θ
−′+′
,
M+′α′ = 12 −γ+′γα′θ + ˇ
M +′α′ ,
∂
∂
∂
M
¯
αa = 1 θγ
+ ˇ
M
γ
+ ˇ
M
2
αγa ∂¯θ
αa
,
M−′α′ = 116 ∂θ +γ−′γα′ ∂¯θ
−′α′
.
(5.5.15)
Finally, we perform the unitary transformations (3.6.13) to find
∂
q0 = δ(a†a)
+ 2γap
∂ ¯
θ
aθ
.
(5.5.16)
The δ now projects out just the OSp(1,1|2)-singlet part of θ (i.e., the usual Lorentz spinor):
∂
q0 =
+ γap
∂ ¯
θ
aθ0
,
0
5.5. SuperBRST
113
∂
∂
∂
= δ(a†a)
,
θ
, ¯
θ
∂ ¯
θ
0 = 2δ(a†a)θ
,
0
= Pδ(a†a) .
(5.5.17)
0
∂ ¯
θ
∂ ¯
θ0
(5.5.15) can be substituted into (3.6.14). We then find the OSp(1,1|2) generators
J+α = ixαp+ ,
J−+ = −ix−p+ , Jαβ = −ix(αpβ) + Mαβ ,
1
J
1
2
β
−α = −ix−pα +
−ix (p + pβp
p
p
α 2
a
β ) + M α
β + Qα
;
(5.5.18a)
+
∂
M αβ = ¯
θa†(αaβ)
+ ˇ
M
∂ ¯
θ
αβ + ˇ
M α′β′
,
∂
∂
Qα = − i1
+ iγ
8
− iγ
∂ ¯
θ
−γ+′ γapaθ
γ+γ−′a†α ∂¯θ
−γ+′ γapaθ
+
ˇ
M
a
ˇ
2
−′α′ + ˇ
M α pa + 1 M
,
(5.5.18b)
2
+′α′ pa
and J
¯
ab = −ix[apb] + 1 θγ
2
ab∂/∂ ¯
θ + ˇ
M ab for the Lorentz generators. Finally, we can
remove all dependence on γ± and γ±′ by extracting the corresponding γ0 factors
contributing to the spinor metric η:
γ
†
0 = −i 1 (γ
,
δ(a†a)γ
2
+ − γ−)(γ+′ − γ−′ ) = γ0
0 = γ0δ(a†a) = δ(a†a)
,
∂
∂
∂
→ γ
= i1γ
,
¯
θ → ¯θγ
¯
θγ
∂ ¯
θ
0 ∂¯θ
2 −γ+′ ∂ ¯θ
0 = i 1
2
+γ−′
,
∂
∂
(γ+, γ−′)
= (γ
(γ
∂ ¯
θ
+, γ−′ )θ = ¯
θ(γ−, γ+′) = ∂θ −, γ+′) = 0 ,
∂ , ¯θ = P(1γ
γ
∂ ¯
θ
2
+γ−)( 1
2 −′ γ+′ )
;
(5.5.19a)
and then convert to the harmonic oscillator basis with respect to these γ’s:
∂
∂
→ ea†+a†−
,
θ → 1ea†+a†−θ ,
∂ ¯
θ
∂ ¯
θ
2
¯
∂
∂
θ → 1 ¯θea+a− ,
ea+a−
;
2
→
∂θ
∂θ
∂
∂
a±
= a
a†
∂ ¯
θ
±θ = ¯
θa†± = ∂θ ± = 0 ;
∂ , ¯θ = P(a
∂ ¯
θ
+a−a†+a†−)
.
(5.5.19b)
All a±’s and a†±’s can then be eliminated, and the corresponding projection operators (the factor multiplying P in (5.5.19b)) dropped. The only part of (5.5.18) (or the
Lorentz generators) which gets modified is
Q
a
ˇ
2
α = − 1 ¯
qa†
p
M
,
2
αd +
ˇ
M −′α′ + ˇ
M α a + 12 +′α′pa
114
5. PARTICLE
∂
∂
q =
+ γap
− γap
∂ ¯
θ
aθ
,
d = ∂¯θ
aθ
,
q0 = δ(a†αaα)q ,
d0 = δ(a†αaα)d .
(5.5.20)
If we expand the first term in Qα level by level in a†α’s, we find a q at each level multiplying a d of the previous level. In particular, the first-level ghost q1α multiplies the physical d0. This means that d0 = 0 is effectively imposed for only half of its spinor components, since the components of q are not all independent.
An interesting characteristic of this type of BRST (as well as more conventional
BRST obtained by first-quantization) is that spinors obtain infinite towers of ghosts.
In fact, this is necessary to allow the most general possible gauges. The simplest
explicit example is BRST quantization of the action of sect. 4.5 for the Dirac spinor quantized in a gauge where the gauge-fixed kinetic operators are all p2 instead of
/
p. However, these ghosts are not all necessary for the gauge invariant theory, or for certain types of gauges. For example, for the type of gauge invariant actions for
spinors described in sect. 4.5, the only parts of the infinite-dimensional OSp(D-1,1|2) spinors which are not pure gauge are the usual Lorentz spinors. (E.g., the OSp(D-1,1|2) Dirac spinor reduces to an ordinary SO(D-1,1) Dirac spinor.) For gauge-fixed, 4D N=1 supersymmetric theories, supergraphs use chains of ghost superfields which
always terminate with chiral superfields. Chiral superfields can be irreducible off-shell representations of supersymmetry since they effectively depend on only half of the
components of θ. (An analog also exists in 6D, with or without the use of harmonic
superspace coordinates [4.12].) However, no chiral division of θ exists in 10D (θ is a real representation of SO(9,1)), so an infinite tower of ghost superfields is necessary for covariant background-field gauges. (For covariant non-background-field gauges,
all but the usual finite Faddeev-Popov ghosts decouple.) Thus, the infinite tower is not just a property of the type of first-quantization used, but is an inherent property of the second-quantized theory. However, even in background-field gauges the infinite tower (except for the Faddeev-Popovs) contribute only at one loop to the effective
action, so their evaluation is straightforward, and the only expected problem would be their summation.
The basic reason for the tower of θ’s is the fact that only 1/4 (or, in the massive case, 1/2) of them appear in the gauge-invariant theory on-shell, but if θ is an irreducible Lorentz representation it’s impossible to cancel 3/4 (or 1/2) of it covariantly.
We thus effectively obtain the sums
1 − 1 + 1 − 1 + · · · = 1 ,
(5.5.21a)
2
5.5. SuperBRST
115
1 − 2 + 3 − 4 + · · · = 1 .
(5.5.21b)
4
(The latter series is the “square” of the former.) The positive contributions rep-
resent the physical spinor (or θ) and fermionic ghosts at even levels, the negative contributions represent bosonic ghosts at odd levels (contributing in loops with the opposite sign), and the 1 or 1 represents the desired contribution (as obtained directly 2
4
in light-cone gauges). Adding consecutive terms in the sum gives a nonconvergent
(but nondivergent in case (5.5.21a)) result which oscillates about the desired result.
However, there are unambiguous ways to regularize these sums. For example, if
we represent the levels in terms of harmonic oscillators (one creation operator for (5.5.21a), and the 2 a†α’s for (5.5.21b)), these sums can be represented as integrals over coherent states (see (9.1.12)). For (5.5.21a), we have:
d2z
str(1) = tr (−1)N =
e−|z|2 z (−1)a†a z
π
d2z
d2z
d2z
=
e−|z|2 z| − z =
e−2|z|2 = 1
e−|z|2
π
π
2
π
= 1
,
(5.5.22)
2
where str is the supertrace; for (5.5.21b), the supertrace over the direct product
corresponding to 2 sets of oscillators factors into the square of (5.5.22). (The corresponding partition function is str(xN ) = 1/(1 + x) = 1 − x + x2 − · · ·, and for 2 sets of oscillators 1/(1 + x)2 = 1 − 2x + 3x2 − · · ·.)
An interesting consequence of (5.5.21) is the preservation of the identity
D′ = 2(D−k)/2
;
D′ = strS(1) ,
D = strV (1) ;
(5.5.23)
upon adding (2, 2|4) dimensions, where D′ and D are the “superdimensions” of a
spinor and vector, defined in terms of supertraces of the identity for that representation, and k is an integer which depends on whether the dimension is even or odd and whether the spinor is Weyl and/or Majorana (see sect. 5.3). k is unchanged by adding (2, 2|4) dimensions, D changes by addition of 4 − 4 = 0, and, because of (5.5.21), D′
changes by a factor of 22 · (1)2 = 1. This identity is important for super-Yang-Mills 2
and superstrings.
Before considering the action for arbitrary supersymmetric theories, we’ll first
study the equations of motion, since the naive kinetic operators may require extra
factors to write a suitable lagrangian. Within the OSp(1,1|2) formalism, the gauge-
fixed field equations are (cf. (4.4.19))
(p2 + M2)φ = 0
(5.5.24a)
116
5. PARTICLE
when subject to the (Landau) gauge conditions [2.3]
Qαφ = Mαβφ = 0 .
(5.5.24b)
Applying these gauge conditions to the gauge transformations, we find the residual
gauge invariance
δφ = −i1
(p2 + M2) +
2 QαΛα
,
−32
Q2 Λα = MαβΛγ + Cγ(αΛβ) = 0 . (5.5.24c)
(In the IGL(1) formalism, sect. 4.2, the corresponding equations involve just the Q+
component of Qα and the M+ and M3 components of Mαβ, but are equivalent, since
M + = M 3 = 0 → M− = 0, and Q+ = M− = 0 → Q− = 0.)
For simplicity, we consider the massless case, and ˇ
M ij = 0. We can then choose
the reference frame where p
+
a = δa p+, and solve these equations in light-cone nota-
tion. (The +’s and −’s now refer to the usual Lorentz components; the unphysical x−, p+, and γ±′ have already been eliminated.) The gauge conditions (5.5.24b) eliminate auxiliary degrees of freedom (as ∂ · A = p+A− = 0 eliminates A− in Yang-Mills),
and (5.5.24c) eliminates remaining gauge degrees of freedom (as A+ in light-cone-
gauge Yang-Mills). We divide the spinors d, q, ∂/∂θ, and θ into halves using γ±, and then further divide those into complex conjugate halves as creation and annihilation operators, as in (5.4.27,29):
d → γ+d, γ−d → da, ¯
da, ∂a, ¯
∂a
,
q → γ+q, γ−q → qa, ¯qa, ∂a, ¯
∂a
,
(5.5.25)
where the “−” parts of d and q are both just partial derivatives because the mo-
mentum dependence drops out in this frame, and d, ¯
d and q, ¯
q have graded harmonic
oscillator commutators (up to factors of p+). (5.5.20) then becomes
Q
¯
¯
α ∼ ¯
qaa†α∂a + qaa†α∂a + ¯
∂aa†αda + ∂aa†αda .
(5.5.26)
Since Qα consists of terms of the form AB, either A or B can be chosen as the con-
straint in (5.5.24b), and the other will generate gauge transformations in (5.5.24c).
We can thus choose either ∂a, or ¯
da and ¯
qa, and similarly for the complex conjugates,
except for the Sp(2) singlets, where the choice is between ∂a and just ¯
da (and simi-
larly for the complex conjugates). However, choosing both d and ¯
d (or both q and ¯
q)
for constraints causes the field to vanish, and choosing them both for gauge generators allows the field to be completely gauged away. As a result, the only consistent constraints and gauge transformations are
¯
daφ = (a
¯
α ¯
qa)φ = ¯
∂aφ = 0 ,
δφ = ∂aλa ,
(5.5.27a)
5.5. SuperBRST
117
subject to the restriction that the residual gauge transformations preserve the gauge choice (explicitly, (5.5.24c), although it’s more convenient to re-solve for the residual invariance in light-cone notation). (There is also a complex conjugate term in φ if it satisfies a reality condition. For each value of the index a, the choice of which oscillator is creation and which is annihilation is arbitrary, and corresponding components of d and ¯
d or q and ¯
q can be switched by changing gauges.) Choosing the gauge
θ−aφ = 0 ,
(5.5.27b)
for the residual gauge transformation generated by ∂a = ∂/∂θ−a, the field becomes
φ(θ+, ¯
θ+, θ−, ¯
θ−) = δ(a
+
+
αθ+)δ(θ−)ϕ(θ0 , ¯
θ0 ) ,
¯
d0ϕ = 0 .
(5.5.28)
ϕ is the usual chiral light-cone superfield (as in sect. 5.4), a function of only 1/4 of the usual Lorentz spinor θ0. This agrees with the general result of equivalence to the light cone for U(1)-type 4+4-extended BRST given at the end of sect. 4.5.
Since the physical states again appear in the middle of the θ expansion (including
ghost θ’s), we can again use (4.1.1) as the action: In the light-cone gauge, from
(5.5.28), integrating over the δ-functions,
S =
dx dθ +
+
0
d¯
θ0
¯
ϕ✷ ϕ ,
(5.5.29)
which is the standard light-cone superspace action [5.25]. As usual for the expansion of superfields into light-cone superfields, the physical light-cone superfield appears in the middle of the non-light-cone-θ expansion of the gauge superfield, with auxiliary light-cone superfields appearing at higher orders and pure gauge ones at lower orders.
Because some of the ghost θ’s are commuting, we therefore expect an infinite num-
ber of auxiliary fields in the gauge-covariant action, as in the harmonic superspace formalism [4.12]. This may be necessary in general, because this treatment includes self-dual multiplets, such as 10D super-Yang-Mills. (This multiplet is superspin 0, and thus does not require the superspin ˇ
M ij to be self-dual, so it can be treated in the
OSp(1,1|2) formalism. However, an additional self-conjugacy condition on the light-
cone superfield is required, (5.4.31b), and a covariant OSp(1,1|2) statement of this condition would be necessary.) However, in some cases (such as 4D N=1 supersymmetry) it should be possible to truncate out all but a finite number of these auxiliary superfields. This would require an (infinite) extension of the group OSp(1,1|2) (perhaps involving part of the unphysical supersymmetries aαq), in the same way that
extending IGL(1) to OSp(1,1|2) eliminates Nakanishi-Lautrup auxiliary fields.
118
5. PARTICLE
The unusual form of the OSp(1, 1|2) operators for supersymmetric particles may
require new mechanics actions for them. It may be possible to derive these actions
by inverting the quantization procedure, first using the BRST algebra to derive the hamiltonian and then finding the gauge-invariant classical mechanics lagrangian.
Exercises
(1) Derive (5.1.2) from (5.1.1) and (3.1.11).
(2) Show that, under the usual gauge transformation A → A + ∂λ, exp[−iq τf dτ .x
τ
·
i
A(x)] transforms with a factor exp{−iq[λ(x(τf )) − λ(x(τi))]}. (In a Feynman
path integral, this corresponds to a gauge transformation of the ends of the prop-
agator.)
(3) Fourier transform (5.1.13), using (5.1.14). Explicitly evaluate the proper-time integral in the massless case to find the coordinate-space Green function satisfying
✷ G(x, x′) = δD(x − x′) for arbitrary D > 2. Do D = 2 by differentiating with
respect to x2, then doing the proper-time integral, and finally integrating back
with respect to x2. (There is an infinite constant of integration which must be
renormalized.) For comparison, do D = 2 by taking the limit from D > 2.
(4) Use the method described in (5.1.13,14) to evaluate the 1-loop propagator correction in φ3 theory. Compare the corresponding calculation with the covariantized
light-cone method of sect. 2.6. (See exercise (7) of that chapter.)
(5) Derive (5.2.1), and find J3. Derive (5.2.2) and the rest of the OSp(1,1|2) algebra.
Show these results agree with those of sect. 3.4.
(6) Quantize (5.3.10) in the 3 supersymmetric gauges described in that section, and find the corresponding IGL(1) (and OSp(1,1|2), when possible) algebras in each
case, using the methods of sects. 3.2-3. Note that the methods of sect. 3.3 require some generalization, since commuting antighosts can be conjugate to the corresponding ghosts and still preserve Sp(2): [Cα, Cβ] ∼ Cαβ. Show equivalence to
the appropriate algebras of sects. 3.4-5.
(7) Derive the tables in sect. 5.3. (Review the group theory of SO(N) spinors, if
necessary.) Use the tables to derive the groups, equivalent to SO(D+,D−) for
D ≤ 6, for which these spinors are the defining representation.
(8) Express the real-spinor γ-matrices of sect. 5.3 in terms of the σ-matrices there for arbitrary D+ and D−. Use the Majorana representation where the spinor is not
Exercises
119
necessarily explicitly real, but equivalent to a real one, such that: (1) for complex representations, the bottom half of the spinor is the complex conjugate of the top
half (each being irreducible); (2) for pseudoreal representations, the bottom half
is the complex conjugate again but with the index converted with a metric to
make it explicitly the same representation as the top; (3) for real representations, the spinor is just the real, irreducible one. Find the matrices representing the
internal symmetry (U(1) for complex and SU(2) for pseudoreal).
(9) Check the Jacobi identities for the covariant derivatives whose algebra is given in (5.4.8). Check closure of the algebra (5.4.10) in D = 10, including the extra
generator described in the text.
(10) Derive (5.4.13).
(11) Write the explicit action and transformation laws for (5.4.16).
(12) Write the explicit equations of motion (5.4.12), modified by (5.4.23), for a scalar superfield for N=1 supersymmetry in D=4. Show that this gives the usual covariant constraints and field equations (up to constants of integration) for the
chiral scalar superfield (scalar multiplet). Do the same for a spinor superfield,
and obtain the equations for the vector-multiplet field strength.
(13) Derive the explicit form of the twistor fields for D=3,4,6. Find an expl