in the world sheet, means reversing σ as well as τ . As a result, all reparametrization-covariant variables with an even number of 2D vector indices are interpreted as string-field coordinates, while those with an odd number are momenta. (See sect. 8.1.) This means that X is a coordinate, while B and C are momenta. Therefore, we should define the string field as Φ[X(σ), G(σ), F α(σ)], where B = iδ/δG and Cα = iδ/δF α. This field is real under a combined complex
conjugation and “twist” (σ → −σ), and Qα is odd in the number of functional plus
σ derivatives. (Note that the corresponding replacement of B with G and C with F
would not be required if the Gi’s had been associated with a Yang-Mills symmetry
rather than general coordinate transformations, since in that case B and C carry no vector indices.)
162
8. BRST QUANTUM MECHANICS
This OSp(1,1|2) algebra can also be derived from the classical mechanics action.
The 2D general coordinate transformations (6.1.2) generated by δ =
dσ ǫm(σ)G
2π
m(σ)
determine the BRST transformations by (3.3.2):
QαX = Cmα∂mX
,
Qαgmn = ∂p(Cpαgmn) − gp(m∂pCn)α ,
QαCmβ = 1Cn(α∂
2
nC mβ) − C αβ Bm
,
QαBm = 1(Cnα∂
2
nBm − Bn∂nC mα)
− 1 2Cnβ(∂
8
nC pα)∂pC mβ + C nβ C pβ ∂n∂pC mα
.
(8.3.9)
We then redefine
Bm = Bm − 1Cnα∂
2
nC mα
→
QαCmβ = Cnα∂nCmβ − CαβBm ,
QαBm = Cnα∂nBm .
(8.3.10)
The rest of the OSp(1,1|2) follows from (3.3.7):
J+α(X, gmn, Cmβ) = 0 ,
J+αBm = 2Cmα ;
J
γ
αβ (X, gmn, Bm) = 0
,
JαβCmγ = δ(α Cmβ) ;
J−+(X, gmn) = 0 ,
J−+Bm = 2Bm ,
J−+Cmα = Cmα .
(8.3.11)
An ISp(2)-invariant gauge-fixing term is (dropping boundary terms)
L
2 1
1 = Qα
η
gmn(∂
2 pqgpq = −ηpq Bp∂mgqm + 1
2
mC pα)(∂nC qα)
,
(8.3.12)
where η is the flat world-sheet metric. This expression is the analog of the gauge-
fixing term Q2 1A2 for Lorentz gauges in Yang-Mills [3.6,12]. Variation of B gives the 2
condition for harmonic coordinates. The ghosts have the same form of lagrangian
as X, but not the same boundary conditions: At the boundary, any variable with
an even number of 2D vector indices with the value 1 has its σ-derivative vanish,
while any variable with an odd number vanishes itself. These are the only boundary
conditions consistent with Poincaré and BRST invariance. They are preserved by the
redefinitions below.
8.3. Lorentz gauge
163
Rather than this Landau-type harmonic gauge, we can also define more general
Lorentz-type gauges, such as Fermi-Feynman, by adding to (8.3.12) a term propor-
tional to Q2 1η
η
2 mnC mαC nα = − 1
2 mnBmBn + · · ·. We will not consider such terms
further here.
Although the hamiltonian quantum mechanical form of Qα (8.3.3) also follows
from (3.3.2) (with the functional derivatives now with respect to functions of just σ
instead of both σ and τ ), the relation to the lagrangian form follows only after some redefinitions, which we now derive. The hamiltonian form that follows directly from (8.3.9,10) can be obtained by applying the Noether procedure to L = L0 + L1: The
BRST current is
Jmα = (gnpCmα − gm(nCp)α)1 (∂
2
nX ) · (∂pX ) + (∂nC qβ )(∂pC qβ )
− Bn∂p(gn[mCp]α − gmpCnα) ,
(8.3.13)
where (2D) vector indices have been raised and lowered with the flat metric. Canon-
ically quantizing, with
1
δ
1
δ
1
δ
P 0 = i
,
B
,
Π
,
(8.3.14a)
α′
δX
α′ m = −iδg0m
α′ mα = iδCmα
we apply
.
1
1
X = −
(P 0 + g01X′) ,
.
Cmα = −
(Πmα + g01Cmα′) ,
(8.3.14b)
g00
g00
to the first term in (8.3.13) and
∂mgmn = 0 ,
g0m∂mCnα = −Πnα ,
(8.3.14c)
to the second to obtain
1
J0α = −
C0α 1 (P 02 + X′2) + (ΠmβΠ
′)
g00
2
mβ + C mβ ′C mβ
g01
+ C1α −
C0α (X′ · P 0 + Cmβ′Π
g00
mβ ) − Bm Πmα + (gm[0C 1]α)′
,
(8.3.15)
where Qα ∼ dσ J0α.
By comparison with (8.3.3), an obvious simplification is to absorb the g factors
into the C’s in the first two terms. This is equivalent to
Cmα → δ m
1
C1α − g0mC0α ,
(8.3.16)
164
8. BRST QUANTUM MECHANICS
and the corresponding redefinitions (unitary transformation) of Π and B. This puts
g-dependence into the BΠ terms,
1
g01
ΠmBm
→
Π0 −
B0 + Π
B0 + · · · ,
(8.3.17)
g00
1
B1 − g00
unlike (8.3.3), so we remove it by the g redefinition
g01 = −g1 ,
g00 = − 1 + 2g0 + (g1)2 ;
1
δ
B
.
(8.3.18)
α′ m = iδgm
These redefinitions give
J0α = C0α 1(P 02 + X′2) + C1αX′
2
· P 0
+ C0α(C0βΠ1β)′ + C1α Cmβ′Πmβ + (C0βΠ0β)′
− C0α(B1 + g1B0)′ + C1α −B0′ + gmBm′ + (g0B0)′
+ C0α 1 Cmβ′C
′ + (g
′